Noisy threshold in neuronal models: connections with the noisy leaky integrateandfire model
Abstract
Providing an analytical treatment to the stochastic feature of neurons' dynamics is one of the current biggest challenges in mathematical biology. The noisy leaky integrateandfire model and its associated FokkerPlanck equation are probably the most popular way to deal with neural variability. Another wellknown formalism is the escaperate 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 socalled agestructured system, a partial differential equation with nonlocal boundary condition famous in the field of population dynamics, where the {\it 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 agestructured model into the solution of the FokkerPlanck 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2015
 arXiv:
 arXiv:1512.03785
 Bibcode:
 2015arXiv151203785D
 Keywords:

 Quantitative Biology  Neurons and Cognition