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SUMMARY:A mean field limit for networks of Integrate and Fire neurons. - T
 om Holding (CCA)
DTSTART:20141029T160000Z
DTEND:20141029T170000Z
UID:TALK55354@talks.cam.ac.uk
CONTACT:Eavan Gleeson
DESCRIPTION:Integrate and Fire neuron models are widely used in computatio
 nal neuroscience as a simpler alternative to the celebrated Hodgkin-Huxley
  model. Despite their simplicity of each neurons\, large neural networks a
 re very hard to analyse\, both mathematically and numerically\, and exhibi
 t many complicated behaviours such as synchronisation\, bifurcations and o
 scillations.\n\nIn the past two decades these difficulties have lead to th
 e introduction of population density models\, in which the state of the ne
 twork is described by a single function satisfying a non-linear partial di
 fferential equation. Equations of this form can be easier to analyse and s
 imulate numerically\, but their rigorous derivation is mathematically chal
 lenging. I will discuss my recent work on proving a mean-field limit for t
 his model\, which to my knowledge\, is the first rigorous proof of such a 
 result for Integrate and Fire neurons.\n\nThis non-linear PDE is an exampl
 e of a mean-field Vlasov equation. In this talk I will start from the basi
 cs\, answering questions such as:\n\nWhat is a Vlasov equation?\n\nIn what
  sense is the Vlasov equation a mean-field model?\n\nHow might one rigorou
 sly justify such a model?\n\nWhat is an Integrate and Fire neuron?\n\nWhat
  issues come up in the mean-field analysis of such neurons?\n\nHow can the
 y be solved?\n\nThe talk will touch on various areas of mathematical analy
 sis and probability\, including PDEs\, SDEs and empirical process theory\,
  but will focus mainly on concepts and ideas\, and these will be introduce
 d in the talk.
LOCATION:MR14\, Centre for Mathematical Sciences
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