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

\begin{equation} \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)} \end{equation}

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.