The Firing of an Excitable Neuron in the Presence of Stochastic Trains of Strong Synaptic Inputs

Abstract

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 with a limiting distribution. This limiting distribution allows for the analytical calculation of the system's probability of firing in response to each input, the expected number of response failures between firings, and the distribution of slow variable values between firings. Moreover, using this approach, it is possible to understand why the system will not have a stationary distribution and why Monte Carlo simulations do not converge under certain conditions. The analytical calculations involved can be performed whenever the distribution of interexcitation intervals and the recovery dynamics of the slow variable are known. The method can be extended to other models that feature a single variable that builds up to a threshold where an instantaneous spike and reset occur. We also discuss how the Markov chain analysis generalizes to any pair of input trains, excitatory or inhibitory and synaptic or not, such that the frequencies of the two trains are sufficiently different from each other. 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. The analysis shows a drastic drop in the likelihood of firing just after inhibitory onset in the case of rhythmic inhibition, relative even to the case of elevated but constant inhibition. This observation provides support for a possible mechanism for the induction of motor symptoms in Parkinson's disease and for their relief by deep brain stimulation, analyzed in Rubin and Terman (2004).

Reducing neuronal networks to discrete dynamics

“…Fast-slow analysis has been used in many studies of neuronal systems. This approach was used in [21], for example, where the analysis of the firing of a single excitable neuron, subject to stochastic input trains, was reduced to a discrete time Markov chain analysis. Finally, reproducible sequence generation in a class of excitatory-inhibitory network with random connections was studied in [16].…”

“…Here, we assume that (21) (Recall that the fixed point along lies at p A = (v A , w A ).) This assumption will be verified later (see inequality (23) or…”

“…First suppose that E i does not receive inhibitory input from an I-cell in I 0 . Then, using (21), (v i , w i ) approaches the fixed point p A and w i (η) remains in J p . Now suppose that E i does receive inhibitory input from an I-cell in I 0 .…”

Analysis of Recurrent Networks of Pulse-Coupled Noisy Neural Oscillators

“…Also, Ermentrout [18] found that the strength of electrical coupling of excitatory neurons causes the asynchronous state to lose stability via an Andronov-Hopf bifurcation. Rubin and Josic analyzed the existence and stability of a steady-state density of a single excitable two-dimensional relaxation oscillator receiving Poisson spike train inputs [48]. Recently, a pulse-coupled stochastic network of discrete state I&F-type model neurons, with enough drive for the neurons to oscillate (i.e., fire without connectivity), was analyzed by DeVille and Peskin [15].…”

Relative spike timing in stochastic oscillator networks of the Hermissenda eye

“…We based our PRC on expected spike time delays; however, investigations involving timescale separation and analysis of stochastic maps derived from biophysical neural models have been analyzed (Rubin and Josić 2007). Theoretical investigations of biophysical neural models in the strong coupling regime have reported similar linear-like PRCs (Oh and Matveev 2009;Maran and Canavier 2008;Butson and Clark 2008a,b).…”