Neuronal Networks: A Discrete Model - Mathematical Concepts and Methods in Modern Biology - page 177

be fixed over time scales of interest. Our focus in this chapter is on how the net-
work connectivity and two intrinsic parameters of individual neurons, the refractory
period and firing threshold, influence the network dynamics, in particular the lengths
and number of attractors, the sizes of their basins of attraction, and the lengths of
transients.
The chapter is organized as follows. Section 6.2 reviews some basic facts about
neuronal networks as well as some experimentally observed phenomena that moti-
vated our work. In Section 6.3 we formally define our discrete models and review
their basic properties. In Section 6.4 we study provable restrictions on the network
dynamics for certain special connectivities, and Section 6.5 reviews some results on
the dynamics of typical networks with given parameters. In Section 6.6 we consider
an alternative interpretation of our models in terms of disease dynamics and discuss
some general issues of choosing an appropriate mathematical model for a biological
system. In Section 6.7 we discuss whether our models may be applicable to actual
problems in neuroscience and review a result that guarantees an exact correspondence
between our discrete models and more detailed ordinary differential equation (ODE)
models of certain neuronal networks. In Section 6.8 we describe how the material
presented here fits into the larger picture of some current research on the cutting edge
of mathematical neuroscience.
We recommend that the reader attempts the exercises included in themain text right
away while reading it. They are primarily intended to help the reader gain familiarity
with the concepts that we introduce. The additional exercises in the online supplement
[ 1 ] are less crucial for understanding the material and can be deferred until later. The
online supplement also contains some additional material that is related to the main
text and, most importantly, several projects for open-ended exploration. Most of these
projects are structured in such a way that they start with relatively easy exercises and
gradually lead into unsolved research problems.
6.2 NEUROSCIENCE IN A NUTSHELL
6.2.1 Neurons, Synapses, and Action Potentials
It is commonly believed that everything the brain does, and therefore everythingwe, as
humans, do—from cognitive tasks such as thinking, planning, and learning to motor
tasks such as walking, breathing, and eating—is the result of the collective electrical
activity of neurons. There are roughly 10 12 neurons in the human brain. Neurons com-
municate with other neurons at synapses and a brain has approximately 10 15 synaptic
connections; that is, on average, each neuron receives input from approximately 1000
other neurons. Whether or not a neuron fires an electrical signal, or action potential ,
depends on many factors. These include electrical and chemical processes within
the neuron itself, properties of the synaptic connections and the underlying network
architecture. A fundamental issue in neuroscience is to understand how these three
factors interact to generate the complex activity patterns of populations of neurons
that underlie all brain functions.