Table 1.

The leaky integrate-and-fire spiking neural network model. Inbetween spikes, the vectors of membrane potentials V and synaptic currents I evolve according to the free dynamics. When some neuron $n\in [1..N]$ crosses the threshold $\vartheta$, the transition condition is fulfilled, causing a spike. This leads to a reset of the membrane potential as well as post-synaptic current jumps. $W\in {\mathbb{R}}^{N\times N}$ is the weight matrix with zero diagonal and $e_n\in {\mathbb{R}}^N$ is the unit vector with a 1 at index n and 0 at all other indices. We use − and $+$ to denote quantities before and after a given spike

Free dynamics	Transition condition	Jumps at transition
$\begin{array}{cc}\hfill {\tau }_{\text{mem}}\frac{\mathrm{d}}{\mathrm{d}t}V& =-V+I\hfill \\ \hfill {\tau }_{\text{syn}}\frac{\mathrm{d}}{\mathrm{d}t}I& =-I\hfill \end{array}$	$\begin{array}{cc}\hfill {(V)}_n-\vartheta & =0\hfill \\ \hfill {(\dot{V})}_n& \ne 0\hfill \\ \hfill \text{for any}& n\hfill \end{array}$	$\begin{array}{cc}\hfill {(V^{+})}_n& =0\hfill \\ \hfill I^{+}& =I^{-}+W{e}_n\hfill \end{array}$