Deterministic stability regimes and noise-induced quasistable behavior in a pair of reciprocally inhibitory neurons

Reciprocal inhibition is a common motif exploited by neuronal networks; an intuitive and tractable way to examine the behaviors produced by reciprocal inhibition is to consider a pair of neurons that synaptically inhibit each other and receive constant or noisy excitatory driving currents. In this work, we examine reciprocal inhibition using two models (a voltage-based and a current-based integrate-and-fire model with instantaneous or temporally structured input), and we use analytic and computational tools to examine the bifurcations that occur and study the various possible monostable, bistable, and tristable regimes that can exist; we find that, depending on system parameters (and on choice of neuron model), there can exist up to 3 distinct monostable regimes (denoted M0, M1, M2), 3 distinct bistable regimes (denoted B, B1, B2), and a single tristable regime (denoted T). We also find that synaptic inhibition exerts independent control over the two neurons - inhibition from neuron 1 to neuron 2 governs the spiking behavior of neuron 2 but has no impact on the spiking behavior of neuron 1 (and vice versa). The excitatory driving current, however, does not exhibit this property - the excitatory current to neuron 1 affects the spiking behavior of both neurons 1 and 2 (as does the excitatory current to neuron 2). Furthermore, we develop a methodology to examine the behavior of the system when the excitatory driving currents are allowed to be noisy, and we investigate the relationship between the behavior of the noisy system with the stability regime of the corresponding deterministic system.