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。     

引起神经元“非周期敏感现象”的分岔机制

实验发现神经元平均发放率变化率在放电脉冲序列串(ISI序列)呈非周期节律时普遍大于ISI呈周期节律时的平均发放率变化率,称为“非周期敏感现象”。以HR神经元模型和胰腺β-细胞模型为例,在合适的参数改变量作用下观察到了“非周期敏感现象”,并进一步讨论了平均发放率变化率与ISI序列动力学性质的关系。发现当ISI序列经历混沌-周期分岔时“非周期敏感现象”表现明显,尤其在ISI序列经历从混沌到嵌入在混沌中的周期窗口的分岔时表现最为显著。进一步的分析表明周期窗口在整个混沌带中所占测度较大,故混沌．周期分岔及从混沌到嵌入在混沌中的周期窗口分岔是引起神经元“非周期敏感现象”的一种重要动力学机制。实验结果支持上述结论。     

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。     

神经放电节律加周期分岔序列中的簇放电到峰放电的转迁

含快慢子系统的神经元数学模型仿真预期,神经放电节律经历加周期分岔序列,可以进一步表现激变,并通过逆倍周期分岔级联进入周期1峰放电。实验调节胞外钙离子浓度,观察到从周期1簇放电开始的带有随机节律的加周期分岔到簇内有多个峰的簇放电,再经激变转迁到峰放电节律的分岔序列,提供了这种分岔序列模式实验证据。实验所见之激变表现为簇放电节律的休止期消失,放电节律变为混沌峰放电和周期峰放电。作者利用随机Chay模型更加逼真地仿真再现了实验所见的分岔序列。该实验结果验证了以前的确定性数学模型的理论预期,并利用随机理论模型仿真了其在现实神经系统的表现;揭示了一类完整的神经放电节律的转换规律。     

一类离散捕食-食饵模型的混沌跟踪控制

针对一类离散捕食食饵系统,采用Lyapunov指数方法、绘制系统分岔图和混沌吸引子等方式验证了混沌现象的存在.为了消除系统的混沌现象,根据控制理论的轨迹跟踪控制原理,设计混沌跟踪控制器将系统控制到任意给定的期望周期轨道上,达到了稳定种群的目的.仿真分析表明了所设计控制器的有效性.     

人心脏低维动力学模型的计算机模拟

心脏窦房结与异位心室搏动非线性相互作用,导致调制并行收缩反常节律。本文从新的低维动力学心脏模型（双参量圆周映射）出发,研究由周期性到不规则、混沌动力学过渡的性质。模型的数值实验揭示出在双参量空间不同区域中,某些特征性步入混沌的道路和总体标度关系。一个明显特征是,在双参量空间宽广范围内观察到无穷层次的嵌套结构,即嵌在混沌带中的周期递加序列。而且,出现周期性的位置或者周期窗口的宽度,均存在普适的标度关系。另一方面,在参量空间一部分,包含相互交叠的不同周期性稳定轨道,有极复杂图象。本文结果显示,运用数值方法计算李雅普诺夫指数谱,可以基本上确定人心脏圆周映射的二维分岔结构和标度性质。     

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。     

一维映射与Monte Carlo方法在生态系统中的应用

自从Feigenbaum的理论经实验验证后,混沌理论的重要性逐渐为人所知。它揭示了一大类非线性系统在外参量逐步变化过程中,会出现从稳态经由倍周期分岔过渡到混沌现象的特征,它也揭示了一个由确定论方程决定的系统也可以出现“内在随机性”的结果。这一理论现已广泛地应用于力、热、光、声、电学乃至医学、生物学等各个领域中。在生态学     

具性别结构时滞捕食系统的Hopf分支,混沌与脉冲控制

建立了具性别结构的时滞捕食系统,研究了平衡点的存在性及局部稳定性,给出了系统发生局部Hopf分支的充分条件,并应用中心流形定理研究了Hopf分支周期解的性质(分支类型,方向及稳定性).数值例子佐证了理论结果,并揭示了系统诸如高倍周期及拟周期振荡,混沌振荡,倍周期分岔等复杂的动力学行为;脉冲控制可以有效的改善系统的稳定性.     

