Synchronization and desynchronization of neural oscillators

Abstract

We have used continuous and discrete-time versions of a neural oscillator model to analyze how various types of synaptic connections between oscillators affect synchronization and desynchronization phenomena. First, we present a synthesis of the mathematical properties of both neural oscillator versions. Then, we show that the choice of parameters leads to a relationship between the two versions. Finally, we achieve the coupling of two oscillators in order to study how synaptic connections affect the phase lag. With this in mind, we state some of the results for the continuous-time model. The second part of this paper deals with the behavior of neural networks comprising connected oscillators, which involves looking at the conditions for desynchronization of a totally synchronized oscillator net. Such a study has been carried out both for a fully and for a sparsely connected network. This leads to the observation that some architectures enable proper desynchronization when the size of the network is large. While searching for the conditions for desynchronization, we have discovered that a macroscopic description of the network is sometimes possible. To conclude, we discuss the advantages and the limitations of this macroscopic approach.

Attractive Periodic Sets in Discrete-Time Recurrent Networks (with Emphasis on Fixed-Point Stability and Bifurcations in Two-Neuron Networks)

“…Saddle xed points appear in the neighborhood of the four sides of the activation square. Unlike in the previous studies (e.g., Blum & Wang, 1992;Borisyuk & Kirillov, 1992;Pasemann, 1993;Tonnelier et al, 1999), we allowed all free parameters of the network to vary.…”

“…Relations between the dynamics of discrete-time and continuous-time networks are considered, for example, in Blum and Wang (1992) and Tonnelier et al (1999). Hirsch (1994) gives a simple example illustrating that there is no xed step size for discretizing continuous-time dynamics that would yield a discrete-time dynamics accurately re ecting its continuoustime origin.…”

“…This is partly due to the lack of mathematical tools for a detailed analysis of higher-dimensional dynamical systems and partly due to the high expressive power of such simple networks, in which many generic properties of larger networks are already present (Botelho, 1999). Moreover, two-neuron networks are sometimes thought of as systems of two modules, where each module represents the mean activity of a spatially localized neural population (Borisyuk & Kirillov, 1992;Pasemann, 1995a;Tonnelier, Meignen, Bosh, & Demongeot, 1999).…”

ϵ-Semantics computations on biological systems

“…A simple continuous model of a single oscillator has been proposed in [20]. The model describes the evolutions of one excitatory neuron (N e ) and one inhibitory neuron (N i ) by mean of the ordinary differential system.…”

Chaotic Dynamics of a Linear Chain of Periodically Stimulated Neurons with Random Synaptic Connections

“…[1,2]), qua sip eri odi c evoluti on [3,4], i nterm i ttency [5,6], synchro ni zati on [3,7] and sto chasti c resonance [ 8{ 10]. A pa rti cul ar ro l e pl ays chaoti c ti m e evol uti on, whi ch m ay o ccur in di˜erent ki nds of ANNs, b ecause it resembl es the ti m e evol uti on of the bra i n [11].…”