神经起步点自发放电节律及节律转化的分岔规律

在神经起步点的实验中观察到了复杂多样的神经放电([Ca^2 ]o)节律模式,如周期簇放电、周期峰放电、混沌簇放电、混沌峰放电以及随机放电节律等。随着细胞外钙离子浓度的降低,神经放电节律从周期l簇放电,经过复杂的分岔过程(包括经倍周期分岔到混沌簇放电、混沌簇放电经激变到混沌峰放电、以及混沌峰放电经逆倍周期分岔到周期峰放电)转化为周期l峰放电。在神经放电理论模型——Chay模型中,调节与实验相关的参数(Ca^2 平衡电位),可以获得与实验相似的神经放电节律和节律转换规律。这表明复杂的神经放电节律之间存在着一定的分岔规律,它们是理解神经元信息编码的基础。     

神经放电加周期分岔中由随机自共振引起一类新节律

当改变实验性神经起步点细胞外[Ca^2 ]时,放电节律表现出从周期1节律转换为周期4节律的加周期分岔序列。其中,周期n节律转换为周期n 1节律的过程中(n=1,2,3)存在一种新的具有交替特征的节律,该新节律为周期n簇与周期n 1簇放电的交替,并且周期n 1簇的时间间隔序列呈现出整数倍特征。确定性神经放电理论模型(chay模型)只能模拟周期n节律直接到周期n 1节律的加周期分岔序列;而随机chay模型可以模拟实验中的加周期分岔过程和新节律。进一步,新节律被确认是经随机自共振机制产生的。这不仅解释了实验现象,也将随机自共振的产生区间从以前认识到的Hopf分岔点附近扩大到加周期分岔点附近,同时扩大了噪声在神经放电和神经编码中起重要作用的参数区间。     

由三态跃迁机制产生的两种神经放电节律

在实验性神经起步点自发放电中,发现了两种三态跃迁节律,其特征为静息、周期n及周期n+1(n=1,2)簇放电随机交替出现。应用随机Chay模型数值仿真,分别得到了与实验模型中相似的两种三态跃迁节律,这两种节律都是在两个紧邻的分岔点附近,由噪声驱动而产生的。理论分析提示,当神经元系统接近从静息经分岔到放电的临界状态,且从静息到周期n的分岔点,与从周期n到周期n+1的分岔点非常接近时,在噪声的作用下,系统运动会在静息、周期n和周期n+1三种状态之间随机跃迁,从而形成了这种三态跃迁节律。基于这种三态跃迁放电的随机共振,还有待进一步深入研究。     

交流外电场下映射神经元放电节律的分析

神经元不同的放电节律承载着不同的刺激信息。文章基于神经元映射模型,研究低频交流电场对神经元放电节律的影响。在外部刺激下映射模型表现出丰富的放电模式,包括周期簇放电、周期峰放电、交替放电和混沌放电。神经元对刺激频率和振幅的变化极为敏感,随着频率的增大,放电节律表现出从簇放电到峰放电和混沌放电的反向加周期分岔序列;在周期节律转迁过程中存在一种新的交替节律,其放电序列为两种周期放电模式的交替,峰峰间期序列具有整数倍特征。外电场的频率影响细胞内、外离子振荡周期,导致神经元放电与刺激信号同步,对放电节律的影响更为明显。研究结果揭示了交流外电场对神经元放电节律的作用规律,有助于探寻外电场对生物神经系统兴奋性的影响和神经系统疾病的致病机理。     

实验性神经起步点产生的整数倍簇放电节律

随机Hindmarsh-Rose模型中产生簇（bursting)放电节律是神经放电中存在随机自共振的一个重要理论证据,但是,该簇放电节律在实验中一直没有被发现。在实验性神经起步点细胞外[Ca^2 ]([Ca^2 ]o)低于周期1节律的[Ca^2 ]o时,发现了一种簇放电节律。其簇簇间期（inter-burst intervals,IBIs)呈现出与随机自共振引起的整数倍峰放电（interger multiple spiking)节律的峰峰间期类似的整数倍特征。随机Hindmarsh-Rose模型中产生的簇（bursting)放电节律也表现出类似的特征。结果验证了随机自共振簇放电的存在性,揭示该簇放电节律的统计特征。此外,该簇放电节律的参数区间以及其与整数倍峰放电节律的区别被揭示,簇放电节律的[Ca^2 ]o低于峰放电律的[Ca^2 ]o。     

