•
CHAOTIC CARDIAC DYNAMICS
VOLUME I
A Thesis
Submitted to the
Faculty of Graduate Studies and Research
In Partial Fulfillment
of the Requirements for the Degree of
Doctor of Philosophy
@ Michael Raymond Guevara
Department of Physiology
McGill University
Montreal
February, 1984
ABSTRACT
Experiments are described in which the electrical activity of
a spontaneously beating aggregate of embryonic chick heart cells is
altered by intracellular injection of a periodic train of current
pulses.
The coupling patterns set up between the electronic
stimul a tor and the aggregate are de se ri bed as a function of the
stimulus amplitude and the stimulation frequency.
Various periodic
and nonperiodic patterns are found, most of which resemble
dysrhythmic patterns seen clinically in the human heart.
The
response of the aggregate to an isolated current pulse delivered at
various phases of its spontaneous cycle is also investigated. 1t
is shown that knowledge of this single pulse response suffices to
explain and even predict the response of the aggregate to periodic
stimulation.
In certain circumstances, the dynamics seen during
periodic stimulation can then be identified as "chaotic ...
0
,
,
RESUME
Nous d~crivons des exp~riences ob 1 'activit& ~lectrique
d'agregats, spontanement actifs,
de
cel1ules cardiaques ct•embryons
de poulets est modifi&e par 1 'application repet&e, au nfveau
intracellulaire, d'fmpulsions electriques.
Les reponses de
couplage etab1ies entre le stimulateur electronique et 1 'agregat
sont decrites
a l'aide
d'une fonction dependant de l'ampl itude et
de la fr&quence de stimulation.
Diverses reponses, periodiques ou
non, sont obtenues, la plupart s'approchant des reponses
arrhythmi ques observees, en cl i ni que, dans 1e coeur humai n.
a une
etudions aussi la reponse d'un agregat
impulsion appliquee a
diverses phases de son cycle spontane.
Cette reponse
stimulation isolee peut alors permettre
d'expliquer~
predire, la reponse de 1 'agr~gat
a la
Nous
stimulation
a la
et meme de
p~riodique.
certains cas, l'activite electrique sous stir.lUlation periodique
peut etre qualifiee de "chaotique11
( traduit par J. Bel air}
•
Dans
To my Mother,
Dorothy Guevara (nee Telfer)
c
0
. ACKNOWLEDGEMENTS
0
I wish to thank Dr. Leon Glass for encouraging me to pursue
graduate studies and for advising on the work presented in this
thesis.
The experimental results were obtained in the laboratory
of Dr. Alvin Shrier, who introduced a novice into the rather arcane
world of cardiac electrophysiology.
Dr. John Clay guided me
through the intricacies of ionic modelling of cardiac cells.
The work recounted in this thesis grew directly out of two
influences in my 1ife.
I must thus first acknowledge the medical
and nursing staff of the emergency room of the Queen Elizabeth
Hospital of Montreal (1975-1978), who encouraged and assisted me in
my early attempts to comprehend cardiac dysrhythmias; I must then
thank Leon Glass· and Michael C.
t~ackey
for showing me that there
existed analytical tools with which one could attack the probl er:1.
The expert laboratory assistance of Diane Colizza, Ken
Rozansky, Glen Ward, Richard Brochu, Robert Lowsky, and Howard
Dubarsky has always been very much appreciated and is gratefully
acknowledged.
I thank John Knowles, Guy Isabel, and Albert
Hagemann for their practical help in matters electronic and
mechanical.
Nelson Publicover encouraged me and gave valuable
advice in the early stages of setting up computer systems.
Dr.
J.S. Outerbridge was always available for consultation concerning
numerical techniques.
A special note of acknowledgement must go to
Peter Krnjevic, whose advice on computers and computing was
0
i
V
invaluable, and at
0
~1
times freely and cheerfully given.
I thank
Sandra James and Christine Pamplin for typing the thesis at very
short notice in their usual careful style.
photographed the figures.
Guy L Heureux expertly
1
I wish to thank Professor Michael C.
Mackey, Dr. Jacques Belair, Diane Colizza, and my fellow graduate
students Carl Graves, Ralf Siegel, and Ehud Isacoff for being
always willing to lend an attentive (if somewhat skeptical!) ear.
Finally, I am again grateful to my wife Diane, this time for her
patience and understanding during the trying circumstances
surrounding the final stages of completion of this manuscript.
I owe a special debt of gratitude to my mother, Dorothy
Guevara, who, by dint of her perseverence and sacrifice, has been
the person most responsible for putting me into a position that
allowed carrying out the work described in this thesis.
During the second half of
my
tenure as a graduate student, I
held a research traineeship awarded by the Canadian Heart
Foundation.
Earlier financial assistance was obtained from
research grants awarded jointly to M.C. Mackey and L. Glass by the
Natural Sciences and Engineering Research Council of Canada, and to
A. Shrier
and
L. Glass
by
the Canadian Heart Foundation.
I once again gratefully acknowledge the support provided me
by
the aforesaid individuals and institutions, without which the work
described in this thesis would not have been possible.
V
FOREWORD
0
t4y
main goal in writing this thesis is to suggest to the
reader that, even though the dynamics of the heart is very complex,
there now exists a mathematical framework which begins to approach
and perhaps even encompass that complexity, and in which that
complexity naturally arises.
mathematics of Chaos
11
11
•
Part of this framework includes the
The term chaos is used here in a technical
sense which does not deviate too much from its everyday meaning of
disorder or confusion.
However, these synonyms are somewhat
misleading, since chaotic dynamics, though highly complex, is
deterministic and has an intricate highl y-order.ed internal
structure.
It may thus be better termed "ordered chaosu.
The layout of the thesis is as follows.
In CHAPTER 1, I
recount some of the early work done in describing the various
patterns of mechanical and electrical activity that can be seen in
the heart.
I next show that these behaviours can be described
mathematically and briefly introduce the topic of chaotic dynamics.
In CHAPTER 2, the experimental response of a cardiac oscillator to
perturbation with single pulses of current is described; in CHAPTER
3, I show that an ionic model assembled from voltage cl amp data
reproduces many of these experimental results.
The experimental
response of the cardiac oscillator to stimulation with a periodic
train of current pulses is next detailed in CHAPTER 4.
0
vi
In CHAPTER
5, I show that the response to single pulses dec ri bed ·; n CHAPTER 2
0
can be used to predict the experimental response to periodic
stimulation that was described in CHAPTER 4.
presents some overall conclusions.
vii
Finally, CHAPTER 6
TABLE OF CONTENTS
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PAGE
CHAPTER
1.
INTRODUCTION:
REGULAR AND IRREGULAR CARDIAC RHYTHMS, THEIR
1-1
MATHEMATICAL DESCRIPTION, AND CHAOTIC DYNAMICS
2.
PHASE RESETTING OF THE RHYTHM OF SPmJTANEOUSLY BEATING
2-1
AGGREGATES OF EMBRYONIC CHICK VENTRICULAR CELLS BY A
CURRENT PULSE OF BRIEF DURATION
3.
THE IONIC BASIS OF SPONTANEOUS ACTIVITY AND PHASE RESETTING
IN THE AGGREGATE:
3-1
NUMERICAL INVESTIGATION OF A PARTIAL
~IODEL
0
4.
PERIODIC STIMULATION OF SPONTANEOUSLY BEATING AGGREGATES OF
EMBRYONIC CHICK VENTRICULAR CELLS
~HTH
4-1
CURREUT PULSES OF
BRIEF DURATION
5.
PREDICTION OF THE RESPONSE OF THE AGGREGATE TO PERIODIC
5-1
STif.1ULATION FROt4 ITS RESPONSE TO STI!'tlULATION WITH SINGLE
PULSES
6.
CONCLUSIONS
6-1
7.
BIBLIOGRAPHY
7-1
viii
0
11
one can perhaps say that the nonlinear field also presents a
kind of a paradise of potential possibilities, once the
11
11
theoretical knowledge of these numerous phenomena is supplemented
by adequate means for their experimental real i zati on.
11
Nicholas Minorsky, 1960
0
ix
0
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CHAPTER 1
INTRODUCTION: REGULAR AND IRREGULAR CARDIAC RHYTHMS,
THEIR MATHEMATICAL DESCRIPTION, AND CHAOTIC DYNAMICS
"Hypothesis has its right place, it forms a working basis; but it
is an acknowledged makeshift, and, as a final expression of
opinion, an open confession of failure, or, at the best, of purpose
unaccompl i shed."
Thomas Lewis, 1920
1-1
The heart is a nonl inear
0
that is, any real ist~c
dev~ce;
mathematical description of it must employ nonlinear mathematics.
Perhaps the best direct evidence for this statement comes from
voltage clamp experiments (of the sort outlined in CHAPTER 3 below)
which show that the processes controlling the flow of ionic
currents through the membrane of space-clamped cardiac cells are
highly nonlinear.
These ionic currents are responsible for the
generation of spontaneous
~lectrJcal
activity in the heart, for the
initiation of mechanical contraction, and (together with the flow
of current from cell to cell) for the spread of electrical and
mechanical activity throughout the heart.
It has become increasingly evident over the last lOO years
that nonlinear systems (both experimental and mathematical) can
display exceedingly complex behaviour.
More
recently~
it has
become cl ear that this dynamics can become so complex that
be described as Chaotic
11
11
•
a
can
In fact, nonperiodic time series
occurring in some deterministic mathematical systems have lead to
the latter being labelled "chaotic".
It has also been known for
about 100 years that the dynamics displayed by the heart can be
quite complex, even nonperiodic.
I show in this thesis that
behaviour approaching the degree of complexity seen in the
~ntact
heart can occur in a very siinple preparation of cardiac origin.
Furthermore, I identify this complex behaviour with chaotic
dynamics.
This introductory chapter is divided into three parts.
1-2
In the
first part, I describe patterns of electrical activity that are
c
commonly seen in the diseased heart.
There is still no consensus
as to the mechanisms by which these various dysrhythmias occur,
even though most of them were originally described before the year
1900.
In the second part of this chapter, I indicate how one may
begin to think of the electrical activity of the heart in a
mathematical way.
The applicability of mathematical concepts such
as equilibrium points, limit cycles, quasiperiodic orbits, and
strange attractors to the description of cardiac events is
outlined.
The first three of these constructs are treated in a
cursory way;
the last one is examined in greater detail.
Since
strange attractors occur in chaotic systems, in the third part of
this chapter I next describe what is meant by chaotic dynamics, and
recount the evidence- accumulated largely within the past five
years - for the existence of chaotic dynamics in several different
systems. The mathematical and physical literature is stressed,
since there is at present little hard evidence for the existence of
chaotic dynamics in biological systems.
1-3
PART A. CARDIAC DYSRHYTHMIA$
0
(i)
Some Patterns of Activity Seen in Cardiac Tissue
a.
1 :1 Pattern
During normal sinus rhythm, all myocytes in the heart are
subjected to a periodically arriving electrical stimulus.
For
cells situated outside the dominant centre of the sinoatrial node,
which is the pacemaker of the heart (Bleeker et al., 1980}, this
stimulation is due to the conducted cardiac impulse that originates
in and spreads from the dominant centre.
Even the cells forming
the dominant centre are subjected to periodic stimulation, since
the activity of the sinoatrial node can be affected by the
Q
activation and contraction of the ventricles (see CHAPTERS 2, 4).
During nonnal sinus rhythm, there is a one-to-one
(1
:1}
synchronization between the electrical activity in any two
particular areas of the heart, with a more or less fixed time delay
or latency between the activation of cells in the two spatially
separated locations.
While there are indeed a few cardiac dysrhythmias that do not
necessarily result in the loss of 1:1 synchronization (e.g. sinus
bradycardia or tachycardia, atrial or junctional or ventricular
escape rhythm or tachycardia, first degree atrioventricular (AV)
heart block, pre-excitation syndrome}, a loss of 1:1
synchronization is seen in most dysrhythmias.
1-4
This
desynchronization is generally due to one of two factors:
i)
0
the emergence of a subsidiary, ectopic, or triggered
pacemaker or the establishment of a micro-reentrant
circuit whose activity may compete and interfere with the
output of the sinoatrial node (e.g. parasystole, premature
contractions};
ii)
block of conduction of the propagated cardiac impulse
(e.g. sinoatrial exit block, AV block, bundle branch
block).
When loss of 1:1 synchronization occurs, the temporal pattern of
activation seen at any given location in the heart may
rema~n
periodic (e.g. 2:1 AV block), or it may become nonperiodic (e.g.
ventricular fibrillation).
I now detail some of the patterns that
can be seen in the heart when overall 1:1 synchronization
lost.
~s
Most of the patterns mentioned below were originally described in
the late nineteenth century.
b.
n:l Patterns
Loss of 1 :1 synchronization is often seen at high heart rates.
Bowditch in 1871, Kronecker and Stirling in 1874, and von Basch in
1879 all reported that sufficiently rapid stimulation of the frog
ventricle with a train of single induction shocks produced a 2:1
pattern of block, where only every second stimulus would provoke a
contraction {Gaskell, 1900}.
Higher grades of n:l block (n
0
1-5
~
3,
where n is an integer) were also observed.
0
Schiff had earlier
reported in 1850 that mechanical stimulation of the surface of the
heart with a needle at a high rate would produce a contraction only
after a definite number of stimuli (Hoff, 1941-42}.
These n:l (n) 2} patterns can also be seen at more normal
heart rates.
Gaskell (1883) slit the atrium of the tortoise so as
to leave a thin bridge of connecting tissue between the sinus
venosus and the ventricle.
There was a greater than normal delay
in the propagation of contraction across the damaged atrium
(analogous to first degree AV block).
to 2:1 block;
Extension of the section led
still further continuation of the slit produced 3:1,
4:1, 5:1, 6:1 and even higher grades of n:l block (Gaskel1, 1883,
1900}.
Finally, if the slit was made long enough, complete block
resulted;
0
with the emergence of a ventricular pacemaker,
atrioventricular dissociation was established.
Application of a screw-cl amp to the atrio-ventricular groove
in the frog or to the atrio-ventricular bundle in the dog ( Gaskell ,
1882; Kent, 1893; Gaskel1, 1900; Erlanger, 1906; Lewis, 1920},
cooling of the atrioventricular region (Zahn, 1912}, stimulation of
the vagus (Gaskell, 1882; McWilliam, 1888a; Lewis, 1920), infusion
of toxic materials such as digitalis or aconitine (Cushny, 1897;
Cushny, 1909-10; Lewis, 1920), asphyxia (Lewis and Mathison, 19101911; Lewis, 1920), or ligation of the coronary arteries (Cohnheim
and van Schulthess-Rechberg, 1881; See, Bochefontaine, and Roussy,
1881) can produce atrioventricular block.
1-6
The final common pathway
of all these interventions is presumably a decrease in the ability
0
of tissue in or below the atrioventricular junction to conduct the
cardiac impulse.
Clinically and experimentally, 2:1 and 4:1 block are the most
common forms of n:l AV heart block; 3:1 and 5:1 patterns are rarely
seen (Lewis, 1920; Besoain-Santander, 'Pick, and Langendorf, 1950;
S1ama et al., 1978).
In certain cases, there appears to be a
direct transition from a 2:1 pattern to a 4:1 pattern of block,
without going through an intermediate pattern of 3:1 block (e.g.
Urthaler et al., 1974; James, Isobe, and Urthaler, 1979).
c.
n+l :n and Associated Patterns
Occasional drcpping or skipping of beats can also be observed
during the
transi~ion
from a 1:1 to a 2:1 pattern of block
(Gaskell, 1882, 1900; Engelmann, 1896, 1896-97; Wenckebach, 1899;
von Kries, 1902;
Erlanger, 1906; Hay, 1906;
Mobitz, 1924).
In
the case of AV block, each skipped ventricular beat is preceded by
n conducted beats, which display a gradually increasing PR interval
on the electrocardiogram.
These n+l :n Wenckebach (or Mobitz type
I) cycles can recur in a periodic pattern (i.e.
~ienckebach
repeated n+l :n
cycles with n fixed) or in a nonperiodic fashion.
While
n:l patterns had been demonstrated in response to fast driving of
cardiac tissue in the nineteenth century, it appears that it was
not until the early years of this century that n+l :n AV b1 ock was
1-7
found in response to fast atrial pacing (van Kries, 1902; Erlanger,
0
1906).
The n+l :n Wenckebach pattern can also be observed in
cardiac tissues other than the atrioventricular node, such as
strips of ventricular muscle (Trendelenburg, 1903).
Other patterns that have been associated with the n+l :n
Wenckebach pattern have been described.
These include n+2:n and
n+3:n patterns (Cushny, 1899-1900), the 2n-1 :n patterns of reverse
Wenckebach (Roberge and Nadeau, 1969), and the 2n+2:n and 2n+l :n
patterns of alternating Wenckebach types A and B respectively
( Sl ama et a1 • , 1979).
d.
2n:2m Patterns
A pattern seen only rarely during the progression of first
0
degree AV block to 2:1 AV block is a 2:2 pattern, in which there is
one
ventri~ular
beat for each atrial beat, but with an alternation
of the PR interval back and forth between two fixed values (Lewis
and Mathison, 1910-1911).
Electrical alternans, an alternation
from beat to beat in the morphology of the electrocardiographic
complexes, was also described around 1910 (Hering, 1908, l910a;
Kahn and Starkenstein, 1910; Lewis, 1910-1911).
A
2:2 or alternans
pattern had been described much earlier in the behaviour called
pulsus al ternans, which is an alternation in the strength of the
peripheral arterial pulse (Traube, 1872).
1-8
I call patterns
displaying the alternans phenomenon 2:2 patterns, since the basic
0
unit that repeates in time consists of 2 stimuli and 2 responses.
Other patterns of the form 2n:2m have also been seen in
cardiac tissue.
For example, a 4:2 pattern can be seen in the
periodically stimulated sinoatri al node ( Kerr and Strauss, 1981 )
and in periodically stimulated Purkinje fibre (Jalife and Moe,
1979a};
a 6:2 pattern is not infrequently seen in cases in
atrioventricular block during atrial flutter (Besoain-Santander,
Pick, and Langendorf, 1950; Slama et al., 1978; Slama et al.,
1979).
e.
N:M Patterns with N < M
All of the N:M patterns (N,M positive integers) listed above
0
had
N~f-1
tissue.
and in most instances occured in presumably quiescent
However, they can also be found in situations in which
non-quiescent tissue is involved.
For example, periodic
stimulation of the atrium can lead to 1:1, 2:2, and n+l :n patterns
in the sinoatrial
nod~
(Kerr and Strauss, 1981; Bonke et al.,
1982).
In some bigeminal rhythms, an atrial, junctional, or
ventricular extrasystole occurs for each impulse of sinoatrial
origin, with a fixed coupling interval between each impulse of
sinoatrial origin and the coupled extrasystole (1 :1 pattern).
While the origin of extrasystoles has not been firmly established
(enhanced automaticity vs. micro-reentrant circuit vs. triggered
automaticity), recent work has shown that many extrasystolic
'1-9
patterns similar to those observed clinically can be generated by
0
periodic subthreshold input to a slow parasysto1 ic" focus U'loe
11
et al., 1977; Jalife and t·loe,
1979a).
In particular,
N:t~
patterns
with N < M occur when there is escape of the driven site if the
frequency or the strength of the driving stimulus falls to below a
critical value.
I shall not go into the historical aspect of these·
patterns, since this has been adequately covered in the
encyclopaedic tome of Scherf and Schott (1973).
It suffices to say
at this point that patterns of the form 1:1, 2n:2m, 1:n, and n:n+l
have been described •
. f.
Nonperi odic Patterns
Many of the above-mentioned patterns (e.g. n:l, n+l :n, 1 :n)
can occur in a periodic fashion.
For example, maintained 2:1 AV
block is quite common in cases of atrial flutter (Lewis, 1920).
More complex periodic patterns that may be described as "mixtures"
of the more basic patterns have also been documented.
For
instance, alternation of a 2:1 and a 3:1 pattern leads to a
periodic 5:2 pattern, and alternation of a 5:4 and a 4:3 pattern
leads to a periodic 9:7 pattern.
However, nonperiodic patterns are often seen in the heart.
One of the earliest irregular cardiac phenomena to be described was
ventricular fibrillation.
This pattern of activity was first
described by Erichsen in 1842 and was provoked by faradic
0
1-10
stimulation of the heart by Hoffa and Ludwig in 1850.
0
Another
early description of irregular behaviour was by Bowditch in 1871.
who found irregularly dropped beats during periodic electrical
stimulation of the frog ventricle (Hoff, 1941-42).
However,
Kronecker and Stirling repeated Bowditch's experiments in 1873, and
could find no irregular responses.
They ascribed the irregular
responses seen by Bowditch to oxide buildup on the contacts of a
relay (Hoff, 1941-42).
Nevertheless, irregular response of the
ventricle to direct electrical pacing or to input arising in
the sinoatrial node does occur.
For example, in textbooks on
cardiac dysrhythmias, one invariably comes across
electrocardiograms in which there are irregular mixtures of various
n+l:n or n:l cycles in a single clinical tracing {e.g.
1920;
Lewis,
Katz, 1946; Bell et, 1971; Phillips and Feeney, 1973; Mandel,
1 980; Se hamroth, 1980; Chung, 1983) •
(ii)
Spatiotemporal Considerations
Since the heart is a spatially distributed structure, the
dysrhythmic patterns described above involve both space and time.
For example, during 2:1 AV block, cells within either the atrial
muscle or within the ventricular muscle may be said to be
responding in a 1:1 fashion to the input presented to them.
However, while this is occurring there are celis within the
atrioventricular node in which there is a 2:2 pattern (Watanabe and
0
1-11
Dreifus, 1980).
0
Thus, different temporal patterns of activity may
coexist in different regions of the heart.
One can perhaps then
speak of spatial bistability or mu1tistability in instances where
there are two or more coexisting stable periodic patterns in
different areas of the heart.
Another example of this kind of behaviour is bundle branch
alternans, in which there is 2:1 conduction in each bundle branch,
but with conduction to the ventricle occurring alternately through
one bundle branch and then the other.
of activation in the ventricle.
This produces a 2:2 pattern
A simple biological analogue of
this situation was constructed by t·1ines ( 1914), who mechanically
produced longitudinal dissociation of the conducting pathway.
A
pattern of 2:1 conduction in one bundle branch and a 1:1 pattern of
conduction in the other would also produce a 2:2 pattern in the
ventricle (Bandura and Brody, 1974; Cohen et al., 1977).
Gaskell
{1882} proposed that alternans of the ventricle would result if
there were two populations of cells in the ventricle, one
responding in a 1:1 fashion, the other in a 2:1 fashion.
To
further complicate matters, the population of cells with the 2:1
rhythm may be composed of two subpopulations, responding on
alternate stimuli (Mines, 1914).
"Localized fibrillationu can occur in a circumscribed area,
while contraction proceeds relatively normally elsewhere (Garrey,
1924; Moe, Harris, and Wiggers, 1941; Harris and Guevara Rajas,
1943; Downar, Janse,and Durrer, 1977).
1-12
The existence of different
dynamics in different parts of the heart can usually be ascribed to
0
the existence of spatial inhomogeneities in the heart.
Spatial
asyrrmetry, a special form of spatial inhomogeneity, may also be
playing a role.
For example, periodic stimulation of cardiac
tissue can lead to 1:1 conduction for antegrade propagation, but
block of retrograde conduction (Engelmann, 1894;
Cranefield,
Klein, and Hoffman, 1971 }.
( i'f i) Bi stability
It is thus possible to see more than one temporal pattern of
activity in different areas of the heart at the same time.
The
heart is also capable of displaying more than one pattern of
overall activation without altering its basic physiol agical state.
The most dramatic example of this is ventricular fibri11ation,
which can be made to appear or disappear more or less
instantaneously (so that the change in dynamics cannot be ascribed
to a change in the physiological condition of the heart) by
delivery of a single pulse of mechanical (McWilliam, 1887;
Pennington, Taylor, and Lown, 1970;
Yakaitis and Redding, 1973;
Befeler and Aranda, 1977; Forester, 1978; Lawn, Verrier, and Blatt,
1978) or electrical (Prevost and Bate1li, 1900;
Wiggers and Wegria, 1940;
energy.
activity:
f,1ines, 1914;
Kouwenhoven and Mi1nor, 1954-1955)
In this case the heart displays one of two modes of
the stable periodic activity of normal sinus rhythm or
the maintained chaotic activity of ventricular fibrillation.
1-13
The heart can also be shown to be capable of supporting two
0
different modes of periodic activity ( bistability").
11
For example,
ventricular tachycardia is routinely converted to normal s'inus
rhythm by a precordial thump or by electrical cardioversion.
Another example is given in the early work of Hines (1913a), who
demonstrated that induction of a single properly timed extrasystole
could convert a 2:1 pattern into a 1:1 pattern, and that
intermission of two or three stimuli in a train of stimuli could
convert a l :1 pattern into a 2:1 pattern.
11
•••
In the words of t4ines:
over a quite considerable range of frequencies of excitation,
there exists two possible equilibria, stable so long as the heart
continues beating regularly and without interruption."
Bistability of two different periodic behaviours has also been
0
seen in diseased human ventricular myocardium.
Injection of a
stimulus during periodic activity demonstrating
11
bimodal" action
potentials can convert the action potentials to ones showing only a
unimodal component (Gilmour et al., 1983).
Similar behaviour can
be seen in ionic models of cardiac tissue (Guevara, unpublished).
Finally, there can be bistability between the stable quiescent
condition of asystole and the stable periodic condition of normal
sinus rhythm or ventricular tachycardia.
mechanical (e.g.
A single pulse of
YakaHi s and Reddi ng, 1973) or el e<;trical {e.g.
Cranefield, 1977) energy can cause spontaneous activity to commence
in a quiescent cardiac system.
l-14
(iv)
0
Unified Theories of Cardiac Oysrhythmias
In summary, a great plethora of patterns can be seen
heart.
~n
the
If anyone needs to be convinced of this fact, I suggest
that they simply open any textbook on clinical electrocardiography;
these books are detailed compendia of the modes of electrical
activity available to the heart.
The patterns range from the
regularity of normal sinus rhythm to the irregularity of atrial or
ventricular fibrillation.
Often, a specific mechanism is
identified with a particular dysrhythmia.
Attempts to provide a
unifying hypothesis for patterns with apparently different origins
have been previously made.
Most notable have been those of Oecherd
and Ruskin (1946), Roberge and Nadeau (1969), El-Sherif, Scherlag,
and Lazzara (1975), and Moe et a1. {1977).
The work described in
this thesis is an effort to extend these earlier endeavours.
The approach of three of the above four groups of
investigators (Oecherd and Ruskin, 1946; Roberge, Bhereur and
Nadeau, 1971; Moe et al., 1977) included using the response of
cardiac tissue to a single stimulus to predict the response to
periodic delivery of that same stimulus.
Indeed, this approach
seems to have been first enunciated by Cushny and Matthews (189})
in their report concerning electrical stimulation of the heart.
Cushny and Matthews used electrical stimulation to mimic
irregularities in the cardiac rhythm that Cushny had previously
0
1-15
seen in response to administration of digitalis or aconitine
0
{Cushny, 1897; Cushny, 1899-1900).
11
Cushny and Matthews stated that
in order to gain any real insight into them [i.e. the
irregularities, M.G.], it was absolutely necessary to study first
the comparatively simple
dev~ ati ons
caused by single stimuli •
11
Thus, I now turn to consideration of the effect of a single
st~mulus
on the heart.
(v) The Response of Cardiac Tissue to Premature Stimulation
The history of direct electrical stimulation of the heart goes
back to at least the early nineteenth century, when Aldini and a
handful of his
cont~Jporaries
stimulated the hearts of animals
{including decapitated criminals) using Galvanic current provided
by the colur.m of Volta (Aldini, 1803, 1804).
Electrical
stimulation could provoke a cardiac contraction and so Aldini
(1803) cl ai rvoyantly suggested:
11
Gal vani sm affords very powerful
c.
means of resus)tation in cases of suspended animation under common
circumstances.
& c.
The remedies already adopted in asphyxia, drowning,
when combined with the i nf1 uence of Gal vi ni sm, wi 11 produce
much greater effect than either of them
separately.~~
Subsequently,
more systematic investigations by Bowditch, Kronecker and Stirling,
and Marey established that cardiac tissue has a refractory period
following a contraction, during which stimulation cannot produce a
second contraction {Hoff, 1941-42).
1-16
Furthermore, the response to a
stimulus is seemingly all-or-none.
0
(According to Hoff (1941-42),
the Abbe Fontana had already realized these facts by 1785.)
If the ventricle is prematurely stimulated outside of its
refractory period, a disturbance in the rhythm of contraction of
the ventricle results: either an interpolated beat (Wenckebach,
1903; Laslett, 1909-1910) or a compensatory pause {Engelmann, 1895;
Cushny and Matthews, 1897) occurs.
Similar behaviours are seen if
there is a spontaneous premature ventricular contraction of
endogenous origin {Lewis, 1920).
A more complex response is seen if the atrium rather than the
ventricle is prematurely stimulated.
The spontaneous cyclic
activity of the sinoatrial node is affected in such a way that the
returning atrial cycle can be fully compensatory, partially
compensatory, or fully reset (Engelmann, 1896-97; Cushny and
Matthews, 1897; Henckebach, 1903; Lewis, 1920).
More recent v-10rk
has shown that the perturbation to the rhythm of the sinoatrial
node caused by a prenature atrial contraction depends on the phase
in the sinus cycle at which the premature contraction is induced
{Bonke, Bouman, and van Rijn, 1969; Bonke, Bouman, and Schopman,
1971; Klein, Singer, and Hoffman, 1973; Strauss et al., 1973;
Miller and Strauss, 1974; Steinbeck et al., 1978; Kerr et al.,
1980).
Similar phasic effects on the sinoatrial node are found if
its spontaneous activity is perturbed by a subthreshold pulse of
current {Sano, Sawanobori, and Adaniya, 1978; Jalife et al., 1980}.
0
1-17
This "phase resetting" of the spontaneous activity of the
0
sinoatrial node also occurs in response to delivery of a single
vagal volley (Brown and Eccles, 1934a,1934b; Dong and Reitz, 1970;
Levy et al., 1969; Greco and Clark, 1976; Jalife and Hoe, 1979b;
Spear et al., 1979; Jalife et al., 1983}.
Phase resetting has
also been described in several other spontaneously active cardiac
tissues (Weidmann, 1951; Klein, Cranefield, and Hoffman, 1972;
DeHaan and Fozzard, 1975; Jalife and Moe, 1976; Scott, 1979;
Ferrier and Rosenthal, 1980; Guevara, Glass, and Shrier, 1981; Ypey,
van
~1eerwijk,
~~eerwij k
.
and DeHaan, 1982; Clay, Guevara, and Shrier, 1984; van
et al • , 1984).
If the atrioventricular node is presented with an earlierthan-expected input arising from a premature activation of the
atrium, it will conduct the cardiac impulse to the ventricle with a
velocity that is slower than normal.
The response (i.e. the
conduction time through the AV node} can be systematically
investigated as a function of the degree of prematurity of
activation of the AV node.
In this way, the AV nodal recovery
curve can be constructed (Mobitz, 1924; Ashman, 1925; Decherd and
Ruskin, 1946).
Thus, the response of several different parts of the heart to
premature stimulation with a single stimulus (a propagated wave of
contraction or an electrical pulse) has been known for some time.
I call the response to delivery of a single current pulse the
single-pulse response.
I show in CHAPTER 5 that knowledge of the
single-pulse response is indeed indispensAble in understanding the
0
response to periodic stimulation of the particular cardiac system
1-18
that I have been investigating.
c
Not only are many of the patterns
described earlier in this section seen in this cardiac preparation,
but their existence is predicted from the single-pulse response.
Thus, it appears that the suggestion of Cushny and Matthews made in
1897 was indeed a valuable one.
The experimental work described below in CHAPTERS 2 and 4 is
carried out on spontaneously oscillating cardiac tissue.
Within
this century, a considerable body of literature has been built up
on the mathematical description of oscillating (or just excitable)
systems, and on the effects of periodic stimulation on such
systems.
In the next section I first show how one may begin to
look· at the h1:!art from a mathematical perspective.
I then give a
sampling of very recent results (both theoretical and experimental)
that demonstrate that Chaotic dynamics can result from the
11
11
periodic stimulation of systems which are excitable or which
oscillate spontaneously.
1-19
0
PART B.
(i)
MATHEMATICAL DESCRIPTION OF SOME CARDIAC PHENOMENA
Equilibrium Points. Limit Cycles, Quasiperiodic Dynamics, and
Strange Attractors in Cardiac Electrophysiology
The experimental work I report on in this thesis is carried
out on a spontaneously oscillating cardiac preparation.
Since the
preparation is effectively isopotential (i.e. space-clamped),
~t
can be described by a system of ordinary differential equations.
Equations describing the electrical properties of space-clamped
cardiac tissue are high-dimensional, nonlinear systems of ordinary
differential equations, which are formulated using results from
voltage-clamp experiments (e.g. Noble, 1962; f'.t::Allister, Noble, and
Tsien, 1975; Beeler and Reuter, 1977; Yanagihara, Noma, and
Irisawa, 1980;
Bristow and Clark, 1982; Irisawa and Noma, 1982;
Clay, Guevara, and Shrier, 1984). A distributed system such as the
heart, where variables such as the transmembrane potential and
activation and inactivation variables of the various ionic currents
are functions of both time and space, must be modelled by a syster.1
of partial differential equations.
In an N-dimensional system of ordinary differential equations
there are N variables x1 , x2 ,
••• ,
xN.
At any one point tin time,
the state of the system is completely specified by the values of
these N variables at that time.
The path traced out by the N-
dimensional state-point (x 1 (t), x2 (t}, ••• ,xN(t)) as time
1-20
0
progresses can be thought of as a trajectory or an orbit in the Ndimensional phase space of the system.
Essentially four types of asymptotic (i.e. t
~ oo)
behaviours
can evolve in a bounded, dissipative system of ordinary
differential equations.
The state-point of the system may
generically tend to ( i) an attractor of dimension zero (a stable
equilibrium point, fixed point, singular point, or steady state};
{ii) a periodic orbit of dimension unity (a stable limit cycle);
(iii) an orbit lying on a toroidal hypersurface (a quasiperiodic
orbit); or {iv) an attractor whose fractal or Hausdorff dimension
01ori, 1980) is greater than its topological dimension (a strange
attractor).
The state point may wander around in an apparently
random fashion in a system in which there are no stable periodic
orbits, but rather an infinity of unstable orbits (Pikovskii and
Rabi novich, 1978; Ueda, 1979, 1980a, 1980b).
Asymptotic approach to a stable equilibrium point is well
known in cardiac electrophysiology.
For example, it generally
occurs after a single action potential is ·induced in quiescent
tissue.
An ionic model of quiescent tissue must therefore possess
at least one stable equilibrium point (e.g. Beeler and Reuter,
1977).
A limit cycle was defined by Poincare in 1881 to be a closed
curve in the phase space of a system of ordinary differential
equations (Minor sky, 1962).
t·1ovemen t of the state point of the
system along the limit cycle trajectory results in a periodic
1-21
.o
time series for the system variables.
Trajectories with initial
conditions sufficiently close to an asymptotically stable 1imit
cycle approach the cycle as t • ""·
Therefore, no trajectory
sufficiently close to the 1imit cycle is also a closed trajectory.
The physicist and engineer van der Pol (1926, 1940) was probably
the first person to think of the cardiac cycle as a relaxation
oscillator (a special case of a limit-cycle oscillator).
