Complicated Dynamic Regimes in a Neural Network of Three Oscillators with a Delayed Broadcast Connection

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. Two remaining pulsed 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 are 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. Dynamics of the problem is described on this manifold. 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 link between the oscillators is weakened. Moreover, a cascade of bifurcations associated with the same type of phase transformations was discovered, where a self-symmetric stable cycle alternately loses symmetry with the appearance of two symmetrical to each other cycles. A cascade of bifurcations of period doubling occurs with each of these cycles with the appearance of symmetric chaotic regimes. With further decreasing 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 computed to study chaotic attractors of the system.