Quasi-static approximation of the interspike interval distribution of neurons driven by time-dependent inputs

TL;DR

This paper derives a closed-form expression for the inter-spike interval distribution of neurons driven by slowly varying inputs, enhancing understanding of neural variability under realistic, time-dependent stimuli.

Contribution

It provides a novel analytical solution for the inter-spike interval distribution in quasi-static conditions, applicable to integrate-and-fire neurons with time-dependent inputs.

Findings

Quasi-static distribution accurately describes neural response for slow stimuli.

Derived expression matches numerical simulations across various input currents.

Results extend to other stochastic systems with explicit first passage time solutions.

Abstract

Variability in neural responses is an ubiquitous phenomenon in neurons, usually modeled with stochastic differential equations. In particular, stochastic integrate-and-fire models are widely used to simplify theoretical studies. The statistical properties of the generated spikes depend on the stimulating input current. Given that real sensory neurons are driven by time-dependent signals, here we study how the inter-spike interval distribution of integrate-and-fire neurons depends on the evolution of the stimulus, in a quasi-static limit. We obtain a closed-form expression for this distribution, and we compare it to the one obtained with numerical simulations for several time-dependent currents. For slow inputs, the quasi-static distribution provides a very good description of the data. The results obtained for the integrate-and-fire model can be extended to other non-autonomous stochastic systems where the first passage time problem has an explicit solution.