Here is an audio recording synchronized to slides of my talk a week and a half ago in Pittsburgh. I noticed some places where I said the wrong thing such as conflating neuron with synapse. I also did not explain the learning part very well. I should point out that we are not applying a control to the network. We train a set of weights so that given some initial condition, the neuron firing rates follow a specified target pattern. I also made a joke that implied that the Recursive Least Squares algorithm dates to 1972. That is not correct. It goes back much further back than that. I also take a pot shot at physicists. It was meant as a joke of course and describes many of my own papers.
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.