Negative self-feedback induces enhancement and transition of spiking activity for class 3 excitability

Post-inhibitory rebound (PIR) spike, which has been widely observed in diverse nervous systems with different physiological functions and simulated in theoretical models with class 2 excitability, presents a counterintuitive nonlinear phenomenon in that the inhibitory effect can facilitate neural firing behavior. In this study, a PIR spike induced by inhibitory stimulation from the resting state corresponding to class 3 excitability that is not related to bifurcation is simulated in the Morris-Lecar neuron. Additionally, the inhibitory self-feedback mediated by an autapse with time delay can evoke tonic/repetitive spiking from phasic/transient spiking. The dynamical mechanism for the PIR spike and the tonic/repetitive spiking is acquired with the phase plane analysis and the shape of the quasi-separatrix curve. The result extends the counterintuitive phenomenon induced by inhibition to class 3 excitability, which presents a potential function of inhibitory autapse and class 3 neuron in many neuronal systems such as the auditory system.

Orbital stability analysis of trajectories of highly nonlinear dynamic systems with feedback coupling

Orbital stability and qualitative study of the oscillations of a highly nonlinear dynamic system with feedback coupling are considered. For a highly nonlinear dynamic system with feedback coupling that satisfies Liénard’s theorem (on the existence and uniqueness of a periodic solution), a complete study of the phase pattern of the system is conducted. Applying the Poincaré criterion, the conditions for the existence of limit cycles and their Lyapunov stability are determined. The diagrams of phase trajectories are constructed numerically using the Mathcad 15 software package. Limit cycles are established, which are consistent with the limit cycles obtained by the Poincaré method. The behavior of trajectories outside the limit cycles is investigated. Recurrent homogeneous Pfaff equations are obtained, which determine the behavior of the systems “at infinity”. It was determined that the infinitely distant point of the horizontal axis is the only singular point for these equations. Linear approximations of recurrent homogeneous equations are obtained, which make it possible to determine the nature of the singular points. It was found that the trajectories then wind like a spiral on the limit cycles. Images of trajectories on the phase plane outside the limit cycles for the cases of degrees of nonlinearity under consideration are constructed.

On Truly Nonlinear Oscillator Equations of Ermakov-Pinney Type

In this paper we present a general class of differential equations of Ermakov-Pinney type which may serve as truly nonlinear oscillators. We show the existence of periodic solutions by exact integration after the phase plane analysis. The related quadratic Lienard type equations are examined to show for the first time that the Jacobi elliptic functions may be solution of second-order autonomous non-polynomial differential equations.

Modelling the coupling of the M-clock and C-clock in lymphatic muscle cells

Lymphoedema develops due to chronic dysfunction of the lymphatic vascular system which results in fluid accumulation between cells. The condition is commonly acquired secondary to diseases such as cancer or the therapies associated with it. The primary driving force for fluid return through the lymphatic vasculature is provided by contractions of the muscularized lymphatic collecting vessels, driven by electrical oscillations. However, there is an incomplete understanding of the molecular and bioelectric mechanisms involved in lymphatic muscle cell excitation, hampering the development and use of pharmacological therapies. Modelling in silico has contributed greatly to understanding the contributions of specific ion channels to the cardiac action potential, but modelling of these processes in lymphatic muscle remains limited. Here, we propose a model of oscillations in the membrane voltage (M-clock) and intracellular calcium concentrations (C-clock) of lymphatic muscle cells. We modify a model by Imtiaz and colleagues to enable the M-clock to drive the C-clock oscillations. This approach differs from typical models of calcium oscillators in lymphatic and related cell types, but is required to fit recent experimental data. We include an additional voltage dependence in the gating variable control for the L type calcium channel, enabling the M-clock to oscillate independently of the C-clock. We use phase-plane analysis to show that these M-clock oscillations are qualitatively similar to those of a generalised FitzHugh-Nagumo model. We also provide phase plane analysis to understand the interaction of the M-clock and C-clock oscillations. The model and methods have the potential to help determine mechanisms and find targets for pharmacological treatment of lymphoedema.

