Computational Analysis and Bifurcation of Regular and Chaotic Ca2+ Oscillations

This study investigated the stability and bifurcation of a nonlinear system model developed by Marhl et al. based on the total Ca2+ concentration among three different Ca2+ stores. In this study, qualitative theories of center manifold and bifurcation were used to analyze the stability of equilibria. The bifurcation parameter drove the system to undergo two supercritical bifurcations. It was hypothesized that the appearance and disappearance of Ca2+ oscillations are driven by them. At the same time, saddle-node bifurcation and torus bifurcation were also found in the process of exploring bifurcation. Finally, numerical simulation was carried out to determine the validity of the proposed approach by drawing bifurcation diagrams, time series, phase portraits, etc.

Effect of Static Bifurcation on Logical Stochastic Resonance in a Symmetric Bistable System

In previous research works, logical stochastic resonance (LSR) was reported to frequently occur in an asymmetric bistable system, where the bias parameter is the key factor to make the LSR appear. In this work, we investigate the effect of different anharmonic periodic signals on the pitchfork and saddle-node bifurcations in a symmetric bistable system. We focus on the relationship between the static bifurcation and LSR. We use both numerical and circuit simulations to analyze some interesting phenomena. Like the bias parameter, some anharmonic periodic signals also break the symmetry of the symmetric bistable system and lead to the saddle-node bifurcation. The anharmonic periodic signal with a constant term in its expanded Fourier series induces a reliable LSR in the symmetric bistable system. The key factor of LSR is the saddle-node bifurcation which implies the asymmetry of the system. Here, we replace the bias parameter by choosing an anharmonic periodic signal and make the LSR occur in a different way.

An Extension of Two Species Lotka-Volterra Competition Model

Limiting resource is a angular stone of the interactions between species in ecosystems such as competition, prey-predators and food chain systems. In this paper, we propose a planar system as an extension of Lotka-Voterra competition model. This describes? two competitive species for a single resource? which are affected by intra and inter-specific interference. We give its complete analysis for the existence and local stability of all equlibria and some conditions of global stability. The model exhibits a rich set of behaviors with a multiplicity of coexistence equilibria, bi-stability, tri-stability and occurrence of global stability of the exclusion of one species and the coexistence? equilibrium. The asymptotic behavior and the number of coexistence equilibria are shown by a saddle-node bifurcation of the level of resource under conditions on competitive effects relatively to associated growth rate per unit of resource.Moreover, we determine the competition outcome in the situations of Balanced and Unbalanced intra-inter species competition effects. Finally, we illustrate results by numerical simulations.

Critical behavior in the artificial axon

The Artificial Axon is a unique synthetic system, based on biomolecular components, which supports action potentials. Here we examine, experimentally and theoretically, the properties of the threshold for firing in this system. As in real neurons, this threshold corresponds to the critical point of a saddle-node bifurcation. We measure the delay time for firing as a function of the distance to threshold, recovering the expected scaling exponent of −1/2. We introduce a minimal model of the Morris-Lecar type, validate it on the experiments, and use it to extend analytical results obtained in the limit of ”fast” ion channel dynamics. In particular, we discuss the dependence of the firing threshold on the number of channels. The Artificial Axon is a simplified system, an Ur-neuron, relying on only one ion channel species for functioning. Nonetheless, universal properties such as the action potential behavior near threshold are the same as in real neurons. Thus we may think of the Artificial Axon as a cell-free breadboard for electrophysiology research.

Multiple and Complex Codimension-2 Bifurcations underlying Post-inhibitory Rebound Spike and Excitability Transition

