New dynamics in cerebellar Purkinje cells: torus canards

We describe a transition from bursting to rapid spiking in a reduced mathematical model of a cerebellar Purkinje cell. We perform a slow-fast analysis of the system and find that-after a saddle node bifurcation of limit cycles-the full model dynamics temporarily follow a repelling branch of limit cycles. We propose that the system exhibits a dynamical phenomenon new to realistic, biophysical applications: torus canards.     

An elementary model of torus canards

We study the recently observed phenomena of torus canards. These are a higher-dimensional generalization of the classical canard orbits familiar from planar systems and arise in fast-slow systems of ordinary differential equations in which the fast subsystem contains a saddle-node bifurcation of limit cycles. Torus canards are trajectories that pass near the saddle-node and subsequently spend long times near a repelling branch of slowly varying limit cycles. In this article, we carry out a study of torus canards in an elementary third-order system that consists of a rotated planar system of van der Pol type in which the rotational symmetry is broken by including a phase-dependent term in the slow component of the vector field. In the regime of fast rotation, the torus canards behave much like their planar counterparts. In the regime of slow rotation, the phase dependence creates rich torus canard dynamics and dynamics of mixed mode type. The results of this elementary model provide insight into the torus canards observed in a higher-dimensional neuroscience model.     

Bursting Ca2+ Oscillations and Synchronization in Coupled Cells

A mathematical model proposed by Grubelnk et al. [Biophys. Chem. 94 （2001） 59] is employed to study the physiological role of mitochondria and the cytosolic proteins in generating complex Ca^2＋ oscillations, lntracellulax bursting calcium oscillations of point-point, point-cycle and two-folded limit cycle types are observed and explanations are given based on the fast/slow dynamical analysis, especially for point-cycle and two-folded limit cycle types, which have not been reported before. Furthermore, synchronization of coupled bursters of Ca^2＋ oscillations via gap junctions and the effect of bursting types on synchronization of coupled cells are studied. It is argued that bursting oscillations of point-point type may be superior to achieve synchronization than that of point cycle type.     

Symmetric bursting of focus-focus type in the controlled Lorenz system with two time scales

By employing a special feedback controlling scheme, a hyperchaotic Lorenz system with the structure of two time scales is constructed. Two kinds of bursting phenomena, symmetric fold/fold bursting and symmetric sub-Hopf/sub-Hopf bursting, can be observed in this system. Their respective dynamical behaviors are investigated by means of slow-fast analysis. In particular, symmetric fold/fold bursting is of focus-focus type, namely, both the up-state and the down-state are stable focus, which is different from the usual fold/fold bursting; Symmetric sub-Hopf/sub-Hopf bursting is also of focus-focus type, which has not been reported in previous work. Furthermore, phase plane analysis has been introduced to explore the evolution details of the fast subsystem for symmetric sub-Hopf/sub-Hopf bursting. With the variation of the parameter, symmetric sub-Hopf/sub-Hopf bursting can evolve to symmetric chaotic bursting or even hyperchaos.     

Canards in a rheodynamic model of cardiac pressure pulsations

This paper reports on the canard phenomenon occurring in a rheodynamic model of cardiac pressure pulsations. By singular perturbation techniques the corresponding parameter value at which canards exist is obtained. The physiological significance of canards in this model is given.     

Controlling coherence of noisy and chaotic oscillators by a linear feedback

We analyze the dynamics of a noisy limit cycle oscillator coupled to a general passive linear system. We analytically demonstrate that the phase diffusion constant, which characterizes the coherence of the oscillations, can be efficiently controlled. Theoretical analysis is performed in the framework of linear and Gaussian approximations and is supported by numerical simulations. We also demonstrate numerically the coherence control of a chaotic system.     

Direct transition from a stable equilibrium to quasiperiodicity in non-smooth systems

The purpose of this Letter is to show how a border-collision bifurcation in a piecewise-smooth dynamical system can produce a direct transition from a stable equilibrium point to a two-dimensional invariant torus. Considering a system of nonautonomous differential equations describing the behavior of a power electronic DC/DC converter, we first determine the chart of dynamical modes and show that there is a region of parameter space in which the system has a single stable equilibrium point. Under variation of the parameters, this equilibrium may collide with a discontinuity boundary between two smooth regions in phase space. When this happens, one can observe a number of different bifurcation scenarios. One scenario is the continuous transformation of the stable equilibrium into a stable period-1 cycle. Another is the transformation of the stable equilibrium into an unstable period-1 cycle with complex conjugate multipliers, and the associated formation of a two-dimensional (ergodic or resonant) torus.     

