Controlling the onset of Hopf bifurcation in the Hodgkin-Huxley model

Abstract

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.

Spike-frequency adaptation of a two-compartment neuron modulated by extracellular electric fields

“…The different interactions between stable and unstable critical manifolds could switch V S from subthreshold oscillation to suprathreshold, and then the neu- Fig. 6 Typical one-parameter bifurcation diagrams of the neuron with I M (left panels) or the I AHP (right panels) current in different ranges of the parameter p. The abscissa of the bifurcation diagram is the electric field E, the ordinate is the steady-state V S at the fixed point or at the max/min of the limit cycle as parameter E increases ron generates MMO (Brøns et al 2006;Rubin and Wechselberger 2007;Xie et al 2008;Desroches et al 2012). The relation between the canard phenomenon and MMO was first proposed by Milik et al (1998), and the detailed descriptions can be found in Rubin and Wechselberger (2007) and Brøns et al (2006).…”

“…That is, the oscillation in the MMO pattern is quickly switched between small and large amplitude limit cycles. This sudden change is known as canard or canard explosion, which is closely related to the generation of the MMO pattern (Rubin and Wechselberger 2007;Xie et al 2008;Desroches et al 2012). The subthreshold limit cycle grows through a series of canard cycles to a suprathreshold oscillation.…”

Controlling bifurcation in power system based on LaSalle invariant principle

“…It is proved in previous work that power systems can easily satisfy Lipschitz condition (leastwise content local Lipschitz condition), so the uncontrolled solution of (1) is bounded [21]. Note that for the traditional bifurcation controller, the following assumptions are necessary: (1) the position of the desired steady state (namely, the equilibrium points of systems) is known a priori; (2) the system parameters are deterministic; (3) all the variables of system are accessible [12,19,22]. However, in our work the controller is independent of the steady state and parameters of system; in other words, it does not require precise knowledge of the parameter values and the position of the desired steady state a priori.…”

Entrainment and Control of Bacterial Populations: An in Silico Study over a Spatially Extended Agent Based Model

“…In cell biology dynamical models of control are more common in the field of neuroscience. In such models, control in the form of external intervention may alleviate symptoms of specific neurological disorders, such as Parkinson's and essential tremors [3][4][5][6], for example by disrupting the synchrony of coupled oscillating neurons through electrical stimulation [7][8][9][10]. The development of microfluidics technologies over the recent years [11] has allowed researchers to apply control in other cellular systems as well, such as microbe and yeast cells, in order to affect their metabolic states usually by manipulating gene expression.…”