Awakening dynamics via passive coupling and synchronization mechanism in oscillatory cellular neural/nonlinear networks

Abstract

SUMMARYWe have studied synchronization mechanism in locally coupled nonlinear oscillators. Here, synchronization takes place by passive coupling based on a reaction-diffusion process. We will compare this mechanism with basic synchronization techniques, showing their similarities and specific properties. In addition to synchronization, passive and local coupling can also 'awaken' non-oscillating cell circuits and trigger oscillation, provided that cells are locally active. This result resembles Turing's and Smale's works showing that locally communicating simple elements can produce very different patterns even if separate elements do not show any activity. This property will be demonstrated for two second-order cells and also for a large ensemble of oscillatory cells. In latter case, the network of oscillatory cells exhibits very sophisticated spatio-temporal waves, e.g. spiral waves.

Coupling-induced oscillations in two intrinsically quiescent populations

“…The fast subsystem (9), which is nothing but the classical LVG model, has four quasi-steady states-three boundary and one interior-denoted by Q (the slow variables are deemed to be frozen):…”

“…Since that time, triggered by Smale's seminal work, a number of plausible models have been proposed in which coupling of identical nonoscillating cells of concrete nature could generate synchronous oscillations. The majority of these models concern neural cells with excitable membrane [7,8,9]. Szatmári and Chua [9] suggested an apt term "awakening dynamics" for the phenomenon.…”

“…The majority of these models concern neural cells with excitable membrane [7,8,9]. Szatmári and Chua [9] suggested an apt term "awakening dynamics" for the phenomenon.…”

Synchronization of identical neural networks and other systems with an adaptive coupling strength

“…Hitherto, many approaches and criteria to ensure complete synchronization have been derived, see [15][16][17][18][19][20][21][22][23][24][25][26]. For example, in [16], the authors found that certain subsystems of nonlinear, chaotic systems can be made to synchronize by linking them with common signal, which may have the application on the neural processes; in [17,18] the authors presented a master stability function based on the transverse Lyapunov exponents to study local synchronization; in [19,20], a distance from synchronization manifold to each state was defined to study the global synchronization; in [21,22], the left eigenvector corresponding to the zero eigenvalue of the diffusive coupling matrix is utilized to investigate the global synchronization; Grassi and Mascolo [23] applied the concept of observer from system theory to synchronizing high-order oscillators; Wu et al [24] investigated chaos synchronization of the master-slave Chua's circuits by a general linear state error feedback controller with propagation delay; Jalili et al [25] investigated the synchronization of dynamical networks by using the connection graph stability method, which regarded the network topology as a graph; and Szatmári and Chua [26] studied synchronization mechanism among cells in reactiondiffusion systems, showed the similarities to basic pulse synchronization technique, and presented that the passive coupling among completely stable cells might produce very interesting dynamical behavior.…”

On the VLSI Implementation of Adaptive-Frequency Hopf Oscillator

“…Theoretically, nonlinear oscillators has been used for decades to model diverse natural phenomena such as neuronal signaling models [3], Central Pattern Generation (CPG) [4], [5], associative memory [6], [7] and beat perception [8], [9]; to engineering applications such as universal machines [10], [11], image processing [12], logic computation, [13] and robotics [14], [15]. The main challenge, which has limited a generalized practical application of nonlinear oscillators, is the mathematical complexity inherent in their model.…”