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1.1 Temporal evolution of the transmembrane voltage of a bullfrog action
potentials shown. The pacing period is the time between pacing
stimuli (shown at the bottom). The action potential duration is the
time between the upward deflection of the signal to the downward
deflection. The diastolic interval is the time from the end of the
previous action potential to the beginning of the next one. . . . . .
4
1.2 The panels A-E show the temporal evolution of a normal heart beat
on the right atria of a sheep heart. The wave originates in the sinus
node (off plaque in the upper left) and propagates across the atrial.
Each of the panels reprensents 6 ms of time. The black dots show
where the leading edge of activation was during that time slice. The
plaque is approximately 7.5 cm 3.5 cm and the wave of excitation
travels arcoss it in 31ms. Each of these waves is separated in time by
about 1 sec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.3 The panels A-I show the temporal evolution of a reentry circuit during
fibrillation on the right atria of a sheep heart. The wave circulates
around the surface of the atria. Each of the panels represents 12 ms
of time. The black dots show where the leading edge of activation
was during that time slice. The wave of excitation travels around the
plaque in 120 ms. While this behavior is slower than normal sinus
rhythm it is continuous. . . . . . . . . . . . . . . . . . . . . . . . .
7
1.4 The bifurcation diagram of the response of small pieces frog ventricle
muscle to periodic pacing. The solid circles are generated as the
pacing period is decreasing, and the open circles are generated as the
pacing period is subsequently increasing. The solid arrows are added
to indicate the time course of the experiment. . . . . . . . . . . . .
12
1.5 A beat by beat illustration of a TDAS control experiment. The action
potential duration is plotted against beat number. While control is
off (the black points ) the tissue displys alternans. When control is
turned on (the gray points) the alternans behavior is supressed and
the normally unstable period one behavior is stabilized. . . . . . . .
14
xii
1.6 A 2:1 1:1 transition can be caused by a single injected stimuli. The
pacing sequence is shown at the bottom of the figure. The pacing
period (PP) is marked. The extra stimuli is T ms after the regular
stimulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.1 Circular process of feedback controllers. The state of the system to
be controlled is measured with the transducer. A calculation of the
difference between the current state and the desired state is used to
generate an error signal. The actuator adjusts some system parameter in order to minimize this difference. . . . . . . . . . . . . . . . .
18
2.2 Cardiac muscle tissue is the system that produces the unstable dynamics in the feedback loop. . . . . . . . . . . . . . . . . . . . . . .
20
2.3 A drawing of a cardiac membrane. The membrane consists of a lipid
bilayer with embedded proteins. The lipid bilayer keeps the solution
that fills the cellular structure separate from the solution that surrounds the tissue. The ions are transported across the membrane
wall by proteins that act as ion-selective voltage-sensitive channels
and pumps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.4 Temporal evolution of the transmembrane potential in a bullfrog cardiac cell in response to a brief stimulus. The transient depolarization
of the membrane (black curve) is called an action potential. The
stimulus is shown at the bottom of the figure (gray curve). . . . . .
23
2.5 An equivalent circuit model of cardiac cell. The ion flows across the
cellular membrane can be modeled by a simple circuit. The batteries
represent the Nerst potential of the resting concentration of each type
of ion. The variable conductances represent the protein gates that
open and close in response to the various voltage levels. . . . . . . .
24
2.6 An experimentally restitution curve I recorded from bullfrog cardiac
muscle relating the action potential duration to the proceeding diastolic interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.7 Electrodes fullfill the role of transducers in the feedback loop used to
control cardiac muscle tissue. . . . . . . . . . . . . . . . . . . . . .
28
xiii
2.8 The temporal evolution of a microelectrode signal. The signal is
recorded from a small peice of periodically paced bullfrog ventricle
muscle. The pacing signal (period 950 ms) is shown at the bottom
of the figure. The transmembrane voltage was amplified by a factor
of ten by the differential amplifier and then digitized with 12-bit
resolution over 2 to -2 volts range at 2000 samples/sec. . . . . . . .
29
2.9 The temporal evolution of the signal from a MAP electrode. The
signal is recorded from a small peice of periodically paces bullfrog
ventricle muscle. The pacing signal (period 800 ms) is shown at the
bottom of the figure. The signal was amplified by arbitrary gain and
then digitized with 12-bit resolution over 2 to -2 volts range at 2,000
samples/sec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.10 A comparison between a signal from a microelectrode (A) and a signal
from an extracellular electrode (B). Both these signals were recorded
simultaneously from the same frog ventricle and both are relative to a
ground wire in the physiological solution. The pacing signal (period
950) is shown at the bottom of each panel. The microelectrode signal
was ampilified by a factor of 10, while the extracellular signal was
amplified arbitrarily to approximately 2 volts peak to peak. . . . . .
33
2.11 An example of how the control and actuation stage of the control
process is exemplified by a simple model of a cardiac muscle cell
dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2.12 A simple model of cardiac behavior can be derived from the restitution relation (solid line) and the pacing relation (dashed line). The
restituion curve describes how the ? ?!Q + $ !
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48
3.3 A flow chart of the temporal sequence of operations of the labview
pacing program. The program flow is broken up into two parts. The
first records the response of the periodically pace tissue to decreasing
periods. The second part of the program records the response of the
tissue to increasing periods. . . . . . . . . . . . . . . . . . . . . . .
49
3.4 A signal from a small piece of periodically paced bullfrog ventricle
muscle obtained using a microelectrode. The pacing signal is shown
at the bottom of the figure. The pacing period was 950 ms. The
signal was amplified by a factor of ten by the differential amplifier
and then digitized with 12-bit resolution over 2 to -2 volts range at
2000 samples/sec. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
xv
3.5 Time evolution of transmembrane potential during a typical hysteresis experiment. The stimulus waveform is shown at the bottom of
each diagram. (a) The tissue exhibits 1:1 dynamics for slow pacing
during a downsweep. This time series is taken from the data
used to generate points on Fig. 3.6 (open circles) at a of 600 ms.
(b) The tissue exhibits 2:1 dynamics (closed triangles in Fig. 3.6) for
the same used in (a), but during an upsweep. Arrows indicate
measured . Vertical dotted lines illustrate the correspondence
between stimulus intervals in (a) and (b). . . . . . . . . . . . . . . .
52
3.6 A bifurcation diagram showing 2:1 1:1 hysteresis. Open circles are
measurements as the is swept downward (i.e., the tissue is
driven faster); closed triangles show during the upsweep.
53
3.7 Observations of alternans, bistability and hysteresis from tissue taken
from a different animal than that used to generate the data in Fig.
3.6. Thick arrows show the direction of the sweep; thin arrows
indicate the 2:2 2:1 and 2:1 1:1 transitions. Note that several data
points appear almost on top of each other for each , representing
slight variations in measured . . . . . . . . . . . . . . . . . . .
54
3.8 Comparison between the predictions of the simple mapping model
and the experimental observations shown in Fig. 3.6. The solid line
is obtained by iterating (3.1) where ( ) is of the form (3.2), and
parameters $( = 616 ms, = 313 ms, = 207 ms, and =
86 ms. The upward arrow indicates the 1:1 2:1 transition; the short
downward arrow indicates the 2:1 1:1 transition in the observed
data and the one-dimensional model. . . . . . . . . . . . . . . . . .
57
3.9 Action potential duration as a function of diastolic interval for the
data shown in Fig. 3.6. Open circles indicate downsweep; closed triangles indicate upsweep. Data separates into two distinct branches,
one corresponding to = 1 and the other to = 2. The solid and
dotted lines were added to guide the eye. Vertical line at =285
ms indicates the diastolic interval at which the slope of the = 1
branch becomes greater than unity. The solid (dotted) curve guide
the eye along the = 1 ( = 2) branch. . . . . . . . . . . . . . . .
60
xvi
3.10 2-D model fit to the bifurcation diagram shown in Fig. 3.6. The line
indicates a fit using the two-dimensional mapping model (3.9), with
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of the vertical dashed line the system is unstable, however, under the
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the simple system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.15 A schematic of the PIC1677 based Projective TDAS controller. The
signal from one of the electrode on the surface of the atria is differentiated. A negative threshold detector is employed to detect activations.
