8.2 Cell assemblies: a binary integer programming problem

Module goal

1. Provide biological motivation for finding cell assemblies in a network of neurons
2. Understand how mathematics and graph theory can be used to locate cell assemblies in a neural network
3. Understand how a minimal k -core can be used to find a k -assembly
4. Learn how to use a bulit-in MATLAB function, bintprog , to aid in the finding of cell assemblies

Introduction

Ever since psychologists and neuroscientists began studying the physiological inner workings of the brain, they have been puzzled by many questions. How are concepts stored and recalled within our brains? How does learning and memory occur? In 1949, D.O. Hebb tried to explain the answers to these questions in terms of cell assemblies in his book The Organization of Behavior . Hebb asserts that a cell assembly is a group of neurons wired in a specific manner such that when a sufficient amount of neurons in this group become excited, the entire group becomes excited in a synchronized manner. Hebb went on to explain that these cell assemblies form via synaptic plasticity. He claims that if neuron A repeatedly fires neuron B, some metabolic activity occurs increasing the efficiency in which neuron A fires neuron B. This phenomenon is more commonly known as “cells that fire together, wire together.” Hebb postulates that the ignition of a series of these groups of neurons, or cell assemblies, can explain how concepts are stored and recalled within our brains, thus allowing learning and memory to occur.

Mathematics of cell assemblies

In 1989, G. Palm was the first mathematician to give a mathematical definition of cell assemblies in his article Towards a Theory of Cell Assemblies . By finding a connection between graph theory and Palm's mathematical definition of cell assemblies, we have found a method for translating Palm's mathematical definition of a cell assembly into a binary integer programming problem. This has allowed us to find at least one cell assembly in a network of neurons and also gives us hope for finding more cell assemblies in the same networks. If we let G be a graph with a set of vertices $\{v_i:i=1{,}_{\cdots },n\}$ and a set of edges $\{w_i:i=1{,}_{\cdots },p\}$ , we can take the following from Palm to help us locate a cell assembly in a network of neurons:

Adjacency matrix ( A _d )

An adjacency matrix( A _d ) is a matrix of binary elements representing the connectivity of a given network of neurons such that if $A_d(n,m)=1$ there exists a connection between neurons n and m and conversely, if $A_d(n,m)=0$ then no connection exists between neurons n and m