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On Dynamics of Integrate-and-Fire Neural Networks with Adaptive Conductances

Bruno Cessac 1, 2, 3 Thierry Viéville 4
1 ODYSSEE - Computer and biological vision
DI-ENS - Département d'informatique de l'École normale supérieure, CRISAM - Inria Sophia Antipolis - Méditerranée , ENS Paris - École normale supérieure - Paris, Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech
4 CORTEX - Neuromimetic intelligence
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
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, which 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.
Complete list of metadatas

https://hal.inria.fr/inria-00338369
Contributor : Thierry Viéville <>
Submitted on : Wednesday, November 12, 2008 - 10:42:28 PM
Last modification on : Tuesday, September 22, 2020 - 3:47:51 AM

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  • HAL Id : inria-00338369, version 1

Citation

Bruno Cessac, Thierry Viéville. On Dynamics of Integrate-and-Fire Neural Networks with Adaptive Conductances. Frontiers in Neuroscience, Frontiers, 2008, 2 (2). ⟨inria-00338369⟩

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