CONTROLLABILLITY OF EXCITABLE SYSTEMS
M. PERNAROWSKI
Abstract. Mathematical models of cell electrical activity typically consist of a current balance
equation, channel activation (or inactivation) variables and concentrations of regulatory agents. These
models can be thought of as nonlinear filters whose input is some applied current I (possibly zero)
and output is a membrane potential V. A natural question to ask is if the applied current I can be
deduced from the potential V. For a surprisingly large class of models the answer to this question is
yes. To show this, we first demonstrate how many models can be imbedded into higher dimensional
quasilinear systems. For quasilinear models, a procedure for determining the inverse of the nonlinear
filter is then described and demonstrated on two models: 1) the FitzHugh-Nagumo model and 2)
the Sherman-Rinzel-Keizer (SRK) model of bursting electrical activity in pancreatic beta-cells. For
the latter example, the inverse problem is then used to deduce model parameter values for which the
model and experimental data agree in some measure. An advantage of the correlation technique is
that experimental values for activation (and/or regulatory) variables need not be known to make the
estimates for these parameter values.
Key words. excitable systems, control, parameter identification, bursting
1. Introduction. Mathematical models of electrical activity in single cells
comprise of a set of differential equations, one of which represents a balance
of transmembrane ionic currents. An external electrical stimuli is modelled by
including an applied current in the current balance equation. Chemical stimuli are
often modelled either by varing concentrations of relevant agents or by varying
parameters which are believed to be correlated to the stimulating chemical.
If the transmembrane potential of a cell at time is
, the model equations
have the form:
# "$ % '&
(1.2)
!
The first of these equations represents a balance of capacitive currents (left side),
currents through
ionic channels (right
side) and the applied current . The pa
(1.1)
rameter
is the total capacitance,
are the transmembrane currents through
ionic channels (e.g. voltage-gated sodium, calcium inactivated potassium), and
are variables which describe gating and other regulatory effects for each ionic
channel of type . Often, many of these variables evolve slowly.
A common experiment in electrophysiology is to isolate a cell, inject a known
current
into the cell and then measure the resulting membrane potential
.
The aim in such experiments is to deduce what ionic currents are relevant for the
given cell and hopefully deduce quantitatively accurate functional forms for .
(
) * This work was supported by the National Science Foundation grant DMS-97-04-966
Department of Mathematical Sciences, Montana State University, Bozeman, Montana 59717.
Email: pernarow@math.montana.edu
1
2
CONTROL OF EXCITABLE CELLS
)
In this regard the cell may be regarded a nonlinear filter in which the input
illicits a response
. In this study we address the inverse problem, namely, if
is known is there a way to deduce what the stimulus
must have been?
Surprisingly, the answer to this question is yes for a great many models.
The approach we take is direct. Given a model we outline a procedure for
explicitly deducing the inverse problem. This procedure requires that occur
linearly in the given model (1.1)-(1.2). To be specific, the model equations must
be of the form
)
,+
/ + 1024 3 +
.- 5
(1.4)
4 + 76 + !
/ 5
:
8
for
9some
scalar function , vector valued functions , "7$ %
and matrix "
6
$%
. Since occurs linearly in these equations but the potential + does not,
we shall refer to (1.3)-(1.4) as quasilinear.
&>&?& cell models, (1.4) is an appropriate form since the variables
In <many
; 2= excitable
@ often obey relaxation equations:
BA A + 2A ED IH: &>&?&KJL&
(1.5)
C A+ GF
(1.3)
In vector form, this can be written as (1.4). Such relaxation variables, however,
do not always occur linearly in (1.3).
In section 2, we show that despite the fact that many models (1.1)-(1.2) are
not quasilinear, they can be embedded in a higher dimensional system which is
quasilinear. This embedding is explicit and we demonstrate it on the HodgkinHuxley model for electrical activity in the squid giant axon.
Then, in section 3, we outline the procedure to invert quasilinear models
by deriving a single differential equation for which depends only on . This
procedure is demonstrated on the FitzHugh-Nagumo model ([2, 5]) where the
is solved to derive an explicit expression
associated differential equation for
for the current which illicits a given arbritrary (smooth) response
.
Lastly, in section 4, we demonstrate how the inversion procedure can be used
to estimate model parameters from experimental data. As with most parameter identification problems, the problem associated with the parameter estimation
is ill-posed. This is discussed, a regularization scheme is introduced and then
demonstrated on a model of electrical activity in the pancreatic cell.
