Neuronal firing in the presence of stochastic trains of strong synaptic inputs

We consider a fast-slow excitable system subject to a stochastic excitatory input train, and show that under general conditions its long term behavior is captured by an irreducible Markov chain. In particular, the firing probability to each input, expected number of response failures between firings, and distribution of slow variable values between firings can be obtained analytically from the distribution of interexcitation intervals. The approach we present immediately generalizes to any pair of input trains, excitatory or inhibitory and synaptic or not, with distinct switching frequencies. We also discuss how the method can be extended to other models, such as integrate-and-fire, that feature a single variable that builds up to a threshold where an instantaneous spike and reset occur. The Markov chain analysis guarantees the existence of a limiting distribution and allows for the identification of different bifurcation events, and thus has clear advantages over direct Monte Carlo simulations. We illustrate this analysis on a model thalamocortical (TC) cell subject to two example distributions of excitatory synaptic inputs, in the cases of constant and rhythmic inhibition.

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