Analytical and simulation results for the stochastic spatial Fitzhugh-Nagumo model neuron.

Abstract

For the Fitzhugh-Nagumo system with space-time white noise, we use numerical methods to consider the generation of action potentials and the reliability of transmission in the presence of noise. The accuracy of simulated solutions is verified by comparison with known exact analytical results. Noise of small amplitude may prevent transmission directly, whereas larger-amplitude noise may also interfere by producing secondary nonlocal responses. The probability of transmission as a function of noise amplitude is found for both uniform noise and noise restricted to a patch. For certain parameter ranges, the recovery variable may be neglected to give a single-component nonlinear diffusion with space-time white noise. In this case, analytical results are obtained for small perturbations and noise, which agree well with simulation results. For the voltage variable, expressions are given for the mean, covariance, and variance and their steady-state forms. The spectral density of the voltage is also obtained. Numerical examples are given of the difference between the properties of nonlinear and linear cables, and the validity of the expressions obtained for the statistical properties is investigated as a function of noise amplitude. For given parameters, analytical results are in good agreement with simulation until a certain critical noise amplitude is reached, which can be estimated. The role of trigger zones in increasing the reliability of transmission is discussed.

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LETTER
Communicated by Adrienne Fairhall
Analytical and Simulation Results for the Stochastic Spatial
Fitzhugh-Nagumo Model Neuron
Henry C. Tuckwell
tuckwell@mis.mpg.de
Max Planck Institute for Mathematics in the Sciences, Leipzig, D-04103 Germany
For the Fitzhugh-Nagumo system with space-time white noise, we use
numerical methods to consider the generation of action potentials and
the reliability of transmission in the presence of noise. The accuracy
of simulated solutions is verified by comparison with known exact
analytical results. Noise of small amplitude may prevent transmission
directly, whereas larger-amplitude noise may also interfere by producing
secondary nonlocal responses. The probability of transmission as a
function of noise amplitude is found for both uniform noise and noise
restricted to a patch. For certain parameter ranges, the recovery variable
may be neglected to give a single-component nonlinear diffusion with
space-time white noise. In this case, analytical results are obtained
for small perturbations and noise, which agree well with simulation
results. For the voltage variable, expressions are given for the mean,
covariance, and variance and their steady-state forms. The spectral
density of the voltage is also obtained. Numerical examples are given
of the difference between the properties of nonlinear and linear cables,
and the validity of the expressions obtained for the statistical properties
is investigated as a function of noise amplitude. For given parameters,
analytical results are in good agreement with simulation until a cer-
tain critical noise amplitude is reached, which can be estimated. The
role of trigger zones in increasing the reliability of transmission is
discussed.
1 Introduction
Many models of action potential initiaton and propagation are formulated
as systems of partial differential equations involving spatial diffusion.
Such models are also called reaction-diffusion systems. Examples are the
Hodgkin-Huxley system (Hodgkin & Huxley, 1952) and the Fitzhugh-
Nagumo (FN) model (Nagumo, Arimoto, & Yoshizawa, 1962; Fitzhugh,
1969). In this letter, we focus on the FN model, which in its deterministic
form has been investigated from both an analytical and numerical point
Neural Computation 20, 3003–3033 (2008)
C ?2008 Massachusetts Institute of Technology