+
+ )
M
2. Transformation to Quasilinear Form. Here we discuss how some common nonlinearities occuring in excitable cell models (1.1)-(1.2) can be eliminated
by introducing new variables and embedding solutions of (1.1)-(1.2) in a quasilinear system (1.3)-(1.4) with degree
.
Argueably the most common nonlinearity which occurs in excitable cell
models are those which arise from modelling voltage activation (and/or inactiva-
NLO
J
M. PERNAROWSKI
3
tion) in ionic channels. Two such examples occur in the Hodgkin-Huxley model:
(2.1)
(2.2)
RQBS J
RW
RY
U
V
T
X
N
P#
Z
J
\[ [_^ ' [
`
`
[
[
c
M
'
] C>` Z ba
ITdeN
QBS W
Y
,
and
are voltage-gated sodium, voltage-gated potassium and
where
leakage currents, respectively. The voltage-gated currents have the form
(2.3)
(2.4)
RQBS QBS Jhg
QBS
X
T
W f W
W f Nji, Q2S W
QBS W
U
f f
J
; = g
BA J g I TdeNX
;
=
; =
T
Ni k (2.5)
f Z [ml where is some function, N is an integer and the activation variable [ satisfies
f
the relaxation equation (2.2). Define the variables
n A [ A ED UHo &>&?& N &
(2.6)
F
Differentiating npA in and using (2.2) we find
npA D [ A)q ; \[ D ` n Aq ; DM ` n A
(2.7)
a
&>&?&
s
n
;
nt; np= &?&>& n these
:
H
U
o
r
N
for D
. The equation for is (2.2). Defining n l
equations can be written
n 5 Z + un (2.8)
L6
?
&
?
&
&
5
`
where '
a Zwx Uvo ev,@ and
{}||
xx M `
v
?
0
>
0
0
?
0
>
0
z
0
?
0
>
0
0
v
||
xx H ` HsM ` v
?
0
>
0
z
0
?
0
>
0
0
v
&
a
r a ` rtM ` v 0?0>0
v |
(2.9)
6 Z .v
..
..
..
y ..
.
.
.
v
v ` ~
`
v
0?0?0
0?0>0 v N a NjM
are (constant) maximal conductances and
are sodium
where
and potassium Nernst potentials, respectively.
Defining
this model fails to be quasilinear
occur in a nonlinear fashion in (2.1). However, if we defined new varisince
ables
and
, (2.1) would be linear in
. But
then, (2.2) would not be linear in . To circumvent this difficulty we introduce
several new variables. First we consider the nonlinearities of the type in (2.4).
Assume
has the form
4
CONTROL OF EXCITABLE CELLS
n
f 9 l
n
l
Thus,
is linear in
and (2.8) has the form (1.4). It is important to note that although the resulting system has
new variables, the initial conditions on the are dependent. In particular,
. For the
is the potassium activation variable.
Hodgkin-Huxley model,
Now we turn to currents
of the form
nA
N .
n F A v€ tn ; v, A
[ ) N
V
[
:
l

9
(2.10)
J f
where is some function, N1
and the activation variables [ ) and
f relaxation equationsareof integers
 satisfy
the form (2.2). Define the new variables
&?&>&
&>&?&ˆJ 3
&
A „
‚
)
A
ƒ
„
[

D v… F N1‡† vo F ‰D † Š v
(2.11)
‚
Again, we differentiate Aƒ „ in and use the relaxation equations (2.2) for [ and
 to find
‚ A)ƒ „
` ‚ Aq ; ƒ „ )DŒM ` 3 †,MVd ‚ Aƒ „3 †  ‚ Aƒ „Rq ; &
(2.12)
D
‹ a
a
‚
A
ƒ
„
From
(2.12) it is apparent that the derivative of
depends linearly on
‚Ž ƒ  variables.
‚ two other
This dependence is illustrated in Figure 1 where each A)ƒ „ variable
‚
‚
is
by circle. The derivative of A)q ; ƒ „ will in turnJ depend on Aq = ƒ „ ,
‚ Aindicated
‚
q ; ƒ „ and Aq ; ƒ „Rq ; .‚ It isƒ evident that by introducing )N 3 ‚ F ARƒ „ 3 F cF variables,
will be linear in
l and the resulting system for will be of the form
(1.4).
