27 December 1999 Ž . Physics Letters A 264 1999 289–297 www.elsevier.nlrlocaterphysleta Phase synchronization in two coupled chaotic neurons Jian-Wei Shuai ) , Dominique M. Durand Department of Biomedical Engineering, Case Western ReserÕe UniÕersity, CleÕeland, Ohio 44106, USA Received 19 August 1999; received in revised form 10 November 1999; accepted 11 November 1999 Communicated by C.R. Doering Abstract Chaotically-spiking dynamics of Hindmarsh–Rose neurons are discussed based on a flexible definition of phase for chaotic flow. The phase synchronization of two coupled chaotic neurons is in fact the spike synchronization. As a multiple time-scale model, the coupled HR neurons have quite different behaviors from the Rossler oscillators only having single ¨ time-scale mechanism. Using such a multiple time-scale model, the phase function can detect synchronization dynamics that cannot be distinguished by cross-correlation. Moreover, simulation results show that the Lyapunov exponents cannot be used as a definite criterion for the occurrence of chaotic phase synchronization for such a system. Evaluation of the phase function shows its utility in analyzing nonlinear neural systems. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 05.45.qb; 87.10.qe Keywords: Neuron; Chaos; Synchronization; Phase 1. Introduction Cooperative behavior is a major focus of study in neuronal systems. The selective synchronization of neural activity has been suggested as a mechanism for binding spatially distributed features into a coher- w x ent object 1–3 and too much synchrony may cause w x dynamical diseases, such as in epilepsy 4,5 . For a periodic neuronal system with the concept of phase, different aspects of coherence, such as phase syn- chronization, phase-locking, frequency-locking be- wx haviors have been discussed in detail 6 . For cou- pled chaotic neuronal oscillators, synchronization be- havior is detected with the cross-correlation function ) Corresponding author. Tel.: q216 368 8534; fax: 216 368 4872; e-mail: jxs131@po.cwru.edu w x 1–3 . Recently, the concept of phase, as well as phase synchronization, has been generalized to the w x study of chaotic oscillators 7–12 . Phase synchro- nization has been observed in nonlinear neural, car- w x diac and ecological systems 13–17 . The oscilla- w x tions between respiratory and cardiac rhythms 14 or between brain activity and the signals from the flexor w x muscle 15 show the behavior of phase synchroniza- tion. The subthreshold chaotic oscillation of electri- cally-coupled inferior olivary neurons in Õitro has also been studied in the view of phase synchroniza- w x tion 16 . Phase synchronization may play an impor- tant role in revealing communication pathways in w x brain 18 . Due to the potential application of phase synchro- nization in the brain, it is important to study the characteristics of phase dynamics in the neural sys- 0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. Ž . PII: S0375-9601 99 00816-6