Numerical
simulation suggests that asymptotically stable limit cycles exist
in ionic models of spontaneously active cardiac cells (e.g. Noble,
1962; Scott and Kang, 1974; McAllister, Noble, and Tsien, 1975;
Yanagihara, Noma, and.Irisawa, 1980; Bristow and Clark, 1982;
I ri sawa and Noma, 1982) •
Quasiperiodic motion is nonperiodic motion, whose Fourier
spectrum has a finite number of characteri stk frequencies which
are rationally independent of each other C' i ncommensurate11
) •
The
trajectory followed by the state-point of the system can be thought
of as lying in a toroidal hypersurface embedded in a higherdimensional space.
Although the time series of a variable
undergoing quasiperiodic dynamics is nonperiodic, any two
trajectories that start out with initial conditions close to one
another do not diverge from each other very rapidly.
the two trajectories start out within a distance
In fact, if
o of each
other, a point in time is eventually reached when the two
trajectories return to within a distance e of each other, with e
q.
However, a system with quasiperiodic dynamics can be
1-22
~
c
structurally unstablet with the quasiperiodic motion being
destroyed by an infinitesimally small change of the system
parameters {e.g. Moser, 1969).
Other systems showing quasiperiodic
dynamics seem to be structurally stable (e.g. Franceschini, 1983;
Thoulouze-Pratt, 1983).
There do not seem to be any published reports labelling any
phenomenon seen in the heart as a manifestation of quasiperiodic
dynamics.
I present evidence in CHAPTER 4 below that complete
heart block with atrioventricular dissociation is indistinguishable
from quasiper1od1c dynamics.
f11ovement
of the state-point of the system along
d
strange
at tractor results in a nonperi odic time series for any of the
system variables (e.g. the transmembrane potential).
Although the
state-point of the system never returns to a location in phase
space previously visited, its motion takes place in an invariant
volume of the phase space.
Unlike the case of quasiperiodic
dynamics, there is "sensitive dependence on initial conditions.. ,
with trajectories that are initially close diverging away from each
other exponentially with time (Guckenheimer, 1979b; Ruelle, 1979).
This exponential divergence means that at least one of the Liapunov
exponents is positive (Benettin, Galgani, and Strelcyn, 1976;
Nagashima and Shimada, 1977;
Shimada and Nagashima, 1978, 1979;
Geisel, Nierwetberg, and Keller, 1981 ).
Furthermore, the fractal
or liausdorff dimension of a strange attractor is generally nonintegral, and is less than its topological dimension {Mori, 1980;
0
1-23
0
Packard et al., 1980; Russel, Hanson, and Ott, 1980; Froehling et
al., 1981; Farmer, 1982; Greenside et al., 1982; Grassberger and
Procaccia, 1983;
Termonia and Alexandrowicz, 1983).
Systems that
admit strange attractors may or may not be structurally stable
(Guckenheimer and Holmes, 1983).
Indeed, some strange attractors
are stable, others not {Kaplan and Yorke, 1979).
Note that exactly
. what is meant by a strange attrac tor (or even just an attrac tor) is
a matter of definition which is presently the subject of some
debate (e.g.
Ruelle, 1980, 1981;
Guckenheimer and Holmes, 1983}.
The concept of Strange attractot··" was initially introduced by
11
Ruelle and Takens (1971 ).
As is the case for quasiperiodic dynamics, there do not appear
c
to be any published reports linking the presence of irregular
dynamics in a cardiac system to the existence of a strange
attractor in the phase space of a model of that system.
In CHAPTER
4 below, I show experimental tracings from a periodically
stimulated cardiac oscillator that are nonperiodic in time.
In
CHAPTER 5, I present theoretical and numerical ar·guments which
suggest that these tracings reflect the presence of a strange
attractor.
(ii) Electrical and Electronic Models of the Heartbeat
In their pioneering study, van der Pal and van der
r~ark
(1928,
1929) modelled the heart as a system of three coupled oscillators -
1-24
0
one each for the sinoatrial node, the atrium, and the ventricle.
They wired together three neon bulb relaxation oscillators with
unidirectional coupling, so that the "sinoatrial node" could affect
the
u
atrium' (but not vice versa), and so that the "atrium" would
affect the "ventricle" (but not vice versa}.
This simple circuit
produced patterns remarkably reminiscent of normal sinus rhythm,
first degree heart block, 3:2, 2:1, 5:2, 3:1 (and higher grade n:l)
heart block, and·complete heart block.
This approach of van der Pol and van der t1ark to modelling the
heartbeat stemmed from earlier experimental and theoretical work on
driving an electronic or electrical oscillator with a sine wave
generator (Appleton, 1923; van der Pal, 1927; van der Pal and van
der
r~ark,
1927).
It was found that the driven oscillator could be
made to modify its spontaneous activity so as to synchronize with
or lock onto the driving sinusoid.
In this circumstance, for each
cycle of the sine wave generator, there is one cycle of the driven
neon bulb or triode valve oscillator, with a fixed phase difference
("phase angle") between the wavefonns of the driving and driven
oscillators (_1 :1 synchronization, entrainment, or phase-locking).
As the ratio of the frequency of the sine-wave generator to the
intrinsic frequency of the driven oscillator would be changed, the
phase·angle would also change.
Eventually, as the driving
frequency became large with respect to the instrinsic frequency of
the driven oscillator, the one-to-one pattern of synchronization or
entrainment would be lost, and would be replaced by periodic n:l
0
1-25
0
coupling patterns {"frequency demul tip 1icati on" ) or nonperi odic
("quasiperiodic
der
~1ark,
11
)
dynamics (van der Pol, 1927; van der Poland van
1927, 1929; t1inorsky, 1962;
Hayashi, 1964).
Thus, the normal sinus rhythm and first degree block observed
in the electrical model of the heart of van der Pol and van der
tvlark corresponds to 1 :1 synchronization, the 2:1, 3:1, and higher
grades of n:l block correspond to frequency demultiplication, and
complete heart block with atrioventricular dissociation corresponds
to quasiperiodic dynamics or periodic dynamics with a very long
period {see CHAPTERS 4 and 5).
The approach of using electrical and electronic analogues of
the heart to model normal cardiac activity and cardiac dysrhythmias
0
has continued through the years down to the present day (Bethe,
l940-41a, 1940-4lb; Grant, 1956; Chebotarev, 1968; Roberge, Nadeau,
and James, 1968;
Roberge and
Nadeau, 1971;
Li~ko
Nadeau, 1971;
Sideris, 1976;
~~oul
opoul os, 1977;
~~adeau,
1969; Bhereur, Roberge, and
and Landahl, 1971;
Roberge, Bhereur, and
Padmanabhan, 1977;
Keener, 1983a).
Sideris and
The work of Roberge,
~Jadeau,
and Bhereur is expecially noteworthy, since they brought to bear on
the problem a two-pronged attack that combined electronic modelling
with physiological experimentation.
The richness of the electrical
modelling approach is underscored by the fact that a model
consisting of eight neon bulb oscillators produces behaviours
similar to a score of phenomena- both normal and pathological experimentally observed in the heart (Sideris, 1976;
0
1-26
Sideris and
0
~1oul
opoul os, 1977).
The various periodic patterns described in these analogues
(e.g. normal sinus rhythm with or without first
de~ree
heart block,
3:2 block, n:l block) presumably correspond to the existence of an
asymptotically stable limit cycle in the phase space of the
equations describing the particular analogue.
The way in which
limit cycles are born and die. as the various patterns come and go
as a parameter (e.g. atrial frequency or degree of atrioventricular
coupling) is changed was not investigated. A single limit cycle
can arise or disappear via a Hopf bifurcation or a reverse Hopf
bifurcation respectively (a translation of Hopf's original 1942
paper is found in Howard and Kopell, 1976); a pair of limit cycles
0
can arise de novo or coalesce via a saddle-node bifurcation
(Minorsky, 1962; Sotomayor, 1973); a pre-existing stable limit
cycle can become unstable producing a new stable 1imit cycle of
approximately twice the original period in its immediate vicinity
via a period-doubling bifurcation (Brunovsky, 1971;
Ruelle, 1973).
More complicated behaviours of a more global nature can also occur
( Ruell e and Takens, 1971; Guckenheimer, 1979a; Guckenheimer and
Holmes, 1983}.
With the possible exception of the study of Bethe
(1940-41a, 1940-41b}, period-doubling bifurcations were apparently
not observed in the studies on electrical and electronic analogues
listed above.
This thesis shows that period-doubling bifurcations
can exist in periodically stimulated cardiac tissue.
0
1-27
0
(iii)
Bistability and Hysteresis
Bistability occurs in a system when there are two coexisting
stable attractors.
For example, two stable equilibrium points, one
stable equilibrium point and one stable limit cycle, or two stable
limit cycles can simultaneously exist;
distinct fonns of bistabil ity.
these are thus three
Experimental and theoretical work
by Appleton and van der Pol (1922) and by van der Pol (1922) showed
that the latter two types of bistability could exist in a triode
valve oscillator.
In the case of the co-existence of a stable equilibrium point
and a stable 1imit cycle,
11
oscillation hysteresis11 is seen.
In
this phenomenon, as a parameter is changed in one direction and
then in the reverse direction, oscillation appears and disappears
at two different values of the parameter.
Experimental evidence
for oscillation hysteresis was. found by Appl eton and van der Pol
(1922) in a triode valve oscillator, and has been recently found in
a neural membrane (Guttman, Lewis, and Rinzel, 1980).
Two other related phenomenon can also be seen when a stable
equilibrium point and a stable 1imit cycle coexist.
If the system
is not oscillating, it can be made to do so by injection of a
stimulus of sufficiently high amplitude.
This procedure was
successfully carried out by Appleton and van der Pol {1922) in a
triode valve circuit by using an electromagnetically-induced
electromotive force.
In this case one speaks of an oscillator with
0
1-28
0
11
hard self-excitation" {e.g. Minorsky, 1962).
Hard self-excitation
has also been seen in cardiac tissue where it has been termed
"triggered activiti• (e.g. Cranefield, 1977; Jalife and
Antzelevitch, 1980).
Conversely, if the system is oscillating, the
osciilation can be annihiliated by a single well-timed stimulus
(Winfree, 1980).
This has been recently seen in three different
cardiac tissues- the sinoatrial node (Jalife and Antzelevitch,
1979), Purkinje fibre (Jalife and Antzelevitch, 1979,1980), and
diseased human ventricular myocardium (Gilmour et al., 1983).
As mentioned earlier, van der Pol (1922) found experimental
evidence for the coexistence of two different periodic solutions in
an unforced triode valve oscillator.
0
In addition, investigation of
the sinusoidal fore i ng of the equations that van der Pal ( 1926)
devel aped to model his electronic oscillator {Cartwright and
Littlewood, 1945; Hayashi, Shibayama, and Nishikawa, 1960;
Littlewood, 1960;
Grasman, Veling, and Willems, 1976; Flaherty and
Hoppensteadt, 1978; Guckenheimer, 1980b; Levi, 1981) and of a
piecewise linear approximation to them (Levinson, 1949) have
revealed that two stable limit cycles can coexist at one set of
parameter values.
When two stable periodic orbits exist
simultaneously, one of two different periodic patterns will appear,
depending on the particular initial conditions chosen.
In an
experimental situation, where one parameter such as the forcing
frequency or amplitude is gradually increased or decreased, this
bistability could manifest itself as hysteresis.
0
1-29
(An experimental
0
system that demonstrates hysteresis or memory is one in which the
behaviour seen at a particular set of parameters depends on how
that parameter set was approached.)
In a system of ordinary
differential equations, it seems to me that hysteresis must be due
to some form of bistability.
Bistability of two periodic orbits can also be seen in
numerical simul ati ons of a forced Ouffi ng-van der Pol .osc ill.ator
(Kawakami, 1982) and in the forced Brusselator.(Kai and Tomita,
1979).
Evidence for the coexistence of two attracting periodic
orbits has been found in several physical systems, including
experiments involving Rayleigh-Benard convection {Gollub and
Benson, 1978) and experiments using lasers {Atecchi et al., 1982).
0
t,1odel s of 1aser systems also show coexistence of two stable 1 imi t
cycles (Antoranz et a1 ., 1982; Arecchi et al., 1982), as do
simplified models of convective fluid flows (e.g. Fowler and
McGuinness, 1982).
Hysteresis where one or the other of two periodic patterns can
appear depending upon the prior history of the system has been
documented in cardiac muscle (Mines, 1913a; Moulopoulos, Kardaras,
and Sideris, 1965; El-Sherif et al., l977a;
Guevara et al., unpublished: see CHAPTER 4).
Bando et al., 1979;
It also occurs in an
electronic model of a cardiac pacemaker cell {Roberge, Bhereur, and
Nadeau, 1971) and in many other analogue systems {Hayashi, 1964).
It is unclear to me at this time whether or not the hysteresis seen
in all five of the above cardiac studies can be ascribed to
0
1-30
0
bistability in a set of ordinary differential equations.
Bistability of two periodic orbits 1n cardiac systems has however
definitely been seen in two separate cases; in each instance, a
periodic pattern was converted into a different periodic pattern by
application of a brief stimulus (lv1ines, 1913a; Gi'lmour et al.,
1983). There is also evidence for tristability in the human heart:
two successive precordial thumps converted ventricular tachycardia
of one morphology first into ventricular tachycardia of another
morphology and then into normal sinus rhythm (Pennington, Taylor,
and Lown, 1970:
Fig. 4}.
Tristability has also been seen in a
model of a laser (Arecchi et al., 1982).
Finally, if the basins of attraction of the two periodic
orbits are very much intertwined, behaviour that appears aperiodic
may result, since small perturbations ("noise11 ) in the system will
make the state-point of the system skip back and forth from the
basin of attraction of one limit cycle to that of the other
(Flaherty and Hoppensteadt, 1978).
A similar hoppingu mechanism
11
operating between two attracting domains can apparently produce a
broadband power spectrum that decays algebraically (Arecchi and
Lisi, 1982; Ben-Jacob et al., 1982).
Coexistence of two stable equilibrium points leads to a third
form of bistability.
In the heart, this produces the phenomenon of
.. two stable states of resting potenti al
11
(
Wiggi ns and Cranefi el d,
1974, 1976; Shrier and Clay, unpublished), in which a brief
perturbation can cause the state-point to move from one equilibrium
1-31
0
point to the other.
This bistability of two equilibrium points has
been seen in physical systems (e.g. Gibbs, r•1cCall, and Venkatesan,
1976).
Tristability of three equilibrium points has been seen in
optical systems (e.g. Cecchi et al., 1982), but is unl ike1y to
exist in cardiac tissue, since curr.ent-vol tage characteristics have
a simple N-shape (see CHAPTER 2).
Quasiperiodic and periodic motions can also coexist in
Rayleigh-Benard convection (Gollub and Benson, 1978} and in models
of Rayleigh-Benard convection (Curry, 1979).
I know of no obvious
cardiac anal ague of this behaviour.
Coexistence of a periodic orbit and a strange attractor is
also possible (e.g. Grebogi, Ott, and Yorke, l983b; Guckenheimer
and Holmes, 1983), as is the coexistence of one or more stable
equilibrium points with a strange attractor (e.g. Nagashima and
Shimada, 1977).
As outlined earlier in this chapter, one can
generally repeatedly fibri11 ate and defibrillate a healthy heart at·
this fact may imply such a coexistence.
will:
Coexistence of two strange attractors can also occur (e.g.
Leven and Koch, 1981; Grebogi, Ott, and Yorke, 1982; Arecchi and
Li si, 1982; Arecchi et al., 1982) and can lead to the production of
a time series whose power spectrum falls off in a 1/fa fashion,
with
a
a positive real number (Arecchi and Lisi, 1982; however, see
also Beasley, D'Humieres, and Huberman, 1983;
Voss, 1983).
It is
interesting to note in this context that strange attractors often
have a similar falloff in spectral content, and that l/f
0
1-32
0
fluctuations are seen in many biological membranes and in the beatto-beat interval of the heart (Kobayashi and Musha, 1982).
Recently, formulation of the problem of periodic stimulation
of the van der Pal and other oscillators in terms of one- and twodimensional maps has yielded many new insights.
Bistability turns
out to be due to the existence of two stable periodic orbits on
such return maps (Guckenheimer, 198Gb; Levi, 1981; Glass and Perez,
1982; Perez and Glass, 1982; Guckenheimer and Hol mes, 1983;
Guevara et al., 1983:
see CHAPTER 6; Glass
0
1-33
et~.,
1984).
0
PART C.
CHAOTIC DYNAMICS
In this section, I give a brief survey of the field of chaotic
dynamics.
I stress two areas:
( i) the experimental observation of
dynamics that has been described as chaotic; and (ii} the numerical
investigation of mathematical models of experimental systems.
The works surveyed below come largely from the physics
literature.
Note that almost no mention is made of chaotic
dynamics in conservative systems, since I am interested in this
thesis in a biological system that is dissipative.
The review
articles by Chirikov {1979) and by He11eman (1980b), and the book
by Lichtenberg and Lieberman (1983) can be consulted with regard to
chaotic dynamics in Hamiltonian systems.
the mathematical literature.
I also 1argely neglect
The books (and the references
contained therein) by Collet and Eckmann (1980}, Gumowski and 11ira
(1980), and Guckenheimer and Holmes (1983) can be used as entry
points into that literature.
The review articles of Feigenbaum
(1980b, 1983}, Eckmann (1981 ), Hofstadter (1981), Ott (1981 ), and
Swinney (1983} and the conference proceedings edited by Gurel and
Ross1er (1979}, Hel1eman (1980a), and Campbell and Rose (1983) may
also be of some general interest to the uninitiated.
In the last two paragraphs, I have used the adjective
"chaotic" and the term 11 Chaotic dynamics11
11
•
The tenn 11 Chaos11 or
Chaotic dynamics 11 is 1oosel y used in the mathematical and
scientific literature and has different meanings to different
0
1-34
0
people.
Perhaps its most common usage is to indicate the presence
of a nonperiodic time series of some variable measured in an
experiment or generated in a numerical simulation.
In this case,
the investigator often assumes that the observed nonperiodic
dynamics is the reflection of a determi ni stically nonperi odic orb1t
in the phase space of the system being studied.
For example, in a
numerical study of a system of ordinary differential equations,
this would mean the appearance of a strange attractor in the phase
space of the system.
(Although a quasi periodic orbit will also
generate a nonperi odic time series, the term chaotic dynamics is
not usually applied in this case.)
Note that the concepts of chaos
and randomness still give rise to some confusion in the literature
0
{e.g. Kozak, Musho, and Hatlee, 1982 versus Karney, 1983).
Simply based on examination of its appearance, a computergenerated time series can never be said to be nonperiodic, since
this would involve carrying out the simulation for an infinitely
1ong time.
However, it can be proven mathematically that some
dynamical systems, for certain well-chosen values of the parameters
and initial conditions, have periodic solutions of arbitrarily long
V
period and even solutions which are nonperiodic (e.g. Sarkovskii,
1964; Li and Yorke, 1975; "Stefan, 1977; Pikovskii and Rabinovich,
1978).
Numerical simulation of these systems on an ideal digital
computer would theoretically result in a periodic time series for
the system variables, since a digital computer is a finite-state
machine (e.g. Stein and Ulam, 1964; Mayer-Kress and Haken, 198la;
1-35
0
Conrad and Rossler, 1982; Levy, 1982; Lichtenberg and Lieberman,
1983}.
However, the effects of "noise", either
(e.g.
~nternal
quantum fluctuations in semiconductor devices, bombardment of semiconductor junctions by disintegration products of radioactive
elements present in trace quantities) or external {e.g. cosmic ray
bombardment) to the machine, may very well convert this finitestate-induced periodicity into nonperiodicity.
However, the
characteristic time-seal e of the computer-induced artifact is
generally much different from the dominant frequency of the
nonperiodic or chaotic oscillation.
Numerical calculation of the
Liapunov exponents and of the fractal dimension is often carried
out to demonstrate the presence of chaotic dynamics.
0
Numerical artifact also occurs duri ~g the simulation of an
orbit that can be mathematically proven to be periodic.
For
example, since a digital computer has finite resolution, the
computed state-point of a set of differential equations will seldom
be exactly at any point on the (mathematically) correct limit_cycle
orbit.
Instead, the simulation will produce a periodic orbit that
will be an approximation to the limit cycle, with finitely many
points on the orbit.
In addition, starting with initial conditions
off of the limit cycle, the state-point will end up on this finitedifference approximation to the real limit cycle in finite time.
In the case of the real (i.e. mathematical) 1imit cycle, the
approach to the 1imit cycle is only asymptotic for initial
conditions not on the limit cycle itself.
0
1-36
0
Some experimental systems are also said to be chaotic or to
display chaotic dynamics because an experimental time series
appears to be nonperiodic.
Again, there are several fundamental
problems in stating that a physical or biological system generates
a deterministically nonper'iodic output.
I shall only state what I
see as the most fundamental objection.
All physical and biological
systems are made up of molecules, atoms, and more fundamental
constituent particles that are governed by the laws of quantum
mechanics.
If one accepts the statistical interpretation of
quantum mechanics, then deterministic nonperi odic dynamics cannot
occur in experiments.
I illustrate this statement with an example
taken from cardiac e1ec trophys i o1ogy.
Close inspection of the potential dHference measured across a
supposedly quiescent cardiac membrane reveals that the membrane is
not truly quiescent:
there is low-amplitude voltage noise present
(e.g. DeFelice and DeHaan, 1977; DeHaan and DeFelice, 1978a,
1978b).
Similarly, observation of the cyclic output from any
cardiac oscillator shows that the activity is not.strictly
periodic:
there are slight variations in the interbeat interval
from cycle to cycle (e.g. Bouman et al., 1982).
If one accepts the
fact that the transmembrane potential is generated by the opening
and closing of single ionic channels, there are at least four
levels at which one can think of a stationary cardiac membrane in
the absence of environmental fluctuations:
1-37
0
(i)
One can construct a deterministic set of Hodgkin-Huxley
like differential equations and then add a stochastic
term to represent membrane noise.
( ii)
One can build an inherently stochastic model
us~ng
the
statistical properties of single channels (experimentally
measured or deduced from the macroscopic rate constants)
and then reconstruct the macroscopic action potential
with its now 1nherent microscopic voltage fluctuations
(e.g. Cl ay and DeFel ice, 1983; DeFel ice and Cl ay, 1983).
(iii}
In principle, one could build a quantum mechanical model
of each of the several ionic channels, obtain the
statistical properties of the currents from the wave
functions of the ions flowing through the channels, and
0
then reconstruct the macroscopic behaviour.
(iv)
One could replace the (Copenhagen) probabilistic
interpretation of quantum mechanics with a deterministic
interpretation that provides a statistical description at
the molecular level for the single channel behaviour, and
then reconstruct the macroscopic action potential
(incidentally winning a Nobel prize!).
Only in case (iv) above can one truly say that the irregular,
nonperi odic dynamics displayed by the cardiac system is
deterministic.
Nevertheless, it may well be that, in the same way
that one speaks of cardiac oscillators, understanding that some
0
1-38
0
microscopic voltage f1 uctuations are present, one can also speak of
cardiac strange attractors.
In mathematical systems, deterministically nonperiodic
dynamics generally occurs at one particular combination of the
various parameters in the system.
An infinitesimal change in any
of the system parameters can change the qualitative form of the
dynamics.
Thus, systems of differential equations are often
structurally unstable at values of the parameters where a
quasiperiodic orbit or a strange attractor exists (e.g. see t·1oser,
1969; Guckenheimer and Holmes, 1983). This fact therefore also
adds another twist to the question:
"Should deterministically
nonperiodic dynamics be observable in physical and biological
systems?n
0
Problems with measurement also cloud the issue.
Since
measurements can be made only to a finite number of decimal places,
can nonperiodic sequences· ever really be demonstrated in
experimental work? The old conundrum of whether the length of a
particular stick is rational or irrational again raises its head.
Again, this measurement problem is a fundamental one involving
quantum mechanics (in particular, the uncertainty principle).
While nonperiodic dynamics with sensitive dependence on
initial conditions may be taken as the litmus test for chaotic
dynamics, chaotic dynamics has several other features that can
be illustrated with the simple example that I present in section
(i)b
below; and which are more amenable to experimental
1-39
observation.
0
I have found it convenient to divide up the systems that
display chaotic dynamics into six main classes.
I now describe
experimental and numerical evidence for the existence of chaotic
dynamics in each of these six classes of systems.
(1) Chaotic Dynamics in Periodically Forced Oscillating Systems
a.
Chaotic Dynamics in Mathematical Models of Periodically Forced
Limit-Cycle Oscillators
Periodic forcing of limit-cycle oscillators can lead to
complicated dynamics which can even be nonperiodic.
Among the
earliest studies in this area were those of Cartwright,
L~ttlewood,
and Levinson previously mentioned in which the van der Pol
oscillator was forced with a high-frequency sinusoid.
In fact, the
behaviour of the periodically forced van der Pol oscillator is
chaotic ( Hol mes and Rand, 1978; Levi, 1981; Guckenheimer and
Holmes, 1983).
~Jumerical
studies using digital computers have revealed the
existence of chaotic dynamics in several other periodically-forced
limit-cycle oscillators.
(i)
These include:
the Brusselator- a two dimensional model of an
oscillating chemical reaction (Tomita and Kai, 1978a,
1978b; Kai and Tomita, 1979; Tomita and Kai, 1979;
1-40
Broomhead,
0
(ii)
~tCread~e,
and Rowlands, 1981;
Ka~,
1981)
a Duffing-van der Pal oscillator (Coullet, Tresser, and
Arneodo, 1980; Kawakami, 1982);
{iii)
the Bonhoeffer-van der Pal (BVP} or Fitzhugh-Nagumo (FN)
equations - a simplified two-dimensional model for
excitable biological membranes modified to produce
spontaneous activity (Guevara et al., 1983);
(iv)
simple impulsively-forced, two-dimensional limit-cycle
oscillators (Zaslavsky, 1978; Guevara and Glass, 1982;
Hoppensteadt and Keener, 1982);
(v)
the Hodgkin-Huxley equations- a four-dimensional model
of quiescent squid axon modified to produce spontaneous
activity (Guevara et al., 1983);
(vi)
the 14cAll ister-Nobl e-Tsien equations - a nine-dimensional
model of spontaneously active cardiac Purkinje fibre
(Guevara, unpublished).
One of the earliest studies to find very complicated behaviour in a
periodically forced oscillator was that of Ueda, Hayashi, and
Akarmatsu {1973), who simulated a sinusoidally forced resonant
circuit cantai ni ng a negative-resistance element on an anal oglie
computer.
0
1-41
b.
~
Chaotic Dynamics in a Prototypical Peri odical1 y Forced Lim1tCycle Oscillator
I will now present and discuss the behaviour of a simple
mathematical system that illustrates several features of chaotic
dynamics.
A more complete description of this work can be found in
Guevara and Glass (1982).
Investigation of this system has also
been carried out by Hoppensteadt and Keener (1982).
The system
considered is a mathematical model of a simple two-dimensional
limit-cycle oscillator that is periodically forced by a train of
impulses.
There are striking similarities between the behaviour of
this periodically forced oscillator and that of the periodically
forced cardiac system described in CHAPTER 4.
0
consideration of the dynamics displayed by
Indeed,
th~s
model preceded and
thus has proven invaluable in guiding' the experimental work
detailed in CHAPTER 4.
The oscillator is described in polar coordinates by the
equations:
d~
at
= 2'lf
( 1-1)
dr
(ff
=
ar(l-r),
where 9 is the angular coordinate {-=
< ~ < ~},
coordinate and a is a positive real number.
1-42
r is the radial
The unit circle forms
a limit cycle that is globally attracting for all initial
0
conditions except for the equilibrium point at the origin.
Note
that the unperturbed oscillator has unit period and its state can
be parameterized by an angular coordinate $
q, =
\P
Z1r (modulo 1 ).
( 1 -2)
The variable$ is called the phase of the oscillation (0 ( q,
1 ).
<
Perturbation away from the limit cycle results in relaxation
back to the stable limit cycle at a rate that depends on the
parameter a.
I consider the limiting case of a
In this case,
+ ~.
following perturbation away from the limit cycle, there is an
instantaneous relaxation back to the 1imi t cycle along a radial
direction.
In what follows I consider perturbations consisting of
impulses of magnitude b which are directed parallel to the x-axis
(Fig. 1-1 ).
Thus, the effect of a single impulse is to
instantaneously reset the phase of the oscillator.
Calling
q,
the
old phase of the oscillator immediately preceeding the
perturbation, and e the new phase of the oscillator immediately
following the perturbation one has
0
= g( q, 'b) '
1-43
( 1 -3)
0
Figure 1-1.
The limit-cycle oscillator and the effect on the oscillator
of stimulation with an impulse of magnitude b.
The unit
circle forms a limit cycle which is globally attracting for
all initial conditions except for the origin.
The stimulus
of amplitude b instantaneously resets the phase of the
oscillator from a phase$ prior to stimulation to a phase
following the perturbation.
Identifying the x-axis variable
with membrane potential , stimuli with b
0
>
0 may be regarded
as being analogous to depolarizations, stimuli with b
are analogous to hyperpolarizations.
<
0
1-44
0
Figures (1-1} to (1-3)
are slightly modified from figures originally drafted by
Leon Glass and Burt Gavin.
0
0
11
-I~
If
11
-IN
11
where the function g is called the phase transition curve or PTC
0
(Kawato and Suzuki, 1978; Kawato, 1981).
trigonometry, an analytic expression for
(Guevara and Glass, 1982; eqn. (5)).
5 different values of
Using some simple
g(~,b)
can be obtained
Figure l-2A shows the PTC for
o.
Consider now the effect on the oscillator of delivering a
periodic train of impulses, with a timeT between stimuli.
Let
~4
I
be the phase of the oscillator immediately preceding delivery of
the ;tll stimulus.
The phase of the oscillator immediately
preceding delivery of the next stimulus will be given by
<jli+l = g( cf>pb) +
T
( 1 -4)
•
This equation is a one-dimensional finite-difference equation,
depending on the two parameters b and r.
Thus, the two assumptions
of impulsive perturbation and of infinitely fast relaxation back to
.
the limit cycle following a perturbation allow the investigation of
the dynamics of the periodically forced two-dimensional system to
be reduced from a two-dimensional to a one-dimensional problem.
Equation (1-4} describes a map
T:~;+~i+l
which is called the
Poincare map (also called the first return map or the phase advance
1-46
0
Figure 1-2.
A.
The phase transition curve (PTC) for five
different values of b•
B.
.
The Poincare map for three different values of
held fixed at b
=
-1 .30.
0
1-47
<
with b
0
1·0 . . . . - - - - - - - - - -
A
e o·5
b = -1·1
b= -0·9
0·5
0·0
1·0
cp
0
B
T=0·35
4't+t
0·5
0·0
0·5
cpl
1·0
map).
0
The dynamics in response to periodic stimulation (i.e. the
evolution of the
<P;
starting with some
found by iterating eqn. (1-4).
init~al
value
$ )
0
can be
For fixed b, the Poincare map is
obtained by vertically translating (modulo 1} the PTC by an amount
r, as shown in Fig. 1-28.
Numerical iteration of the Poincare map at many different
combinations of band r yields Fig. l-3A.
In each "phase-locking
zone" 'in the (b,r) parameter space that is labelled
~J:f~,
periodic
dynamics occurs in which for every N impulses, the state-point of
the oscillator makes Mcomplete revolutions about the origin.
Note
that most of the area of the (b,r) parameter plane is filled with
periodic dynamics.
Figure l-3B shows a small region of Fig. 1-3A at an expanded
scale.
In the unlabelled region is found periodic and nonperiodic
dynamics.
Indeed, in this region, there are an infinity of
phase-1 ocki ng zones of very suall extent, in
.
~1hich
dynamics of arbitrarily high period can be found.
~~=i~
periodic
There are also,
theoretically, some values (b,-r) in the unlabelled region at which
nonperiodic behaviour with sensitive dependence on initial
conditions occurs.
Contiguous with and lying below the 4:M zones
of Fig. 1-3B are zones of the form
8:~1.
In fact, there are
infinitely ma.ny 2":t•1 zones with arbitrarily high n.
More recent
computations have shown that there is also a 3:3 zone in the
unlabelled region (Glass and Belair, unpublished).
Between the
accumulation boundary of the 2°:M zones and the 3:3 zone is a
1-49
0
Figure 1-3.
Phase-locking zones resulting from periodic delivery of
impulses of magnitude b at a frequency .- 1 • The areas not
labelled contain dynamics that is phase-locked and dynamics
that is not.
A.
Phase-locking zones for N ~ M, N ~ 3.
The borders
between the 1 : 0 and 1 : l , 1 : 0 and 2: 0 , 2: 0 and 2: 1 , 2: 1
and 2:2, and 2:2 and 1:1 zones are determined
analytically, as are the borders of the 1:0 and 1:1
zones down to b
0
= 0.
The other borders are found
numerically.
B.
Phase-1 ocking zones for N ~ t4, N " 4 over a more
limited region of (T,bl parameter space than shown in
panel A.
The upper borders of the 3:2 and 4:3 zones
are uncertain, as indicated by the interrupted lines
(same for panel A).
In panel A, the borders of the 3:1, 3:2, and 2:1 (for
b "1.05) zones were determined numerically by Lean
Glass, as were the borders of the 3:2 and lower 4:3·
zones in panel B.
0
l-50
2 5
....----.----..,.. ------r----,
A
0
1·5
b
1·0
0·5
00
0·25
0·50
T'
0
B
1·3
b
1·2
1·1
10
region which is sometimes called the chaotic zone or chaotic region
0
in which phase-locking zones of the form 2n:M and 2n+l :M are found,
as well as regions of "banded chaos"," ser:1ipedodici ty", or "noisy
periodicity'' (Lorenz, 1980).
Bistability of periodic orbits also
occurs in the unlabelled region, with phase-locking zones
overlapping so that two different N:M patterns can be found at the
same point in the (b,T) parameter plane (Hoppensteadt and Keener,
1982; Glass and Belair, unpublished).
In contrast, for b<l, only periodic and quasiperiodic
dynamics exist.
In addition, bistabil ity and sensitive dependence
on initial conditions
do not occur for b<l.
that the dynamics is chaotic only for b>l.
Thus, one might say
Note however, that for
b sufficiently large, only periodic dynamics is found (Fig.
1-3).
Thus, in this model, chaotic dynamics exists only at intermediate
0
stirnul us amp 1 i tu des.
The phenomena 1 ogy seen in different systems undergoing chaotic
dynamics shows certain common features.
I now identify six of
these features, illustrating them with reference to the particular
case of a periodically forced 1 imi t-cycl e asci 11 a tor.
( i)
The dynamics is deterministic.
The mathematical
equations describing the system contain no stochastic
terms:
once the initial conditions are set, the
evolution of the dynamics is predestined.
(ii)
The dynamics is densely packed.
c
1-52
An exceedingly small
change in any of the properties of the oscillator itself
0
or in the frequency or amplitude of the periodic
stimulation can lead to qualitative changes
(" bifurcations
(iii)
11
}
in the dynamics.
The dynamics can display "sensitive dependence on initial
conditions".
Exceedingly small changes in initial
conditions (such as the exact phase in the spontaneous
cycle at which the driving oscillator is turned on) can
lead to one of two or more different behaviours, without
any alterations to either the forcing or the forced
oscillator.
However, these different behaviours are
often unstable.
(iv)
0
Bistability can be present.
