Biol. Cybern. 72, 93-101 (1994) Biological cybernetics 9 Springer-Verlag 1994 A modified radial isochron clock with slow and fast dynamics as a model of pacemaker neurons Global bifurcation structure when driven by periodic pulse trains T. Nomura I, S. Sato 1, S. Doi t , J.P. Segundo 2'*, M.D. Stiber 3 I Department of Biophysical Engineering, Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan 2 Department of Anatomy and Cell Biology, Brain Research Institute, University of California, Los Angeles, CA 90024-1763, USA 3 Computer Science Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Received: 29 March 1994 / Accepted in revised form: 28 July 1994 Abstract. A simple mathematical model of living pace- maker neurons is proposed. The model has a unit circle limit cycle and radial isochrons, and the state point moves slowly in one region and fast in the remaining region; re- gions can correspond to the subthreshold activity and to the action potentials of pacemaker neurons, respectively. The global bifurcation structure when driven by periodic pulse trains was investigated using one-dimensional maps (PTC), two-dimensional bifurcation diagrams, and skeletons involv- ing stimulus period and intensity. The existence of both the slow and the fast dynamics has a critical influence on the global bifurcation structure of the oscillator when stimulated periodically. 1 Introduction Neurons or muscle cells which spontaneously discharges spikes at practically constant time intervals are called "pace- makers." They are common in living organisms, and their oscillations may play important roles in the temporal behav- ior of many systems. Spontaneous rhythmic activities are often modulated by external periodic influences from, say, other cells in neural networks (Perkel et al. 1964; Perkel and Bullock 1968) or the environment in circadian rhythm (Win- free 1980,1987). Inhibitory influences are involved when bursts of vagal impulses arrive at pacemakers in the heart sino-arterial node where oscillatory pacemaker activities be- come synchronized with their inputs (Dong and Reitz 1970). Recent investigations (Segundo et al. 1991a,b) of periodic synaptic inhibition of pacemaker neurons in slowly adapting stretch receptor organs (SAOs) of crayfish showed that en- trained behaviors could be complicated. Depending on the average period or frequency of the inputs, it could be either n : m locked, intermittent, messy erratic or messy stam- mering; messy behaviors may be chaotic (Sugihara et al., manuscript in preparation). Cells with excitable membranes can be modeled at sev- eral levels of complexity (e.g., Hodgkin and Huxley 1952; * Supported by Trent H. Wells Jr. Inc. Correspondence to: T. Nomura FitzHugh 1961; Nagumo et al. 1962; Morris and Lecar 1981; Koch and Segev 1989; Hayashi and Ishizuka 1992; Kepler et al. 1992). Pacemaker neurons submitted to periodic inputs can be modeled using coupled master and slave oscillators (Aihara et al. 1984; Stiber 1992; Stiber and Segundo 1993). Such periodically driven oscillators are analyzed mainly via phase equations and/or one-dimensional maps such as phase transition curves (PTCs) or Poincar6 maps (e.g., Ermen- trout 1981; Prrez and Glass 1982; Jensen 1983; Keener and Glass 1984; Kiemel and Holmes 1987; Parlitz and Lauter- born 1987). Pacemaker cells can be classified simply in the oscillatory dynamical systems with a unique stable limit cy- cle. They should differ from neurons as regular bursters that also are periodic, but where each cycle involves many spikes. This implies the important question of which are the main features that separate neuronal pacemakers from other gen- eral oscillators; in other words, what should a modeled pace- maker cell be like? Earlier papers investigated the global bifurcation structure of the Bonhoeffer van der Pol model (BVP) or FitzHugh-Nagumo model with self-sustained os- cillations when driven by periodic pulse trains (Nomura et al. 1993,1994). The analysis of simulated data revealed in- teresting features. These were comparable and complemen- tary to those in living preparations (Segundo et al. 1991a,b) and allowed qualitative interpretations of neuronal behav- iors. It was strongly speculated that not only the oscillatory property but also the excitability of neurons which have a stable equilibrium point (resting state) with both slow and fast changes of the membrane potential, respectively, in the sub- and suprathreshold regions are responsible for the fact that neuronal oscillator models such as the BVP can mimic living pacemakers' responses to external stimuli (Nomura et al. 1994). In this paper, we propose a simple mathematical model that has radial isochrons, the state point moving slowly in one part of a cycle and fast in the rest. The dynamics in each part are called "slow" and "fast," respectively. The propor- tion of the cycle occupied by each part and the velocities of the state point in each depend on specific control parameters. As a consequence, the modeled membrane potential has a variety of time courses, i.e., can generate various waveforms. For appropriate parameters, it exhibits the essential feature