Complex Investigations of a Piecewise-Smooth Remanufacturing Bertrand Duopoly Game

This paper considers a Bertrand competition between two firms whose decision variables are derived from a quadratic utility function. The first firm produces new products with their own prices while the second firm re-manufactures returned products and sells them in the market at prices that may be less than or equal to the price of the first firm. Dynamically, this competition is constructed on which boundedly rational firms apply a gradient adjustment mechanism to update their prices in each period. According to this mechanism and the nature of the competition, a two-dimensional piecewise smooth discrete dynamic map was constructed in order to study the complex dynamic characteristics of the game. The phase plane of the map was divided into two different regions, separated by border curve. The equilibrium points of the map, in each region on where they are defined, were calculated, and their stability conditions were investigated. Furthermore, we conducted a global analysis to investigate the complex structure of the map, such as closed invariant curves, periodic cycles, and chaotic attractors and their basins, which cause qualitative changes as some parameters are allowed to vary.

Direct yaw-moment control of vehicles based on phase plane analysis

In view of the problems related to vehicle-handling stability and the real-time correction of the heading direction, nonlinear analysis of a vehicle steering system was carried out based on phase plane theory. Subsequently, direct yaw-moment control (DYC) of the vehicle was performed. A four-wheel, seven-degree-of-freedom nonlinear dynamic model that included the nonlinear characteristics of the tire was established. The stable and unstable regions of the vehicle phase plane were divided, and the stable boundary model was established by analyzing the side slip angle–yaw rate ([Formula: see text]) and side slip angle–side slip angle rate [Formula: see text] phase planes as functions of the vehicle state variables. In the unstable region of the phase plane, taking the instability degree as the control target, a fuzzy neural network control strategy was utilized to determine the additional yawing moment of the vehicle required for stability restoration, which pulled the vehicle back from an unstable state to the stable region. In the stable region of the phase plane, a fuzzy control strategy was utilized to determine the additional yawing moment so that the actual state variables followed the ideal state variables. In this way, the vehicle responded rapidly and accurately to the steering motion of the driver. A simulation platform was established in MATLAB/Simulink and three working condition was tested, that is, step, sine with dwell, and sine amplification signals. The results showed that the vehicle handling stability and the instantaneous heading-direction adjustment ability were both improved due to the control strategy.

Analysis and Synthesis of a Fuzzy Controller by the Phase Plane Method

The article is devoted to solving the problem of analysis and synthesis of a control system with a fuzzy controller by the phase plane method. The nonlinear transformation, built according to the Sugeno fuzzy model, is approximated by a piecewise linear characteristic consisting of three sections: two piecewise linear and one piecewise constant. This approach allows us to restrict ourselves to three sheets of phase trajectories, each of which is constructed on the basis of a second-order differential equation. Taking this feature into account, the technique of "stitching" of three sheets of phase trajectories is considered and an analytical base is obtained that allows one to determine the conditions for "stitching" of phase trajectories for various variants of piecewise-linear approximation of the characteristics of a fuzzy controller. In view of the specificity of the approximated model of the fuzzy controller used, useful analytical relations are given, with the help of which it is possible to calculate the time of motion of the representing point for each section with the involvement of the numerical optimization apparatus. For a variant of the approximation of three sections, a technique for synthesizing a fuzzy controller is proposed, according to which the range of parameters and the range of input signals are determined, at which an aperiodic process and a given control time are provided. On the model of the automatic control system of the drive level of the mechatronic module, it is shown that the study of a fuzzy system by such an approximated characteristic of a fuzzy controller gives quite reliable results. The conducted studies of the influence of the degree of approximation on the quality of control show that the approximated characteristic of a fuzzy controller gives a slight deterioration in quality in comparison with the smooth characteristic of a fuzzy controller. Since the capabilities of the phase plane method are limited to the 2nd order of the linear part of the automatic control system, the influence of the third order on the dynamics of the system is considered using the example of a mechatronic module drive. It is shown that taking into account the electric time constant leads to overshoot within 5-10 %. Such overshoot can be eliminated due to the proposed recommendations for correcting the static characteristic of the fuzzy controller.