For the Hodgkin-Huxley sodium current, [ and  J are current activation and
inactivation variables, g respectively, with N
r and F so that the current
equation is linear in [  . The inflated quasilinear system corresponding to this
current requires the addition of 6 new variables. Defining
(2.13)
the system for
(2.14)
‚
‚
‚ ; ƒ ‘ ‚ >‘ ƒ ; ‚ ; ƒ ; ‚ = ƒ ‘ ‚ = ƒ ; ‚ g ƒ ‘ ‚ g ƒ ;
@
has the form
‚ 5 ' ' ‚ ’
76
wxx M `
v
v
v
v
v
v { |||
xx v
v
v
v ||
xx  Mm ` Mv ` 3 Mm vv
v
v
v ||
6 Z xx “ aH a ` v“a v `
HsM `` 3
v
v
v |
v
v
H a
HsM ` Mm ”\
v `
v
y v
v
v
r a
v ` rtM ` 3
v ~
v
v
v
v
r
t
r
M
M


a
a
5
`
and
‚ note the initial conditions are depen a a !ev…evoUvoUvoev€K@ . Again we
dent on one another via the definition of A)ƒ „ .
where
5
M. PERNAROWSKI
—
—
D ˜F †
—
DI †
–
DI † ˜F
•
™mšœ› 
™mšœ› 
™mšœ› sž Ÿ¡d¢]£ š)¤: (located at cross)
™m¥ › ¦ variables indicated by solid
F IG . 1. Figure showing the dependence of the derivative of
on other
variables. The derivative of
depends linearly on
circles.
In summary, the methods outlined in this section can be used to embedd the
Hodgkin-Huxley model in an 11-th order quasilinear system. Given the dependence of the initial conditions on one another, we see that solutions of the 4-rth
order Hodgkin-Huxley model are in fact projections of trajectories on a differentiable manifold embedded in the 11-th order solution space of the corresponding
quasilinear system.
+ ‘ 3. Controllability. In this section we outline a procedure for inverting initial value problems of the form (1.3)-(1.4). We define the initial condition
and assume that for the applied current
2¨ ;
+ v€ v€@ "§$ %
)
;
q
?& &>&
@
) ª© q ;d«
(3.1)
is continuous. Lastly, we let ‘
v€ .
With these definitions
and
assumptions,
we can
view (1.3)-(1.4) as a nonlinear filter whose input is u+ ‘ and output is + )R ‘ , i.e., there is some operator
6
¬®­ t+ ‘ §
¯° + ‘ .
CONTROL OF EXCITABLE CELLS
‘
In words, if the input current and initial cell state are
known, the model yields an output voltage . And since
is known, can be
found.
A natural question to ask is whether has an inverse. That is to say, if the
output
is known, can
be predicted given ? Surprisingly, the answer
to this question is yes in a great many instances. To demonstrate this fact, we
explicitly compute the inverse operator. Owing largely to the linearity of (1.3)(1.4) in , repeated differentiations of (1.3) and substitutions using (1.4) results in
a procedure whereby can be eliminated resulting in a linear differential equation
for
involving
and its derivatives.
Solving (1.3) for we have
+ )
¬
+
‘
+ ) ‘ + ,+ « 3 ² ‘ + 102 u+ + 3 / + !0B &
³7P± © Differentiating (3.2) in results in the calculations
´I´U·!¶ ¸ 3 ´!´UWW ¹ ¶ ¸ 3 ² ‘ ´ ´U¶ ¹º
´I´Uµ¶
´I´U·!¶ ¸ 3 @@ ´!´U¹ ¶ ¸ 3 ² W‘ @@ 5 ´I´U·!¶ ¸ 3 ² ‘ @ 5 3 @ ´´U¹ ¶ ¸ 7W ² 6 ‘ @ (3.3)
´I´U·‰¶ ¸ 3 ² ‘ @ 5 ¼ 3 @ ´1´U¹ ¶ ¸ 6 @ ² ‘ ¼
»
» 76
(3.2)
which can be written in the form
=
u
+
+=½« 3 ² ; + ,+ « 0B ;
+
(3.4)
b± © © ²
for some functions ; and ; . Proceeding inductively, repeated differentiations
±
of this equation in and using the same substitutions it is easily seen that one can
&?&>& l ¨ ¨ ; + 3 ²
,
+
,+ &?&>& l + « 0“ l
X
;
«
+
+
(3.5)
l © l
l &?P
l
?& &± J l © where N
F F . In matrix notation ² (3.2),(3.5) can be written
¾
3 (3.6)
&?&?& q ; ±
² ".$ % 98:
has ² e N
;
‘
where&?&?&J
and the matrix
@
l ± So ± long as ² is nonsingular,
vo F ±¿ 4F ± , as rows.
