Mean Time-to-Fire for the Noisy LIF Neuron - A Detailed Derivation of the Siegert Formula

TL;DR

This paper provides a detailed, pedagogic derivation of the Siegert formula for the mean time-to-fire in a noisy leaky integrate-and-fire neuron modeled as an Ornstein-Uhlenbeck process, aimed at students and newcomers.

Contribution

It offers a clear, accessible derivation of the classical Siegert formula, reducing mathematical complexity for educational purposes.

Findings

Derivation of the Siegert formula for firing rate as a function of injected current

Pedagogic approach simplifies complex mathematical background

Clarification of the Ornstein-Uhlenbeck process in neuron modeling

Abstract

When stimulated by a very large number of Poisson-like presynaptic current input spikes, the temporal dynamics of the soma membrane potential $V(t)$ of a leaky integrate-and-fire (LIF) neuron is typically modeled in the diffusion limit and treated as a Ornstein-Uhlenbeck process (OUP). When the potential reaches a threshold value $\theta$, $V(t) = \theta$, the LIF neuron fires and the membrane potential is reset to a resting value, $V_0 < \theta$, and clamped to this value for a specified (non-stochastic) absolute refractory period $T_r \ge 0$, after which the cycle is repeated. The time between firings is given by the random variable $T_f = T_r+ T$ where $T$ is the random time which elapses between the "unpinning" of the membrane potential clamp and the next, subsequent firing of the neuron. The mean time-to-fire, $\widehat{T}_f = \text{E}(T_f) = T_r + \text{E}(T) = T_r + \widehat{T}$, provides a measure $\rho$ of the average firing rate of the neuron, \[ \rho = \widehat{T}_f^{-1} = \frac{1}{T_r + \widehat{T}} . \] This note briefly discusses some aspects of the OUP model and derives the Siegert formula giving the firing rate, $\rho = \rho(I_0)$ as a function of an injected current, $I_0$. This is a well-known classical result and no claim to originality is made. The derivation of the firing rate given in this report, which closely follows the derivation outlined in the textbook by Gardiner, minimizes the required mathematical background and is done in some pedagogic detail to facilitate study by graduate students and others who are new to the subject. Knowledge of the material presented in the first five chapters of Gardiner should provide an adequate background for following the derivation given in this note.