Аннотация
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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