² q ; (3.7)
±
obtain a set of equations of the form:
which when substituted into (1.3) yields
(3.8)
,+
/ + !0 ² q ; 3 &
+
À- ±
M. PERNAROWSKI
‘
‘ ‘
+ ¯Á t+ ) J
7
GF
¬ q; ­
This is a linear differential equation for
of degree
. Given the initial condition
and
the solution
defines the inverse operator
.
Before explaining the significance of this result we first illustrate the procedure on a simple model. For this we choose the FitzHugh-Nagumo model ([2, 5])
+ )
+ Â u + À.- + 43 Â (3.10)
à + ÅÄ where + - + BF + ? Æ + ”: and ”V Ä Ã are parameters. Differentiating (3.9)
in we find
 4 3 Â
+
+
+
(3.11)
-_Ç + + Â + 3 Â
-_Ç ÈÃ ÈÄ
Solving for we find
Æ
 3 +  ¼ &
F +
+
+
(3.12)
ÃpÄ » É- Ç
à Substituting this into (3.9) weÆ obtain Æ
&
 3 Â
Â
Â
3
3
(3.13)
ÃpÄ PÊ + + + R + 7- Ç + + à + ÃpÄ + É- + ‘ the solution of this first-order differential
Given an initial condition v,
equation is:
Æ
¶ ÌÍsÏ
;
q
V
q
K
Ì
Í
¶
Â
‘B34Î ‘
«
(3.14)
) ¬ + ‘ bË
©
Ë Ê + + + RÐÑ Ð ¬
demonstrating explicitly a formula
=
for ‘ the inverse of . For example, (3.14) can
+
be used to compute the stimulus ) ,
which yields an output
v
PÒeÓÕÔ .
Performing this calculation explicitly 1 we plot the corresponding ) in Figure 2
for a fixed set of
= parameter values. In this example, the cell is initially at rest since
for + bÒeÓÕÔ , + v, + Ç v, v…ev€ .
Now we explain the significance of these results.
It is clear that given the
solution + 1 of (1.3)-(1.4) one can easily find ) by simply solving (1.3).
Therefore, the elimination of from the model equations seems irrelevant to
¬
defining the inverse of . However, in most experimental situations it is often
difficult or impossible to measure + and simultaneously. Voltage measurements,
on the other hand are always made. And, if the sampling rate for this measurement
is sufficiently fast, estimates of + and its derivatives are easily obtained. In this
regard, the elimination of is paramount to developing a technique for estimating
Done symbollically with MapleÖ:×
(3.9)
1
8
CONTROL OF EXCITABLE CELLS
2
1.5
1
)
0.5
0
−0.5
−1
0
5
10
F . 2. The input ØsžÕŸ¡
ž átâã:âäI¡m¢]žåeæIçÑâeåâåeæ>åèR¡
IG
15
20
25
30
35
40
for the FitzHugh-Nagumo which yields an output
45
ÙsžÕŸ¡Ú¢ÜÛÝßÞpàoŸ
50
for
+ model parameters and unknown input currents when only
and its derivatives
can be measured. These techniques we describe in the next section.
Before we conclude this section, however, we cast the inverse problem as a
control theory problem. In [1], Casti defines a variety of different control problems. One which most closely mimics our system is the “realization-identification”
problem. For example, consider the system
ë é é )R 3ê f é )e
) ½T é R
ë
ê
é
where ) is an input, is a state and ) is the output. For the “realizationë
identification”
problem associated with this system one tries to find and T so
- f
that ) is known in terms of ê ) . The control problem associated with (1.3)(1.4) is a minor modification
If we de ofé the realization-identification
é ; &>&?& éVî ¨ ; )eK@ problem.
fine
, ê , )
)
R
,
v
@
§
)
K@
ì
f
í
F
é
j
é
;
and T½ our problem is one of finding - given the input and output.
(3.15)
(3.16)
9
M. PERNAROWSKI
4. Parameter Identification. We now use the formulation of the inverse
problem to describe a method for estimating unknown model parameters given
that the output potential
and input current
are known. We define two
classes of problems:
P A model (1.3)-(1.4) for a given electrophysiological experiment is suspect but that estimates for all but parameters
are
known. In the experiment, an input current
and measured potential
are found for a discrete set of times
. The problem is
then finding so that the model most closely agrees with the measured
output .
