The Dynamic Range of Bursting in a Model Respiratory Pacemaker Network

Abstract

A network of excitatory neurons within the pre-Bötzinger complex (pre-BötC) of the mammalian brain stem has been found experimentally to generate robust, synchronized population bursts of activity. An experimentally calibrated model for pre-BötC cells yields typical square-wave bursting behavior in the absence of coupling, over a certain parameter range, with quiescence or tonic spiking outside of this range. Previous simulations of this model showed that the introduction of synaptic coupling extends the bursting parameter range significantly and induces complex effects on burst characteristics. In this paper, we use geometric dynamical systems techniques, predominantly a fast/slow decomposition and bifurcation analysis approach, to explain these effects in a two-cell model network. Our analysis yields the novel finding that, over a broad range of synaptic coupling strengths, the network can support two qualitatively distinct forms of synchronized bursting, which we call symmetric and asymmetric bursting, as well as both symmetric and asymmetric tonic spiking. By elucidating the dynamical mechanisms underlying the transitions between these states, we also gain insight into how relevant parameters influence burst duration and interburst intervals. We find that, in the two-cell network with synaptic coupling, the stable family of periodic orbits for the fast subsystem features spike asynchrony within otherwise synchronized bursts and terminates in a saddle-node bifurcation, rather than in a homoclinic bifurcation, over a wide parameter range. As a result, square-wave bursting is replaced by what we call top hat bursting (also known as fold/fold cycle bursting), at least for a broad range of parameter values. Further, spike asynchrony is a key ingredient in shaping the dynamic range of bursting, leading to a significant enhancement in the parameter range over which bursting occurs and an abrupt increase in burst duration as an appropriate parameter is varied.

Emergent Bursting in Small Networks of Model Conditional Pacemakers in the pre-Bötzinger Complex

“…Increasing g syn-e , however, pushes the homoclinic point to smaller h values, such that a larger g tonic-e is needed to attain the bursting-to-spiking transition ( Fig. 2; Best et al 2005). These findings can also be cast in terms of I NaP .…”

“…Moreover, changes in g tonic-e and g syn-e induce complex variations in burst characteristics (Butera et al 1999b). The key to these results (Best et al 2005) is the observation that both g tonic-e and g syn-e affect the bifurcation structure of the pBC cell's fast subsystem. Consideration of a single self-coupled cell gives a first approximation to the effect of g syn-e .…”

“…1-3), corresponding to quiescence, occurs where this nullcline intersects a stable part of S. Left panel used with permission from J. Best et al (2005), SIAM J. Appl. Dyn.…”

“…This input mitigates the downstroke, which prevents deinactivation of I NaP , such that the cell is prevented from becoming tonically active. A full analysis of the coupled cell pair requires treatment of a system of two slow variables and is given elsewhere (Best et al 2005; see also Butera et al 2005). One outcome of this analysis was the discovery of four different modes of activity in the coupled pBC network, namely two types of bursting (symmetric and asymmetric) and two types of spiking (symmetric and asymmetric), with corresponding differences in burst characteristics.…”

Dynamics of in-phase and anti-phase bursting in the coupled pre-Bötzinger complex cells

“…Bifurcation structures of bursting oscillations in cells has been studied in detail Marhl 2003, 2007;), for example. Best et al studied a two-cell model network with synaptic coupling and they found that the square-wave bursting is replaced by the top hat bursting (also known as fold/fold cycle bursting) over a wide parameter range (Best et al 2005). The transition from bursting to spiking has been studied in a variety of neuronal models (Bi et al 2014;Duan et al 2012;Rinzel 1985;Izhikevich 2000;Jia et al 2012;Gu and Xiao 2014).…”