Synapses showing a preferred frequency in a reciprocally inhibitory neuronal network

Recent data from our lab indicates that inhibitory synapses in the crab pyloric central pattern generator (freq ~ 1 Hz) show resonance with a preferred presynaptic frequency of ~ 0.5 Hz (Fig. 1A). Additionally, it is known that the bursting pacemaker neurons and some bursting follower neurons in this network have maximum impedance frequencies in response to periodic current injection (membrane resonance), all within the range of the network frequency [4]. To explore the potential role of synaptic preferred frequency, we use a model consisting of two neurons coupled with reciprocally inhibitory synapses that display preferred frequencies (Fig. 1B). This model represents the coupling between the pyloric pacemaker neurons AB/PD and the follower neuron LP. The effect of preferred frequencies in our model is explored analytically as well as with simulations. In a network of two neurons coupled with resonance synapses, the frequency of each presynaptic neuron affects the strength of synapse and therefore the frequency of the postsynaptic neuron. Due to the recurrence of the network, these effects are reciprocal. We show that, regardless of cell type, the dynamics of each cell can be described by a logistic map which is composed with two functions: one characterizing the dependence of the synaptic conductance on the previous cycle period of presynaptic neuron, and the other determining the cycle period of postsynaptic neuron depending on this synaptic conductance. We show that the synapses with preferred frequencies can lead to complex network outputs, even in a simple two-cell network. In fact, the activity state of the two-cell network, as a function of the maximal synaptic strength (m) is characterized by a bifurcation diagram that involves a classical period-doubling cascade leading to chaotic dynamics (Fig. 1C). The analytical prediction leading to the logistic-map bifurcation diagram can be reproduced in simulations, as we show using Morris-Lecar model neurons coupled with resonance synapses.