Biology Reference
In-Depth Information
There is a nice generalization of the second part of Exercise
6.15
to arbitrary
networks
N
,
1
p
constant: These networks have a periodic attractor
if and only if the connectivity
D
has a directed cycle of length at least
p
∗
+
=
D
,
p
with
1 (see
=
1, then the existence of any directed
cycle in
D
implies the existence of a periodic attractor in
N
. But the proof is rather
more difficult than Exercise
6.15
.
Exercise 6.35 [
1
] shows that the straightforward generalization of this result to
networks with unequal refractory periods fails. When we allow for firing thresholds
th
i
=
th
Theorem 14 of [
9
]). In particular, if
p
1, then for many interesting classes of digraphs the existence of a directed
cycle of any length no longer will be sufficient for the existence of a periodic attractor
in
N
,evenif
>
1. The problem of characterizing all networks that have at least
one periodic attractor under the most general assumptions on
p
=
p
and
th
remains open.
Project 4 [
1
] invites the reader to embark on an open-ended exploration of this problem,
starting with some interesting special cases.
The second problem we want to discuss here is finding optimal bounds for the
maximal length of attractors. For the classes of connectivities we have studied so far,
all attractors were relatively short (the maximal length is 1 for acyclic connectivities,
n
for directed cycle graphs; for complete digraphs when
th
=
1itis
lcm
{
p
i
+
:
1
∈[
]}
i
, which is typically much less than
n
when there are only few distinct types
of neurons and
n
is large). But in general, attractors can be much longer than
n
;some
examples are given in Exercise 6.17 and Project 5 [
1
]. What is the upper bound on
attractor length in the class of all networks
N
n
,
th
with
n
nodes,
p
∗
=
D
,
p
P
and
th
∗
1.
Project 5 [
1
] reviews some partial results and gives guidance for readers who wish to
explore this exciting problem.
Th
, where
P
,
Th
are fixed? This is an open question even for
P
=
Th
=
6.4.5
Discussion: Advantages and Limitations
of the Approach in this Section
The results of this section and the corresponding parts of [
1
] show that the study
of relations between network connectivity and network dynamics is a fertile ground
for exciting mathematical problems on the intersection of several areas of discrete
mathematics, including graph theory, enumerative combinatorics, and number theory.
The problems in our exercises and projects range from very easy ones to unsolved
research problems. One can see that the problem of completely characterizing all
possible dynamics becomes quickly very hard if we allow connectivities beyond the
simplest ones. This may be good news for mathematicians, because solving these
questions may well require development of new mathematical tools. It has even been
claimed that this type of investigation will lead to an entirely new kind of science
[
10
]. While the latter claim seems exaggerated, the hyperbole with which it has been
promoted should not deter mathematicians' efforts to develop new tools for studying
the dynamics of discrete-time finite-state networks.
The question arises to what extent the approach taken in this section is likely to
yield results that are relevant to neuroscience. After all, with very few exceptions
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