tures. The literature on this subject is too vast to be exhaustively recalled here. A comprehensive
review including an outline of the appropriate mathematical techniques can be found in Riccia-
rdi 1995 and in Ricciardi et al. 1999 and references therein.
Among the proposed neuronal diffusion models the most famous is undoubtedly the so called
Ornstein – Uhlenbeck OU model. This includes the presence of the exponential decay of the neu-
ron’s membrane potential that occurs between successive postsynaptic potentials — missing in
Gernstein – Mandelbrot model — at the price, however, of a great increase of analytical com-
plexity. As a consequence, related FPT problems within such a model must be approached by a
variety of subtle techniques and, in general, the firing pdf is only obtained as the numerical solu-
tion of a second kind Volterra integral equation by
implementation of
a fast-converging
al- gorithms developed by some of us Buonocore et
al., 1987. As is well-known, diffusion neuronal models
rest on the strong Markov assumption, which allows one to use various analytic methods for the
FPT pdf evaluation that models the neuronal firing pdf. However, it is conceivable that, partic-
ularly if a neuron is subject to strongly correlated inputs, the Markov assumption may turn out to
be inappropriate, so that models based on non- Markov stochastic processes ought to be consid-
ered. Some attempts along such directions may be found in Di Nardo et al. 1998a. In the present
paper, we assume that the processes underlying the neuron’s model belong to the rather general
Gauss – Markov class. For such processes, a new computationally simple, speedy and accurate
method has been developed to obtain FPT pdf’s through time-dependent boundaries Di Nardo et
al., 1998b. Here, we report some results obtained by use of such a method to evaluate FPT pdf’s
and
cumulative distributions
for a
Gauss – Markov neuronal model — that will be further
investigated in the future — representing a time- inhomogeneous generalization of OU model, in-
cluding a time-dependent external input. Such results are compared with the corresponding ones
for the OU model to pinpoint qualitative and quantitative diversities.
2. Neuronal models
Consistently with a frequent approach in theo- retical neurobiology literature, here we assume
that the neuron’s membrane potential is described by a continuous scalar stochastic process Xt
that represents changes in the membrane potential between two consecutive neuronal firings. The
threshold potential, denoted by S = St, with St
\ x , is assumed to be a deterministic func-
tion of time. Our interest focuses on the proper- ties of the FPT random variable
T = inf{t ] t :Xt \ St},
with Xt = x
B St
. Indeed, T is suitable to describe neuronal interspike intervals. The recip-
rocal relationship between the firing frequency and the interspike interval naturally leads to the
problem of determining the pdf of T, namely the firing probability density
g[St, t x
, t ] =
t P{T 5 t}.
When this function cannot be obtained analyti- cally which is almost the rule, the analysis re-
sorts to
methods based
on computational
approximations, or on asymptotic estimates of the FPT pdf, or on the simulation of the sample paths
of Xt.
2
.
1
. Diffusion approach A customary assumption is that process Xt is
a regular diffusion process; i.e. a Markovian pro- cess generated by the stochastic differential
equation
dXt = A
1
[Xt, t]dt + A
2
[Xt, t]dBt, 1
where Bt is the standard Brownian motion and A
1
x, t and A
2
x, t are real-valued functions of their arguments satisfying certain regularity con-
ditions cf., for instance, Karlin and Taylor, 1981. It can then be seen that the pdf of Xt
conditional on Xt = x
, given by fx, t
x , t
= x
P{Xt 5 x Xt
= x }
satisfies the Fokker – Planck equation
f t
= − x
[A
1
x, tf ] + 1
2
2
x
2
[A
2
x, tf ] 2
where A
1
x, t and A
2
x, t are the ‘drift’ and ‘infinitesimal variance’ of the process defined as
A
k
x, t = lim
t ¡0
1 t
E{[Xt + t − Xt]
k
Xt=x} for k = 1, 2. To determine f via the diffusion
equation the following initial condition must be considered:
lim
t ¡t
fx, t x
, t = dx − x
. However, as the delta initial condition is not
always sufficient to determine uniquely the transi- tion pdf, suitable boundary conditions may have
to be imposed cf., for instance, Karlin and Tay- lor, 1981.
The neuronal models based on the applications of diffusion processes are predominantly time-ho-
mogeneous and thus the infinitesimal moments A
1
and A
2
do not depend on t. Among the time-ho- mogeneous diffusion models, ranks the above re-
called OU model, characterized by infinitesimal moments
A
1
x = − 1
q x − r,
A
2
= s
2
. 3
A diffusion approximation leading to this model and starting from a neuronal model char-
acterized by an arbitrary number of excitatory and inhibitory inputs is given in Ricciardi 1976.
In the absence of randomness i.e. when s = 0, for the OU model one has
dx dt
= − 1
q x − r,
xt = x
. 4
This equation expresses the spontaneous expo- nential decay of the membrane potential towards
the resting potential r. Hence, q can be viewed as the time-constant of the neuron’s membrane ap-
proximately 5 ms.