非线性动力学在脑电分析中的应用

非线性科学于20世纪60年代发展起来,被誉为20世纪自然科学的“第三次革命”,己广泛应用于生物、物理、经济、通讯及天文学等领域。脑电图(EEG)反映了作为非线性系统的大脑的电活动,体现出混沌行为。在癫痫病症的EEG研究中,混沌特性得到了很好的证明。在精神分裂症和老年痴呆等病症的EEG研究中,混沌的作用也体现得越来越明显。本文综述了近年来非线性动力学在脑电信号分析中应用的进展,以期获得在健康和疾病状态下对大脑神经动力系统的更好理解。     

Non-linear analysis of the electroencephalogram in Creutzfeldt-Jakob disease

Creutzfeldt-Jakob disease is a rare, neurological, dementing disorder characterised by periodic sharp waves in the electroencephalogram (EEG). Non-linear analysis of these EEG changes may provide insight into the abnormal dynamics of cortical neural networks in this disorder. Babloyantz et al. have suggested that the periodic sharp waves reflect low-dimensional chaotic dynamics in the brain. In the present study this hypothesis was re-examined using newly developed techniques for non-linear time series analysis. We analysed the EEG of a patient with autopsy-proven Creutzfeldt-Jakob disease using the method of non-linear forecasting as introduced by Sugihara and May, and we tested for non-linearity with amplitude-adjusted, phase-randomised surrogate data. Two epochs with generalised periodic sharp waves showed clear evidence for non-linearity. These epochs could be predicted better and further ahead in time than most of the irregular background activity. Testing against cycle-randomised surrogate data and close inspection of the periodograms showed that the non-linearity of the periodic sharp waves may be better explained by quasi-periodicity than by low-dimensional chaos. The EEG further displayed at least one example of a sudden, large qualitative change in the dynamics, highly suggestive of a bifurcation. The presence of quasi-periodicity and bifurcations strongly argues for the use of a non-linear model to describe the EEG in Creutzfeldt-Jakob disease. Received: 28 October 1996 / Accepted in revised form: 8 July 1997     

The mechanism of verbal memory and scenario of the development of speech V-rhythms

It was shown that the Feigenbaum scenario for the transition of bifurcation of period doubling to chaos, which explains the singularities of V-rhythm disorders in the neighborhood of the critical point, is a good model of the development of phonetics in children speech. It was also shown that the singularities of the dynamics of V-rhythms in the bifurcation lacuna in the zone of chaos intrinsic to the Pomeau-Manneville scenario for route of bifurcations of period 3 to chaos are capable, basically, to describe some features of both remembrance at the extraction from memory during speech and remembering in memory during perception of speech.     

Dynamics of period-doubling bifurcation to chaos in the spontaneous neural firing patterns

Period-doubling bifurcation to chaos were discovered in spontaneous firings of Onchidium pacemaker neurons. In this paper, we provide three cases of bifurcation processes related to period-doubling bifurcation cascades to chaos observed in the spontaneous firing patterns recorded from an injured site of rat sciatic nerve as a pacemaker. Period-doubling bifurcation cascades to period-4 (π(2,2)) firstly, and then to chaos, at last to a periodicity, which can be period-5, period-4 (π(4)) and period-3, respectively, in different pacemakers. The three bifurcation processes are labeled as case I, II and III, respectively, manifesting procedures different to those of period-adding bifurcation. Higher-dimensional unstable periodic orbits (UPOs) can be detected in the chaos, built close relationships to the periodic firing patterns. Case III bifurcation process is similar to that discovered in the Onchidium pacemaker neurons and simulated in theoretical model-Chay model. The extra-large Feigenbaum constant manifesting in the period-doubling bifurcation process, induced by quasi-discontinuous characteristics exhibited in the first return maps of both ISI series and slow variable of Chay model, shows that higher-dimensional periodic behaviors appeared difficult within the period-doubling bifurcation cascades. The results not only provide examples of period-doubling bifurcation to chaos and chaos with higher-dimensional UPOs, but also reveal the dynamical features of the period-doubling bifurcation cascades to chaos.     

