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A mean field limit for networks of Integrate and Fire neurons.

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If you have a question about this talk, please contact Eavan Gleeson.

Integrate and Fire neuron models are widely used in computational neuroscience as a simpler alternative to the celebrated Hodgkin-Huxley model. Despite their simplicity of each neurons, large neural networks are very hard to analyse, both mathematically and numerically, and exhibit many complicated behaviours such as synchronisation, bifurcations and oscillations.

In the past two decades these difficulties have lead to the introduction of population density models, in which the state of the network is described by a single function satisfying a non-linear partial differential equation. Equations of this form can be easier to analyse and simulate numerically, but their rigorous derivation is mathematically challenging. I will discuss my recent work on proving a mean-field limit for this model, which to my knowledge, is the first rigorous proof of such a result for Integrate and Fire neurons.

This non-linear PDE is an example of a mean-field Vlasov equation. In this talk I will start from the basics, answering questions such as:

What is a Vlasov equation?

In what sense is the Vlasov equation a mean-field model?

How might one rigorously justify such a model?

What is an Integrate and Fire neuron?

What issues come up in the mean-field analysis of such neurons?

How can they be solved?

The talk will touch on various areas of mathematical analysis and probability, including PDEs, SDEs and empirical process theory, but will focus mainly on concepts and ideas, and these will be introduced in the talk.

This talk is part of the Cambridge Analysts' Knowledge Exchange series.

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