= 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, put
$$
\tau_{\rm eff}=\frac{C_m}{g_L+g_E},\qquad V_\infty=\frac{g_LV_L+g_EV_E+I_e}{g_L+g_E}.
$$
Then
$$
\boxed{V(t)=V_\infty+[V(0)-V_\infty]e^{-t/\tau_{\rm eff}}}.
$$
If 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 $V_\infty\leq V_\theta$ 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.
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