Neuronal Networks: A Discrete Model - Mathematical Concepts and Methods in Modern Biology - page 178

FIGURE 6.1
A schematic drawing of a neuron. More pictures of neurons can be found at [ 2 ].
A schema of a neuron is shown in Figure 6.1 . Most neurons consist of dendrites,
an axon, and a cell body (or soma). The dendrites spread out from the cell body in
a tree-like manner and detect incoming signals from other neurons. In response to
these incoming signals, the neuron may (or may not) generate an action potential
(or nerve impulse) that propagates away from the soma along the axon. Many axons
develop side branches that help bring information to several parts of the nervous
system simultaneously.
All living cells are surrounded by a cell membrane that maintains a resting potential
between the outside and the inside of the cell. In response to a signal, the membrane
potential of a neuron may undergo a series of rapid changes, corresponding to the
action potential. In order to generate an action potential, the initial stimulus must be
above some threshold amount. Properties of the nerve impulse, including its shape and
propagation velocity, are often independent of the initial (superthreshold) stimulus.
Once a neuron fires an action potential, there is a so-called refractory period .During
this time, it is impossible for the neuron to generate another action potential.
6.2.2 Firing Patterns
The firing pattern of even a single neuron may be quite complicated. An individual
cell may, for example, fire action potentials in a periodic fashion, or it may exhibit
bursting oscillations inwhich periods of rapid firing alternatewith periods of quiescent
behavior. More complicated patterns, including chaotic dynamics, are also possible.
There has been a great deal of effort among mathematicians in trying to classify the
types of firing patterns that can arise in single neuron models. For references, see [ 3 ].
In this chapter, we are primarily concerned with the collective behavior of a
neuronal population. Examples of population rhythms include synchrony, in which
every cell in the network fires at the same time and clustering, in which the entire
population breaks up into subpopulations or clusters; every cell within a single cluster