8.5 Cell assembly enumeration in random graphs  (Page 4/4)

It should be noted that throughout the book, only scale-free graphs are allowed either:

• Edges with weights greater than 1 (That is, there may exist more than one edge $uv\in E\left(G\right)$ where $u$ and $v$ are vertices in $V\left(G\right).$ )
• "Loops" which connect nodes to themselves (That is, edges of the form $uu.$ )

We allow these exceptions so that our graphs will be in full compliance with the Cooper-Frieze model.

Data

After generating a number of random graphs, we investigated a number of relationships between cell assemblies vs. other features in random graphs. We speculate on a few of these relationships here.

When we vary the probability that any given pair of vertices has an edge between them over a number of undirected, Bernoulli random graphs, we can see a number of distinct phases in the number of cell assemblies that these graphs will have, on average ( [link] ).

First, edge probability is insufficient to form even one assembly, then the number of assemblies increases to a peak as the graph becomes more dense. Past that peak, the closure of all tight sets converges to a single cell assembly. It would be interesting to find a method for predicting the probability at which this peak occurs, and then comparing that value to biology.

The Cooper-Frieze scale-free graph has no parameter to directly adjust edge density, but varying the related parameter, $\alpha ,$ does have similar effects.

A fairly promising, although ostensibly not very accurate, indicator of the number of k-cores is the maximum eigenvalue of a graph's adjacency matrix ( [link] ).

On undirected Bernoulli random graphs, the two variables appear to have a nontrivial, positive correlation. However, here we see that the method of graph construction is extremely important to any such observations, since on directed scale-free graphs, we see a relationship that is not nearly as clear ( [link] ).

Future work

• On commodity hardware, the k-core enumeration algorithm terminates in a few minutes on graphs on the order of 10 nodes. It can process larger graphs given a low density of edges or a large k. Consequently, it is not clear that strict enumeration will be useful on human-brain-sized graphs with approximately 100 billion neurons and 1000 connections per neuron, but it may be useful for smaller graphs. For instance, a honey bee has fewer than 1 million neurons [link] .
• If strict enumeration fails even in smaller cases, it would perhaps be possible to take advantage of certain knowledge about the structure of brains in order to speed up enumeration by restricting the types of graphs that need to be enumerated. For instance, if we could safely use a bipartite model for a brain, the algorithm could likely be optimized for bipartite graphs.
• Also, cell assembly enumeration could proceed further from a statistical standpoint, as hinted at in the "Data" section. Perhaps some function could be devised that maps some more easily discerned information about a graph to the probability that that graph will contain a given number of cell assemblies.
• The definition of cell assemblies, as presented here, is fairly cumbersome. If certain aspects of the definition, such as closure, could be simplified, perhaps an easier approach to cell assemblies may become apparent.
• Cell assemblies should theoretically have far more function than simply existing as static structures in random graphs. Studying the interaction of cell assemblies and the learning processes that create cell assemblies may yield interesting insights into neuroscience in general.

Conclusion

Our approach to enumeration of cell assemblies in arbitrary graphs probably runs insufficiently fast to explore the problem as an end in itself. However, it does give us a useful tool to help us understand cell assemblies. We can now find an unlimited number of examples of cell assemblies which we may use as tools to explore general trends and gain insight into the structure of cell assemblies. Perhaps with enough work on the subject, we may find a viable way to understand the workings of the brain through random graph theory.