A Family of Mechanisms Controlling Bursting Activity and Pulse-triggered Responses of a Neuron Model

Abstract

Central pattern generating neuronal networks coordinate and control rhythmic movements. In many such networks, the phase relations of activities of neurons are conserved over a range of values of cycle period. How temporal characteristics are controlled in oscillatory networks is an open question. We model cell intrinsic mechanisms that control temporal characteristics and produce phase maintenance in neuronal networks. We focus on the coregulation of a potassium current (I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</sub> ) and a hyperpolarization-activated current (I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</sub> ). The dynamics of this model are governed by a codimension-2 bifurcation: the cornerstone bifurcation. The bifurcation satisfies conditions for the saddle-node bifurcation on invariant circle (SNIC) and the blue sky catastrophe [1]. For parameter values close to the bifurcation, we achieved control over the burst duration and interburst interval by varying the voltages of half activation of I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</sub> and I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</sub> . Similarly we were able to control the latency to spiking after inhibition in a spiking neuron and the duration of single evoked bursts in a silent neuron. We constructed a series of network motifs for central pattern generators typical for motor control and demonstrated how control of bursting activity on the level of individual cells controls phase constancy.