On the dynamics of electrically-coupled neurons with inhibitory synapses

We study the dynamics and bifurcations of noise-free neurons coupled by gap junctions and inhibitory synapses, using both delayed delta functions and alpha functions to model the latter. We focus on the case of two cells, as in the studies of Chow and Kopell (2000) and Lewis and Rinzel (2003), but also show that stable asynchronous splay states exist for globally coupled networks of N cells dominated by subthreshold electrical coupling. Our results agree with those of Lewis and Rinzel (2003) in the weak coupling range, but our Poincaré map analysis yields more information about global behavior and domains of attraction, and we show that the explicit discontinuous maps derived using delayed delta functions compare well with the continuous history-dependent, implicitly-defined maps derived from alpha functions. We find that increased bias currents, super-threshold electrical coupling and synaptic delays promote synchrony, while sub-threshold electrical coupling and fast synapses promote asynchrony. We compare our analytical results with simulations of an ionic current model of spiking cells, and briefly discuss implications for stimulus response modes of locus coeruleus and for central pattern generators.