Modular Neurodynamics and Its Classification by Synchronization Cores

It is assumed that the cause of cognitive and behavioral capacities of living systems is to be found in the complex structure-function relationship of their brains; a property that is still difficult to decipher. Based on a neurodynamics approach to embodied cognition this paper introduces a method to guide the development of modular neural systems into the direction of enhanced cognitive abilities. It uses formally the synchronization of subnetworks to split the dynamics of coupled systems into synchronized and asynchronous components. The concept of a synchronization core is introduced to represent a whole family of parameterized neurodynamical systems living in a synchronization manifold. It is used to identify those coupled systems having a rich spectrum of dynamical properties. Special coupling structures—called generative—are identified which allow to make the synchronized dynamics more “complex” than the dynamics of the isolated parts. Furthermore, a criterion for coupling structures is given which, in addition to the synchronized dynamics, allows also for an asynchronous dynamics by destabilizing the synchronization manifold. The large class of synchronization equivalent systems contains networks with very different coupling structures and weights allsharing the same dynamical properties. To demonstrate the method a simple example is discussed in detail.

Stability of The Synchronization Manifold in An All-To-All Time LAG- Diffusively Coupled Oscillators

we consider a lattice system of identical oscillators that are all coupled to one another with a diffusive coupling that has a time lag. We use the natural splitting of the system into synchronized manifold and transversal manifold to estimate the value of the time lag for which the stability of the system follows from that without a time lag. Each oscillator has a unique periodic solution that is attracting.

STABILITY OF THE CONTROLLED SYNCHRONIZATION MANIFOLD IN A RING OF MUTUALLY COUPLED CHAOTIC SYSTEMS

The active control of the unstable synchronization manifold in a shift-invariant ring of N mutually coupled chaotic oscillators is investigated. After deriving the bifurcation structures and chaotic states in the single oscillator, we find the regime of coupling parameters leading to stable and unstable synchronization phenomena in the ring, using the Master stability function approach with the transverse Lyapunov exponents. The active control technique is applied on the mutually coupled chaotic systems to suppress unstable synchronization states. We derive the range of control gain parameters which leads to a successful control and the stability of the control design. The effects of the amplitude of the parametric perturbations on the stability boundaries of the controlled unstable synchronization process are also studied.

SYNCHRONIZATION IN NETWORKS OF SLIGHTLY NONIDENTICAL ELEMENTS

We study synchronization processes in networks of slightly nonidentical chaotic systems, for which a complete invariant synchronization manifold does not rigorously exist. We show and quantify how a slightly dispersed distribution in parameters can be properly modeled by a noise term affecting the stability of the synchronous invariant solution emerging for identical systems when the parameter is set at the mean value of the original distribution.