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Noisy threshold in neuronal models: connections with the noisy leaky integrate-and-fire model

Gregory Dumont 1 Jacques Henry 2, 3 Carmen Oana Tarniceriu 4
1 Group for Neural Theory [Paris]
LNC - Laboratoire de Neurosciences cognitives, IEC - Institut d'étude de la cognition
3 CARMEN - Modélisation et calculs pour l'électrophysiologie cardiaque
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest, IHU-LIRYC
Abstract : Providing an analytical treatment to the stochastic feature of neu-rons' dynamics is one of the current biggest challenges in mathematical biology. The noisy leaky integrate-and-fire model and its associated Fokker-Planck equation are probably the most popular way to deal with neural variability. Another well-known formalism is the escape-rate model: a model giving the probability that a neuron fires at a certain time knowing the time elapsed since its last action potential. This model leads to a so-called age-structured system, a partial differential equation with non-local boundary condition famous in the field of population dynamics, where the age of a neuron is the amount of time passed by since its previous spike. In this theoretical paper, we investigate the mathematical connection between the two formalisms. We shall derive an integral transform of the solution to the age-structured model into the solution of the Fokker-Planck equation. This integral transform highlights the link between the two stochastic processes. As far as we know, an explicit mathematical correspondence between the two solutions has not been introduced until now.
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Submitted on : Tuesday, December 13, 2016 - 6:38:22 PM
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Gregory Dumont, Jacques Henry, Carmen Oana Tarniceriu. Noisy threshold in neuronal models: connections with the noisy leaky integrate-and-fire model. Journal of Mathematical Biology, Springer Verlag (Germany), 2016, 73, pp.1413 - 1436. ⟨10.1007/s00285-016-1002-8⟩. ⟨hal-01414588⟩

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