We introduce a procedure to infer the interactions among a set of binary variables, based on their sampled frequencies and pairwise correlations. The algorithm builds the clusters of variables contributing most to the entropy of the inferred Ising model, and rejects the small contributions due to the sampling noise. Our procedure successfully recovers benchmark Ising models even at criticality and in the low temperature phase, and is applied to neurobiological data.Understanding the correlated activity of complex, nonhomogeneous multi-component systems is of fundamental importance in physics, biology, sociology, finance, ... A natural issue is to separate direct correlations (due to direct interactions) from network-mediated correlations. The Ising model, of ubiquitous importance in statistical physics, provides a natural framework to extract interactions from correlations [1], and was recently used for the analysis of neurobiological data [2][3][4]. It is indeed the least constrained model capable of reproducing the individual and pairwise frequencies of a set of, say, N binary-valued variables, σ i = 0, 1. In practice, these frequencies, p i and p ij , are often estimated through empirical averages over a number of sampled configurationsThe task then consists in inferring the parameters (fields h i and interactions J ij ) of the Ising model reproducing those data. From a mathematical point of view, one has to solve the 1 2 N (N + 1) implicit equations p i = σ i and p ij = σ i σ j for the fields and interactions, where · denotes the Gibbs average with Boltzmann factor exp [9]. Despite their specificities, those methods have in common to be efficient when the correlations, c ij = p ij − p i p j , are weak, and to perform badly when most pairs (i, j) are strongly correlated, e.g. when the data are generated by a critical Ising model. Those examples seem to suggest that fast algorithms cannot infer BMs with long-range correlations [10].However, the existence of a relationship between the presence of strong correlations in the 'direct' model and the intrinsic hardness of the inverse problem is questionable [11]. Let p = {p i , p ij }, J = {h i , J ij }, σ = { σ i , σ i σ j } be the 1 2 N (N + 1)-dimensional vectors of, respectively, the measured frequencies, the interaction parameters and the Gibbs frequencies. We define the susceptibility and the inverse susceptibility matrices through, respectively,χ is attached to the direct model, and quantifies how the frequencies respond to a small change in the interaction parameters. χ −1 , which gives the response of the BM interaction parameters to a small change in the frequencies, is a natural characterization for the inverse problem. An essential point, which has received little attention in the context of BM so far, is that χ −1 is generally much sparser and shorter-range than χ; evidence for this claim is reported below. Even if strong responses (and correlations) pervade the system, each BM interaction parameter may mostly depend on a small (compared to N ) number ...
Complexity of neural systems often makes impracticable explicit measurements of all interactions between their constituents. Inverse statistical physics approaches, which infer effective couplings between neurons from their spiking activity, have been so far hindered by their computational complexity. Here, we present 2 complementary, computationally efficient inverse algorithms based on the Ising and ''leaky integrate-and-fire'' models. We apply those algorithms to reanalyze multielectrode recordings in the salamander retina in darkness and under random visual stimulus. We find strong positive couplings between nearby ganglion cells common to both stimuli, whereas long-range couplings appear under random stimulus only. The uncertainty on the inferred couplings due to limitations in the recordings (duration, small area covered on the retina) is discussed. Our methods will allow realtime evaluation of couplings for large assemblies of neurons.inference and inverse problems ͉ multielectrode recordings ͉ neural couplings A vertebrate retina is a structured, complex network of interacting neurons that process visual input stimuli at the photoreceptors into an output pattern of action potentials of the retinal ganglion cells (1-2). It is now a well-established fact that retinal cells process information in a collective fashion: The firing of one ganglion cell is correlated with the firing pattern of other cells (3-4). Multielectrode recordings have made accessible hours-long, simultaneous spiking activity of tens of retinal ganglion cells and thus have become a powerful tool to investigate the information processing performed by a vertebrate retina (5-7). The analysis of pairwise correlations in the activity has revealed different patterns of synchrony between 2 cells that have been related to different retina circuits (7).Analyzing the concerted activity of all of the recorded cells is, however, a very challenging task. Recently Schneidman et al. (8) and Shlens et al. (9) pointed out that correlations in the firing activity of cell populations can be reconstructed from the average firing rates, f i , and 2-cell correlations, c ij , alone. The theoretical model, which has been used to generate the frequencies of all possible 2 N spiking configurations for a system of N neurons, is the well-known Ising model. It is characterized by a reduced set of ϷN 2 parameters: N ''fields,'' h i , experienced by individual cells, and N(N Ϫ 1)/2 ''couplings,'' J ij , between pairs of cells. Computing the parameters h i and J ij from the firing patterns can be viewed as an example of the inverse statistical physics method.The existence of a low-dimensional parameterization of the retinal activity (8-11) is an interesting and encouraging result. Naively, one may be tempted to assign to the inferred parameters a simple interpretation: The fields could represent the external stimuli, and the couplings could reflect the physiological interactions between the cells. However, because most of the neural circuitry (all cells in intermediat...
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.