Synchronized neural activity is thought to underlie many aspects of cognition, including recognition, recall, perception, and attention. Coordinated neural activity in central pattern generators (CPGs) located in the spinal cords of vertebrates or the ganglia of invertebrates also underlies repetitive motor activity. Pathological synchrony is associated with epilepsy and tremor. In order to understand how synchrony can be established and maintained, we have focused on synchronization that is a result of interactions between neurons that oscillate intrinsically in the absence of coupling. Neurons are excitable cells that emit pulsatile electrical signals called action potentials. An equivalent RC circuit for a neuron includes nonlinear resistors and can be mathematically described as a system of coupled nonlinear ordinary differential equations. In certain parameter regimes, neurons can fire repetitively in an oscillatory manner.

Our approach to understanding coordinated network activity is to characterize each isolated oscillator by how its cycle length is lengthened or shortened by an input from another oscillator, depending upon at what point in the cycle the pulsatile input is received. Since this information can be obtained experimentally, we do not require the intermediate step of constructing an equivalent circuit model. Using the measured changes in cycle length, we can then assume a network architecture and formulate coupled systems of nonlinear recurrence relations for activity in this network. We develop algorithms to find the fixed points of these relations. The fixed points of the recurrence relations correspond to periodic phase locked modes of the network. We then linearize about these fixed points to obtain difference equations that give the stability of the phase locked modes.