Chaos in neurons and its application: perspective of chaos engineering.

We review our recent work on chaos in neurons and its application to neural networks from perspective of chaos engineering. Especially, we analyze a dataset of a squid giant axon by newly combining our previous work of identifying Devaney's chaos with surrogate data analysis, and show that an axon can behave chaotically. Based on this knowledge, we use a chaotic neuron model to investigate possible information processing in the brain.

[1]  H. Poincaré,et al.  Erratum zu: Etude des surfaces asymptotiques , 1890 .

[2]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1952, The Journal of physiology.

[3]  F. Takens,et al.  On the nature of turbulence , 1971 .

[4]  James A. Anderson,et al.  A simple neural network generating an interactive memory , 1972 .

[5]  Kaoru Nakano,et al.  Associatron-A Model of Associative Memory , 1972, IEEE Trans. Syst. Man Cybern..

[6]  Teuvo Kohonen,et al.  Correlation Matrix Memories , 1972, IEEE Transactions on Computers.

[7]  Robert M. May,et al.  Simple mathematical models with very complicated dynamics , 1976, Nature.

[8]  Y. Ueda Randomly transitional phenomena in the system governed by Duffing's equation , 1978 .

[9]  Philip Holmes,et al.  A magnetoelastic strange attractor , 1979 .

[10]  J J Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[11]  K Aihara,et al.  Periodic and non-periodic responses of a periodically forced Hodgkin-Huxley oscillator. , 1984, Journal of theoretical biology.

[12]  G. Matsumoto,et al.  Periodic and Nonperiodic Responses of Membrane Potentials in Squid Giant Axons during Sinusoidal Current Stimulation , 1984 .

[13]  Kazuyuki Aihara,et al.  An alternating periodic-chaotic sequence observed in neural oscillators , 1985 .

[14]  J. Rinzel,et al.  Bursting, beating, and chaos in an excitable membrane model. , 1985, Biophysical journal.

[15]  K. Aihara,et al.  Structures of attractors in periodically forced neural oscillators , 1986 .

[16]  D. Ruelle,et al.  Recurrence Plots of Dynamical Systems , 1987 .

[17]  K. Aihara,et al.  Chaos and phase locking in normal squid axons , 1987 .

[18]  W. Freeman,et al.  How brains make chaos in order to make sense of the world , 1987, Behavioral and Brain Sciences.

[19]  B. LeBaron,et al.  Nonlinear Dynamics and Stock Returns , 2021, Cycles and Chaos in Economic Equilibrium.

[20]  K. Aihara,et al.  Chaotic neural networks , 1990 .

[21]  James Theiler,et al.  Testing for nonlinearity in time series: the method of surrogate data , 1992 .

[22]  Walter J. Freeman,et al.  TUTORIAL ON NEUROBIOLOGY: FROM SINGLE NEURONS TO BRAIN CHAOS , 1992 .

[23]  Kazuyuki Aihara,et al.  Deterministic prediction and chaos in squid axon response , 1992 .

[24]  M. Bernhard Introduction to Chaotic Dynamical Systems , 1992 .

[25]  M. Rosenstein,et al.  A practical method for calculating largest Lyapunov exponents from small data sets , 1993 .

[26]  M. B. Kennel,et al.  Statistical test for dynamical nonstationarity in observed time-series data , 1995, chao-dyn/9512005.

[27]  Kazuyuki Aihara,et al.  Chaos engineering in Japan , 1995, CACM.

[28]  Schreiber,et al.  Improved Surrogate Data for Nonlinearity Tests. , 1996, Physical review letters.

[29]  Kaplan,et al.  Subthreshold dynamics in periodically stimulated squid giant axons. , 1996, Physical review letters.

[30]  Kazuyuki Aihara,et al.  Associative Dynamics in a Chaotic Neural Network , 1997, Neural Networks.

[31]  Jens Timmer,et al.  Power of surrogate data testing with respect to nonstationarity , 1998, chao-dyn/9807039.

[32]  Gen Matsumoto,et al.  Type-III Bifurcation to Chaos in Self-Oscillating States of Squid Giant Axons , 1998 .

[33]  Rajesh P. N. Rao,et al.  Predictive coding in the visual cortex: a functional interpretation of some extra-classical receptive-field effects. , 1999 .

[34]  T. Schreiber,et al.  Surrogate time series , 1999, chao-dyn/9909037.

[35]  Maurice Courbage,et al.  Anti-phase regularization of coupled chaotic maps modelling bursting neurons , 2001 .

[36]  N. Rulkov Regularization of synchronized chaotic bursts. , 2000, Physical review letters.

[37]  I. Tsuda Toward an interpretation of dynamic neural activity in terms of chaotic dynamical systems. , 2001, The Behavioral and brain sciences.

[38]  Nikolai F Rulkov,et al.  Modeling of spiking-bursting neural behavior using two-dimensional map. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  Kazuyuki Aihara,et al.  Chaos engineering and its application to parallel distributed processing with chaotic neural networks , 2002, Proc. IEEE.

[40]  Yoshito Hirata,et al.  Estimating topological entropy via a symbolic data compression technique. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[41]  A. Grinvald,et al.  Spontaneously emerging cortical representations of visual attributes , 2003, Nature.

[42]  J. Nagumo,et al.  On a response characteristic of a mathematical neuron model , 1972, Kybernetik.

[43]  K. Judd,et al.  Estimating a generating partition from observed time series: symbolic shadowing. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  Kazuyuki Aihara,et al.  Sensitive Response of a Chaotic Wandering State to Memory Fragment Inputs in a Chaotic Neural Network Model , 2004, Int. J. Bifurc. Chaos.

[45]  Kazuyuki Aihara,et al.  Characterizing chaotic response of a squid axon through generating partitions , 2005 .

[46]  Jürgen Kurths,et al.  Recurrence plots for the analysis of complex systems , 2009 .

[47]  Kazuyuki Aihara,et al.  Chaos in neurons , 2008, Scholarpedia.

[48]  Kazuyuki Aihara,et al.  Devaney's chaos on recurrence plots. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[49]  Kazuyuki Aihara,et al.  Associative dynamics of color images in a large-scale chaotic neural network , 2011 .

[50]  Kazuyuki Aihara,et al.  Timing matters in foreign exchange markets , 2012 .