%0 Journal Article
%T Statistics of spike trains in conductance-based neural networks: Rigorous results
%A Bruno Cessac
%J The Journal of Mathematical Neuroscience
%D 2011
%I Springer
%R 10.1186/2190-8567-1-8
%X Neural networks have an overwhelming complexity. While an isolated neuron can exhibit a wide variety of responses to stimuli [1], from regular spiking to chaos [2,3], neurons coupled in a network via synapses (electrical or chemical) may show an even wider variety of collective dynamics [4] resulting from the conjunction of non-linear effects, time propagation delays, synaptic noise, synaptic plasticity, and external stimuli [5]. Focusing on the action potentials, this complexity is manifested by drastic changes in the spikes activity, for instance when switching from spontaneous to evoked activity (see for example A. Riehle¡¯s team experiments on the monkey motor cortex [6-9]). However, beyond this, complexity may exist some hidden laws ruling an (hypothetical) ¡°neural code¡± [10].One way of unraveling these hidden laws is to seek some regularities or reproducibility in the statistics of spikes. While early investigations on spiking activities were focusing on firing rates where neurons are considered as independent sources, researchers concentrated more recently on collective statistical indicators such as pairwise correlations. Thorough experiments in the retina [11,12] as well as in the parietal cat cortex [13] suggested that such correlations are crucial for understanding spiking activity. Those conclusions where obtained using the maximal entropy principle[14]. Assume that the average value of observables quantities (e.g., firing rate or spike correlations) has been measured. Those average values constitute constraints for the statistical model. In the maximal entropy principle, assuming stationarity, one looks for the probability distribution which maximizes the statistical entropy given those constraints. This leads to a (time-translation invariant) Gibbs distribution. In particular, fixing firing rates and the probability of pairwise coincidences of spikes lead to a Gibbs distribution having the same form as the Ising model. This idea has been introduced by S
%U http://www.mathematical-neuroscience.com/content/1/1/8