Disordered Oscillations in a Neural Network of Three Oscillators with a Delayed Broadcast Connection

Abstract

A model of neural association of three pulsed neurons with a delayed broadcast connection is considered. It is assumed that the parameters of the problem are chosen near the critical point of stability loss by the homogeneous equilibrium state of the system. Because of the broadcast connection the equation corresponding to one of the oscillators can be detached in the system. The two remaining impulse neurons interact with each other and, in addition, there is a periodic external action, determined by the broadcast neuron. Under these conditions, the normal form of this system is constructed for the values of parameters close to the critical ones on a stable invariant integral manifold. This normal form is reduced to a four-dimensional system with two variables responsible for the oscillation amplitudes, and the other two, defined as the difference between the phase variables of these oscillators with the phase variable of the broadcast oscillator. The obtained normal form has an invariant manifold on which the amplitude and phase variables of the oscillators coincide. The dynamics of the problem on this manifold is described. An important result was obtained on the basis of numerical analysis of the normal form. It turned out that periodic and chaotic oscillatory solutions can occur when the coupling between the oscillators is weakened. Moreover, a cascade of bifurcations associated with the same type of phase rearrangements was discovered, where a self-symmetric stable cycle alternately loses symmetry with the appearance of two symmetrical cycles. A cascade of bifurcations of doubling occurs with each of these cycles with the appearance of symmetric chaotic regimes. With further reduction of the coupling parameter, these symmetric chaotic regimes are combined into a self-symmetric one, which is then rebuilt into a self-symmetric cycle of a more complex form compared to the cycle obtained at the previous step. Then the whole process is repeated. Lyapunov exponents were calculated to study chaotic attractors of the system.