## Are neuronal networks that vicious ? or only their models ? Neural computation (2007)

Citations: | 2 - 1 self |

### BibTeX

@MISC{Cessac07areneuronal,

author = {B. Cessac and T. Viéville},

title = {Are neuronal networks that vicious ? or only their models ? Neural computation},

year = {2007}

}

### OpenURL

### Abstract

We present a mathematical analysis of a network with integrate and fire neurons, taking into account the realistic fact that the spike time is only known within some finite precision. This leads us to propose a model where spikes are effective at times multiple of a characteristic time scale δ, where δ can be mathematically arbitrary small. We make a complete mathematical characterization of the model-dynamics for conductance based integrate and fire models. We obtain the following results. The asymptotic dynamics is composed by finitely many periodic orbits, whose number and period can be arbitrary large and diverge in a region of the parameters space, traditionally called the “edge of chaos”, a notion mathematically well defined in the present paper. Furthermore, except at the edge of chaos, there is a one-to-one correspondence between the membrane potential trajectories and the spikes raster plot. This shows that the neural code is entirely “in the spikes ” in this case. As a key tool, we introduce an order parameter, easy to compute numerically, and closely related to a natural notion of entropy providing a relevant characterization of the computational capabilities of the network. This allows us to compare the computational capabilities of leaky and integrate and fire models and conductance based models. The present study considers networks with constant input, and without time-dependent plasticity, but the framework has been designed for both extensions.