Towards a theory of cell assemblies.

The term cell assembly, first introduced by D. O. Hebb, is defined in the framework of graph theory. This definition leads to some beautiful problems concerning the number and size of cell assemblies in large graphs. Some approaches to solve these problems are presented. In particular, the graphs Kn X Km are constructed that have n . m points, n + m - 2 connections per point, and at least 2n + 2m - 4 assemblies. Several new notions of connectivity in directed graphs are introduced and their relationships are investigated. The insight into these notions and their relationships will be helpful for further construction of graphs with many assemblies and/or high connectivity. The resulting graphs are not only important for the idea of cell assemblies in the content of neurodynamics, they may also find applications in the construction of communication networks and associative memories.