Modelling populations of integrate and fire neurons: a Fokker-Planck approach to population density dynamics

Much of the phenomenology of interest in the field of neuroscience arises from the interaction of large populations of densely interconnected neurons (~\(10^5\) neurons per mm³ of mammal cortex, averaging \(10^4\) connections per neuron). Different levels of abstraction may be adopted when modelling such systems, and these need to be well suited with regards to the phenomena one is interested in studying. Here we aim at the study of the (sparse) synchronization of neurons observed during electrophysiologically recorded fast oscillatory behavior of networks of large populations. For that we use a ubiquitous simplified neuronal model - the conductance based leaky integrate and fire neuron. This model may be described by a one dimensional stochastic differential equation. Under mean field assumptions we may describe, using a linear Fokker-Planck equation, the behavior of a single population of uncoupled neurons with a PDE. The coupling of different populations will render this Fokker-Planck equation strongly non-linear. In this presentation I will also explore some details of such modelling approaches, namely: the non-natural boundary conditions generated by the neuronal firing mechanism and the numerical scheme used to deal with the brittleness from there ensued. I will also present results on the types of behavior, data, and statistics that such modelling approach is able to predict, e.g. neuronal (a)synchrony, neuronal input currents, firing rates, inter spike intervals, etc... This type of approach allows for a computationally tractable and scalable study of networks of populations of neurons. In the future we plan to implement parameter estimation algorithms to this family of models.