148
Z. Zheng
as oscillations in gene-regulatory networks [46, 49–53], neural networks and brains
[54–59].
The exploration of the key determinants of collective oscillations is a great challenge. It is not an easy task to directly apply the slaving principle in synergetics
to dig out the order parameters in governing the self-sustained oscillations in this
complex system. Recent progresses revealed that some fundamental topologies or
sub-networks may dominate the emergence of sustained oscillations. Although the
collective self-sustained oscillation emerges from the organization of units in the
system, only a small number or some of key units form typical building blocks
and play the dominant role in giving rise to collective dynamics. Thus some key
topologies that are composed of a minor proportion of units may lead to a collective
oscillation and most other units play the role of slaves. We may call these organizing
centers the self-organization core or the oscillation source.
Theoretically, diverse self-sustained oscillatory activities and related determining
mechanisms have been reported in different kinds of excitable complex networks.
It was discovered that one-dimensional Winfree loops may support self-sustained
target group patterns in excitable networks [60, 61]. Moreover, it was also revealed
the center nodes and small skeletons to sustain target-wave-like patterns in excitable
homogeneous random networks [62–64]. The mechanism of long-period rhythmic
synchronous firings in excitable scale-free networks has been explored to explain the
temporal information processing in neural systems [65].
On the other hand, node dynamics on biological networks depends crucially
on different systems. For example, the gene dynamics is totally different from
excitable dynamics, and the network structures are also different. It was found
some fundamental building blocks in gene-regulatory networks can support sustained
oscillations, and the interesting chaotic dynamics and its mechanism were studied
[66–68].
Revealing the key topology of the organizing center, oscillation cores and further
the propagation path is the dominant mission in this section. We first propose two
useful methods, i.e. the dominant phase-advanced driving method and the functional weight approach, and then apply them to analyze and further pick up the key
topologies from dynamics.
4.3.2 The Dominant Phase-Advanced Driving (DPAD)
Method
An important subject in revealing the coordination of units is to explore the core structure and dynamics in the organization process of a large number of non-oscillatory
units. DPAD is a dynamical method that can find the strongest cross driving of
the target node when a system is in an oscillatory state. Here, we briefly recall the
dynamical DPAD structure [60–64].
Z. Zheng
as oscillations in gene-regulatory networks [46, 49–53], neural networks and brains
[54–59].
The exploration of the key determinants of collective oscillations is a great challenge. It is not an easy task to directly apply the slaving principle in synergetics
to dig out the order parameters in governing the self-sustained oscillations in this
complex system. Recent progresses revealed that some fundamental topologies or
sub-networks may dominate the emergence of sustained oscillations. Although the
collective self-sustained oscillation emerges from the organization of units in the
system, only a small number or some of key units form typical building blocks
and play the dominant role in giving rise to collective dynamics. Thus some key
topologies that are composed of a minor proportion of units may lead to a collective
oscillation and most other units play the role of slaves. We may call these organizing
centers the self-organization core or the oscillation source.
Theoretically, diverse self-sustained oscillatory activities and related determining
mechanisms have been reported in different kinds of excitable complex networks.
It was discovered that one-dimensional Winfree loops may support self-sustained
target group patterns in excitable networks [60, 61]. Moreover, it was also revealed
the center nodes and small skeletons to sustain target-wave-like patterns in excitable
homogeneous random networks [62–64]. The mechanism of long-period rhythmic
synchronous firings in excitable scale-free networks has been explored to explain the
temporal information processing in neural systems [65].
On the other hand, node dynamics on biological networks depends crucially
on different systems. For example, the gene dynamics is totally different from
excitable dynamics, and the network structures are also different. It was found
some fundamental building blocks in gene-regulatory networks can support sustained
oscillations, and the interesting chaotic dynamics and its mechanism were studied
[66–68].
Revealing the key topology of the organizing center, oscillation cores and further
the propagation path is the dominant mission in this section. We first propose two
useful methods, i.e. the dominant phase-advanced driving method and the functional weight approach, and then apply them to analyze and further pick up the key
topologies from dynamics.
4.3.2 The Dominant Phase-Advanced Driving (DPAD)
Method
An important subject in revealing the coordination of units is to explore the core structure and dynamics in the organization process of a large number of non-oscillatory
units. DPAD is a dynamical method that can find the strongest cross driving of
the target node when a system is in an oscillatory state. Here, we briefly recall the
dynamical DPAD structure [60–64].