ON THE FIRING ACTIVITY OF A STATE-DEPENDENT STOCHASTIC NEURONAL MODEL

Abstract

We address the problem of studying the number of excitatory stimuli producing a spike for a Stein-type stochastic model which includes a multiplicative state-dependent effect. Some results on the probability distribution of the number M of excitatory stimuli triggering a spike are obtained. We also evaluate the distribution of M conditional on the firing time, and disclose some properties of its mode. Finally, some results on the probability generating function of the conditional distribution are given. 1 Introduction A mathematical model based on Stein stochastic differential equation has been recently proposed in (5) to describe the firing activity of a neuronal unit. Similarly to Stein approach (15), such model includes the arrival of excitatory stimuli according to a Poisson process with an exponential decay in absence of stimuli. As an innovative feature, the model is characterized by a multiplicative state-dependent effect, since the depolariza- tions are random and depend on the voltage level at the stimulus time. This state-dependent effect is suitable to describe a behavior sometimes observed in neuronal dynamics (see, for instance, (14) where noisy stochastic conductance components are multiplicatively coupled to the membrane potential). Closed-form expressions for the distribution of the membrane potential level and for the firing density (in the case of homogeneous Poisson inputs) have been recently obtained in (5) for our model. The state-dependent Stein-type model is considered here to study the distribution of the number of excitatory stimuli able to yield a neuronal firing. This provides a basic characterization of the relationship between environmental stimuli and neural response. For instance, it is well-known that spontaneous quantal transmitter release from the motor nerve endings follows the Poisson distribution and generates miniature potentials which contribute to the neuron postsynaptic potential (7). The study performed in the present paper can be useful to face the question of what determines the post-synaptic response to the release of the quantal transmitter (see, for instance, (6) where an analysis of how retinal ganglion cells re-encode the received information is present). More generally, the obtained results are suitable to characterize the firing activity of a postsynaptic cell in networks of Stein-type neuronal units under ad hoc assumptions about the dynamics of presynaptic neurons (see (2), (3) and (4) where a simulation-based approach is used to investigate the synchronization between interacting neurons). The mathematical background of the present contribution is that of classical treatises on stochastic neuronal models, such as (12) and (13).