Sir Alan Lloyd Hodgkin OM , KBE , PRS Sir Andrew Fielding Huxley OM

s (Listed in alphabetically order of author’s surname) Kazuyuki Aihara Institute of Industrial Science, University of Tokyo CHAOS AND BIFURCATIONS IN THE HODGKIN-HUXLEY EQUATIONS AND SQUID GIANT AXONS Chaos, or deterministic chaos, is ubiquitous in nonlinear dynamical systems of the real world, including biological systems. Nerve membranes have their own nonlinear dynamics which generate and propagate action potentials. Such nonlinear dynamics of nerve membranes has been intensively studied experimentally with squid giant axons and theoretically with the Hodgkin-Huxley equations. In fact, chaos and various routes, or bifurcations to chaos in nerve membranes were found both with the Hodgkin-Huxley equations and with squid giant axons by our group (1, 3-7, 9, 13-15). Moreover, a chaotic neuron model that qualitatively reproduces chaos in nerve membranes observed in squid giant axons and the Hodgkin-Huxley equations can be described by a simple 1-dimensional map (8, 9). Chaotic neural networks composed of the chaotic neurons generate various kinds of spatio-temporal dynamics with abilities of parallel distributed processing (2), and are implemented by integrated electronic circuits for possible applications (10-12). This talk will be to confirm the great importance of the deteministic nonlinear dynamics in squid giant axons, which was formulated by the Hodgkin-Huxley equations, through analysis of neural chaos and its bifurcations which can contribute even for development of new technology and engineering(2). (1) Aihara, K.(1995). Chaos in axons, in The handbook of brain theory and neural networks, 1st edition, edited by M.A. Arbib, The MIT Press, Cambridge, Massachusetts, pp.183-185. (2) Aihara, K.(2002). Chaos engineering and its application to parallel distributed processing with chaotic neural networks, Proc. IEEE 90(5), 919-930. (3) Aihara, K., and Matsumoto, G.(1982). Temporally coherent organization and instabilities in squid giant axons,J. Theor. Biol. 95(4), 697-720. (4) Aihara, K., and Matsumoto, G.(1986). Chaotic oscillations and bifurcations in squid giant axons, in Chaos, edited by A.V. Holden, Manchester University Press, Manchester, and Princeton University Press, Princeton, pp.257-269. (5) Aihara, K., Matsumoto, G., and Ichikawa, M.(1985). An alternating periodic-chaotic sequence observed in neural oscillators, Phys. Lett. A 111(5), 251-255. (6) Aihara, K., Matsumoto, G., and Ikegaya, Y.(1984). Periodic and non-periodic responses of a periodically forced Hodgkin-Huxley oscillator, J. Theor. Biol. 109, 249-269. (7) Aihara, K., Numajiri, T., Matsumoto, G., and Kotani, M.(1986). Structures of attractors in periodically forced neural oscillators, Phys. Lett. A 116, 313-317. (8) Aihara, K., Takabe, T., and Toyoda, M.(1990). Chaotic neural networks, Phys. Lett. A 144(6/7), 333-340. (9) Aihara, K., and Suzuki, H.(2010). Theory of hybrid dynamical systems and its applications to biological and medical systems, Phil. Trans. Roy. Soc. A 368(1930), 48934914. (10) Horio, Y. and Aihara, K. (2003). Neuron-synapse ic chip-set for large-scale chaotic neural networks, IEEE Transactions on Neural Networks 14(5),1393-1404. (11)Horio, Y. and Aihara, K. (2008). Analog computation through high-dimensional physical chaotic neuro-dynamics,Physica D: Nonlinear Phenomena 237(9),1215-1225. (12) Horio, Y., Ikeguchi, T. and Aihara, K. (2005). A mixed analog/digital chaotic neurocomputer system for quadratic assignment problems, Neural Networks 18(5-6),505-513. (13) Matsumoto, G., Aihara, K., Hanyu, Y., Takahashi, N., Yoshizawa, S.,and Nagumo, J.(1987). Chaos and phase locking in normal squid axons,Phys. Lett. A 123, 162-166. (14) Matsumoto, G., Aihara, K., Ichikawa, M., and Tasaki, A.(1984). Periodic and nonperiodic responses of membrane potentials in squid giant axons during sinusoidal current stimulation, J. Theor. Neurobiol. 3, 1-14. (15) Mees, A., Aihara, K., Adachi, M., Judd, K., Ikeguchi, T., and Matsumoto, G.(1992). Deterministic prediction and chaos in squid axon response, Phys. Lett. A 169, 41-45.