## On dynamics of integrate-and-fire neural networks with adaptive conductances (2008)

Venue: | Frontiers in Neuroscience |

Citations: | 21 - 11 self |

### BibTeX

@INPROCEEDINGS{Cessac08ondynamics,

author = {B. Cessac and T. Viéville},

title = {On dynamics of integrate-and-fire neural networks with adaptive conductances},

booktitle = {Frontiers in Neuroscience},

year = {2008}

}

### OpenURL

### Abstract

We present a mathematical analysis of a networks with Integrate-and-Fire neurons with conductance based synapses. Taking into account the realistic fact that the spike time is only known within some finite precision, we propose a model where spikes are effective at times multiple of a characteristic time scale δ, where δ can be arbitrary small (in particular, well beyond the numerical precision). We make a complete mathematical characterization of the model-dynamics and obtain the following results. The asymptotic dynamics is composed by finitely many stable periodic orbits, whose number and period can be arbitrary large and can diverge in a region of the synaptic weights 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 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.