Chaos in three species food chains

We study the dynamics of a three species food chain using bifurcation theory to demonstrate the existence of chaotic dynamics in the neighborhood of the equilibrium where the top species in the food chain is absent. The goal of our study is to demonstrate the presence of chaos in a class of ecological models, rather than just in a specific model. This work extends earlier numerical studies of a particular system by Hastings and Powell (1991) by showing that chaos occurs in a class of ecological models. The mathematical techniques we use are based on work by Guckenheimer and Holmes (1983) on co-dimension two bifurcations. However, restrictions on the equations we study imposed by ecological assumptions require a new and somewhat different analysis.     

Nonlinearity in normal human EEG: cycles,temporal asymmetry,nonstationarity and randomness,not chaos

Two-hour vigilance and sleep electroencephalogram (EEG) recordings from five healthy volunteers were analyzed using a method for identifying nonlinearity and chaos which combines the redundancy–linear redundancy approach with the surrogate data technique. A nonlinear component in the EEG was detected, however, inconsistent with the hypothesis of low-dimensional chaos. A possibility that a temporally asymmetric process may underlie or influence the EEG dynamics was indicated. A process that merges nonstationary nonlinear deterministic oscillations with randomness is proposed for an explanation of observed properties of the analyzed EEG signals. Taking these results into consideration, the use of dimensional and related chaos-based algorithms in quantitative EEG analysis is critically discussed. Received: 25 September 1994 / Accepted in revised form: 10 July 1996     

Time-dependent regimes of a tourism-based social–ecological system: Period-doubling route to chaos

The period-doubling route to chaos has occupied a prominent position and it is still object of great interest among the different complex phenomena observed in nonlinear dynamical systems. The reason of such interest is that such route to chaos has been observed in many physical, chemical and ecological models when they change over from simple periodic to complex aperiodic motion. In interlinked social–ecological systems (SESs) there might be an apparent great ability to cope with change and adapt if analysed only in their social dimension. However, such an adaptation may be at the expense of changes in the capacity of ecosystems to sustain the adaptation and it could affect the quality of ecosystem goods and services since it could degrade natural renewable and non-renewable resources and generate traps and breakpoints in the whole SES eventually leading to chaotic behaviour. This paper is rooted in previous results on modelling tourism-based SESs, only recently object of theoretical investigations, focusing on the dynamics of the coexistence between mass-tourists and eco-tourists. Here we describe a finer scale analysis of time-dependent regimes in the ranges of the degradation coefficient (bifurcation parameter), for which the system can exhibit coexistence. This bifurcation parameter is determined by objective changes in the real world in the quality of ecosystem goods and services together with whether and how such changes are perceived by different tourist typologies. Varying the bifurcation parameter, the dynamical system may in fact evolve toward an aperiodical dynamical state in many ways, showing that there could be different scenarios for the transition to chaos. This paper provides a further evidence for the period-doubling route to chaos with reference to tourism-based socio-ecological models, and for a period locking behaviour, where a small variation in the bifurcation parameter can lead to alternating regular and chaotic dynamics. Moreover, for many models undergoing chaos via period-doubling, it has been showed that structural perturbations with real ecological justification, may break and reverse the expected period-doublings, hence inhibiting chaos. This feature may be of a certain relevance also in the context of adaptive management of tourism-based SESs: these period-doubling reversals might in fact be used to control chaos, since they potentially act in way to suppress possibly dangerous fluctuations.     