One of two stable periodic
behaviours occurs, depending on i nit~ al conditions.
(v)
The dynamics which displays sensitive dependence on
initial conditions is nonperiodic.
For certain
combinations of stimulation frequency and amplitude, the
driven oscillator displays a nonperiodic time series;
that is, one that does not repeat in finite time.
Also,
periodic sequences of arbitrarily long length can be
found, which are impossible to distinguish from aperiodic
sequences in numerical and experimental work.
Quasiperiodic dynamics- which also has a nonperiodic
time series- can also be found.
(vi)
The dynamics becomes chaotic by fo11 owing one of a few
1-53
well~characterized
0
"routes to chaos" (Eckmann, 1981 ).
That is, there is some stereotyped sequence of
bifurcations from one type of periodic or quasiper1odic
behaviour to another that precedes the onset of
· nonperiodic or turbulent behaviour.
For example, as one
moves vertically down through the (T,b) plane of Figs. 1
-3A and 1-38 for T = 0.65 starting from b
= 2.0, the
period-doubling route to chaos is found; zones of the
form 2°:M with an increasingly higher value of n are
found, until finally the chaotic region is entered.
The
three major routes to chaos that have been described to
date are the quasiperiodic route (Ruelle and Takens,
1971;
~lewhouse,
Ruell e, and Takens, 1978), the period-
doubling route 01ay, 1976;
0
Feigenbaum, 1978), and the
intermittent route (Mannevi11e and Pomeau, 1979).
routes to chaos have also been described (e.g.
instability" : Langford et al • , 1980;
1980; "crises":
11
Other
Soft-mode
Iooss and Langford,
Grebogi, Ott, and Yorke, 1982;
Ikezi,
deGrassie, and Jensen, 1983}.
Thus, when I speak of chaotic dynamics, I mean complex,
tightly-compressed dynamics that displays several or all
of the above-named features, the most striking of which
are perhaps the nonperiodicity and sensitive dependence
on initial conditions.
1-54
c.
Experimental Work on Periodically Forced Oscillators
There have not been very many experiments that have shown
chaotic dynamics in a periodically forced osc ill ati ng system
(unless one regards the simulation of a forced oscillator on an
analogue or digital computer as an experiment on a very complex
electronic osci11 ator).
One of the earliest experiments to show what is now generally
regarded as chaotic dynamics was one involving Rayleigh-Benard
convection.
This form of hydrodynamic flow occurs when a fluid
layer that is held between two horizontal plates is heated from
below in the presence of a gravitational field.
induced at a critical Rayleigh number.
Periodic motion is
Periodic modulation of the
temperature difference between the two plates (a parametric
modulation) leads to nonperiodic dynaQics (Gollub and Benson,
1978).
Curry {1978) developed a model for Rayleigh-Benard
convection in which the system equations (a system of partial
differential equations derived from the Navier-Stokes equations for
incompressible fluid
flow
and the heat conduction equation) were
reduced by using separation of variables and truncation of higherorder Fourier modes to a 14-dimensional system of ordinary
differential equations.
Small amplitude modulation of the
dimensionless Rayleigh number in this model yields results
qualitatively similar to those seen in the experiments of Gollub
0
1-55
and Benson (Curry, 1979}.
0
Other experiments
~n
which nonperiodic time series have been
found have been carried out on periodically forced neural (Hayashi,
Nakao, and Hirakawa, 1982; Hayashi et al., 1982) and cardiac
(Guevara, Glass, and Shrier, 1981;
·oscillators.
Glass et ai., 1984)
The study of Gollub and Benson (1978) showed the
quasiperiodic route to chaos;
Guevara, Glass, and Shrier (1981)
showed the period-doubling route to chaos; Hayashi, Nakao, and
Hirakawa (1982) based their identification of chaos on the theorem
of Li and Yorke (1975); Glass et al. (1984) showed the perioddoubling and intermittent routes to chaos.
In certain circumstances, experimental work (Glass et al.,
1984: see CHAPTER 5) or numerical simulation (e.g. Tomita and Kai,
0
1978a, 1978b, 1979; Guevara et ·al., 1983) shows that the
periodically-forced oscillator yields a 11 Single-humped11 map of the
type that is known to be capable of generating chaotic behaviour
"
(Sarkovskii,
1964; Li and Yorke, 1975; Collet and Eckmann, 1980).
An alternative approach to direct numerical simulation of the
trajectory in phase space of the forced oscillator is to
numerically study the behaviour of one-or two-dimensional
difference equations that can be obtained from the system equations
by making certain approximations (Zaslavsky, 1978;
Coullet,
Tresser, and Arneodo, 1980; Guckenheimer, 1980b; Broomhead,
r~1cCreadie,
and Rowlands, 1981; Levi, 1981; Alseda, Llibre, and
Serra, 1982;
0
Glass and Perez, 1982; Perez and Glass, 1982; Guevara
1-56
and Glass, 1982;
0
Belair and Glass, 1983).
obtain a one-dimensi anal
Shrier, 1981;
m~p
It is also possible to
in experimental { Guevara,
Glass et al., 1984:
Glass~
and
see CHAPTER 5) and modelling
(Guevara et al., 1983) work which allows prediction of the response
of an oscillator to periodic delivery of a pul satil e stimulus given
its response to that same stimulus delivered at various phases in
its cycle.
0
0
1-57
{ii)
Chaotic Dynamics in Periodically Forced Quiescent Systems
As indicated in the last section, chaotic dynamics can result
from periodic forcing of a system that intrinsically oscillates.
Chaotic dynamics can also be produced by periodically forcing a
quiescent system.
In this case, there is generally a sequence of
bifurcations that takes place as the stimulus intensity or
frequency is changed, first leading to the appearance of various
periodic or quasiperiodic behaviours and then to the appearance of
a strange attractor.
(A caveat:
in the physics 1iterature, the
term n oscillator" is often applied to systems which are not
spontaneously oscillating but which nevertheless have a
characteristic frequency.
These are often systems that have a
single stable equilibrium point with complex eigenvalues;
a damped
oscillatory return to the equilibrium point occurs if the system is
perturbed from rest.
.
The classic example of this kind of system is
the damped simple harmonic oscillator.)
Physical systems which do not oscillate spontaneously and
which apparently show chaotic dynamics experimentally in response
to periodic fore i ng include
( i)
(ii)
{iii}
{iv)
the damped pendul urn (Koch et al • , 1983);
the buckled beam {Moon and Holmes, 1979; Moon, 1980);
the cantilivered rod (Moon, 1980);
acoustical systems ( Lauterborn and Cramer, 1981; Smith,
Tejwani, and Farris, 1982; Smith and Tejwani, 1983};
1-58
0
(v)
the varactor-LRC resonator (Linsay, 1981; Hunt, 1982;
Jeffries and Perez, 1982; Perez and Jeffries, l982a,
1982b; Testa, Perez, and Jeffries, 1982;
Brorson,
Oewey, and Li nsay, 1983; Cascai s, Di 1ao,
and Noronha da
-
Costa, 1983; Ikezi; deGrassie, and Jensen, 1983);
(vi)
the Josephson junction 0·1iracky, Clarke, and Koch,
1983);
{vi 1)
electronic analogues of the Josephson junction ( Yeh and
Kao, 1982) and of the damped pendulum (D'Humieres et
al., 1982}.
I know of no experimental work on the periodic forcing of a
0
quiescent biological system which has
dynamics.
demonstl~ated
chaotic
Chaotic behaviour has also been found in studies carried
out on numerical models of periodically forced quiescent systems.
These include models of
(i)
the damped pendulum (Leven and Koch, 1981; Mclaughlin,
1981; Arneodo et al., 1983);
( ii)
a particle lying in an anharmonic potential well
(Huberman and Crutchfield, 1979; Crutchfield and
Huberman, 1980;
Arecchi and Lisi, 1982; Steeb, Erig,
and Kunick, 1983);
(iii)
a particle in a spatially periodic potential (Huberman,
Crutchfield, and Packard, 1980);
1-59
0
( iv)
the Josephson junction (Huberman, Crutchfiel d,. and
Packard, 1980; Ben-Jacob et a.l., 1982; Yeh and Kao,
1982; Miracky, Clark, and Koch, 1983 (and references
therein));
(v}
(v'i)
(vii)
the buckled beam (Holmes, 1979};
the cantilevered rod (Moon, 1980);
the varactor LRC resonator (Brorson, Dewey, and Linsay,
1983; Rollins and Hunt, 1983};
(vi i i)
(ix)
neural membranes (Jensen et al., 1983};
excitable membranes (Rossler, Rossler, and Landahl,
1978; Keener, 1981 a, 1981 b).
0
Work on the forced Duffing equation was initially carried out by
Ueda, Hayashi, and Akamatsu (1973) and later extended by Ueda
(1979, 1980a, 1980, 1981 ) , Ueda and Akamatsu (1981), Hol mes and
associates (Ho1mes, 1979; Guckenheimer and Ho1mes, 1983; Holmes and
Whitley, 1983 {and references contained therein)), and Sate, Sano,
and Sawada {1983}.
In particular, chaotic dynamics arises due to
the presence of transversal homoclinic orbits {Guckenheimer and
Ho1mes, 1983).
As in the case of periodically forced oscillators,
one-dimensional maps are a great aid in understanding the very
intricate behaviours observed when a quiescent system is
periodically forced (e.g. Perez and Jeffries, 1982; Rollins and
Hunt, 1982; Turschner, 1982; Holmes and Whitley, 1983).
Keener (l98la, 198lb) has shown that a simple two-factor
1-60
(excitation-refractoriness) model of the quiescent but excitable
atrioventricular node can produce chaotic dynamics under periodic
stimulation.
Unlike most of the reports cited above, this study
was an analytical one that relied on the iterative properties of a
class of one-dimensional maps (Keener, 1980).
Iteration of
human atrioventricular nodal recovery curves yields the same class
of maps and therefore a similar kind of chaotic behaviour (Guevara,
unpub 1 i shed.).
(iii) Chaotic Dynamics in Systems That Are Not Periodically Forced
A third class of systems in which chaotic dynamics has been
exp~rimentally
forced.
demonstrated are systems which are not periodically
In these systems there is a sequence of
bifurcat~ons
to
various periodic or quasiperiodic behaviours culminating in
nonperiodic dynamics as some parameter in the system is changed.
This parameter might be the Rayleigh number in experiments
involving the dynamics of fluids or the flow rate of reactants into
the reaction chamber in experiments involving chemical reactions.
Hydrodynamic systems belonging to this third class of systems
were among the first systems in which chaotic dynamics was
demonstrated.
One hydrodynamic system which has been studied a
great deal is circular Couette flow, where a f1 uid rotates in the
space between two concentric cylinders, the outer one being hollow
and usually fixed in space, the inner one solid and rotating.
the bifurcation parameter- the Rayleigh number- is increased,
1-61
As
0
there is a sequence of
bifurca~ions
.
involving periodic and
quasiperiodic motions that eventually culminates in a nonperiodic
motion, as evidenced by an elevation in the base-line of the
velocity power spectrum C'broadband noise11
).
Since the mid-1970's
there have been many studies on this system looking at nonperiodic
dynamics (Gollub and Swinney, 1975; Sw.inney and Gollub, 1978;
Fenstermacher, Swinney, and Go11ub, 1979; Fenstermacher et a1 .,
1979; Gorman, Reith, and Swinney, 1980; L'vov and Predtechensky,
1981; Swinneys 1983 (and references contained therein)).
Modelling
of circular Couette flow produces power spectra similar to those
seen experimentally (Shennan and Mclaughl in, 1978) and can produce
complicated periodic and nonperiodic dynamics (L'vov and
Predtechensky, 1981 ) •
Another hydrodynamic system which has been investigated is
Ray1 eigh--:Benard convection taking p1 ace in water ( Fenstemacher et
al., 1979; Gollub.and Benson, 1979; Gollub and Benson, 1980;
Gollub, Benson, and Steinman, 1980; Giglio, Musazzi, and Perini,
1981 ) , in oi·l ( Berge et al • , 1980; Duboi s and Berge, 1981 ; Duboi s,
Berge, and Croquette, 1982}, in liquid helium (Ahlers and
Behringer, 1978a, 1978b; Libchaber and Maurer, 1978; Maurer and
Libchaber, 1979, 1980; Arneodo et al., 1983; Haucke and Maeno,
1983), and in mercury (Fauve, Laroche, and Libchaber, 1981;
Libchaber, Laroche, and Fauve, 1982; Libchaber, Fauve, and Laroche,
1983).
In many of the above studies, the transition to turbulent
behaviour was identified with the onset of chaotic dynamics.
0
1-62
0
Deterministic numerical models of Rayleigh-Benard convection
have been successfully investigated.
As in experiment, there are
successive bifurcations to various periodic or
before nonperiodic dynamics sets in.
quasiperiod~c
orbits
One of the earliest models
ever to demonstrate deterministic nonperiodic dynamics was that of
Lorenz (1963).
This reduced three-dimensional model for Rayleigh-
Benard convection was generated in a manner similar to that
described earlier for another model of Rayleigh-Benard convection
(Curry, 1978).
There has been intensive theoretical and numerical
investigation carried out on the Lorenz model; much has been
discovered but a full description of the dynamics of the model is
yet to be attained {see for example:
Shimizu, 1978; Shimizu and
0
~lorioka,
Rossler, 1977; Morioka and
1978;
t·~anneville
and Pomeau,
1979; Shimada, 1979; Yorke and Yorke, 1979; Franceschini, 1980;
Sparrow, 1982 (and references contained therein)).
Extensions to
the original simplistic model of Lorenz were subsequently made by
Nclaughlin and
~1artin
(1974, 1975), by Curry (1978, 1979), and by
Franceschini ( 1983); these resulted in more complex models which
also demonstrated chaotic dynamics.
On the other hand, Rossler
(1976a, 1976d} found that simplification of the Lorenz equations so
that there was only one nonlinear cross term instead of two
produced a system which was still capable of generating chaotic
behaviour.
It is perhaps surprising that the Lorenz equations
arise in models of the segmental disc dynamo {Knobloch, 1981 ), the
laser (Haken, 1975; Graham, 1976), and the water wheel (Sparrow,
1-63
1982).
One final cautionary note:
a recent paper proposes that
some of the features of Rayleigh-Benard convection can be accounted
for by a stochastic (i.e. not deterministic) model {Greenside et
a1 • , 1982}.
Other early observations of chaotic dynamics were made in
chemical systems, most notably the Belousov-Zhabotinsky reaction
{Olsen and Oegn, 1977; Schmitz, Graziani, .and Hudson, 1977; Rossler
and Wegmann, 1978; Hudson, Hart, and Marinko, 1979; Schmitz,
Renola, and Garrigan, 1979; Yamazaki, Oono, and Hirakawa, 1978,
1979;
Vid~
et al., 1980; Pomeau et al., 1981; Roux et a1 ., 1981;
Turner et al., 1981; Simoyi, Wolf, and Swinney, 1982; Epstein,
1983;
Roux, 1983; Roux, Simoyi, and Swinney, 1983; Swinney, 1983;
other references can be found in the last four references cited}.
Modelling of the reaction steps and numerical simulation of the
resulting equations leads to nonperiodic dynamics (Tomita and
Tsuda, 1979; Turner et al., 1981 ).
Even very simplified reaction
kinetics can produce· dynamics similar to that seen experimentally
{Rossler, 1976b; Kuramoto, 1978;
1983).
Pikovsky, 1981; Rinzel and Tray,
Dynamical behaviours resembling those seen in these
chemical systems have been seen in a molluscan neuron treated with
the potassium-channel blocker 4-aminopyridine (Holden, Win1ow, and
Haydon, 1982);
similar behaviours can also be seen in ionic models
of cardiac tissue (Guevara, unpublished).
Once again, in many of the studies mentioned in this section,
the dynamics can be {at least partially) understood by
0
1-64
0
consideration of a one-dimensional map (e.g. hydrodynamic
turbulence: Lorenz, 1963; Yorke and Yorke, 1979; Berge et al.,
1980; Sparrow, 1982; chemical reactions: Pikovsky, 1981; Pomeau et
al., 1981; Simoyi, Wolf, and Swinney, 1982; Rinzel and Troy, 1983;
Roux, 1983; Roux, Simoyi, and Swinney, 1983; cardiac systems:
Guevara, unpublished).
(iv) Chaotic Dynamics 1n Systems of Two or More Coupled
Oscillators
A fourth situation in which chaotic dynamics can result is
when two oscillators are coupled together bi di recti onally.
Unlike
the case of unidirectional coupling considered in section {i)
above, with bidirectional or mutual coupling there is no driven
oscillator and no driving oscillator; instead, the activity of each
oscillator influences the other.
Chaotic activity has been found experimentally in a system of
two mutually coupled electronic oscillators (Gollub, Brunner, and
Danly, 1978; Ito et al., 1983) and in a system of two coupled
chemical reactors (Schreiber, Kubicek, and Marek, 1980).
It can a1 so be seen in rnodel s of two coupled e1 ectronic
oscillators (Gollub, Romer, and Socolar, 1980; Ita et·al., 1983;
Thoulouze-Pratt, 1983), two or more coupled chemical cells
(Rossler, l976c; Fujisaka and Yamada, 1978; Yamada and Fujisaka,
1978; Schreiber, Kubicek, and
i~arek,
0
1-65
1980; Schreiber and 1•1arek,
0
1982a, 1982b), and two coupled cardiac cells (Guevara,
unpublished).
Chaotic dynamics also exists in an analytic
treatment of two coupled van der Pol oscillators {Belair, 1982),
and in numerical modelling of two coupled BVP oscillators {Guevara,
unpublished).
(v)
Chaotic Dynamics in Systems With Time Delays
The four sets of systems considered so far have been modelled
by systems of ordinary differential equations.
The evolution of
the behaviour of the state-point of the system simply depends on
its present position in phase space.
0
However, if there are time
delays in the system, this is no longer the case and time-delay
differential equations must be used to model the system.
Chaotic dynamics has been experimentally demonstrated in an
acoustical system (Kitano, Yabuzaki, and Ogawa, 1983), in optical
systems (Gibbs et al., 1981; Hopf et al., 1982;
Nakatsuka et al.,
1983} and in electronic systems (Ikezi, deGrassie, and Jensen,
1983), all of which possess time delays.
In fact, in each
instance, the time delay is said to be crucial to the generation of
the chaotic dynamics.
Modelling work suggests that time delays can induce chaotic
behaviour in physiological systems (Mackey and Glass, 1977; Glass
and
~1ackey,
1979b; an der Heiden, t1ackey, and Wal ther, 1981; r·1ackey
and an der Heiden, 1982),
-~n
optical ring cavities ( Ikeda, 1979;
0
1-66
0
Ikeda, Da1do, and Akimoto, 1980; Hopf et al., 1982; Ikeda and
Akimoto, 1982; Ikeda, Kondo, and Akimoto, 1982; Nakatsuka et al.,
1983}, and in acoustical feedback systems (Kitano, Yabuzaki, and
Ogawa, 1983). Simple models, in which one-dimensional maps again
play a role, provide insight into the complex dynamics (an der
Heiden, f'.1ackey, and Walther, 1981; an der Heiden and t·1ackey, 1982;
Ikeda, Kondo, and Akimoto, 1982).
It is interesting to note that
the Lorenz equations for Rayleigh-Benard convection can be
reinterpreted as the equations of a particle subjected to a force
which, at any given time, depends both on the present position of
the particle and its position at all previous times {Shimi zu and
f.torioka, 1978).
Note that even though a system with a time delay
is of infinite dimension, there can be finite dimensional
structures such as limit cycles and strange attractors.
The
fractal dimension of the strange attractor can be as low as 2.13
{Farmer, 1982).
(vi)
Chaotic Dynamics in Spatially-Distributed Systems
Chaotic dynamics can also occur in spatially-distributed
systems where the variables are tunctions of both time and space.
In fact, several of the systems mentioned so far (e.g.
hydrodynamical systems and the heart) are distributed systems which
should properly be modelled by partial differential equations.
Chaotic dynamics has been shown to exist in models of systems
1-67
0
formulated as partial differential equations (e.g. Moon and Holmes,
1979; Keener, 1981a; Bishop et al., 1983a, 1983b; Moon, Huerre, and
Redekopp, 1983).
Interesting phenomena such as spatial period-
halving bifurcations and the suppression of temporal chaos by
spatial structure can be seen (Bishop et al., 1983a).
The converse
phenomenon of spatial chaos and temporal periodicity has also been
seen in modelling work (Kopell, 1980).
{vii) Concluding Remarks Concerning Chaotic Dynamics
Within the last five or so years, chaotic dynamics has been
seen experimentally in many systems:
0
physical (hydrodynamic,
el ectrohydrodynamic, acoustical, electronic, and optical),
chemical, and biological (neural and cardiac).
i~athematically,
chaotic dynamics can be produced by one- or two-dimensional finite
difference equations, systems of three or more ordinary
differential equations, time-delay differential equations, integrodifferential equations, and partial differential equations.
Chaotic dynamics can be seen in mathematical models of many systems
in which it is experimentally observed.
It is also seen in models
of systems where the experimental determination is yet to be made.
For example, the Einstein field equations admit chaotic solutions
in the mixmaster cosmological model (Chernoff and Barrow, 1983}.
This thesis grew out of the conviction that much of the
irregular dynamics appearing in the clinical electrocardiogram
might be the result of chaotic dynamics.
In what follows, I set
out to con vi nee the reader that this might indeed be so.
1-68
0
CHAPTER 2
PHASE RESETTING OF THE RHYTHM OF SPONTANEOUSLY BEATING AGGREGATES
OF EMBRYONIC CHICK VENTRICULAR CELLS BY A
CURRENT PULSE OF BRIEF DURATION
0
u
in order to gain any real insight ••• , it was absolutely
necessary to study first the comparatively simple deviations
caused by single stimuli •
11
A. Cushny and S. Matthews, 1897
1.
c
INTRODUCTION
A single premature beat can dramatically affect the rhythm of
the heartbeat.
An appropriately timed stimulus delivered to a
normal healthy heart can abolish rhythmicity by inducing
ventricular fibrillation {Mines, 1914), as can a premature beat of
endogenous origin {Smirk, 1949). A single premature stimulus can
annihiliate spontaneous activity in the isolated sinoatrial node,
in depolarized Purkinje fibre, or in diseased human ventricular
myocardium (Jalife and Antzelevitch, 1979; Jalife and
Antzelevitch, 1980; Gilmour et al., 1983); it can also terminate
triggered activity in canine Purkinje fibres or in fibres of the
simian mitral valve (Cranefield and Aronson, 1974;
Cranefield, 1976).
Wit and
However, the effect of a premature stimulus is
usually not quite so drastic; the result more generally seen is a
transient alteration in the cardiac rhythm, followed by a
reestablishment of the rate and rhythm initially present.
As mentioned in CHAPTER 1, it has been known for a long time
that a premature beat resets the phase.of the oscillatory activity
of the sinoatrial node or a subsidiary pacemaker to an extent that
depends upon the prematurity of the beat.
For instance, a
premature atrial contraction that arrives sufficiently late in the
cycle of the sinoatrial node will invade and capture it, thus
shortening the sinus cycle, whereas one arriving early enough
encounters refractoriness and entrance block and will instead
0
2-1
produce an electrotonic depolarization in the node that will delay
0
the time of appearance of the next spontaneous action potential
(Klein, Singer, and Hoffman, 1973; Kerr et al., 1980}.
A similar
situation occurs in pacemakers other than the sinoatrial node
{Klein, Cranefield, and Hoffman, 1972; Ferrier and Rosenthal,
1980; Gilmour et al., 1983).
Within the last decade, there has been renewed interest in
the phase-resetting of spontaneously active cardiac tissue produced
by premature stimulation with current pulses.
Sano, Sawanobori,
and Adaniya (1978) demonstrated that the cycle length of the rabbit
sinoatrial node could be either prolonged or shortened by
extracellular injection of a depolarizing current pulse.
Jalife et
al. (1980) studied the response of strips of kitten sinoatrial node
subjected to pulses of current delivered across a sucrose gap.
The
sign and magnitude of the change produced in the sinus cycle length
depends on the phase of the cycle at which the pulse is injected,
as well as on the polarity, amplitude, and duration of the current
pulse.
For example, a depolarizing pulse delivered early in the
cycle prolongs the cycle length, while the same pulse delivered
later on in the cycle shortens the cycle length.
Weidmann (1951}
injected current pulses into Purkinje fibres during diastolic
depolarization, and was the first to demonstrate clearly that a
depolarizing input could prolong the cycle length of spontaneously
active tissue, while a hyperpolarizing input could shorten the
cycle length.
0
Jalife and Moe (1976) later carried out a more
2-2
systematic study using canine false tendons in a sucrose gap
0
apparatus.
The above experiments have two factors in common which
complicate the analysis, interpretation, and modelling of the
results.
Firstly, the pacemaker is a distributed structure
consisting of a heterogeneous population of cells in which
propagation effects contribute to produce an asynchronous
activation of the individual cells of the pacemaker.
For example,
in the sinoatrial node, there are transitional cells between the
pacemaker proper and the surrounding quiescent atrial myocardium
(Pevet-Masson, 1979).
Secondly, the stimulus is a broad invading
wavefront of depolarization which does not affect all cells
equally.
For example, electrotonic shortening of action potential
duration can occur in some cells of the sinoatrial node in response
0
to a relatively late premature atrial contraction that is blocked
in the peri nodal fibres ( Mil 1er and Strauss, 1974; Dorticos et al.,
1978;
Steinbeck et al., 1978}.
A premature beat that is early
enough to encounter entrance block has a fractionated wavefront
which invades and captures only a minority of the cells in the
pacemaker.
Other cells display a subthreshold depolarization, the
magnitude of which varies from cell to cell.
Furthermore, shifts
in the site of the dominant pacemaker can occur in response to
premature stimulation (Bonke, Bouman, and van Rijn, 1969; Bonke,
Bouman, and Schopman, 1971;
Klein, Singer, and Hoffman, 1973).
These complications remain in studies of the sinoatrial node {Sano,
2-3
Sawanobori, and Adaniya, 1978; Jalife
0
et~.,
1980} and of
spontaneously active Purkinje fibre (Jalife and Moe, 1976) in which
extracellular injection of current was employed.
Even in cases
where intracellular injection of current was carried out
( sinoatrial node:
IJshiyama and Brooks, 1974; Purkinje fibre:
Weidmann, 1951), population and conduction effects necessarily
complicate the response.
Finally, ionic modelling of propagated
action potentials requires the numerical integration of a partial
differential equation.
Computer programs to carry this out are
not readily available, as well as being very time consuming (and
therefore also expensive) to run.
To obviate these complicating factors, I have decided to
investigate the-phase resetting of a cardiac oscillator in which
the presence of a heterogeneous cellular population and propagation
effects do not appear to play a significant role.
The preparation
employed is the embryonic chick ventricular heart cell aggregate,
whose phase-resetting behaviour has been hitherto studied (DeHaan
and Fozzard, 1975; Scott, 1979; Guevara, Shrier, and Glass, 1980;
Ypey, van Meerwijk, and DeHaan, 1982; van Meerwijk et al., 1984).
Since the cells that make up the aggregate are virtually
isopotential {DeHaan and Fozzard, 1975; OeFelice and DeHaan, 1977;
DeHaan and DeFelice, 1978a, 1978b; Ebihara et al., 1980; Mathias et
al., 1981), the aggregate can be modelled by a system of ordinary
differential equations {Guevara et al., 1982;
Shrier and Clay,
1982; Shrier et al., 1983; Clay, Guevara, and Shrier, 1984:
0
2-4
see
CHAPTER 3 of this thesis).
0
The aggregate may be viewed as an
analogue of the dominant centre of the sinoatrial node, which is
the pacemaker of the heart and consists of about 5000 cells
displaying virtually simultaneous electrical activity (Sleeker et
al • , 1980).
In what follows, I first outline the methods used to prepare
the aggregate and to study its response to premature electrical
stimulation.
Next, I summarize its basic el ectrophysiol ogical
properties and describe its response to intracellular injection of
a single pulse of current.
Finally, I discuss the experimental
observations, offer some interpretations, and indicate the
implications of this work for normal and abnormal electrical
activity in the intact heart.
0
2.
METHODS
(1)
Tissue Culture:
a.
Technique
Aggregates were prepared following the techniques described
in McDonald, Sachs, and DeHaan (1972) and in Sachs and DeHaan
(1973) with minor modifications.
Dissociation of the parent tissue
into single cells is obtained· by a multiple cycle procedure
2-5
(DeHaan, 1967, 1970), and reaggregation of single cells into heart
0
cell aggregates is obtained by a gyration process ( ~1oscona, 1961,
1965; Fischman and Moscona, 1971}.
White Leghorn chick embryos were incubated for 7 days at a
temperature of 37°C and a relative humidity of 85%.
the embryos were between stage 29 and stage 31.
At this point,
Each embryo was
removed from its shell and decapitated under sterile conditions.
The thorax was opened and the heart {together with the attached
great vessels) gently removed.
The heart was transected by a
transverse cut made just below the level of the atria.
The atria
were discarded and the apical portions of the ventricles of four to
twelve enbryos were then snipped into small fragments with a fine
iridectomy scissors.
0
The fragments were then transferred into a 25
ml Erlenmeyer f1 ask containing 10 ml of di ssoc i ati on medi urn at 37° C
and a magnetic stirring bar; the flask was then tightly sealed with
a silicone stopper.
The f1 ask containing the medium, ventricular fragments, and
magnetic stirring bar was then placed on a magnetic stirring table
(Thermolyne: Type 7200).
After ten minutes of combined mechanical
agitation and chemical digestion at ambient temperature, the
supernatant, which contains few viable myocytes (Josephson and
Sperelakis, 1982}, was discarded.
A second seven-minute
dissociation cycle was then carried out in the same way, using 3.3
ml of fresh dissociation medium, but the supernatant (now
containing many isolated myocytes) was poured into a 50 ml
2-6
centrifuge tube containing 20 ml of enzyme-inactivating medium
0
maintained at 37°C.
Two more dissociation cycles were carried out,
each time with 3.3 ml of fresh dissociation medium.
Before
transferring the supernatant to the centrifuge tube at the end of
each cycle the contents of the flask were gently agitated by
sucking the contents up and down in a pipette in order to help
break up the surviving ventricular fragments.
At the end of the last dissociation cycle, the entire
contents (including undigested fragments) of the flask were
transferred to the centrifuge tube.
The contents of the centrifuge
tube were then poured into the barrel of a 20 ml syringe, the end
.of which had been previously fitted with a polycarbonate filter
with 12.0 llm-diameter pores OJuclepore: No. 110616).
Gentle
pressure applied to the plunger of the syringe yielded a suspension
of single cells, including myocytes, red blood cells, and
fibroblasts.
The suspension of single cells was then compacted
into a pellet by centrifugation at about 170 g for 15 minutes.
The
pellet was resuspended in 1 ml of pre-gassed (10% 02 , 5% C0 2 , 85%
N2 ) maintenance medium.
After the cells were evenly suspended, a
Pasteur pipette was used to place an aliquot of the single cell
suspension on a haemacytometer slide.
After the cell density in
the suspension was determined, appropriate volumes of the
suspension were aliquoted out into two to six flasks, each
containing 3 ml of pre-gassed maintenance medium.
The resultant
number of eel 1 s in each flask was from 5 x 10 5 to 7 x 10s. The
2-7
f1
asks were then placed on a gyratory table (New Brunswick
Scientific Co: Model G2) which rotated at 70 r.p.m. with a stroke
of about
1~
cm and which was enclosed in an incubator maintained at
37°C.
b. Media
The dissociation medium consisted of 5.25 x l0- 5 g/ml
crystalline lyophilized trypsin (Worthington Biochemical, 245 U/mg)
and 5 x l0- 6 g/ml deoxyribonuclease I (Worthington, 9.1 x 10 4 U/mg)
in a ca++-Mg++-free, phosphate-buffered, balanced salt solution:
(millimolar) NaCl 116.0, KCl 5.4; NaH 2 P0 4 0.44, Na 2 HP0 4 0.95,
dextrose 5.6.
In some cultures the trypsin was replaced with
·collagenase 5 x 10-s g/ml (Sigma, Type 1, 146 U/mg).
Aggregates
formed from trypsin-dissociated cells will be referred to as
trypsin-dissociated aggregates; those formed from collagenasedissociated cells will be called collagenase-dissociated
aggregates.
rhe pH of the dissociation medium was adjusted to 7.3
with either 1 N HC1 or 1 N NaOH.
The maintenance medium, a modification of medium 818A {OeHaan
and Fozzard, 1975) consisted of 2% horse serum (Kansas City
Biological), which was heat-inactivated by placing in an oven for
30 minutes at 57°C, 4% fetal bovine serum {Grand Island Biological
(GIBCO}), and 20% medium 199 {GIBCO) in a bicarbonate-buffered
balanced salt sol uti on.
0
The final concentrations (millimolar) were
2-8
approximately: NaCl 116.0, KCl 1.3, CaC1
0
0.9, NaHC0 3 20.0, dextrose 5.5.
{Schering:
2
1.8, MgSO'+ 0.8, NaH 2 P0 4
The antibiotic gentamicin sulphate
Garamycin, 10 mg/ml) was also added to the medium to
yield a final concentration of 5 x 10-s g/ml.
The enzyme-inactivating medium was the same as the
maintenance medium, but with the following exceptions:
0% fetal
bovine serum, 10% horse serum, and approximately 4 mM KCl.
All
solutions were filtered with a sterile filter having a 0.45
~m­
diameter pore size (Nalgene: No. 245-0045).
Preparation of aggregates and formulation of solutions was
carried out by the technical staff mentioned earlier in the
acknowledgements.
(11)
Electrophysiology
After two to four days of gyration culture, the 3 ml of
maintenance medium, now containing several score reaggregates of
cardiac cells, was poured into a 35 mm x 10 mm plastic tissue
culture dish (Becton, Oickinson: Falcon 3001}. The aggregates
firmly adhered to the bottom of this dish within 20 minutes.
The
dish was placed in the well of a heater plate lying on the stage of
a dissecting microscope.
The temperature was continuously
monitored by a thermistor probe (YSI:
Model 43TD) and was
maintained to within one degree of 36°C by a switching regulator
built around a temperature regulator integrated circuit (National:
0
2-9
LM3911N}.
0
Slight adjustment of the controller was needed at the
beginning of and occasionally during each experiment.
Non-toxic
mineral oil (Witco Chemical: Klearol) was layered out on top of the
medium to prevent evaporation.
The medium was gassed from above by
a toroidal gassing ring at a flow rate of 200 ml/min with a gas
mixture of 5% C0 2 , 10% 02 , and 85% N2 • The bicarbonate buffer in
the medium maintained the pH at about 7.2 or 7.3.
Phenol red in
the maintenance medium (0.04 mg/ml} provided a rough continuous
estimate of the pH.
Under these conditions, more than 95% of the
aggregates in a dish will beat spontaneously
(r~Donald,
Sachs, and
OeHaan, 1972; Sachs and OeHaan, 1973).
Mean aggregate diameter was taken to be the mean of the minor
and major axes in the horizontal plane.
0
These were measured with
an optical graticul e placed in the microscope eyepiece.
Oi ameters
could be estimated accurately and repeatedly to within one half of
a minor division of the graticule
(±
19
~m).
The volume of an
aggregate was calculated assuming it to be a sphere with a diameter
equal to the computed mean diameter.
