Slides for second SIAM talk

Here are the slides for my SIAM talk on generalizing the Wilson-Cowan equations to include correlations.  This talk was mostly on the paper with Michael Buice and Jack Cowan that I summarized  here.  However, I also contrasted our work with the recent work of Paul Bressloff who uses a system size expansion of the Markov process that Michael and Jack proposed as a microscopic model for Wilson-Cowan in their 2007 paper.  The difference between the two approaches stems from  the interpretation of what the Wilson-Cowan equation describes.  In our interpretation, the Wilson-Cowan equation describes the firing rate or stochastic intensity of a Poisson process.  A Poisson distribution is notable because all cumulants are equal to the mean.  Our expansion is  in terms of factorial cumulants (we called them normal ordered cumulants in the paper because we didn’t know there was a name for them), which are deviations from Poisson statistics.  Bressloff, on the other hand, considers the Wilson -Cowan equation to be the average population firing rate of  a large population of neurons.  In the infinite size limit, there are no fluctuations.  His expansion is in terms of regular cumulants and the inverse system size is the small parameter.  In our formulation, the expansion parameter  is related to the distance to a critical point where the expansion would break down.   In essence, we use a Bogoliubov  hierarchy of time scales expansion where the higher order  factorial cumulants decay to steady state much faster than the lower order ones.