慢变参数激励Duffing系统中的延迟分岔现象及其诱发的簇发振荡

研究了慢变参数激励下Duffing系统的动力学行为.由于慢变参数激励可以周期性地穿越叉型分岔点,周期性的延迟分岔行为可能会发生.探讨了延迟分岔的动力学特性,尤其是基于此产生的簇发振荡.指出了延迟分岔现象所形成的稳定慢流形之间的滞后环是系统中可以观测到簇发振荡主要原因.此外,还分析了初始时间对延迟行为的延迟时间的影响.研究表明,首次延迟行为依赖于系统的初值.但是,随后发生的延迟现象与系统初值无关.     

耦合Pre-Botzinger复合体中神经元的反相簇放电模式及同步研究

本文以pre-B(o)tzinger复合体中两个耦合兴奋性神经元为研究对象,分别研究了耦合神经元反相簇放电类型及其同步转迁.当钠电导参数在一定范围内变化时,耦合神经元分别表现为“sup-Hopf/fold cycle”型和“fold/fold cycle”型反相簇放电.通过计算耦合神经元的簇放电内的峰相位差,进而判定反相簇放电中的峰同步行为,并且研究了反相放电的类型与同步的关系.     

Minimal model for intracellular calcium oscillations and electrical bursting in melanotrope cells of Xenopus laevis

A minimal model is presented to explain changes in frequency, shape, and amplitude of Ca2+ oscillations in the neuroendocrine melanotrope cell of Xenopus Laevis. It describes the cell as a plasma membrane oscillator with influx of extracellular Ca2+ via voltage-gated Ca2+ channels in the plasma membrane. The Ca2+ oscillations in the Xenopus melanotrope show specific features that cannot be explained by previous models for electrically bursting cells using one set of parameters. The model assumes a KCa-channel with slow Ca2+-dependent gating kinetics that initiates and terminates the bursts. The slow kinetics of this channel cause an activation of the Kca-channel with a phase shift relative to the intracellular Ca2+ concentration. The phase shift, together with the presence of a Na+ channel that has a lower threshold than the Ca2+ channel, generate the characteristic features of the Ca2+ oscillations in the Xenopus melanotrope cell.     

不同时间尺度的慢变量引起的簇放电研究

本文基于两个重要的慢负反馈机制给出了一个组合型的胰腺β-细胞模型.在这个模型中,不同簇放电模式对快、中、慢的振荡周期具有鲁棒性,这样可以通过快振荡周期簇放电模式的快慢动力学分析得到所有簇放电的动力学机理和拓扑类型.对于快振荡周期的簇放电,较慢的慢变量α几乎为常数,较快的慢变量Cer对快子系统没有影响,因此只要考虑慢变量...     

Spiking and bursting of a fractional order of the modified FitzHugh-Nagumo neuron model

This paper reports bursting behavior and related bifurcations in a fractional order FitzHugh-Nagumo neuron model, by adding sub fast-slow system. We classify different bursters of the system consisting fold/Hopf via a fold/fold hysteresis loop, homoclinic/homolininc cycle-cycle, fold/homoclinic, homoclinic/Hopf via homoclinic/fold hysteresis loop. We determine stability and dynamical behaviors of the equilibria of the system by numerical simulations.     

On a model of three-dimensional bursting and its parallel implementation

A mathematical model for the simulation of three-dimensional bursting phenomena and its parallel implementation are presented. The model consists of four nonlinearly coupled partial differential equations that include fast and slow variables, and exhibits bursting in the absence of diffusion. The differential equations have been discretized by means of a second-order accurate in both space and time, linearly-implicit finite difference method in equally-spaced grids. The resulting system of linear algebraic equations at each time level has been solved by means of the Preconditioned Conjugate Gradient (PCG) method. Three different parallel implementations of the proposed mathematical model have been developed; two of these implementations, i.e., the MPI and the PETSc codes, are based on a message passing paradigm, while the third one, i.e., the OpenMP code, is based on a shared space address paradigm. These three implementations are evaluated on two current high performance parallel architectures, i.e., a dual-processor cluster and a Shared Distributed Memory (SDM) system. A novel representation of the results that emphasizes the most relevant factors that affect the performance of the paralled implementations, is proposed. The comparative analysis of the computational results shows that the MPI and the OpenMP implementations are about twice more efficient than the PETSc code on the SDM system. It is also shown that, for the conditions reported here, the nonlinear dynamics of the three-dimensional bursting phenomena exhibits three stages characterized by asynchronous, synchronous and then asynchronous oscillations, before a quiescent state is reached. It is also shown that the fast system reaches steady state in much less time than the slow variables.     

