Understanding the dynamics and synchronization of inhibitory neurons is a question of fundamental importance in neuroscience, since inhibitory networks play a crucial role in rhythmogenesis, both in invertebrate motor pattern generators  and in the mammalian hippocampus and neocortex . Invertebrate CPGs in particular often contain simple two-cell inhibitory sub-networks that play a crucial role in the control of rhythmic motor behaviors. Therefore, characterizing the dynamics of two-cell inhibitory networks is relevant for a better understanding of the rhythmic dynamic activity produced by central pattern generators and other inhibitory circuits.
Here we describe the activity states in a network of two cells with type-I excitability coupled by reciprocal inhibition. Weak coupling analysis is very successful in the study of phase-locked activity in such a network, and predicts synchronous or anti-synchronous dynamics, depending on the time scale of inhibition and the intrinsic cell properties . However, it is known that an increase in coupling strength can destabilize synchrony in many neural circuits . In particular, recent work by Maran and Canavier  has shown that non-weak coupling leads to alternating-order firing (termed "leap-frog" spiking by G.B. Ermentrout) in an inhibitory network of two Wang-Buzsáki model neurons, whereby the order of spiking of the two cells alternates in each cycle of the oscillation. Here we show that such activity is a generic property of an inhibitory network of oscillators of type-I excitability class. In particular, we demonstrate that leap-frog spiking can also be obtained in a two-cell network of simpler Morris-Lecar model cells, as shown in Figure 1.
Figure 1. Alternating-order (leap-frog) activity in an inhibitory network of two Morris-Lecar cells with type-I excitability. The membrane potentials of the two model cells are shown as black and red traces, respectively.
We examine the phase-plane geometry of such order non-preserving dynamics, and find that it arises when the inhibition is sufficiently strong to allow a presynaptic cell to transiently suppress the postsynaptic cell below its excitability threshold (saddle-node on an invariant cycle bifurcation) in each cycle of the oscillation. Therefore, non-zero synaptic decay time is crucial for obtaining leap-frog spiking in a continuous system. However, we show that alternating-order spiking can also be obtained in an appropriately modified pulse-coupled integrate-and-fire network. Following the approach of Maran and Canavier , we show that the entire activity state bifurcation profile of the two-cell Morris-Lecar network can be completely characterized using the phase-resetting properties of each of the two cells, and describe the conditions on the phase resetting curve that lead to the loss of synchrony and the emergence of leap-frog spiking.
This work is partially supported by the National Science Foundation grant DMS-0417416
J Computational Neurosci 1998, 5:5-16. Publisher Full Text
J Math Biol 1991, 29:195-217. Publisher Full Text