A scalar Poincaré map for struggling in Xenopus tadpoles

Abstract

AbstractShort-term synaptic plasticity is widely found in many areas of the central nervous system. In particular, it is believed that synaptic depression can act as a mechanism to allow simple networks to generate a range of different firing patterns. The locomotor circuit of hatchling Xenopus tadpoles produces two types of behaviours: swimming and the slower, stronger struggling movement that is associated with rhythmic bends of the whole body. Struggling is accompanied by anti-phase bursts in neurons on each side of the spinal cord and is believed to be governed by a short-term synaptic depression of commissural inhibition. To better understand burst generation in struggling, we study a minimal network of two neurons coupled through depressing inhibitory synapses. Depending on the strength of the synaptic conductance between the two neurons, such a network can produce symmetric n – n anti-phase bursts, where neurons fire n spikes in alternation, with the period of such solutions increasing with the strength of the synaptic conductance. Relying on the timescale disparity in the model, we reduce the eight-dimensional network equations to a fully explicit scalar Poincaré burst map. This map tracks the state of synaptic depression from one burst to the next, and captures the complex bursting dynamics of the network. Fixed points of this map are associated with stable burst solutions of the full network model, and are created through fold bifurcations of maps. We derive conditions that describe period increment bifurcations between stable n – n and (n + 1) – (n + 1) bursts, producing a full bifurcation diagram of the burst cycle period. Predictions of the Poincaré map fit excellently with numerical simulations of the full network model, and allow the study of parameter sensitivity for rhythm generation.