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Propagation of periodic and chaotic action potential trains along nerve fibers. (English) Zbl 0890.92007
Summary: We report the findings that the action potential trains transmitted along the nerve fibers are encoded not only periodically but also chaotically. First, spontaneous action potentials along a single fiber of injured sciatic nerves in the anesthetized rat were recorded. Then, the data were divided into two groups and analyzed with different methods. Phase space representation, spectral analysis and the calculation of correlation dimension were used for the first group of data sampled with constant frequency. Due to the serious influence of the measurement noise, no reliable conclusion can be drawn from them.
For the second group of data of the interspike intervals (ISI) which seem to convey more rich and important information, nonlinear forecasting method, the surrogate data and the plot of ISI(n+1) vs. ISI(n) were used in the analysis. Good results have been obtained which confirm with those from the \(\beta{}\)-cell model. The largest Lyapunov exponent (LLE) was calculated not only to further support our findings of chaos but also to quantitatively determine the degree of chaos.

92C20 Neural biology
37N99 Applications of dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
Full Text: DOI
[1] Hodgkin, A.L.; Huxley, A.F., A quantitative description of membrane current and its application to conduction and excitation in nerve, J. physiol. (London), 117, 500, (1952)
[2] Scott, A.C., The electrophysics of a nerve fiber, Rev. modern phys., 47, 487, (1975)
[3] Chay, T.R., Electrical bursting and intracellular calcium ion oscillations in excitable cell models, Biol. cybern., 63, 15, (1990)
[4] Lansky, P.; Rospars, J.P., Ornstein-Uhlenbeck model neuron revisited, Biol. cybern., 72, 397, (1995) · Zbl 0819.92005
[5] Bruner, L.J.; Harvey, J.R., The spike generation zone of the ampullary electroreceptor. I. stimulus-response characteristics of a relaxation oscillator circuit model, Biol. cybern., 72, 371, (1995) · Zbl 0825.92038
[6] Aihara, K.; Matsumoto, G., Chaotic oscillations and bifurcations in squid giant axons, (), 257
[7] Matsumoto, G.; Aihara, K.; Utsunomiya, T., A spatially ordered pacemaker observed in squid giant axons, J. phys. soc. Japan, 51, 942, (1982)
[8] Aihara, K.; Matsumoto, G.; Ichiwaka, M., An alternating periodic-chaotic sequence observed in neural oscillations, Phys. lett. A, 111, 251, (1985)
[9] Rapp, P.E.; Zimmerman, I.D.; Albano, A.M.; Deguzman, G.C.; Greenbaun, N.N., Dynamics of spontaneous neural activity in the Simian motor cortex: the dimension of chaotic neurons, Phys. lett. A, 110, 335, (1985)
[10] Elbert, T.; Zbigniew, W.J.; Kowalik, J.; Skinner, J.E.; Graf, K.E.; Birbaumer, N., Chaos and physiology: deterministic chaos in excitable cell assemblies, Physiol. rev., 74, 1, (1994)
[11] Packard, N.H.; Crutchfield, J.P.; Farmer, J.D.; Shaw, R.S., Geometry from a time series, Phys. rev. lett., 45, 712, (1980)
[12] Takens, F., Detecting strange attractors in turbulence, (), 366
[13] Grassberger, P.; Procaccia, I., Characterization of strange attractors, Phys. rev. lett., 50, 346, (1983)
[14] Grassberger, P.; Procaccia, I., Measuring the strangeness of strange attractors, Physica D, 9, 189, (1983) · Zbl 0593.58024
[15] Ben-Mizrachi, A.; Procaccia, I.; Grassberger, P., The characterization of experimental (noisy) strange attractor, Phys. rev. A, 29, 975, (1984)
[16] Grassberger, P., Do climatic attractors exist?, Nature, 323, 609, (1986)
[17] Osborne, A.R.; Provenzale, A., Finite correlation dimension for stochastic systems with power-law spectra, Physica D, 35, 357, (1989) · Zbl 0671.60030
[18] Sugihara, G.; May, R.M., Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series, Nature, 344, 734, (1990)
