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Fast and reliable methods for extracting functional connectivity in large populations

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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
figure1

(a) R2 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.

References

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Correspondence to Yasser Roudi.

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Open Access This article is published under license to BioMed Central Ltd. This is an Open Access article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Keywords

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