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

The leaky integrate-and-fire model treats the membrane as a capacitance $C_m$. The leak conductance $g_L$ drives voltage towards its resting reversal $V_L$; the excitatory conductance $g_E$ drives it towards its reversal $V_E$; and $I_e$ is injected current, positive inward. A sufficiently high $V_E$ makes increasing $g_E$ depolarizing over the relevant voltage range, while also increasing total conductance and shortening the integration time.

For constant input and conductances, putThenIf this trajectory reaches $V_{\theta }$, register a spike and reset the voltage to $V_0<V_{\theta }$, optionally enforcing a specified neuronal refractory period. The reset makes repeated threshold crossings possible under a constant suprathreshold drive. If $V_{\infty }\le V_{\theta }$ and the initial voltage is below threshold, no finite-time crossing occurs; equality gives asymptotic approach. For time-varying currents or conductances, integrate the subthreshold differential equation until the next crossing.

Use this model when spike timing, spike counts and network integration matter more than the detailed waveform: it is inexpensive for large recurrent networks or comparisons of current-driven and conductance-driven input. It omits ionic spike initiation, channel-specific adaptation and many dendritic effects, for which a Hodgkin-Huxley model or a spatial model may be needed.