Using a micromanipulator, one cell of an aggregate was
impaled with a machine-pulled (Sutter Instrument: Model P-77
Brown-Flaming) glass microelectrode filled with 3t4 KCl (20-100 Pl\1
resistance).
Transmembrane potential was registered using a
negative capacitance compensated amplifier which had a gain of x50.
In these experiments, the capacity compensation was adjusted by
bringing the amplifier just to the edge of oscillation.
2-10
The medium
in the dish was maintained at virtual ground by being coupled to a
current-to-voltage converter (10-100 mV/nA) through an agar salt
bridge and a chlorided silver wire.
Pulses of current were
injected into the aggregate through the same microelectrode used
for recording the membrane voltage and their amplitudes measured by
the current-to-voltage converter.
nearest~
nA.
Currents were measured to the
A programmable stimulator (Frederich Haer: Pulsar
4i) was used in conjunction with a 22 Mn resistor to deliver
current pulses.
In some experiments, current pulses were injected
in a chopped fashion, at a frequency of 1kHz.
Within several minutes of the initial impalement, the
11
Sealing in process (Draper and Weidmann, 1951; Shrier and Clay,
11
1982} was complete, and spontaneous activity with stationary action
potential parameters could be ·obtained.
Voltage and injected
current waveforms were monitored on a digital oscilloscope
{Nicolet: Model 206) and recorded on an FM instrumentation recorder
at 3% ips {Hewl ett Packard: Hodel 3964A; 3dB frequency response at
3% i ps: OC-1250 Hz) for 1ater offl i ne analysis.
Impal ements could
· sometimes be maintained for several hours.
The protocol for investigating the response of spontaneous
beating to perturbation with a pulse of current was as follows.
A
pulse of current was delivered at a certain coupling interval after
a spontaneous beat.
This current pulse altered the interbeat
interval of the perturbed cycle.
After allowing ten beats to take
place to allow recovery back to control activity, another pulse was
0
2-11
injected at a coupling interval that was automatically incremented
(by a multiple of 1 msec).
cycle could be scanned.
In this way, the entire spontaneous
In some experiments, the coupling interval
was not systematically incremented, but was instead randomly
varied.
The above protocol could then be repeated for a different
current amplitude, duration, or polarity.
(iii) Data Analysis
Onl ine and some offl ine analysis was carried out using the
digital oscilloscope to measure i nterbeat intervals and coupling
intervals.
The sampling frequency was usually set to 1kHz, and
since the scope memory contained 4096 samples, approximately 4
seconds of data could be displayed on the oscilloscope screen.
Interbeat intervals could therefore be reproducibly measured with a
precision of
±
1 msec.
The bulk of the offline analysis was performed by an
automated system.
(i.e.
1:4
Magnetic tapes were played back at 15/16 ips
real time), 1ow-pass filtered at 500 Hz, and the voltage
waveform then sampled at 250 Hz by a Z80-based microprocessor
system (Cromenco: System III; California Data Corporation:
12 bit analogue-to-digital converter).
AD-100
The digitized waveform was
transferred over an RS-232 serial line at 9600 baud to a
minicomputer (Hewlett-Packard: Model HP1000 Series F) and stored on
digital magnetic tape.
Interbeat intervals were extracted out of
2-12
the digitized waveform by a pattern recognition program.
Raw data
and processed curves were output on a video terminal (Lear-Siegler:
r1odel ADM-3A fitted with a graphics board (Digital Engineering:
Model RG-512}} or on a digital plotter (Hewlett-Packard: Model
9872$).
With the exception of Fig. 2-lA, all of the experimental
voltage traces in this thesis were obtained by playing back the
tape-recorded signal to the digital osc ·ill oscope through a 1ow-pass
filter (Krohn-Hite:
Model 3323: 4-pole RC filter, 24dB/octave
attenuation) with a cutoff frequency set at twice the oscilloscope
sampling frequency.
The contents of the oscilloscope memory were
then reconverted to an analogue signal using a digital-to-analogue
converter internal to the oscilloscope and sent to an analogue X-Y
plotter (Hewlett-Packard:
3.
RESULTS
(f)
Spontaneous Actfvity
l~odel
7015B}.
In this study. stable spontaneous electrical activity was
recorded from 60 trypsin-dissociated and 12 collagenase-dissociated
aggregates, with mean diameters ranging from 95
~m
to 228
~m.
Aggregates in this size range are composed of approximately 6008,000 cells (Clay, DeFelice, and DeHaan, 1979}, 75-85% of which are
2-13
myocytes (Sachs and DeHaan, 1973}.
Sachs and DeHaan (1973) found
that extracellular space amounted to 20.4 ml/100 gm of
aggregate tissue, while Clapham (1979) found that it amounted to
3.6 ml/100 ml of tissue volume.
than 95
~m
Aggregates with diameters of less
were not used, since it is known that beat-to-beat
fluctuation in interbeat interval increases dramatically with
decreas.ing size for aggregates containing fewer than 125 cells
(Cl ay and DeHaan, 1979).
{An aggregate containing 125 cells is
about 60
Aggregates with diameters larger than
228
~m
in diameter.}
were not studied, because of the possible existence of
~m
voltage inhomogeneity and interstitial potassium accumulation in
such large preparations.
A 200
~m
diameter aggregate has about 2/3
of its cells in the outermost 3 cell layers (Clay and Shrier,
0
198la).
Necrotic cells are not seen in aggregates with diameters
of less than 250
~
{Williams and DeHaan, 1981).
Fig. 2-1A shows a typical recording of spontaneous electrical
activity.
In a separate study carried out on other aggregates
obtained from the same cultures as used in these experiments, the
visually monitored beat rate (mean
±
S.D.) of 104 trypsin-
dissociated aggregates was found to be 53.3
Guevara, and Shrier, 1983).
±
32.3 min-1 (Colizza,
While larger aggregates tend to beat
more slowly than smaller aggregates, there is a large scatter in
the beat rates of different aggregates of the same size.
The
action potentials of 14 trypsin-dissociated aggregates were found
c
2-14
Figure 2-1.
A.
Tracing of transmembrane voltage recorded during
spontaneous unperturbed activity in a trypsindissociated aggregate, illustrating the regularity
of beating.
All experimental traces in this
thesis are from trypsin-dissociated aggregates
unless otherwise labelled.
Since this impalement
lasted for several hours, and since there are
drifts in the recording system, the zero of
transmembrane potential difference can only be
estimated.
This consideration also holds true for
most of the other voltage tracings in this
thesis.
0
B. A single action potential from the same period of
spontaneous ac ti vi ty as shown in pane 1 A, with OS
•
= 31 mV, MOP = -88 mV, APO = 145 msec, V
max = 72
V sec- 1 • Panel A was obtained by playing back the
tape recorder (at !.i real time) directly onto an
analogue X-Y plotter.
Panel B was obtained by
transferring the contents of the digital
oscilloscope memory (sampling frequency 10kHz)
through a digital-to-analogue converter (internal
to the scope) to the analogue X-Y plotter.
(Aggregate #1: diameter = 114 urn. )
2-15
0
lsec
B
]
O.lsec
0
V
-som
•
to have (mean± S.D.) maximum upstroke velocity ( Vmax) of 120.9 ±
0
35.4 V sec-1, maximum overshoot potential (OS) of 28.0
action potential duration (APO:
diastolic potential) of 171.5
±
±
4.1 mV,
time from upstroke to maximum
27.6 msec, and maximum diastolic
potential {MOP} of -90.8 ± 6.3 mV {Colizza, Guevara, and Shrier,
1983}.
Fig. 2-18 shows an action potential demonstrating these
typical parameters.
The fast-channel blocker tetrodotoxin (TTX} at
a concentration of 10- 7 g/ml or the slow channel blocker D600 at a
·concentration of 10- 5 g/ml abolishes spontaneous activity in these
aggregates, as does i ne rea sing the external potassi urn concentration
to 4.5 mM (Col izza, Guevara, and Shrier, 1983}.
A general review
of other properties of the trypsin-dissociated aggregate
preparation can be found in OeHaan and OeFelice {1978b).
0
Aggregates formed from collagenase-dissociated cells have
been shown to have action potential parameters which are not too
different from those listed above for aggregates formed from
•
trypsin-dissociated cells: the one major exception is Vmax (mean±
s.o.,
25 aggregates), which is 24.3 ± 13.4 V sec- 1 (Colizza,
•
Guevara, and Shrier, 1983). Reduced values of Vmax have also been
found in collagenase-dissociated aggregates in two other studies
(Mackenzie and Standen, 1982; van Meerwijk et al., 1984).
Furthermore, TTX {10- 7 g/ml} did not block spontaneous activity in
six out of fourteen collagenase-dissociated preparations studied
(Colizza, Guevara, and Shrier, 1983). Thus, the fast sodium
0
2-16
channel does not appear to be functionally present in these
0
collagenase-dissociated aggregates to the extent
tha~
it is in
trypsin-dissociated aggregates, nor does it appear to be necessary
for spontaneous action potential generation.
Furthermore,
collagenase-dissociated aggregates tend to beat faster than
trypsin-dissociated aggregates of the same size (Colizza, Guevara,
and Shri er, 1983).
Figure 2-2A shows the i~ interbeat interval IBI; plotted as
a function of the interval number i for 826 consecutive beats
recorded during a period of unperturbed spontaneous activity.
The
interbeat interval (or cycle length) is the time between crossings
of 0 mV on the upstrokes of two successive action potentials.
For
these 826 beats, the mean interbeat interval was 686 msec and the
standard deviation (S.D.) was 10.2 msec, giving a percentage
coefficient of variation C (= 100 S.D./mean} of 1.48%.
shows the interval histogram.
Fig 2-28
Irregularly beating aggregates {as
judged by observing the f1 uc tuati ons in i nterbeat interval on the
oscilloscope) were occasionally encountered, but were not used to
obtain phase-resetting data.
One such aggregate had C = 4.92%.
Thus, the beat-to-beat regularity of the 7-day embryonic chick
ventricular aggregate (C
~
1-5%) appears to be better than that of
a single isolated embryonic chick ventricular cell, for which C"'
20% {Clay and DeHaan, 1979), but is not quite as good as that of
the isolated rabbit sinoatrial node, for which C < 1% (Jongsma and
0
2-17
Figure 2-2.
A.
Interbeat interval IBI 4 plotted vs. interval
number i from a period of unperturbed activity.
Number of beats = 826.
an interbeat interval is
B..
The precision in measuring
±
1 msec.
Unsmoothed interval histogram obtained from the
data shown in panel A.
Intervals are sorted into
bins and the number of intervals n in a given bin
is plotted against the interbeat interval IBI in
msec.
Number of interbeat intervals = 825, mean
interbeat interval
= 686
msec, standard deviation
= 10.2 msec, coefficient of variation = 1.48%, bin
size = 2 msec. This aggregate appeared to have a
typical beat-to-beat variability, as judged by
triggering on the upstroke of the action potential
and observing successive interbeat intervals on
the oscilloscope.
This ·was done in all impale-
ments to obtain a rough estimate of the
variability of the interbeat interval.
2-18
0
c.
Scattergram for the same data used in panel A
showing each of 824 interbeat intervals plotted as
a function of the immediately preceding interbeat
interval.
The straight line plotted through the
data is a least-squares !it to the data; it has a
slope of 0.92 and a coefficient of determination
(r2)
o.
of 0.922.
Plot of the first 51 serial correlation
coefficients Rj for the data shown in panel A.
E.
Plot of the Rj (0(j,50) for the same data, but
with the IBI.; randomly shuffled.
Note the· 1ack of
'
any carrel ati on for j ;Jrl.
Adjacent points ( j, Rj)
and. {j +1, Rj +l) are connected together by straight
lines.
0
(Aggregate #2:
2-19
diameter= 105
~m.)
A
0
750
IBii
675
(m sec)
B
D
100
2 r
I;
l
I
t
n
R·J
50
Q
0
650
c
IBI
(msec)
E
750
/
IBii+l
700
(m sec)
1
R· 0
J
,,
IBii
(msec)
0
0
750
750
0
50
Tsjernina, 1982).
0
Figure 2-2C is a scattergram showing each of 824 interbeat
intervals plotted against the previous interbeat interval.
Thus,
an interbeat interval that is longer (shorter) than average tends
to be followed by one that is also longer (shorter) than average,
but less so.
A straight line fit through the data resulted in a
line of slope less than 1 in each of four trypsin-dissociated
aggr.egates. whose spontaneous activity was analyzed.
The serial
correlation coefficients (Perkel, Gerstein, and Moore, 1967;
Jongsma et al., 1975)
R.J
=----------------------------
were calculated, where Rj is the jth serial correlation
coefficient, N is the total number of interbeat intervals, IBii is
the ith interbeat interval, and 11IT is the mean interbeat interval.
The coefficients Rj (1 < j < 50) were all positive and fell more or
less monotonically with increasing j (Fig. 2-20).
Random shuffling
of the IBI; destroys the dependence of IBI 1 on IBii-l and reduces
the Rj ( l<<j <50) to values close to zero (Fig. 2-2E).
Most aggregates displayed occasional "bursts" (sudden
2-20
increases in the spontaneous beat rate).
c
of bursts of varying
le~gths
Figure 2-3 shows examples
from four different aggregates.
The
increase in intrinsic frequency is accompanied by a marked increase
in the rate of diastolic depolarization;
there is typically a
small change in the maximum diastolic potential and little or no
change in the threshold potential.
Some bursts developed and died
away gradually (Fig. 2-3A,B); qthers were more paroxysmal (Fig.
2-JC,D).
Bursting could encompass anywhere from 3-5 action
potentials (Fig. 2-3A} to approximately 100 action potentials (Fig.
2-30).
(Bursting was not present during collection of the data
used to make up Fig. 2-2.)
(ii)
0
The Response to Perturbation with Single Depolarizing Pulses
Phase-resetting data was obtained at one and only one
d~polarizing
current amplitude in 9 trypsin-dissociated aggregates,
and at two. or more amplitudes in another 17 trypsin-dissociated
aggregates; these aggregates were taken from 20 cultures.
Since
the phase-resetting responses of only 8 collagenase-dissociated
aggregates were studied, I will not here report on their response.
Figure 2-4 shows the effect of injecting a single depolarizing
current pulse of 20 msec duration and 6.5 nA amplitude into a
spontaneously be·ati ng aggregate.
Depending on the phase of the
cycle at which it falls, this brief pulse of current can either
prolong (Fig. 2-4, upper trace} or shorten (Fig. 2-4, lower trace)
0
2-21
0
Figure 2-3.
11
Bursti ng
behaviour recorded in four different
11
aggregates.
Only the first half of the burst that
occurred in panel D is shown.
These voltage tracings
were obtained by transferring the contents of the
digital oscilloscope memory to the analogue X-Y
plotter.
Since the interval between samples was 2
msec (panels A,B) or 5 msec (panels C,O), the
upstrokes of the action potentials do not appear to be
as rapidly rising as they really are (contrast with
Fig. 2-lA, which was obtained without any digitization
process).
This artifact is present in many of the
experimental voltage tracings following in this
thesis.
(A:
aggregate #1, diameter = 114 11m; B:
#3, diameter = 190 11m; C: aggregate #4,
aggregate
diameter =
152 11m; 0: aggregate #5, diameter= 95Jlm.)
2-22
0
A
0
]mv
-50
1sec
B
D
0
0
Figure 2-4.
Effect on the spontaneous rhythm of an aggregate
produced by injecting a 6.5 nA amplitude current pulse
of 20 msec duration.
The stimulus artifact is the
off-seal e rapid vertical deflection in the tracing and
can be used as a marker for the time during which
current is injected.
The number to the 1eft of these
and other voltage traces in this chapter is the
coupling interval in milliseconds. ·
0
Upper trace:
T1 = 527 msec.
T0 = 469 msec, tc = 170 msec,
The perturbed cycle length T1 is
greater than the control cycle length T0 •
Lower Trace:
msec.
T0 = 467 msec, tc = 180 msec, T1 = 305
The perturbed cycle length T1 is less than the
control cycle length T0 •
Note that in both cases the
stimulus has 1i ttl e effect on the post-stimulus
cycle.
(Aggregate #1: diameter= 114
2-23
~m.)
0
PULSE
AMPLITU DE
6.5nA
COUPLIN G
INTERVA L
(m sec)
0
170
] mV
180
]
-50
1 sec
the interbeat interval of the perturbed cycle.
0
I call this
interbeat interval the perturbed interbeat interval, and denote it
by T1 • The coupling interval ( tc) is defined to be the time from 0
mV on the upstroke of the action potential immediately preceding
the stimulus to the beginning of the stimulus; the interbeat
interval of the spontaneous cycle immediately preceding the
perturbed cycle is denoted by T0 •
I refer to the cycle immediately
preceding (following) the perturbed cycle as the control (poststimulus) cycle.
Note that there is almost no residual effect of
the stimulus on the interbeat intervals of the cycles following the
perturbed cycle (however see section (vii) below).
Figure 2-5 (left panel) shows the effect of delivering
depolarizing pulses of 5 nA amplitude and 20 msec duration at
0
coupling intervals that are systematically incremented in 10 msec
steps.
The current waveform remained constant as the coupling
interval was changed.
Ten spontaneous beats are allowed between
trials in order to give the preparation enough time to recover back
to its control activity.
Stimuli delivered at coupling intervals
of less than about 100 msec had little effect on the interbeat
interval due to the low membrane resistance during phase 2, and the
voltage tracings are therefore not shown.
A stimulus delivered
later in the cycle {110 msec < tc < 170 msec) causes a prolongation
in the perturbed interbeat interval beyond the control value,
~hile
an identical stimulus delivered still later in the cycle (t) 180
c
msec) causes an abbreviation of the interbeat interval.
0
2-24
The
0
Figure 2-5.
Phase resetting at 3 different current amplitudes
(left panel: 5 nA; middle panel: 6.5 nA; right panel:
8 nA) produced by a 20 msec duration current pulse.
The coupling interval tc is
steps.
(e.g. tc
incre~ented.
in 10 msec
For stimuli sufficiently late in the cycle
~
200 msec in the right panel) the upstroke
of the stimulated action potential is obscured by the
stimulus artifact.
Note that the transition from
prolongation of cycle length to abbreviation of cycle
0
length occurs abruptly in the right panel (between
tc
=
160 msec and tc
= 170
msec).
One can also see
from the rightmost panel that the stimulus produces
little effect on the interbeat interval of the poststimulus cycle (tc = 170, 180, 190 msec).
aggregate as in Figs. 2-4, 2-6, 2-7, 2-8.
(Aggregate #1:
diameter
= 114
2-25
0
~m.}
Same
0
PVLS £
A.MPLJTI-'~!:
~
,.
...
6.5 nA
I
J_"'o"
CONTROL
"
~~
\
'
100m tu
COUP LING
ltHE RVA L
fm•.c l
110
j~~L-~~
120
130
r~J
.J
i
_j~~t
140
150
160
110
\J\ ___]'\
J\Jl ___~__J\ _
___0JL_r'L
_j
\_
L_ _
180
I !H)
200
210
I'\
_)1\\JI!L__o \._..-...-
_ i\ J lJ '\ _
_f \ IU\
.
~
/:
220
230
___ j\ _
_J\j~
1\Jl
1\~
j \\
_J
I
_ j\ J L _ f\__
S\JU\\__-_I\ju\___
_i\ju"\__11'\
--'
/f\__
\_
_f\J~L~--j
_}\jf\____
.J\Jf\
ii \~-__,f\uU
_j\j~L--
_j\J~\-~~-
_j\J~-~-
J \j
-
J\J~L -_ j
2<0
250
_ j\ /~\
~
\._. .--.- ----
i"
----'
IL
_J~
i
·.__.. .!
I
prolongations are associated with a decrease in the slope of
0
diastolic depolarization following the end of the current pulse.
This decrease becomes progressively greater as tc increases for
approximately tc
(T 1/T 0
= 1.12)
~
180 msec; however, the maximum prolongation
occurs for tc
= 150
msec.
If one thinks of the spontaneous activity as being generated
by diastolic depolarization to a fixed threshold voltage, there are
at 1east two other factors that influence the time to threshold
following a stimulus:
{i)
the take-off potential becomes more
positive with increasing \;
increases with increasing \·
(ii) the membrane slope resistance
These two factors conspire to make
the voltage attained at the end of a pulse more positive with
increasing tc.
Thus, for tc increasing in the range 150 msec " tc
( 180 msec, progressively decreasing interbeat intervals can be
obtained despite a progressively decreasing slope of diastolic
depolarization, since the membrane voltage is being brought
increasingly closer to threshold at the end of a current pulse.
Stimuli falling before the occurrence of MOP can prolong the action
potential duration (time from upstroke to t4DP); this is a separate
cause for prolongation of the interbeat interval.
Figure 2-5 {middle panel) demonstrates that increase in the
amplitude of the current pulse to 6.5 nA causes an increase in the
maximal prolongation and abbreviation observed, as well as a
decrease in the coupling interval at which prolongation of cycle
length changes into abbreviation of cycle length.
c
2-26
The transition
from prolongation to abbreviation also occurs more abruptly (i.e.
0
over a shorter range of coupling intervals) at this higher stimulus
intensity.
These effects are even more pronounced as the amplitude
of the current pulse is further increased to 8 nA, as shown in Fig.
2-5 (right panel).
At this current level, the transition from
prolongation of cycle length to abbreviation of cycle length takes
place w;th a change in tc of 1ess than 10 msec.
Figure 2-6A shows 10 randomly .selected superimposed cycles
fran a control period of unperturbed activity, while Fig. 2-68
shows superimposed tracings taken from ten repeated trials with a
16 nA amplitude pulse at each of two fixed coupling intervals.
At
this current level, the transition from prolongation to
abbreviation of cycle length is very abrupt, and can occur when
c
tc is changed by as little as 1 msec (see section (vi) below).
The responses shown in Fig. 2-68 are highly repeatable, in that
there is only a slight degree of variability apparent in the
response from trial to trial.
The same holds true for current
amplitudes of less than about 5 nA in this aggregate.
However, at
current amplitudes greater than about 5 nA but below about 10 nA,
there can be considerable scatter in the response in repeated
trials carried out at a fixed coupling interval, if tc is in the
range where the response is rapidly changing
fr~~
maximal
prolongation to maximal abbreviation of cycle length. This effect
is considered further in section (vi) below.
For a much larger current amplitude, the above effects
0
2-27
0
Figure 2-6.
A.
10 superimposed voltage tracings randomly selected
from a period of unperturbed control activity
immediately preceding the collection of the data
shown in panel B.
The upstrokes of the first of
the two action potentials occurring in each trace.
are synchronized.
B.
10 superimposed voltage tracings of perturbed
activity in response to a single 16 nA amplitude,
20 msec duration current pulse at each of two
different coupling intervals (tc = 130 msec and
tc = 140 msec).
(Aggregate #1:
diameter= 114
2-28
0
~m.)
0
200 msec
A
B
CONTROL
0
]mv
-50
COUPLING
INTERVAL
(m sec)
130
140
continue to become more pronounced, except that the size of the
c
maximum prolongation obtainable decreases and graded action
potentials (Kao and Hoffman, 1958) can be elicited (Fig. 2-7).
Prolongation in cycle length by an early stimulus (e.g. Fig. 2-7:
\ = 80 msec} is now attained by an increase
in action potential
duration, which is partially nullified by a decrease in the
diastolic time (time from MOP to next upstroke) of the perturbed
cycle. This decrease in di as tolie tii1Je is associated with an
~~DP
of the perturbed action potential that is more negative than
control, and with an increase in the slope of diastolic
depolarization.
The diastolic time of the cycle following the
perturbed cycle is increased for stimuli falling sufficiently late
in the cycle.
Figure 2-8 demonstrates that the current pulse
waveforrfl does not change as the coupling interval is increased.
Note that this current amplitude ( 24 nA) is small in comparison to
the peak sodium current, which is about 3000 nA (see CHAPTER 3}.
Effects qualitatively similar to those I have just described are
seen if the pulse duration is increased instead of the current
ampl i tude.
(iii) The Response to Perturbation with Single Hyperpolarizing
Pulses
Figure 2-9 shows the response to hyperpolarizing input at two
different pulse amplitudes.
These responses were qualitatively
2-29
0
Figure 2-7.
Phase resetting at a higher current amplitude (24 nA)
than that shown in Figs. 2-5 and 2-6.
Stimuli
delivered earlier than about 100 msec now have an
appreciable effect on the action potential duration.
Note also the appearance of graded action potentials
at, for example,
tc = 120 or 130 msec.
Thus, the
coupling interval which marks the border between
prolongation of. cycle length and abbreviation of cycle
1ength is now at about tc = 120 msec.
{Aggregate# 1:
diameter= 114
0
2-30
~m.}
0
/~',,
'
'
CO NT RO L
--=-=--
1-s0o'"\(
I
20 0m ," "
COUPLING
IN TU V A l
(m sec}
40
/i
;'"'-'
50
-- -'
60
70
J
80
)f\_
r11\
'
... ... .-'
N
______;
90
Q
lOO
\'- --- "-
no
12 0
~I\
I
__ _. )
'~
J\_
130
14 0
IS O
loo
17 0
18 0
'~
J
\
\
c
Figure 2-8.
Superimposed current pulse waveforms (unfiltered) from
the first five trials of the experimental series shown
in Fig. 2-7.
A negative deflection corresponds to
injection of a depolarizing current.
current
waveform{±~
The noise in the
nA peak-to-peak) originates
largely in the virtual ground current measurement
circuitry and in the tape recorder.
The current
waveform is typical of that seen in most experiments;
a time of about 5 msec is required for the current to
0
rise to or fall from its steady-state injected value.
(Aggregate# 1:
diameter= 114 JJm.)
2-31
0
0
20
0
m sec
. similar to those seen in 8 other aggregates.
0
The response is the
opposite to that produced by a depolarizing input, in that an early
stimulus now abbreviates the cycle while a later one lengthens the
cycle.
For the smaller stimulus strength (Fig. 2-9, left panel),
the abbreviation obtained is negligible(- 1%).
With increasing
stimulus intensity, the maximal prolongation and shortening
observed increase, and the transition from prolongation to
shortening of cycle length {which takes place in the region of the
action potential upstroke) moves to a larger value of tc and
becomes more abrupt {Fig. 2-9, right panel: tc
tc
=
560 msec and
= 0 msec). At the smaller stimulus intensity (Fig.
panel), the maximal prolongation occurs at about tc
=
2-9, left
520 msec
{T/To
= 1.09); lesser prolongations are obtained for\ ) 520
msec.
At the higher stimulus intensity, the degree of prolongation
obtainable continues to increase for stimuli delivered right up to
the end of the cycle.
.
A stimulus occurring before MOP is attained abbreviates the
cycle length largely by producing a decrease in action potential
duration (e.g. Fig. 2-9, right panel: tc
= 80
msec), while one
falling soon after MOP shortens the cycle by increasing the slope
of diastolic depolarization beyond the control value (e.g. Fig. 29, right panel:
tc
=
200 msec).
This abbreviation of cycle length
occurs even though the potential at the end of the current pulse is
more negative than it would have been otherwise (i.e. in the
absence of the pulse).
A pulse delivered later in the cycle (e.g.
2-32
0
Figure 2-9.
The effect of delivering a 20 msec duration
hyperpolarizing pulse on spontaneous rhythmic activity
in an aggregate.
Determinations with coupling
intervals 20 msec apart were made at two current
levels (left panel:
4.5 nA; right panel: 23 nA);
every second trial is shown.
The latest pulse in each
trial (tc = 560 msec) is delivered very close to the
end of the spontaneous cycle.
For the higher current
amplitude, there is an abrupt transition from
prolongation of cycle length at tc = 560 msec to
shortening of cycle length at tc
0
=0
msec.
The sharp
vertical deflection on the 11 of'P' of the stimulus
artifact in some traces in the right panel is due to
oscillation or "ringing" in an overcompensated
amplifier.
Since the chopped mode of current
injection was used during this experiment, the
effective values of the currents are quoted.
(Aggregate# 6:
diameter= 228
2-33
~m.)
4. S oA
0
C O N TR
l~mv
ot
C O U P liN
(\
\
)
_
_
\
_ ; \._
r
L
___;
G
t:Rv.} -. l
INTtts
ee
H1
0
40
80
12 0
160
200
0
-fL_}\L
~fLJ\_
~~l
_j}-J\_
~1\~
_j),_J\_
J~
i \-
r'\L
nl
_;~L.........-'
(\
___.)''jI f L
_1\]r--I'l
L
240
280
J2 o
360
400
440
1\
f\
----- ~
J
!I
480
L
-'~rA
_}\~((\_
_j\~~l
_ ! \ \. .. ._ -, r-J\-' L
1/
0
I
/\
'
~
y
\
{\
' \
'--
I
_!'~~'\_
Fig. 2-9, right panel: \
0
= 520 msec) produces a prolongation even
though the slope of diastolic depolarization is increased beyond
control following the pulse and the take-off potential is more
positive than at smaller coupling intervals, since the change in
voltage produced by the pulse is greater (due to the i ne rea se in
slope resistance).
(iv)
The Perturbed Cycle Length Function
The phase resetting data of Figs. 2-5 and 2-7 can be plotted
in a normalized fashion as shown in Fig. 2-lOA.
perturbed interbeat interval T1/T
normalized coupling interval t/T
0
is plotted as a function of the
0
0
The normalized
•
{The normalized coupling
interval is also called the (old) phase and is denoted by cp.)
I
call a functional fit through this data the perturbed cycle length
function and denote it by T1/T 0
respect to T
0
,
= g(cp).
Normalization is with
since the intrinsic interbeat interval of the
perturbed cycle would have been approximately equal to T in the
0
absence of stimulation (see Fig. 2-2C), and since R1 is the largest
serial correlation coefficient Rj for j)l (Fig, 2-20}.
In CHAPTER
5, I show that the response of the aggregate to periodic
stimulation can be predicted given the function g{!J>).
Figure 2-108 shows that increasing the stimulus duration has
qualitatively the same effect as increasing the stimulus amplitude
(different experiment from Fig. 2-lOA).
0
2-34
Since the stimulus
Figure 2-10. Plots of the normalizea phase-resetting data:
c
the
normalized perturbed interbeat interval T1 /T 0 plotted
vs. the normalized coupling interval tc/T 0 •
A.
The effect of i ne re as i ng the amp1i tu de of a fixed
duration {20 msec) depolarizing pulse from 5 nA
(e)
to 8 nA ( +) and then to 24 nA ( • ) • Pot nts
obtained at 8 nA for tc
plotted.
~
130 msec are not
For\ sufficiently large, the point
(tc/T 0 , T1/T 0 ) lies somewhere in the stippled
region between the two diagonal lines.
This data
is from the same experiment as that shown in Figs.
2-4 to 2-8.
0
B.
The effect of increasing the duration of a fixed
amplitude (10 nA) depolarizing pulse from 10 msec
(e)
to 40 msec (•).
The points 1n this panel and
in panels C and D were obtained by stimulating at
random phases in the cycle.
Not all data points
obtained have been plotted.
Same aggregate as in
panels B and D.
The intrinsic interbeat interval
of this aggregate was about 470 msec.
c.
Symmetry in the response to a depolarizing
hyperpolarizing
(0)
(e)
and
pulse of duration 10 msec and
ampl i tu de 10 nA.
0
2-35
c
D.
Asymmetry in the response to a hyperpolarizing
(D.)
and depolarizing ( _.) pulse of duration 20
msec and amplitude 10 nA.
Same aggregate as in
panels Band C.
(A:
Aggregate # 1, diameter = 114 wm;
Aggregate# 7, diameter= 228 wm.)
2-36
B-D:
1.5
1.5
A
..
..•..•
c
•
1.0
4-::::
1.0
·-· ...·. '· .. ......
I
") . . .
•
•••
•
0.5
0.5
•
20
M$eC
5
1.0
0.5
0.0
10nA 10 msec
8 nA
• 24
+
• D
H
0.5
0.0
1.0
Jj_
To
1.5
1.5
B
1.0
····~--=
D
•
••
..... '-:. . ..
•
1.0
...
•
0.5
0.5
lOnA
o.o
0
0.5
10
• 40
m sec
1.0
lOnA
0.0
0.5
20m$eC
• D
H
1.0
artifact obscures the ac.tion potential upstroke when the stimulus
0
is delivered late enough in the cycle to be a threshold stimulus
(e.g. Fig. 2-68, lower panel), the interbeat interval of the
perturbed cycle can only be estimated to within one half of the
pulse duration for such cycles.
Thus, for tc/T 0 sufficiently
large, the point (t /T , T1/T 0 ) falls somewhere in the stippled
c 0
regions of Fig. 2-10A,B.
Figures 2-10C and 2-100 contrast the response to depolarizing
and hyperpolarizing inputs.
For stimuli of small enough strength
(i.e. amplitude or duration), the response to a hyperpolarizing
stimulus is close to being the mirror image of the response to a
depolarizing stimulus (Fig. 2-lOC).
However, for stimuli of higher
amplitude or longer duration the mirror symmetry is lost (Fig. 2-
0
100).
(v)
Long Delays; Triggered Activity
The phase-resetting data presented in Figs. 2-4 to 2-7 were
all taken from one aggregate.
These results were typical of the
response to depolarizing current pulses seen in 14 other
aggregates studied at two or more pulse amplitudes.
However, much
longer prolongations of interbeat interval could be obtained in 2
other aggregates, both of which had spontaneous eye 1 e 1eng ths of
greater than 1 second.
(Only 3 preparations studied had interbeat
intervals of greater than 1 second.)
0
2-37
Figure 2-11 shows part of a
phase-resetting experiment in one of these 2 slower-beating
0
aggregates.
There is a range of coupling intervals for which
prolongations of greater than 50% above the control cycle length
{i.e. T1
>
1.5 T0 ) can be produced; in contrast, prolongations of
greater than 41% were never seen in any of the other 15 aggregates
which had intrinsic periods of less than 1 second.
prolongation at tc
The very long
= 570 msec is accompanied by oscillatory
behaviour of the transmembrane voltage in the subthreshold range of
potentials.
Long delays with subthreshold oscillatory activity can only
be produced in a narrow range of coupling intervals and stimulus
amplitudes.
An increase or decrease in the coupling interval of as
little as 10 msec can destroy the effect (Fig. 2-11).
0
The response
is variable, in that the long prolongation is of a different length
and might be seen at a slightly different coupling interval if the
phase-resetting run is repeated keeping the stimulus amplitude and
duration fixed.
Indeed, there can be significant fluctuation in
the response when repeated trials at a fixed coupling interval are
carried out:
threshold is attained at one or other of the maxima
of the subthreshold oscillation (Figs.
(tc
(T 1
= 620 msec)
= 5T 0 ), and
2-12,2-13).
Figure 2-14
shows one of the longest prolongations observed
shows that more than one cycle of subthreshold
oscillation can occur before threshold is attained.
Even though I have been able to obtain prolongations
amounting to several cycle lengths (e.g. Fig. 2-14: tc
0
2-38
= 620
msec),
0
Figure 2-11. Phase resetting in a preparation with a long intrinsic
interbeat interval (i.e.
>
1 sec) produced by a 9 nA
amplitude, 20 msec duration depolarizing pulse.
Note
the 1ong prolongation produced at tc = 570 msec that
is accompanied by oscillatory activity in the
pacemaker range of potentials.
(Aggregate# 8:
diameter= 170
2-39
0
~m.)
PULSE
AMPLITUDE
0
9nA
,......_.....,
lsec
COUPLI NG
• ~;,:;t· /~UllliJ Ls}·
7
520
wu m
560
570
ill~
580
J
590
600
610
0
620
Jll
W lllu
vL
UJJuJJL
JL}
~
'-'
0
Figure 2-12. Ten repeated trials at a fixed coupling interval of
570 msec.
