Neuronal Networks: A Discrete Model - Mathematical Concepts and Methods in Modern Biology - page 194

the connectivity of neuronal networks in actual organisms is not known. The one
organism for which we have a complete wiring diagram of its neuronal network is the
little roundworm Caenorhabditis elegans . Its hermaphrodite form has 302 neurons.
The website [ 11 ] has a lot of fascinating information about this organism and its
nervous system. For other organisms we have wiring diagrams of important parts
of the nervous system, such as the stomatogastric ganglia of some crustaceans [ 12 ]
which contain between 20 and 30 neurons.
The connectivities of networks that have been mapped are much more compli-
cated than acyclic, directed cycle, or complete digraphs, and our results do not apply
to them directly. However, let us assume that we have a trustworthy discrete-time
finite-state model for the dynamics of such a model (we will discuss this assumption
in some detail in Section 6.7 ). If the network is very small (like in the ganglia of
crustaceans), one could use computer simulations as in the present section to com-
pletely characterize the dynamics. This will no longer be possible in a network like
the one for C. elegans with its 302 neurons, since there would be at least 2 302 states
to consider. But simulations still could be used to explore the dynamics for a large
sample of initial states. Theoretical investigations like the ones in this section are
still useful for this type of exploration: Let, for example,

( )

s 0 be an initial state in a

, 1 , 1 =

network N , where D is strongly connected and has a few hundred nodes.

Suppose that in simulations of the trajectory for, say, 1000 steps, we do not find (yet)
an attractor, but we discover a node for which the result of Exercise 6.13 guarantees
that it will be minimally cycling. While extending the simulation until the trajectory
visits the same state twice may not be computationally feasible, by Proposition 12 of
[ 7 ] we would know already that all nodes in the attractor will be minimally cycling
and the attractor will have length 2.
What if the neuronal network in the organism of interest is too large to map all
the connections? In that case, we cannot directly investigate the dynamics of the true
network, but we can explore the “typical” dynamics of a network that is randomly
drawn from the class of all networks that share certain properties for which we do have
empirical confirmation. Two such properties that are often known are an estimate of
the total number n of neurons and an estimate of the average number of connections
that a given neuron makes to other neurons. In the next section and Projects 6 and
7[ 1 ], we will introduce some methods for exploring “typical” dynamics. As it will
become clear, such methods rely on theoretical results of the kind that we proved in
the present section.

D

6.5 EXPLORING THE MODEL FOR SOME RANDOM
CONNECTIVITIES
6.5.1 Erd os-Rényi Random Digraphs
The notion of a “typical” connectivity can bemademathematically rigorous by assum-
ing that D is a digraph with vertex set

[ n ] that is randomly drawn from a given