The method of moments is applied to an underdamped bistable oscillator driven by Gaussian white noise and a weak periodic force for the observations of stochastic resonance and the resulting resonant structures are compared with those from Langevin simulation. The physical mechanisms of the stochastic resonance are explained based on the evolution of the intrawell frequency peak and the above-barrier frequency peak via the noise intensity and the fluctuation-dissipation theorem, and the three possible sources of stochastic resonance in the system are confirmed. Additionally, with the noise intensity fixed, the stochastic resonant structures are also observed by adjusting the nonlinear parameter.

Previous experimental work has shown that the firing rate of multiple time-scales of adaptation for single rat neocortical pyramidal neurons is consistent with fractional-order differentiation, and the fractional-order neuronal models depict the firing rate of neurons more verifiably than other models do. For this reason, the dynamic characteristics of the fractional-order Hindmarsh-Rose (HR) neuronal model were here investigated. The results showed several obvious differences in dynamic characteristic between the fractional-order HR neuronal model and an integer-ordered model. First, the fractional-order HR neuronal model displayed different firing modes (chaotic firing and periodic firing) as the fractional order changed when other parameters remained the same as in the integer-order model. However, only one firing mode is displayed in integer-order models with the same parameters. The fractional order is the key to determining the firing mode. Second, the Hopf bifurcation point of this fractional-order model, from the resting state to periodic firing, was found to be larger than that of the integer-order model. Third, for the state of periodically firing of fractional-order and integer-order HR neuron model, the firing frequency of the fractional-order neuronal model was greater than that of the integer-order model, and when the fractional order of the model decreased, the firing frequency increased.

It is a challenging problem to establish safe and simple therapeutic methods for various complicated diseases of the nervous system, particularly dynamical diseases such as epilepsy, Alzheimer's disease, and Parkinson's disease. From the viewpoint of nonlinear dynamical systems, a dynamical disease can be considered to be caused by a bifurcation induced by a change in the values of one or more regulating parameter. Therefore, the theory of bifurcation control may have potential applications in the diagnosis and therapy of dynamical diseases. In this study, we employ a washout filter-aided dynamic feedback controller to control the onset of Hopf bifurcation in the Hodgkin-Huxley (HH) model. Specifically, by the control scheme, we can move the Hopf bifurcation to a desired point irrespective of whether the corresponding steady state is stable or unstable. In other words, we are able to advance or delay the Hopf bifurcation, so as to prevent it from occurring in a certain range of the externally applied current. Moreover, we can control the criticality of the bifurcation and regulate the oscillation amplitude of the bifurcated limit cycle. In the controller, there are only two terms: the linear term and the nonlinear cubic term. We show that while the former determines the location of the Hopf bifurcation, the latter regulates the criticality of the Hopf bifurcation. According to the conditions of the occurrence of Hopf bifurcation and the bifurcation stability coefficient, we can analytically deduce the linear term and the nonlinear cubic term, respectively. In addition, we also show that mixed-mode oscillations (MMOs), featuring slow action potential generation, which are frequently observed in both experiments and models of chemical and biological systems, appear in the controlled HH model. It is well known that slow firing rates in single neuron models could be achieved only by type-I neurons. However, the controlled HH model is still classified as a type-II neuron, as is the original HH model. We explain that the occurrence of MMOs can be related to the presence of the canard explosion where a small oscillation grows through a sequence of canard cycles to a relaxation oscillation as the control parameter moves through an interval of exponentially small width.

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