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4
4
4
A.
B.
C.
2
2
2
0
0
0
0
1000
2000
0
1000
2000
0
1000
2000
time
time
time
4
4
D.
E.
2
2
0
0
0
1000
2000
0
1000
2000
time
time
4
4
F.
T
1
T
2
G.
T
1
T
2
T
3
T
4
2
2
0
0
0
1000
2000
0
1000
2000
time
time
Fig. 3.9. Dierent oscillatory clusters for a system of N = 8 oscillators. A: In-phase oscillations
for
1
= 3; d = 0:005; d
e
= 1. B, C: Anti-phase oscillations with dierent distributions of the
oscillators between the clusters, for
1
= 3:3, d = 0:001. D, E: Asymmetric solution with dierent
distribution of the oscillators, for
1
= 2:868, d = 0:001. F: Three oscillatory clusters for
1
= 3:3,
d = 0:00105. G: Five oscillatory clusters for
1
= 3:3, d = 0:001.
We discuss here two main phenomena. First, we show the existence of dierent
possible modes of organized collective behavior in the system of globally coupled
relaxation genetic oscillators. We distinguish between two dierent types of clusters:
(i) steady-state clusters, and (ii) oscillatory clusters. Second, for each separate
cluster formation, we demonstrate how the dependence on initial conditions can
lead to dierent distributions of the oscillators between the clusters. In general, a
system consisting of N oscillators can exhibit N1 dierent distributions of the
oscillators among the clusters.
When the cells are identical, the coupled system is symmetric and identical
behavior of the cells is a solution (Fig. 3.9A), though not necessarily stable. The
inhibitory coupling and the presence of multiple time scales, as previously discussed,
create the possibility for multistability and multirhythmicity, resulting in the gen-
eration of various dynamical regimes, among which oscillatory clusters are formed.
For d < 0:01, the system can exhibit anti-phase oscillations, with oscillators dis-
tributed between the two oscillatory clusters (Fig. 3.9B,C). An important feature to
be mentioned is the characterization of dierent distributions with dierent periods
of the limit cycle, providing more complex dynamics with dierent rhythms: com-
