8.5 Cell assembly enumeration in random graphs

Introduction

History

Models of individual neurons vary in complexity, but in general, neurons tend to behave like this:

• A neuron is either excited or not excited.
• When excited, a neuron will stimulate neurons to which it has an outgoing connection. Otherwise, it will not stimulate other neurons.
• A neuron becomes excited when it receives a sufficient amount of stimulation from other neurons.

When modeling the brain, see seek to model the collective behavior of neurons. The fundamental type of collective behavior is, by the Donald Hebb model of the brain, the cell assembly.

First introduced by Hebb, the cell assembly is, "a diffuse structure comprising cells... capable of acting briefly as a closed system, delivering facilitation to other such systems and usually having a specific motor facilitation" [link] . A cell assembly is a particular arrangement of a group of neurons with certain properties. The most salient of these properties is that a certain fractional portion of the assembly will excite the entire assembly.

By Hebb's proposal, a cell assembly represents a single concept in the brain. For instance, Hebb proposes that the corner of an abstract triangle may be represented by a cell assembly [link]

Since Hebb first discussed the concept of a cell assembly, there has been some amount of biological research supporting his ideas. For instance, the work of György Buzsáki suggests that groups of cells that fire during a given time period are correlated [link] .

Definitions

Gunther Palm defined Hebb's assembly in the concrete language of graph theory [link] . In brief, Palm's discussion constructs, or depends upon, the following definitions:

• graph: A graph $G$ has a vertex set, $V\left(G\right),$ and a set of edges, $E\left(G\right)\subseteq V\left(G\right)\times V\left(G\right).$ If $u,v\in V\left(G\right)$ and $uv\in E\left(G\right),$ we say that the graph $G$ has an edge directed from the vertex $u$ toward the vertex $v.$ Vertices are labeled with integers by convention.
• neighborhood: We define the neighborhood of some vertex $v\in V\left(G\right)$ with respect to $G,$ call it $N(v,G),$ as $\{u:\exists uv\in E(G\left)\right\}$ ( [link] ).
• degree: The degree of a vertex $v\in V\left(G\right)$ with respect to $G,$ call it $D(v,G),$ is $\left|N\right(v,G\left)\right|.$
• subgraph: A graph $g$ is a subgraph of $G$ iff $V\left(g\right)\subseteq V\left(G\right)$ and $E\left(g\right)\subseteq E\left(G\right).$ Further, $g$ is an induced subgraph of $G$ iff $g$ is a subgraph of $G$ and $\forall e\in E\left(G\right)\cap \left(V\right(g)\times V(g\left)\right),e\in E\left(g\right).$
• k-core: A subgraph $g$ of $G$ is a $k$ -core iff $\forall v\in V\left(g\right),\left|X\right|\ge k,$ where $X=N\left(v\right)\cap V\left(g\right)$ ( [link] ).
• minimal k-core A subgraph $g$ of $G$ is a minimal k-core iff it has no induced subgraphs which are k-cores.
• maximum k-core A subgraph $g$ of $G$ is a maximum k-core iff $G$ contains no k-cores $h$ for which the $\left|V\right(h\left)\right|>\left|V\right(g\left)\right|.$
• activation: We say that a vertex $v\in V\left(G\right)$ can be either active or inactive. We generally say that, initially, an arbitrary subset of vertices $M\subseteq V\left(G\right)$ are active and the rest inactive. We further define a map $f_k(M,G):(\text{sets}\text{of}\text{vertices},\text{graphs})\to \left(\text{sets}\text{of}\text{vertices}\right)$ which performs the following operation:
  • Take a graph $G,$ and a set of vertices, $M,$ where $M\subseteq V\left(G\right).$
  • $\forall v\in V\left(G\right)$ :
    • let $Y=M\cap N(v,G)$
    • iff $\left|Y\right|\ge k,$ then $v\in R$
  • Return $R.$
  For convenience, we add a superscript $f_k^n(M,G),$ where $f_k^2(M,G)=f_k(f_k(M,G),G),f_k^3(M,G)=f_k(f_k(f_k(M,G),G),G),$ etc.
• closure: We say that the closure of a set of active nodes $M$ with respect to the graph $G,$ call it $c{l}_k(M,G)$ , is equal to $f_k^{\infty }(M,G).$ If $f_k^n(M,G)$ does not converge for some sufficiently large $n,$ then the closure of $M$ is undefined ( [link] ).
• k-tight: A k-core $T,$ which is an induced subgraph of $G,$ is k-tight iff it satisfies the following condition:
  • $\forall K$ where $K$ is an induced subgraph of $T$ and a k-core:
    • $cl\left(V\right(K),G)\supseteq V\left(T\right),$ or,
    • $cl\left(V\right(T)\setminus V(K),G)=\varnothing$
• k-assembly: An induced subgraph $A$ of $G$ is a k-assembly iff $V\left(A\right)=cl\left(V\right(T),G)$ where $T$ is a k-tight induced subgraph of $G.$