A threshold crossing causes a pulse to be sent to the microcontroller.
This microcontroller in software implements the PTDAS algorithm
and output diagnostic data the a computer as well as sending a pulse
to a constant current source to cause an activation. . . . . . . . . . 126
6.16 A flowchart of the PTDAS implimentation by the microcontroller.
An activation or a timer over flow causes an interupt. The type of
interupt is determined. If it is an activation the next control time is
computed. If it is a timer overflow the tissue is paced an then the
next control time is computed. . . . . . . . . . . . . . . . . . . . . . 128
6.17 Histograms from positive gain PTDAS control experiments on fibrillating sheep atria . The grey bars are from the middle 100
electrode from the plaque. The striped regions are from the electrode the controller uses to observe the dynamics and a neighboring
electrode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.18 Histograms from negative gain PTDAS control experiments on fibrillating sheep atria . The grey bars are from the middle 100
electrode from the plaque. The striped regions are from the electrode the controller uses to observe the dynamics and a neighboring
electrode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.19 A time series showing that some pacing pulses are ignored by the tissue. The stimulus is marked with an arrow. The time series from the
control observation electrode (A) does not show any active response
to the stimulus. This can also be seen in the derivative signal (B). . 133
C.1 A drawing of a core conductor along with the relevant the voltages
and currents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
xxiii
D.1 A drawing of a microelectrode. A microelectrode is a glass pipette
with a sharp tip (approximately 1m in diameter). It is filled with
a 3M KCl solution. The solution is contained and the silver wire is
held in place by some wax placed in the end. . . . . . . . . . . . . . 151
D.2 Microelectrode signal. This signal is from a small peice of periodically
paced bullfrog ventricle muscle that was pinned down. The pacing
signal is shown at the bottom of the figure. The pacing period was
950 ms. The signal was amplified by a factor of ten by the differential
amplifier and then digitized with 12-bit resolution over 2 to -2 volts
range at 2000 samples/sec. . . . . . . . . . . . . . . . . . . . . . . . 152
D.3 An unstable microelectrode. The microelectrode is entering the cell.
Notice that the size of the observed action potential varies. In addition the baseline of the signal changes as a microelectrode enters of
leaves the cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
D.4 A drawing of the microelectrode mount. The microelectrode is held
in place by the tension in the silver wire. It is kept vertical by a large
tube that allows free vertical movement. . . . . . . . . . . . . . . . 154
D.5 Comparison between an transmembrane signal (A) and a extracellular signal (B). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
D.6 A diagram of the electrode layout of the plaque electrode. The plaque
electrode consists of two pieces. One piece is attached to the left atria
and the other to the right atria. The individual silver electrodes are
set in a staggered pattern that is equivalent to a square lattice with
side spacing of 40 mm rotated by 45. . . . . . . . . . . . . . . . . . 156
D.7 MAP electrode. The catherter MAP electrode (A) is used to measure
on the inside (endocardium) of the heart. The tip presses on the
tissue to damage it and a differential measurement is made verses the
reference elctrode on its side. The suction MAP electrode (B) is used
to measure signals from the outside surface of the tissue (epicardium).
The suction throught the inner electrode damages the tissue and the
outer one is the reference. . . . . . . . . . . . . . . . . . . . . . . . 157
D.8 Signal recorded from a MAP electrode . . . . . . . . . . . . . . . . 158
xxiv
F.1 A flowchart of the PTDAS implimentation by the microcontroller.
An activation or a timer over flow causes an interupt. The type of
interupt is determined. If it is an activation the next control time is
computed. If it is a timer overflow the tissue is paced an then the
next control time is computed. . . . . . . . . . . . . . . . . . . . . . 163
xxv
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Bifurcation
1000
APD (ms)
800
600
400
Forward
Bifurcation
Subcritical
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200
200
400
600
800
1000
PP (ms)
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(1.2)
thus ? 0 and the pacing period is only shortened. This type of control is discussed
theoretically by Gauthier and Socolar [45].
Figure 1.5 shows the versus beat number from a control experiment using
TDAS. When control is off (black points on left) the response of the tissue to
periodic pacing alternates and the unstable 1:1 behavior is stabilized when control
is switched on (gray points). Once control is turned off again (black points on right),
the system returns to alternans.
In Chapter 5, I present a control experiment on the behavior most commonly
observed: bistability. I show that it is possible to cause transitions between the two
stable states by injecting a single pulse when the system of paced cardiac muscle
is exhibiting bistability. I observe that it is possible to induce two types of 1:1-to13
Action Potential Duration (ms)
380
Control
On
360
340
320
300
280
260
240
5
15
25
35
Beat Number
2$: A beat by beat illustration of a TDAS control experiment. The action
potential duration is plotted against beat number. While control is off (the black
points ) the tissue displys alternans. When control is turned on (the gray points)
the alternans behavior is supressed and the normally unstable period one behavior
is stabilized.
2:1 transitions observed, as well as two types of 2:1-to-1:1 transitions. One type
of the 2:1-to-1:1 transitions is shown in Fig. 1.6. The transmembrane potential is
shown at the top of the figure, while, the pacing signal is shown at the bottom of the
figure. An extra stimulus is inserted between to regular periodic stimuli. I show that
most of these transitions can be understood through use of the simplest mapping
model of the tissue dynamics from Chapter 3, while an explanation of the other
two transitions requires a fundamentally different type of theoretical model. These
studies of the dynamics and control of small pieces of cardiac tissue are important
for understanding the local mechanisms of cardiac arrythmias that may contribute
to eventual control of conduction blocks, tachyrhythmia, and atrial and ventricular
14
Transmembrane Potential
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fibrillation.
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Electrode Difficulties
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Stable 1:1
600
500
400
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Unstable 1:1
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0
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700
750
800
Pacing Period (ms)
Figure 2.13: A bifurcation diagram of the simple model of point for pacing period
greater than 631 ms. For periods less than that the 1:1 dynamics is unstable and
the system alternates.
A feedback controller can stabilize the fixed point of the map (the 1:1 state) and
suppress alternans. Here, I illustrate control mathematically using proportional
feedback. The pacing period serves as the accessible system parameter that the
controller actuates to suppress the alternans.
Proportional control seeks to minimize the difference between the current state
of the system and a reference state. To stabilize the fixed point of the mapping
(2.3), I use the unstable fixed point
9 as the reference state.
The algorithm
generates an error signal defined by,
? = F? 9G
F1EG
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/
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! + ! !
J L ! 4 '& J@ J +
$
&
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(
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J
F ∗)
Expanding around the fixed point and eliminating higher order terms gives
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Control
Off
550
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Off
450
400
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On
350
300
15
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5
0
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$ D?
Chapter 3
Dynamics of small pieces of cardiac
muscle
As discussed in the introduction to this dissertation, the long term goal of this research program is to devise methods for controlling or suppressing spatio-temporal
disorganization occurring in human hearts during fibrillation. The general approach
is to investigate methods developed by the nonlinear dynamics community for controlling chaos using small perturbations. In this chapter and the next two, I try
to understand the dynamics of small pieces of tissue where spatial complexity is
not important. This work has implications for controlling whole heart dynamics.
For example, I found a correlation between windows of bistability and unsustained
reentry in whole sheep atria
, during initial experiments conducted by my
collaborators in biomedical engineering and myself [65]. In addition, Karma
[24] has found a connection between the initiation spatio-temporal complexity and a
local bifurcation to alternans (a period doubling bifurcation) using a mathematical
model of cardiac muscle.
In the sections that follow, I present an exploration of tissue dynamics during
rapid pacing. The experimental setup consists of a small piece of frog heart tissue
(ventricular myocardium) and various support systems. The bullfrog ventricular
muscle is periodically paced with an external electrical stimulator. This test-bed
gives an initial assessment of the importance of parameter variation during experiments and animal-to-animal variation as well as experience in cardiac electrophysiology. In addition, I use this test bed system to determine types and prevalence of
local tissue bifurcations. This information is important because control protocols
44
need to be designed to deal with commonly encountered behaviors. This chapter
ends with the analysis of simple mathematical models that give some insight into
the observations and develops a framework for later use.
