Neuronal Networks: A Discrete Model - Mathematical Concepts and Methods in Modern Biology - page 192

It is easy to see that Proposition 6.10 implies the following:
Corollary 6.12. , 1

Let N = D max ( n ), p for some n. Then every periodic attractor

is fully active and minimal.
As long as p = ( p ,..., p ) is constant, Exercise 6.12 (b) implies that every periodic

attractor has length p 1, which is the minimum possible length of any periodic

attractor in networks with constant refractory periods and arbitrary connectivity. The
latter observation explains why we call such attractors minimal. If

+

p is not constant
though, as the next exercise shows, minimal attractors are longer than that.
Exercise 6.14.

, 1 Let N = D max ( n ), p for some n . Show that the length of any

periodic attractor AT is equal to lcm

One can deduce from the above that in contrast to Proposition 6.7 for directed
cycle graphs, there does not exist a universal upper bound for the attractor length of
networks with complete loop-free connectivities. Exercise 6.20 [ 1 ] gives more details
and also shows that the same remark applies to the number of different attractors.
If

{ p i + 1 : i ∈[ n ]} .

th

= 1, then it may no longer be true that every periodic attractor is minimal
or fully active, see Exercise 6.21 [ 1 ]. The example in this exercise is somewhat
pathological though: For networks N

, th =

D max (

),

with n sufficiently large
relative to p ∗ and th ∗ , a typical initial state will still belong to a fully active minimal
attractor. Moreover, the same is true for typical network N

n p

th

whenever
the connectivity digraph has sufficiently many arcs. We will prove these results and
spell out a precise meaning of “typical” in Section 6.5 .

= D , p ,

6.4.4 Other Connectivities
There are a number of other interesting classes of digraphs for which one might
explore the possible dynamics of corresponding networks. This leads to more chal-
lenging problems than for the classes of acyclic, directed cycle, and complete loop-free
digraphs that we have discussed so far. This territory remains largely unexplored. The
Wikipedia page [ 8 ] contains a gallery of named graphs, and the reader may want to
explore these questions for selected connectivities that are digraph counterparts of
some of these graphs.
In this subsection we attempt to give a flavor of this area of research and suggest
projects for open-ended exploration. As a sample from the large menu of possible
research directions we will focus on two problems.
When does the existence of a directed cycle in the network connectivity imply the
existence of a periodic attractor in the network? By Lemma 6.5 the existence of a
directed cycle in D is necessary. But it is not all by itself sufficient; the directed cycle
must also be long enough.
Exercise 6.15. , 1 . Deduce from the proof of Proposition 6.7

that N has a periodic attractor only if the length n of the directed cycle is at least
p ∗ +

Let N = D c ( n ), p

1. Show that the latter condition is also sufficient for the existence of a periodic
attractor in this type of networks.