Chapter 21 - A Heuristic Approach to Stochastic Models of Single Neurons

Abstract

This chapter discusses heuristic approximation methods for the firing times obtained from several stochastic one-compartment or lumped models of a neuron. The first-passage times through four neural models include Ornstein–Uhlenbeck process, Stein's model, Stein's model with a reversal potential, and stochastic after-hyperpolarization model. The models considered in the method are of the voltage threshold type; specifically, when the voltage across the cell membrane at a particular spatial location, called the trigger zone, exceeds a voltage threshold, then a brief electrical pulse known as an action potential or spike is generated. Noise is considered to be unwanted in a system and to be eliminated by some appropriate averaging or filtering. The rate parameter of the limiting exponential distribution has been obtained in three ways for the Ornstein–Uhlenbeck process and interaction among the three approaches—asymptotic series in Seigert's approach, perturbation of dynamic systems via the Wentzell–Freidlin method, and Poisson nature of up-crossings from extreme value theory—appears to be a rich area for gaining insight.