# $
I investigate
the response of small pieces of bullfrog (
G
0
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!G 5 4 F54 G $ !
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PA?Q PACQ P)1@Q . ,
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D C + - ! ) !I=
?,
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$ .
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+ & !
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Transmembrane Potential (V)
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
0
500
1000
1500
Time (ms)
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,1 ! 1@@@ I
5
$ FAPD), the duration of the deflection
of the transmembrane potential away from its rest state in response to a stimulus.
The APD is determined at 70% of peak height (which I find gives less scatter in our
experimental data than the more common cutoff of 90%) using a custom software
routine. I also measure the diastolic interval (DI ), defined as the interval between
the repolarization after an action potential and the start of the next response.
This experimental setup allows the exploration of rate dependent dynamics of
small pieces of cardiac tissue. This exploration is important when trying to un50
derstand what causes the breakup of cardiac arrythmia into the spatio-temporal
complex dynamics of fibrillation.
%
A typical method for investigating the dynamics of small pieces of cardiac tissue is
to measure the time for cells to repolarize after depolarization by a brief electrical
stimulus (this time interval is known as the action potential duration, APD) as the
interval between stimuli (pacing period,
PP ) is varied slowly.
When the tissue is
APD equal (1:1
Faster pacing results in a decrease in both DI and APD, a property of
paced slowly, each stimulus elicits one action potential, with all
pattern).
cardiac muscle known as restitution.
As
PP
is decreased, there is a minimum diastolic interval
θ where the tissue
becomes unable to recover fast enough after each stimulus to respond to the next.
The combination of
θ and APD is defined as the refractory period, during which
the tissue apparently does not respond to applied paces. If the tissue is unable to
respond to every stimulus, it then enters a 2:1 pattern in which one response occurs
for every two stimuli. Occasionally, a 2:2 pattern (responses of alternating duration)
intervenes between the 1:1 and 2:1 states [85]. I observe hysteresis between these
dynamical states as the pacing interval is swept up and down.
The coexistence of two stable states at the same
PP
is seen in Figs. 3.5a and
3.5b, where I show the temporal evolution of the transmembrane potential for a PP
PP
3.5b is recorded after the PP
of 600 ms in the same tissue preparation. Figure 3.5a is recorded after the
is
slowly varied from 1200 ms to 600 ms, and Fig.
is
slowly varied from 300 ms to 600 ms. This observation demonstrates that the tissue
can display two different behaviors at the same PP .
A bifurcation diagram is a convenient way to summarize observations of the
51
Transmembrane Voltage
25 mV
(a)
(b)
0
2000
4000
6000
8000
Time (ms)
Figure 3.5: Time evolution of transmembrane potential during a typical hysteresis
experiment. The stimulus waveform is shown at the bottom of each diagram. (a)
The tissue exhibits 1:1 dynamics for slow pacing during a PP downsweep. This
time series is taken from the data used to generate points on Fig. 3.6 (open circles)
at a PP of 600 ms. (b) The tissue exhibits 2:1 dynamics (closed triangles in Fig.
3.6) for the same PP used in (a), but during an upsweep. Arrows indicate measured
APD. Vertical dotted lines illustrate the correspondence between stimulus intervals
in (a) and (b).
PP is varied slowly. I obtain such a
diagram by recording APD while adjusting PP across a wide range of physiological
long-term dynamical behavior of the tissue as
values, from 1200 to 300 ms in 100 or 50 ms intervals (downsweep), and then from
300 to 1200 ms, again in 100 or 50 ms steps (upsweep). For each PP , the response of
the tissue to the first 5-10 stimuli is discarded in order to eliminate transients and the
subsequent behavior is recorded for up to 10 seconds. After discarding transients,
the width of each action potential is determined at 70% of full repolarization and
plotted at each
PP .
52
800
APD (ms)
700
2:1 (N=2)
600
500
1:1 (N=1)
400
300
200
400
600
800
1000
1200
PP (ms)
Figure 3.6: A bifurcation diagram showing 2:1
)) 3
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300
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400
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$ ! ! ?
4
)) 11 &
J )
1) J 1 + . 8 1 6 ! ! PB)Q F? G J 4@ !?
$
F?1G
≡ N ∗PP −APD? , ! ! !
$ &
$ APD"( A τ ,
+ '
' $
6! ?A . ,
→
→
( ! )) 1) 1) )) FPP , APD G FPP , APD G &
.
. ( ! 11 ! $
'
θ DI PP − APD J θ
PP − APD J θ
APD J f FPP − APDG
EA
F??G
F?DG
F?EG
800
APD (ms)
700
600
500
400
300
200
400
600
800
1000
1200
PP (ms)
2% 8
$
! &
$
6! ?A + ! F?)G $ F G F?1G max J A)A
J ?)? J 1@B J CA + $ $ )) 2:1 transition; the short downward arrow indicates the 2:1 1:1 transition in
the observed data and the one-dimensional model.
= (2 )
() = 1
(3.6)
(3.7)
Combining Eqs. (3.3)-(3.6) gives,
=
=
=
=
Using (3.7) to solve (3.2) for
(2 ) ( )
( + ) ()
3
(1 3 _
)
gives,
= 57
(3.8)
and that along with (3.8) gives,
= (1 ! d
)
2$ 7
. ( '
! (
)) + '
( ' (
!
( 6 ,( F G ! 0
( $
9 9 (
+ !
J
− 9
P
FF
9
G
GQ
9 ! J
F G $ (
,
+ - / + !
! ! !
$ $ $ $
!
J L !
where the step size is selected to be as large as a possible without the process losing
stability.
Figure 3.8 shows a comparison between our experimental observations shown
in Fig. 3.6 and the theoretically predicted bifurcation diagram generated using
the simple mapping given by Eqs. (3.1)-(3.2). While it is seen that bifurcations
occur at the correct
values, there is a significant discrepancy in the observed
and predicted values of the . &( ! !
EC
)@N 11 1)
+ ! ! ,
! $
! ! ! . ! $
,
'
'
? J ? '
J ) ! .
. 1) 0 + 6! ?* $ $ ? ! $ 0
! $ J ) F
G J 1 F G + $ 6! ?A . 1) !, 6 )) $ ' J ) F G ! %( F?)G,F?1G + )) !
$
! 2$ )) $
$ !
+ + 0
,0
8
P)1)Q "
! !, + ' ! 0
0
(
! !
.
(
!
E*
800
slope > 1
N=1
slope < 1
APD (ms)
700
600
N=2
500
400
300
100
300
500
700
DI (ms)
2* $
6! ?A 3
$
$M ! $ $ ! J )
J 1 + $ ! > J1CE $ J ) ! + FG ! ! J ) F J 1G $ . !
. !
0 '
?
"?n
$
J F)
J
"n GP) N ∗ − ? −? 9 ! ? ?
P) L F"
)G
Q
Q
Q
F?*G
( ' '& F?*G J ) F
6! ?AG 6! ?)@
A@
+ ( '
! ! + 0
" 2$ . '
! '&
"? + $ '& " 9 F9 G !
!
F?*G "?+1 (
"9 J ( 9 !APD9 /τ (1 !∗
G#F) ∗ −9 −9
G
)
+ (
' (
F?*G !
!
9 J FF9 G FQ Q GG
F?*G + ( F G 0
'
J
[ P
9
9
FF
G
F 1
2
1
2
2 GGQ
8,( 2 - ! !
J ) 2$ !
$
J 1 J 1 J ) ! &
.
+ ' 4J) 0
$
4J) 4J1 + !! !
'
&
"
.
. , &
! . $ ,
! & + ! !
A)
1100
1000
APD (ms)
900
800
700
600
500
400
300
200
400
600
800
1000
1200
PP (ms)
Figure 3.10 1, ' ! $
6! ?A + ' ! $,
! F?*G $ Q J)?E* J1AD QJE) J)@C@ J )?A + $ $ )) 1) M ! $
$ $ 1) )) $,
! )) 11 1) ,
$ 0
! !
!
! ,
. M )) 11 1) + ,,
. &
- $
$ 2$ &
- . !
5 !
'
'
!
+@?@ U@|itMi@?@
! ! ,
!
