International Journal of Bifurcation and Chaos, Vol. 17, No. 10 (2007) 3523–3528 c  World Scientific Publishing Company FORCED SYNCHRONIZATION IN MORRIS–LECAR NEURONS HIROYUKI KITAJIMA Department of Reliability-Based Information Systems Engineering, Kagawa University, 2217-20 Takamatsu, Kagawa 761-0396, Japan kitaji@eng.kagawa-u.ac.jp J ¨ URGEN KURTHS Institute of Physics, The University of Potsdam, Am Neuen Palais 10, D-14469 Potsdam, Germany juergen@agnld.uni-potsdam.de Received October 6, 2005; Revised March 29, 2006 We investigate forced synchronization between electrically coupled Morris–Lecar neurons with class I and class II excitability through numerical bifurcation analysis. We find that class II neu- rons have wider parameter regions of forced synchronization. However, the bifurcation structure and patterns of spikes for class II are complicated; there exist period-doubling bifurcations, inter- esting two-periodic oscillations and irregular bursting spikes with high values of the coefficient of variation of the interspike interval. Keywords : Class I and Class II neurons; synchronization; coupled neurons; bifurcation. 1. Introduction A neuron, or the elementary processing units in the central nervous system, generates various temporal patterns of spikes. Among such firing patterns, syn- chronous firing of neurons in connection with neural signal processing has attracted much interest (see [Fujii et al., 1996] and references therein). Many studies confirm that oscillatory dynamics of neu- ral activity and its synchronization play an impor- tant role in the models of information processing in the brain [Pikovsky et al., 2001]. The oscillation mechanisms of neuron models are classified into two by their bifurcations: class I (saddle-node bifurca- tion) and class II (Hopf bifurcation). The oscilla- tion of the former and the latter has almost zero frequency and a finite frequency at the bifurcation point, respectively. We study forced synchronization in electrically cyclic-coupled Morris–Lecar (ML) neurons with both class I and class II excitability. The reasons of using ML neurons are as follow: • Recently, Tsumoto et al. showed that these excitabilities are switched by only one parameter value in the ML model [Tsumoto et al., 2006]. • Patel proposed the analogue VLSI ML neuron model and showed the advantage of using the ML neuron model for considering large-scale systems of coupled neural oscillators [Patel & DeWeerth, 1997]. The aim of this paper is to compare forced syn- chronization of a class I neuron with that of a class II neuron. For mutual synchronization of two neurons, it is clarified by using the phase reset- ting curve (PRC) that class II neurons are easy to achieve synchronization [Ermentrout, 1996; Rinzel & Ermentrout, 1998; Ermentrout et al., 2001]. Also for large number of neurons with random connec- tions, class II neurons present a good level of syn- chronization regardless of the connection topology 3523