Observance of period-doubling bifurcation and chaos in an autonomous ODE model for malaria with vector demography

We illustrate that an autonomous ordinary differential equation model for malaria transmission can exhibit period-doubling bifurcations leading to chaos when ecological aspects of malaria transmission are incorporated into the model. In particular, when demography, feeding, and reproductive patterns of the mosquitoes that transmit the malaria-causing parasite are explicitly accounted for, the resulting model exhibits subcritical bifurcations, period-doubling bifurcations, and chaos. Vectorial and disease reproduction numbers that regulate the size of the vector population at equilibrium and the endemicity of the malaria disease, respectively, are identified and used to simulate the model to show the different bifurcations and chaotic dynamics. A subcritical bifurcation is observed when the disease reproduction number is less than unity. This highlights the fact that malaria control efforts need to be long lasting and sustained to drive the infectious populations to levels below the associated saddle-node bifurcation point at which control is feasible. As the disease reproduction number increases beyond unity, period-doubling cascades that develop into chaos closely followed by period-halving sequences are observed. The appearance of chaos suggests that characterization of the physiological status of disease vectors can provide a pathway toward understanding the complex phenomena that are known to characterize the dynamics of malaria and other indirectly transmitted infections of humans. To the best of our knowledge, there is no known unforced continuous time deterministic host-vector transmission malaria model that has been shown to exhibit chaotic dynamics. Our results suggest that malaria data may need to be critically examined for complex dynamics.     

Hysteresis Dynamics,Bursting Oscillations and Evolution to Chaotic Regimes

This article describes new aspects of hysteresis dynamics which have been uncovered through computer experiments. There are several motivations to be interested in fast-slow dynamics. For instance, many physiological or biological systems display different time scales. The bursting oscillations which can be observed in neurons, beta-cells of the pancreas and population dynamics are essentially studied via bifurcation theory and analysis of fast-slow systems (Keener and Sneyd, 1998; Rinzel, 1987). Hysteresis is a possible mechanism to generate bursting oscillations. A first part of this article presents the computer techniques (the dotted-phase portrait, the bifurcation of the fast dynamics and the wave form) we have used to represent several patterns specific to hysteresis dynamics. This framework yields a natural generalization to the notion of bursting oscillations where, for instance, the active phase is chaotic and alternates with a quiescent phase. In a second part of the article, we emphasize the evolution to chaos which is often associated with bursting oscillations on the specific example of the Hindmarsh-Rose system. This evolution to chaos has already been studied with classical tools of dynamical systems but we give here numerical evidence on hysteresis dynamics and on some aspects of the wave form. The analytical proofs will be given elsewhere.     

Explosive route to chaos through a fractal torus in a generalized Lotka-Volterra model

The behavior of a model that generalizes the Lotka-Volterra problem into three dimensions is presented. The results show the analytic derivation of stability diagrams that describe the system's qualitative features. In particular, we show that for a certain value of the bifurcation parameter the system instantly jumps out of a steady state solution into a chaotic solution that portrays a fractal torus in the three-dimensional phase space. This scenario, is referred to as the explosive route to chaos and is attributed to the non-transversal saddle connection type bifurcation. The stability diagrams also present a region in which the Hopf type bifurcation leads to periodic and chaotic solutions. In addition, the bifurcation diagrams reveal a qualitative similarity to the data obtained in the Texas and Bordeaux experiments on the Belousov-Zhabotinskii chemical reaction. The paper is concluded by showing that the model can be useful for representing dynamics associated with biological and chemical phenomena.     

Multiple attractors,catastrophes and chaos in seasonally perturbed predator-prey communities

A classical predator-prey model is considered in this paper with reference to the case of periodically varying parameters. Six elementary seasonality mechanisms are identified and analysed in detail by means of a continuation technique producing complete bifurcation diagrams. The results show that each elementary mechanism can give rise to multiple attractors and that catastrophic transitions can occur when suitable parameters are slightly changed. Moreover, the two classical routes to chaos, namely, torus destruction and cascade of period doublings, are numerically detected. Since in the case of constant parameters the model cannot have multiple attractors, catastrophes and chaos, the results support the conjecture that seasons can very easily give rise to complex populations dynamics.