On synchronization and control of coupled Wilson-Cowan neural oscillators. (English) Zbl 1077.34040

The paper investigates the dynamics of coupled Wilson-Cowan oscillators \[ \dot x_i = -\alpha x_i + f(ax_i -b y_i +\rho_x +\delta_x (x_{i-1}+x_{i+1})), \]
\[ \dot y_i = -\beta y_i + f(cx_i -d y_i +\rho_y -\delta_y (y_{i-1}+y_{i+1})), \]
\[ i=1,\dots\pmod n, \] where \(\alpha>0,\) \(\beta>0\), and \(a,b,c,d,\rho_x,\rho_y,\delta_x\) and \(\delta_y\) are constants. \(f\) is defined as \[ f(x)=1/(1+\exp{(-\varepsilon x})), \quad \varepsilon>0. \] Each oscillator in this model is described by the interaction of an excitatory and an inhibitory neuron.
The study is mainly focused on the case \(n=2\). First, the authors study the stability of periodic solutions and obtain corresponding bifurcation diagrams. Finally, they discuss how to stabilize an unstable periodic orbit embedded in the chaotic attractor by using continuous state feedback.


34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
34C25 Periodic solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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