Neuronal Networks: A Discrete Model - Mathematical Concepts and Methods in Modern Biology - page 186

cycles of length bigger than one in the state transition digraph D N . Moreover, for the
two examples with periodic attractors that we have explored so far, we also found
directed cycles in the network connectivity D . Is the existence of directed cycles in D
necessary for the existence of periodic attractors? Is it sufficient? Note that in essence
these are questions about how the network connectivity D is related to the network
dynamics. We will study this type of question in the next two sections. In Section
6.4 we will try to derive provable bounds on certain features of the dynamics such as
maximumpossible lengths of transients and attractors when D belongs to some special
classes of digraphs. In Section 6.5 we will derive bounds that are true “on average”
for connectivities D that are randomly drawn from certain probability distributions.
Numerical simulations are another powerful tool for exploring network dynamics.
For a given network one could in principle use software to track the trajectory of
every possible initial state until it visits the same state twice and record the length
of all transients, all attractors, their lengths, and sizes of their basins of attraction.
This is possible in practice when the number of nodes n is relatively small, but
quickly becomes computationally infeasible since the number of states grows at least
exponentially in n .
Exercise 6.6.

, th = ,

Suppose N D p is a network with n nodes. Show that the

size of the state space is

n

| St N |=

1 (

p i + 1 ). (6.1)

i =

n , and for arbitrary

Conclude that if p = ( p ,..., p ) is constant, then | St N |= ( p + 1 )

2 n .

p we always have | St N |

= 1 contain about a billion different
states. For larger networks one can still explore the trajectories of a few randomly
chosen initial states, but it may be computationally infeasible to run the simulation
until the same state will be reached twice. Exercise 6.17 of the online supplement
[ 1 ] gives an illustration that even networks with very few nodes can have very long
transients and attractors. Thus rigorous results about the dynamics of all but the
smallest networks will require proving theorems. However, computer simulations
can be very useful for generating conjectures for such results and giving us insight
into the mechanisms that are responsible for interesting features of the dynamics.
The online supplement for this chapter contains a number of projects that involve
such numerical explorations.

Thus even networks with 30 nodes and p

6.4 EXPLORING THE MODEL FOR SOME SIMPLE
CONNECTIVITIES
Recall that in Exercise 6.5 we explored the following numerical characteristics of
network dynamics: lengths of the attractors, number of different attractors, sizes of
their basins of attraction, and maximum lengths of transients. In this section we will