Abstract
Neural networks generate a variety of rhythmic activity patterns, often involving different timescales. One example arises in the respiratory network in the pre-Bötzinger complex of the mammalian brainstem, which can generate the eupneic rhythm associated with normal respiration as well as recurrent low-frequency, large-amplitude bursts associated with sighing. Two competing hypotheses have been proposed to explain sigh generation: the recruitment of a neuronal population distinct from the eupneic rhythm-generating subpopulation or the reconfiguration of activity within a single population. Here, we consider two recent computational models, one of which represents each of the hypotheses. We use methods of dynamical systems theory, such as fast-slow decomposition, averaging, and bifurcation analysis, to understand the multiple-timescale mechanisms underlying sigh generation in each model. In the course of our analysis, we discover that a third timescale is required to generate sighs in both models. Furthermore, we identify the similarities of the underlying mechanisms in the two models and the aspects in which they differ.
Highlights
Many years of experimental work have elucidated a range of properties of the neuronal circuits involved in breathing
To understand the mechanisms underlying the generation of sighs, we considered two distinct single-compartment models for respiratory neurons in the pre-BötC that have the ability to generate sigh-like activity
In some cases with class membership changing depending on the location of the trajectory in phase space, we used a non-rigorous geometric singular perturbation theory (GSPT) approach to elucidate the roles of various timescale-based subsystems and their bifurcation structures in producing sigh-like dynamics
Summary
Many years of experimental work have elucidated a range of properties of the neuronal circuits involved in breathing. In that we assume that there are abrupt transitions between timescales and we do not prove any results; we will sometimes make approximations, such as treating a nullcline that only weakly depends on a parameter as fixed under variations of that parameter This approach has a long history of providing powerful insights, for example, in work ensuing from the classical dissection of minimal bursting models by [16] and in the study of coupled neuronal oscillators and bursters (e.g, [17,18,19,20] and many others since). 5, we compare these two sighing solutions to highlight the similarities and differences in the mechanisms and timescale interactions involved in producing them, and we conclude with a discussion in Sect.
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