Dynamics Analysis and Transition Mechanism of Bursting Calcium Oscillations in Non-Excitable Cells

A one-pool model with Ca^2＋-activated inositol-trisphosphate-concentration degradation is considered. For complex bursting Ca^2＋ oscillation, point-cycle bursting of subHopf-subHopf type is found to be in the intermediate state from quasi-periodic bursting to point-point bursting of subHopf-subHopf type. The fast-slow burster analysis is used to study the transition mechanisms among simple periodic oscillation, quasi-periodic bursting, point-point and point-cycle burstings. The dynamics analysis of different oscillations provides better insight into the generation and transition mechanisms of complex intra- and inter-cellular Ca^2＋ signalling.     

Synchronization in coupled nonidentical incommensurate fractional-order systems

Synchronization of fractional-order nonlinear systems has received considerable attention for many research activities in recent years. In this Letter, we consider the synchronization between two nonidentical fractional-order systems. Based on the open-plus-closed-loop control method, a general coupling applied to the response system is proposed for synchronizing two nonidentical incommensurate fractional-order systems. We also derive a local stability criterion for such synchronization behavior by utilizing the stability theory of linear incommensurate fractional-order differential equations. Feasibility of the proposed coupling scheme is illustrated through numerical simulations of a limit cycle system, a chaotic system and a hyperchaotic system.     

计算相位响应曲线的方波扰动直接算法

通过相位响应曲线可对具有极限环周期运动的动力系统的性质有更为深入的理解.神经元是一个典型的动力系统,因此相位响应曲线提供了一种研究神经元重复周期放电行为的新思路.本文提出一种求解相位响应曲线的方法,即方波扰动的直接算法,通过Hodgkin-Huxley,Fitz Hugh-Nagumo,Morris-Lecar和Hindmarsh-Rose神经元模型验证该算法可计算周期峰放电、周期簇放电的相位响应曲线.该算法克服了其他算法在运用过程中的局限性.利用该算法计算结果表明:周期峰放电的相位响应曲线类型是由其分岔类型所决定;在Morris-Lecar模型中发现一种开始于Hopf分岔终止于鞍点同宿轨道分岔的阈上周期振荡,其相位响应曲线属于第二类型.通过大量的相位响应曲线的计算发现相位响应的相对大小及正负性仅取决于扰动所施加的时间,而且周期簇放电的相位响应曲线比周期峰放电的相位响应曲线更为复杂.     

Burst synchronization of electrically and chemically coupled map-based neurons

Burst synchronization and burst dynamics of a system consisting of two map-based neurons coupled through electrical or chemical synapses are discussed. Some basic characteristic quantities are introduced to describe burst synchronization and burst dynamics of neurons. It is observed that excitatory coupling leads to in-phase burst synchronization but inhibitory coupling results in anti-phase one. By using the basic characteristics of burst dynamics, the effects of the intrinsic bursting properties and the coupling schemes on complex bursting behaviors are also presented for both inhibitory and excitatory couplings. The results are instructive to identify bursting behaviors through experimental data.     

Small-World Connections to Induce Firing Activity and Phase Synchronization in Neural Networks

We investigate how the firing activity and the subsequent phase synchronization of neural networks with smallworld topological connections depend on the probability p of adding-links. Network elements are described by two-dimensional map neurons （2DMNs） in a quiescent original state. Neurons burst for a given coupling strength when the topological randomness p increases, which is absent in a regular-lattice neural network. The bursting activity becomes frequent and synchronization of neurons emerges as topological randomness further increases. The maximal firing frequency and phase synchronization appear at a particular value of p. However, if the randomness p further increases, the firing frequency decreases and synchronization is apparently destroyed.     

State-to-State Transitions in a Hindmarsh-Rose Neuron System

We investigate the dynamical response of the neuron system to a feeble external signal by using the Hindmarsh-Rose model, when the system is tuned below the first bifurcation point, which corresponds to the period-1 bursting state, and an external signal with a fixed period of about 170s is introduced to the system. It is found that to respond to the outside signal, the system changes from the period-1 state to a period-2 one with variation of the signal amplitude, indicating the occurrence of state-to-state transition （SST）. Moreover, when a signal with different fixed periods is introduced, we can also find a similar transition between other states. Furthermore, the effect of the frequency of the signal on the transition is also discussed. These results may imply that SST plays a constructive role in information processing in neuron systems.     

兴奋性自突触引起神经簇放电频率降低或增加的非线性机制

兴奋和抑制性作用分别会增强和压制神经电活动,这是神经调控的通常观念,在神经信息处理中起重要作用.本文选取了放电簇和阈下振荡相交替、放电簇谷值小于阈下振荡谷值的Homoclinic/Homoclinic型簇放电,研究发现时滞和强度合适的兴奋性自突触电流作用在放电簇的谷值附近时,能引起簇内放电个数降低,并进而导致平均放电频率降低,这是不同于通常观念的新现象.进一步,用快慢变量分离获得的分岔和相轨迹,揭示了阈下振荡和放电簇分别对应快子系统的阈下和阈上极限环,兴奋性自突触电流引起阈上极限环向阈下极限环的转迁导致放电提前结束是频率降低原因.并与近期在Fold/Homoclinic簇放电报道的兴奋性自突触诱发的簇内放电个数降低但放电频率增加的现象和机制进行了比较.研究结果丰富了神经电活动的反常现象并揭示了背后的非线性机制,给出了调控簇放电的新手段,揭示了兴奋性自突触的潜在功能.     

