Single-Parameter Analytic Solution for Modeling Local Pattern Distributions

The cluster variation method (CVM) offers a means for the characterization of both 1-D and 2-D local pattern distributions. The paper referenced at the end of this post provides neuroscientists and BCI researchers with a CVM tutorial that will help them to understand how the CVM statistical thermodynamics formulation can model 1-D and 2-D pattern distributions expressing structural and functional dynamics in the brain. 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.

Local Patterns Contribute to Entropy Formation

The following Figure 1 shows a set of three patterns, each arranged as a 1-D zigzag chain. If we use only our classic entropy expression, which depends only on the distribution of units into on and off (or A and B) states, then the entropy for each of these patterns would be the same. In fact, for this illustration, the entropy for each would be at a maximum, since there are equal numbers of A and B units in each case.

Appealing to our intuitive sense about the nature of entropy, we immediately recognize that the top and bottom patterns in this figure are substantially not at maximal entropy; each of these two patterns expresses a high degree of order (albeit with a small pattern discontinuity in each). Our understanding of the concept of entropy tells us that neither pattern A (topmost) nor pattern C (bottom-most) is achieving a maximal distribution among all available states. (Obviously, the notion of states here has to be broadened to include patterns as well as distribution into A and B states.)

It is the middle pattern (B) of Figure 1 that intrigues us the most at this moment. As with both the top (A) and bottom-most (C) patterns, it is not entirely a maximal distribution among possible states; we see that the same eight-unit pattern is repeated four times. However, despite this regularity, we notice that there is a greater diversity of local patterns in this middle example than in either the topmost or bottom-most examples.

The key to expressing this diversity among the types of local patterns, and to noting their resulting impact on the entropy term, is to consider the local configuration variables. For more, see “The Cluster Variation Method: A Primer for Neuroscientists,” link provided in Reference Link at the end of this post.

 

Reference

• Maren, A.J. (2016) The Cluster Variation Method: A Primer for Neuroscientists. Brain Sciences, 6(4), 44. doi:10.3390/brainsci6040044 PDF