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The simplest maximum entropy model for collective behavior in a neural network
Gasper Tkacik, Olivier Marre, Thierry Mora, Dario Amodei, Michael J. Berry II, William Bialek
TL;DR
Pairwise maximum-entropy models have described neural activity [10] [13], but the retina’s total spike-count distribution departs strongly from independence despite weak pairwise correlations. This paper models that global activity distribution and examines its large-network thermodynamics. The entropy–energy slope approaches 1, consistent with an unusual critical point.
Problem
Method
They construct a maximum-entropy model constrained by the distribution of how many neurons spike in a time bin, then examine its large-N thermodynamics.
Source: p. 2Results
β0 = 0.999 ± 0.004: the entropy–energy relation approaches slope 1, consistent with a critical point where specific heat diverges.
Source: p. 3Takeaways & Limitations
The large-N thermodynamic limit depends on how the global activity distribution evolves with network size, rather than rescaling measured pairwise correlations.
Source: pp. 2–3
Abstract
from arXiv · showhide
Recent work emphasizes that the maximum entropy principle provides a bridge between statistical mechanics models for collective behavior in neural networks and experiments on networks of real neurons. Most of this work has focused on capturing the measured correlations among pairs of neurons. Here we suggest an alternative, constructing models that are consistent with the distribution of global network activity, i.e. the probability that K out of N cells in the network generate action potentials in the same small time bin. The inverse problem that we need to solve in constructing the model is analytically tractable, and provides a natural "thermodynamics" for the network in the limit of large N. We analyze the responses of neurons in a small patch of the retina to naturalistic stimuli, and find that the implied thermodynamics is very close to an unusual critical point, in which the entropy (in proper units) is exactly equal to the energy.