Biology Reference
In-Depth Information
4.2.4 Definition of the Structural Stability with Respect
to Updating Modes
A dynamical system is called structurally stable with respect to updating modes, if
no disturbance in the updating schedule, e.g., by passing from the sequential mode
(in which nodes of the network are updated by the state transition rule one after the
other in a given order) to the parallel one (in which all the nodes of the network are
updated by the state transition rule at the same time), can change the number or
nature of its attractors.
4.2.5 Definition of the Resistance to Boundary Perturbations
(Resilience)
A dynamical system having a frontier separating them from its environment is
resistant to boundary perturbations, if no perturbation in state or architecture of the
environmental elements can provoke a change in number or nature of its attractors.
4.2.6 Definition of the Robustness (Resilience)
A dynamical system whose trajectories
x
p
(
a
,
t
) depend on a parameter
p
is said to be
robust (or resilient), if all of its trajectories are asymptotically stable and if it is
boundary resistant and structurally stable with respect to any parameter or updating
schedule perturbation. We will give in the next Section some examples of robust
and non-robust biological networks at different levels, genetic, metabolic,
and physiologic. These networks can be decomposed following their dynamical
typology and we can distinguish between four categories of dynamics, whose
definition will be given hereafter:
- Potential (or gradient, or purely dissipative)
- Hamiltonian (or conservative)
- Mixed potential-Hamiltonian (MPH)
- MPH with principal potential part.
4.2.7 Definition of a Potential Dynamics
A dynamical system has a potential dynamics if the velocity along its trajectories is
equal to the gradient of a scalar potential
P
defined on the state space
E
. If the
system is governed by a differential equation defining its state transition rule, we
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