Analysis of Nonlinear Noisy Integrate\&Fire Neuron Models: blow-up and steady states

Details Analysis of Nonlinear Noisy Integrate\&Fire Neuron Models: blow-up and steady states Analysis of Nonlinear Noisy Integrate\&Fire Neuron Models: blow-up and steady states

2010-10-22

arxiv.org/abs/1010.4683v1

Biology

Quantitative Biology

Neurons and Cognition

Scientific paper

Nonlinear Noisy Leaky Integrate and Fire (NNLIF) models for neurons networks can be written as Fokker-Planck-Kolmogorov equations on the probability density of neurons, the main parameters in the model being the connectivity of the network and the noise. We analyse several aspects of the NNLIF model: the number of steady states, a priori estimates, blow-up issues and convergence toward equilibrium in the linear case. In particular, for excitatory networks, blow-up always occurs for initial data concentrated close to the firing potential. These results show how critical is the balance between noise and excitatory/inhibitory interactions to the connectivity parameter.

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