Dynamics of networks of leaky-integrate and fire neurons

The dynamics of pulse-coupled leaky-integrate-and-fire neurons is dis- cussed in networks with arbitrary structure and in the presence of delayed inter- actions. The evolution equations are formally recasted as an event-driven map in a general context, where the pulses are assumed to have a finite width. The final structure of the mathematical model is simple enough to allow for an easy imple- mentation of standard nonlinear dynamics tools. We also discuss the properties of the transient dynamics in the presence of quenched disorder (and _-like pulses). We find that the length of the transient depends strongly on the number N of neurons. It can be as long as 106-107 inter-spike intervals for relatively small networks, but it decreases upon increasing N because of the presence of stable clustered states. Finally, we discuss the same problem in the presence of randomly uctuating synap- tic connections (annealed disorder). The stationary state turns out to be strongly affected by finite-size corrections, to the extent that the number of clusters depends on the network size even for N 20, 000.