Maximum Entropy Principle for Neuronal Network Dynamics

We address the issue of why the second order maximum entropy model, by using only firing rates and second order correlations of neurons as constraints, can well capture the observed distribution of neuronal firing patterns in many neuronal networks. We derive an expression for the effective interactions of the n-th order maximum entropy model using all orders of correlations of neurons as constraints and explore a possible dynamical state in which the strengths of higher order interactions always smaller than the lower orders. This provides a possible mechanism underlying the success of the second order maximum entropy model.

Maximum entropy principle (MEP) analysis with few non-zero effective interactions has been shown to successfully characterize the statistical distribution of neuronal network states. To understand its underlying mechanism, we establish a mapping between the dynamical structure, i.e., effective interactions in MEP analysis, and the anatomical coupling structure of integrate-and-fire networks. Our work helps to understand how a sparse coupling structure could lead to a sparse coding by effective interactions.

The implementation of MEP analysis often requires a sufficiently long-time data recording. By investigating relationships underlying the probability distributions, moments, and effective interactions in the MEP analysis, we show that, with short-time recordings of network dynamics, the MEP analysis can be applied to reconstructing probability distributions of network states that is much more accurate than the one directly measured from the short-time recording. We verify our results using data from both simulations of Hodgkin-Huxley neuronal networks and electrophysiological experiments.