P* A model (1.3)-(1.4) for a given experiment is suspect but since the stimilus is chemical,
. In this case one may still be interested in
estimating unkown parameters defining the model.
Though these problems may seem different to experimentalists, the correlation
technique illustrated here is the same.
From (3.8), we define
+ )
ð
) N
(4.1)
ñ
ò 0ò
ð
ï
v
&>&?&
;
ï
ï
Iï l ?ð
; )= ?& &>& Q
ï
ñ ïX + u + + 3 / + !0 ² q ; ó ò ³7 ± ò
where
is some norm (perhaps simply absolute value). Then for problem P
one might attempt to minimize 2
Q
ñ ð >ð UïV ñ ïdI ð ) A >ð ) A AÕôj‘
ñ
in
the agreement were
ð exact, one would have v . For a given model,
ñ ï is. Ifknown
but since ) is only known at discrete times and noise is often
Ž is an issue in itself.
present, estimation of the derivatives evaluated at Before addressing the issues of derivative estimation and data smoothing,
ñ
there is a more fundamental theoretical issue of well posedness. In general,
will not have a unique minimum. To illustrate this, consider problem õ
for
which
ñ ïX + u+ + / + 10 ² q ; (4.3)
óò ³7- ±ò
ñ
will be identically zero.
If + ) is any solution of the differential equation (3.8),
It is often the case that the model equations possess a rich bifurcation structure
in parameter space. For some ï , (3.8) may possess stable fixed points while for
other ï the modelñ may have stable periodic solutions or other attractors. Thus, a
minimization of in ï may limit on a value of ï corresponding to a fixed point
where the output + ) is clearly periodic (and vice-versa).
To illustrate this, we
Æ
v
note that for the FitzHugh-Nagumo
model with
in (3.13),
&
ñ
Â
Â
3
3
+
+
+
+
+
+
(4.4)
óò É- Ç Ã ÃpÄ 7- e ò
For problem P*, the only difference is that ØKö‰¢hè .
(4.2)
2
10
CONTROL OF EXCITABLE CELLS
To correlate periodic solutions of this model to data one might introduce phase
and scaling variables for time and the potential:
C
¶ [ ¶ ê P÷ + [ ÷> ø ¶ [ ¶ ø [ "h$ % i &
ï P
÷ ÷>ø
ø
3
d
ê
ù
¶
+
[
Then, with
ø
÷?ø , as ÷ ¯v and ÷?ø ¯zú ,
´I´ÿ ¼ 3 +
ñ
¶ ´I´ þÿ þ 3 +
;
q
¯ òhûˆûˆý ü »Ñ÷ [ ÃÑħ[ 7- Ç
à È
ÃÑÄd- + ò
(4.8)
ÃpÄÈòB- ø ÅÄ ø ò
v
if +
[ ø is a steady state of the model. In other ñwords, if [ ø is a steady state,
the parameter search ï]¯‡voúPUvo [ will drive to a minimum regardless of
ø
what the data is. For this reason, a penalty function should be introduced to
(4.5)
(4.6)
(4.7)
avoid parameter searches which yield undesirable solutions such as steady states.
In general, if is a solution
of the model equations for a given parameter
value , a penalty function
should be introduced and then the
functional
V)
ð
I mUïV
ð >ð
ñ ð >ð UïV 3 ð UmUïV
(4.9)
U
mUïV
should be minimized in ï . The penalty function should be chosen so that it is
numerically larger at ï for which is an undesirable solution (such as a steady
ï
state) while smaller at other solutions (such as periodic solutions). In the next
section we illustrate this procedure using the penalty function which is an apon the time interval over
proximation of the norm which the sampling of the data
was measured.
