Volume 10 Supplement 1

## Eighteenth Annual Computational Neuroscience Meeting: CNS*2009

# Fast and reliable methods for extracting functional connectivity in large populations

- Yasser Roudi
^{1}Email author, - Joanna Tyrcha
^{2}and - John Hertz
^{1, 3}

**10(Suppl 1)**:O9

https://doi.org/10.1186/1471-2202-10-S1-O9

© Roudi et al; licensee BioMed Central Ltd. 2009

**Published: **13 July 2009

The simplest model for describing multi-neuron spike statistics is the pairwise Ising model [1, 2]. To start, one divides the spike trains into small time bins, and to each neuron *i* and each time bin *t* assigns a binary variables *s*_{
i
}(*t*) = -1 if neuron *i* has not emitted any spikes in that time bin and 1 if it has emitted one or more spikes. One then can construct an Ising model, *P*(s) = *Z*^{-1}exp{h's+s'Js} for the spike patterns with the same means and pair correlations as the data, using Boltzmann learning, which is in principle exact. The elements *J*_{
ij
}, of the matrix J can be considered to be functional couplings. However, Boltzmann learning is prohibitively time-consuming for large networks. Here, we compare the results from five fast approximate methods for finding the couplings with those from Boltzmann learning.

*N*up to 200 excitatory neurons chosen randomly from the 800 in the simulated network. We studied the following methods: A) a naive mean-field approximation, for which J is equal to the negative of the inverse covariance matrix, B) an independent-pair approximation, C) a low rate, small-population approximation (the low-rate limit of (B), which is valid generally in the limit of small

*Nrt*, where

*r*is the average rate (spikes/time bin) and

*t*is the bin width [3], D) inversion of the TAP equations from spin-glass theory [4] and E) a weak-correlation approximation proposed recently by Sessak and Monasson [5]. We quantified the quality of these approximations, as functions of

*N*, by computing the RMS error and

*R*

^{2}, treating the Boltzmann couplings as the true ones. We found, as shown in figure 1, that while all the approximations are good for small

*N*, the TAP, Sessak-Monasson, and, in particular, their average outperform the others by a relatively large margin for

*N*. Thus, these methods offer a useful tool for fast analysis of multineuron spike data.

## Authors’ Affiliations

## References

- Schneidman E, Berry MJ, Segev R, Bialek W: Weak pairwise correlations imply strongly correlated network states in a neural population. Nature. 2006, 440: 1007-1012. 10.1038/nature04701.PubMed CentralPubMedView ArticleGoogle Scholar
- Shlens J, Field GD, Gauthier JL, Grivich MI, Petrusca D, Sher A, Litke AM, Chichilnisky EJ: The structure of multi-neuron firing patterns in primate retina. J Neurosci. 2008, 28: 505-518. 10.1523/JNEUROSCI.3359-07.2008.View ArticleGoogle Scholar
- Roudi Y, Nirenberg S, Latham P: Pairwise maximum entropy models for large biological systems: when they can and when they can't work. arXiv:0811.0903v1 [q-bio.QM].Google Scholar
- Tanaka T: Mean-field theory of Boltzmann machine learning. Phys Rev E. 1998, 58: 2302-2310. 10.1103/PhysRevE.58.2302.View ArticleGoogle Scholar
- Sessak V, Monasson R: Small-correlation expansions for the inverse Ising problem. J Phys A. 2009, 42: 055001-10.1088/1751-8113/42/5/055001.View ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd.