Surprising Result Modeling a Simple Scale-Free Brain Network Using the Cluster Variation Method

One of the primary research thrusts that I suggested in my recent paper, The Cluster Variation Method: A Primer for Neuroscientists, was that we could use the 2-D Cluster Variation Method (CVM) to model distribution of configuration variables in different brain network topologies. Specifically, I was expecting that the h-value (which measures the interaction enthalpy strength between nodes in a 2-D CVM grid) would change in a continuum, depending on the network topology used.

Last week, I constructed a very simple scale-free-like model, to see what the h-value would be for this simplest of network types.

Simply expressed, a scale-free network is one in which a pattern repeats itself over different scales of resolution. For example, a fractal is scale-free. We can look at it at any resolution level, and the same pattern appears.

Three different network topologies have been suggested for the brain, over the past decade or more. These are:

• Scale-free,
• Small-world, and
• Rich club.

Of these three, the scale-free seemed likely to have the least connectivity between nodes in like states. (Let’s call these nodes the A nodes, or those in the “on” state. The others will be B nodes, or “off” state. We’re working with a bistate model.)

To model a scale-free topography, I created a 2-D CVM grid of 256 nodes, arranged as a 16-layer grid of 16 nodes each. (The nodes are offset by the diagonal, so I wouldn’t exactly call this a square grid, but close enough.)

I created the left and right sides to be 180-degree maps of each other; that is, a simple dihedral group D2 (using group theory notation). This was just to keep things simple. Also, I wanted to have a system where the pattern structure was replicated at least once.

In a 256-node system, with dihedral symmetry, the right and the left replicate the patterns in each other. So, the right side would contain 128 nodes, and the left would have the same number of nodes, in the same patterns.

I want to use the 2-D CVM for which there is an analytic solution; this means that we need to have equal numbers of nodes in states A and B. This would give us, on each side, 64 A nodes, and 64 B nodes.

I decided to create my “pattern” using the A nodes. The key thing about the pattern is that it has to replicate through multiple scales. If my basic “pattern” was a slightly elongated, slightly paisley-shaped cluster of 16 nodes, then I would have 64 – 16 = 3×16 potential patterns of the same size and shape left. Of those, I split one into two clusters of 8 nodes each. That gave 2 sets of 16 units each remaining. I split one of those into 4 sets of 4 units each, which left one set of 16 units remaining. I split that one into 8 sets of 2 nodes each.

This, at least, was my design.

Trying to fit all of these patterns into the grid structure was another matter entirely. The following figure shows what I was actually able to accomplish.

Once I had the patterns arranged, I computed (by hand, as I’m still evolving my CVM program from 1-D to 2-D) the configuration variables Z1 and Z3. These would give, once divided by the total number of triplets, my configuration variables z1 and z3.

{To be continued … }

{To be continued … }

 

Related Cluster Variation Method Posts

• The Cluster Variation Method: A Primer for Neuroscientists (Blogpost and Link)
• Visualizing Configuration Variables Using the 2-D Cluster Variation Method

 

References

• Maren, A.J. (2016) The Cluster Variation Method: A Primer for Neuroscientists. Brain Sciences, 6(4), 44. doi:10.3390/brainsci6040044 pdf
• V. M. Eguluz, D.R. Chialvoy, G. Cecchiz, M. Balikiy and A.V. Apkariany (2005). Scale-free brain functional networks. Physical Review Letters, 94 (1), (February) 018102. pdf
• Biyu J. He and John M. Zempel and Abraham Z. Snyder and Marus Raichle (2010). The Temporal Structures and Functional Significance of Scale-free Brain Activity. Neuron, 66 (33), May) 353-69. doi:10.1016/j.neuron.2010.04.020 pdf Note to self: check PubMed code
• Xiaolin Liu, B. Douglas Ward, Jeffrey R. Binder, Shi-Jiang Li, and Anthony G. Hudetz (2014). Scale-Free Functional Connectivity of the Brain Is Maintained in Anesthetized Healthy Participants but Not in Patients with Unresponsive Wakefulness Syndrome. PLOS One, March 19, 2014. doi:10.1371/journal.pone.0092182. pdf
• Carlo Nicolini & Angelo Bifone (2016) Modular structure of brain functional networks: breaking the resolution limit by Surprise. Nature: Scientific Reports, 6, 19250. doi:10.1038/srep19250. pdf