Stationary and dynamical properties of finite N-unit Langevin models subjected to multiplicative noises

We have studied the finite N-unit Langevin model subjected to multiplicative noises, by using the augmented moment method (AMM), as a continuation of our previous paper [H. Hasegawa, J. Phys. Soc. Japan 75 (2006) 033001]. Effects of couplings on stationary and dynamical properties of the model have been investigated. The difference and similarity between the results of diffusive and sigmoid couplings are studied in details. Time dependences of average and fluctuations in local and global variables calculated by the AMM are in good agreement with those of direct simulations (DSs). We also discuss stationary distributions of local and global variables with the use of the Fokker–Planck equation (FPE) method and DSs. It is demonstrated that stationary distributions show much variety when multiplicative noise and external inputs are taken into account.

[1]  Hideo Hasegawa,et al.  Dynamical mean-field theory of spiking neuron ensembles: response to a single spike with independent noises. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  H. Brand,et al.  Multiplicative stochastic processes in statistical physics , 1979 .

[3]  L Schimansky-Geier,et al.  Macroscopic limit cycle via pure noise-induced phase transitions. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Hideo Hasegawa,et al.  Augmented moment method for stochastic ensembles with delayed couplings. I. Langevin model. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  J. Elgin.  The Fokker-Planck Equation: Methods of Solution and Applications , 1984 .

[6]  Włodarczyk,et al.  Interpretation of the nonextensivity parameter q in some applications of tsallis statistics and Levy distributions , 2000, Physical review letters.

[7]  L. Monteiro,et al.  Analytical results on a Wilson-Cowan neuronal network modified model. , 2002, Journal of theoretical biology.

[8]  Celia Anteneodo,et al.  Multiplicative noise: A mechanism leading to nonextensive statistical mechanics , 2002 .

[9]  Van den Broeck C,et al.  Noise-induced nonequilibrium phase transition. , 1994, Physical review letters.

[10]  H. Tuckwell,et al.  Statistical properties of stochastic nonlinear dynamical models of single spiking neurons and neural networks. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  Hideo Hasegawa,et al.  Augmented moment method for stochastic ensembles with delayed couplings. II. FitzHugh-Nagumo model. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[13]  Hideo Hasegawa,et al.  Dynamical mean-field approximation to small-world networks of spiking neurons: from local to global and/or from regular to random couplings. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[15]  Mean-field limit of systems with multiplicative noise. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Fluctuation Dissipation Relation for a Langevin Model with Multiplicative Noise , 2001, cond-mat/0108393.

[17]  Hideo Hasegawa.  Dynamical mean-field theory of noisy spiking neuron ensembles: application to the Hodgkin-Huxley model. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Rates of convergence of non-extensive statistical distributions to Lévy distributions in full and half-spaces , 2000, cond-mat/0009399.

[19]  C Anteneodo,et al.  Additive-multiplicative stochastic models of financial mean-reverting processes. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Nonextensive aspects of small-world networks , 2005, cond-mat/0506301.

[21]  J. García-Ojalvo,et al.  Effects of noise in excitable systems , 2004 .

[22]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[23]  H. Hasegawa.  Synchronizations in small-world networks of spiking neurons: diffusive versus sigmoid couplings. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Critical behavior of nonequilibrium phase transitions to magnetically ordered states. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Raúl Toral,et al.  Nonequilibrium phase transitions induced by multiplicative noise , 1997 .