Neuronal Networks: A Discrete Model - Mathematical Concepts and Methods in Modern Biology - page 199

(a stochastic process) over millions of years, so we should expect that the connectiv-
ity is at least somewhat random. But there is no absolute notion of randomness; this
mathematical concept is always based on some distributions. However, the distribu-
tions

D n that define Erdos-Rényi random digraphs may not be the most realistic ones.
Recall our derivation of the formula ( 6.9 ) for the average indegrees and outdegrees
in Erdos-Rényi digraphs. These were based on sums of independent randomvariables.
Thus it follows from the Central Limit Theorem that for large n the ratio between
the smallest indegree and the largest indegree of any node will be very close to
1 with probability arbitrarily close to 1; similarly for outdegrees. This is not what
we observe in many real-world networks. In particular, some empirical studies have
found evidence that the degree distributions in some actual neuronal networks are so-
called power law or scale-free distributions, with a small number of highly connected
neurons [ 14 , 15 ]. Erdos-Rényi digraphs are the wrong kind of random digraphs for
modeling such networks.
Project 6 [ 1 ] gives a brief introduction to power law distributions and methods
for randomly generating such digraphs with certain parameters. It invites the reader
to explore the dependence of P for

networks with the resulting connectivities on the parameters of the degree distribution.

( NS ), P ( MNODE ), P ( AU T O ), P ( FAMA )

6.6 ANOTHER INTERPRETATION OF THE MODEL:
DISEASE DYNAMICS
While our models are motivated by a problem in neuroscience and while we refer to
our models N as “neuronal networks,” there is nothing inherently “neuronal” about
these structures. They are just mathematical objects. In general, a given mathematical
object, such as a dynamical system, may serve as a model for any number of very
different natural systems.
Imagine an infectious disease that spreads in a population of n individuals. At
any given time, an individual can be in one of three categories: infected and prone
to infect others (in the set I ), healthy but susceptible to the infection (in the set S ),
or recovered and (temporarily or permanently) immune to the disease (in the set R ).
Infectionmay result froma contact between an infectious and a susceptible individual.
These assumptions lead to the classical SIR models of disease dynamics (see [ 16 ]fora
review). Notice that infectiousness is similar to the firing of a neuron: It can induce the
same state at a subsequent time in another individual. In this section we will explore
whether our networks might be suitable models for the dynamics of certain diseases.
The correspondence between a mathematical model and the underlying natu-
ral system is never perfect; mathematical modeling is always based on simplifying
assumptions. The so-called art of mathematical modeling is in essence a knack for
making simplifying assumptions that lead to models which are simple enough to
allow exploration either by computer simulations or mathematical methods and yet
incorporate enough detail to make realistic predictions about a natural system. The
first question a modeler needs to address is which aspects of the natural system the