$ 7
H& $ $ A1
. $ ! ! ! . '
! ! ! ), PB) A?Q ' $ " ,
1, 8
i| @*
P)1)Q $ 4J) F)) G 4J1
F1) G + 1) 0
A?
+,$ ! PABQ ,
$ '
.
, & '
PA)Q + ! $ ,
PA) A1Q $ PDD D@Q
! !
! & |i4TLh@* & , P1? DA C DBQ !
, & , ( $ P1? DDQ ,
8, $ $
6! D) |i4TLh@**)
& = PDCQ ,
PC?Q + ! ! , &
! $ $ ! 8, - ! /
+ + 0
$
AD
"2 6 '& F
$
G !
$ ! + -
F+G $
! PCBQ + ( $
.
+ 0 ! &
+ $ ! P*E BCQ + H
PC?Q 83
PCCQ > PD@Q ! $
$ 1, PDDQ
.
. + ! 8 ? .
' AE
. &
6! D) .
& . + + 6! D)
!
!
+ ! / !
6
. &
4.1
Experimental Setup
+ . ' $ + $
! !!
4 PB)Q ! $ & $
& + &
, 8 ? + 0
$
6! D1
+ (
&
$ 0
8 ? + ∼ )@@@ 1@ ! $ ! + ! &
8 ? F ! 54 $ 5 4 0
G
+ ! $ .
! !
11 ! !
11 F
G $ ! + AA
"2 %&
+ ! &
8 ? + / 0
$ !
!
!
!,,
+ !
0 ! ! AB
! 6! D? $ ! ! $ )@@@ ?@@ E@ +
)@ ' E,)@ ! ! $
)) $ 5 ! ! ,
F G
)) 1) 2$ ! )) ! $ .
'! )) 11
! E@@ .
!
1) ! ?E@ 400
a
1:1
APD (ms)
2:1
300
2:2
200
100
200
300
400
500
600
700
Pacing Period (ms)
"2 ! ! $
! + $ ,
6 ! ! FE@@ G ))
$
E@@ ?E@ ?@@ 11 1) AC
. - ! ,
F5G + 5 1) & % "
5 $
1,? +
!
!
! F
!G F !G ' )> ! +
&
E114 0
' ! + , 5 & % . '
5
!
- $ ! !
5
! ! $ FG ! ,
'
H
$ +
- + &
! + ! '& ! ( + ! +
! $
6! DD + !
F$ G ! & E@N ! "
!
!
$ !
.
++=
F
!G !
! !
!
+ $ ! ! ! ,,! $
6! DE + !,
A*
"2" ! + !
F> G
' ' F
! !!G + &,
E@N ! ! ! $ + $ -
O !
.
F88G
-
B@
!
! !
! ! + !
,
! ! ! ,
, !
+ ,
, ! −Q.
Figure 4.5: A circuit diagram of the time to voltage converting circuit. The sample
and hold integrated circuits provide this circuit with a two level stack memory.
A fixed voltage is integrated while the action potential is being detected. The
integrated voltage is set by the voltage divider. During all of these experiments this
voltage was set to 109 mV, which gives an integration slope of 11.5 mV/ms. At
each clock pulse (C,C̄), the output voltage of the integrator is propagated down a
line of sample-and-holds.
"
.
. + + 6 ! F? G + /
! ! !
$ B)
! !
! !,, $
6! D1 + ! !
!
J L !Q .
4.2.1
Time Delay Autosynchronization
+ !
!
ε + ( $
+ !
$ !
? J F? ?−QG
!
F? G F?−QG .
0 &
$
? J ?−Q J ∗ $ 4 !
! - !
?
J ?−Q M 0 "
0
$
$ ! %&
+ + $ ! ! ?
?−Q ,,!
F6! DEG + 0
! ' 0 $ $ ! !
! !
!
B1
Figure 4.6 + ! +$ ! ,
! ' FA1@G
0
' γ $ $ ,
,
+ ! ? ! $ ! !,, + ! !
γ ( ))KF
!
,
G ! !,, @)*) I>
% " #
$
" ! ( + ,
! . $
(
$ in
vivo
+ in
vivo
! !
:
$ &
& , .
$ ! '
! %
! + / ! . !
+ . $ 7 !
! + ! ! ( 2$ ! ! /,
& / ! + $ / !
B?
! + . + ! + +: !
+ & ,
! " @ !
? J
F? ?!QG
0
if
? ?!Q
n n!Q
thus n 0 and the pacing period is only shortened. This restriction transforms the
linear TDAS into a nonlinear controller.
The restricted version of TDAS is similar to TDAS in several respects. The error
signal goes to zero if the cardiac muscle is displaying the desired dynamics, because
n=n!Q.
In addition, the restricted TDAS controller is also able to track
changes in the dynamics of the cardiac tissue.
The implementation of the restricted TDAS is similar to TDAS. The two voltages
corresponding to n and n!Q from the time-to-voltage converter (Fig. 4.5)
are subtracted from one another. This difference voltage is amplified by an amount
and buffered. The resulting voltage is checked by a comparator to see if it is
negative. Only if the error signal is negative is it added to the computer generated
pacing voltage.
"
( )
. ! ! ! 6! DC $ + &
5 ! F6! DCG !
F6! DCG 6! DC $ !
! !
+ JE@@ + $ 0
BD
"2 + ! + ! 0
'
$ $ + ! ! $ - ! ! + ! ! $ ! !,, + ! !
(
))KF
! ,
G ! !,, ,
@)*) I>
+ !
J )ED " 0 F
G ?
$
& ))@ )C@ F
$
G ! 1@N . 6! DC ! $ )) $ ∗J)?1 + !
? @EN !
F G + )) 0 F ! G 0 ! BE
)) & ? J ?−Q
$
? J?,9
FD)G
6( -
! ! $
0 $
)) F G (
FD)G ? J FG? FD1G
J ,)? '
! ' 0
(
FD1G 4 ) )) . 8 1
3 )C D@ $ .
1 )C $
7 &
.
)D
)A . $ $ + !
)) ? $ 1N ! ? )EN ! ? 6 ?
! )) ! & 1N ! ? . '
)) - ! ! !
$ ( - ! 9
: + ! ! !
γ - PP + + ! $
6! D* .
!
! ! PP ! ! !
+ $
BA
Error Signal
% of Pacing Period
MAP Electrode
Voltage (arb. units)
Control Off
Control Off
a
Control On
0.4
0.2
0.0
-0.2
b
20
10
0
-10
-20
0
3000
6000
9000
12000 15000 18000
Pacing Period (ms)
Time (ms)
600
c
570
540
510
480
450
420
0
5
10
15
20
25
30
35
40
Beat Number
"2% + 5 !
FG !
FG !
FG + &
+ !
$ $ + ! ! + ! E@@ !
γ J )ED
BB
)) F 6( ! !G
PP +
&
$ $ priori
a
$! !
400
a
APD (ms)
1:1
2:1
2:2
300
200
100
4.0
b
3.2
Gain
Uncontrolled
2.4
1.6
Controlled
0.8
0.0
200
300
400
500
600
700
Pacing Period (ms)
"2* ! + ! + ! FG $
PP J E@@ + FG $ $ !
!
+ )) ! BC
H
!
! + + ! & 6! D)@ .
'! 0 !
APD ! &
! &
$ $ + ! 0 F G ! ! + ! 0 % $ !
$ γ J 1* + APD9 )) ??C 6!
D)@ ?@@ 6! D)@ '
1*) 6! D)@ + - ! $
?1*,?E1 !
$
1EE,?EE ! $
1CE,1** .
6( )) $ )1) )1C )E) +! &
$
$ )) ""
( ( )
.
! $ ! + ! 0
. .
)1 )E . - )) ! + $ )D )A + $
! ! + 6! D)? + !
$
6! D)? $ ( - + 0 ! + . $ B*
Action Potential Duration (ms)
a
360
c
b
340
320
300
280
260
240
0
10 20 30
895 905 915 2890 2905 2920
Beat Number
Figure 4.10 ! &
$
!
+ ! + ! $ ED@
!
$ @AA
? )) + PDEQ + + ), !
!
