Oscillatory networks are a special class of neural networks where each neuron exhibits time periodic behavior.
They represent bio-inspired architectures which can be exploited to model biological processes such as the binding
problem and selective attention. In most of situations, each neuron is assumed to have a stable limit cycle as
the unique attractor. In this paper we investigate the dynamics of networks whose neurons are hard oscillators,
namely they exhibit the coexistence of a stable limit cycle and a stable equilibrium point. We consider a constant
external stimulus applied to each neuron, which influences the neuron's own natural frequency. We investigate
the bifurcations in the neuron's dynamics induced by the input. We show that, due to the interaction between
different kind of attractors, as well as between attractors and repellors, new interesting dynamics arises, in the
form of synchronous oscillations of various amplitudes. We also show that neurons subject to different stimuli
are able to synchronize if their couplings are strong enough.
Many studies in neuroscience have shown that nonlinear dynamic networks represent a bio-inspired models for information and image processing. Recent studies on the thalamo-cortical system have shown that weakly connected oscillatory networks have the capability of modelling the architecture of a neurocomputer. In particular they have associative properties and can be exploited for dynamic pattern recognition. In this manuscript the global dynamic behavior of weakly connected cellular networks of oscillators are investigated. It is assumed that each cell admits of a Lur'e description. In case of weak coupling the main dynamic features of the network are revealed by the phase deviation equation (i.e. the equation that describes the phase deviation due to the weak coupling). Firstly a very accurate analytic expression of the phase deviation equation is derived via the joint application of the describing function technique and of Malkin's Theorem. Then it is shown that the total number of periodic limit cycles with their stability properties can be estimated through the analysis of the phase deviation equation.