Ergodicity and spike rate for stochastic FitzHugh–Nagumo neural model with periodic forcing

We discuss ergodicity on a Poincaré section and estimate the average spike rate for a time periodically forced stochastic FitzHugh–Nagumo model with degenerate noise. Stochastic FitzHugh–Nagumo (SFHN) model is a prototype stochastic neural oscillator, describing the generation and propagation of action potentials or spikes in an excitable neuron at the intracellular level. Neuronal spikes play significant role in neural information coding of various nervous systems, they are described in terms of an infinitesimal probability that spikes occur, known as spike rate. Estimation of this spike rate is a subtle task for time continuous stochastic processes such as solutions of SFHN model and, in particular, time periodically forced SFHN model. Using the regularity of the ergodic periodic measure, we estimate the average spike rate in terms of the probability density of two-point motions of the membrane potential via Rice's formula.