耦合pre-B tzinger复合体神经元中混合簇放电的多时间尺度动力学分析

Pre-B tzinger复合体是新生哺乳动物呼吸节律起源的关键部位,是呼吸节律产生的中枢.本文以pre-B tzinger复合体中两个耦合的神经元为研究对象,并考虑钙离子动力学的耦合神经元模型.利用多时间尺度动力学、快慢尺度分解和分岔分析,研究混合簇同步放电模式及其产生机制,并研究了耦合神经元同相和反相簇放电类型及其同步转迁.结果表明钙离子的周期性波动对混合簇放电模式的产生有极大的影响,钙离子波动导致的时间尺度变化及分岔曲线相对位置的改变是混合簇放电产生的主要原因.本文的研究对认识pre-B tzinger中大规模网络的动力学有着重要的意义,为进一步探索呼吸节律的产生机制提供了一些有益的思考和见解.     

Bursting without slow kinetics: a role for a small world?

Bursting, a dynamical phenomenon whereby episodes of neural action potentials are punctuated by periodic episodes of inactivity, is ubiquitous in neural systems. Examples include components of the respiratory rhythm generating circuitry in the brain stem, spontaneous activity in the neonatal rat spinal cord, and developing neural networks in the retina of the immature ferret. Bursting can also manifest itself in single neurons. Bursting dynamics require one or more kinetic processes slower than the timescale of the action potentials. Such processes usually manifest themselves in intrinsic ion channel properties, such as slow voltage-dependent gating or calcium-dependent processes, or synaptic mechanisms, such as synaptic depression. In this note, we show rhythmic bursting in a simulated neural network where no such slow processes exist at the cellular or synaptic level. Rather, the existence of rhythmic bursting is critically dependent on the connectivity of the network and manifests itself only when connectivity is characterized as small world. The slow process underlying the timescale of bursting manifests itself as a progressive synchronization of the network within each burst.     

Modeling alternation to synchrony with inhibitory coupling: a neuromorphic VLSI approach

We developed an analog very large-scale integrated system of two mutually inhibitory silicon neurons that display several different stable oscillations. For example, oscillations can be synchronous with weak inhibitory coupling and alternating with relatively strong inhibitory coupling. All oscillations observed experimentally were predicted by bifurcation analysis of a corresponding mathematical model. The synchronous oscillations do not require special synaptic properties and are apparently robust enough to survive the variability and constraints inherent in this physical system. In biological experiments with oscillatory neuronal networks, blockade of inhibitory synaptic coupling can sometimes lead to synchronous oscillations. An example of this phenomenon is the transition from alternating to synchronous bursting in the swimming central pattern generator of lamprey when synaptic inhibition is blocked by strychnine. Our results suggest a simple explanation for the observed oscillatory transitions in the lamprey central pattern generator network: that inhibitory connectivity alone is sufficient to produce the observed transition.     

A-type K(+) current can act as a trigger for bursting in the absence of a slow variable

Models of bursting in single cells typically include two subsystems with different timescales. Variations in one or more slow variables switch the system between a silent and a spiking state. We have developed a model for bursting in the pituitary lactotroph that does not include any slow variable. The model incorporates fast, noninactivating calcium and potassium currents (the spike-generating mechanism), as well as the fast, inactivating A-type potassium current (I(A)). I(A) is active only briefly at the beginning of a burst, but this brief impulse of I(A) acts as a burst trigger, injecting the spike trajectory close to an unstable steady state. The spiraling of the trajectory away from the steady state produces a period of low-amplitude spiking typical of lactotrophs. Increasing the conductance of A-type potassium current brings the trajectory closer to the unstable steady state, increasing burst duration. However, this also increases interburst interval, and for larger conductance values, all activity stops. To our knowledge, this is the first example of a physiologically based, single-compartmental model of bursting with no slow subsystem.     

一类分段线性神经元系统的动力学行为分析

针对一类平面上三分段连续线性神经元模型,研究了边界平衡点的持续性分岔(persistence)、非光滑折分岔(non-smooth fold)的存在条件及一类跨边界周期解.最后,通过施加缓慢变化的周期外激励研究其对边界分岔和系统周期放电的影响.     

Mechanisms for Frequency Control in Neuronal Competition Models

We investigate analytically a firing rate model for a two-population network based on mutual inhibition and slow negative feedback in the form of spike frequency adaptation. Both neuronal populations receive external constant input whose strength determines the system's dynamical state-a steady state of identical activity levels or periodic oscillations or a winner-take-all state of bistability. We prove that oscillations appear in the system through supercritical Hopf bifurcations and that they are antiphase. The period of oscillations depends on the input strength in a nonmonotonic fashion, and we show that the increasing branch of the period versus input curve corresponds to a release mechanism and the decreasing branch to an escape mechanism. In the limiting case of infinitely slow feedback we characterize the conditions for release, escape, and occurrence of the winner-take-all behavior. Some extensions of the model are also discussed.     