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3004H. Tuckwell
of view as a nerve cell model by many authors (Hastings, 1975; Scott,
1975; Sleeman & Tuma, 1985; Zhou & Bell, 1993; Horikawa, 1995; Romeo &
Jones, 2000; Slavova & Zecca, 2003) and in contexts such as heart dynamics
(Rabinovitch, Aviram, Gulko, & Ovsyshcher, 1999), chemical engineering
(Armaou, Kevrekidis, & Theodoropoulos, 2005), and spreading depression
(Dahlem, Schneider, & Scholl, 2008).
The FN system is sometimes studied by omitting the recovery variable.
The resulting scalar equation is referred to as the reduced FN system,
or Nagumo’s equation (Slavova & Zecca, 2003; Browne, Momoniat, &
Mahomed, 2008; see Tuckwell, 1988a, for earlier references). For a neuron,
this reduced model can represent the effects of blocking potassium chan-
nels by, for example, tetraethylammonium chloride (Tasaki & Hagiwara,
1957) so that there is no recovery. Exact analytical expressions have been
obtained in some cases for solutions of the reduced equation (Kawahara &
Tanaka, 1983; Li & Guo, 2006).
Stochastic FN models with spatial diffusion (Tuckwell, 1986), which are
examples of systems of stochastic partial differential equations (SPDEs),
have been less studied because of their complexity. However, the advent
of readily available efficient computation has recently led to their study
by simulation, although sometimes without the correct representation of
the noise, in diverse applications (Armaou et al., 2005; Tessone & Wio,
2007; Gosak, Marhl, & Perc, 2007). An analytical approach was briefly
reported for stimulation by small-amplitude two-parameter white noise
(Tuckwell, 1988b). In Neiman, Schimansky-Geier, Cornell-Bell, and Moss,
(1999), discrete networks of coupled FN model neurons were found to
be synchronized by two-parameter gaussian white noise. The review by
Lindner, Garcia-Ojalvo, Neiman, and Schimansky-Geier (2004) discusses
these and many related systems. Some progress was made with an ana-
lytical approach to the reduced FN model with stochastic input (Tuckwell,
1987, 1992). Although most of the above applications have been nonlinear,
related linear SPDE models have been studied analytically in connection
with nerve-membrane voltage fluctuations and information processing
(Tuckwell & Walsh, 1983; Manwani & Koch, 1999; Gagne & Gagne,
2005).
In this letter, we first consider the two-component FN model and
proceed by simulation to study the effects of noise on the instigation and
propagation of action potentials for various sets of parameters. In some
cases, the second (recovery) variable can be neglected, which facilitates an
analytic approach for determining the statistical properties of subthreshold
responses. The analytical and simulation results are compared, and when
the analytical approach is accurate, various statistical properties of the
voltage variable are found and analyzed for small noise amplitudes. Note
that we use subscripts throughout to denote partial differentiation, so that,
for example, Vxt≡∂2V(x,t)
∂x∂t.

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Stochastic Spatial Fitzhugh-Nagumo Model Neuron3005
2 Stochastic FN Equations and Solutions by Simulation
In general form, for one space dimension, the FN model can be written,
with voltage variable u(x,t) and a recovery variable v(x,t), as
ut= D1uxx+ κu(u − a)(1 − u) − λv + I(x,t)
vt= D2vxx+ ??[u − pv + b],
(2.1)
(2.2)
where x ∈ (0, L)isthespacevariableandt > 0istime.Initialandboundary
conditions must also be specified. The quantities κ, λ, ??, p, D1, and D2are
positive and usually taken to be constants, although they could vary with
both x andt.Theparameterb canbepositive,negative,orzero.Theapplied
current or input signal I(x,t) may be due to external or intrinsic sources. In
the majority of applications, D1= 1 and D2= 0, which will be the case in
this letter.
In the model of Fitzhugh (1969), the parameterization was
ut=uxx+ u −u3
vt=0.08(u − 0.8v + 0.7).
3
− v + I(x,t)(2.3)
(2.4)
This system can be transformed to the form of equations 2.1 and 2.2 by
setting, for example,
˜ u = Au + B,
to yield
˜ v = Cv + D
˜ ut= ˜ uxx+ 4˜ u
?
˜ u −1
2
?
(1 − ˜ u) + A(D − ˜ v + I(x,t))(2.5)
˜ vt=0.08
A
(˜ u − 0.8A˜ v),
(2.6)
where A= (2√3)−1, B =1
inal parameters of Fitzhugh, 1961, whereby ut= 3(u −u3
vt= −(u − 0.7 + 0.8v)/3, do not yield traveling wave solutions with D1= 1
and D2= 0.)
2, C = 1, and D =β−0.7A
0.8A. (Note that the orig-
3− v + I) and
2.1 Two-Parameter White Noise. The stochastic input current terms
of this letter contain the two-parameter random white noise process w =
{w(x,t)}, where, by definition,
w(x,t) = Wxt,