=
ð
V
ð
ò ò
v
4.1. An Example: Parameter estimation in the SRK model. In this section we demonstrate how to use experimentally measured potential time series to
estimate parameters in Sherman-Rinzel-Keizer (SRK) model of -cell electrical
activity (Sherman et al. 1988). The model equations are:
M
S q
W
W q S
, +
+
+
e
X
N
+ ?R
N N ^ + N (4.11)
C l + &
S q S
+
>
(4.12)
- “a
The dependent variables + , N , and are the transmembrane potential, the activa(4.10)
tion variable for the voltage-gated potassium channel, and the intracellular (free)
calcium concentration, respectively. Equation (4.10) is a current balance equation
11
M. PERNAROWSKI
Sq
W
where the terms on the right side represent the transmemebrane currents due to
voltage-gated calcium channels (
), voltage-gated
potassium channels ( ),
and calcium-activated potassium channels (
). Formulae for these currents
and standard parameter values used in [8] are tabulated in Appendix I. The nondimensionalization of this model reveals [7] that the calcium-activated potassium
current can be approximated by
W q S
(4.13)
f
W qS
W q S
W q S &
f ² ´ We adopt this approximation here and then note that (4.10)-(4.12) is linear in
and has the form (1.3)-(1.4). Therefore, by the procedure outlined in
section 2, and can be eliminated from the equations. The resulting (equivalent)
third-order equation for
is derived in Appendix II:
)N1 ?N + ) = = u + =+
± ©+
ñ
(4.14)
g
=
+g « 3 ² = + , + +=‰« 0,)N @ v &
© A sample of experimentally measured values for the membrane potential
were obtained from David Mears (National Institutes for Health). This data is
shown in Figure 3 (top trace). The potential
was obtained from a single
mouse pancreatic -cell in an intact islet of Langerhans bathed in a 10mM glucose concentration. The sampling rate for the data was 150Hz which translates to
6.7msec. The membrane potential for this expera measured value every iment (type ) is seen to lie roughly in the range of 20-30mV. Superimposed on
this figure (bottom trace) is a numerical integration of the SRK model using the
“standard” parameter values given in [8] 3 .
Though the experimental data and theoretical prediction depict bursting oscillations, both are badly out of phase, have different periods and voltage ranges.
Since these are the dominant differences we introduce into the model equations
scaling and phase parameters as defined in (4.5)-(4.6). All other parameters in
the model equations were fixed at their original values. To define the measure
in (4.9) we use computed in (4.14) and a penalty function equal to the square
of the norm
M
ð
õ
' ñ
=
(4.15)
ñ
ð
) ï
=Y ‘>ƒ Î @ ð
=
ð
þ
)
V
)
V) ‘
ó ò
ò @ where was chosen as 30 seconds (4 burst durations in the experimental data).
To evaluate using experimental data, we approximated derivatives using a series
of smoothing and discrete differentiation procedures.
First, data was
smoothed using a sliding window average.
Given the sam
pling times are with for all , a smoothed time
series of the data
was computed using
&>&?& ¬
Ž
Ž¨ ; Ž
Q
U
o
H
'
q
ð
ð
ð
Ž
B F Ž ô ; Ž ¨
) Ž ¨ &%(' ¶
AÕô ð
ð A
ð
Ž
Î$Ž '#
(4.16)
F
)
"
!
F
) Ž
¶
A ô
Z
3 tabulated
in Appendix I
12
CONTROL OF EXCITABLE CELLS
−20
−30
V
−40
−50
−60
−70
0
10
20
30
t (sec)
40
50
60
F IG . 3. Membrane potential of a pancreatic ) -cell in *,+ . The top trace shows the membrane
potential measured experimentally when the cell was bathed in a 10mM glucose solution. The bottom
trace shows a theoretical prediction using the Sherman-Rinzel-Keizer model with parameter values
chosen from [8]. For these standard parameter values, theory and experiment differ greatly. The
object is to find parameter values for which theory and experiment agree in some measure.
where is an averaging parameter indicating the width of the window over which
the averaging is performed.
A discrete differentiation operation - was used to compute dervatives. For
example,
𠎨 ;
ð Ž
ð &
Ž
)
)
-] !
(4.17)
'
ñ at a given ï , ´ ´U/¶ . was approximated by 0-1R ð , ´I´Uþ ¶ /þ . by
To computeð
2-34-1? and so forth. ñ
Unlike the computations of from the exerimental data, the computation of
ð
the penalty function requires a (more computational costly) numerical integration
of the model equations. However, this computation needs only be performed once
when minimizing with respect to the phase and scaling parameters defined in
(4.7). After has been found by numerically integrating the model equations
¸ )
ï
13
M. PERNAROWSKI
ï ï ‘ F F ev…ev€ )
ï
for
, for other is easily computed from the relations
(4.5)-(4.6) without further integrations. As a result, however, the time grid for
in the numerical integration may not match the experimental time grid
making it necessary to interpolate (via cubic splines) values at
so
that the penalty function defined above may be approximated via
Ž
Ž
=Y >‘ ƒ ð Ž
ð
Ž e = ' &
þ
!
