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When stimulated by a very large number of Poisson-like presynaptic currentinput spikes, the temporal dynamics of the soma membrane potential $V(t)$ of aleaky integrate-and-fire (LIF) neuron is typically modeled in the diffusionlimit and treated as a Ornstein-Uhlenbeck process (OUP). When the potentialreaches a threshold value $\theta$, $V(t) = \theta$, the LIF neuron fires andthe membrane potential is reset to a resting value, $V_0 < \theta$, and clampedto 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 bythe random variable $T_f = T_r+ T$ where $T$ is the random time which elapsesbetween 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 ofthe 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-knownclassical result and no claim to originality is made. The derivation of thefiring rate given in this report, which closely follows the derivation outlinedin the textbook by Gardiner, minimizes the required mathematical background andis done in some pedagogic detail to facilitate study by graduate students andothers who are new to the subject. Knowledge of the material presented in thefirst five chapters of Gardiner should provide an adequate background forfollowing the derivation given in this note.

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