J %?!Q n!Q
F? ?−QG ? ?!Q n
@
? ?−Q
$ 6( !: '& % ∗ J @
%?
L
FD?G
J
+
$ + ! $
! + + FD?G $
6! D)) + !
+ !
+ 4 ), FD?G C@
! + $ + gain
6
Region B
00111
4
Region A
0111
0011
2
001
011
01
TDAS Region
0
1
2
3
4
Floquet Multiplier
"2 + + '& 4 + + '! + PDEQ $ 0
(
! + .
! F!
6! D))G $ + $ ! (
9@):
9@@): $ 9): ! F? @G 9@:
F?
F!
@G + & ! 6! D))G (
9@@)): 9@)):
%&
. (
!
6! D)) $
. + !
.
? )) $ + . !
C)
Microelectrode
Voltage (arb. units)
Error Signal
% of Pacing Period
Control Off
Control On
0.4
0.2
0.0
-0.2
0
-2
-4
10101010
-6
-8
-10
-12
-14
0
2000
4000
6000
8000
Time (ms)
Figure 4.12 !
5 FG !
+ FG ! &
+
EBE + !
DD "
F G )) + ! $ 4 (
9@):
C1
Microelectrode
Voltage (arb. units)
Control Off
Control On
0.6
a
0.4
0.2
0.0
-0.2
Error Signal
% of Pacing Period
-0.4
001 1 0 0 1 1 0
b
0
-2
-4
-6
-8
-10
0
2000
4000
6000
8000
10000 12000
Time (ms)
Figure 4.13 !
5 FG !
+ FG ! &
+
EBE + !
DD "
F G )) + ! $ 4 (
9@@)):
C?
! ' PDEQ !!
! !
$ . + (
$ &
$
6! D)1 6! D)? 6! D)1 )1 )E $ !
6! D)) + (
'! 9@): .
&
. & 9@): 9@@): (
6! D)? (
. 4 $ 9@@)): (
!
6! D)) + ), ! 2$ $ 4.5
Summary
+ ! ! ! '
. $ $ .
,
$ '
PABQ ! , & PA)Q +,
. & 0
$ + !
8 ?
. + $ )D )A .
. + $ !
! & ,
+ ( !
& ! . ! + ! ! !
'
0
.
)1 )E . $ CD
$ + + !
( $ /
! ! .
. $ PDEQ
CE
!
+ ! ! , & '
+ $ , & P)* 1D A?Q + ! & $ &
'
2$ . &
!
.
. ! &
$ %
!
! !
& $ $ ! $
$ PAEQ
!! $
&
$ =
PACQ $
$ ! .
5
P?Q PB@Q )) 1) ! ! 5
1) )) $ / + &
! $
! .
' . &
.
& CA
. -
! &
. '
/
! )) 1) 1) )) .
. ! . '
! )) 6 1) )) . $ ! ( $ / $ 5.1
Experimental Proto + &
, 8 ?
+ $ & $ ,
& + !
0
8 ? + $ ! + ' - ? $ ! $
$ 3
! ! ,
$
. ! !
$
/
! & + ! ! 8 1 & % 8 ?
$ 8 ? JAA@
$
$ $
6! E? 3
$
F G PE*Q )) 1) F $,
! $G + & 1)
J?@@ / H
?) F ,
,$
$G . !
CB
$2 + &
! $
-
&
8 ? + !
! ! ,
.
+ & !
! ! & $ ! +
! !
, '
!
! F !G + 1) $ !
& 1) JBAE $
! )) F! $G + &
$
AA@ BAE & )) 1) ! ! ! )) 1) ,
. $
$
$ + )) 1) $
! $
! $
$ 3
))
1) . / & CC
! $2 + H$ =>$ ! /
&
+ ! ! ,
E / & $ + ! $ ! $
!
& C*
800
APD (ms)
700
2:1
3:1
600
500
1:1
400
300
400
500
600
700
800
900
1000
PP (ms)
$2 ! $
! *1@ ?@@ E@,EE F
G ?@@ *1@ E@,EE F !G
$ ! $ ! +
$
$M $
J) J1
$
AA@ BAE + & $
!
& ! . $ + F G $
+ E@ E@ E@ $
! 1@ + &
6 ,
*@
$
$ 6
! ! $ ! .
! ,
F1) ))G $ + ,
/ &
, &
/ $ *
(
. &
F G 0
$
+ $ )) 1)
3
& ! ! ! 2$ ,
0
! .
$ 1) )) F+ .
+ ..G + 0
3 )1 $ &, $ ! *
& %&
$ $
! &
$
.
' ! & & $
+
& )) 1) ' & 1) )) *)
! & & .
. $ )) 1) + / ! + $ 0
!
+ ! $ $ + +
/ 20 mV
Transmembrane Potential
+ 0
)) 1) 6! ED 6! EE
APD
T
Stimulus
PP-T
PP
PP
Extra Stimulus
0
2000
4000
6000
time (ms)
$2" + $
! +
→
)) 1) + $
'! & $ $
→
6! ED $ + )) 1) .
)) B@@ F$
$
$G 1? $
'! . & JE*E $ + 0 2$ (
/ +
*1
$
! ! 1) + / ! . ! ! !
20 mV
Transmembrane Potential
$ Stimulus
APD
T
PP-T
PP
PP
Extra Stimulus
0
1000
2000
3000
4000
5000
time (ms)
$2$ + $
! +
)) 1) + $
'!
& $ $
6! EE $ +
)) 1) 0
+ &
F $ $G ! +
/ ! & + 0 + / &
: ! ! & ! 0 !
! ( 1
*?
$ ! 1) T (ms)
Type A
Type B
1) $2# 3 ! / ))
! % 0
! + ' + + + $ $ E@ ,E@ )@ E@ A@@
6! EA $ F G ! )) 1)
+ ' + 0
0
3
$ & $ F G 4 +
+ + !-
! & / !
*D
( . ! $ +
$ $ $ !
$ + / + )) 1) !
+ ! $ +
!- !
! + $ ! & , ! & & .
' . $ 1)
)) . $ 1) )) $ $
6! EB 6! EC
! ! & + . 1) )) 6! EB + AE@ 1) + & F
$ $G E J)@@ "
& / 7
&
& 2$ & ! !
& ! !
! )) 6! EC + .. 1) )) + !
1)
$ B@@ $
$
$ .
& JE*@ + ! + & ! (
$
*E
20 mV
Transmembrane Potential
BCL
BCL-T
BCL
Stimulus
T
Extra Stimulus
0
2000
4000
6000
8000
10000
$2 + $
! +
. 1) )) + $
%& $ $ + ! ' $ - $
! + $
! $ 1) $ )) JB@@ 6! E* $ F G ! )) 1)
+ ' + 0
0
3
$ & stimuli delivered at
small , and lie in the lower half of the ( ) plane.
96
20 mV
Transmembrane Potential
T
BCL-T
BCL
Stimulus
BCL
Extra Stimulus
0
2000
4000
6000
8000
10000
time (ms)
$2%: Temporal evolution of the transmembrane potential showing a Type II
→
2:1 1:1 transitions. The stimulus train is shown at the bottom of each time series.
Extra stimuli are indicated with arrows.
*
)
.
. $ 8 1 ? ! / 1) ))
+ ,
2$ !
)) 1) ! APD? APD?n !
+ '
$
! J) J1 $ ! ??
+ ! ! 1))) ( ! *B
1000
800
T (ms)
Type II
600
400
200
Type I
0
400
500
600
700
800
900
PP (msec)
$2* 3 ! / 1) )) ! % 0
! + ' + .
+ .. + $ $ E@ ,E@ )@ E@ A@@
8 1 F,
!G &
! & ?n ? PB) A?Q + ? L ? J ∗ $ !
∗ − ? !
?n J F? G
! ! ? ! ! F)G . '
*C
?n J F ∗ − ? G
FE)G
J) !
! ))
11 F
G ! "
J1 "
F1) G ( $
$ %&
?n ? &
F G J 4@ − &F−# G
FE1G
? ?n +
( ' "
4@ + $
J ,
0
6& $
9=n=nn. The fixed point stability is governed by the slope of
(9) at a given . For some , there exist two stable fixed points 9
and 29 , which lie on the =1 and =2 branches respectively. This gives rise
+
! FE)G,FE1G '
! to bistability: two stable solutions coexist for the same parameter values. Note that
no unstable state lies between the two stable steady states due to the discontinuity
of the map.
