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 givesThe 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 $T(I)+t_{\mathrm{ref}}$ and makes the firing rate saturate rather than grow without bound.