The pulse amplitude was 9 nA and the pulse
duration 20 msec.
Threshold is attained either at the
first crest of the subthreshold oscillatory activity
(the group of 7 action potentials labelled "a") or at
the second crest (the group of 3 action potentials
labelled 11 b11 ) .
Same aggregate as in Figs. 2-11, 2-13,
2-14.
(Aggregate# 8:
diameter= 170 um.)
2-40
0
PULSE
AMPLITUDE
9nA
COUPLING
INTERVAL
E
a
)
( b >
(msec:)
l
0
mV
570
-so
lsec:
0
0
Figure 2-13. Thirty-one
cons~cutive
trials at a fixed coupling
interval of 570 msec;
consecutive trials are
vertically displaced.
The current pulse amplitude was
9 nA and the pulse duration 20 msec.
indicate the stimulus artifact.
The arrows
Again, the action
potentials following the stimulus cluster into two
groups, labelled nau and 11 b11
•
The tracings in Fig.
2-12 were taken from this group.
(Aggregate# 8:
0
diameter= 170
2-41
~m.)
0
PULSE
AMPLITUDE.
9nA
+
COUPLING
I N TERVAL
(ms ec)
570
t
0
lsec
<
a
)
( b )
Figure 2-14. Three trials from a phase-resetting experiment in
which the coupling interval was changed in 10 msec
steps.
The pulse amplitude was 6.5 nA, the pulse
duration 20 msec.
The action potential following
delivery of the stimulus fires on the first (tc
msec), second ( tc
=
570 msec), or third ( tc
=
msec) crest of the subthreshold oscillation in
c
membrane voltage.
(Aggregate# 8:
diameter= 170
2-42
0
~m.)
= 600
620
0
PULSE
AMPLITUDE
6.5nA
COUPLING
INTERVAL
(ms ec)
600
570
620
)
/
i..-..J
lsec
0
I have not been able to completely abolish spontaneous activity in
0
the aggregate with a single 20 msec duration depolarizing current
pulse at an external potassium concentration of 1.3 mM.
This is in
spite of systematic searches in many aggregates changing the
coupling interval by as little as 1 msec and changing the stimulus
amplitude by as little as a fraction of a nanoampere.
However, there were a few atypical cultures in which most
aggregates were found to be not beating or in which impalement of a
beating aggregate caused it to become quiescent.
Upon irnpal ement,
most of these aggregates were found to be resting at a potential in
the plateau range of potentials.
Injection of a large enough pulse
of current into such an aggregate could provoke sustained triggered
ac t i vi ty ( Fi g • 2-15 ) •
Triggered activity could also be obtained (but rarely) if
spontaneous activity in a normal preparation was stopped by ·
injecting a constant depolarizing bias current.
Triggered activity
can however often be seen in aggregates which are quiescent during
the transient seal i ng-i n process which immediately fall ows
11
impalement.
11
During this process in such aggregates, the membrane
potential hyperpolarizes with time, until a point is reached where
spontaneous cyclic action potential generation commences.
The
sealing-in process can be modeled as a gradual diminuition of a
depolarizing bias current (Shrier and Clay, 1982).
Theory predicts that a single well-timed current pulse should
be capable of extinguishing triggered activity (Best, 1979; Jalife
0
2-43
Figure 2-15. Sustained activity triggered by a 13 nA amplitude, 20
msec duration depolarizing pulse in a spontaneously
beating aggregate that bee ame quiescent upon
impalement.
activity.
A 4 nA pulse was too small to trigger
Triggered activity was also obtained in
five other aggregates successfully
~npaled
in this
experiment (all aggregates from the same culture).
(Aggregate# 9:
diameter= 200
2-44
0
~m.)
0
L-....1
lsec
c
and Antzelevitch, 1979, 1980; Winfree, 1980).
This was not
attempted in the experiments in which triggered activity was found
in the aggregate.
(vi} Apparent Discontinuities in the Phase-Resetting Response
As stimulus intensity is increased, the transition between
prolongation of cycle length and abbreviation of cycle length
becomes more and more abrupt for either a depolarizing (Fig. 2-5)
or a hyperpolarizing (Fig. 2-9) pulse.
For example, Fig. 2-68
showed an abrupt transition from prolongation to shortening as tc
was changed from 130 to 140 msec.
To investigate this transition
further, a stimulus of arnpl i tu de 27 nA and duration 20 msec was
delivered 11 times at a fixed coupling interval of 142 msec into
another aggregate.
Figure 2-16. shows that one of two responses
occurred ( all-or-none depolarization
11
11
):
one at a prolonged
interbeat interval, and the other at a shortened interbeat
interval.
Responses with intermediate values of interbeat interval
were not observed.
In other experiments in which as many as 50
trials at a fixed coupling interval were carried out. a similar
dichotomyin the response was observed.
Since there is spontaneous
fiuctuation in the unperturbed interbeat interval of several
milliseconds from beat to beat (as well as in the shape of the
action potential), the stimulus is actually being delivered in a
narrow range of phases even though the coupling interval tc is
2-45
Figure 2-16. Superimposed tracings from 11 repeated trials with a
27 nA amplitude, 20 msec duration current pulse at a
fixed coupling interval of 142 msec.
Eight of these
trials produced an advance in the time of occurrence
of the next beat, while the other 3 produced a delay.
5/5 trials attempted at tc = 141 msec produced only
delays, while 7/7 at tc = 143 msec produced only
advances.
The unperturbed interbeat interval of this
aggregate was about 615 msec.
Thus the discontinuity
in the response T1 is about 2/3 of the spontaneous
cycle length.
(Aggregate# 10:
diameter= 149
2-46
~m.)
0
PULSE
AMPLITUDE
27nA
COUPLING
INTERVAL
(m sec)
0
142
] mV
-50
200msec
c
fixed from trial to trial.
There is also a small degree of
fluctuation in the trigger point on the action potential upstroke,
which will serve to enhance this effect.
For the experiment shown
in Fig. 2-16, only prolongation of interbeat interval was seen at
tc
= 141
msec; at tc
= 143
interval was observed.
msec, only abbreviation of interbeat
Figure 2-17 shows that in some
circumstances a change in tc of as 1i ttl e as 1 msec can suffice to
transform prolongation into abbreviation.
The coupling interval
where this rapid transition occurs is just a bit less than the
action potential duration.
I thus say that there is an
experimentally observed discontinuity in the response T1 as a
function of the coupling interval tc, and thus also in the
perturbed cycle length function
g(~).
However, if the stimulus amplitude used in Fig. 2-16 or Fig.
2-17 is reduced, intermediate values of T1 can be obtained, with a
large scatter from trial to trial at a fixed coupling interval.
Figure 2-18 shows an example:
over a range of coupling intervals
of 20 msec, almost all intermediate values of T1 could be obtained.
Thus, the perturbed interbeat interval T1 appears to be a
continuous function of the coupling interval tc at this level of
stimulation.
Increase in stimulus amplitude leads to a response
similar to the apparently discontinuous response shown in Fig. 2-6
or Fig. 2-17.
The amplitude at which the transition from cycle delay to
cycle advance (in response to a 20 msec duration depolarizing
0
2-47
c
Figure 2-17. Superimposed tracings from 10 repeated trials at a
coupling interval of 183 msec (upper trace) and 184
msec ( 1ower trace).
The stimulus amplitude was 11 nA,
and the stimulus duration 20 msec.
Note the absence
of intermediate values of T1 , even though T0 is
fluctuating substantially from trial to trial at a
fixed coupling interval.
Here, the discontinuity in
T1 is about 0.9 of the spontaneous cycle length.
The
upstrokes of the stimulus artifacts have been
retouched in this figure, since misadjustment of the
balance control on the amplifier produced a biphasic
stimulus artifact.
(Aggregate# 11:
diameter= 132 urn.)
2-48
0
.
PULSE
AMPLITUDE
11 nA
COUPLING
INTERVAL
(msec)
183
]
m~
-50
184
500
msec
Figure 2-18. Ten superimposed voltage tracings at each of three
coupling intervals for a current pulse of amplitude 5
nA and duration 20 msec.
Notice the large scatter in
the response at a given coupling interval.
The
current waveform was constant from trial to trial at a
fixed coupling interval.
A burst occurred during
collection of the data at tc = 230 msec, as evidenced
0
by the decrease in T0 for three trials.
(Aggregate# 2:
diameter= 105
2-49
0
~m.)
PULSE
AMP LITU DE
5 nA
COU PLIN G
INTE RVA L
(m sec)
220
0
230
240
.5sec
c
pulse) no longer appeared to be continuous was generally found to
lie between about 5 nA and about 15 nA.
This variability is not
unexpected, due to the different sizes and presumably differing
electrophysiological properties of different aggregates.
For
faster-beating aggregates, the coupling interval at which the
transition occurs is smaller than for the 2 more slowly beating
aggregates in which long delays could be obtained.
For the faster
beating aggregates, the transition occurs at about the maximum
diastolic potential (e.g.
Figs. 2-6, 2-16, and 2-17); for the
slower beating aggregates, it can occur much further into phase 4
{Fig. 2-11).
Figure 2-19 shows that with high-amplitude
hyperpolarizing input, the· apparent discontinuity appears during
the plateau phase, just after the action potential upstroke - all11
or-none repolarizatiorl' (Weidmann, 1951, 1956; Cranefield and
Hoffmann, 1958; Noble and Hall, 1963; Vassale, 1966; McAllister,
Noble,and Tsien, 1975; Beeler and Reuter, 1977).
The slower-
beating aggregates that display a subthreshold oscillation can have
more than one apparently discontinuous jump in the plot of the
perturbed cycle length function, since they only fire on one or
another of the maxima of the subthreshold oscillatory activity
(Fig. 2-14); action potential s are not seen between groups 11 a" and
11
0
b in Figs. 2-12 and 2.13.
11
2-50
·-
Figure 2-19. Phase-resetting in response to a high-amplitude (64
nA), 20 msec duration hyperpolarizing current pulse.
There can thus be an abrupt abolition of the action
potential by a strong hyperpol ari zi ng stimulus
delivered during the plateau of the action potential.
A slight change in coupling interval is sufficient to
induce the effect ("all-or-none repolarization" }.
0
(Aggregate# 12:
diameter= 132
2-51
~m.)
-
PULSE
AMPLITUDE
64nA
~·
COUPLING
I N"rERVAL
(msec)
6
8
38
46
500 m sec
(vii) Effects on the Post-Stimulus Cycles
A cycle that is shortened by a depolarizing stimulus falling
1ate enough in its cycle tends to be followed by a postextrasystolic cycle that is longer than control, while a cycle that is
prolonged by an early stimulus tends to be followed by a poststimulus cycle that is briefer than control.
These effects can
last for several cycles following stimulation, but with a gradually
decreasing effect on cycle length.
At lower stimulus strengths
(producing responses such as those shown in Fig. 2-5), for a
maximal prolongation of approximately 20% (i.e. T1/T 0
= 1.2), the
post-stimulus cycle is typically shortened by about 2-5%, while for
a shortening of about 50% (i.e. T1/T 0
= 0.5), the post-stimulus
cycle is typically lengthened by about the same amount.
In the
former (latter) case, the decrease (increase) in the duration of
the post-stimulus cycle is associated with a decrease (increase) in
its diastolic period.
At a stimulus amplitude high enough to
produce a graded action potential (e.g. Fig. 2-7:
tc
= 120 msec),
a shortened cycle also has a post-stimulus cycle with an increased
diastol ic time; however, the decrease in the duration of the graded
action potential can cause the actual post-stimulus cycle length
APO + diastolic time) to decrease.
(=
The cycle following the post-
stimulus cycle has a normal APD, but an increased diastolic time,
and thus is lengthened beyond control.
Appreciable decreases in
post-stimulus cycle length are also apparent when a very long delay
is produced (e.g. Fig. 2-14).
2-52
(vfi} Pulse-induced Rapid Repetitive Activity
Rapid repetitive activity in response to a depolarizing
current pulse was a rare phenomenon (incidentally seen in only 2
aggregates from two different cultures subjected to high-amplitude
current pulses}.
I present results from the aggregate in which the
effect was more dramatic.
Figure 2-20A shows the induction of
repetitive activity at a rate faster than the spontaneous beat rate
by a single 20 msec duration pulse.
The increase in rate is
mediated by an increase in the slope of diastolic depolarization.
The beat rate gradually declines back to normal.
The exact form of
the response varied from trial to -trial carried out at fixed tc.
The effect was seemingly not dependent on the coupling interval of
the stimulus, in that it could be elicited at phases throughout the
spontaneous cycle.
It is unlikely that this effect is due to
dislodging of the electrode and a subsequent speeding up of the
.
rate due to the resultant "1 eakage", si nee the maximum di astolic
potential remains relatively unaltered.
In some trials, there was
a s11ght decrease in the absolute value of the MOP.
The two
aggregates in which this phenomenon was seen came from cultures in
which aggregates appeared to be contracting normally under the
microscope, and which had the usual high percentage of
spontaneously beating aggregates.
Furthermore, the action
potential parameters measured from the aggregates in which the
rapid activity was seen were all within the normal range.
2-53
Figure 2-20. Rapid repetitive activity induced by a single 20 msec
duration current pulse.
A.
Depolarizing pulse of amplitude greater than 100
nA, with tc = 130 msec.
The exact current is not
known due to saturation of the current measurement
circuitry.
The rapid activity was not seen at a
current amplitude of two thirds of this amplitude;
only a single extrasystole occurred.
(Aggregate# 13:
0
B.
diameter= 190 urn.)
Hyperpolarizing pulse of amplitude 37 nA, with
tc = 8 msec.
Upstroke of stimulus artifact
retouched.
(Aggregate# 11: diameter= 132 urn.)
2-54
A
B
1 sec
Rapid repetitive activity was also .seen in two aggregates in
response to a high-amplitude hyperpolarizing stimulus (Fig. 2-208).
However, in these cases, the phenomenon could only be provoked by
stimuli ,falling during the action potential.
The phenomenon was
accompanied by a decrease in the absolute values of the overshoot
potential and the maximum diastolic potential, both of which
gradually recovered back to normal {Fig. 2-208}.
4.
DISCUSSION
(i)
B1phasic Nature of the Response to Premature Stimulation
It is not surprising that a brief depolarizing input can
temporarily accelerate a cardiac oscillator.
However, it is less
commonly appreciated that a subthreshold depolarizing pulse, if
.
delivered early in the spontaneous cycle, can transiently prolong
the interbeat interval or that an early hyperpolarizing pulse can
shorten the interbeat interval.
This is so in spite of the fact
that the biphasic nature of the response to a current pulse has
been demonstrated in the sinoatrial node (Ushiyama and Brooks,
1974; Sano, Sawanobori, and Adaniya, 1978; Jalife et al., 1980), in
Purkinje fibre (Weidmann, 1951; Klein, Cranefield, and Hoffman,
1972; Jalife and Moe, 1976), in diseased human ventricle {Gilmour
et al., 1983}, and in heart cell aggregates (DeHaan and Fozzard,
0
2-55
1975;
Scott, 1979; Guevara, Shrier, and Glass, 1980;
Glass, and Shri er, 1981; Ypey, van
Meerwij k et al., 1984).
r~eerwij
Guevara,
k, and DeHaan, 1982; van
Electrotonic depol ari zati on of a pacemaker
produced by a blocked premature beat can produce a prolongation in
the cycle length.
This has been found in the sinoatrial node {Kerr
et al., 1980) and in a focus produced by depolarization of the
right
bun~le
branch {Ferrier and Rosenthal, 1980).
A biphasic
response of the sinoatrial node to ventricular activity during
atrioventricular dissociation has also been documented {Roth and
Kisch, 1948; Rosenbaum and Lepeschkin, 1955).
In addition, the
sinoatrial node has been shown to have a biphasic response to vagal
stimulation {Jalife and Moe, 1979b), and Moe et al. {1977) have·
inferred a biphasic response of a junctional parasystol ic focus in
a human being from analysis of the electrocardiogram.
Several invertebrate neural pacemakers also have a biphasic
response to premature stimulation {Winfree, 1977; Ayers and
Selverston, 1977,1979; Guttmann, Lewis, and Rinzel, 1980).
Winfree
{1980) has pointed out that the response to premature stimulation
of nearly all biological oscillators that have been studied is
biphasic.
Ionic models for cardiac oscillators (McAllister, Noble,
and Tsien, 1975; Bristow and Clark, 1982; Difrancesco and Noble,
1982a; Guevara et al., 1982;
Clay, Guevara, and Shrier, 1984; see
also CHAPTERS 3 and 6 of this thesis) and for neural oscillators
{Best, 1979;
1983:
Guttmann, Lewis, and Rinzel, 1980; Guevara et al.,
see CHAPTER 6 of this thesis) display a biphasic response to
2-56
premature stimulation, as do simpler 1imi t cycle models ( Greco and
c
Clark, 1976; Scott, 1979; Winfree, 1980; Guevara and Glass, 1982;
Hoppensteadt and Keener, 1982; Barbi and Holden, 1983; Guevara et
al., 1983:
see also CHAPTER 6 of this thesis)and electronic models
of cardiac pacemakers (Roberge and Nadeau, 1965; Roberge, 1968;
Roberge, Bhereur, and Nadeau, 1971).
Furthermore, the integrate-
and-fire (slow pacemaker depolarization to a fixed threshold
voltage) model that is often used to explain spontaneous phase 4
depolarization of cardiac pacemakers does not yield a biphasic
response to either a depolarizing or a hyperpolarizing stimulus
( Wi nfree, 1980) •
Many of the electrophysiological studies listed above show
that the transition point from prolongation to delay occurs at
earlier coupling intervals as the intensity of a depolarizing
stimulus is increased.
Scott {1979) offers a model for this using
the Bonhoeffer-van der Pol {BVP) equations (FitzHugh, 1961), while
I describe the ionic basis below in CHAPTER 3.
I also indicate in
the following section how the biphasic nature of the response to
premature stimulation may be responsible for a number of
interesting synchronization phenomena.
(ii)
The Synchronism of Spontaneous Activity in the Aggregate:
Mutual Synchronization
Electrical activity occurring in widely separated cells
within an aggregate is virtually synchronous.
2-57
During spontaneous
.·-·
activity, the action potential upstrokes in two cells separated by
165
~m
occur within 45
~sec
of each other (Clapham, 1979).
Based
upon the response of the aggregate to injection of sma11 amplitude
current pulses, DeHaan and Fozzard (1975) could find no evidence to
contradict the assumption that the aggregate could be considered to
be a spherical synctium.
Mathi as et al. ( 1981) came to a similar
conclusion using impedance measurements.
For events with frequency
content below about 10Hz, Clapham (1979) and Clay, OeFelice, and
DeHaan ·(1979) could find no experimental evidence of deviation from
isopotentiality.
Furthermore, membrane voltage noise or
subthreshold oscillations recorded from two widely separated cells
in aggregates made quiescent by the addition of TTX is virtually
identical (DeFelice and DeHaan, 1977; DeHaan and DeFelice, 1978a,
0
1978b).
Visual observation of the aggregate carried out on
numerous occasions shows no evidence of gross disorganization in
the spread of contractile activity following perturbation with a
single current pulse.
In light of the above facts, I feel
justified in treating the aggregate, for
~
purposes, as a single
card i ac osc i1 1ator.
Isolated
~ocytes
are found to have a large range of mean
interbeat intervals, with many cells beating irregularly (DeHaan
and Hirakow, 1972; Jongsma et al., 1975; Jongsma and Tsjernina,
1982).
The variability in the interbeat interval of an isolated
myocyte is much greater than that of a single cell in a monolayer
or of a single cell in the isolated sinoatrial node (Jongsma and
0
2-58
Tsjernina, 1982).
When assembled together to form an aggregate,
the single cells synchronize their individual rates to achieve a
single common rate.
As the number of cells in the aggregate
increases from 1 to 125, the beat-to-beat variability in interbeat
interval decreases, with the coefficient of variation C dropping
from about 20% to about 3% {Clay and DeHaan, 1979).
connection, it is noteworthy that the
dom~nant
In this
centre of the
sinoatrial node is made up of a population of approximately 5000
cells {Sleeker et al., 1980).
In the·aggregate, as in the dominant
centre of the sinoatrial node, each cell is controlled by the
overall activity of the other cells, but each cell can be presumed
to play as important a role in determining the overall activity as
any other cell •
0
The flo·w of current from cell to cell is presumably the
mechanism by which this mutual synchronization occurs within the
aggregate.
This has been shown to be the case in the sinoatrial
node {Sano, Sawanobo~i, and Adaniya, 1978; Tuganowski, Kopee, and
Tarnowski, 1981).
Two isolated myocytes (DeHaan and Hirakow, 1972)
or aggregates {Scott, 1979; Ypey, Clapham, and DeHaan, 1979;
Clapham, Shrier, and DeHaan, 1980; Clapham and DeHaan, 1982; Ypey,
van Meerwijk, and DeHaan, 1982) will synchronize their rates if
coupled together.
In the 1atter case, there is a rapid fall in the
coupling resistance between the two aggregates within a few minutes
of coupling, presumably as a result of insertion of gap junctions
into the membrane area common to the two aggregates {Clapham,
2-59
.
Shrier, and DeHaan, 1980).
.,.-...,
'-"'
Indeed, calculation suggests that a
single gap junction should be more than sufficient to allow mutual
synchronization of two single cells (Clapham, Shrier, and DeHaan,
1980; Noble, in discussion appended to DeHaan, 1982).
Normally,
each cell within an aggregate has of the order of 10 6 nonj uncti anal
membrane particles and approximately 10 4 particles clustered into
identifiable nexuses (Williams and DeHaan, 1981).
Upon treatment
with cycloheximide, an inhibitor of protein synthesis, organized
gap junctions cannot be demonstrated ultrastructurally, but yet
mutual synchronization is maintained (Will iams and DeHaan, 1981).
These authors conclude that the enduring synchronization is due to
the existence of
11
isolated channels scattered throughout the area
of closely apposed plasma membranes
r~echanical,
11
•
as well as electrical , effects may contribute to
the process of mutual synchronization.
It has been known for over
two hundred years that mechanical stimulation of the heart can
provoke a premature contraction (Hoff, 1941-42).
For example,
gentle prodding of an aggregate with a blunt glass probe can induce
a contraction.
Stretch of the sinoatrial node in mammals (James
and Nadeau, 1963; James, 1967; James et al., 1970; Brooks and Lu,
1972; Ushiyama and Brooks, 1977) and of other cardiac pacemakers in
a wide range of phyla (Jensen, 1971) influences the rate of
spontaneous activity.
Diederich (1969) has concluded that
acceleration of the sinoatrial node can be produced by stretch of
the node as a result of contraction of the right atrium against a
2-60
closed tricuspid valve.
c
Ushiyama and Brooks (1977) have
demonstrated that sinusoidal stretching of strips of right atrium
containing the sinoatrial node can lead to entrainment of the node.
Hashimoto et al. (1967) and James (1967, 1973) have proposed that
the arterial pulse in the sinoatrial nodal artery may act as the
mechanical feedback arm of a servomechanism in which sinoatrial
activity is modulated by the ventricular output.
The extent to which the mechanical stimulation of any one
cell in the aggregate by the contraction of the rest of the
aggregate is responsible for synchronization has not been
investigated.
Segers (1946b) showed that mechanical coupling with
a non-conducting paraffin bridge of two separate pieces of
contracting ventricle could result in synchronization.
0
Pollack
(1977) has suggested than stretch of sinoatrial cells contributes
to their mutual synchronization.
However, in one study in which
aggregates were treated with cytochalasin B (which blocks
mechanical but not electrical activity} no effect on the regularity
of beating was reported (Sachs, McDonald, and Springer, 1974}.
Finally, the onset of mechanical contraction follows the action
potential upstroke with a delay that is not inconsiderable.
The spontaneous rate of the upper half of the sinoatrial node
is generally faster than that of the lower half, with the rate of
the intact node being intermediate to the rates of the two halves
(Lu, Lange, and Brooks, 1965; Sano, Sawanobori, and Adaniya, 1978;
0
2-61
Bouman et al., 1982).
Similar results are obtained if a functional
b1ock between the two ha1 ves is created by p1 ac i ng the node in a
sucrose-gap apparatus (Tuganowski, Kopec, and Tarnowski, 1981).
However, coupling together of two isolated myocytes can produce a
synchronized state with a final common rate that is higher than,
intermediate to, or lower than the two initial rates (OeHaan and
Hi rakow, 1972).
The experiments described in this paper suggest one mechanism
by which mutual entrainment might occur.
simpler case of two coupled cells.
mutually synchronized.
Consider first the
Assume that they have become
That is, not only do they share the same
common rate, but there is also a fixed latency between the upstroke
of one cell and that of the other, and the action potential
durations are approximately equal.
0
Assume also that cell A is
intrinsically faster than cell B (Fig. 2-21A,8) and that, upon
synchronization, the intrinsically faster cell (cell A) 1eads the
intrinsically slower cell (cell 8} and that the final common rate
is intermediate to the two uncoupled rates.
Si nee the up stroke of
the action potential of A occurs late in the-cycle of B, cell A
sources a depolarizing current to cell B that advances the time of
occurrence of the next action potential of B.
Conversely, the
occurrence of the upstroke of the action potential of B early in
the cycle of A produces an early depolarizing stirnul us that results
in a slowing down of A.
Cell A also sources a hyperpolarizing
current to cell 8 during the repol ari zati on phase of its action
potential, which will tend to speed up the activity in B.
2-62
Thus, A
0
Figure 2-21. Schematic diagram illustrating synchronization of two
cardiac oscillators.
One oscillator (panel A} has an
intrinsic frequency higher than that of the other
{panel B).
The action potential duration of the
slower cell is longer.
Upon coupling (panel C), the
two oscillators synchronize to an intermediate rate,
.with the intrinsically faster oscillator (labelled A)
leading the intrinsically slower oscillator
(labell~d
B). The time between upstrokes of A and B is
exaggerated for clarity of presentation.
2-63
0
A
0
c
may be said to be overdriving B, while at the same time, B is
underdriving A.
The reader is cautioned that the experimental coupling of two
cells does not always result in a final common rate that is
intermediate to the two initial rates (OeHaan and Hiral<ow, 1972).
Ionic modelling studies also suggest that an intermediate rate does
not necessarily have to occur upon synchronization (Berkinblit et
al., 1975).
However, it is possible to use the phase-resetting
behaviours in response to single pulses to spell out how these more
complicated cases may occur (Ypey and van Meerwijk, 1980; Ikeda,
1982).
During synchronized activity, some cells in an aggregate
presumably have intrinsic rates slower than the aggregate rate,
others faster.
However, there is an essential difference between
overdrive or underdrive spontaneously occurring in an aggregate,
and that which results in response to a periodic train of current
pulses (see CHAPTER 4).
In the former case, the perturbation
occurs throughout the entire cycle, and not just for a brief period
of time as· in the latter case.
Due to the great degree of
synchronism present in an aggregate (Clapham, 1979), large
potential differences between neighbouring cells can only occur
during the upstroke and rapid repolarization phases of the action
potential.
During the slower plateau and diastolic depolarization
phases, only small potential differences can exist.
Thus, large
currents will fl a.-~ from cell to cell for brief periods of time
2-64
during phase 0 and phase 3, while smaller currents will flow for
0
longer periods of time during phase 2 and phase 4.
To further
complicate matters, gap junctions formed between coupled amphibian
embryonic blastomers show increased resistance with increased
transjunctional potential (Spray, Harris, and Bennett, 1979).
Thus, several different effects may all act to improve the beat-tobeat regularity of an aggregate; at the present time, it remains
unclear to me whether or not current flow during one particular
phase of the cycle may be said to be mainly responsible for mutual
synchronization.
Figure 2-20 shows that there is a roughly linear dependence
of any i nterbeat interval on the immediately preceding one.
Random
shuffling of the intervals destroys this dependence (Fig. 2-2E).
Furthermore, si nee the Rj fall monotonically, this suggests that
the effect is brought about by a
action within a few beats.
mec~anism
that can alter its
An interbeat interval that is longer
(shorter) than average is followed by one that is also longer
(shorter) than average, but less so.
It is interesting to note
that an interbeat interval that is artificially prolonged
(shortened} by application of a current stimulus is followed by a
post-stimulus cycle that is shorter (longer) than control.
Perhaps the same mechanisms are operating in the two cases.
The bursting phenomenon shown in Fig. 2-3 occurred even in
aggregates which had been successfully impaled for several hours.
The most likely explanation is that there is a temporary partial
2-65
loss of impalement,
~ith
subsequent resealing of the membrane
around the microelectrode tip.
Partial loss of impalement leads to
a small depolarizing leakage current which can presumably be
sufficient to significantly increase the beat rate {Shrier and
Clay, 1982).
Athias et al. (1979) found 2 types of periodically
occurring irregularities in the beat rate of neonatal rat heart
cells.
In their type II arrhythmian, the beat-to-beat variation
11
was due to alteration in the level of the threshold voltage, and
not to alteration in the rate of diastolic depolarization, as
found in the aggregate.
Phenomena similar to that shown in Fig.
2-3 also occur in the sinoatrial node, and have been attributed to
shifts in pacemaker site within the node or to the effect of a
premature stimulus of unknown origin (Sano, 1966).
One alternative hypothesis that might explain the bursting
seen in these experiments is that it is a normal phenomenon that
one might expect in any population of many coupled oscillators.
Indeed, in his firefly machine.. that consisted of 71 coupled neon
11
bulb oscillators, Wi nfree ( 1980) found that .. it is not unusual to
see small groups of neon oscillators temporarily escape
entrainment ......
Thus, occasional bursting in the cells that form
the aggregate may be a periodic revolution against conformism that
is somehow necessary to allow that conformism to become reestablished.
0
2-66
(iii) Apparent Discontinuity of the Phase-Resetting Response
For a sufficiently strong stiQulus, be it depolarizing or
hyperpolarizing, there is an abrupt transition between prolongation
of the perturbed cycle length and shortening of the cycle length
(Figs. 2-16, 2-17, 2-19).
This effect has been previously seen in
the sinoatrial node in response to vagal (Oong and Reitz, 1970;
Iano, Levy, and Zieske, 1972; Greco and Clark, 1976; Jalife and
Moe, 1979b) and electrical (Sano, Sawanobori, and Adaniya, 1978;
Jalife and Moe., 1979b; Jalife et al., 1980) stimulation, as well
as in Purkinje fibres (Jalife and i•1oe, 1976; Jalife and Moe,
1979a}, a focus in the right bundle branch (Ferrier and Rosenthal,
1980), and heart cell aggregates (Scott, 1979; Guevara, Glass, and
Shrier, 1981) in response to electrical stimulation.
I have found,
in agreement with the studies of these other investigators, that
the transition from prolongation to abbreviation of cycle length is
continuous at a low stimulus level and occurs at increasingly
smaller coupling intervals as the depolarizing pulse intensity is
increased.
At higher stimulus levels, the response becomes
apparently discontinuous, in that intermediate values of T1 are not
seen.
There are fundamental problems in ascertaining whether or not the
perturbed cycle length T1 is truly a discontinuous function of the
coupling interval tc, as is suggested by Figs. 2-16 and 2-17.
If
the response is continuous but very steeply changing in the range
of coupling intervals lying between those at which maximal
2-67
prolongation and maximal shortening of the interbeat interval are
produced~
trials.
intermediate values may not be seen, even after very many
If they were seen only very rarely, the possibility of
artifactual causes of the response such as a spontaneous
fluctuation in the properties of the aggregate or a partial loss of
impalement could not be ruled out.
This difficulty is compounded
by the fact that the i nterbeat intervals would be expected to
fluctuate over a wide range of values from trial to trial in the
critical range of coupling intervals if the response were indeed
continuous (as in Fig. 2-18, but over a range in tc of 1 msec and
not 20 msec).
However~
the fact that intermediate responses were
not seen in Fig. 2-16, even in the presence of considerable
fluctuation in T0 , can be taken as circumstancial evidence for the
0
discontinuity of the response.
The waveforms shown in Figs. 2-16 and 2-17 are very similar_
to those seen in quiescent cardiac tissue {e.g. Myerburg, Gelband,
and Hoffman, 1971; Sasyniuk and Mendez, 1971), when a change in the
coupling interval of 1 msec will produce either a propagated
response or a subthreshold event in response to premature
stimulation.
This similarity is not surprising, since the ionic
currents responsible for the phenomenon in the aggregate are also
present in quiescent cells; the currents involved in generating
spontaneous activity in the aggregate are not involved in this
response (see CHAPTER 3).
Investigation of the response of an ionic model of the
2-68
-
aggregate to perturbation with current pulses suggests that the
response is fundamentally continuous, but steeply changing, at
lower stimulus amplitudes (Clay, Guevara, and Shrier, 1984:
CHAPTER 3 of this thesis for further details).
see
Although the
response is also continuous in the (noise-free) model at
intermediate amplitudes, the transition from prolongation to
abbreviation can be very steep.
In fact, at a sufficiently high
amplitude, increments in the coupling interval of as little as 1
~sec
are not sufficient to demonstrate continuity.
This is largely
due to the threshold-like behaviour of the fast sodium current INa'
which produces a quasi-threshold phenomenon (FitzHugh, 1955;
FitzHugh, 1960).
Investigation of the quiescent giant axon of the
squid at 35°C (Cole, Guttman, and Bezanilla, 1970) and of the
Hodgkin-Huxley equations appropriate for squid axon at 35°C {Cole,
1958; Cole, Guttman, and Bezanilla, 1970; Clay, 1977) show similar
continuous responses, demonstrating that depolarization is not an
all-or-nothing affair for that particular non-propagated action
potential.
The addition of realistic membrane noise to ionic
models of either the squid axon (at 6°C) or the aggregate (at 37°C)
would presumably reproduce the apparent all-or-none threshold
phenomenon that occurs experimentally in both these preparations.
(iv)
Depression of Cardiac Pacemakers
The disturbance in the arterial pulse that we now attribute
2-69
to the presence of an extrasystole has been known to medical
0
practitioners for a long time.
The compensatory pause following a
premature ventricular contraction had been described and explained
by the late nineteenth century.
It was also known by then that the
returning cycle following a premature atrial contraction could be
equal in duration to the spontaneous sinus cycle length, fully
compensatory~
or of i ntennedi ate duration ( Lewi s, 1920).
The three
zones of reset, incomplete interpolation, and complete
interpolation are routinely encountered in the course of clinically
determining the sinoatrial conduction time (see for example Dhingra
et al • , 197 5).
The reset zone results from invasion and capture of
the sinoatrial node by a late premature contraction.
The zone of
incomplete interpolation results from an electrotonic subthreshold
0
depolarization which delays the time of the next sinoatrial
activation (Kerr et al., 1980).
The perturbation is subthreshold,
since it results from an early premature beat that encounters
entrance block.
This del ay or depressi on
11
11
of spontaneous impulse
formation in the sinoatrial node had been earlier clinically
inferred (Pick et al., 1951;
Dressler, 1966).
Similar depression
of subsidiary pacemakers can also be seen, both clinically and
experimentally (Pick et al., 1951;
1972;
Klein, Cranefield, and Hoffman,
Klein et al., 1973; Goel, Han, and Rogers, 1974;
and Lane, 1978;
Loeb et al., 1979).
Kennelly
Rarely, oscillations in
membrane potential in the pacemaker range of potentials similar to
c
2-70
those of Figs. 2-11 to 2-14 have been observed experimentally
(Klein, Cranefield, and Hoffman, 1972; Scott, 1979).
