I gave a talk at the Center for Scientific Computing and Mathematical Modeling at the University of Maryland today. My slides are here. I apologize for the excessive number of pages but I had to render each build in my slides, otherwise many would be unreadable. A summary of the work and links to other talks and papers can be found here.
I’m currently in Göttingen, Germany at the Bernstein Sparks Workshop: Beyond mean field theory in the neurosciences, a topic near and dear to my heart. The slides for my talk are here. Of course no trip to Göttingen would be complete without a visit to Gauss’s grave and Max Born’s house. Photos below.
Carson C. Chow and Michael A. Buice. Path Integral Methods for Stochastic Differential Equations. The Journal of Mathematical Neuroscience, 5:8 2015.
Abstract: Stochastic differential equations (SDEs) have multiple applications in mathematical neuroscience and are notoriously difficult. Here, we give a self-contained pedagogical review of perturbative field theoretic and path integral methods to calculate moments of the probability density function of SDEs. The methods can be extended to high dimensional systems such as networks of coupled neurons and even deterministic systems with quenched disorder.
I’m currently in Banff, Alberta for a Festschrift for Jack Cowan (webpage here). Jack is one of the founders of theoretical neuroscience and has infused many important ideas into the field. The Wilson-Cowan equations that he and Hugh Wilson developed in the early seventies form a foundation for both modeling neural systems and machine learning. My talk will summarize my work on deriving “generalized Wilson-Cowan equations” that include both neural activity and correlations. The slides can be found here. References and a summary of the work can be found here. All videos of the talks can be found here.
Addendum: 17:44. Some typos in the talk were fixed.
Addendum: 18:25. I just realized I said something silly in my talk. The Legendre transform is an involution because the transform of the transform is the inverse. I said something completely inane instead.
I’m currently at the National Center for Theoretical Sciences, Math Division, on the campus of the National Tsing Hua University, Hsinchu for the 2013 Conference on Mathematical Physiology. The NCTS is perhaps the best run institution I’ve ever visited. They have made my stay extremely comfortable and convenient.
Here are the slides for my talk on Correlations, Fluctuations, and Finite Size Effects in Neural Networks. Here is a list of references that go with the talk
M.A. Buice and C.C. Chow, `Correlations, fluctuations and stability of a finite-size network of coupled oscillators’. Phys. Rev. E 76 031118 (2007) [PDF]
M.A. Buice, J.D. Cowan, and C.C. Chow, ‘Systematic Fluctuation Expansion for Neural Network Activity Equations’, Neural Comp., 22:377-426 (2010) [PDF]
C.C. Chow and M.A. Buice, ‘Path integral methods for stochastic differential equations’, arXiv:1009.5966 (2010).
M.A. Buice and C.C. Chow, `Effective stochastic behavior in dynamical systems with incomplete incomplete information.’ Phys. Rev. E 84:051120 (2011).
MA Buice and CC Chow. Dynamic finite size effects in spiking neural networks. PLoS Comp Bio 9:e1002872 (2013).
MA Buice and CC Chow. Generalized activity equations for spiking neural networks. Front. Comput. Neurosci. 7:162. doi: 10.3389/fncom.2013.00162, arXiv:1310.6934.
Here is the link to relevant posts on the topic.
Michael Buice and I have a new paper in Frontiers in Computational Neuroscience as well as on the arXiv (the arXiv version has fewer typos at this point). This paper partially completes the series of papers Michael and I have written about developing generalized activity equations that include the effects of correlations for spiking neural networks. It combines two separate formalisms we have pursued over the past several years. The first was a way to compute finite size effects in a network of coupled deterministic oscillators (e.g. see here, here, here and here). The second was to derive a set 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. The first formalism adapted kinetic theory of plasmas to coupled dynamical systems. The second used ideas from field theory (i.e. a two-particle irreducible effective action) to compute self-consistent moment hierarchies for a stochastic system. This paper merges the two ideas to generate generalized activity equations for a set of deterministic spiking neurons.
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.