BioSystems 58 2000 59 – 65
New parameter relationships determined via stochastic ordering for spike activity in a reversal potential neural
model
Laura Sacerdote
a,
, Charles E. Smith
b
a
Department of Mathematics, Uni6ersity of Torino, Via Carlo Alberto
10
,
10123
Torino, Italy
b
NCSU Department of Statistics,
2700
Stinson Dri6e Campus Box
8203
,
110
Cox Hall, Raleigh, NC
27695
-
8203
, USA
Abstract
Purpose of this work is to study the dependence of interspike interval distribution on the model parameters when use is made of the Feller diffusion process to describe the subthreshold membrane potential of a neuron. To this aim
we make use of a new approach, namely the ordering of first passage times. The functional dependence among the model parameters e.g. membrane time constant, reversal potential, etc. resulting from the ordering criteria employed
and from the study of mean trajectory plots is analyzed into detail for four different scenario. © 2000 Elsevier Science Ireland Ltd. All rights reserved.
Keywords
:
Feller model; Stochastic order; Interspike interval distribution www.elsevier.comlocatebiosystems
1. Introduction
The Feller process has been proposed as a model of subthreshold membrane behavior by
various authors cf. Hanson and Tuckwell, 1983; La´nsky´ and La´nska´, 1987; Giorno et al., 1988;
La´nska´ et al., 1994. A realistic advantage of this model with respect to the Ornstein – Uhlenbeck
process is that the introduction of this particular state-dependent changes in depolarization con-
strains the membrane potential values to a finite interval. No procedure exists to obtain closed
form expressions for the firing distribution, so both models have been investigated in many ways.
Numerical methods have been used to study the firing times corresponding to different choices of
the parameters characterizing the two models in Ricciardi and Sacerdote, 1979; Giorno et al.,
1988 while in La´nsky´ et al. 1995 interspike interval ISI distributions obtained from the
Feller model are compared with those obtained from the Ornstein-Uhlenbeck model. However
numerical methods are of little help if one is interested in determining the functional depen-
dence
of ISI
distribution on
the model
parameters. The features of the Ornstein-Uhlenbeck model
can help the intuition if one wishes to understand
Corresponding author. E-mail addresses
:
sacerdotedm.unito.it L. Sacerdote, bmasmithstat.ncsu.edu C.E. Smith.
0956-566300 - see front matter © 2000 Elsevier Science Ireland Ltd. All rights reserved. PII: S0303-26470000107-6
the effect of a parameter value on the ISI distribu- tion. On the other hand, this approach cannot be
employed in the case of the Feller model since the complexity of the underlying mathematical process
limits to a few instances the use of this method. Indeed, as we illustrate in Section 2 by means of
some examples, the use of mean trajectory plots is useful in understanding the dependences of the
Feller model on the parameters only if spikes correspond to the so called ‘deterministic crossings’
cf. Smith, 1992.
Here we propose a new approach to the study of the parameter dependence of ISI distributions. We
make use of a theorem proved in Sacerdote and Smith, 1999 to order first passage times FPT
corresponding to different processes. In Section 3 we re-state this theorem in the particular instance
of the comparison of first passage times for two Feller processes characterized by two different set
of values of the parameters.
In Section 4 we analyze, with the help of the aforementioned theorem, the dependence of ISI
intervals on the values of the inhibitory reversal potential, the membrane spontaneous decay time
constant, the noise characterizing the model and the net excitation impinging upon the neuron.
Furthermore we determine different sets of values for the parameters corresponding to the same ISI
distribution. For example we can prove in this way that a percent change of the coefficient describing
the noise of the membrane can be balanced by a corresponding percent variation of the values of the
boundary, of the reversal potential and of the net excitation.
2. The model
