Seung Kee Han, Seon Hee Park, Tae Gyu Yim, Seunghwan Kim, and Seunghwan Kim

Chaotic bursting behavior of coupled neural oscillators
Seung Kee Han , Seon Hee Park , Tae Gyu Yim ,
Seunghwan Kim , and Seunghwan Kim
Research Dept., ETRI, Yusong-Gu, Taejon, 305-600, Korea
Dept. of Physics, Chungbuk National University, Cheongju, Chungbuk, 360-763, Korea
Dept. of Physics, Pohang University of Science and Technology, Pohang, 790-784, Korea

Usually, diffusive coupling of nonlinear oscillators in one dynamical variable leads to synchronization of oscillators. We studied a model of coupled neural oscillators in which diffusive coupling in voltage, unexpectedly, leads to dephasing of oscillators [1,2]. We examined the general conditions under which dephasing through diffusive interaction will occur.

Using the Morris-Lecar [3] system, we showed [2] that such systems with dephasing limit cycles lead to a new bursting behavior: oscillators switch between high and low oscillation amplitude. This occurs because the interaction is such that oscillators tend to synchronize for sufficient small oscillation amplitude, while they tend to desynchronize once their oscillation amplitude has become large. It is noted that the single neuron is regularly firing but bursting when coupled with other neurons. Thus the mutual coupling between neurons plays the role of the additional slow variables in the conventional bursting mechanism [4], where the additional slow variables switch the fast dynamics between the steady state and oscillatory state.

To analyze this behavior, we studied diffusively coupled neural oscillators. For a large coupling constant , diffusively coupled neural oscillators are perfectly synchronized on the limit cycle of a single oscillator. While for a small , the oscillators are homogeneously distributed on the limit cycle because of dephasing between oscillators. For the intermediate values of , the bursting behavior occurs, which shows various chaotic behavior depending on . To characterize the chaotic bursting behavior, we introduced a set of mean activities. >From Poincare sections, we found [5] a period-doubling route to chaos for 3-diffusively coupled neural oscillators. We observed that the chaotic behavior arises from the competition between the property of limit cycle of the single oscillator and that of the diffusive coupling among the oscillators.

1. A. Sherman and J. Rinzel, Proc. Nat. Acad. Sci. (USA), 89 (1992) 2471.
2. S. K. Han, C. Kurrer and Y. Kuramoto, Phys. Rev. Lett., 75 (1995) 3190.
3. C. Morris and H. Lecar, Biophys. J., 35 (1981) 193.
4. T. R. Chay and J. Rinzel, Biophys. J., 47 (1985) 357.
5. S. K. Han, S. H. Park, T. G. Yim, S. Kim, and S. Kim, preprint (1995).