I ascribe
these oscillations to the presence of a spiral point in the
pacemaker range of potentials {see section (viii) below).
al.
Loeb et
{1979) have recorded dramatic depression of spontaneous
activity in ectopic atrial foci in response to a single premature
atrial contraction.
However, it is not clear from the above
reports whether or not the effect is phase-dependent, as it is in
the case of the aggregate.
My studies extend these other observations that spontaneous
activity in tissue of ventricular origin is subject to temporary
depression or suppression by premature subthreshold stimuli.
Prolongations in cycle length of as much as approximately thirty
percent can be seen in all aggregates for an appropriately chosen
pulse strength, while much longer-lasting suppression was seen in
2 out of 3 more slowly-beating aggregates studied (Figs. 2-11 to 214).
While alterations in conduction can sometimes be invoked to
explain the cases of depression seen in the intact heart
(Parkinson, Papp, and Evans, 1941) , this cannot be the explanation
in our case given the experimental circumstances.
It is not generally appreciated, but has been explained by
Winfree (1980).that the theory of nonlinear oscillators guarantees
that temporary or permanent suppression can be seen for any limit
cycle oscillator given a stimulus of the appropriate size and
timing.
This theorem of nonlinear dynamics hinges upon the fact
that every limit-cycle oscillation has associated with it one or
2-71
more equilibrium points (also called singular points, fixed points,
or steady states).
Perturbation of the state-point of the system
into a neighbourhood of the singular point by a subthreshold
stimulus can lead to permanent cessation of firing (if the
equilibrium point is stable) or to a temporary suppression of
generation of action potentials (if it is unstable).
However,
there is no guarantee that the appropriate combination of
stimulation strength and timing will necessarily occur if the
stimulus is a premature contraction of endogenous origin.
Suppression of automatic impulse generation is usually seen
in patients with evidence of cardiac pathology or in the
hypodynamic state in the experimental setting.
This is in
agreement with what theory would predict, since a subthreshold
0
input to the pacemaker (as would occur if there is generalized
depression of conduction) is theoretically needed to elicit the
phenomenon.
Furthermore, the sinoatrial node and parasystolic foci
are known to display entrance block for sufficiently premature
stimulation, leading to a subthreshold electrotonic depolarization
of the central region of the pacemaker.
However, my experiments
show that the phenomenon of suppression is only seen in a limited
range of combinations of coupling interval and stimulus strength,
perhaps accounting for the clinical rarity of the phenomenon.
In the sick sinus syndrome, abnormally long values of
sinoatrial conduction time {SACT) are often reported.
The point
has been previously made that abnormally large values of SACT may
2-72
be not in fact a reflection of prolonged conduction times into
and out of the node, but may rather be evidence of a depression
of nodal automaticity directly attributable to the premature
stimulus (Breithardt and Seipel, 1976).
reinforce this point.
My observations
Since sinus arrest is a hallmark of the
sick sinus syndrome, one could speculate that some instances of
temporary sinus arrest could be due to suppression caused by
subthreshold depolarization of the node following a premature
atrial contraction blocked in the perinodal fibres.
The activity
shown in the two lower panels of Fig. 2-14, if it occurred in nodal
cells, would be clinically interpreted from the electrocardiogram
as evidence of sinoatrial arrest or sinoatrial exit block.
Furthermore, oscillations in the subthreshold range of potential
0
.
seem to occur quite readily in tissue of sinus origin (Bozler,
1943;
West, 1961; Lu, Lange, and Brooks, 1965).
However, it must be noted that one attempt to produce
.
suppression of activity in the in vivo sinoatrial node of healthy
dogs using a single stimulus was unsuccessful (Loeb et al.,
1979), although another study (Jalife and Antzelevitch, 1979)
demonstrated that subthreshold stimuli could temporarily suppress
{and even permanently stop) activity in an in vitro sinoatrial
preparation.
2-73
( v)
Anni·hn ation of Spontaneous Activity; Triggered Automaticity
It has been reported that a single subthreshold stimulus of
the right size delivered within a narrow range of coupling
intervals can abolish spontaneous activity in the kitten sinoatrial
node (Jalife and Antzelevitch, 1979).
The same effect could be
produced in canine and bovine Purkinje fibres made to fire
spontaneously by placing in a low-potassium solution containing
epinephrine (Jalife and Antzelevitch, 1979; Jalife and
Antzelevitch, 1980).
However, it appears that a constant bias
depolarizing bias current had to be passed in at least one case to
permit observation of the effect (Jalife and Antzelevitch, 1980).
A more recent report shows that spontaneous activity in diseased
c
human ventricular myocardium can be annihilated by a critically
timed stimulus (Gilmour et al., 1983}.
Earlier reports indicated
that triggered activity in fa1 se tendons exposed to
acetylstrophanthidin (Ferrier, Saunders, and Mendez, 1973), in
Purkinje fibres exposed to a low-sodium, high-calcium, TEAcontaining solution (Cranefield and Aronson, 1974), and in fibres
of the simian mitral value exposed to solutions containing
catecholamines and EDTA (Wit and Cranefield, 1976) could be stopped
by appropriately timed premature stimuli.
Theory predicts that in
the case of other preparations displaying triggered activity (e.g.
Segers, 1939; Reid and Hecht, 1967; Mary-Rabine et al., 1980;
Gilmour et al., 1983), a single critically-timed stimulus should
c
also be capable of annihilating the triggered activity.
Guttman, Lewis, and Rinzel (1980) showed that repetitive
2-74
c
activity in a non-cardiac membrane, the giant axon of the squid
(which can be made to oscillate by placing in a calcium-poor medium
and passing a constant depolarizing current), could be abolished by
a single critically timed hyperpolarizing or depolarizing pulse of
current.
This phenomenon had been earlier predicted by a numerical
analysis of the Hodgkin-Huxley equations (Best, 1979).
Another
numerical study, this time of Noble's earlier (1962) equations
describing Purkinje fibre, demonstrated triggered activity
(Krinskii and Kokoz, 1973). An examination of an ionic model of
two coupled cells showed that triggered activity occurring at a
high frequency and at depolarized voltage levels could be induced
(Kokoz, Krinskii, and Mornev, 1974). A later modelling study
coupled together two simplified cells, one of which showed
c
spontaneous activity, the other not.
Upon coupling, the pair was
quiescent, but triggerable (van Capelle and Durrer, 1980}.
The extent to which the sinoatrial node or any other
intrinsic
focus responsible for pacing the heart can be actually
stopped in vivo by premature stimuli arising elsewhere in the heart
is unknown.
However, the question may be somewhat academic, since
if the sinoatrial node were stopped due to the arrival of such
an impulse, it would presumably resume spontaneous activity upon
receipt of a retrograde threshold wave of depolarization arising in
a subsidiary pacemaker.
The same would also hold true for a
primary subsidiary pacemaker, assuming that a secondary subsidiary
pacemaker would take over and (again) assuming no retrograde block
2-75
of conduction.
Adams-Stokes' attacks in cases of complete heart
block can be sometimes attributed to the temporary cessation of
ventricular activity due to the extinction of spontaneous impulse
formation in the subsidiar-Y junctional or idioventricular focus
responsible for driving the ventricle.
There are instances on
record of this extinction being immediately preceded by a single
premature contraction (e.g.
Parkinson~
Papp, and Evans, 1941: Fig.
7).
Despite strenuous attempts to do so, I have not been able to
abolish spontaneous activity in the aggregate with a single 20 msec
duration pulse.
A similar result has recently been found by van
Meerwijk et al. {1984) using
hea~t
cell aggregates.
This probably
indicates that the steady state{s) of the system are unstable.
0
This conclusion is backed up by the ionic modelling studies of the
aggregate described in the next chapter.
Investigation of ionic
models of sinoatrial node and Purkinje fibre demonstrates that
spontaneous activity in these systems cannot be annihilated, unless
modifications are made to the equations (Guevara, unpublished).
It remains to be seen, however,·whether measures such as injection
of a constant current, alteration of external ionic concentrations,
or addition of pharmacological agents may reproducibly convert an
unstable steady state into a stable steady state, thus allowing
annihilation of the spontaneous activity of the aggregate with a
single pulse.
c
2-76
c
{vi)
Ionic Mechanisms
The response of an ionic model of the aggregate to premature
stirnul ati on has recently undergone some i nvesti gati on ( Guevara et
al., 1982; Shrier et al., 1983; Clay, Guevara, and Shrier, 1984}.
The model is a partial one, since it simulates only phases 3, 4,
and 0 of the spontaneous activity.
The parameters of the model are
derived from voltage- clamp measurements of the ionic currents
underlying spontaneous activity in the aggregate preparation.
The
currents included are the fast sodium current INa(Ebihara and
Johnson, 1980), the time-dependent pacemaker current IK (Clay and
2
Shrier, 198la,198lb), the background sodium and potassium currents
INa,b and IK (Clay and Shrier, ·1981a, 198lb), and the plateau
1
potassium current Ix (Shrier and Clay, 1982}.
I have also
investigated the phase resetting behaviour of the MNT equations
(McAllister, Noble, and Tsien, 1975), which are a more complete
model for spontaneously active cardiac Purkinje fibre.
The results
of these investigations are described in CHAPTERS 3 and 6.
For the
moment, it suffices to say that ionic mechanisms can explain many
of the phenomena described in this chapter.
2-77
{vii) The Vulnerable Period, Repetitive Extrasystoles, and
0
Ventricular Fibrillation
Perhaps the most dramatic effect of the delivery of a single
current pulse to the
activity.
~ocardium
is the induction of fibrillatory
If the ventricle is not in its refractory period, a
single threshold current pulse leads to a propagated wave of
depolarization.
Within a narrow
11
VU1
nerable period
11
,
increasing
the amplitude of the stimulus produces, in addition, a second,
non-driven extrasystole.
Further increase in intensity results in
multiple or repetitive extrasystoles (RES), which decrease in rate
and die away, eventually returning control of the ventricle to the
supraventricular input.
Yet another increase in stimulus amplitude
(to something of the order of ten times diastolic threshold) leads
to sustained repetitive activity (ventricular tachycardia) which
accelerates and degenerates into ventricular fibrillation (Matta,
Verrier, and Lown, 1976).
The mechanism underlying the above sequence of behaviours
remains controversial, seventy years after the demonstration by
r4ines that a single shock from an induction coil, "if properly
timed', would induce fibrillation in the ventricles of an isolated
cooled rabbit heart (Mines, 1914).
Mines offered two mechanisms,
which in today's terminology, would be called {i} reentry, and
{ii} echo (e.g. Reshetilov, Pertsov, and Krinskii, 1979) or
reflection (Antzelevitch, Jalife, and Moe, 1980; Jalife and Moe,
1981).
In fact, reentry has been implicated in the generation of
repetitive activity in response to electrical stimulation in many
2-78
c
preparations of ventricular origin.
These include an isolated
papillary muscle- false tendon preparation {Sasyniuk and Mendez,
1971), focally cooled ventricle (Wallace and 11iynone, 1966), and in
situ ventricle (El-Sherif et al., 1977a,l977b; de Bakker, Henning,
and Merx, 1979; Janse et al., 1980; Wit
et~.,
1982).
However, in many studies, single pulse stimulation apparently
cannot induce fibri11ation, and a two or three pulse protocol must
be employed.
It may well be that had higher intensities been used,
fibrillation would have been elicited by single pulses in these
studies (van Tyn and
~1acLean,
1961; Spielman et al., 1978).
As has been previously stated, there is generally thought to
be a narrow window of vulnerability for the provocation of
repetitive ventricular activity (Ferris et al., 1936; Wiggers and
0
Wegria, 1940). Clinically, this window is identified with the
R-on-T phenomenon (Smirk, 1949).
However, transthoracic shock,
when applied to dogs, has been shown to be capable of producing
repetitive extrasystoles and ventricular fibrillation at times
scattered throughout the cardiac cycle (Milnor, Knickerbocker, and
Kouwenhoven, 1958).
Evidence has been accumulating over the last
fifteen years that episodes of ventricular tachycardia are more
often initiated by late premature ventricular contractions than by
early extrasystoles in the in-hospital and post-hospitalization
ambulatory phases of myocardial infarction (Ahuja, Gutierrez, and
Manning, 1968; Bleifer et al., 1973; De Soyza et al., 1974; Kleiger
et al., 1974; Winkle, Derrington, and Shroeder, 1977; Chou and
0
2-79
Wenzke, 1978; Roberts et al., 1978).
Lie et al. (1975} found that
many episodes of fibrillation were associated with late premature
beats.
However, induction of ventricular tachycardia and
fibrillation does occur via the R-on-T pathway, and has been
documented in at least four ambulatory patients showing
el ectrocardi agraphic signs of acute myocardial ischaemia ( Gradman,
Bell, and DeBusk, 1977; Hinkle et al., 1977;
Reichenbach et al.,
1977; Wei et al., 1979). Three of these four cases occurred in an
out-of-hospital setting, and resulted in .. sudden cardiac death
11
•
I have described pulse-induced repetitive activity in
preparations that were less than 200
~m
in diameter (Fig. 2-20).
Similar observations were previously made in quiescent·and in
spontaneously active aggregates (Parshintsev, 1973).
c
I corroborate
the observation made in this earlier study that the phase of the
·cycle at which the depolarizing stimulus is injected is not
critical in inducing the phenomenon and that a stimulus amplitude
of many times diastolic threshold is needed.
The mechanism
underlying the behaviour cannot be classical reentry, since the
size of the preparation is too small to· support a long-loop
reentrant pathway.
Pulse-induced repetitive discharge has also
been described in small strips of atrial muscle in which it is
11
most improbable that a circus movement can exist' {Dawes and Vane,
1
1951).
Alternative mechanisms to explain the rapid activity include
"refl ection11 , which has been recently experimentally demonstrated
c
2-80
(Antzelevitch, Jalife, and Moe, 1980; Jalife and
0
11
echo" (a 1imi ti ng case of
11
r~oe,
1981), and
reverberation") which has been shown
to be possible in an ionic model of two coupled cells (Reshetilov,
1974; Reshetilov, Pertsov, and Krinskii, 1979).
It is not likely
that the activity I see is triggered, since it occurs in the normal
range of membrane potential.
One alternative which cannot
immediately be discarded is that the depolarizing current pulse
decoupl ed the cell in which the mic roel ectrode was embedded from
its neighbours by increasing the resistance of the gap junctions
that are presumably responsible for the synchronous activity
normally observed (Spray, Harris, and Bennett, 1979).
However, it
is unreasonable to expect that, in all cases seen both by us and by
Parshintsev (1973), the decoupled cell had an intrinsic rate faster
than the overall rate of the aggregate.
Furthermore, as already
stated, one cell in an aggregate apparently has of the order of
10 6 more gap junctions than needed for electrical synchronization.
Transjunctional potentials of the order of 40 mV are the maximum
possible in the response of Fig. 2-20 and probably lead to at most
a tenfold increase in gap-junctional resistance (Spray, Harris, and
Bennett, 1979). Thus, the possibility of cellular decoupling
appears slight.
Also, repeated visual observation of the aggregate
through the microscope showed no sign of disorganized mechanical
activity.
Finally, the tracings in Fig. 2-20A show no evidence of major
decrease in the magnitude of the maximum diastolic potential, as
2-81
would be expected if the electrode were slipping out of the cell
11
0
11
(the converse of the sealing i n phenomenon de se ri bed and modelled
11
in Shrier and Clay (1982)):
11
the increased rate is due to an
increase in the rate of diastolic depolarization.
However, the
voltage traces shown in Parshintsev {1973) show significant
decrease in the amplitude of the maximum diastolic potential which
might indicate a "slipping out.. of the electrode.
This would cause
a small leakage cur.rent which would be then responsible for a
speeding up of the beat rate.
Subsequent"sealing in" would
gradually restore the originally prevailing situation.
The fact
that I only saw the phenomenon four times may be due to the fact
that the large currents needed to elicit the phenomenon could not
be obtained with most of the microelectrodes used in this study.
0
Rapid repetitive activity also can result if a monolayer of
chick embryonic heart cells is subjected to a countershock-type
electric field stimulation, and has been ascribed to a prolonged
depolarization of the membrane potential probably due to
electromechanical deformation of the cell membrane (Jones et al.,
1978).
This was not seen in the aggregate.
The existence of any
connection between the repetitive activity seen twice in response
to a depolarizing stimulus in a ventricular tissue culture
preparation and that seen in the intact ventricle and associated
with paroxysmal ventricular tachycardia and fibrillation is purely
speculative.
Another finding of this study which might have more to do
c
2-82
with paroxysmal tachycardia and fibrillation than the phenomenon
0
described above is the abrupt trans1tion shown in Figs. 2-6, 2-16,
and 2-17.
If a single stimulus is applied to the ventricle with
effectively the same timing and amplitude as shown in these
figures, there will be a great deal of asynchrony produced, since
some cells will immediately fire, while others will only suffer a
subthreshold depolarization.
The
respo~se
in neighbouring cells
will differ because the properties of adjacent cells, as well as
the timing and strength of the current stimulus delivered to these
cells, will be slightly different.
This induced spatiotemporal
inhomogeneity will predispose the ventricle to reentrant circuit
formation.
Stimulation of isolated strips of cardiac muscle can indeed
0
produce repetitive firing via a mechanism resembling the one
outlined above, with the activity in two neighbouring cells being
approximat~ly
1970:
180° out of phase with each other (Lu and Brooks,
Fig. 3}.
In this case, the immediate responses of the two
cells to the current pulse, which must not be too high nor too low
in amplitude and which must be correctly timed, are very similar to
the two responses shown in Fig. 2-16.
This experimental fact fits
in well with the proposed mechanism, since the discontinuous
response of Fig. 2-16 is not seen at smaller amplitudes {e.g. Fig.
2-5) or at higher amplitudes (Fig. 2-7).
Thus, the discontinuous
response found in the aggregate only at intermediate amplitudes may
well explain why multiple extrasystoles, tachycardia, and
0
2-83
fibrillation are often only seen at intermediate stimulus
0
intensities with critically-timed pulses delivered towards the end
of the effective refractory period (e.g. Matta, Verrier, and Lawn,
1976).
However, this mechanism offers no obvious explanation for
the experimental fact that, as the pulse amplitude is increased,
first only single non-driven extrasystoles, then multiple
extrasystoles, and finally fibrillation make their appearance.
The
explanation may well have to do with the fact that, with increasing
stimulus intensity, the point of discontinuity moves to a shorter
coupling interval, and so the duration of the evoked action
potential decreases.
It is interesting to note in this context that a numerical
model consisting of 1000 interconnected quiescent cells shows rapid
0
repetitive activity in which there are two suboooulations of cells
activated approximately 180° out of phase with respect to each other
{Herschl eb, var1 der Twee1, and :·!eijl er,
19~2).
These authors note
that the characteristic frequency of ventricular fibrillation is
about twice that of ventricular tachycardia; they also report that
spectral analysis of electrocardiograms recorded during ventricular
fibrillation reveals a subharmonic at one half of the dominant
fibrillation frequency
in~
of the 874 cases analyzed.
0
2-84
(viii} Topological Considerations
0
a.
Type 1 and Type 0 Phase Resetting
In phase-resetting experiments such as those described above
for the aggregate, a stimulus of fixed intensity and waveform is
delivered at different points in the spontaneous cycle.
perturbation caused by the stimulus
(gener~lly)
The
produces a
transient change in the instantaneous frequency of the oscillator.
The magnitude of the change depends both on the strength of the
stimulus and on the phase in the spontaneous cycle at which it is
injected.
The instantaneous frequency is defined to be the
reciprocal of the time between two consecutive reference or marker
0
events of the process.
In these experiments, I have taken the
marker event to be the zero-crossing on the action potential
upstroke.
When a stimulus is injected, the next event is taken to
be the first positive-going zero-crossing that occurs during the
stimulus or following the termination of the stimulus.
As t+®, the
instantaneous frequency recovers back to its unperturbed value, but
the oscillator is left phase-shifted with respect to an unperturbed
control (Fig. 2-22).
The number of cycles needed for the effect of
the stimulus to effectively wear off varies considerably from one
biological oscillator to another.
The ventricular heart cell
aggregate generally recovers from the effects of a perturbation
within one cycle following stimulation (e.g. Fig. 2-22).
2-85
0
Figure 2-22. A typical phase-resetting experiment in the aggregate.
The heavy vertical bars represent the occurrence in
time of the marker event -
.:~
zero-crossing of
potential on the action potential upstroke.
trace is an unperturbed control.
The upper
The 1 ower trace
shows the effect of a single depolarizing stimulus
delivered at a coupling interval tc.
(i~}
In this case,
the depolarizing stimulus is delivered relatively
early in the cycle, producing a phase delay; thus the
~Ti
are negative.
2-86
0
~Ti
is positive if the phase of the oscillator is
advanced, negative if it is delayed.
c
Note that
0
0
Figure 2-22 shows an example of phase-resetting in the
0
A plot of the phase shift
aggregate.
~T,
•
M.;'
( modul o
•
( 2-1)
1)
as a function of the old phase
( 2-2)
i~ transient phase response curve, which 1s denoted
is called the
by PRC.; (Kawato and Suzuki, 1978;
Kawato, 1981).
In the limit
'
0
the curve is called the steady-state phase response curve
;~,
(PRC )
(lC)
. are
6~~
and~~~ (i~)
•
is called the steady-state phase shift.
functions not only
parameters.
of~.
The
but also of the stimulus
Note that several other ways of plotting phase-
resetting data are also termed phase response curves {Winfree,
1980).
.th- trans1ent
•
h
A p1ot of th e 1
p ase
~ 1. '
=~ +
I
<P;
6c/> ,• ' ( mod
1)
( 2-3)
as a function of q, is called the i~ phase transition curve, ~1hich
is denoted by PTC.; (Kawato and Suzuki, 1978;
'
c
the limit
1~,
Kawato, 1981).
In
q,i• is called the new phase, the eventual phase, or
the latent phase, and the corresponding plot of
2-87
~i
'(i+oo) vs~
is
c
called the steady-state phase transition curve (PTC ) or the new
00
phase-old phase curve (Winfree, 1980).
Note that the eventual
phase is not necessarily defined for all values
of~
(see below).
Hypersurfaces of dimension N-1 connecting points in the Ndimensional phase space of a limit cycle model which all have the
same value of eventual phase are called isochrons.
The one-
dimensional locus of points to which the state-point of the system
is taken immediately following a perturbation (for all
can be called the perturbed cycle.
~.
0<$,1)
It is the intersection of the
perturbed cycle (for a particular set of stimulus parameters) with
the isochronal surfaces that gives a particular PTC • There has
00
been some mathematical work concerning the topology of isochronal
surfaces (Guckenheimer, 1975; Kawato and Suzuki, 1978; Kawato,
0
1981; Winfree, 1980); these surfaces can have a very tortuous
geometrical structure (Winfree, 1980; Glass and Winfree, 1984).
In a system of ordinary differential equations with only one
equilibrium point and one asymptotically stable limit cycle, PTCoo
is defined and continuous for all
strengths· (Kawato, 1981}:
itself.
~at
almost all stimulus
PTCoo it is a mapping of the circle into
The winding number or topological degree of PTC can be
00
defined to be its average slope (Winfree, 1980}.
Examination of
the available biological phase-resetting data 16d Winfree (1977,
1980) to conclude that only two types of phase resetting are to be
found experimentally:
PTCoo is either of topological degree one or
of topological degree zero.
c
2-88
0
Figure 2-23 shows the change in the form of PTC 2 for the
aggregate as the stimulus amplitude is increased.
Due to the very
fast recovery following perturbation, the curve PTC 2 is an
excellent approximation to PTC
00
•
At the two lower stimulus
intensities (5 nA and 6.5 nA), type 1 phase resetting is found in
the aggregate, whilst type 0 phase resetting is found at the
highest stimulus intensity (24 nA).
At a stimulus current of 8 nA, the response to a single pulse
is almost at the point at which it becomes discontinuous (Fig. 2-5,
right panel ; Fig. 2-6).
Thus, for a pulse ampl i'tude somewhere
between 8 nA and 16 nA in this aggregate, PTC 2 (as calculated from
eqn. (1-3)) becomes discontinuous.
This is probably due to a
breakdown in the assumption that spontaneous activity in the
aggregate is a reflection of the existence of a limit cycle in the
phase space of a deterministic set of differential equations that
model the behaviour of the aggregate (see CHAPTER 3).
A stochastic
term must be added to the model; better still, the model should be
completely reformulated along the lines indicated in CHAPTER 1 as
an inherently stochastic one based on the behaviour of single
channels in the membrane.
Thus, use of the term PTC is called into
question at this level of stimulation, since the assumptions under
which it was derived (a limit cycle existing in a continuous system
of ordinary differential equation) probably no longer hold
in the
range of stimulus amplitudes at which this discontinuous transition
c
2-89
c
Figure 2-23. Second transient phase transition curve (PTC 2 ) at four
different pulse amplitudes:
C. 8 nA
D. 24 nA.
A. 5 nA
B. 6.5 nA
Data taken from the experimental
traces shown in Figs. 2-5 and 2-7.
Type 1 phase
resetting occurs in panels A and B, while type 0 phase
resetting occurs in panel D.
region
~~0.4
Repeated trials in the
reveal that the curve is continuous in
panel B as indicated by the dashed lines.
Experimental determinations for $larger than about
0
0.5 were not carried out in this experiment.
dashed lines for •
~
0.5 are approxfmations based on
the results of other experiments.
(Aggregate# 1:
diameter= 114 urn.)
2-90
0
The
0
B
A
........
.... ····
••• •
•
•
•
•
•
•
• •••••
•
5nA
0
6.5 nA
0
0
0
0
c
......................................
•••••
D
8 nA
0
0
0
24 nA
0
0
from prolongation to shortening of cycle length occurs.
The
difference in T1 between the two responses seen can be
significantly less than one spontaneous cycle length (e.g. Fig. 2In addition, the lengths of the post-stimulus cycles are very
16}.
similar in the two cases.
Thus, in PTC;
(i~2),
discontinuity of condiderably less than 1:
there is also a
type 0 phase resetting
is not occurring.
In the aggregate, as stimulus strength is increased, PTCoo is
first type 1 and then type 0.
In the limit of zero stimulus
strength, there is no effect of the stimulus on the rhythm,
I
8~~•
+ 0, and so PTC <» approaches the diagonal line
1
$ 00
= ~.
At
finite but suffic·iently small stimulus strength (e.g. Fig. 2-lOC),
PTCoo is type 1 and monotone increasing.
For a higher stimulus
strength, it remains type 1 but becomes non-monotonic (e.g. F1g. 2238}.
As the stimulus strength increases, ·the dip in the curve
responsible for the non-monotonicity deepens (e.g. Fig. 2-25A,B,C).
Eventually, type 0 phase resetting occurs (Fig. 2-230).
This
sequence {type 1 (monotonic) +type 1 (non-monotonic) +type 0}
occurs in several different periodically forced limit-cycle
oscillators (see CHAPTER 6).
There is reason to believe that the preparation from which
the curves shown in Fig. 2-23 were obtained has only one
equilibrium point which is unstable and which lies in the plateau
range of potentials (Clay, Guevara, and Shrier, 1984; Fig. 2-24A).
·As the stimulus amplitude is increased from low levels which give
type 1 phase resetting, the perturbed cycle eventually intersects
0
2-91
the stable manifold ("null space.. ) of this point.
c
At this one
particular amplitude, there is one value of tc for which the
trajectory of the system would (in a noise-free situation)
asymptotically approach the equilibrium point and spontaneous
oscillation would cease.
Further increase in amplitude results in
type 0 phase resetting.
Preliminary investigation of ionic models of space-clamped
cardiac tissue that possess only one equilibrium point in the
plateau potential range (Guevara, unpublished) suggests that the
voltage at the end of a current pulse must be very close to the
voltage of this equilibrium point in order for the state point of
the system to be close to the stable manifold of the equilibrium
This fact is borne out in the experiments in faster-beating
point.
aggregates, since the membrane potential must be pushed into the
plateau range of potentials (Fig.
2-7) before type 0 phase
resetting (Fig. 2-230) is seen.
When behaviour such as that shown in Figs. 2-11 to 2-14
occurs, the situation is more complex.
Ionic modelling indicates
that slower-beating preparations may have an equilibrium point of a
spiral nature (i.e. with complex eigenvalues) in the pacemaker
range of potentials (Clay, Guevara, and Shrier, 1984; Fig. 224B,C).
If this equilibrium point is the only equilibrium point
in the system (Fig. 2-24C), then type 0 phase resetting occurs for
a much smaller amplitude depolarizing pulse than required if the
only equilibrium point in the system were to lie in the plateau
2-92
c::J
Figure 2-24. Schematic steady-state current-voltage (IV) relations
for three different aggregates.
The curves are
shifted in a hyperpolarizing direction as one moves
from panel A to panel C, corresponding to a decrease
in the beat rate.
The intersection of the IV
characteristic with the horizontal axis I=O gives the
equilibrium point(s) of the system.
The middle
equilibrium point in panel B is a saddle point.
Perturbation of the state point of the system into a
neighbourhood of the equilibrium point in panel C
would lead to subthreshold oscillatory activity in the
pacemaker range of potentials if that point had
0
complex eigenvalues.
The same holds true for the most
negative equilibrium point in panel B.
In panels A
and C, the isochrons of the single unstable
equilibrium point foliate the entire phase space.
In
panel B, the separatrix associated with the middle
equilibrium point, which is a saddle, presumably winds
into the other equilibrium points.
Thus, the topology
of the isochronal hypersurfaces in this case is
considerably more complex than in the simpler cases
shown in panels A and C.
2-93
I
A
0
/
B
/
c
I
I
I
/
----·------.
___....
',
I
range of potentials (Fig. 2-24A}.
However, this equilibrium point in the pacemaker range of
potentials can coexist with two other equilibrium points:
one
equilibrium point is the previously mentioned point in the plateau
range of potentials, the other is a saddle point (Fig. 2-24B}.
In
this case, for i ntennedi ate stimulus amplitudes, a topological
degree cannot be assigned to the phase-resetting, since T1 becomes
unbounded as tc approaches one critical value which results in
perturbation of the state-point onto the separatrix hypersurface
which forms the stable manifold of the saddle point (Clay, Guevara,
and Shrier, 1984; Glass and Winfree, 1984}.
Thus, one cannot say
with certainty that the type of behaviour shown in Fig. 2-11 lies
close to the border between type 1 and type 0 phase-resetting.
0
For
example, in the other slowly beating preparation in which long
delays were seen (not the one whose phase-resetting response is
shown in Fig. 2-11}, long delays were produced with a monotonic
approach to threshold, showing no evidence of subthreshold
oscillatory activity in the pacemaker range of potentials.
This
suggests the existence of three equilibrium points, one being a
saddle.
The considerations outlined in this section probably also
apply to the phase-resetting ·study of van Meerwij k et al • ( 1984}.
Phase resetting experiments carried out on sinoatrial node
( Dong and Reitz, 1970; Levy, Iano, and Zieske, 1972; Greco and
Clark, 1976; Sano, Sawanobori, and Adaniya, 1978; Jalife and Moe,
1979b; Jal ife et al., 1980 ), a focus in the right bundle branch
2-94
(Ferrier and Rosenthal, 1980), Purkinje fibre (Jal ~fe and Moe,
1976; Jal ife and t1oe, 1979a), and heart cell aggregates ( Scott,
1979; Guevara, Glass, and Shrier, 1981; van r1eerwijk et al., 1984)
all show apparent discontinuities when T1/T 0 is plotted as a
function of tc/T 0 •
However, in the majority of these reports,
increments in tc of substantially greater than 1 msec were
employed.
Thus, some of these curves might actually have been
continuous had finer increments in tc been used.
In most of these
experiments, the discontinuity in T1/T 0 is less than 1, and voltage
tracings showing graded action potentials in response to a
depolarizing pulse are not shown.
These graded action potentials
appear to be necessary for type 0 phase resetting 1n ionic models
with only one equilibrium point in the plateau range of potentials
0
(Guevara, unpublished).
In light of the above considerations, I
caution against the i nferral of the ex1 stence of Type 0 phase
resetting simply based on an examination of curves of T1/T 0 plotted
against t/To (Scott, 1979;
Winfree, 1981).
It is unclear from
Fig. 1D of Jalife and Antzelevitch (1979) whether type 0 phaseresetting is seen in the sinoatrial node; however, one curve in
Winfree (1983b) plotted from data obtained by Jalife and Salata
shows type 0 phase resetting.
It is unclear to me whether type 0
phase resetting has been seen in Purkinje fibre (Jalife and
Antzelevitch, 1979, 1980}.
2-95
b.
Discontinuities in Phase Resetting
Using the definition of the phase transition curve given
. {i.;;:oo)
earlier, real discontinuties can appear in plots of PTC.;
obtained from systems of ordinary differential equations possessing
only one equilibrium point, even when PTC
00
is continuous.
SiDple
two-dimensional limit-cycle oscillators can be used to demonstrate
this point (Kawato and Suzuki, 1978; Kawato, 1981;
Holden, 1983).
Barbi and
These discontinuities arise when the perturbed
cycle intersects the event surface.
Using My definition of an
•
event, the event surface is the demi-hyperplane V~= 0, V 0 ~0. Fig.
'··
r.
2-250 shows PTC 1 corresponding to PTC 2 of Fig. 2-23D, demonstrating
this kind of discontinuity. Note that this discontinuity appears
even though the behaviour of the preparation is smoothly changing
(Fig. 2-7); there is no real physical discontinuity.
A second kind of discontinuity is the threshold-type
behaviour shown in Figs. 2-16 and 2-17 and discussed in section
{iii) above and in CHAPTER 3 below.
I reiterate that this
discontinuity of all-or-none depolarization appears to be real, and
that I believe it arises from the stochastic nature of the
behaviour of single channels in the cardiac membrane (see CHAPTER
3).
A similar statement can probably be made with regard to all-
or-none repolarization.
A third kind of discontinuity is evident when the perturbed
cycle passes through regions of phase space wherein lie the
tightly-coiled trajectories of an unstable spiral point or unstable
2-96
Figure 2-25. First transient phase transition curve {PTC 1 ) for four
pulse amplitudes:
D. 24 nA.
A. 5 nA
B. 6.5 nA
C. 8 nA
The dashed lines in the upper right hand
corners of all four panels are approximations to the
data, necessary because the stimulus artifact obscures
the action potential upstroke for stimuli delivered
sufficiently late in the cycle.
The dashed line in
the middle of panel B indicates that the response is
continuous in this region.
The discontinuity in panel
0 is a Kawato-Suzuki type discontinuity, and is
discussed in the text.
in Fig. 2-230.
PTC 2 for these data is shown
The increased scatter in the data
points of Fig. 2-23 is due to the inherent
fluctuations in the interbeat interval of the
aggregate.
(Aggregate# 1:
diameter= 114
2-97
~m.)
A
.~
•
••
, .....
B
•
•
•••
6.5 nA
5 nA
0 e---------------------------~
0
0
0
0
..............
c
D
•
..
•
8 nA
24 nA
0
0
0
0
small-amplitude limit cycle.
In this case, there are apparent
discontinuities in the perturbed cycle length function (T 1 /T 0 vs.
\/T 0 ), since the action potential fires on one or other of the
crests of the subthreshold oscillatory activity (Figs. 2-13, 2-14).
Glass and Winfree (1984) have shown in a simple two-dimensional
limit cycle model (motivated by the results of these experiments)
that real discontinuities in the perturbed cycle length function
can occur if a threshold is included in the model.
