Dynamics of an adaptive higher-order Cohen–Grossberg model

In this paper, we study the dynamical behavior of an adaptive higher-order Cohen–Grossberg model and choose a biologically plausible rule specifying how the connection weights will vary in time, i.e., we incorporate an unsupervised Hebbian-type learning rule with a higher-order Cohen–Grossberg model. By constructing several Lyapunov functions, some sufficient conditions for the asymptotic and exponential stability of the equilibrium are derived. Furthermore, we also study how a temporally varying, in particular, a periodic environment, can influence on the dynamics of this model, i.e., the neuronal parameters, synaptic weights, and gains can either be temporally uniform or be periodic with same period as that of the stimulus. Sufficient condition for the existence of a globally attractive periodic solution associated with a given periodic external stimulus is also derived. Some numerical examples are employed to illustrate our theoretical results.