Neuronal Networks: A Discrete Model - Mathematical Concepts and Methods in Modern Biology - page 176

CHAPTER

6

Neuronal Networks:
A Discrete Model

Winfried Just ∗ , Sungwoo Ahn † and David Terman ‡
∗ Department of Mathematics, Ohio University, Athens, OH 45701, USA
† Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis,
IN 46202, USA
‡ Department of Mathematics, Ohio State University, Columbus, OH 43210, USA

6.1 INTRODUCTION AND OVERVIEW
What do brains, ant colonies, and gene regulatory networks have in common? They are
networks of individual agents (neurons, ants, genes) that interact according to certain
rules. Neurons interact through electric and chemical signals at synapses, ants interact
by means of olfactory and tactile signals, and products of certain genes regulate the
expression of other genes. These interactions may cause a change of the state of an
agent. It seems often possible to model such networks by considering only finitely
many states for each agent. A neuron may fire or be at rest, an ant may be in the mood
to forage, defend the nest, or tend to the needs of the queen, and a gene may or may not
be expressed at any given time. If one also assumes that time progresses in discrete
steps, then one can build a discrete dynamical system model for such networks.
Anyone who has ever closely looked at an ant colony will realize that, in general,
there may be a significant amount of randomness in the agents' interactions, and
for this reason one may want to conceptualize many biological networks as stochas-
tic dynamical systems, as in the agent-based models of Chapter 4 in this volume.
But if there is sufficient predictability in the interactions, a mathematically simpler
deterministic model may be adequate. In this chapter we study in detail one such
deterministic model for networks of neurons; more examples of models of this type
are considered in several other chapters of this volume.
In a deterministic discrete dynamical system, the state of each agent at the next
time step is uniquely determined by its current state and the current states of all agents
it interacts with, according to the rules that determine the dynamics. The resulting
sequence of network states is called a trajectory . If the system has only finitely many
states, each trajectory must eventually enter a set of states that it will visit infinitely
often. This set is called its attractor. The sequence of states visited prior to reaching
the attractor is called the transient part of the trajectory, and the set of all initial states
from which a given attractor will be reached is called its basin of attraction.
The network connectivity specifies who interacts with whom. In neuronal networks
the connectivity corresponds to the wiring of the brain and can often be assumed to