Strategies to Maximize Burst Lengths in Rhythmic Anti-Phase Activity of Networks with Reciprocal Inhibition

We consider repetitive activity patterns in which a pair of oscillators take turns becoming active, motivated by anti-phase bursting activity in neuronal networks. In our framework, when one unit is active, it inhibits the other, as occurs with inhibitory synaptic coupling of neurons; when the inhibition is strong enough, the inhibited unit is prevented from activating. We assume that the coupling resources available to each oscillator are constrained and allow each unit to select the amount of input that it provides to the other each time that it activates. In this setting, we investigate the strategies that each oscillator should utilize in order to maximize the number of spikes it can fire (or equivalently the amount of time it is active), corresponding to a burst of spikes in a neuron, before the other unit takes over. We derive a one-dimensional map whose fixed points correspond to periodic anti-phase bursting solutions. We introduce a novel numerical method to obtain the graph of this map and we extend the analysis to select solutions that achieve consistency between coupling resource use and recovery. Our analysis shows that corresponding to each fixed point of the map, there is actually an infinite number of related strategies that yield the same number of spikes per burst.