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On the inverse Ising problem: application to neural networks data

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Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)

Abstract

S. Cocco, S. Leibler, R. Monasson On the inverse Ising problem: application to neural networks data The Ising model is a paradigm in statistical physics. Here we address the inverse problem: given a set of local magnetizations $m(i)$ and correlation $c(i,j)$ measured in a system of N spins what is the set of magnetic fields $h(i)$ and couplings $J(i,j)$ defining an Ising model giving the same $m(i)$ and $c(i,j)$ ? We show that this problem can be solved by the knowledge of the Ising model free energy at fixed magnetizations and correlation $A(\m(i)\,\c(i,j)\)$. We construct $A(\m(i)\,\c(i,j)\)$ through its high temperature expansion extending the procedure of 1, and show how to resum numerically full classes of diagrams corresponding to the subsets of interacting spins. We have applied the inference technique to data of the spiking activity of populations of N neurons (up to N=60) in the retina 2,3. We discuss the quality of the inference as a function of the amount of data, the size N of the network and the biological relevance of the inferred fields and couplings. References\\ 1) A. Georges , J.Yeidida J. Phys. A (24), 2173 (1991). \\ 2) E Schneidman, MJ Berry II, R Segev, W Bialek Nature 440, 1007,(2006).\\ 3) M.Schnitzer,M. Meister Neuron 37,499 (2003).

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