4-氨基吡啶敏感钾通道对神经自发放电节律转迁历程的调节作用

为进一步研究损伤神经放电节律的分岔转迁规律,以实验性神经起步点模型为研究对象,在联合改变胞外的钙离子和钾离子浓度的条件下,记录神经单纤维的放电节律转迁方式。选取4-氨基吡啶(4-aminopyridine,4-AP)作为条件参数,Ca2+浓度作为分岔参数,观察了实验性神经起步点自发放电节律的分岔规律。28例实验结果中,有21例神经对本文所取的条件参数变化不敏感,7例实验性神经起步点的自发放电节律会在不同的条件参数下出现不同类型的分岔序列结构。在不同的4-AP浓度下,随着Ca2+浓度的降低,同一实验性神经起步点会表现出不同的放电节律模式的分岔序列,不同实验性神经起步点,双参数分岔序列是不同的。以上结果说明,不同参数配置下的神经放电节律的变化规律是不同的,而且分岔序列结构是认识放电节律转迁规律的基础。     

神经放电节律转化的分岔序列模式

神经元接受到的外界信号是动态变化的,神经放电节律模式则会依据一定的规律动态转化来反映这种变化,以往确定性理论模型(如Chay模型和Rose-Hindmarsh模型)模拟出了部分神经放电模式转化的整体分岔规律。利用Chay模型仿真,通过调节具有生理学意义的参数,模拟出了神经元放电的一系列分岔序列,同时在神经起步点的实验中,应用与模型对应的参数进行调节,观察到了与仿真结果整体上一致的分岔序列,印证了数值模拟的结果,展现了真实的神经元放电整体分岔结构的基本规律,为理解具体的生理调节活动中神经放电节律的转化提供了理论基础。     

心肌细胞团搏动整数倍节律的非线性动力学机制

利用心肌细胞耦合模型研究心肌整数倍节律的动力学机理。确定性模型仿真揭示了心肌细胞团同步搏动加周期分岔的节律变化规律;随机模型仿真发现在加周期分岔序列中分岔点附近会出现整数倍节律,其中,0-1整数倍节律产生于从静息到周期1的Hopf分岔点附近,1-2整数倍节律产生于周期1和周期2极限环间的加周期分岔点附近;对系统相空间轨道的分析进一步揭示出整数倍节律是由系统运动在相邻的两个轨道之间随机跃迁形成的。上述分析结果不仅阐明了心肌整数倍节律的机理,并且揭示了各种整数倍节律与加周期分岔序列中相邻节律的内在联系,为重新认识心律变化的规律开辟了新的途径。     

混沌放电的可兴奋性细胞对外界刺激反应敏感的动力学机制

在大鼠损伤背根节神经元受到去甲肾上腺(NE)、四乙基胺(TEA)和高浓度钙等剌激的实验中,观察到非周期放电的神经元明显地比周期放电的神经元对外界刺激的反应敏感程度高。现有的结果表明许多非周期放电的神经元实际上表现为确定性的混沌运动,比如混沌尖峰放电、混沌簇放电以及整数倍放电等。以修正的胰腺B细胞Chay模型为例,通过对其分岔结构的分析和对构成混沌吸引子的基本骨架的不稳定周期轨道的计算,揭示了分岔、激变和混沌运动对参数敏感依赖性是该现象产生的动力学机制。同时指出以往使用平均发放率来刻划可兴奋性细胞放电活动存在的缺陷,提出了一种新的利用周期轨道信息的刻划方法。     

