The Cluster Variation Method: A Primer for Neuroscientists

Abstract

Effective Brain–Computer Interfaces (BCIs) require that the time-varying activation patterns of 2-D neural ensembles be modelled. The cluster variation method (CVM) offers a means for the characterization of 2-D local pattern distributions. This paper provides neuroscientists and BCI researchers with a CVM tutorial that will help them to understand how the CVM statistical thermodynamics formulation can model 2-D pattern distributions expressing structural and functional dynamics in the brain. The premise is that local-in-time free energy minimization works alongside neural connectivity adaptation, supporting the development and stabilization of consistent stimulus-specific responsive activation patterns. The equilibrium distribution of local patterns, or configuration variables, is defined in terms of a single interaction enthalpy parameter (h) for the case of an equiprobable distribution of bistate (neural/neural ensemble) units. Thus, either one enthalpy parameter (or two, for the case of non-equiprobable distribution) yields equilibrium configuration variable values. Modeling 2-D neural activation distribution patterns with the representational layer of a computational engine, we can thus correlate variational free energy minimization with specific configuration variable distributions. The CVM triplet configuration variables also map well to the notion of a M = 3 functional motif. This paper addresses the special case of an equiprobable unit distribution, for which an analytic solution can be found.

Lifespan Development of the Human Brain Revealed by Large-Scale Network Eigen-Entropy

“…These results may partly reveal the architectural transition of the human brain shift from a relatively anatomically driven configuration to a functional well-organized configuration with age. Furthermore, the current findings seem consistent with the notion of free energy minimization in the brain, as the entropy-like term (NEE) did indeed decrease with aging subjects [54][55][56]. Considering the notion of entropy in the context of free energy minimization may extend the value of this study to the broader community.…”

Neuronal message passing using Mean-field, Bethe, and Marginal approximations

“…convergence to a limit cycle or improbable configurations of the marginal posterior) and anomalous free energy estimates. To mitigate these issues, higher order approximations have been proposed that are based on cluster variational methods and the Kikuchi approximation 59,62,63 .…”

The 2-D Cluster Variation Method: Initial Findings and Topography Illustrations

“…The work by Maren has used a CVM architecture where the fundamental units were nearest-neighbors, next-nearest neighbors, and triples; as the base unit of a triple has been deemed appropriate for neural modeling. (See discussion in Maren (2016) [3]. )…”

“…The notion that the brain operates to minimize free energy is not new; there is a long and robust history of efforts to advance this approach. (See Maren (2016) for a substantial discussion [3].) There is vigorous discussion within the neural dynamics modeling community as to exactly how free energy minimization plays its role.…”

“…The Cluster Variation Method (CVM) was originally devised by Kikuchi in 1951 [1], and then further advanced by Kikuchi and Brush (1967) [2]. Maren (2016) suggested that the CVM would be appropriate for modeling brain topographies [3], and presented initial computational results for the 1-D CVM. More recently, Maren (2019a) investigated how the free energy minimization process influenced resultant 2-D topographies represented by a 2-D CVM grid [4].…”