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.

Firing patterns and bifurcation analysis of Purkinje cell dendrite model

This paper mainly studied firing patterns and related bifurcations in the Purkinje cell dendrite model. Based on the methods of equivalent potentials and time scale analysis, the initial six-dimensional (6D) dendrite model is reduced to a 3D form to facilitate the calculation. We numerically show that the dendrite model could exhibit period-adding bifurcation and four bursting patterns for several vital parameters. Then the bifurcation mechanisms and transition of these four bursting patterns are discussed by phase plane analysis, and two-parameter bifurcation analysis of the fast subsystem, respectively. Moreover, we computed the first Lyapunov coefficient to determine the stability of Hopf bifurcation. Ultimately, we analyzed the codimension-two bifurcation of the whole system and gave a detailed theoretical derivation of the Bogdanov–Takens bifurcation.

Dissipative electrostatic wave modulation in warm multi-ion dusty plasmas with superthermal electrons

A theoretical investigation has been carried out to explore the modulational instability (MI) of electrostatic waves in a warm multi-ion dusty plasma system containing positive ions, negative ions and positively or negatively charged dust in presence of superthermal electrons. With the help of the standard perturbation technique, it is found that the dynamics of the modulated wave is governed by a damped nonlinear Schrödinger equation (NLSE). Regions of MI of the electrostatic wave are precisely determined and the analytical solutions predict the formation of dissipative bright and dark solitons as well as dissipative first- and second-order rogue wave solutions. It is found that the striking features (viz., instability criteria, amplitude and width of rogue waves, etc.) are significantly modified by the effects of relevant plasma parameters such as degree of the electron superthermality, dust density, etc. The time dependent numerical simulations of the damped NLSE reveal that modulated electrostatic waves exhibit breather like structures. Moreover, phase plane analysis has been performed to study the dynamical behaviors of NLSE by using the theory of dynamical system. It is remarked that outcome of present study may provide physical insight into understanding the generation of several types of nonlinear structures in dusty plasma environments, where superthermal electrons, positive and negative ions are accountable (e.g. Saturn’s magnetosphere, auroral zone, etc.).

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.

Mechanistic Model of Telomere Length Homeostasis in Yeast

Cells maintain homeostatic telomere length, and this homeostatic disruption leads to various types of diseases. Presently, it is not clear how telomeres achieve homeostasis. One of the prevailing hypotheses is a protein-counting model with a built-in sensor mechanism that counts proteins that directly regulate the telomeric length. However, it does not explain telomere length regulation at the mechanistic level. Here, we present a mathematical model based on the underlying molecular mechanisms of length regulation needed to establish telomere length homeostasis in yeast. We perform both deterministic and stochastic simulations to validate the models with the experimental data of Teixeira et al., rate-balance plot, and phase plane analysis to understand the nature of dynamics exhibited by the models. For global analysis, we constructed bifurcation diagrams. The model explains the role of negative and positive feedback loops and a delay between telomerase and telomere-bound proteins, leading to oscillations in telomere length. We map these in-silico results to proposition by Teixeira of telomeres making a transition between extendible and non-extendible states.

Travelling Wave and Turing Patterns in Subdiffusive Autocatalytic Systems

In this work, we analyse autocatalytic reactions in complex and disordered media which are governed by subdiffusion. The mean square displacement of molecules here scale as tγ where 0<γ<1. These systems are governed by fractional partial differential equations. Two systems are analysed i) in the first a logistic growth expression is used to represent the growth kinetics of bacteria. Here the system dynamics is governed by a single variable. ii) the second system is a two variable cubic autocatalytic system in a porous media. Here each reactant is involved in the autocatalytic generation of the other. These systems have multiple steady states. They exhibit traveling waves moving from an unstable steady state to a stable steady state. The minimum wave velocity has been obtained from phase plane analysis analytically for the first system. In addition, the two variable system also shows Turing patterns in selected regions of parameter space. The stability boundary for Turing patterns for subdiffusive system is found to be the same as that for regular diffusive systems obtained by Seshai et al. [1]. System behaviour as predicted by the stability analysis is verified using a robust implicit numerical method based on L1 scheme.

Research on the Potential of Front Wheel Steering Control for Vehicle Dynamics Control

The development of active steering control technology not only provides key actuators for intelligent vehicle motion control, but also expands vehicle stability and safety. This paper studies the potential control ability of the front-wheel steering control to the vehicle plane dynamics, and the controllable area boundary is designed on the phase plane of side slip angle and yaw rate. Previous studies have defined a dynamics stable area on the vehicle states phase plane, in which the vehicle state can autonomously return to a stable equilibrium point. The area outside the stable area are divided into the controllable area and the uncontrollable area in this paper. In the controllable area, the front-wheel steering control has the ability to pull the vehicle states back towards the stable area. Considering actuator constraints and model errors, based on the principle of safety design, a band-shaped critical area is designed to separate the controllable area from the uncontrollable area, and the linear mathematical model of the controllable area boundary is designed. In order to verify the rationality of the controllable area definition, nonlinear model predictive controller is designed to control the vehicle outside the dynamics stable area. The controller uses the high-fidelity nonlinear vehicle model and the magic formula tire model as the state equation constraints, and the practical steering actuator constraints are used as the control input constraints, and the nonlinear numerical optimization solver is used to solve the optimal steering input sequence. The phase plane analysis of the controlled vehicle verifies the rationality of the controllable area defined in this paper.