SIAM J. APPLIED DYNAMICAL SYSTEMS c  2005 Society for Industrial and Applied Mathematics Vol. 4, No. 4, pp. 1107–1139 The Dynamic Range of Bursting in a Model Respiratory Pacemaker Network ∗ Janet Best † , Alla Borisyuk † , Jonathan Rubin ‡ , David Terman § , and Martin Wechselberger † Abstract. A network of excitatory neurons within the pre-B¨otzinger complex (pre-B¨otC) of the mammalian brain stem has been found experimentally to generate robust, synchronized population bursts of activity. An experimentally calibrated model for pre-B¨otC 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. Key words. square-wave bursting, fast/slow decomposition, synaptic coupling, averaged equations, bifurcation analysis, respiratory pacemaker AMS subject classifications. 34C15, 34C29, 37G15, 37N25, 92C20 DOI. 10.1137/050625540 1. Introduction. The inspiratory phase of the respiratory rhythm is believed to originate in a group of neurons in a region of the brain stem referred to as the pre-B¨ otzinger com- plex (pre-B¨ otC) [28]. Within the pre-B¨ otC, when coupling among cells is removed, there are silent cells, cells that spike continuously, and intrinsically bursting cells that generate groups of spikes separated by pauses [28, 12, 14]. Cells in all of these classes seem capable of reg- ∗ Received by the editors February 28, 2005; accepted for publication (in revised form) by M. Golubitsky July 29, 2005; published electronically November 18, 2005. This research was supported by the National Science Foundation, through awards DMS-0414023 to JR, DMS-0414057 to DT, and DMS-0112050 to the Mathematical Biosciences Institute. http://www.siam.org/journals/siads/4-4/62554.html † The Mathematical Biosciences Institute, The Ohio State University, Columbus, OH 43210 (jbest@mbi.ohio-state. edu, borisyuk@mbi.ohio-state.edu, wm@mbi.ohio-state.edu). ‡ Department of Mathematics and Center for the Neural Basis of Cognition, University of Pittsburgh, Pittsburgh, PA 15260 (rubin@math.pitt.edu). § Department of Mathematics, The Ohio State University, Columbus, OH 43210 (terman@math.ohio-state.edu). 1107