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In this paper, the condition of completely nearest-neighbor couplings is introduced into the coupled Hindmarsh-Rose neurons in two-dimensional arrays. It is found that the stable rotating spiral wave can be developed and the transition of spiral wave in the coupled Hindmarsh-Rose neurons are investigated. The factor of synchronization is defined to investigate the development and instability of the spiral wave. Furthermore, the external injected current, coupling coefficients and other decisive bifurcation parameter r and χ, are endowed with different values to study the transition of spiral wave by analyzing the factor of synchronization and the snapshots of the activator. It is found that the critical sudden change points in the curve for factor of synchronization often indicates sudden transition of spiral wave, the instability or death of the spiral wave. The snapshots are also plotted to confirm the results from the curve of the factor of synchronization. Finally, the noise-induced instability and chaotic logistic map-induced instability of spiral wave are investigated and discussed. [ABSTRACT FROM AUTHOR]

Collective behaviors of spiral waves in the networks of Hodgkin-Huxley neuron are investigated. A stable rotating spiral wave can be developed to occupy the quiescent areas in networks of neurons by selecting appropriate initial values for the variables in the networks of neurons. In our numerical studies, most neurons are quiescent and finite (few) numbers of neurons are selected with different values to form a spiral seed. In this way, neurons communicating are carried by propagating spiral wave to break through the quiescent domains (areas) in networks of neurons. The effect of membrane temperature on the formation of spiral wave is investigated by selecting different fixed membrane temperatures in the networks, and it is found that a spiral wave cannot be developed if the membrane temperature is close to a certain threshold. A quantitative factor of synchronization is defined to measure the statistical properties and collective behaviors of the spiral wave. And a distinct phase transition, which indicates the critical condition for spiral survival, is observed in the sudden changing point of the factors of synchronization curve vs. certain bifurcation parameter. Internal noise is introduced into ion channels (channel noise) with the Langevin method. It is found that a stable rotating spiral wave is developed and the spiral wave is robust to weak channel noise (the membrane patch is not small). The spiral wave can not grow up and the stable rotating spiral wave encounters instability in presence of strong channel noise. Coherence resonance-like behavior is observed in calculating the factors of synchronization in presence of channel noise. [ABSTRACT FROM AUTHOR]

Abstract: Transition of spiral wave in the regular networks of Hodgkin–Huxley (H–H) neurons is simulated and discussed in detail when the effect of membrane temperature and forcing current is considered. Neurons are distributed in the sites of two-dimensional array, neurons are connected with complete nearest-neighbor connections, no-flux boundary conditions, appropriate initial values and physiological parameters are used to develop a stable rotating spiral wave. A statistic factor of synchronization is defined to discuss the transition and development of spiral wave in the two parameters space (membrane temperature T and forcing current I), and it is found that spiral wave keeps alive due to positive current forcing and the spiral wave can be removed completely when the temperature is increased to a threshold about T =22.3°C at a fixed current intensity. Periodical forcing current is imposed on the networks of neurons globally and locally, respectively. It is found that spiral wave could be suppressed by the new generated traveling wave or target wave when periodical forcing current is imposed on the border of networks of neurons, and the most effective frequency of the external forcing current is close to the intrinsic frequency of the spiral wave of the networks. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the synchronization of certain degenerate optical parametric oscillators is investigated in detail. Complete and/or partial synchronization can be reached when linear controller, constructed by the real part or imaginary part of the subharmonic mode, is imposed on the chaotic degenerate optical parametric oscillators with appropriate coupling intensity. The Lyapunov exponents under different coupling coefficients are calculated to find the critical condition for complete synchronization. Transition from complete synchronization to partial synchronization is observed when nonlinear coupling is introduced into the two chaotic oscillators. It is found that synchronization of chaotic oscillators keeps robust when the intensity of the nonlinear coupling is less than the intensity of the linear coupling; the complete synchronization state is destructed and transient period for partial synchronization is in certain delay when the intensity of the nonlinear coupling is beyond the intensity of the linear coupling. [Copyright &y& Elsevier]