Phase locking between similar or dissimilar clusters (synchronized groups) of neurons may be widespread in the nervous system [1-3]. We examine two reciprocally coupled clusters of pulse-coupled oscillatory neurons. Neurons within each cluster are presumed to be identical and identically coupled but not necessarily identical to neurons in the other cluster. We construct a discrete map using Phase Response Curves (PRCs) for a firing pattern in which the neurons within each cluster are synchronized but the two clusters fire out of phase with respect to each other. We extend this map to include a perturbation of a single neuron within one cluster and linearize about the fixed point of the original map. We derive expressions that give stability of the phase-locked cluster solution using only the slopes of the PRC at the locking points. We give an example of a cluster of inhibitory Type II excitable neurons that cannot synchronize in isolation because the absolute value of the eigenvalue that determines synchrony in the isolated cluster is greater than one. The reciprocal coupling with another cluster scales this eigenvalue such that it becomes less than one, guaranteeing stability (Figure 1). These results suggest a mechanism by which local synchronization can be induced through reciprocal coupling between brain regions via the feedback loop.
Figure 1. A1,A2) Cluster 1 isolated and after coupled to cluster 2. B1) Neurons in cluster 1 firing out of phase B2) Cluster 1 neurons firing synchronously. (C) Examples of characteristic shapes of first and second order phase resets with Type II inhibition. D) Qualitative stability results for two clustered solution. Black bar(observed) obtained by integrating full system of differential equations for a range of conductance values. Compared to analysis in the current study denoted by purple bar. E) Maximum eigenvalues vs. conductance. greater than one for isolated cluster (blue dots) and less than one for the coupled cluster (purple dots).