)
) ó ò
ò 5@ Ž
6
A sliding window “width” of
v was used to compute the measure ïm
from the data and a numerical integration of the SRK model. The functional
ïV was then minimized using the Nelder-Mead simplex algorithm ([3, 6]) us8& 7
& the
9
:& 7
9…& yielded
algorithm
ing the initial value ï
F F
UvoUv, . After several iterations,
6
¶
¶
the parameter values ï
ï ð ÷ ÷ ø [ [ ø F H:UH H r€Hu . In
Figure 4, the experimental data is plotted against the scaled and phased theoretical prediction /; ) . It is apparent that this choice of ï has significantly
(4.18)
improved the correlation of phase and scaling (compared to Figure 3).
5. Discussion. As has already been demonstrated, a large class of excitable
cell models are controllable in the sense that there exists an input
which
illicits any (smooth) response
. This does not mean that these models are simultaneously controllable in the other dependent variables . Nevertheless, the
implications are still broad and interesting. For example, the predicted controllability in the FitzHugh-Nagumo model implies that there is an input
such
that
equals the first component of the Lorenz attractor. It is therefore not a
surprise that forced excitable cell models exhibit such a rich variety of behaviors.
At this early stage, it is not clear how (or if) the invertibility of these models can
be used as a tool to examine these behaviors.
From the results in section 4, however, it is evident that the inversion can be
used for parameter estimation purposes. What is most relevant in the procedure
described and demonstrated there is that
need not be measured experimentally to obtain parameter estimates. It is very probable that there are several improvements which can be made in the procedure implemented on the SRK model.
There are many ways to de-noise data, minimize functionals and make derivative
estimates from data. All of these play a role and the crude approach taken on the
SRK model was primarily demonstrative. A future goal for work in this area will
be find an integral formulation for so that derivate estimates can be avoided.
Lastly, although de-noising will always be an issue, it may be less so in neuronal
systems since electrical measurements on neurons tend to be cleaner than those
made on endocrine cells.
+ + )
ñ
)
)
14
CONTROL OF EXCITABLE CELLS
−15
V (mV)
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−25
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−35
0
5
10
15
t (sec)
20
25
30
F IG . 4. Comparsion of experimental data and SRK model prediction for different model parameters. The scaling and phase parameters used to make the theoretical prediction agree better with
>=@? 0=@A B? CA
ED FG HG/D IJ KLG/D F KLI/D EG . This choice was
the experimental trace when <
determined using the algorithm described in the text.
¢bž eâ t⣠K⣠>¡¢.žå Ñâ pâ >â ç R¡
6. Appendix I. I: SRK function definitions
(6.1)
(6.2)
(6.3)
(6.4)
(6.5)
(6.6)
(6.7)
J
S
S Mq S
^
^
+
+
+
+
+
T
R
RW
f Ž N + + W +
X
N
f W S
R W q S
W
+ ? f q ² ´ 3 + + J
^ + 3 exp + F + ù F
^T + 3 exp + F +BN ù N F
N ^ + 3 exp + F + ù l C l
&
C + F
S
l
l exp Õ + B+ O ù 3 exp + +BO ù O 15
M. PERNAROWSKI
Standard paramater values in SRK model:
S
Parameter
+W
+
S
W
W f f q S
f 𠏏
² ´
S
q;
+- O
B
+BN
+
+ lS
q; J g q;
2P )
&
6
vBR€HTS F v
.001
(mV)
(mV)
(mV)
(mV)
(mV)
(mV)
(mV)
(mV)
(mV)
(ms)
O
N
SRK value
110
-75
1400
2500
30000
1150
5310
100
8Q
.03 U
J g
a
(units)
(mV)
(mV)
(pS)
(pS)
(pS)
(P
)
(fF)
(P
)
(ms )
(mole coul
C l
l
q
-75
-10
4
-15
65
20
10
14
5.6
37.5
ù H, ð   / where / is Faraday’s cona# F
7. Appendix II. Elimination of N1 ? from SRK Model
In this section we reduce the modified SRK model equations
third
to S aqMsingle
/
+
order differential equation in + ) . For simplicity, set + . The
In the SRK model, the parameter
stant.
