= Constant-input firing rate of a leaky integrate-and-fire neuron
For $C_m\dot V=-g_L(V-V_L)+I$, reset $V_0<V_\theta$ and no explicit refractory interval, define $I_c=g_L(V_\theta-V_L)$ and $\tau_m=C_m/g_L$. Constant current gives
$$
r(I)=\begin{cases}0,&I\leq I_c,\\\bigl[\tau_m\log((I-g_L(V_0-V_L))/(I-I_c))\bigr]^{-1},&I>I_c.\end{cases}
$$
The threshold is approached only asymptotically at $I=I_c$. At large current, $r\sim I/[C_m(V_\theta-V_0)]$. Adding an absolute <neuronal refractory period> changes the denominator to $T(I)+t_{\rm ref}$ and makes the <firing rate> saturate rather than grow without bound.
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