On a Nonlinear Master Equation and the Haken-Kelso-Bunz Model

A nonlinear master equation (NLME) is proposed basedon general information measures.Classical and cut-off solutions of the NLME are considered.In the former case, the NLME exhibits uniquely defined stationary distributions. In the latter case, there are multiple stationary distributions.In particular, for classical solutions, it is shown that transient solutions converge to stationary distributions that maximize information measures (H-theorem). Cut-off distributions arestudied numerically for the Haken-Kelso-Bunz model. The Haken-Kelso-Bunz modelis known to describe multistable human motor control systems. It is shownthat a stochastic Haken-Kelso-Bunz model based on a NLME can exhibit multiplestationary cut-off distributions.In doing so, we illustrate that multistability in stochastic biological systems can beestablished by means of cut-off distributions.

[1]  N. Kampen,et al.  Stochastic processes in physics and chemistry , 1981 .

[2]  A. Compte,et al.  Non-equilibrium thermodynamics and anomalous diffusion , 1996 .

[3]  H. Haken,et al.  Impacts of noise on a field theoretical model of the human brain , 1999 .

[4]  Till D. Frank.  On nonlinear and nonextensive diffusion and the second law of thermodynamics , 2000 .

[5]  H. Haken,et al.  Towards a comprehensive theory of brain activity: coupled oscillator systems under external forces , 2000 .

[6]  C. Tsallis.  Generalized entropy-based criterion for consistent testing , 1998 .

[7]  Solomon Kullback,et al.  Information Theory and Statistics , 1970, The Mathematical Gazette.

[8]  Angel Plastino,et al.  Nonlinear Fokker–Planck equations and generalized entropies , 1998 .

[9]  P. T. Landsberg,et al.  Is equilibrium always an entropy maximum? , 1984 .

[10]  E. K. Lenzi,et al.  Statistical mechanics based on Renyi entropy , 2000 .

[11]  Andreas Daffertshofer,et al.  H-theorem for nonlinear Fokker–Planck equations related to generalized thermostatistics , 2001 .

[12]  Andreas Daffertshofer,et al.  Nonlinear Fokker-Planck equations whose stationary solutions make entropy-like functionals stationary , 1999 .

[13]  Celia Anteneodo,et al.  Maximum entropy approach to stretched exponential probability distributions , 1999 .

[14]  Lisa Borland,et al.  Microscopic dynamics of the nonlinear Fokker-Planck equation: A phenomenological model , 1998 .

[15]  C. Tsallis.  Possible generalization of Boltzmann-Gibbs statistics , 1988 .

[16]  Peter J. Beek,et al.  Dynamical Models of Movement Coordination , 1995 .

[17]  M. Shiino.  H-Theorem with Generalized Relative Entropies and the Tsallis Statistics , 1998 .

[18]  Sumiyoshi Abe,et al.  A note on the q-deformation-theoretic aspect of the generalized entropies in nonextensive physics , 1997 .

[19]  H. Haken,et al.  A theoretical model of phase transitions in human hand movements , 2004, Biological Cybernetics.

[20]  H. Haken.  Principles of brain functioning , 1995 .

[21]  E. Jaynes.  Information Theory and Statistical Mechanics , 1957 .

[22]  C. Tsallis,et al.  Statistical-mechanical foundation of the ubiquity of Lévy distributions in Nature. , 1995, Physical review letters.

[23]  C. Tsallis,et al.  Information gain within nonextensive thermostatistics , 1998 .

[24]  Viktor K. Jirsa,et al.  The HKB model revisited: How varying the degree of symmetry controls dynamics , 2000 .

[25]  A. Opstal.  Dynamic Patterns: The Self-Organization of Brain and Behavior , 1995 .

[26]  F. Reif,et al.  Fundamentals of Statistical and Thermal Physics , 1965 .

[27]  H. Haken,et al.  A stochastic theory of phase transitions in human hand movement , 1986, Biological Cybernetics.

[28]  A. G. Bashkirov,et al.  Information entropy and power-law distributions for chaotic systems , 2000 .

[29]  C. Tsallis.  Nonextensive statistics: theoretical, experimental and computational evidences and connections , 1999, cond-mat/9903356.

[30]  Solomon Kullback,et al.  Information Theory and Statistics , 1960 .

[31]  Anomalous diffusion with absorption: exact time-dependent solutions , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[32]  T. D. Frank,et al.  Generalized Fokker–Planck equations derived from generalized linear nonequilibrium thermodynamics , 2002 .

[33]  Shiino.  Dynamical behavior of stochastic systems of infinitely many coupled nonlinear oscillators exhibiting phase transitions of mean-field type: H theorem on asymptotic approach to equilibrium and critical slowing down of order-parameter fluctuations. , 1987, Physical review. A, General physics.

[34]  P. Landsberg,et al.  Distributions and channel capacities in generalized statistical mechanics , 1998 .

[35]  Tsallis,et al.  Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[36]  A. Wehrl.  General properties of entropy , 1978 .

[37]  Interpretation of Lagrange multipliers of generalized maximum-entropy distributions , 2002 .

[38]  Andreas Daffertshofer,et al.  A dynamical model for mirror movements , 1999 .

[39]  Ernesto P. Borges,et al.  A family of nonextensive entropies , 1998 .

[40]  A. Plastino,et al.  The nonlinear Fokker-Planck equation with state-dependent diffusion - a nonextensive maximum entropy approach , 1999 .

[41]  C. Tsallis,et al.  The role of constraints within generalized nonextensive statistics , 1998 .

[42]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[43]  A. Daffertshofer,et al.  Impacts of statistical feedback on the flexibility-accuracy trade-offin biological systems , 2002, Journal of biological physics.

[44]  A. R. Plastino,et al.  Non-extensive statistical mechanics and generalized Fokker-Planck equation , 1995 .

[45]  Andrés R R Papa.  On one-parameter-dependent generalizations of Boltzmann-Gibbs statistical mechanics , 1998 .