Michael Buice and I have just published a review paper of our work on how to go beyond mean field theory for systems of coupled neurons. The paper can be obtained here. Michael and I actually pursued two lines of thought on how to go beyond mean field theory and we show how the two are related in this review. The first line started in trying to understand how to create a dynamic statistical theory of a high dimensional fully deterministic system. We first applied the method to the Kuramoto system of coupled oscillators but the formalism could apply to any system. Our recent paper in PLoS Computational Biology was an application for a network of synaptically coupled spiking neurons. I’ve written about this work multiple times (e.g. here, here, and here). In this series of papers, we looked at how you can compute fluctuations around the infinite system size limit, which defines mean field theory for the system, when you have a finite number of neurons. We used the inverse number of neurons as a perturbative expansion parameter but the formalism could be generalized to expand in any small parameter, such as the inverse of a slow time scale.
The second line of thought was with regards to the question of how to generalize the Wilson-Cowan equation, which is a phenomenological population activity equation for a set of neurons, which I summarized here. That paper built upon the work that Michael had started in his PhD thesis with Jack Cowan. The Wilson-Cowan equation is a mean field theory of some system but it does not specify what that system is. Michael considered the variable in the Wilson-Cowan equation to be the rate (stochastic intensity) of a Poisson process and prescribed a microscopic stochastic system, dubbed the spike model, that was consistent with the Wilson-Cowan equation. He then considered deviations away from pure Poisson statistics. The expansion parameter in this case was more obscure. Away from a bifurcation (i.e. critical point) the statistics of firing would be pure Poisson but they would deviate near the critical point, so the small parameter was the inverse distance to criticality. Michael, Jack and I then derived a set of self-consistent set of equations for the mean rate and rate correlations that generalized the Wilson-Cowan equation.
The unifying theme of both approaches is that these systems can be described by either a hierarchy of moment equations or equivalently as a functional or path integral. This all boils down to the fact that any stochastic system is equivalently described by a distribution function or the moments of the distribution. Generally, it is impossible to explicitly calculate or compute these quantities but one can apply perturbation theory to extract meaningful quantities. For a path integral, this involves using Laplace’s method or the method of steepest descents to approximate an integral and in the moment hierarchy method it involves finding ways to truncate or close the system. These methods are also directly related to WKB expansion, but I’ll leave that connection to another post.
Scientific Clearing House 
i wonder how this relates to s grossberg. looks similar—isn’t the wkb almost a taylor’s expansion (n g van kampen on markov/nonequilibrium stuff—-and wkb is almost a path integral)
if you quantize it, you get to t’hooft etc.
kinchin i think did the second law via steepest descents; or maybe fowler (beyond the von neumann / weiner proofs of the ergodic theorem—CLT).
the finite systems escape the clt.
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p.s. i notice you answered my question in the first paragraph. i’m going to have to look up wkb again on wikipedia. “j statistical mechanics’—-one of my favorite journals when i used the library (i also liked j theor biology). i also like general view like arxiv.org/1008.1405 and arxiv.org/abs/hep-ph/0301197 though i dont understand much of it. sometimes people even use feynman diagrams to find solutions (i saw this in the mayer condensation theory ) its spring.
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The Grossberg-Grosberg equation is like the Wilson-Cowan equation – they’re both mean field theories. Ishi, what is your field? Your interests are quite eclectic.
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Right now i don’t really have a field (i’m unemployed); i’ve been trying to do music (but my ‘music’ is also eclectic, so it doesn’t really sell). That can be a real mean field, as well as a complex one. (One thing i have tried but mostly failed doing was to try to combine something fun—eg danceable type pop music (including go-go music, rock, fold,and hip-hop) — with something less enjoyable at times (taking classes), so at my show you get three hours of musical excercize and also the equivalent, via the lyrics) of 3 undergrad or grad lectures in, say, quantum mechanics. Unfortunately its hard to put solving math problems into music —though there is ‘stochastic music’ (which translates equations and alogorithms into sound.)
I’m also into ‘social and ecological justice’ (though maybe i just want to get paid for beig viewed as morally superior than a criminal or hedge fund manager; equally I like being out in nature alot and have been to many mountain regions and wanted to be paid while wandering around rather than being in a computer lab or office). The central problem there i think is consumerism or efficiency—for example, how many congesspersons does it take to screw in a lightbulb; what % of US GDP should be spent on health care? (I recently have been tried to get on the DC low income health care plan, and if you are poor you don’t set an appointment time and show up; instead you show up at 6:30 AM ; wait 2 hours in the cold for the place to open, wait 12 hours for your apppointment, then get told ‘we’re closing so come back tomorrow’, so you do and then get told you’ll have to come back again so we can review your situation.
I have some science training and had 2 research projects in that time—one was revising the Flory-Stockmayer theory of antibody-antigen aggregation away from the infinite (essentially mean field) limit, to look at finite size effects (when the central limit theorem/calculus of variations (maximum entropy) approach does not apply; the idea was simply to take Julian Gibbs’ theory of ‘polyciondensation’ and consider the case when you have m,ore than one kind of condensing monomer—he was studying phase transition such as water or glass ).
The other was to try to predict RNA secondary structure from its nucleotide sequence—essentaially one tried to generate the ensemble of all the possible ways RNA could fold up, and then cluster them into types (which i did using information theory) and then attempt to say which would be the one found in nature. (My approach completely failed because I was trying for a ‘fact free science’ approach and didn’t use any chemical characteristics of the nucleotides. I was hoping for a Wigner style miracle such that I might be able to find ‘primes’ which simply exist logically, and all the others were just numbers which were epiphenomena. )
I am a large part self-educated (which meant when i was a student or employed) i spent most of my time exploring the library rather than doing what i was supposed to do. For awhile I was really interested in quantum biology (and also wrote an article on possible health effects of EM radiation) and still am somewhat. Once I read Prigogine and ‘complexity’ fields (and noted that the classical fokker-planck equation looks like a schrodinger, and a feynman path integral looks like the chapman-kolmogov-smolukowski equation, i realzed in a way all the fields are the same—which is why now one even has ‘gauge theories of finance’ and such. (of course Jack Cowan found goldstone bosons involved in vision, though these evolved from quantum fields, and Uzemeza (sic) of canada also had a quantum brain theory. Turing’s theory of morphogenesis—bifurcations as representing developmental stages also shows such connections (and there are also those who have suggested ‘quantum jumps’ are phase transtions).
I guess that is my sunday sermon. ola
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Pretty fascinating. You should write a book.
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i thought books were out–nowadays some have evolved to the web (or even i pods/iphones which i don’t understand). I’m doing 3 coursera courses now (i did one already on ‘social contxt of mental health’ —got 75% which should make me eligible for several noble prizes (all on 2 hours/week—only about 600G/wk—-minimum wage) . quantum physics (some people from u md.), dan ariely’s bheaviroal econ, keith devlin’s math (already missed 3 weeks).
i mostly do the homework; if i need answers i look at wikipeida.
the lectures are quite good though. beats rush limbaugh.
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[…] of generalized Wilson-Cowan equations that includes correlation dynamics (e.g. see here, here, and here ). Although both formalisms utilize path integrals, they are actually conceptually quite different. […]
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