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  • Oral presentation
  • Open Access

Fast and reliable methods for extracting functional connectivity in large populations

BMC Neuroscience200910 (Suppl 1) :O9

  • Published:


  • Functional Connectivity
  • Ising Model
  • Spike Train
  • Simulated Network
  • Excitatory Neuron

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-1exp{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.

We used data from a simulated network of spiking neurons operating in a balanced state of asynchronous firing with a mean rate of ~10 Hz for excitatory neurons. Employing a bin size of 10 ms, we performed Boltzmann learning to fit Ising models for populations of size 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 R2, 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.
Figure 1
Figure 1

(a) R 2 and (b) RMS error for various approximate methods. Green (dashed dotted), naive mean-field; Purple (dashed double-dotted) low-rate, small N; Gray (dotted) independent-pair; Red (dashed), TAP; Black (dashed), Sessak-Monasson; Blue, average of TAP and Sessak-Monasson.

Authors’ Affiliations

NORDITA, Roslagstullsbacken 23, 10691 Stockholm, Sweden
Dept of Mathematical Statistics, Stockholm University, 10691 Stockholm, Sweden
The Niels Bohr Institute, Copenhagen University, 2100 Copenhagen, Denmark


  1. 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
  2. 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
  3. 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
  4. Tanaka T: Mean-field theory of Boltzmann machine learning. Phys Rev E. 1998, 58: 2302-2310. 10.1103/PhysRevE.58.2302.View ArticleGoogle Scholar
  5. 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


© Roudi et al; licensee BioMed Central Ltd. 2009

This article is published under license to BioMed Central Ltd.