A first-passage-time analysis of the periodically forced noisy leaky integrate-and-fire model

Abstract

We present a general method for the analysis of the discharge trains of periodically forced noisy leaky integrate-and-fire neuron models. This approach relies on the iterations of a stochastic phase transition operator that generalizes the phase transition function used for the study of periodically forced deterministic oscillators to noisy systems. The kernel of this operator is defined in terms of the the first passage time probability density function of the Ornstein Uhlenbeck process through a suitable threshold. Numerically, it is computed as the solution of a singular integral equation. It is shown that, for the noisy system, quantities such as the phase distribution (cycle histogram), the interspike interval distribution, the autocorrelation function of the intervals, the autocorrelogram and the power spectrum density of the spike train, as well as the input-output cross-correlation and cross-spectral density can all be computed using the stochastic phase transition operator. A detailed description of the numerical implementation of the method, together with examples, is provided.

A phase transition in the first passage of a Brownian process through a fluctuating boundary with implications for neural coding

“…Finding the very first such time, τ ¼ infft j W ðtÞ ¼ 1g; known as the "first passage" of the process through the boundary B = 1, is easier said than done, one of those classical problems whose concise statements conceal their difficulty (1-4). For general fluctuating random processes, the first-passage time problem is both extremely difficult (5-9) and highly relevant, due to its manifold practical applications: it models phenomena as diverse as the onset of chemical reactions (10)(11)(12)(13)(14), transitions of macromolecular assemblies (15)(16)(17)(18)(19), time-to-failure of a device (20)(21)(22), accumulation of evidence in neural decision-making circuits (23), the "gambler's ruin" problem in game theory (24), species extinction probabilities in ecology (25), survival probabilities of patients and disease progression (26)(27)(28), triggering of orders in the stock market (29)(30)(31), and firing of neural action potentials (32)(33)(34)(35)(36)(37).…”

A result on the first-passage time of an Ornstein–Uhlenbeck process

“…Bulsara et al, 1996;Lansky, Sacerdote and Tomasetti, 1995;Ricciardi and Sacerdote, 1979;Shimokawa et al, 2000;Tuckwell, Wan and Rospars, 2002), in survival analysis the model has been applied by Aalen and Gjessing (2004), and also in mathematical finance it has found applications (see e.g. Jeanblanc and Rutkowski, 2000;Leblanc and Scaillet, 1998;Linetsky, 2004).…”

Neuron firing in driven nonlinear integrate-and-fire models

“…In the case of periodic driving, the waiting time density and the first passage time density depend on the phase of the driving force at the starting time. In order to find the interspike interval density the first passage time density has to be averaged over the starting times resulting from firing times [20][21][22] and refractory periods. If the refractory periods last only short time compared to the typical interspike interval times the density of firing times is given by the firing rate [23].…”