The transition pdf for the OU model with infinitesimal moments 3, is obtained by solving
the Fokker – Planck equation with the delta initial condition. One thus finds that at each time t \ t
the transition pdf fx, t x
, t is normal with
conditional mean Mt
t = r[1 − e
− t − t
q
] + x e
− t − t
q
and conditional variance Vt
t =
s
2
q 2
[1 − e
− 2t − t
q
]. We note that the conditional mean Mt
t identifies with the solution of deterministic Eq.
4. Unfortunately, even though the transition pdf
is easily obtained, there is a lack of closed form solutions to the firing density for the OU neuronal
model, as well as for other diffusion neuronal models. Hence, aiming to obtain accurate numeri-
cal evaluations for the FPT pdf g in the general case of time-varying thresholds and of arbitrary
time-homogeneous
one-dimensional diffusion
models viz. not necessarily of the OU type, an efficient procedure has been developed in Buono-
core et al. 1987, Giorno et al. 1989. This is based on the numerical solution of the following
second-kind Volterra integral equation:
g[St,t x
, t ]
= − 2c[St,t
x , t
] +
2
t t
g[St,t x
, t ]c[St,t
St,t] dt, x
B St
5 where St is a sufficiently smooth threshold func-
tion and C
[St, t y, t]
= A
2
[St] 2
fx, t y, t
x
x = St
+ 1
2 dSt
dt −
A
1
x 2
+ 3
8 dA
2
x dx
n
x = St
× f[St, t
y, t]. Figs. 1 and 2, obtained by means of such
numerical procedure, show the firing pdf’s for OU model when t
= 0, x
= r = −
70 and s
2
= 2,
with suitable choices of q and of the constant threshold S.
2
.
2
. Gaussian approach By analogy with OU model, we shall now refer
to correlated Gaussian neuronal models, the Gaussian nature being conceivably the result of
the superposition of a very large number of synaptic inputs, as indicated in Ricciardi 1976.
Along such direction, in Kostyukov 1978, Kostyukov et al. 1981 the problem of single
neuron modeling was approached by restricting the analysis to Gaussian stationary processes. In
the present paper, we shall refer to Gaussian processes as well. However, we shall not require
stationarity, but will instead assume that their nature is Markov.
To approach our theme, let us recall some necessary notions on Gauss – Markov processes.
Let {Xt, t I}, where I is a continuous parame- ter set, be a real continuous Gauss – Markov pro-
cess with the following properties:
1. mt E[Xt] is continuous in I; 2. the covariance
cs, t E{[Xs − ms][Xt − mt]} is continuous in I × I;
3. Xt is non-singular except possibly at the end points of I, where it could be equal to mt
with probability one. We recall that a Gaussian process is Markov if
and only if its covariance satisfies cs, u =
cs, tct, u ct, t
6 for all s 5 t 5 u, with s, t, u belonging to I.
Well-behaved solutions of Eq. 6 are of the form
cs, t = h
1
sh
2
t, s 5 t,
7 where
rt h
1
t h
2
t 8
is a monotonically increasing function by virtue of the Cauchy – Schwarz inequality, and h
1
th
2
t \ 0 because of the assumed non-singularity of the
process on I. Any Gaussian process with covari- ance as in Eq. 7 can be represented in terms of
the standard Wiener process {Wt, t ] 0} as
Xt = mt + h
2
tW[rt], 9
and is therefore Markov. The conditional pdf fx, t
x , t
of a Gauss – Markov process is a normal density characterized,
respectively, by conditional mean and variance Mt
t = mt +
h
2
t h
2
t [x
− mt
] Vt
t = h
2
t h
1
t − h
2
t h
2
t h
1
t
n
, 10
with t, t I, t
B t. It satisfies equation of type Eq.
2 with A
1
x, t and A
2
x, t respectively, given by
A
1
x, t = mt + [x − mt] h
2
t h
2
t A
2
x, t = h
2 2
trt. 11
Nevertheless, in general corresponding FPT densities are not analytically obtainable. How-
ever, in analogy with the result originally pro-
Fig. 1. Plot of the firing pdf for the OU model with t =
0, x
= r = −
70, s
2
= 2 and S = − 60, and with q = 5 bottom
and q = 10 top.
Fig. 2. Same as in Fig. 1, for S = − 65. In the case q = 5 the firing pdf exhibits the lower peak.
posed for time-homogeneous diffusion processes and mentioned in the previous subsection, in a
previous paper cf. Di Nardo et al., 1998b it has been proved that the FPT density of a Gauss –
Markov process can be obtained by solving the simple, non-singular, Volterra second kind inte-
gral Eq. 5, with St C
1
[t , and
c [St, t
y, t]= St − mt
2 −
St − mt 2
h
1
th
2
t − h
2
th
1
t h
1
th
2
t − h
2
th
1
t −
y − mt 2
h
2
th
1
t − h
2
th
1
t h
1
th
2
t − h
2
th
1
t ×
f[St, t y, t].
By making use of this result, in Di Nardo et al. 1998b an efficient numerical procedure based on
a repeated Simpson’s rule has been proposed to evaluate
FPT densities
of Gauss – Markov
processes. In the following, we present a special non-sta-
tionary Gauss – Markov neuronal model and make use of such numerical procedure to analyze
the corresponding firing pdf’s.
3. A special Gauss – Markov neuronal model