There are two separate routes from the 2:1 to the 1:1 regime. For a Type
I transition, the extra stimulus must occur after the periodically skipped pace,
thereby giving the relation
+ 29 '(
FE?G
$ 9 J1 . **
$
! '( FEDG
$ J F'(L WG $ ' $
! & 6 + .. !
( '( FEEG
FE?G 6 + . + .. 1
'( '( L $
J F1
FEAG
FEBG
'( G
8
FE?G FEDG FEAG
FEBG $ !
6! E)@ $ + . 1) ))
8
FEEG,FEBG !
6! E)@
$ + .. 1) )) 6! E)@ $ F G ! 1) )) 3
$
6! ?G + $ + .. 1))) +
! &!
$ + . 1) )) F
0
0
+ ! + .. 1) )) '
! + .. + . .
6! E)@ !
!
F' ! $ G + -
!
$ + . !
$ + .. +
!
1) )) )@@
1000
T (ms)
800
Type II
600
400
Type I
200
0
400
500
600
700
800
900
PP (msec)
T $
P P 8 &
+ F$G !
$
+ .. F+ .G 1) )) + &!
+ . 6! ? + . + .. !
' !
Figure 5.10 + !
& !
,,
6 & !
$
6! E)@ !
' ! ! *"
.
$ $ .
8
. $ ! / $
! )@)
. '
! / 1) )) $
! !
$ 3
,
&
! !
+ )) 1) %( FE1G
!
!
$
6! E)@ ' ! $ &
+ %( FE1G !
'
+ !
$ 7
H& ,,
$ 0
+ $ ( ,
.
$ ! , + / ! P??Q !
( ( P?* D@ ? B@Q + $ ( P?* D@Q ! TU $,
! )@1
"
#
+ ! ( &
, ! '
. ! !
!
!
& , !
-
P1? DA C DBQ " &
! P1? DDQ &
P1?Q !! '
! ! + $ ! 3S P1*Q .
&
$ $ ( ! ! . $ $ , ! ( 2$ $ $ .
'
$ ! + '
F6G
! ! ! + $ !
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Atrial Fi$
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The mammalian heart is a complex and nonlinear mechanical system, designed to
pump blood. It has four chambers, divided left and right, atria and ventricle. The
right two chambers take blood returning from the body and pump it to the lungs.
Subsequently, the left two chambers pump the blood coming from the lungs out
to the body. The atria act as priming pumps to the ventricles (the main pumping
chambers). The pumping action of the heart is accomplished by the muscle tissue
that makes up the walls of the various chambers. Its mechanical contractions are
mediated by waves of electro-chemical excitation that sweep through specialized
conduction systems and the muscle of the heart. These waves normally originate
in the sinus node (the main pace-maker of the heart). Under normal and healthy
conditions, these waves of excitation cause a coordinated contraction of the muscle.
In some situations, this orderly procession of waves can devolve into a spatially
complex dynamical state known as fibrillation [1], in which the waves of excitation
and consequent contractions of the heart are not coordinated, causing a loss of blood
pressure and death. While both the atria and ventricles can display fibrillation,
atrial fibrillation is studied in my research because it can be sustained for some
time without causing death, whereas ventricular fibrillation is often fatal [1].
#2: A Lead II electro-cardiogram of normal sinus rhythm. The deflection
of the trace corresponds to the phases of a heart beat. The P complex corresponds
to the depolarization of the atria. The QRS complex part of the signal corresponds
to ventricular depolarization, and the T-wave is the repolarization of the ventricles.
105
Normal sinus rhythm (NSR) is a relatively regular process. Figure 6.1 shows a
Lead II electro-cardiogram (ECG) time series [96] from a sheep during NSR. This
signal is obtained by measuring the voltage differences between the left leg and right
shoulder of the animal. It is important to note that this signal is a summation of
the electrical activity of the whole heart, so it is not possible to distinguish the
spatial patterning on the surface of the heart. The feature denoted P is associated
with excitation of the atria and the QRS feature corresponds to excitation in the
ventricles. The feature labeled T is associated with the repolarization of the ventricle. Notice that the separation between individual QRST shapes is about 1 second
while the duration of the P-wave is less than 100 ms. This implies the atria has
about 900 ms to recover between beats (see Table A.9).
The spatio-temporal electrical activity on the surface of the atria during can be
observed with the plaque electrodes and mapping system (described in Sec. 6.3).
The temporal pattern of activation during NSR is show in the series of panels in
Fig. 6.2. Each panel corresponds to a 4.5 ms time slice of a single time series. The
gray and black dots in each panel represent the spatial location of the electrodes.
The black dots illustrate at which electrodes an activation was observed during that
time slice. Arrows have been added to guide the eye through the time course of the
wave. The wave of excitation shown in this figure corresponds to the P-wave part
of the ECG and repeats approximately once a second. The total propagation time
across the plaque (33 mm) is about 31 ms.
Atria fibrillation is very different from the orderly and periodic NSR. Figure 6.3
shows a Lead II electro-cardiogram (ECG) time series from a sheep experiencing
atria fibrillation. There is no longer a clear P-wave and the QRS complex comes at
what seems to be random intervals. In addition, notice that there is no time when
activity ceases on the heart.
106
0-5 ms
A
5.5-10 ms
D
20.5-25 ms
25.5-30 ms
F
E
30.5-35 ms
G
C
B
15.5-20 ms
10.5-15 ms
35.5-40 ms
40.5-45 ms
I
H
#2: Normal sinus rhythm in a sheep stria measured with a plaque of
electrodes. The panels A-E show the temporal evolution of a normal heart beat on
the right atria of a sheep heart. The wave originates in the sinus node (off plaque
in the upper left) and propagates across the atria. Each of the panels represents 4.5
ms of time. The black dots show where the leading edge of activation was during
that time slice. The plaque is approximately 7.0 cm 3.2 cm and the wave of
excitation travels across it in 31 ms. Each NSR wave is separated in time by about
1 second from the next.
The spatio-temporal evolution of atrial fibrillation from the same sheep that
produced this ECG chart is shown in the series of panels in Fig. 6.4. These panels
are similar to the ones shown in Fig. 6.2, except that each panel corresponds to a
9.5 ms time slice of a single time series. A longer width of the time slice was used
to generate these plots than in Fig. 6.2, because the waves propagate more slowly.
During atrial fibrillation, it seems that there is no point of origin for the waves of
excitation and that the observed activity is self-sustaining.
During atrial fibrillation, the shape of the wave is uneven and its speed varies
widely from location to location, while during normal sinus rhythm the shape of
107
#2 : A Lead II electro-cardiogram during atrial fibrillation or flutter. There
is no longer a clear PQRST shape. The atria is observed to be generating a seemingly
random signal. The ventricle is firing at a more normal and slower rate. The AV
node, which is an electrical connection between the atria and ventricle protects the
ventricle by only allowing wave of excitations to pass with a low frequency.
the wave of electrical excitation is very simple. The ultimate goal of this research
is to develop methods to convert the fibrillation state into the NSR state. In this
experiment, however, I only attempt to stabilize a periodic state of the fibrillating
tissue, which may or may not correspond to NSR.
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are maintained by adjusting the ventilator and by IV injection of electrolytes.
In order to observe directly the dynamics of the atria, it is necessary to attach
the plaque of electrodes to the surface of the atria. To accomplish this, the chest
is opened through a median sternotomy and the heart is suspended in a pericardial
cradle. Two plaques described in Sec. 6.3 are affixed to the epicardium of the right
109
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Right Atrium Plaque
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The recording system works in conjunction with a plaque placed on the atria
with 528 individual extracellular electrodes. The two plaques measure the spatio-
temporal dynamics on both the right and left atria. The left atrium was monitored
to compare with the dynamics of the right atrium and observe large scale reentrant
113
Voltage (mV)
100
A
50
0
-50
-100
-150
5 point
derivative (mV/ms)
-200
10
B
5
0
-5
-10
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Detected
Activation
C
0
200
400
600
800
1000
Time (ms)
#2: A set of time series showing the voltage from a single plaque electrode
(A), the five point derivative (B), and detected activations (C). The top figure is
the voltage measurement over 1100 ms from an electrode very near the center of
the plaque on fibrillating sheep atria. The derivative signal is tested for threshold
crossings to determine activations.
