Journal of ComputationalNeuroscience3, 199-223 (1996) © 1996 KluwerAcademicPublishers. Manufacturedin The Netherlands. Dissection and Reduction of a Modeled Bursting Neuron R.J. BUTERA, JR. AND J.W. CLARK, JR. Dept. of Electrical and Computer Engineering, MS 366, Rice University, Houston, TX 77251-1892 rbutera@ece.fice.edu J.H. BYRNE Dept. of Neurobiology and Anatom.3; University of Texas Medical School, Houston, TX 77030 jbyrne@nbal9.med.uth.tmc.edu Received September 5, 1995; Revised February 1, 1996; Accepted February 13, 1996 Action Editor: John Rinzel Abstract. An 11-variable Hodgkin-Huxley type model of a bursting neuron was investigated using numerical bifurcation analysis and computer simulations. The results were applied to develop a reduced model of the underlying subthreshold oscillations (slow-wave) in membrane potential. Two different low-order models were developed: one 3-variable model, which mimicked the slow-wave of the full model in the absence of action potentials and a second 4-variable model, which included expressions accounting for the perturbational effects of action potentials on the slow-wave. The 4-variable model predicted more accurately the activity mode (bursting, beating, or silence) in response to application of extrinsic stimulus current or modulatory agents. The 4-variable model also possessed a phase-response curve that was very similar to that of the original 11-variable model. The results suggest that low-order models of bursting cells that do not consider the effects of action potentials may erroneously predict modes of activity and transient responses of the full model on which the reductions are based. These results also show that it is possible to develop low-order models that retain many of the characteristics of the activity of the higher-order system. Keywords: nonlinear dynamics, bifurcation, bursting, model reduction 1. Introduction The endogenous oscillations in the membrane po- tential of excitable membranes have been the sub- ject of extensive analyses using mathematical mod- eling and computer simulation approaches. Among the most detailed models are those of cells whose os- cillations in membrane potential exhibit bursting ac- tivity: a slow oscillation in membrane potential that alters between a silent state and a bursting state. The bursting state is characterized by the firing of a suc- cession of action potentials. The most extensively modeled bursting cells are pancreatic fl-cells (Chay and Keizer, 1983; Chay, 1990; Keizer and Man- gus, 1989; Sherman et al., 1988) and neuron R15 in Aplysia (Bertram, 1993; Butera et al., 1995; Canavier et al., 1991; Plant and Kim, 1976). Electrophysi- ological investigations over the past several decades have revealed a rich repertoire of biophysical mech- anisms. Recent models simulate not only endoge- nous activity and changes in behavior produced by extrinsic currents, but also simulate changes in en- dogenous activity as the concentration of extrinsic modulatory agents are varied (Bertram, 1993; Butera et al., 1995; Keizer and Mangus, 1989). To achieve this level of accuracy and predictability the models