神经起步点产生的一种新型簇放电节律———阵发周期1节律

实验中发现了神经起步点产生的一种新型的簇放电节律－－阵发周期1节律。其特征如下：连续周期1放电与休止期（quiescence）轮流出现;非周期性,连续放电持续期、连续放电次数以及休止期有较大变异性;位于周期1节律和静息状态之间。具有较长周期的伪单色噪声激励的FHN（FizHugh-Nagumo）模型可以产生类似的阵发周期1节律。模型和实验中的阵发周期1节律的统计特征、变化规律和所处的参数区间相类似。这表明：阵发周期1节律是由与伪单色噪声类似的长时程振荡激励引起的。     

A basic bifurcation structure from bursting to spiking of injured nerve fibers in a two-dimensional parameter space

Two different bifurcation scenarios of firing patterns with decreasing extracellular calcium concentrations were observed in identical sciatic nerve fibers of a chronic constriction injury (CCI) model when the extracellular 4-aminopyridine concentrations were fixed at two different levels. Both processes proceeded from period-1 bursting to period-1 spiking via complex or simple processes. Multiple typical experimental examples manifested dynamics closely matching those simulated in a recently proposed 4-dimensional model to describe the nonlinear dynamics of the CCI model, which included most cases of the bifurcation scenarios. As the extracellular 4-aminopyridine concentrations is increased, the structure of the bifurcation scenario becomes more complex. The results provide a basic framework for identifying the relationships between different neural firing patterns and different bifurcation scenarios and for revealing the complex nonlinear dynamics of neural firing patterns. The potential roles of the basic bifurcation structures in identifying the information process mechanism are discussed.     

State-Dependent Effects of Na Channel Noise on Neuronal Burst Generation

We explore the effects of stochastic sodium (Na) channel activation on the variability and dynamics of spiking and bursting in a model neuron. The complete model segregates Hodgin-Huxley-type currents into two compartments, and undergoes applied current-dependent bifurcations between regimes of periodic bursting, chaotic bursting, and tonic spiking. Noise is added to simulate variable, finite sizes of the population of Na channels in the fast spiking compartment.During tonic firing, Na channel noise causes variability in interspike intervals (ISIs). The variance, as well as the sensitivity to noise, depend on the model's biophysical complexity. They are smallest in an isolated spiking compartment; increase significantly upon coupling to a passive compartment; and increase again when the second compartment also includes slow-acting currents. In this full model, sufficient noise can convert tonic firing into bursting.During bursting, the actions of Na channel noise are state-dependent. The higher the noise level, the greater the jitter in spike timing within bursts. The noise makes the burst durations of periodic regimes variable, while decreasing burst length duration and variance in a chaotic regime. Na channel noise blurs the sharp transitions of spike time and burst length seen at the bifurcations of the noise-free model. Close to such a bifurcation, the burst behaviors of previously periodic and chaotic regimes become essentially indistinguishable.We discuss biophysical mechanisms, dynamical interpretations and physiological implications. We suggest that noise associated with finite populations of Na channels could evoke very different effects on the intrinsic variability of spiking and bursting discharges, depending on a biological neuron's complexity and applied current-dependent state. We find that simulated channel noise in the model neuron qualitatively replicates the observed variability in burst length and interburst interval in an isolated biological bursting neuron.     

Biological Experimental Observations of an Unnoticed Chaos as Simulated by the Hindmarsh-Rose Model