Detecting phase synchronization between coupled non-phase-coherent oscillators

We compare two methods for detecting phase synchronization in coupled non-phase-coherent oscillators. One method is based on the locking of self-sustained oscillators with an irregular signal. The other uses trajectory recurrences in phase space. We identify the pros and cons of both methods and propose guidelines to detect phase synchronization in data series.     

Dynamical role of the degree of intraspecific cooperation: A simple model for prebiotic replicators and ecosystems

We present a simple mean field model to analyze the dynamics of competition between two populations of replicators in terms of the degree of intraspecific cooperation (i.e., autocatalysis) in one of these populations. The first population can only replicate with Malthusian kinetics while the second one can reproduce with Malthusian or autocatalytic replication or with a combination of both reproducing strategies. The model consists of two coupled, nonlinear, autonomous ordinary differential equations. We investigate analytically and numerically the phase plane dynamics and the bifurcation scenarios of this ecologically coupled system, focusing on the outcome of competition for several degrees of intraspecific cooperation, σ, in the second population of replicators. We demonstrate that the dynamics of both populations can not be governed by a limit cycle, and also that once cooperation is considered, the topology of phase space does not allow for coexistence. Even for low values of the degree of intraspecific cooperation, for large enough autocatalytic replication rates, the second population of replicators is able to outcompete the first one, having a wide basin of attraction in state space. We characterize the same power law dependence between the outcompetition extinction times, τ, and the degree of intraspecific cooperation for both populations, given by τ∼c_iσ⁻¹. Our results suggest that, under some kinetic conditions, the appearance of autocatalysis might be favorable in a population of replicators growing with Malthusian kinetics competing with another population also reproducing exponentially.     

Bursting neurons with coupling delays

A pair of Hindmarsh–Rose neurons with delayed coupling is studied. Bifurcations due to time-lag and coupling and stability of the stationary state that corresponds to the quiescence behavior are analyzed. Bursting is created by coupling and its properties strongly depend on the time-lag. In particular, there is a domain of values of time-lags which renders the bursting of the two neurons exactly synchronous.     

Novel routes to chaos through torus breakdown in non-invertible maps

The paper describes a number of new scenarios for the transition to chaos through the formation and destruction of multilayered tori in non-invertible maps. By means of detailed, numerically calculated phase portraits we first describe how three- and five-layered tori arise through period-doubling and/or pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. We then describe several different mechanisms for the destruction of five-layered tori in a system of two linearly coupled logistic maps. One of these scenarios involves the destruction of the two intermediate layers of the five-layered torus through the transformation of two unstable node cycles into unstable focus cycles, followed by a saddle-node bifurcation that destroys the middle layer and a pair of simultaneous homoclinic bifurcations that produce two invariant closed curves with quasiperiodic dynamics along the sides of the chaotic set. Other scenarios involve different combinations of local and global bifurcations, including bifurcations that lead to various forms of homoclinic and heteroclinic tangles. We finally demonstrate that essentially the same scenarios can be observed both for a system of nonlinearly coupled logistic maps and for a couple of two-dimensional non-invertible maps that have previously been used to study the properties of invariant sets.     

Routes to bursting in a periodically driven oscillator

The dynamical behaviors of a periodic excited oscillator with multiple time scales in the form that order gap exists between the frequency of the excitation and the natural frequency, are investigated in this Letter. By regarding the whole excitation term as a parameter, bifurcation sets are derived, which divide the generalized parameter space into several regions corresponding to different kinds of dynamics. Different types of bursting phenomena, such as fold/Hopf bursting, fold/Hopf/homoclinic bursting and Hopf/homoclinic bursting, are presented, the mechanism of which is obtained based on the bifurcations of the generalized autonomous system as well as the introduction of the so-called transformed phase portraits. Furthermore, the evolution of the bursting is discussed in details, in which one may find that when the two limit cycles caused by the Hopf bifurcations of the two related equilibrium points interact with each other, homoclinic bifurcation may occur, leading to the merge of the two cycles to form a large amplitude cycle. The homoclinic bifurcation may cause the two asymmetric bursters to merge into a symmetric enlarged burster, in which the large amplitude of the spiking state agrees well with the amplitude of the cycle caused by the homoclinic bifurcation.     

Logarithmic frequency scaling of semiconductor oscillations caused by a modified saddle-node bifurcation on a limit cycle

The oscillatory behavior of low-temperature impact ionization breakdown inp-type germanium is investigated experimentally. We explain the anomalous scaling behavior of a saddle-node bifurcation on a limit cycle in terms of a simple model approach. It represents the low-dimensional analog to a new type of intermittency proposed recently.