It remains to
be seen whether the gaps occurring in Fig. 2-13 where action
potentials do not fire would occur (in the absence of noise) in a
higher-dimensional ionic model of continuous differential equations
that does .not possess a discontinuous threshold.
A fourth kind of discontinuity appears when the system has
three equilibrium points (Fig. 2-248).
This occurs when the
perturbed cycle pierces the stable separatrix associated with the
saddle point.
There is little experimental evidence for this
behaviour (one aggregate), but it can be shown to occur in ionic
models (Clay, Guevara, and Shrier, 1984).
The influence of noise
(arising from single channel activity) on this behaviour would
presumably be to smear out the discontinuity.
For example, the
arbitrarily large values of T1 predicted in the model would not
occur in experiments (Lecar and Nossal, 197la, 1971b).
2-98
.-
c.
Annihilation of Spontaneous Activity
It appears to be impossible to abolish spontaneous activity
in the aggregate with a 20 msec duration depolarizing pulse.
Although the type 1- type 0 border can be crossed using a pulse of
this duration, there is no guarantee that a null-space of full
dimensionality, if present, will be encountered.
Thus, it is
conceivable that it might be possible to abolish spontaneous
activity in the aggregate by using a pulse duration different from
20 msec; I have not investigated this possibility.
However, ionic
modelling ·indicates that faster-beating aggregates only have one
equilibrium point (Clay, Guevara, and Shrier, 1984}, which is
unstable (see CHAPTER 3}.
0
Ionic modelling of the sinoatrial node
and Purkinje fibre also indicates that, without any external
intervention, it is impossible to stop spontaneous activity with a
single pulse of current (Guevara, unpublished).
Thus, in these
cases, the stable manifold of the equilibrium point (the phaseless
set or null-space) has dimensionality less than the dimension N of
the phase space of the equations.
d.
Cardiac Rotors
Winfree (1982, 1983a, 1983b) has recently suggested that
there may be a connection between the topology of phase resetting
0
2-99
of cardiac oscillatorsand the induction of paroxysmal tachycardia
or fibrillation by premature stimulation.
I will now give an
example that I think illustrates Winfree•s main points.
Consider the situation shown in Fig. 2-26, where one has a
monolayer of spontaneously oscillating cells.
Let us assume that
these cells have only one equilibrium point, lying in the plateau
range of potentials.
Assume that at the point in time considered
in Fig. 2-26, a wave of activation has just propagated uniformly
across the sheet from right to left.
The electrical activity in
all cells along the same vertical line is synchronous, with the
voltage of a cell at any horizontal coordinate being given by the
corresponding point on the waveform shown at the top of the figure.
Suppose that a premature stimulus is then applied at the location
0
s.
Suppose further that the stimulus is sufficiently strong to
cause type 0 phase resetting in cells sufficiently close to S (e.g.
at location A).
Cells sufficiently distant from S (e.g. at·
location C) will only feel a small electrotonic influence, and will
undergo type 1 phase resetting.
By continuity, there will be a
point B, somewhere between A and C, which will be on the border
between type 1 and type 0 phase resetting; the perturbation takes
the state-point of the cell located at B to the phaseless manifold
of the equilibrium point.
By continuity, there will be cells in a
neighbourhood of B that will have all possible values of eventual
phase.
This sets up optimal conditions for a circulating helical
wave of excitation - a cardiac reverberator or rotor - to come into
0
2-100
o
Figure 2-26. Schematic diagram used in the text to illustrate
Winfree's theory of fibrillation.
The rectangle
represents a monolayer of cardiac cells.
The voltage
of any cell in the monolayer can be found by
projecting a vertical 1 ine up to the waveform at the
top of the figure.
See text for further description.
2-101
!.-...
•
/
I
!
0
i
'--
!
/
u
•
CO
•
<
being.
c
If
the oscillating cells making up the planar medium can be
stopped with a single pulse, it is not difficult to see that a
spatial
11
black hole11 will be created, which will cover a small
area of the plane (Winfree, 1982, 1983a, 1983b).
may even grow in size (Winfree, 1983b).
This black hole
Note that the centre of
the rotor will be located at a point some distance away from the
point of stimulation.
It is of some interest in this connection to
note that in a review article on ventricular fibrillation, Wiggers
( 1940) wrote:
11
The fact that only strong stimuli, applied during
the vulnerable phase, are followed by undulatory waves and
fibrillation can be interpretated to signify that they exert some
influence at a distance, which modifies conduction and permits reentry, or
0
... .
11
The above argument of Wi nfree can be modified to account for
the case of a sheet containing quiescent, but excitable, cells.
This modification is essential if one wishes to make a connection
with tachycardia and fibrillation, since these dysrhythmias
generally occur in cardiac tissues that are taken to be quiescent.
To carry out this modification, one simply has to replace the
concept of phase with that of latency.
The continuity argument
needed is one based on the continuity of latency.
By latency, I
mean the time to the next maximum of voltage in a cell following
delivery of a stimulus.
By continuity of 1 atency, I mean that
this time is a continuous function of the stimulation intensity.
Note that continuity of latency is largely due to continuity in the
function relating the voltage attained following a stimulus to the
0
2-102
strength of that stimulus.
c
That continuity has been demonstrated
in the Hodgkin-Huxley equations for quiescent squid axon (Clay,
1977), and is to
be
expected in ionic models of cardiac tissue
which possess only one equilibrium point (FitzHugh, 1955,1960}.
The trypsin-dissociated ventricular aggregate does not show a
continuous progression from type 1 to type 0 phase resetting as
stimulus intensity is increased; the response is discontinuous in
the neighbourhood of the type 1 - type 0 border (e.g. Fig. 2-16).
As mentioned earlier, many cardiac tissues, both quiescent and
spontaneously active, show this discontinuous all-or-nothing
response to premature stimulation applied close to the end of their
refractory period.
This discontinuity is partly due to a
competition between the depolarizing fast inward sodium current INa
0
and the repolarizing potass'ium current Ix (see CHAPTER 3).
Thus,
the continuity assumption crucial to Winfree's scheme is missing in
the case of tissues demonstrating fast action potentials.
However, in tissues with slow action potentials, the continuity
condition necessary for the Wi nfree mechanism to take place may be
present.
t1anoeuvres which lead to slow action potentials {and slow
conduction) in ventricular muscle uniformly decrease the
ventricular fibrillation threshold; this fact fits in well with
Winfree's hypothesis.
2-103
CHAPTER 3
THE IONIC BASIS OF SPONTANEOUS ACTIVITY AND OF PHASE
RESETTING IN THE AGGREGATE:
NUMERICAL INVESTIGATION OF A PARTIAL MODEL
0
11
The theory is a mess and seems inherently messy."
John Guckenheimer, 1980b
0
INTRODUCTION
In this chapter, I assemble an ionic model of spontaneous
electrical activity in the embryonic chick ventricular heart cell
aggregate and investigate its response to perturbation with brief
duration current pulses.
The model consists of a set of ordinary
differential equations that describe the ionic currents that flow
across the cell membrane.
These equations are obtained from
published analyses of data from voltage clamp experiments.
There have been few models of spontaneously active cardiac
tissue assembled from voltage clamp data:
models for sinoatrial
node (Yanagihara, Noma, and Irisawa, 1980; Irisawa and tJoma, 1982;
Bristow and Clark, 1982) and for Purkinje fibre 0Joble, 1962;
Gul'ko and Petrov, 1970; M::Allister, Noble, and Tsien, 1975;
DiFrancesco and Noble, 1982a).
The model of Beeler and Reuter
(1977) for quiescent ventricular muscle can be made spontaneously
active if a constant depolarizing bias current is applied.
There
appears to have been only one systematic study undertaken on the
phase resetting of an ionic model of spontaneously oscillating
cardiac cells (Bristow and Clark, 1982).
The model investigated in this chapter is a partial one, in
that it only simulates phase 3 of the action potential (the late,
rapid phase of repolarization}, phase 4 (the diastolic or pacemaker
potential), and phase 0 (the upstroke of the action potential).
3-1
This is due to the fact that only five currents are included in the
model:
the fast inward sodium current INq' the plateau current Ix,
the pacemaker current IK , and the background currents INa b and
2
'
IK • Other currents known to exist in the aggregate, such as the
l
slow inward current Isi (Josephson and Sperelakis, 1982), are not
included in the model, since a quantitative description of their
properties is not presently available.
In Section 2 below, I describe the individual currents and
give their mathematical formulation.
Section. 3 briefly outlines
the method used to numerically integrate the system equations.
Section 4 shows the spontaneous activity displayed by the model and
its response to current pulses of brief duration.
Finally, in
Section 5, I relate the activity of the model to the experimentally
0
observed behaviour described in Chapter 1 and discuss the
shortcomings of the model.
2.
FORMULATION OF THE MODEL: THE INDIVIDUAL CURRENT COMPONENTS
Throughout, voltages are in units of millivolts (mV),
currents in nanoamperes ( nA), conductances in microsi ernens ( JJS),
and times in seconds {sec).
All conductances and currents assume
an aggregate diameter of 200 Jlm. ·Inward currents are negative;
outward currents are positive.
c
3-2
The TTX-sensitive, fast inward sodium current INa is both
time (t) and voltage (V) dependent, having both activation (m) and
inactivation (h) variables
( 3-1)
where gNa is the fully activated conductance and ENa is the Nernst
equilibrium potential for the sodium ion.
is thus a linear, gated channel.
The fast sodium channel
The activation and inactivation
variables m and h each obey first-order kinetics
m= a m(1-m) -em
m
( 3-2a)
The overdot denotes differentiation with respect to time.
The rate
coefficients am, sm' ah' and eh are given (in units of inverse
seconds) by (Ebihara and Johnson, 1980)
0
an=
320 (V+47.13)/(1-exp{V+47.13))
I,
( 3-3a)
Bm = 80 exp(-V/11)
{3-3b)
3-3
ah
c
( 3-3c)
= 135 exp (-(V+80}/6.8)
eh= 3.56 x 10 3 exp(0.079V) + 3.1 x 10s exp (0.35V). (3-3d)
Figures 3-lA and 3-lB are plots of these activation and
inactivation rate constants as functions of voltage.
2)
m~
Equation (3-
be rewritten
•
1
(3-4a}
m =-(m
-m)
T
m
QC)
• '1
h =T- ( hoo -h)
h
(3-4b)
'
where -rm( Th} is the time constant of activation (inactivation) and
mQO (hQO) is the asymptotic or steady-state value of m{h) at a fixed
potential
( 3- Sa)
m
QC)
= a m'I( a mm
+e )
( 3-Sb}
( 3-Sc}
( 3-Sd}
3-4
Figure 3-1.
Variables associated with the activation and
inactivation of INa plotted as functions of voltage.
Unmodified equations of Ebihara and Johnson (1980).
0
A.
The rate constants of activation (am and 8m}.
B.
The rate constants of i nac ti vati on (ah and Sh}.
c.
The steady-state values of m and h (m and h ).
D.
The time constants of activation and inactivation
CO
(Tm and
T
00
h).
The scales in panels A and B are chosen to facilitate
comparison with the corresponding curves shown in
Ebihara and Johnson ( 1980). All of these curv.es are
similar to those
~otted
in Ebihara and Johnson
(1980) with the exception of the m curve (see text).
QO
3-5
1 6121121121.
(sec-1)
e 121121121.
121. 121
-3121.121
-6121. 121
5121121. 121
( sec-1)
121. 121
B
25121.121
121. 121
-9121. 121
0
-6121.121
-4121.121
1. 121
• 5
121. 121
-1121121.0
• 1211211214
Tm
(sec)
-5121. 121
121. 0
• 1214
D
• 1211211212
• 02
0
121. 121
0. 121
-10121.121
-50.0
121. 121
Th
(sec)
Figures 3-1C and 3-1D are plots of -r , 'h, m , and h as functions
m
0
""
""
of voltage.
The expressions for INa in eqn. (3-1) and for am, sm, and ah
in eqn. (3-3} have the same functional form as those originally
used to describe the fast inward sodium current in the giant axon
of the squid (Hodgkin and Huxley, 1952).
However, the maximal
specific sodium conductance is much larger in nerve than in the
aggregate (120 mS cm- 2 versus 23 mS cm- 2 ).
This difference is in
part responsible for the smaller maximal rate of rise of the
upstroke in the aggregate.
Even though the rate constants are
larger in the aggregate at 37°C than in the squid axon at 15°C, the
ratio Th/-rm is about the same in both preparations over a wide
range of potentials (Ebihara and Johnson, 1980).
0
In the model for the aggregate, the h"" curve is shifted by
about 12 mV in the hyperpolarizing direction relative to the h
""
curve in the Hodgkin-Huxley model for the squid axon:
there is
little overlap of the m"" and h"" curves in the aggregate model (Fig.
3-1C) and hence theoretically 1ittle window current INa
in the
00
pacemaker range of potentials (-100 mV
~V~
-60 mV).
The window
current INa (Attwell et al., 1979) is the steady state value of
Oil
INa at a potential V and is given by INa = m}hj'Na ( V-E~Ja).
00
Thus, INa would not be expected to play a significant role in the
genesis of phase 4 depolarization.
However, following addition of
tetrodotoxin ( TTX), which specifically blocks INa' the spontaneous
beat rate of an aggregate slows before action potential generation
0
3-6
c
stops.
This slowing is due to a prolongation of the duration of
phase 4 {Colizza, Guevara, and Shrier, 1983: Fig. 6}.
Thus, INa is
experimentally implicated in controlling the duration of phase 4,
if
not its slope.
The window current INa has recently been measured in the
""
aggregate {Brochu, Shrier, and Clay unpublished}, and is much
larger than that predicted in the pacemaker range of potentials
using the equations given above.
The equations for m listed above
were obtained by voltage clamping at potentials more positive than
-40 mV.
Thus, these formulae are not necessarily applicable in the
pacemaker range of potentials.
Furthermore, there are
inconsistencies in the paper of Ebihara and Johnson (1980)
regarding the activation variable m.
0
Using their published
equations for the rate constants am and sm (eqn. (3-3a,b) above), I
obtain curves of am and am as functions of voltage very similar to
their curves (my Fig. 3-lA, their Fig. 4C).
However, using these
same equations, one obtains a curve for m as a function of voltage
""
(Fig. 3-lC) which .is not the same as the one shown in Fig. 4A of
Ebihara and Johnson (1980).
However, both curves fit the data
points well for V more positive than -40 mV;
they are only
different for V more negative than about -40 mV, where there are
anyway no experimental data points.
Another problem arises if one carries out a numerical
simulation of the membrane voltage using the equations given above
for INa and the equations given below for the other currents.
membrane potential slowly becomes more positive, but does not
attain threshold:
the upstroke of the action potential is not
3-7
The
generated.
0
It is reasonable to suspect that this deficiency is
associated with the aforesaid problems with the activation variable
m.
Clay has shown that inserting an extra factor of 0.1
multiplying the term (V+ 47.13) appearing in the exponential in
the denominator of eqn. (3-3a) resolves this problem (Clay,
Guevara, and Shrier, 1984).
Moreover, the predicted window current
is now close to the experimentally measured value (Shrier and Clay,
unpublished).
rm.
Figure 3-2 shows the modified curves for am,
m~.
and
However, these modified curves do not fit the data points of
Ebihara and Johnson {1980) as well as the curves {Fig. 3-1)
resulting from their original unmodified equations.
In what
follows, I nevertheless incorporate the extra factor of 0.1 into
the model.
0
Equation (3-3) for the rate constants is taken from voltage
cl amp data obtained from aggregates prepared from 11-day-ol d chick
embryos {Ebihara and Johnson, 1980; Ebihara et al., 1980). This
equation can be applied to 7-day-old aggregates (i.e. aggregates
prepared from 7-day-old embryos} , since the maximal upstroke
•
velocity Vmax does not change appreciably from day 7 to day 11
(Ebihara et al., 1980; Clay and Shrier, 1981b; Colizza, Guevara,
and Shrier, 1983).
It is unlikely that there could be simultaneous
compensating shifts in opposite directions in the maximal sodium
conductance and in the kinetics between day 7 and day 11 that would
•
leave both Vmax and the potential at which it occurs
(~
- 20 mV)
unchanged.
Ebihara and Johnson (1980} found a maximal specific sodium
conductance of 23 mS cm- 2 in a 75 urn-diameter aggregate that had an
3-8
0
Figure 3-2.
Variables associated with the activation and
inactivation of INa plotted as functions of voltage.
Equations of Ebihara and Johnson (1980}, with am
modified as indicated in the text.
A. The rate constants of activation (am and Sm}.
B. The rate constants of inactivation ( ah and Bh}.
Q
c.
The steady-state values of m and h (m and h ) •
""
(10
D. The time constants of activation and inactivation
(TmandTh).
There is now a significant INa due to the overlap of
00
moo and hoo shown in panel
0
c.
3-9
1 6121121121.
(se c-l)
e 121121121.
121. 121
-6121. 121
5121121. 121
(sec-1)
-3121. 121
121. 121
B
25121.121
121. 121
-8121. 121
0
-6121. 121
-4121. 121
"
1. 121
• 5
121. 121
-1121121. 121
• 1211211214
Tm
(sec)
-5121.121
121. 121
• 1214
D
• 1211211212
• 1212 (
121. 121
121. 121
-1121121. 121
-5121. 121
121. 121
Th
sec )
active cell surface area of 1.47 x 10- 3 cm 2 • Thus, an aggregate of
0
diameter 200
lliD
would have gNa = 641 l-!S in eqn·. (3-1), since
membrane surface area is directly proportional to the cube of the
aggregate diameter (Clay, DeFelice, and
DeH~an,
reversal potential for the sodium current,
E~Ja,
1979).
The
was experimentally
measured to be 29 mV (Ebihara and Johnson, 1980), which agrees well
with 38 mV, the Nernst potential calculated for the sodium ion from
the measured intracellular sodium concentration of 33.5
{McDonald and DeHaan, 1973).
~~
The discrepancy between the two
values may be partly due to the different ages of aggregates and
the different external potassium concentrations used in the two
studies.
In eqn. (3-1), I use ENa
The fast sodium current
0
IN~
=
40 mV.
has been described using voltage
clamp techniques in several cardiac tissues including the
sinoatrial node (Noma, Yanagihara, and Irisawa, 19t7), atrium
(Rougier, Vassort, and Stampfli, 1968), Purkinje fibre (Dudel and
Rildel, 1970; Colatsky and Tsien, 1979; Colatsky, 1980), ventricular
muscle (Lee et al., 1979;
Bodewei et al., 1982), and ventricular
heart cell aggregates (Nathan and DeHaan, 1979; Ebihara and
Johnson, 1980; Ebihara et al., 1980).
INa is not necessary for
spontaneous action potential generation in tissues with slow action
potentials such as the sinoatrial node (Yamagishi and Sano, 1966)
or the atrioventricular node {Zipes and Mendez, 1973); however, in
most tissues with fast action potential s, pharmacological blockade
3-10
of Iua with TTX abolishes spontaneous activity.
Thus, INa can be
involved in generating the action potential upstroke, in
maintaining the action potential duration (Attwell et al., 1979),
or in contributing to spontaneous phase 4 depolarization.
Application of a voltage-clamp step in the pacemaker
potential range (-100 mV < V < -60 mV) yields a current waveform
with an exponential time course.
Such voltage clamp data has been
interpreted (Shrier and Clay, 1980; Clay and Shrier, 198la, 1981b;
Shrier and Clay, 1982; Clay and Shrier, 1983} as evidence for the
existence of a time-dependent, potassium-ion pacemaker current IK
2
0
in the aggregate that is similar to that
e~r1ier
Purkinje fibre (e.g. Noble and Tsien, 1968).
described in
Like the potassium
current in squid axon, IK activates but does not inactivate
2
(3-6)
The activation variable is denoted by s, while fK (V) gives the
2
current flow through the fully activated (i.e. s=l) channels.
Unlike the fast sodium channel in both squid axon and heart, and
unlike the potassium channel in squid axon. the IK channel is not
2
ohmic:
it rectifies in the inward direction, with the dependence
of current on voltage given by the nonl inear function fK (V)
2
3-11
0
.
where
+
p2
= 1/(1
{3-8a}
+ exp (-e(V-EK)/kT))
{3-8b)
Pz
{ 3-8c)
Yz-- •
+
Pz
This formulation for the rectifying channel comes from a
0
si ngl e-fil e knock-on model o.f ion traffic through a channel which
possesses a block1ng particle and which has 2 ion-selective sites
(Hodgkin and Keynes, 1955; Clay and Shlesinger, 1977; Clay and
Shrier, 1981a).
The parameter 62 gives the probability that the
blocking particle will be dislodged if struck by a permeant
+
potassium ion, p2 gives the probability that it will then enter the
channel, t 2 is the average time between collisions, N2 is the total
number of IK channels, EK is the equilibrium potential for the
2
potassium ion, e is the elementary charge, k is Boltzmann S
1
constant, and T is the temperature C'K).
mv.
3-12
ForT= 35°C, kT/e = 26.5
The activation variable s obeys first order kinetics
0
( 3-9)
or
s =-}- (s..,-s),
( 3-10)
s
with
(3-lla}
S 00
0
=a s/(a s + Bs ).
( 3-llb}
The rate constants are given by
as= 1.05 (V+57) I (1-exp(-0.2(V+57))}
(3-12a}
a5
(3-12b)
= 0.095 exp (-0.075 (V+57)).
Note that the rate constants as and as for the activation and
deactivation of IK have the same functional form as those for the
2
potassium current IK in squid axon.
Furthermore, the functional
forms for as and a 5 are the same as those for am and am given
3-13
earlier in eqn. {3-3).
0
The constants in eqn. (3-12) were found by
fitting eqns. (3-11) and (3-12) to the experimentally determined
values of r S and sCO in 15 aggregates {Clay and Shrier, 198la).
Although these 15 experiments were carried out at external
potassium concentrations ranging from 1.3 mM to 4.8 mM, the bestfit through all the data (eqn. (3-12)) can be used, since the
kinetics of IK do not appear to be a function of the external
2
potassium concentration (Clay and Shrier, 1981a).
as, as,
S 00 ,
and •s as functions of V.
Figure 3-3 shows
The curves for sco and rs
agree well with the corresponding ones in Fig. 5 of Clay and Shrier
(198la) obtained from eqns. (3-11) and {3-12).
Voltage clamp steps in the pacemaker range of potentials
produce instantaneous changes in both IK
and the time-independent
2
0
background current Ibg due to changes in driving force.
These
changes are complicated by the rectification present in both
currents.
However, the contaminating effect of Ibg can be removed
by computing the ratio of the time-dependent changes in current at
the "on11 and "off.. of a voltage clamp pulse.
Shrier (1981a), this "ratio
analysi~
(Noble and Tsien, 1968) gives
a value of 1090 nA for N2 et 2 - 1 in eqn. (3-7).
in the model.
According to Clay and
I adopt this value
The calculated Nernst potential for potassium at an
external potassium concentration of 1.3 mM is -124 mV, assuming an
intracellular potassium concentration of 146.1 mM (McDonald and
DeHaan, 1973).
While the reversal potential for IK has not been
2
experimentally obtained at 1.3 mM, it has been obtained at higher
3-14
0
Figure 3-3.
Variables associated with IK plotted as functions of
2
voltage.
0
A.
The rate constants of activation (as and
B.
The steady-state value of s {s }.
c.
The time constant of activation (T 5 ).
CO
3-15
e5 ).
5121. 121
5. 121
A
0
121. 121
r21. 121
-1121121. 121
-5121. 121
121. 121
-5121.121
121. r2l
-5121.121
r21. 121
1. 121
sex>
•5
0
121. 121
-1121121. 121
2. 121
Ts
c
1. 121
(sec)
121. 121
-1121121.121
potassium concentrations, and the experimental values agree very
0
well with the calculated Nernst values (Clay and Shrier, 1981a).
thus use EK
I
= -124 mV in eqn. (3-8a).
The formula for IK
in eqn. (3-6) is similar to that
2
originally used to .describe the potassium current IK in squid axon
(Hodgkin and Huxley, 1952). However, following a voltage clamp step
in the aggregate, there is an exponential (and not a sigmoidal)
change in current.
Thus, the activation variable s in eqn. ( 3-6)
is taken to the first power and not to the fourth power as in the
Hodgkin-Huxley equations.
Also note that the pacemaker current IK
2
is not responsible for the repolarization phase of the action
potential, as is IK in nerve, since it is activated over a much
more negative range of potentials.
0
constants for IK
Not unexpectedly, the rate
are about two orders of magnitude smaller than
2
those for IK in nerve.
The IK current described by eqns. {3-6) to (3-12) is also
2
similar to that originally described in adult Purkinje fibre (Deck
and Trautwein, 1964; Vassale, 1966; Dudel et al., 1967; Noble and
Tsien, 1968; Peper and Trautwein, 1969; Hauswirth, Noble, and
Tsien, 1972a; McAllister, Noble, and Tsien, 1975).
a maximum value of about 1.3 sec at V ~ -80
mv
However, Ts has
in the aggregate
(Fig. 3-3C}, as contrasted with a maximum value of about 2.3 sec at
V
:'::!
-80 mV in Purkinje fibre.
This may have to do with the fact
that the duration of phase 4 is much longer in Purkinje fibre than
in the aggregate, since IK
is intimately tied to the generation
2
0
3-16
of diastolic depolarization (see RESULTS below).
c
Currents similar
to IK apparently do not exist in adult atrial (Brown, Clark, and
2
Noble, 1976a, 1976b) or ventricular (Beeler and Reuter, 1970, 1977)
muscle, in atrial aggregates (Shrier and Clay, 1982) or in the
sinoatrial node (Noma and Irisawa, 1976).
IK also gradually
2
disappears with increasing age in ventricular aggregates fabricated
from tissue taken between day 7 and day 12 of development (Clay and
Shrier, 198lb). Thus, while IK
is not found in quiescent tissues,
2
it is also not necessarily found in spontaneously oscillating
tissues.
Finally, within the last three years, evidence has
accumulated suggesting that the depolarization-activated IK does
2
not exist in Purkinje fibre, and is to be replaced by a
hyperpolarization-activated inward current If carried by both
0
sodium and potassium ions (see DISCUSSION).
(ii1) Ibg
Voltage clamp steps in the pacemaker range of potentials
produce an instantaneous change in current followed by a
time-dependent change with an exponential time course due to
activation or deactivation of IK • Part of this instantaneous
2
change in current is due to a change in the drivi.ng force for
potassium ions through the IK channel, another part derives from
2
the rectification of the IK channel, and a residual part is
2
thought to be a result of alteration in the level of background
3-17
current fiow through time-independent (but voltage-dependent)
0
channels.
Once IK
is determined as outlined in the previous
2
'
section, its steady-state contribution, given by s (V)fK (V), can
00
2
be subtracted from the total current fiowing in the steady state
during a voltage clamp step to a given voltage.
In this manner,
the residual current (i.e. the background current Ibg) can be
determined.
Partly because the reversal potential of the extracted
current component is about -50 mV, it has been modelled
(McAllister, Noble, and Tsien, 1975;
Clay and Shrier, 1981a) as
the sum of a linear inward sodium current (INa,b) and an inwardlyrectifying outward current carried mostly by potassium ions (IK )
l
(3-13)
0
where
(3-14)
and
(3-15a)
where
(3-lSb)
and
3-18
( 3-lSc)
0
with
+
p3
= 1/(l+exp(-e(V-E 3 )/kT)),
(3-16a)
(3-16b}
{ 3-16c)
Figure 3-4 shows plots of INa,b' IK
,
1
IK
,
3
IK,._' and Ibg as
functions of potential.
0
This description of the rectifying IK channel is of the same
3
fonn as that earlier used to describe the fully-activated, inwardly
rectifying IK channel, but assumes four ion-selective sites
2
.
instead of two, since the range of voltage over which rectification
occurs is narrower for Ibg than for IK • The additional term IK
'+
2
must be added to IK
since Ibg' although inwardly rectifying in the
3
pacemaker range of potentials, becomes progressively more outward
as V increases to values more positive than about
4).
Thus, the model used for IK
~70
mV
(Fig. 3-
in the aggregate is similar to
1
that used in models of Purkinje fibre (tt:All ister, Noble, and
Tsien, 1975) and ventricular muscle (Beeler and Reuter, 1977), with
the exception that the functional form of the equation used to
model the rectifying IK component is that of Clay and
0
3
3-19
0
Figure 3-4.
The background current Ibg' its components IK
and
1
0
I~la,b ( Ibg = IK
IK and IK
3
4
( IK
1
1
+ INa,b}, and the IK
= IK 3
subcomponents
+ IK ) as functions of
4
voltage.
0
1
3-20
0
3121.121
I
(nA)
121. 121
JNa,b
0
-3!2! .. 121
-1121121.!2!
-75.121
v(mV)
-5121 .. 121
Shlesinger {1977) and not that of Adrian (1969).
0
By using the IK
subtraction technique outlined above, Clay
2
and Shrier {1981a) found
E3
= -95
mV, E4
= -40
~Na,b
= 0.202
mV, and s3
~S,
= 0.63.
§K
4
= 0.516
~s,
The value given in the
caption of Fig. 11 of Clay and Shrier (198la) for N3 et 3
incorrect (Clay, personal communication).
-1
is
For reasons given in
section (v) below, I set N3 et31 = 303 nA and change §K from 0.516
4
~S
to 0.9
~S.
As stated before, ENa
= 40
mV.
Preliminary voltage clamp data obtained from steps to
voltages more positive than -65 mV reveals that there is a gated
0
channel that is largely responsible for the later repolarization
phase of the action potential in the aggregate (Shrier and Clay,
1982).
Since the fully-activated current is independent of voltage
over a wide voltage range and since the current displays a single
exponential time course in response to a voltage clamp step, it can
be modelled by the equation
( 3-17)
where I~ is the fully-activated current and the activation variable
x has first order kinetics
3-21
0
• -a (1-X) + B X.
X=
X
X
0
( 3-18)
The rate constants for the opening and closing of the gate are
given by
ax = 0.04 (V-10) I (1-exp(-0.1(V-10)))
Bx
~
(3-19a)
(3-19b)
0.01 exp(-0.1(V-10)}.
These equations are the same as those employed by Shrier and Clay
(1982) to describe Ix in 12-day-old aggregates, with the sole
exception that (V-10) occurs in both expressions instead of (V+10).
0
This modification better describes the 7-day-old aggregate, since
there is a shift in the kinetics of Ix with development (Clay,
personal communication).
The functional form of eqn. (3-17) is not
similar to that originally used to describe the pl atea1J currents
Ix and Ix in Purkinje fibre (Noble and Tsien, 1969a, 1969b;
l
2
McAllister, Noble, and Tsien, 1975).
However, ax and Bx have the
same functional form as those of the corresponding rate constants
for m and s given earlier, and were obtained by fitting the
expression
( 3-20)
0
3-22
to the time constants obtained from voltage clamp experiments in
0
7-day-ol d aggregates ( Shri er and Cl ay, unpublished).
In eqn •
.
(3-17), I~= 100 nA (Shrier and Clay, 1982}.
sx, Tx and
x~
Fig. 3-5 shows ax•
plotted as functions of potential.
Note that x and
~
Tx are very small in the pacemaker range of potentials relative to
s~
(v)
and Ts respectively (Fig. 3-3B,C).
Current-Voltage Characteristics
As mentioned earlier, the two parameters N3 et 3 - 1 and gK were
4
modified from their published values (Clay and Shrier, l981a); this
was done in order to make the steady-state current-voltage (IV}
characteristic curve for ITTX
=
ITOT- INa (Fig. 3-6A) reasonably
~
0
close to the experimentally-measured one shown in Fig. 2 of Clay
and Shrier {1981a).
The difference between the two curves for
!TOT and ITTX in Fjg. 3-6A is the window current INa , which has a
<.10
peak value of -25.0 nA at
v·
=-
47.7 mV.
Using the unmodified
equations of Ebihara and Johnson {1980), INa
has a peak value of
00
-3.7 nA at V = -40.2 mV.
Since
s~l
throughout diastolic depolarization (see RESULTS
below), Fig. 3-68 shows the effect on the IV characteristic curves
of setting s = 1 at all potential_s.
Note that ITOT is very small
(( 5 nA in magnitude) for V ( -65 mV, and that INa (the difference
00
between ITOT and ITTX) makes a significant contribution to ITOT for
c
3-23
0
Figure 3-5.
Variables associated with Ix plotted as functions of
voltage.
A.
The rate constants of activation {ax and
B. The steady-state value of x (x=).
0
c.
The time constant of activation (Tx}.
3-24
~\).
• 5
5121121.121
A
0
• 25
121. 121
121. 121
-1121121.121
1. 121
-5121. 121
121. 121
-5121. 121
121. 121
8
0
121. 121
-1121121. 121
1121. 121
Tx
c
5. 121
(sec)
0
121. 121
-1121121.121
-5121.121
0
Figure 3-6.
A.
Steady state current-voltage (IV) characteristics
for IK
2
,
Ibg' ITOT
ITTX = ITOT - INa·
= IK 2
+ Ibg + INa + Ix, and
The contribution of Ix to
ITOT is negligible (( 0.15 nA) over this range of
potentials.
The total current has a local
minimum at V = -48.5 mV, attaining a most
negative value of -20.6 nA.
B.
Steady state IV characteristics for IK
'
Ibg'
2
=1
ITTX' and ITOT with s
curve for fK (V)
2
= IK
at all
poten~ials.
The
(s=l) is a bell-shaped
2
function of potential, attaining a maximum value
of about 26 nA at V = -106.5 mV.
0
3-25
3fZJ.I2l
0
I
(nA)
A
12l.l2l
-3121.121
-1121121.121
-75.12l
-512l. l2l
V(mv)
3121.12l
I
(nA)
S= 1
12l.l2l
-3121.121
-100.121
-75.121
V{mv)
0
-5121.121
V ) -65 mV.
0
3.
METHODS
The total current flowing through the membrane of an
aggregate in this partial model is given by
(3-21)
where Iappl represents any current injected into the aggregate from
an external source.
Considering the cell membrane to be a leaky
capacitor, application of the equation of state for a capacitor and
Kirchoff's first law leads to the equation
0
•
1 ..
V = - !' 1TOP
(3-22)
where C is the membrane capacitance.
aggregate, C = 0.023
~F (Clay~
For a 200
~m-diameter
DeFelice, and DeHaan, 1979).
Thus,
the partial ionic model of the aggregate is formally a fivedimensi anal system of coupled· ordinary differential equations
.
V= i 0 (V,m,h,s,x)
(3-23a)
(3-23b)
c
3-26
.= f
0
h
2 (V,h}
( 3-23c)
( 3-23d)
( 3-23d)
where the f; are nonlinear functions obtained from eqns. (3-1) to
( 3-22) above.
To carry out the simulation, eqn. (3-23) was numerically
integrated in double precision {16.3-16.6 significant decimal
digits) using a fixed time-step implementation of the algorithm due
to Rush and Larsen {1978).
0
The program used was modified from one
kindly supplied by David Clapham.
was 50
~sec
in most instances; a step size of 10
used on occasion.
s
= 1,
x
=
The iteration step size employed
~sec
or 1
~sec
Initial conditions of V = -70 mV, m = 0, h
was
= 1,
1 were set, and the integration carried out until the
threshold voltage {arbitrarily taken as V = -55 mV) was attained on
the upstroke of the action potential.
The initial conditions
approximate the values expected for a complete limit cycle model
once the steady state of oscillation is reached (e.g.