114
circuits. A diagram of the plaques is shown in Fig. 6.6. The individual electrodes
are 0.01 in. diameter silver wires. The electrodes are set in a square grid with a 40
mm spacing. The electrodes are chloridized before each use to minimize the noise
they generate [54].
Extracellular electrograms were recorded from the tips of each electrode in the
plaque with respect to the return electrode placed in the aortic root. The electrograms were band-pass filtered from 0.5 to 500 Hz, sampled at 2 kHz, and saved to
disk. The controller monitors only the electrode marked ‘recording electrode’ in the
figure. The pacing stimulus is a cathodal stimulus current of duration 4 ms and
strength (chosen by the procedure to find the threshold described in Sec. 6.2) in
the range of 0.1-3.5 mA was applied at the pacing electrode. The return electrode
for this current was attached to a chest retractor that remained in place during the
studies. The stimulus current was generated by a constant-current source whose
timing controlled by the control circuit described in Sec. 6.5.
The control circuit is attached to the plaque so that the controller can pace the
tissue from the center-most electrode on the right atrial plaque and monitor the
tissue dynamics from a neighboring electrode (approximately 3.6 mm away). The
pacing and recording electrodes are different because the constant current pulse for
pacing the tissue causes the stimulus electrode to become temporarily polarized.
This charge saturates the amplifiers and make observing the temporal dynamics at
this electrode impossible.
To analyze the electrograms, the signal from each electrode were differentiated
using a five-point algorithm and then displayed on a computer as an array of elements corresponding to the geometry of the mapping plaque. A computer program
using a level crossing routine determines activation times at all plaque electrodes
An illustration of the activation detection process is shown in Fig. 6.7. First, an
115
extracellular electrogram (Fig. 6.7a) from an electrode is differentiated (Fig. 6.7b).
The differentiated signal is compared with a threshold value (the solid horizontal
line in Fig. 6.7b). An activation is considered a threshold crossing. The observed
detected activations are shown as pulses in Fig. 6.7c.
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Under normal conditions, the dynamics of the atria are completely governed by
the driving period of the sinus node. This intrinsic pacemaker causes a wave of
excitation to be generated at the top of the atria. The wave sweeps across the
atrial muscle (as shown in Fig. 6.8) and collides with the boundary at the bottom
of the atria. After the passage of a wave of activation the atria muscle remains
unexcited until the sinus node once again causes an activation wave. The pattern
of activity is very regular and each location experiences essentially the same interactivation interval. Figure 6.9 shows a histogram (normalized to unit area) of the
inter-activation intervals from a 100 electrodes from the center of the right atrial
plaque during NSR. Notice that the inter-activation intervals from all 100 electrodes
fall inside a single 10 ms bin centered at 880 ms.
In experiments where I establish reentrant waves, several complicated spatiotemporal patterns of activity are observed. While every run was different and the
tissue dynamics never repeated, the observed re-entry dynamics can be broadly categorized into two classes: flutter and fibrillation. Flutter is a macroscopic re-entry
pattern of activity that involves most of the atria and is quite periodic. Fibrillation
typically has little or no discernible structure, and appears to be the result of many
small re-entrant circuits.
An example of atria flutter is illustrated in Fig. 6.10. In this figure, a well organized wave pattern is observed crossing the surface of the atria. This wave structure
116
0-5 ms
A
5.5-10 ms
D
20.5-25 ms
25.5-30 ms
F
E
30.5-35 ms
G
C
B
15.5-20 ms
10.5-15 ms
35.5-40 ms
40.5-45 ms
I
H
#2%: The time evolution of a wave of avtivation from NSR. During NSR
the progress of activations is orderly and periodic. The sinus node is near the top
center of the plaque. The arrows are added to guide the eye.
117
Percentage
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
200
400
600
800
1000
Inter-activation Intervals (10 ms bins)
#2*: Histogram of inter-activation interval during NSR of 100 electrodes
from the right atria. The bin sizes are 10 ms. There were 3 activations observed
during the 2 seconds recorded here giving only 2 inter-activation intervals per electrode. Notice, that while the histogram is from inter-activation observed at many
different locations on the atria. All the points fall in the bin centered at 880 ms.
118
0-10 ms
A
10.5-20 ms
D
40.5-50 ms
50.5-60 ms
F
E
60.5-70 ms
G
C
B
30.5-40 ms
20.5-30 ms
80.5-90 ms
90.5-100 ms
I
H
#2': A series of panels showing the temporal evolution of atria flutter. An
organized wave of excitation travels around the atria in a large reentrant circuit.
usually recurs in some form with a period of a few hundred milliseconds. Note that
while the wave seem just as organized as in NSR, it does not originate from the
sinus node and the propagation pattern varies with each recurrence. This orderly
wave structure occasionally breaks up into more complicated spatio-temporal patterns. Figure 6.11 shows a histogram of the inter-activation intervals from both a
single electrode (stripes) and 100 electrodes (solid grey) on the right atrial plaque
during a sheep experiment where I observed flutter. In both the single point and
multi-point histograms most of the inter-activation intervals cluster around 110 ms
and show approximately the same spread in timing. This indicates similar dynamics
take place everywhere on the atria and that this activity can vary in timing by more
that 20%. Notice that the average inter-activation interval is much shorter than the
NSR.
119
0.04
P(Tn)
0.03
0.02
0.01
0.00
0
100
200
300
Inter-activation Intervals (10 ms bins)
#2: A histogram of inter-activation intervals observed during atria flutter.
The solid gray bars are a histogram from the 100 midle electrodes. The diagonal
line bars are the histogram of just the very middle two electrodes. Notice that the
spread in the timing is about the same for both cases.
Atrial fibrillation is illustrated in Fig. 6.12. The activation of the atria does not
seem to follow any clear pattern. At several points (panels A and H), waves break
up into two waves. At other times two waves (not shown) can be observed to collide
an annihilate each other. The single and multi-point histograms (see Fig. 6.13) of
this behavior show about the same average timing as flutter but a larger variation
in the inter-activation intervals.
From these observations, there is a clear difference between NSR and reentry
patterns. However, the difference between atrial fibrillation and flutter is hard to
specify exactly. Often, an atria that is displaying fibrillation-like dynamics with
small reentry patterns will shift into flutter-like macroscopic patterns and then shift
back to small patterns.
120
0-10 ms
A
10.5-20 ms
D
40.5-50 ms
50.5-60 ms
F
E
60.5-70 ms
G
C
B
30.5-40 ms
20.5-30 ms
80.5-90 ms
90.5-100 ms
I
H
#2: A series of panels showing the temporal evolution of atria fibrillation.
With little or no organization wavelets circulate around the atria, sometimes colliding, other times breaking in two. Notice that in panel G a wave seems to appear
from no where. It is likely the wave came from the interior structure of the atria
wall.
121
0.04
P(T n)
0.03
0.02
0.01
0.00
0
100
200
300
Inter-activation Intervals (10 ms bins)
#2 : A histogram of inter-activation intervals observed during atria flib-
rillation. The solid gray bars are a histogram from the 100 middle electrodes. The
diagonal line bars are the histogram of just the very middle two electrodes. Notice
that there are more short period inter-activation intervals than in flutter.
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of the atria. However, it could be that the system produces an activation before
the controller is able to cause an activation. If this happens the controller resets.
Therefore the dynamics of the system plus control is given by the minimum of
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case. It is unclear why this happened.
,1
There are several possible reasons why this controller seems to be ineffective. Implicitly I have assumed that the atria are two dimensional sheets of muscle tissue.
