4 An Introduction to Emergence Dynamics in Complex Systems
153
f (u) =
⎧
⎨
⎩
0,
if u < 1/3,
1 − 6.75u(u − 1)
2
, if
1
3
≤ u < 1,
1,
if u > 1.
(4.37)
The relaxation parameter ε 1 represents the time ratio between the activator
u and the inhibitor v. The dimensionless parameters a and b denote the activator
kinetics of the local dynamics and the ratio u T = b/a can effectively control the
excitation threshold. D is the coupling strength between linking nodes. A = {A i,j } is
the adjacency matrix. For a symmetric and bidirectional network, the matrix is defined
as A i,j = A j,i = 1 if there is a connection linking nodes i and j, and A i,j = A j,i = 0
otherwise.
We study the random network shown in Fig. 4.5a as a typical example. Without
couplings among nodes, each excitable node is non-oscillatory, i.e. they evolve
asymptotically to the rest state u = v = 0 and will stay there perpetually unless
some external force drives them away from this state. When a node is kicked from its
rest state by a stimulus large enough, the unit can excite by its own internal excitable
dynamics.
With the given network structure and parameters, one studies the dynamics of
the system by starting from different sets of random initial conditions. The system
evolves asymptotically to the homogeneous rest state in many cases. However, one
still finds a small portion of tests eventually exhibit global self-sustained oscillations.
The spatiotemporal patterns given in Figs. 4.5b, c are two different examples of these
oscillatory (both periodic and self-sustained) states.
One can unveil the mechanism supporting the oscillations and the excitation propagation paths by using the DPAD approach. In Figs. 4.5d, e, the reduced directed
networks corresponding to the oscillatory dynamics by using the DPAD method
are plotted. For the case with dynamics shown in Fig. 4.5b, the single dynamical
loop plays the role of oscillation source, with cells in the loop exciting sequentially
to maintain the self-sustained oscillation, as shown in Fig. 4.5d. We can observe
waves propagating downstream along several tree branches rooted at various cells
in the loop. If we plot the spatiotemporal dynamics along these various paths by
re-arranging the node indices according to the sequence in the loop, we can find
regular and perfect wave propagation patterns. This indicates that the DPAD structure well illustrates the wave propagation paths. For the case with dynamics shown
in Fig. 4.5c, the corresponding DPAD structure given in Fig. 4.5e is a superposition of two sub-DPAD paths, i.e. there are two organization loop centers, where two
sustained oscillations are produced in two loops and propagate along the trees.
The DPAD structures in Fig. 4.5d, e clearly show the distinctive significance
of some units in the oscillation which cannot be observed in Fig. 4.5b, c, where
units evolve in the homogeneous and randomly coupled network, and no unit takes
any priority over others in topology. Because the unidirectional loop works as the
oscillation source, units in the loop should be more important to the contribution of
the oscillation.
153
f (u) =
⎧
⎨
⎩
0,
if u < 1/3,
1 − 6.75u(u − 1)
2
, if
1
3
≤ u < 1,
1,
if u > 1.
(4.37)
The relaxation parameter ε 1 represents the time ratio between the activator
u and the inhibitor v. The dimensionless parameters a and b denote the activator
kinetics of the local dynamics and the ratio u T = b/a can effectively control the
excitation threshold. D is the coupling strength between linking nodes. A = {A i,j } is
the adjacency matrix. For a symmetric and bidirectional network, the matrix is defined
as A i,j = A j,i = 1 if there is a connection linking nodes i and j, and A i,j = A j,i = 0
otherwise.
We study the random network shown in Fig. 4.5a as a typical example. Without
couplings among nodes, each excitable node is non-oscillatory, i.e. they evolve
asymptotically to the rest state u = v = 0 and will stay there perpetually unless
some external force drives them away from this state. When a node is kicked from its
rest state by a stimulus large enough, the unit can excite by its own internal excitable
dynamics.
With the given network structure and parameters, one studies the dynamics of
the system by starting from different sets of random initial conditions. The system
evolves asymptotically to the homogeneous rest state in many cases. However, one
still finds a small portion of tests eventually exhibit global self-sustained oscillations.
The spatiotemporal patterns given in Figs. 4.5b, c are two different examples of these
oscillatory (both periodic and self-sustained) states.
One can unveil the mechanism supporting the oscillations and the excitation propagation paths by using the DPAD approach. In Figs. 4.5d, e, the reduced directed
networks corresponding to the oscillatory dynamics by using the DPAD method
are plotted. For the case with dynamics shown in Fig. 4.5b, the single dynamical
loop plays the role of oscillation source, with cells in the loop exciting sequentially
to maintain the self-sustained oscillation, as shown in Fig. 4.5d. We can observe
waves propagating downstream along several tree branches rooted at various cells
in the loop. If we plot the spatiotemporal dynamics along these various paths by
re-arranging the node indices according to the sequence in the loop, we can find
regular and perfect wave propagation patterns. This indicates that the DPAD structure well illustrates the wave propagation paths. For the case with dynamics shown
in Fig. 4.5c, the corresponding DPAD structure given in Fig. 4.5e is a superposition of two sub-DPAD paths, i.e. there are two organization loop centers, where two
sustained oscillations are produced in two loops and propagate along the trees.
The DPAD structures in Fig. 4.5d, e clearly show the distinctive significance
of some units in the oscillation which cannot be observed in Fig. 4.5b, c, where
units evolve in the homogeneous and randomly coupled network, and no unit takes
any priority over others in topology. Because the unidirectional loop works as the
oscillation source, units in the loop should be more important to the contribution of
the oscillation.