Bursting induced by excitatory synaptic coupling in nonidentical conditional relaxation oscillators or square-wave bursters

This work explains a mechanism through which the introduction of exci- tatory synaptic coupling between two model cells, one of which is excitable and the other of which is tonically active when uncoupled, leads to bursting in the resulting two-cell network. This phenomenon can arise when the individual cells are conditional relaxation oscillators, in that they can be tuned to engage in relaxation oscillations, or when they are conditional square-wave bursters. The mechanism is illustrated with a model for con- ditional pacemaker neurons in the pre-Botzinger complex as well as with a reduced form of this model. In the relaxation oscillator case, a periodic bursting solution is proved to exist in the singular limit, under a pair of general conditions. These conditions relate the durations of the silent and active phases of the bursting solution to the locations of certain structures in the phase plane, at appropriate synaptic input strengths. Further, additional conditions on the relative flow rates in the silent and active phases are proved to imply the uniqueness and asymptotic stability of the bursting solution.