However, the structure of the atria is complicated by the existence of pectinate muscles on the endocardium (inside wall of the atria). These large cable like structures
are excitable and could provide a path for reentry. This might also explain why my
results differ from those of Ditto [23], because their experiments were conducted on the interior surface of the right atrium. In addition, the lack of success in
this experiment may be a result of the fact that the range of gains explored was not
exhaustive and it is possible that the domain of control is very narrow. In addition,
because the pacing and recording sites were slightly different (3.6 mm) it could be
the control pulses were mis-timed. The most likely explanation of the apparent
131
failure of control is that the algorithm implicitly assumes that every pacing pulse
elicits an activation.
Figure 6.19 illustrates that not every pacing pulse elicits an activation. The
time series shown in Fig. 6.19a was taken from an electrode neighboring the pacing
electrode. The stimulus at G170 ms into the time series does not cause an activation
of the tissue under this electrode. The derivative signal Fig. 6.19b shows that
there is no threshold crossing for more than 20 ms after the stimulus. A possible
explanation for why some paces are ignored is that the stimulus occurs while the
tissue is still refractory from the last action potential. If this explanation is true
then the control algorithm would have to be modified to account for this effect.
132
Stimulus
Voltage (mV)
200
A
100
0
-100
No Response To Pacing
5 Point Derivative
-200
10
B
5
0
-5
-10
-15
0
100
200
300
400
Time (ms)
#2*: A time series showing that some pacing pulses are ignored by the
tissue. The stimulus is marked with an arrow. The time series from the control
observation electrode (A) does not show any active response to the stimulus. This
can also be seen in the derivative signal (B).
133
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Many times a system’s performance is degraded by the occurrence of complex behavior. In the case of cardiac tissue, since the presence of complex behavior either
degrades the heart’s performance or indicates disease, it is valuable from a fundamental as well as a clinical standpoint to investigate both the behaviors that
contribute to the complex dynamics and how to control them.
In the main part of this dissertation, I use small pieces of periodically paced
cardiac tissue as a test bed to study the behavior of local muscle tissue and demonstrate how to control the observed dynamics. Using small pieces of cardiac tissue
allows the study of muscle dynamics exclusively without the complications of spatially complex dynamics or internal cells that stimulate the tissue. This work on
small pieces of tissue has implications for controlling the whole heart, since the
initiation of complex spatio-temporal dynamics is believed to originate with local
tissue instabilities [24].
In Chapter 3, I investigated
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-0.129 mV/ C [54] p 6
+0.213 mV/ C [54] p 6
+0.923 mV/ C [54] p 6
: Electrode Half-Cells Potentials
139
-
∆ 0 )
Stainless Steel
10 mV
[54] p 7
Silver (Ag)
94 mV
[54] p 7
Silver/Silver Chloride (Ag/AgCl) 2.5 mV
[54] p 7
2": Fluctuation in Potential between Electrodes in Saline
1
) )
0.9% Saline
70 Ω-cm
Internal Resistance (Guinea Pig): 200 Ω-cm
Frog Ventricle Internal Resistance R
J ECC V
[54] p 54
[90]
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r
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0=
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0.4
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0
500
1000
1500
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0.4
A
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acing Electrode
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2
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While the microelectrode gives a more accurate reproduction of the transmembrane
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There are two basic types of MAP electrodes: pressure and suction. Schematics
of each are shown in Fig. D.7. The pressure MAP electrode damages the tissue
by poking it. The particular type drawn here is a catheter that is inserted into
the heart via an artery. The suction MAP electrode has two concentric hollow
electrodes separated by an insulator. The suction is applied to the inner electrode
156
2: MAP electrode. The catherter MAP electrode (A) is used to measure
on the inside (endocardium) of the heart. The tip presses on the tissue to damage
it and a differential measurement is made verses the reference elctrode on its side.
The suction MAP electrode (B) is used to measure signals from the outside surface
of the tissue (epicardium). The suction throught the inner electrode damages the
tissue and the outer one is the reference.
and the differential measurement is made between them.
MAP electrodes have several useful properties. In most situations signals they
measure are stable to tissue movement. Therefore, they can be used on both . -
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48
#8
88
MgCl
Glucose
Na HPO
HEPES
NaHCO
))@ 5
1B 5
1.8 mM
1.5 mM
5.6 mM
2.8 mM
1.0 mM
25 mM
/2: Frog Physiological Solution
The concentrations of the ions was developed from a survey of papers describing
physiological solutions used on bullfrogs and from the relevant sections of Burton
161
[69]. The sodium concentration I use seems absolutely standard (110mM). The
potassium concentration is the mean value of the solutions that were used or suggested in the various papers. Calcium is one of the most important ions, the almost
universally agreed on value is 1.8mM. Magnesium seems to be optional, however,
the discussion in Burton [69] and the fact that all ‘modern’ solutions contain it,
caused me to include an average value in this solution (1.5mM). In some instances
5-20 mM 2-3-butanedione monoxime (DAM) is added to reduce the mechanical motion of the tissue due to contractions. The solution is kept at a pH of 7.4±0.1 and
a temperature of 20±2 8
&"
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2 $ ?B 8
! NaCl
KCl
CaCl
MgCl
Glucose
Na HPO
NaHCO
125 mM
5.4 mM
1.8 mM
1.0 mM
5.6 mM
0.4 mM
28.8 mM
/2: Sheep Physiological Solution
The sheep solution needs to be kept at 37 C for the health of the tissue.
162
% *
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(
George Martin Hall was born in Decatur, Ga. on May 28, 1971. He grew up in
Marietta, Georgia. He attended the Walker School, which is a private school in Marietta. After graduating high school, he enrolled at Georgia Institute of Technology
as a President’s Scholar. In August 1992 he recieved a B.S. in Physics. Continuing
on at Ga. Tech. Martin received a M.S. in Applied Mathematics in June 1994.
Then He received the Townes Perkin-Elmer fellowship to attend Duke University in
Physics. In May 1997, he recieved a M.A. in Physics. Finally, in November 1999 he
earned his Ph.D. in Physics with a dissertation entitled ‘Controlling complex behavior in cardiac muscle.’ For the next little while he can be found at The Corporate
Executive Board, Washington, DC.
LIST OF PUBLICATIONS
G.M. Hall and D.J. Gauthier, ‘Controlling alternans in small pieces of rapidly-paced
cardiac muscle,’ Manuscript in preparation (1999).
G.M. Hall, S. Bahar, and D.J. Gauthier, ‘Inducing transitions between bistable
states in cardiac muscle using small electrical stimuli,’ Manuscript in Preparation
(1999).
R.A. Oliver, G.M. Hall, S. Bahar, W. Krassowska, P.D. Wolf, E.G. Dixon-Tulloch,
D.J. Gauthier, ‘Existence of bistability and correlation with arrhythmogenesis in
paced sheep atria,’ Manuscript in preparation (1999).
G.M. Hall, S. Bahar, D.J. Gauthier, ‘Prevalence of rate-dependent behaviors in
cardiac muscle,’ !, %, pp 2995-2998 (1999).
G.M. Hall, S. Bahar, D.J. Gauthier, ‘Experimental control of a chaotic point process
using interspike intervals,’ ', $% 2:1685-1689 (1998).
A. Chang, J.C. Bienfang, G.M. Hall, J.R. Gardner, D.J. Gauthier, ‘Stabilizing
unstable steady states using extended time-delay autosynchronization,’ $
, %
4:782-790 (1998).
LIST OF PRESENTATIONS
215
G.M. Hall, S. Bahar, D.J. Gauthier, ‘Control of alternans in cardiac muscle using time-delay autosynchronization,’ American Physical Society Centennial Meeting
1999, Atlanta, GA.
G.M. Hall, S. Bahar, D.J. Gauthier, ‘Control of Alternans in Bullfrog Myocardium,’
Dynamics Days 1999, Atlanta, GA.
G.M. Hall, S. Bahar, D.J. Gauthier, ‘A Test Bed for Control of Cardaic Tissue,’
Dynamics Days 1998 Chapel Hill, NC.
G.M. Hall, S. Bahar, D.J. Gauthier, ‘Complex dynamical behavior of small pieces
of cardiac tissue,’ Dynamics Days 1997, Scottsdale, AZ.
G.M. Hall and D.J. Gauthier, ‘Experimental control of a chaotic system using interspike intervals,’ 4th Experimental Chaos Conference 1997, Boca Raton, FL.
216