Voltage-stepping schemes for the simulation of spiking neural networks.

The numerical simulation of spiking neural networks requires particular attention. On the one hand, time-stepping methods are generic but they are prone to numerical errors and need specific treatments to deal with the discontinuities of integrate-and-fire models. On the other hand, event-driven methods are more precise but they are restricted to a limited class of neuron models. We present here a voltage-stepping scheme that combines the advantages of these two approaches and consists of a discretization of the voltage state-space. The numerical simulation is reduced to a local event-driven method that induces an implicit activity-dependent time discretization (time-steps automatically increase when the neuron is slowly varying). We show analytically that such a scheme leads to a high-order algorithm so that it accurately approximates the neuronal dynamics. The voltage-stepping method is generic and can be used to simulate any kind of neuron models. We illustrate it on nonlinear integrate-and-fire models and show that it outperforms time-stepping schemes of Runge-Kutta type in terms of simulation time and accuracy.