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3006H. Tuckwell
where W = {W(x,t)}, for values of x ∈ [0,x1] and t ∈ [0,t1], is a two-
parameter Wiener process or Brownian motion. Properties of W and w
parallel those of the usual one-parameter Wiener process or Brownian mo-
tion and the usual gaussian white noise. W(x,t) is a gaussian random vari-
ablewithmeanzeroandvariancext andcovarianceCov[W(x,s),W(y,t)] =
min(x, y)min(s,t) so that
Cov[w(x,s),w(y,t)] = δ(x − y)δ(s − t).
Sample paths of W are (continuous) surfaces, and for a fixed value of one
of the parameters, the process is the usual one-parameter Wiener process.
W can be heuristically obtained as a limit of Poisson processes in the plane
as the jumps get smaller and more frequent (Tuckwell & Walsh, 1983).
Early references on W and its properties, including stochastic integrals, are
ˆCenkov(1956),Park(1970),andZimmerman(1972).Additionaldescription
is given in appendix A.
2.2 Method of Simulation. There are several methods of simulation of
stochastic partial differential equations of the type
ut= Duxx+ f (u) + σw(x,t),
(2.7)
where f is a given function and w(x,t) is two-parameter white noise. We
have found that with a good choice of space and time step, a simple explicit
method works well, as has been verified by comparing with both exact an-
alytical solutions when these are known and with results of other methods,
such as implicit Crank-Nicolson-type methods (Gy¨ ongi & Nualart, 1995;
see also Chapter 8 of Tuckwell, 1988a, for the deterministic case). Suppose
the space interval is 0 ≤ x ≤ L and the time interval is 0 ≤ t ≤ T. Then put
?x = L/m and ?t = T/n, and let xi= (i − 1)?x for i = 0,1,...,m, and let
tj= (j − 1)?t for j = 0,1,...,n. Approximating u at the grid point (xi,tj)
by ui,j, the simulation proceeds by the following scheme:
ui,j=ui,j−1+ δ[ui−1,j−1− 2ui,j−1+ ui−1,j−1] + ?tf (ui,j−1)
+σ??t/?xNi,j,
where
(2.8)
δ =
D?t
(?x)2,
and where the Ni,j’s are independent standard (zero mean, unit variance)
normal random variables, which will be generated by a computer routine.
The method generally works well if δ < 0.5 and particularly well if δ ≈ 0.2.

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Stochastic Spatial Fitzhugh-Nagumo Model Neuron3007
To test the accuracy of the programs, the means and variances for the
SPDE,
ut= −u + uxx+ µ + σw(x,t),
(2.9)
where µ is a constant input, may be computed and compared with the
exact analytical solutions for the moments (Tuckwell & Walsh, 1983).
2.3 Simulation of the Original FN Model with White Noise. We are
interested first in the effects of noise on the propagation of an action poten-
tial so we consider an FN system with the original parameterization as in
equations 2.3 and 2.4 and let
I(x,t) = σ(x)w(x,t),
thatis,driftlesswhitenoisewithamplitudewhichmaydependonposition.
Sealed end conditions will usually be employed. In order to start an action
potential, we apply a current J at x = 0 for 0 ≤ t ≤ t∗. The boundary
conditions are thus
ux(0,t)= J,
ux(0,t)=0,
ux(L,t)=0,
0 < t ≤ t∗,
t > t∗,
t > 0.
For
D1uxx+ f (u,v) + σw(x,t) and vt= D2vxx+ g(u,v), we choose suitable
equilibrium values,
initialconditionsforthe generalsystemof SPDEs,
ut=
u(x,0) = u∗,v(x,0) = v∗,
0 < x < L,
where u∗and v∗satisfy
f (u∗,v∗) = 0,
g(u∗,v∗) = 0.
For the standard FN model, equations 2.3 and 2.4, these equilibrium
values are u∗= −1.1994 and v∗= −.6243, being the unique real solution of
u − u3/3 − v = 0 and 0.08(u − 0.8v + 0.7) = 0.
2.3.1 The Effect of Noise on Reliability of Transmission. A solitary wave so-
lution for equations 2.3 and 2.4 representing an action potential has a speed
of about 0.8 space units per time unit. If a space unit is chosen to represent
0.1mmandatimeunitischosenas0.04msec,thenthespeedofpropagation
is 2 meters/sec, which is close to the value for unmyelinated fibers. We