An unnoticed chaotic firing pattern, lying between period-1 and period-2 firing patterns, has received little attention over the past 20 years since it was first simulated in the Hindmarsh-Rose (HR) model. In the present study, the rat sciatic nerve model of chronic constriction injury (CCI) was used as an experimental neural pacemaker to investigate the transition regularities of spontaneous firing patterns. Chaotic firing lying between period-1 and period-2 firings was observed located in four bifurcation scenarios in different, isolated neural pacemakers. These bifurcation scenarios were induced by decreasing extracellular calcium concentrations. The behaviors after period-2 firing pattern in the four scenarios were period-doubling bifurcation not to chaos, period-doubling bifurcation to chaos, period-adding sequences with chaotic firings, and period-adding sequences with stochastic firings. The deterministic structure of the chaotic firing pattern was identified by the first return map of interspike intervals and a short-term prediction using nonlinear prediction. The experimental observations closely match those simulated in a two-dimensional parameter space using the HR model, providing strong evidences of the existence of chaotic firing lying between period-1 and period-2 firing patterns in the actual nervous system. The results also present relationships in the parameter space between this chaotic firing and other firing patterns, such as the chaotic firings that appear after period-2 firing pattern located within the well-known comb-shaped region, periodic firing patterns and stochastic firing patterns, as predicted by the HR model. We hope that this study can focus attention on and help to further the understanding of the unnoticed chaotic neural firing pattern.     

Dynamics of On-Off Neural Firing Patterns and Stochastic Effects near a Sub-Critical Hopf Bifurcation

On-off firing patterns, in which repetition of clusters of spikes are interspersed with epochs of subthreshold oscillations or quiescent states, have been observed in various nervous systems, but the dynamics of this event remain unclear. Here, we report that on-off firing patterns observed in three experimental models (rat sciatic nerve subject to chronic constrictive injury, rat CA1 pyramidal neuron, and rabbit blood pressure baroreceptor) appeared as an alternation between quiescent state and burst containing multiple period-1 spikes over time. Burst and quiescent state had various durations. The interspike interval (ISI) series of on-off firing pattern was suggested as stochastic using nonlinear prediction and autocorrelation function. The resting state was changed to a period-1 firing pattern via on-off firing pattern as the potassium concentration, static pressure, or depolarization current was changed. During the changing process, the burst duration of on-off firing pattern increased and the duration of the quiescent state decreased. Bistability of a limit cycle corresponding to period-1 firing and a focus corresponding to resting state was simulated near a sub-critical Hopf bifurcation point in the deterministic Morris—Lecar (ML) model. In the stochastic ML model, noise-induced transitions between the coexisting regimes formed an on-off firing pattern, which closely matched that observed in the experiment. In addition, noise-induced exponential change in the escape rate from the focus, and noise-induced coherence resonance were identified. The distinctions between the on-off firing pattern and stochastic firing patterns generated near three other types of bifurcations of equilibrium points, as well as other viewpoints on the dynamics of on-off firing pattern, are discussed. The results not only identify the on-off firing pattern as noise-induced stochastic firing pattern near a sub-critical Hopf bifurcation point, but also offer practical indicators to discriminate bifurcation types and neural excitability types.     

Bursting synchronization dynamics of pancreatic β-cells with electrical and chemical coupling

Based on bifurcation analysis, the synchronization behaviors of two identical pancreatic β-cells connected by electrical and chemical coupling are investigated, respectively. Various firing patterns are produced in coupled cells when a single cell exhibits tonic spiking or square-wave bursting individually, irrespectively of what the cells are connected by electrical or chemical coupling. On the one hand, cells can burst synchronously for both weak electrical and chemical coupling when an isolated cell exhibits tonic spiking itself. In particular, for electrically coupled cells, under the variation of the coupling strength there exist complex transition processes of synchronous firing patterns such as “fold/limit cycle” type of bursting, then anti-phase continuous spiking, followed by the “fold/torus” type of bursting, and finally in-phase tonic spiking. On the other hand, it is shown that when the individual cell exhibits square-wave bursting, suitable coupling strength can make the electrically coupled system generate “fold/Hopf” bursting via “fold/fold” hysteresis loop; whereas, the chemically coupled cells generate “fold/subHopf” bursting. Especially, chemically coupled bursters can exhibit inverse period-adding bursting sequence. Fast–slow dynamics analysis is applied to explore the generation mechanism of these bursting oscillations. The above analysis of bursting types and the transition may provide us with better insight into understanding the role of coupling in the dynamic behaviors of pancreatic β-cells.