## Modeling and simulation

Computational modeling uses standard differential equations like the following

where V is the membrane potential of the neuron. The numerical simulation of such equations is however handicaped by the discontinuities during the spikes. Depending on the modeling formalism (Hodgkin-Huxley or integrate-and-fire), we employ different simulation schemes.

### Time-stepping simulation : sirene

With time-stepping (Fig. 1), time is discretized by arbitrary time-steps. Our time-stepping simulator is *Sirene*.

### Voltage-stepping and event-driven simulation : Mvaspike

With voltage-stepping (Fig. 2), V is discretized by arbitrary voltage-steps. Adaptive time-steps are implicitly defined through discretization of the voltage state-space (Zheng et al.*,* 2009).

With event-driven (Fig. 3), the spike timings are given analytically and calculated with arbitrary precision (Rochel and Martinez, 2003;Tonnelier et al., 2007).

Our event-driven simulator is *Mvaspike*. Recently, we implemented voltage-stepping as a local event-driven strategy within *Mvaspike* (Kaabi et al., 2011).

**Relevant publications**:

Kaabi M.G., Tonnelier A. and Martinez D. (2011) On the performance of voltage-stepping for the simulation of adaptive, nonlinear, integrate-and-fire neuronal networks. Neural Computation, 23, 1187-1204.

Zheng G., Tonnelier A. and Martinez D. (2009) Voltage-stepping schemes for the simulation of spiking neural networks. Journal of Computational Neuroscience, 26:3, 409-423.

Tonnelier A., Belmabrouk H. and Martinez D. (2007) Event-driven simulations of nonlinear integrate-and-fire neurons. Neural Computation, Vol. 19, No. 12, pp. 3226-3238.

Rochel O. and Martinez D. (2003) An event driven framework for the simulation of networks of spiking neurons. European Symposium on Artificial Neural Networks (ESANN), 2003, Bruges, Belgium.