# Leaky Integrate-And-Fire

A Leaky Integrate-and-Fire neuron at layer $$l$$ and index $$i$$ can be described in differential form as:

$$\label{eq:lif} \tau_{\mathrm{mem}} \frac{\mathrm{d} U_{i}^{(l)}}{\mathrm{d} t}=-\left(U_{i}^{(l)}-U_{\mathrm{rest}}\right)+R I_{i}^{(l)}$$

where the terms denote:

$$U_{i}(t)$$
membrane potential
$$U_{\text{rest}}$$
resting potential
$$\tau_{\text{mem}}$$
membrane time constant
$$R$$
input resistance
$$i_{i}(t)$$
input current

$$U_{i}$$ acts as a leaky integrator of the input current $$I_{i}$$. Neurons emit spikes when the membrane voltage reaches firing threshold $$\theta$$, and resets to resting potential $$U_{\text{\rest}}$$.

Equation eq:lif only describes the dynamics of a LIF neuron sub-threshold.

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