[19] Xu Jianxue, Gong Yunfan, Ren Wei, Hu Sanjue and Wang Fuzhou, Application of nonlinear forecasting method to the analysis of interspike intervals in nervous system, Acta Biophysica Sinica, accepted. · Zbl 0895.92008
[20] ()
[21] Cvitanovic, P., Periodic orbits as the skeleton of classical and quantum chaos, Physica D, 51, 138, (1991) · Zbl 0744.58013
[22] Cvitanovic, P., Invariant measurement of strange sets in terms of cycles, Phys. rev. lett., 61, 2729, (1988)
[23] Auerbach, D.; Cvitanovic, P.; Eckmann, J.-P.; Gunaratne, G.; Procaccia, I., Exploring chaotic motion through periodic orbits, Phys. rev. lett., 58, 2387, (1987)
[24] Pierson, D.; Moss, F., Detecting periodic unstable points in noisy chaotic and limit cycle attractors with applications to biology, Phys. rev. lett., 75, 2124, (1995)
[25] Bennett, G.J.; Xie, Y.-K., A peripheral mononeuropathy in rat that produces disorders of pain sensation like those seen in man, Pain, 33, 87, (1988)
[26] Sigeti, D.; Horsthemke, W., High-frequency power spectra for systems subject to noise, Phys. rev. A, 35, 2276, (1987)
[27] Jianxue, Xu; Yunfan, Gong; Wei, Ren; Sanjue, Hu; Fuzhou, Wang, Bifurcation and chaos phenomenon of interspike intervals in nervous system, (), accepted · Zbl 0895.92008
[28] Theiler, J.; Eubank, S.; Longtin, A.; Galdrikian, B.; Farmer, J.D., Testing for nonlinearity in time series: the method of surrogate data, Physica D, 58, 77, (1992) · Zbl 1194.37144
[29] Rinzel, J.; Sherman, A.; Stokes, C.L., Channels, coupling, and synchronized rhythmic bursting activity, (), 29
[30] Broomhead, D.S.; King, G.P., Extracting qualitative dynamics from experimental data, Physica D, 20, 217, (1986) · Zbl 0603.58040
[31] May, R.M., Simple mathematical models with very complicated dynamics, Nature, 261, 459, (1976) · Zbl 1369.37088
[32] Roux, J.-C.; Simoyi, R.H.; Swinney, H.L., Observation of a strange attractor, Physica D, 8, 257, (1983) · Zbl 0538.58024
[33] Swinney, H.L., Experimental observations of order and chaos, (), 232
[34] Moon, F.C., Chaotic and fractal dynamics, (), 110
[35] Sarkovskii, A.N., Coexistence of cycles of a continuous map of a line into itself, Ukrainian J. math., 16, 61, (1964)
[36] Li, T.-Y.; Yorke, J.A., Period three implies chaos, Amer. math. monthly, 82, 985, (1975) · Zbl 0351.92021
[37] Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.A., Determining Lyapunov exponents from a time series, Physica D, 16, 285, (1985) · Zbl 0585.58037
[38] Wolf, A., Quantifying chaos with Lyapunov exponents, (), 273
[39] Fan, Y.S.; Chay, T.R., Generation of periodic and chaotic bursting in an excitable cell model, Biol. cybern., 71, 417, (1994) · Zbl 0805.92005
[40] Gong Yunfan, Xu Jianxue, Ren Wei, Hu Sanjue and Wang Fuzhou, Observing nonlinear phenomena in nervous system: from theory to experimental realization, to be submitted. · Zbl 0895.92008
[41] Gilchrist, J.M.; Perrone, M.; Ross, J., Dynamical analysis of neuromuscular transmission jitter, Muscle-nerve, 18, 685, (1995)
[42] Gong Yunfan, Xu Jianxue, Ren Wei, Hu Sanjue and Wang Fuzhou, Determine the degree of chaos from the analysis of ISI time series in nervous system: a comparison between the correlation dimension and the nonlinear forecasting method, to be submitted. · Zbl 0895.92008
[43] Wales, D.J., Calculating the rate of loss of information from chaotic time series by forecasting, Nature, 350, 485, (1991)
[44] Gong Yunfan, Xu Jianxue, Ren Wei, Hu Sanjue and Wang Fuzhou, Bifurcation: the mechanism underlying the delayed feedback method and its applications to the control of neural chaotic bursting, Physica D, submitted. · Zbl 0890.92007
[45] Gong Yunfan, Xu Jianxue, Ren Wei, Hu Sanjue and Wang Fuzhou, Bifurcation chaos and control in nervous system, Acta Biohysica Sinica, accepted. · Zbl 0895.92008
[46] Bushev, M., Synergetics: chaos, order, self-organization, (), 61
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