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Recent studies of networks of neural oscillators have identified a number of important factors contributing to the synchronizing properties of synaptic interactions (1- 7). These include the time course of excitatory and inhibitory synapses and the type of neural response to small depolarizations. For example, it is found that slow excitatory synapses tend to be desynchronizing but can be synchronizing if the synapses are sufficiently fast (7). On the other hand, noninstantaneous inhibitory synapses tend to be synchronizing (4- 6). Most work on the synchronization of coupled neural oscillators has been carried out in the weak coupling regime where phase- reduction methods can be applied (1,8). However, in the case of integrate-and-fire (IF) and related spiking models it is possible to extend the analysis to the strong coupling regime (3,6,9- 11). So far most of the results obtained from the latter models have been consistent with the basic principles extracted from the weak coupling theory. In this Letter we show that synaptic interactions found to be synchronizing in the weak coupling regime can actu- ally become desynchronizing when the strength of interac- tions becomes large. We proceed by studying a pair of I F (type I) neural oscillators with noninstantaneous synaptic interactions. These are chosen such that the pair is syn- chronized for sufficiently small coupling. Linear stability analysis is carried out by considering perturbations of the oscillator firing times. We show that a discrete Hopf bi- furcation in the firing times can occur at a critical value of the coupling leading to the formation of a non-phase- locked state. In the case of symmetric inhibitory coupling this results in one of the oscillators becoming quiescent (oscillator death). On the other hand, in the presence of a mixture of excitation and inhibition, mode locking is found to occur with the two oscillators exhibiting periodic bursting patterns. We interpret these results in terms of an analog version of the I F model. We begin by considering two coupled IF oscillators whose state variables Uistd, i › 1, 2, evolve according to the pair of equations dUistd

Desynchronization, Mode Locking, and Bursting in Strongly Coupled Integrate-and-Fire Oscillators