Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 84 / 3 / d / Solution

Assume constant current, $g_L>0$, $C_m>0$, and reset below threshold. With $g_E=0$, $V_{\infty }=V_L+I_e/g_L$. A spike occurs in finite time after reset precisely when $V_{\infty }>V_{\theta }$, so the critical boundary isAt equality the voltage approaches threshold asymptotically, so the finite-time firing condition is the strict inequality $I_e>I_c$.

Solve the exponential voltage trajectory from reset to threshold. The interspike interval of this constant-input firing rate of a leaky integrate-and-fire neuron isThusThe logarithm decreases strictly with current, so $r$ increases, starting from zero as $I_e\downarrow I_c$. At large current, its argument increment is small andThe second expression states the leading linear growth, not an exact intercept formula. If an explicit refractory time $t_{\mathrm{ref}}$ is added, the rate instead becomes $1/(T+t_{\mathrm{ref}})$ and approaches $1/t_{\mathrm{ref}}$ at very high current. The printed model contains no such extra interval, which is why its unrestricted high-current rate is linear.