Systematic fluctuation expansion for neural networks

A new paper  “Systematic fluctuation expansion for neural network activity equations“, by Michael Buice, Jack Cowan and myself has just been uploaded to the q-bio arXiv.  The paper arose from a confluence of my desire to adapt moment hierarchy approaches from kinetic theory to studying fluctuations in neural networks and Michael and Jack’s field theory formulation of stochastic neural dynamics (see here).  In this paper, we show that the two approaches are identical and give a systematic scheme to derive the equations.  We give an example for self-consistent equations for the first two moments.

Classically, neural networks have been described either by rate equations, such as the Wilson-Cowan equation of the form \dot{a_i} = -r a_i + f(\sum_{j} w_{ij} a_j + S_i) (and the continuum version) or networks of (more biophysical)  spiking neurons.  Although rate equations average over neural spikes,  they have been extremely successful in describing many neural phenomena.  Wilson and Cowan, Grossberg, Amari, Hopfield, Ermentrout, and many others, have used these types of equations to describe phenomena as diverse as associative memory, working memory, persistent activity, hallucinations, orientation tuning, and neural activity waves.  In fact, the term neural network, has essentially been co-opted to imply a network of rate equations (i.e., multi-layer perceptron) with a back propagation learning rule for the weights to perform supervised learning.

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Why can’t biology or economics be more like physics?

February 12 was the 200th anniversay of Charles Darwin and Abraham Lincoln. I’m not going to comment on either of the two directly in this post (given the large amount of press devoted to them recently) even though their impact on our lives cannot be overstated.  What I do want to talk about is whether or not biology and economics can be more like physics.  I will do this in two parts, with this post focusing on  biology. By “not like physics”, I mean that there is not a more quantitative and unifying approach to biology.  I think many physicists feel that biologists miss the big picture and that much more could be gleaned if they only started to think like physicists.    This attitude is perfectly represented in biophysicist Bob Austin’s letter to Physics Today a decade ago, which can be found here.  I think this view has evolved recently as more physicists work on biology but I still see it.

Although I am a former physicist,  I’m going to take the side of the biologists.  I’m not saying that biology couldn’t be more quantitative and better understood.  I’m also not saying that ideas from physics couldn’t be useful.  These are all probably true.  What I am saying is that the reason biology is not more like physics isn’t because biologists are misguided (or as Bob Austin puts it “can’t reason their way out of a paper bag”) but because biology is different from physics.

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