McAllister, Noble, and Tsien, 1975}.
3-27
see
4.
RESULTS
(i)
Analysis of Spontaneous Activity
0
Figure 3-7A shows the transmembrane voltage computed during
unperturbed activity in the model.
There is an initial rapid ·
repolarization from the starting voltage of -70 mV to an
-97.3 mV.
~1DP
of
This is followed by a slower diastolic depolarization to
Thus, the intrinsic interbeat interval is 0.524 sec,
threshold.
assuming that the time from up stroke to -70 mV on the repol ari zi ng
1 imb of the action potential is 0.150 sec.
Figure 3-7B shows the total ionic current flowing through the
membrane.
The initial rapidly declining outward current, which is
responsible for the repolarization from -70 mV to t1JP, is due to a
fast deactivation of Ix (Fig. 3-7C,G).
Th~
time constant -rx is
very small during this part of the simulation(..-; 0.04 sec).
During.
diastolic depolarization, both INa {Fig. 3-7E) and Ix (Fig. 3-7C)
are effectively zero.
The magnitudes of IK and Ibg decline in a
2
roughly parallel fashion, thus maintaining a small net inward
current IK + Ibg that is responsible for diastolic depolarization
2
(Fig. 3-70).
In the final stages of phase 4, the total current
rapidly increases in the inward direction (Fig. 3-7B) due to
activation of INa (Fig. 3-7E,J).
Throughout most of the period of diastolic depolarization,
the activation variables for IK continually declines (since
2
s
0
< S00 } ,
attaining a minimum value of about 0.7 (Fig. 3-7H). This
deactivation contributes to the shutting down of IK
shown in Fig.
2
3-28
0
Figure 3-7.
Spontaneous unperturbed activity of several variables
in the model as functions of time.
starts at t
= o.
F. V G. x
0
H.
s ,s
00
I.
T
s
J. m,h.
Throughout this simulation, 'm< .13 msec and 'h <
28.2 msec.
Voltages are in mV, currents in nA, and
times and time constants in sec.
size
0
The simulation
=
50 ]..lSec.
3-29
Integration step
sa. a
0
sa. a
A
V -zs.a
V -zs. a
-1aa. a
-1aa. a
1aa. a
1. 11
8
ITOT a. a
X
-1aa. a
taa. a
lx
0
a
1. Gl
sa. m
• !!I
ra. ra
sra.
a. a
11
2. 11
m. m
1. 121
-sa.11
m. ra
G
.s
121.
c
F
121. 121
1. 121
E
J
h
INa -sa. 11
.s
-1aa.a L
0
a. a
• :Z!!I
t
.!!I
ra. a
• :zs
t
.!I
3-70.
0
However, a large part of the decline in IK
is due to the
2
ongoing inward rectification that takes place in the I!( channel
'2
(Fig. 3-68) as the membrane depolarizes from MOP to threshold.
Late in phase 4, both IK and Ibg approach zero, as does their sum
2
IK + Ibg·
Thus, in this model, the properties of IK and Ibg are
2
2
such that the sum IK
+ Ibg just suffices to bring the membrane
2
into the voltage range for activation of INa·
Both IK and Ibg are
2
fully turned off by the time of the action potential upstroke.
However, by the time of the next MOP, both currents are back to
their maximal amplitudes (with s
~
1,
s~ ~
0, Ts
~~sec),
ready to
decline together once again in the subsequent phase of diastolic
depolarization.
0
(11)
The Effect of a Current Pulse on the Voltage Waveform
Figure 3-8 (left panel) shows the effect on the transmembrane
potential difference of delivering a 20 msec duration, 20 nA
amplitude current pulse at various phases of the spontaneous cycle.
A 20 nA current pulse in a·200
~m-diameter
aggregate theoretically
has the same effect as a 2. 5 nA pulse ·r n a 100 urn diameter
aggregate, since membrane surface area increases with the cube of
aggregate diameter (Cl ay, DeFel ice, and DeHaan, 1979).
simulation starts at t
The
= 0. A pulse delivered before t = 6 msec
produces a prolongation in interbeat interval; the same pulse
delivered later than t = 6 msec produces an abbreviation of
interbeat interval.
0
Late in the cycle, the latency from the
stimulus to upstroke decreases with increasing coupling interval
3-30
Figure
3~8.
Effect of delivery of a 20 msec duration depolarizing
current pulse on the spontaneous ac ti vi ty of the
model.
The amplitude of the current pulse is 20 nA
in the left panel, 60 nA in the right panel.
The top
trace in each panel shows unperturbed activity; the
numbers to the left of the other traces indicate the
0
time at which the current pulse is injected.
point t = 0 corresponds to the start of each .
simulation.
aggregate.
0
The
The model is for a 200 JJm-diameter
Integration step size = 50 11sec.
3-31
20 nA
0
-·~·
60nA
[
-2~ a
_::~~~~ •~
O(msec)L , ,
20
40
60
0
80
100
150
200
300
I
1
)
-lam.
~ ..... .
f
~ ..... .
f
1.5
J
u-.--,-,
f
1
I
,
,
,
1.0
I
o
1
Le... ,, ..
2.0
f .
~
2.5
J
...
,
..
l .L .. .. ,
L
I
..... .
.~
J
[
-··~····
• 25
t(sec)
.5
~,,
3.0
1
Ill
O.O(msec)l, '
0.5
m. m
0
(mV)L • .
o
.~ , '
f
f
f
'~'
.-/
•~!
t'\-:'
f
~
'
I
/
..
~..
r
__.-/
~=·~~·
b:. ' ' '~ '
~
r
~:11~''
b.. ~ ... ,
3.5
V. ........ .
4.0
f,
V' '
ra. a
I
I
I
I
• 25
t{sec)
I
I
I
I
• 5
tc.
0
Due to the increase in membrane slope resistance with
increasing diastolic time, the change in voltage that occurs during
the current pulse increases with increasing tc.
Increase in the current pulse amplitude to 60 nA (7.5 nA in
the model of a 100
~m-diameter
aggregate) results in larger maximal
prolongation and abbreviation, and a less gradual transition from
prolongation to abbreviation of cycle length as tc is increased
(Fig. 3-8, right panel).
The range of tc over which the transition
from maximal prolongation to maximal shortening takes place is
about 2 msec wide at this amplitude, as compared with about 70 msec
at a pulse amplitude of 20 nA (Fig. 3-8, left panel).
The
transition from prolongation of cycle length to abbreviation of
cycle length is continuous at an amplitude of 20 nA.
0
Further increase to 80 nA (10 nA in a 100
to a transition that occurs within 1
~sec
~m
aggregate) leads
or less (Fig. 3-9A).
Figure 3-98 shows the trajectories in the current-voltage {IV)
phase plane.
Thus, during the current pulse, both trajectories
follow an almost identical route; however, soon after the stimulus
is turned off, the two trajectories go their separate ways.
Figure 3-10 summarizes the effect on interbeat interval of a
current pulse, showing plots of the normalized perturbed interbeat
T1 /T 0 as a function of the normalized coupling interval tc/T 0 for
three different current amplitudes.
Since the time from the
upstroke of the action potential to -70 mV on the repolarizing limb
of the action potential is assumed to be 0.150 sec, T0 = 0.150 +
0.374
= 0.524
sec.
3-32
Q
Figure 3-9.
A.
Effect on the voltage wavefonn produced by an 80
nA amplitude, 20 msec duration current pulse.
For the trace labelled I, the current pulse is
turned on at t
= 0.770
labelled II, at t
=
msec; for the trace
0.771 msec.
Thus, a change
in tc of 1 JJSec can produce remarkably different
behaviour at this stimulus intensity.
B.
Trajectories in the current-voltage phase plane
for the two trials shown in panel A above.
The
point labelled a is the·starting point (V= -70
mV}.
0
The trajectory moves from b to c when the
current pulse is switched on, from c to d during
the current pulse, and from d toe when the pulse
is switched off.
From e, the trajectories evolve
in two quite different directions:
trajectory I
(pulse applied at t = 0. 770 msec), trajectory II
(pulse applied at t
= 0.771
msec}.
The two
trajectories overlie each other from a toe on a
diagram of this seal e.
The simulations in both panel A and panel B were
terminated at V = -40 mV.
3-33
Integration step size
0
A
-25. 121
V
(m V)
-1121121.121
0. 121
• 25
. 5
t(s ec)
B
10121. 0
lror
(nA)
1
121. 0
-11210.0
-11210. 121
-75. l2l
V(mV)
-512l. l2l
0
Figure 3-10.
Phase-resetting data plotted for three different
pulse amplitudes:
nA ( 0 ).
with T0
10 nA ( 0 ) , 20 nA ( 6 ) , and 80
T1 /T 0 is plotted as a function of t/T
= 0.524 sec. T1 is found by adding
0
,
0.150 sec
to the time computed for threshold to be attained,
since the simulation is started with V= -70 mV.
Similarly,\ is computed by add.ing 0.150 sec to the
time t at which the current pulse is injected.
Points are computed every 50 msec for tc ) 250 msec,
and every 10 msec for 150 msec " \
<:
250 msec.
In
addition, for the 20 nA curve, points are computed
every 2 msec for 150 msec
tc ..: 160 msec, in order
<;
to illustrate the continuity of that curve.
Integration step size
= 50
~sec.
3-34
0
0
~
1
-------
----·---- ----------- -- ---- --- - ---------- --.-
0000
0 00
~
-----------
....
0
0
0
D.
0
0
D.
0
0
0
0
0
o.
0
0
0
10
20 nA
80
1
.o
(111) Analysis of the Response to Current Pulse Perturbation
Figure 3-8 showed the effect of delivering a 20 nA amplitude,
20 msec duration current pulse at several coupling intervals.
effect of a pulse delivered at t
The
= 0 msec (i.e. tc = 150 msec) is
to prolong the i nterbeat interval beyond control (Fig. 3-llA}.
Figure 3-llB shows that during the current pulse, !TOT is more
inward than in control (Fig. 3-7B), so that V falls more slowly
In fact, during the latter half of the pulse, the
than in control.
current is inward and the membrane depolarizes. Since the final
phase of repolarization proceeds more slowly, and since rx is very
small during this time (tx = 5 msec), Ix is maintained at a higher
outward level for a longer period of time (Fig. 3-llC) than in
control (Fig. 3-7C).
Thus, when the current pulse is switched off,
Ix is still non-zero (Fig. 3-llB) in contrast to the control case
( Fi g • 3- 7B) •
The sum IK + Ibg is also outward immediately after the
2
current pulse ends (Fig. 3-110) in contrast to control (Fig. 3-70),
so that ITOT is outward (Fig. 3-llB). This is in part due to a
slight decrease in the rate at which s declines (Fig. 3-llH), which
in turn is due to the doubling in
r
5
from about 0.5 sec to 1.0 sec
produced by the perturbation (Fig. 3-7!, Fig. 3-lli).
However, the
fact that IK + Ibg is outward is primarily a consequence of the
2
fact that the change in membrane voltage produced by the current
pulse causes a bigger decrease in the amplitude of Ibg than in IK
2
0
3-35
0
Figure 3-11. The time course of several variables in the model in
response to a 20 nA amplitude, 20 msec duration
depolarizing current pulse delivered at t
=0
msec.
The arrows at the bottom of panels A and F indicate
the time during which current is being injected.
A. V B. ITOT
C. Ix
F. V G.
H. s,
X
D. IKz' Ibg' IK +Ibg
E. INa
r. Ts
J. m,h.
2
s~
L
.
Voltages are in mV, currents in nA, and times and
time constants in sec.
0
Integration step size
3-36
= 50
~sec.
Sill. Ill
0
5111. Gl
A
V
V -zs.121
-1111(11. Ill
1111121. Ill
ITOT
0
t
1. Ill
B
X
Ill. Ill
G
• !5
Ill. Ill
-lalll. a
lx
-25.111
-1111111.111
t
F
c
laa.
Ill
!1111.
a
• !I
a. 111
Ill. Gl
sa. a
1. Gl
2. 11.1
D
Ts
Ill. Gl
1. 11.1
a.
-sa. Ill
Ill. Ill
111
1. Ill
E
INa -!1111. 121
J
• 5
m
-lllla. 13
0
a.
Ill. 13
111
• zs
t
• 5
Ill. liiJ
. ;zs
t
.s
{Fig.
3-110).
This is because of the different degree of
rectification present in the two currents (Fig. 3-68}.
The effect
of the current pulse is thus to reset the activity backwards in
time:
the latter half of each curve in Fig. 3-11 is virtually
superimposable with the final part of the corresponding curve in
Fig. 3-7.
Figure 3-12 shows the effect of delivering a 20 nA amplitude,
20 msec duration current pulse at a coupling interval of 170 msec.
This leads to a shortening of the interbeat interval {Fig. 3-12A).
The effect of the current pulse is basically to charge the membrane
capacitance in a linear fashion (Fig. 3-12A), since the stimulus
current is much larger than the ionic membrane current {Fig. 312B). The perturbation has little or no effect on Ix (Fig. 3-12C)
or its activation variable x (Fig. 3-12G).
At the end of the current pulse, the sum IK + 1bg is not
2
much different from the control value at that time, since IK and
2
Ibg change by about equal amounts during the current pulse·(Fig. 3120}.
Thus dV/dt is about the same in both instances {Figs. 3-7A
and 3-12A).
However, -r 5 (Fig. 3-121) has been reset to a higher
value {about 1.3 sec) than in control (about 0.6 sec).
As the
membrane depol arfzes, -r 5 (Fig. 3-12!) proceeds along the falling
limb of its bell-shaped curve (Fig. 3-3C) instead of along the
ascending limb as in control (Fig.
3-71).
Thus, immediately after
the end of the current pulse, s falls more slowly than in control,
aided by the fact that
s~
is higher at the end of the current pulse
than in control (Fig. 3-121, Fig. 3-7!, eqn. (3-10)).
0
3-37
0
Figure 3-12. The time course of several variables in the model in
response to a 20 nA amplitude, 20 msec duration
depolarizing current pulse delivered at t
=
20 msec.
The arrows at the bottom of panels A and F indicate
the time during which current is being injected.
0
A. V B. 1TOT . C. IX
F. V G. X
H. s, s
D. IK ' I bg ' I K2 +I bg E. IN a
2
CO
I.
T
s
J. m, h.
Vol tages are in mV, currents in nA, and times and
time constants in sec.
]..I
0
Integration step size = 50
Sec.
3-38
5111.11.1
0
V
S0.1lJ
A
V
-2S.IlJ
-lliJS. 11.1
1011.1. Ill
lror
t
0
t
1. 11.1
B
X
-lllJ0.11.1
lx
-2s.0
-111.111.1. 11.1
11.1. llJ
lllJS.II
F
.s
Ill. 11.1
c
G
1.'------
1. 11.1
SS.IlJ
.s
a. a
11.1. 0
38.11.1
2. 11
Ts
11. 11
I
1. 11.1
lbg
-as. 11
111.11.1
11. s
1.
E
s
.s
INo -ss. s
m
a. s
-1011. 11
0
a. a
• 2!5
t
.!5
s. s
• 2!5
t
.s
At first sight, one would think that resetti n~·
0
; to
a higher
value should produce an increase in outward current and so lead to
a prolongation and not an abbreviation of the interbeat interval.
However, once again the rectification properties of the IK and Ibg
2
channels appear to be more important than the time-dependent
behaviour of the IK channel.
The effect of the pulse is to
2
advance the activity:
most of the traces in Fig. 3-12 for the time
following the delivery of the current pulse are virtually
superimposable with the final parts of the corresponding traces
during unperturbed activity (Fig. 3-7).
5.
DISCUSSION
(f)
Spontaneous Activity
Figure 3-7 shows that diastolic depolarization is not simply
due to a decrease in the outward time-dependent pacemaker current
IK , but is in fact due to the simultaneous decrease in the
2
magnitudes of two large (in comparison to ITOT) oppositely directed
currents, IK and Ibg· A similar situation occurs in the MNT model
2
for Purkinje fibre
(Me All
i ster, Noble, and Tsi en, 1975}.
Furthermore, one cannot say that the decrease in IK is due to a
2
progressive deactivation of that current (i.e. decrease in the
activation variables) as diastolic depolarization proceeds.
In
fact, IK continues to fall in late diastole after the minimum of s
2
0
3-39
is attained, when s is actually increasing (Fig. 3-70,H).
0
In th'is
model, s falls from a maximum of about 1.0 at MOP to a minimum of
about 0.7, as it does in the MNT model.
the aggregate, IK
However, in the model of
falls by a factor of about 7 during this time
2
(Fig. 3-70}.
Thus, it is the rectification properties of IK
and
2
Ibg that play a major role in detennining the evolution of the
transmembrane voltage during the phase of diastolic depolarization
in this model of the aggregate.
model, where the fall in IK
This is in contrast to the MNT
is largely due to and occurs in a
2
fashion parallel with the deactivation of IK
(McAllister, Noble,
2
and Tsien, 1975).
The magnitudes of IK
2
,
Ibg' and IK
2
+ Ibg decline as
pacemaker depolarization progresses (Fig. 3-70).
0
In fact, the sum
IK + Ibg is close to zero towards the final part of phase 4 {Fig.
2
3-70) • Thus, changes in any of the parameters contra11 i ng IK
or.
2
Ibg that lead to an increase in IK
2
or decrease in Ibg of only a
few nanoamperes will abolish spontaneous phase 4 depolarization.
For example, increase of gK
in eqn. {3-lSc) from 0.9
~s
to 1.2
~s
4
abolishes spontaneous activity.
Injection of a constant
hyperpolarizing current of 4.3 nA {i.e. Iappl = 4.3 nA in eqn. (321)) also stops activity.
Injection of a constant hyperpolarizing
current of this magnitude will also stop spontaneous action
potential generation in the aggregate (Guevara, unpublished).
3-40
0
As pacemaker depolarization enters its final phase, the
currents Ibg and IK turn themselves off, since their sum IK
2
+
2
Ibg
serves to depolarize the membrane increasingly into the region of
rectification of both channels (Fig. 3-6B).
As the sum Ibg + IK
2
approaches zero, the depolarization process is taken over by INa
(Fig. 3-7E,J}.
Thus, as in the MNT model, the final part of
pacemaker depolarization is due to the window current INa • This
CO
can be seen in experiments on aggregates in which TTX is added to
the bathing medium:
there is a slowing of beat rate due to
prolongation of the duration of phase 4 (Colizza, Guevara, and
Shrier, 1983), presumably due to a decrease in the window current.
Decrease of gNa in eqn. (3-1) to 44 llS abolished spontaneous
activity.
Thus, in surrmary, it appears that the three currents INa, IK ,
2
and Ibg control the duration of phase 4 in the aggregate.
the window current Isi
However,
may also play a role, since there is also a
CO
slowing in the beat rate of an aggregate when the slow inward
channel blocker 0600 is applied, and since the window current Isi
""
is approximately
~
of the window current INa
in the MNT model of
01)
Purkinje fibre in the pacemaker range of potential s.
The arrangement in which diastolic depolarization, even in
physiological preparations such as Purkinje fibre and the
0
3-41
sinoatrial node, appears to be due to the decline in magnitude of
0
two large currents does not seem to be a very economical one.
An
alternate mechanism would be to have, for example, a single much
smaller outward current which declines in value throughout phase 4.
This would decrease the number of channe1 s needed in the membrane,
as well as significantly decrease the load on the Na+-K+ pump.
One theoretical reason for having two 1arge currents has to do
with control of spontaneous activity.
Many target organs in the
body (e.g. the heart} have a reciprocal sympathetic-parasympathetic
innervation with high levels of resting tone in both branches of
the autonomic system.
Alteration in the state of the system is
accomplished by simultaneously decreasing the level of activity in
one branch and increasing it in the other.
I suggest that there
are many advantages to this mode of control, analogous to the
advantages seen when differential input rather than single-ended
input is used in electronic amplifiers.
The two large currents
that flow during diastolic depolarization in the aggregate may be
analogous in some way to the two high resting levels of neural
tone.
A neurotransmitter or circulating hormone may have
simultaneous effects on more than one current.
For example,
acetylcholine both increases the outward potassium current and
decreases the inward sodium-calcium current in the sinoatrial node.
These two simultaneous changes work in concert to slow the
spontaneous beat rate; there is however a "differential amplifier..
c
3-42
kind of effects in that the "non-inverting" or "excitatory" input
is being decreased, and the,. inverting" or
u
inhibitoryu input
increased.
There has recently been a re-interpretation of the results of
the original voltage clamp experiments carried out in the pacemaker
potential range in Purkinje fibre.
These results were originally
thought to indicate the presence of a depolarization-activated
outward current IK carried by potassium ions (Dudel and Trautweins
2
1958; Trautwein and Kassebaum, 1961;
Dudel et al., 1967;
1969).
Deck and Trautwein, 1964;
Noble and Tsien, 1968; Peper and Trautwein,
Recent work has indicated that this time-dependent
pacemaker current may actually be a hyperpol ari zati on-activated
inward current that is carried by both potassium and sodium ions
DiFrancesco and Noble, 1982a,
(DiFrancesco, 1980, 1981a, 198lb;
1982b}.
This current is termed If, and is in sone respects similar
to the hyperpolarization-activated inward'current If (also called
ih) found in the sinoatrial node (Brown, DiFrancesco, and Noble,
1979; DiFrancesco and Ojeda, 1980;
Yanagihara and Irisawa, 1980;
Irisawa and Noma, 1982).
Arguments can be made both for and against the reinterpretation
of IK
in terms of If in Purkinje fibre (e.g.
Cohen, Falk, and
2
Kline, 1982).
However, algebraic computation (DiFrancesco, 198la)
and numerical simulation (DiFrancesco and Noble, 1982b) suggest that
as far as the voltage and total current waveforms are concerned, it
does not matter whether the IK. or the If description is used.
2
Thus, reformulation of the description of IK given above
2
0
3-43
in eqns. (3-6) to {3-12) in terms of If should not change the
0
phase-resetting behaviour of this model of the aggregate.
things presumably changed are the
deta~ls
The only
of how the currents IK
2
(or If) and Ibg are involved.
The mechanism of spontaneous activity in the model of the
aggregate is quite different fr·om the mechanism demonstrable in two
similar models of the sinoatrial node (Yanagihara, Noma, and
Irisawa, 1980; Irisawa and Noma, 1982).
Three currents change
significantly during the pacemaker depolarization of the sinoatrial
node:
the current is, the slow inward current, which is largely
responsible for the action potential upstroke; the current it'
which is a time-independent leakage current of unknown origin; and
the current iK' which is a potassium-ion current responsible for
the repolarization phase of the action potential.
During pacemaker
depolarization in these sinoatrial node models, there is a
progressive movement of is in a more inward direction, and a
progressive movement of ii in a more outward direction.
There is
a1 so a small decline in the magnitude of \, which remains outward
throughout phase 4.
Note that ih (:If) is not involved in
generating spontaneous phase 4 depolarization.
An alternative,
less-physiologically based ionic model of the sinoatrial node has a
pacemaker potential that is generated by an IK
type of mechanism
2
similar to that involved in the pacemaker phase of Purkinje fibre
or of the aggregate (Bristow and Clark, 1982).
Periodic action potential generation can be elicited in models
0
3-44
of quiescent ventricular myocardium by inj ecti nq a constant
0
depolarizing current (Beeler and Reuter, 1977).
Since this bias
current is much 1 arger than the sum of the membrane-generated
currents during diastole, there is a linear phase 4 depolarization
until threshold is attained.
Thus, the mechanism is similar to
that involved in action potential generation in the Hodgkin-Huxley
model for squid axon subjected to a constant depolarizing bias
current.
Another case in which spontaneous phase 4 depolarization can
take place without the primary involvement of a time-dependent
current is in an ionic model of atrial aggregates {Shrier and Cl ay,
1982). Atrial aggregates fabricated from 10- to 14-day-old embryos
do not demonstrate a time-dependent change in current fall owing a
0
voltage clamp step made in the pacemaker range of potentials.
Thus, IK or If is not present; there is only a background current
2
apparent.
Yet, there is a slow pacemaker depolarization to
threshold; the rate at which this depolarization proceeds is
largely controlled by the membrane time constant {Shrier and Clay,
1982).
Thus, there are several different arrangements of ionic
currents that can 1ead to periodic activity in cardiac cells.
The
mechanisms can be quite different from system to system as
illustrated above.
However, the periodic activity is presumably
due to the presence of a 1 imit cycle in the phase space of the set
of differential equations describing the system.
0
3-45
Ionic modelling
indi'cates· that this limit cycle can arise from one or more
0
bifurcations as a parameter in the system is changed (Guevara,
unpublished).
In the latter case, a sequence of complex periodic
and aperiodic phenomena are seen, which are analogous to those seen
experimentally.
(ii) Comparison of the Phase-Resetting Behaviour of the Model with
Experiment
The phase-resetting response of the ionic model shown in Figs.
3-8 and 3-9A and summarized in Fig. 3-10 is similar to the
experimental response detailed earlier in CHAPTER 2 in the
fall owing ways.
(i)
The response is biphasic:
an early depolarizing pulse
prolongs the cycle length; a late depolarizing pulse
shortens the cycle length.
(ii)
As stimulus intensity increases, the maximum possible
prolongation and abbreviation increase, the neutral
point at which T1/T 0 equals unity moves to a small er
coupling interval, and the transition from prolongation
to shortening becomes more abrupt.
(iii)
The transition from prolongation to shortening can
occur with a change in coupling interval of less than 1
0
3-46
msec at a sufficiently high pulse amplitude (e.g.
0
compare Fig. 3-9A with Fig. 2-16 or
Fig~
2-17).
Moreover, the transition occurs at about the time of
MOP in both instances.
(iv)
The current amplitude needed to elicit a certain
behaviour in the model is in the same range as that
seen experimentally, keeping in mind that the model is
for a 200
~m-diameter
aggregate, and that aggregate
membrane area increases as the cube of the aggregate
diameter.
The most notable discrepancy between the experimental and
modelling results is at lower pulse amplitudes.
Experimentally,
prolongation is produced at small amplitudes when the slope of the
pacemaker potential is reset to a value less than in control {e.g.
Fig. 2-5, middle panel).
The model does not respond to a small
current pulse with a change in the slope of diastolic
depolarization (Fig. 3-8, left panel}.
that if
~ight
It can be however shown
changes in some of the parameters in the model are
made (which leave the steady-state IV and the currents IK and Ibg
2
within the range of experimentally determined values}, there
results a more linear pacemaker potential that does tend to respond
to a small current pulse with a change in its slope (Clay, Guevara,
and Shrier, 1984}. However, even with these modifications,
c
prolongations of at most 10% are produced.
3-47
This does not agree·
with the experimental results, where prolongations
0
{i.e. 1.2 ( T1/T 0
(
of~
20-30%
1.3) can be routinely obtained in all
aggregates (Fig. 2-10A,B).
Another discrepancy is that the long delays shown in Figs. 211 to 2-14 are not reproduced by the model.
However, this is not
unexpected, since long delays were only seen in more slowly beating
aggregates, and the model investigated in this chapter is one for a
faster-beating preparation with an intrinsic period of 524 msec.
It can be shown that oscillatory activity in the pacemaker range of
potentials in response to a current pulse can be produced in a
model differing slightly from the one developed in this chapter
(Clay, Guevara, and Shrier, 1984).
This involves shifting the IV
characteristic of ITOT shown in Fig. 3-6A in the hyperpolarizing
direction so as to sl 0~1 the beat rate; the modification creates an
equilibrium point lying in the pacemaker range of potentials.
~1oreover,
since this equilibrium point is unstable, spontaneous
.
activity cannot be annihilated with a single current pulse.
11
This
Saddle-node bifurcation also results in the creation of an
11
additional equilibrium point which is a saddle-point {Fig. 2-24).
(iii} Ionic Mechanisms Involved in Phase Resetting
The above analysis of a partial model shows that the response
to a current pulse is very complex, with ,the membrane voltage and
the various currents feeding back on one other in a very
complicated manner.
However, I will now summarize what I see the
c
3-48
main mechanisms of phase resetting to be ·as revealed by study of
c
ionic models of the aggregate.
For a very small current amplitude, essentially only IK and
2
Ibg are involved in the phase-resetting behaviour.
because:
This is
{i} the pulse amplitude is too small in comparison to Ix
to affect events during the repolarizing limb of the action
potential ; and ( i i} the pulse amplitude is too small to bring the
membrane potential into the voltage range for activation of INa
except for times very late in the cycle.
There is a resetting of
the activity of IK and Ibg forward or backward in time, producing
2
abbreviation or prolongation of cycle length respectively (Figs. 311, 3-12). This behaviour hinges upon the rectifying properties of
the IK and Ibg channels.
2
0
The maximum prolongation and
abbreviation of cycle length seen will be quite small at this
stimulus intensity.
At the other extreme, for a very large current pulse,
essentially only INa and Ix are involved in the phase resetting
behaviour.
If such a pulse is delivered after
r~DP
occurs, it will
depolarize the membrane into the range of activation of INa; the
membrane will immediately come to threshold and there will be an
abrupt shortening of the cycle length.
If the pulse is delivered
not too much earlier than the time of occurrence of MOP, the same
kind of behaviour will be seen.
If however the pulse is delivered
even earlier, Ix will still be sufficiently 1arge so as to nullify
most of the depolarizing effect of the pulse; repolarization will
continue at a reduced rate, producing an increase in action
3-49
potential duration, a more positive t>10P, and .an increased cycle
1 ength.
At intermediate current amplitudes, all four currents come
into play.
For example, a pulse delivered at about the time of MOP
(when Ix is quite small) can affect Ix, .rK
2
and Ibg·
The same
pulse delivered later in the cycle can affect IK and Ibg' as well
2
as INa to a lesser extent.
(iv) Discontinuity in the Phase-Resetting Response
There is a coupling interval somewhere just before r1DP occurs
where a large current pulse will take the membrane voltage very
close to the threshold voltage.
0
Due to the very fast rate at which
INa becomes activated in this range of potential s {Fig. 3-2A) and
the relatively large value of the conductance gNa' one of two
behaviours will result (Fig. 3-9A}. The behaviour is very
delicate:
for some time immediately following termination of the
current pulse, the difference in potential between traces I and II
in Fig. 3-9A is only 7
~v.
I have not investigated whether the response shown in Fig. 39A is continuous; that is, whether delivery of pulses with 150.770
msec
~
tc
~
150.771 msec would produce action potentials with
upstrokes falling in between those of the two traces shown in Fig.
3-9A.
However, in the Hodgkin-Huxley model for squid axon, it has
been shown that a response presumably equivalent to that shown in
c
3-50
Fig.
0
3-9A is in fact continuous.
In that case, the voltage
arrived at following a stimulus must be changed in increments of 1
femtovolt in order to reveal the continuity (Clay, 1977).
kind of "quasi-threshold
phenomena~
This
(FitzHugh, 1955) cannot be
easily investigated numerically on a computer that has at most only
16-17 significant decimal digits.
The equilibrium point lying in
the plateau range of potential s may have some sort of saddl e-1 ike
geometry associated with it.
If such be the case, then there might
indeed be a true all-or-none phenomenon in this model {FitzHuqh
1955, 1960).
While the response shown in Fig. 3-9A is probably continuous,
in real life the situation is somewhat different (Figs. 2-16, 2.-
17).
We know that the Hodgkin-Huxley approach to the description
of ionic currents {formulation as a system of continuous ordinary
differential equations) is incorrect.
A more precise description
of ionic currents is a stochastic one in which currents arise from
the random openings and closings of a population of single channels
(e.g. Clay and DeFelice, 1983). A single ionic channel passes a
current of roughly 2 pA and opens for about 5 msec.
A single
opening and closing of this channel thus transfers a charge of 10
fC across the membrane.
For a 200
capacitance of 0.023
this results in a change in the membrane
potential of
~F,
about~ ~V.
~m-diameter
aggregate with a
Thus, the difference in the two responses
shown in Fig. 2-9A can be produced if 14 channels open in one case
during the pulse and do not open in the other case.
Intermediate
responses would only be seen if fewer than 14 channels change their
0
3-51
activity during the current pulse.
0
Since many thousands of
channels open and close during the current pulse, this eventuality
is unlikely.
Even if less than these 14 critical channels open or
close during the pulse so that the voltage at the end of the pulse
changes by less than 7 uV, there is no guarantee that the
intermediate responses theoretically predicted in continuous models
will be found.
Thus, only the two behaviours shown in Fig. 3-9A
will be seen experimentally.
the
s~ngle
This situation will be reinforced if
channels do not operate independently of each other;
that is, if there is some form of cooperative phenomenon in the
membrane {Changeux et al., 1967; Carnay and Tasaki, 1971).
The perturbed cycle lengths of traces I and II in Fig.
differ by about 0.8 of the spontaneous cycle length.
0
3~9
Thus, type 0
phase resetting is not present, unless a further phase shift
difference of 0.2 can be produced during the post-stimulus cycles.
This eventuality cannot be tested, since the partial model does not
generate repetitive activity.
In summary, combined experimental and modelling work
indicates that the action potential in the spontaneously beating
aggregate at 35°C is indeed all-or-none, provided that stimulation
is carried out in a narrow range of potentials near MOP.
(v)
Suggestions for Future Ionic Modelling Work
The partial model for the electrical activity of the aggregate
0
3-52
developed and investigated in this chapter is a preliminary one in
many respects.
currents.
There appear to be problems with all of the
These problems were revealed by comparing the predicted
with the experimental phase-resetting response, and incidentally
illustrate the value of using the phase-resetting behaviour of an
ionic model as a test of its validity.
(i)
INa:
The issues are as follows:
The problem with rm was mentioned earlier.
The
modification made to am is not entirely satisfactory,
since it produces a poorer fit to the 'm data than the
original equation of Ebihara and Johnson (1980).
Perhaps modifications to Sm, am, or an should be made
instead.
0
( ii)
IK
,
Ibg:
The model does not account very well for the
2
response to a small amplitude pulse which leaves the
membrane potential in the pacemaker range of
potential s.
In particular, prol ongati ons of cycle
length of more than about 10% are not produced.
This
probably indicates a problem with IK or Ibg (or both).
2
(iii)
Ix:
The formulation of Ix may need to be modified,
si nee recent work shows that Ix in the atrial aggregate
actually has two components, I
xl
and I
x2
, which are
activated in two different ranges of potential
c
and Shrier, unpublished}.
3-53
(Cl~y
The major deficiency of the model is that, being an incomplete
0
model, it does not produce cyclic activity.
At a minimum, I s.~
would have to be added to generate realistic cyclic activity. In
such a full model, graded action potentials (Fig. 2-7) would
presumably be produced in response to an early high-amplitude
depolarizing stimulus.
Furthermore, the change in the form of the
PTC could be investigated as a function of stimulus strength.
example, it
~ould
For
qe possible to see if the model would produce the
sequence {type 1 (monotonic) +type 1 (non-monotonic) +type a} in
PTC~
23).
as the amplitude of a depolarizing pulse is increased (Fig. 2This sequence is seen in the MNT model for Purkinje fibre
which, being a complete limit cycle model, generates cyclic
activity (Guevara, unpublished:
see CHAPTER 6}.
There has been only one systematic investigation published to
date concerning the phase-resetting properties of an ionic model of
cardiac tissue (Bristow and Clark, 1982).
Although the ionic
mechanisms of spontaneous activity and thus of phase-resetting may
be
different in different cardiac oscillators, the topological
properties of the phase-resetting should be similar.
Only
systematic investigation of several different ionic models can test
this hypothesis.
3-54
0
0
0
,......,..
'-"'