BioSystems 58 2000 19 – 26
On some computational results for single neurons’ activity modeling
A. Di Crescenzo
a
, E. Di Nardo
a
, A.G. Nobile
b
, E. Pirozzi
c
, L.M. Ricciardi
d,
a
Dipartimento di Matematica, Uni6ersita` della Basilicata, Potenza, Italy
b
Dipartimento di Matematicae Informatica, Uni6ersita` di Salerno, Baronissi, SA, Italy
c
Dipartimento di Informatica, Matematica, Elettronica e Trasporti, Uni6ersita` di Reggio Calabria, Reggio Calabria, Italy
d
Dipartimento di Matematica e Applicazioni, Uni6ersita` di Napoli
‘
Federico II
’
, Via Cintia,
80126
Naples, Italy
Abstract
The classical Ornstein – Uhlenbeck diffusion neuronal model is generalized by inclusion of a time-dependent input whose strength exponentially decreases in time. The behavior of the membrane potential is consequently seen to be
modeled by a process whose mean and covariance classify it as Gaussian – Markov. The effect of the input on the neuron’s firing characteristics is investigated by comparing the firing probability densities and distributions for such
a process with the corresponding ones of the Ornstein – Uhlenbeck model. All numerical results are obtained by implementation of a recently developed computational method. © 2000 Elsevier Science Ireland Ltd. All rights
reserved.
Keywords
:
Firing distribution; Diffusion; Gauss – Markov model www.elsevier.comlocatebiosystems
1. Introduction
Within the framework of the study of single neuron’s activity a privileged role has historically
been played by mathematical models based on continuous Markov processes, better known as
‘diffusion processes’. The theoretical description of the firing process has thus been viewed as a
first passage time problem FPT through a time- dependent threshold function, by generally assum-
ing that the membrane potential after each attainment of the threshold value is reset to a
unique well-specified
value. Here,
we limit
ourselves to pointing out that over 35 years have elapsed since the beginning of the history of neu-
ronal models based on diffusion processes cf. Gerstein and Mandelbrot, 1964. Indeed, it was
then shown that for some intracellular recording the interspike intervals histograms could be fitted
to an excellent degree of approximation by means of the first passage time probability density func-
tion pdf of a Wiener process. Ever since, alterna- tive
stochastic diffusion
models have
been proposed in the literature, aiming at refinements
and embodiments of other neurophysiological fea-
Corresponding author. Fax: + 39-081-675665. E-mail address
:
luigi.ricciardiunina.it L.M. Ricciardi. 0303-264700 - see front matter © 2000 Elsevier Science Ireland Ltd. All rights reserved.
PII: S 0 3 0 3 - 2 6 4 7 0 0 0 0 1 0 2 - 7
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
