Delayed coupling between two neural network loops

Coupled loops with time delays are common in physiological systems such as neural networks. We study a Hopfield-type network that consists of a pair of one-way loops each with three neurons and two-way coupling ( of either excitatory or inhibitory type) between a single neuron of each loop. Time delays are introduced in the connections between loops, and the effects of coupling strengths and delays on the network dynamics are investigated. These effects depend strongly on whether the coupling is symmetric ( of the same type in both directions) or asymmetric ( inhibitory in one direction and excitatory in the other). The network of six delay differential equations is studied by linear stability analysis and bifurcation theory. Loops having inherently stable zero solutions cannot be destabilized by weak coupling, regardless of the delay. Asymmetric coupling is weakly stabilizing but easily upset by delays. Symmetric coupling ( if not too weak) can destabilize an inherently stable zero solution, leading to nontrivial fixed points if the gain of the neuron response function is not too negative or to oscillation otherwise. In the oscillation case, intermediate delays can restabilize the zero solution. At the borderline of the weak coupling region ( symmetric or asymmetric), stability can change with delay ranges. When the coupling strengths are of the same magnitude, the oscillations of corresponding neurons in the two loops can be in phase, antiphase ( symmetric coupling), or one quarter period out of phase ( asymmetric coupling) depending on the delay.