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19). Neurophysically, the wave propagation is realized by the flow of electrons along the axon which is caused by the change of membrane potential and a mathematical model, called the Hodgkin-Huxley equation, is presented for this dynamic phenomenon by Hoi3gkin and Huxley (1952). Starting from this Hodgkin-Huxley equation, Fitzhugh (1969) obtained the following non-linear dynamical system model for the dynamics of the potential V: dV . 59) tit where ( V - E o ) B + = ( V - E o ) 3 for V>~Eo and ( V - E o ) 3 - O for V < E o , E 0 < E 1% E 2 and E 0, E 1 and E 2 are ionic equilibrium potentials determined by the sodium and potassium ion and some other ion.

Non-linear time series models and dynamical systems 63 W(X) ~(Yt) 2 5 Fig. 8. Fig. 7. Yt, where Yt+at = 6(Yt)Y, + X/~te,+at. 58) 2 1 2 2 qb(yt) = 3 + 5 exp(--30. 2xtyt). 59) T h e figure of the ~b function is shown in Fig. 8. EXAMPLE 3. 5X 3 - X5. 60) has five zero points, so0 = 0, sc~ = ~22, ~:~ =-Xf~-~, sc~ = 2 and ( ~ = - 2 (see Fig. 9). T h e y are called singular points of the dynamical system. 60) is one of the five singular points, then x(t) stays at x 0 for any t > 0. 5x 3 - x 5~ o'n(t).

5. If this dynamical system is Wlxl 0 Fig. 3. × Fig. 4. 62 T. Ozaki driven by a white noise of variance 0"2, we have 2 = - x 3 + o'n(t). 51) The Fokker-Planck equation of the process x is _013 _ = _ _0 [x3p ] + _102 [0-2p]. 54) = _0-2y3. The associated potential function (see Fig. 6) is V(y) 0-1 y4 . 56) 0"2 W ( x ) . Then the distribution W ( x ) is given by (see Fig. 57) where W 0 is a normalizing constant. V{y) 0 Fig. 5. Fig. 6. Non-linear time series models and dynamical systems 63 W(X) ~(Yt) 2 5 Fig.