Neuron exhibits nonlinear dynamics such as excitability transition and post-inhibitory rebound (PIR) spike related to bifurcations, which are associated with information processing, locomotor modulation, or brain disease. PIR spike is evoked by inhibitory stimulation instead of excitatory stimulation, which presents a challenge to the threshold concept. In the present paper, 7 codimension-2 or degenerate bifurcations related to 10 codimension-1 bifurcations are acquired in a neuronal model, which presents the bifurcations underlying the excitability transition and PIR spike. Type I excitability corresponds to saddle-node bifurcation on an invariant cycle (SNIC) bifurcation, and type II excitability to saddle-node (SN) bifurcation or sub-critical Hopf (SubH) bifurcation or sup-critical Hopf (SupH) bifurcation. The excitability transition from type I to II corresponds to the codimension-2 bifurcation, Saddle-Node Homoclinic orbit (SNHO) bifurcation, via which SNIC bifurcation terminates and meanwhile big homoclinic orbit (BHom) bifurcation and SN bifurcation emerge. A degenerate bifurcation via which BHom bifurcation terminates and fold limit cycle (LPC) bifurcation emerges is responsible for spiking transition from type I to II, and the roles of other codimension-2 bifurcations (Cusp, Bogdanov-Takens, and Bautin) are discussed. In addition, different from the widely accepted viewpoint that PIR spike is mainly evoked near Hopf bifurcation rather than SNIC bifurcation, PIR spike is identified to be induced near SNIC or BHom or LPC bifurcations, and threshold curves resemble that of Hopf bifurcation. The complex bifurcations present comprehensive and deep understandings of excitability transition and PIR spike, which are helpful for the modulation to neural firing activities and physiological functions.

Sliding Dynamics and Bifurcations of a Filippov System with Nonlinear Threshold Control

Considering the effectiveness of introducing the change rate of viral loads into the threshold setting policy for triggering interventions, we propose an immune-virus Filippov system with a nonlinear threshold. By developing new analytical and numerical methods, we systematically studied the rich dynamical behaviors and bifurcations of the proposed system, including the existence of three sliding segments and three pseudo-equilibria, boundary-center bifurcation, boundary-saddle bifurcation, pseudo-saddle-node bifurcation and tangency bifurcation. We further showed that the proposed system can exhibit virous structures in the coexistence of multiple steady states. Phenomena include bistability of two pseudo-equilibria, tristability and multiplestability of two pseudo-equilibria with regular equilibria or touching cycles. The modeling methods, as well as the analytical and numerical methods, can be widely applied to many other fields.

Bifurcation Analysis of a Two-Dimensional Neuron Model under Electrical Stimulation

The two-dimensional neuron model can not only reproduce abundant firing patterns, but also satisfy the research of dynamical behavior because of its nonlinear characteristics. It is the most simplified model that includes the fast and slow variables required for neuron firing. In this paper, the dynamic characteristics of two-dimensional neuron model are described by both analytical and numerical methods, and the influence of model parameters and external stimuli on dynamic characteristics is described. The firing characteristics of the Prescott model under external electrical stimulation are studied, and the influence of electrophysiological parameters on the firing characteristics is analyzed. The saddle-node bifurcation and Hopf bifurcation characteristics are studied through the distribution of equilibrium points. It is found that there are critical saddle-node bifurcation and critical Hopf bifurcation in the Prescott model. And the value range of the key parameters that cause the critical bifurcation of the model is obtained by analytical methods.

Bifurcation analysis in a predator–prey model with strong Allee effect

In this paper, strong Allee effects on the bifurcation of the predator–prey model with ratio-dependent Holling type III response are considered, where the prey in the model is subject to a strong Allee effect. The existence and stability of equilibria and the detailed behavior of possible bifurcations are discussed. Specifically, the existence of saddle-node bifurcation is analyzed by using Sotomayor’s theorem, the direction of Hopf bifurcation is determined, with two bifurcation parameters, the occurrence of Bogdanov–Takens of codimension 2 is showed through calculation of the universal unfolding near the cusp. Comparing with the cases with a weak Allee effect and no Allee effect, the results show that the Allee effect plays a significant role in determining the stability and bifurcation phenomena of the model. It favors the coexistence of the predator and prey, can lead to more complex dynamical behaviors, not only the saddle-node bifurcation but also Bogdanov–Takens bifurcation. Numerical simulations and phase portraits are also given to verify our theoretical analysis.

A new heterogeneous traffic flow model based on lateral distance headway

Based on a density gradient model proposed recently by Imran and Khan, a new heterogeneous traffic flow model considering time and lateral distance is proposed. The type and stability of the equilibrium solution of the model are discussed by using the differential equation theory, and the global distribution structure of the trajectory in the phase plane is analyzed. In addition, the density wave stability conditions and saddle-node bifurcation conditions of the model are studied, and the solitary wave solutions of the KdV equation in the metastable region are derived by using the reduced perturbation method. The numerical results show that the new model cannot only reproduce the spatiotemporal oscillation phenomenon when walking and stopping, but also describe the sudden change behavior of traffic near the critical point of saddle-node bifurcation. It is shown that the model can reproduce some complex traffic phenomena qualitatively.