Blue sky catastrophe in systems with nonclassical relaxation oscillations

The feasibility of the well-known blue sky bifurcation in a class of three-dimensional singularly perturbed systems of ordinary differential equations with one fast and two slow variables is studied. A characteristic property of the considered systems is that so-called nonclassical relaxation oscillations occur in them. The same name is used for oscillations with slow components, which are asymptotically close to some time-discontinuous functions and a δ-like fast component. Cases when the blue sky catastrophe results in a stable relaxation cycle or a stable two-dimensional invariant torus are analyzed. The problem of the appearance of homoclinic structures is also considered.     

周期激励下四维非线性系统的簇发共存现象

在一个周期激励的四维非自治系统中,当激励的频率远小于系统的固有频率时,系统表现出了两时间尺度的动力学行为.将激励项定义为慢变参数,激励系统可以转化为广义自治系统.分析了广义自治系统平衡点的稳定性及其分岔条件.应用快慢分析法和转换相图,探讨了系统对应于不同初始条件的簇发现象及其产生机制,并对其中多种簇发共存的形成机理进行了讨论.同时,由于慢过效应的存在,簇发振荡的激发态和沉寂态的连接点和理论分析中的分岔点相比存在一定的滞后现象.     

Simulating Ramsey-Type Fringes in a Pulsed Microwave-Driven Classical Josephson Junction

We present evidence for a close analogy between the nonlinear behaviour of a pulsed microwave-driven Josephson junction at low temperature and the experimentally observed behaviour of Josephson systems operated below the quantum transition temperature under similar conditions. We specifically address observations of Ramsey-type fringe oscillations, which can be understood in classical nonlinear dynamics as results of slow transient oscillations in a pulsed microwave environment. Simulations are conducted to mimic experimental measurements by recording the statistics of microwave-induced escape events from the anharmonic potential well of a zero-voltage state. Observations consistent with experimentally obtained Ramsey-type oscillations are found in the classical model. An erratum to this article can be found at     

不同耦合类型的三维Hybrid神经元同步动力学

神经元网络的信息传递与多个神经元间的耦合同步密切相关.大量研究表明,神经元耦合系统的同步化问题是研究大脑处理信息的关键.本文基于三维混合神经元模型研究了神经元的复杂放电,该模型由Wilson模型的快子系统和H-R模型的慢子系统组成.该模型能够重现大脑皮层神经元一系列包括规则峰放电、快速峰放电和簇放电等神经动力学行为.本文基于三维混合神经元模型,探讨了三维混合神经元在电突触耦合、化学突触耦合和磁通耦合条件下的同步放电行为,模拟膜电位序列图、相位差图等探讨恒同及非恒同下耦合强度对神经元同步放电的影响.本研究将为人们进一步了解神经系统疾病的发病机制提供指导和帮助,并为神经科学领域提供可能的研究思路.     

Cross-coupling errors of micromachined gyroscopes

A model is developed for the response of a micromachined, rotary gyroscope subject to a general input motion. The governing equations are formulated for weakly nonlinear oscillations of the rotor, suspended above the moving substrate via elastic beams. The method of multiple scales is used to separate the slow and fast responses. This approach allows quick computation of the long-term behavior of the rotor without the need to integrate over fast oscillations. The power of the model to evince cross-coupling errors is demonstrated through examples.     

Phase precession through synaptic facilitation

Phase precession is a relational code that is thought to be important for episodic-like memory, for instance, the learning of a sequence of places. In the hippocampus, places are encoded through bursting activity of so-called place cells. The spikes in such a burst exhibit a precession of their firing phases relative to field potential theta oscillations (4-12 Hz); the theta phase of action potentials in successive theta cycles progressively decreases toward earlier phases. The mechanisms underlying the generation of phase precession are, however, unknown. In this letter, we show through mathematical analysis and numerical simulations that synaptic facilitation in combination with membrane potential oscillations of a neuron gives rise to phase precession. This biologically plausible model reproduces experimentally observed features of phase precession, such as (1) the progressive decrease of spike phases, (2) the nonlinear and often also bimodal relation between spike phases and the animal's place, (3) the range of phase precession being smaller than one theta cycle, and (4) the dependence of phase jitter on the animal's location within the place field. The model suggests that the peculiar features of the hippocampal mossy fiber synapse, such as its large efficacy, long-lasting and strong facilitation, and its phase-locked activation, are essential for phase precession in the CA3 region of the hippocampus.     

分段线性脉冲神经元模型的簇放电分析

应用一种二维分段线性脉冲神经元模型的简单计算特性,对其簇放电特性进行了详细研究。发现该模型表现出强迫和内禀两类簇放电模式,并分析了内禀簇放电的产生机理及分岔现象。在实验中,应用该模型对一类具有簇放电特性的皮层神经元进行了模拟。