Oral presentation
The simplest model for describing multineuron 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 timeconsuming 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 meanfield approximation, for which J is equal to the negative of the inverse covariance matrix, B) an independentpair approximation, C) a low rate, smallpopulation approximation (the lowrate 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 spinglass theory [4] and E) a weakcorrelation 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, SessakMonasson, 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. (a) R^{2 }and (b) RMS error for various approximate methods. Green (dashed dotted), naive meanfield; Purple (dashed doubledotted) lowrate, small N; Gray (dotted) independentpair; Red (dashed), TAP; Black (dashed), SessakMonasson; Blue, average of TAP and SessakMonasson.
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