Abstract
We propose a minimalistic model called the 2θ-burster due to two slow phase characteristics of endogenous bursters, which when coupled in 3-cell neural circuits generate a multiplicity of stable rhythmic outcomes. This model offers the benefits of simplicity for designing larger neural networks along with an acute reduction in the computation cost. We developed a dynamical system framework for explaining the existence and robustness of phase-locked states in activity patterns produced by small rhythmic neural circuits. Several 3-cell configurations, from multifunctional to monostable, are considered to demonstrate the versatility of the proposed approach, allowing the network dynamics to be reduced to the examination of 2D Poincaré return maps for the phase lags between three constituent 2θ-bursters.
Highlights
Neural networks called central pattern generators (CPGs) [1,2,3,4,5,6,7,8] produce and control a great variety of rhythmic motor behaviors, including heartbeat, respiration, chewing, and locomotion
Our ultimate goal is to use the top-down approach to single out the fundamental principles of the rhythm formation in small networks that can be systematically generalized and applied for understanding larger network architectures
Due to the rhythmic nature of the bursting patterns, we employed Poincaré return maps defined on phases and phase lags between burst initiations in the interneurons
Summary
Neural networks called central pattern generators (CPGs) [1,2,3,4,5,6,7,8] produce and control a great variety of rhythmic motor behaviors, including heartbeat, respiration, chewing, and locomotion. The concept of the new model, called the 2θ-burster due to the driving term cos2θ in its ODE description, is inspired by the dynamics of endogenous bursters (like ones shown in Figure 1) with two characteristic slow phases: depolarized tonic spiking and hyperpolarized quiescent. These phases are referred to as “on” or active and “off” or inactive depending on whether the membrane voltage is above or below some synaptic threshold. We recall that a similar saddle-node bifurcation controlling the duration of the tonic-spiking phase, and the number of spikes is associated with the codimension-one bifurcation known as the blue-sky catastrophe [23, 29,30,31,32]
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