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“ Population density for an excitatory network of integrate-and-fire neurons ”

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November 15, 2012 14:00 - 15:30


In this talk we will consider a population of excitatory neurons where all the cells of the network follow the integrate-and-fire model. Each neuron of the population is assumed to be stochastically driven by an independent Poisson spike train and the synaptic interaction between neurons is modeled by a potential jump at the reception of an action potential. The point of view is given by the population density approach that has been introduced more than a decade ago in order to facilitate the simulation of a large assembly of neurons, see [1] and [2]. More precisely, this approach gives a partial differential equation that describes the density of neurons in the state space that is the set of all admissible potential of a neuron. First of all, we will give a mathematical framework for the equation that arises from this the population density approach. Then we will discuss the existence and the possible blow up in finite time of the solution [3] and [4]. Interpreting the blow up of the solution as the presence of a Dirac mass in the firing rate of the population, we will relate the blow up of the solution to the occurrence of synchronization of neurons. We will discuss how the consideration of more realistic modelling assumptions, as the refractory period and the delay between the emission and the reception of an action potential can stop the blow up of the solution and give a well posed model. But, as we will see, with this additional features the model will generate some periodical solutions that can also be related to the synchronization of the population.

[1] Wulfram Gerstner and Werner M. Kistler, Spiking Neuron Models Single Neurons, Pop-ulations, Plasticity Cambridge University Press, 2002 [2] Omurtag, A., Knight, B.W. and Sirovich, L. On the simulation of large populations of neurons J. Comp. Neurosci., 8:51-63, 2000 [3] Dumont Gregory, Henry Jacques, Population density models of integrate-and-fire neurons with jumps, Well-posedness. Journal of Mathematical Biology, 2012 [4] Dumont Gregory, Henry Jacques, Synchronisation of an excitatory integrate-and-fire neural network, submitted.

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BSI Private Event
Taro Toyoizumi [Taro Toyoizumi, Neural Computation and Adaptation ]