Synchronized Activity and Loss of Synchrony Among Heterogeneous Conditional Oscillators

The inspiratory phase of the respiratory rhythm involves the synchronized bursting of a network of neurons in the brain stem. This paper considers activity patterns in a reduced model for this network, namely, a system of conductance-based ordinary differential equations with excitatory synaptic coupling, incorporating heterogeneities across cells. The model cells are relaxation oscillators; that is, no spikes are included. In the continuum limit, under assumptions based on the disparate time scales in the model, we derive consistency conditions sufficient to give tightly synchronized oscillations; when these hold, we solve a fixed point equation to find a unique synchronized periodic solution. This solution is stable within a certain solution class, and we provide a general sufficient condition for its stability. Allowing oscillations that are less cohesive but still synchronized, we derive an ordinary differential equation boundary value problem that we solve numerically to find a corresponding periodic solution. These results help explain how heterogeneities among synaptically coupled oscillators can enhance the tendency toward synchronization of their activity. Finally, we consider conditions for synchrony to break down.