Statistical Physics of Pairwise Probability Models by Various Authors

Author:Various Authors
Language: eng
Format: epub
Tags: pairwise model, inverse ising problem, maximum entropy distribution, inference
Published: 2015-04-30T16:00:00+00:00

Extensions of the Binary Pairwise Model

In the previous sections, we described various approximate methods for fitting a pairwise model of the type of Eq. 2. We also studied how good a model it will be for spike trains, using analytical calculations and computer simulations. As we describe below, there are two issues with a model of the type of Eq. 2 that lead to new directions for extending the pairwise models studied here.

The first issue is the use of binary variables as a representation of the states of the system. For fine time bins and neural spike trains, the binary representation serves its purpose very well. However, in many other systems, a binary representation will be a naive simplification. Examples of such systems are modular models of the cortex in which the state of each cortical module is described by a variable taking a number of states usually much larger than 2. In Section “Extension to Non-binary Variables” we briefly describe a simple non-binary model useful for modelling the statistics of such systems.

The second issue is that by using Eq. 2 in cortical networks, one is essentially approximating the statistics of a highly non-equilibrium system with asymmetric physical interactions, e.g. a balanced cortical network, by an equilibrium distribution with symmetric interactions. This manifests itself in a lack of a simple relationship between the functional connectives to real physical connections. In our simulations we observed that there was no obvious relation between the synaptic connectivity and the inferred functional connections. Second, as we showed here and in our previous work, for large populations, the model quality decays. Although one can avoid this decay by decreasing δt as N grows, eventually one will get into the regime of very fine δt, where the assumption of independent bins used to build the model does not hold any more and one should start including the state transitions in the spike patterns (Roudi et al., 2009a). In fact, Tang et al. (2008) showed that even in the cases that the pairwise distribution of Eq. 2 is a good model for predicting the distribution of spike patterns, it will not be a good one for predicting the transition probabilities between them. These observations encourage one to go beyond an equilibrium distribution with symmetric weights. In the second extension, described in Section “Extension to Dynamics and Asymmetric Interactions,” we propose one such model, although a detailed study of the properties of such model is beyond the scope of this paper.

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