potential equation
h,+
/ + W N + + Ž W S t + + Ž À
f
È
f
È
(7.1)
is equivalent to
(7.2)
where
(7.3)
(7.4)
² ‘
N1? , ‘
±²
‘
‘ u+ « 3 ² ‘ +
± ©+
² ‘ ; ² ‘ = K@ and
u+ 3 / + W W W
q
f + + f
!0B v
S
W
+ + e 16
CONTROL OF EXCITABLE CELLS
v,
; u+ ‰=+ « 3 ² ; + ,+ « 0B v
(7.5)
± ©+
© ²
² ;U; ² ;K=
where ;
= K@ and W
W q S/
+= 3 / ,+ 3 f N ^ + + W (7.6) ;
+ C
Ç
7-_aXf
± W
l
W u+
W W q S u+
W q S S
²
+
+
R
+ (7.7) ;
f
C
7-mf
ª©\f l f
²
² =; ² =e=
To find = and =
= Kg @ in
±
=
+
+
,
+
u
+
²
g
=
3
=
+ +=‰« 02 v
=
«
+
(7.8)
± © © differentiate (7.5) in :
± ; 3 ² ; 024 3 ² ; 0 N ^ N / S > «
(7.9)
© C l - “a Thus,
²
= ± ; 3 ;UC ; N ^ ² ;= /
(7.10)
± ² l ;e; -_S a
²
;
² ;= «
² = (7.11)
© Cl where
=
; g +g 3 / = + 3 /
u
+
(7.12) ±
Ç =
ÇÇ + © «
W q S/
W u+
W q S / ,+
+
+
+
(7.13)
?
Ç
<-_Wadf
W ³7-_aXf
W
W
W
^
N
N
^
3
C
(7.14)
f© C Ç + + f C = l Ç + + f C N ^ «
l
l
l
and
² ; V Ž ´I´Uþ¶ þø 3XWYZ þ C + + W Y\ ´ ´Uø¶
[]
[
(7.15)
 f W q S ´I´Uþ¶ þø l Ç W q S S ´ ´Uø¶ ^
f
É-mf
Next, solve for )N1 ? @ using (7.2),(7.5) to find
² ‘ = ; ² ;= ‘
N N ± ² ±
(7.16)
² ;e; ‘ Ë ² ‘ ; ;
(7.17)
± ² ±
Ë
Using the procedure discussed in section 2 with
=
W
+ W
+ «
,+
M. PERNAROWSKI
17
where
Ë ² ² ‘ ; ² ;K= ² ‘ = ² ;U;
Substituting these into (7.8), one obtains a single third order equation for + ) :
= g
=
_
ñ
,
+
+
+
u
+
+=‰« 0uN @ v
²
g
=
=
3
=
«
+
+
(7.19)
© ± © (7.18)
8. Acknowledgment. I would like to express my deep regret and sadness
at recent passings of Teresa Chay and Joel Keizer. Their influence on my work
has been immense. Indeed, even in this manuscript Joel Keizer’s influence can be
noted in the references.
Also, I would like to thank David Mears at the National Institutes for Health
for sharing his data on pancreatic beta-cells so that I could complete the last section of this paper.
REFERENCES
[1] Casti, J. L. (1985) Nonlinear System Theory, Academic Press, Inc., Orlando, FL.
[2] FitzHugh, R. (1961) Impulses and physiological states in theoretical models of nerve membranes. Biophys. J. 1:445-466.
[3] Himmelblau, D.M. (1972) Applied Nonlinear Programming, McGraw-Hill Book Company,
New York, NY.
[4] Hodgkin, A.L., Huxley, A.F. (1952) A quantitative description of membrane current and its
application to conduction and excitation in nerve. J. Physiology (Lond.). 117:500-544.
[5] Nagumo, J.S., S. Arimoto, S. Yoshizawa. (1962) An action pulse transmission line simulating
nerve axon. Proc. IRE, pp 2061-2071.
[6] Nelder, J., R. Mead. (1965) A simplex method for function minimization, Comput. J. 7:308313.
[7] M. Pernarowski, Miura, R.M., Kevorkian, J. (1991) The Sherman-Rinzel-Keizer model for
bursting electrical activity in the pancreatic ) -cell, In “Differential Equation Models in
Population Dynamics and Physiology,” Lecture Notes in Biomathematics, S. Busenberg
and M. Martelli (editors), Springer-Verlag, Berlin, 92:34-53.
[8] Sherman, A., Rinzel, J., Keizer, J. (1988) Emergence of organized bursting in clusters of pancreatic ) -cells by channel sharing. Biophys. J.54:411-425.