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。     

Bifurcation of orbits and synchrony in inferior olive neurons

Inferior olive neurons (IONs) have rich dynamics and can exhibit stable, unstable, periodic, and even chaotic trajectories. This paper presents an analysis of bifurcation of periodic orbits of an ION when its two key parameters (a, μ) are varied in a two-dimensional plane. The parameter a describes the shape of the parabolic nonlinearity in the model and μ is the extracellular stimulus. The four-dimensional ION model considered here is a cascade connection of two subsystems (S(a) and S(b)). The parameter plane (a - μ) is delineated into several subregions. The ION has distinct orbit structure and stability property in each subregion. It is shown that the subsystem S(a) or S(b) undergoes supercritical Poincare-Andronov-Hopf (PAH) bifurcation at a critical value μ(c)(a) of the extracellular stimulus and periodic orbits of the neuron are born. Based on the center manifold theory, the existence of periodic orbits in the asymptotically stable S(a), when the subsystem S(b) undergoes PAH bifurcation, is established. In such a case, both subsystems exhibit periodic orbits. Interestingly when S(b) is under PAH bifurcation and S(a) is unstable, the trajectory of S(a) exhibits periodic bursting, interrupted by periods of quiescence. The bifurcation analysis is followed by the design of (i) a linear first-order filter and (ii) a nonlinear control system for the synchronization of IONs. The first controller uses a single output of each ION, but the nonlinear control system uses two state variables for feedback. The open-loop and closed-loop responses are presented which show bifurcation of orbits and synchronization of oscillating neurons.     

Autonomous Navigation for Unmanned Aerial Vehicles Based on Chaotic Bionics Theory

In this paper a new reactive mechanism based on perception-action bionics for multi-sensory integration applied to Un-manned Aerial Vehicles (UAVs) navigation is proposed.The strategy is inspired by the olfactory bulb neural activity observed inrabbits subject to external stimuli.The new UAV navigation technique exploits the use of a multiscroll chaotic system which isable to be controlled in real-time towards less complex orbits,like periodic orbits or equilibrium points,considered as perceptiveorbits.These are subject to real-time modifications on the basis of environment changes acquired through a Synthetic ApertureRadar (SAR) sensory system.The mathematical details of the approach are given including simulation results in a virtual en-vironment.The results demonstrate the capability of autonomous navigation for UAV based on chaotic bionics theory in com-plex spatial environments.     

Hindmarsh-Rose神经模型的混沌控制

应用稳定性准则的混沌控制方法控制单个Hindmarsh-Rose神经元模型的混沌发放峰序列和混沌爆发运动。通过对膜电压的非线性连续-时间反馈干扰的输入,将混沌运动控制到5峰／爆发（5spikes/burst）轨道上,该轨道嵌入在混沌吸引子内。数值模拟结果显示该方法在控制HR神经元模型方面是有效的。     

Low-Dimensional Dynamics in Sensory Biology 1: Thermally Sensitive Electroreceptors of the Catfish

We report the results of a search for evidence of periodic unstableorbits in the electroreceptors of the catfish. The function of thesereceptor organs is to sense weak external electric fields. Inaddition, they respond to the ambient temperature and to the ioniccomposition of the water. These quantities are encoded by receptorsthat make use of an internal oscillator operating at the level of themembrane potential. If such oscillators have three or more degreesof freedom, and at least one of which also exhibits a nonlinearity,they are potentially capable of chaotic dynamics. By detecting theexistence of stable and unstable periodic orbits, we demonstratebifurcations between noisy stable and chaotic behavior using theambient temperature as a parameter. We suggest that the techniquedeveloped herein be regarded as an additional tool for the analysisof data in sensory biology and thus can be potentially useful instudies of functional responses to external stimuli. We speculatethat the appearance of unstable orbits may be indicative of a stateof heightened sensory awareness by the animal.     

Analysis of chaotic oscillations induced in two coupled Wilson–Cowan models

Although it is known that two coupled Wilson–Cowan models with reciprocal connections induce aperiodic oscillations, little attention has been paid to the dynamical mechanism for such oscillations so far. In this study, we aim to elucidate the fundamental mechanism to induce the aperiodic oscillations in the coupled model. First, aperiodic oscillations observed are investigated for the case when the connections are unidirectional and when the input signal is a periodic oscillation. By the phase portrait analysis, we determine that the aperiodic oscillations are caused by periodically forced state transitions between a stable equilibrium and a stable limit cycle attractors around the saddle-node and saddle separatrix loop bifurcation points. It is revealed that the dynamical mechanism where the state crosses over the saddle-node and saddle separatrix loop bifurcations significantly contributes to the occurrence of chaotic oscillations forced by a periodic input. In addition, this mechanism can also give rise to chaotic oscillations in reciprocally connected Wilson–Cowan models. These results suggest that the dynamic attractor transition underlies chaotic behaviors in two coupled Wilson–Cowan oscillators.     

Canards for a reduction of the Hodgkin-Huxley equations

This paper shows that canards, which are periodic orbits for which the trajectory follows both the attracting and repelling part of a slow manifold, can exist for a two-dimensional reduction of the Hodgkin-Huxley equations. Such canards are associated with a dramatic change in the properties of the periodic orbit within a very narrow interval of a control parameter. By smoothly connecting stable and unstable manifolds in an asymptotic limit, we predict with great accuracy the parameter value at which the canards exist for this system. This illustrates the power of using singular perturbation theory to understand the dynamical properties of realistic biological systems.     

An artificial neural network that utilizes hip joint actuations to control bifurcations and chaos in a passive dynamic bipedal walking model

Chaos is a central feature of human locomotion and has been suggested to be a window to the control mechanisms of locomotion. In this investigation, we explored how the principles of chaos can be used to control locomotion with a passive dynamic bipedal walking model that has a chaotic gait pattern. Our control scheme was based on the scientific evidence that slight perturbations to the unstable manifolds of points in a chaotic system will promote the transition to new stable behaviors embedded in the rich chaotic attractor. Here we demonstrate that hip joint actuations during the swing phase can provide such perturbations for the control of bifurcations and chaos in a locomotive pattern. Our simulations indicated that systematic alterations of the hip joint actuations resulted in rapid transitions to any stable locomotive pattern available in the chaotic locomotive attractor. Based on these insights, we further explored the benefits of having a chaotic gait with a biologically inspired artificial neural network (ANN) that employed this chaotic control scheme. Remarkably, the ANN was quite robust and capable of selecting a hip joint actuation that rapidly transitioned the passive dynamic bipedal model to a stable gait embedded in the chaotic attractor. Additionally, the ANN was capable of using hip joint actuations to accommodate unstable environments and to overcome unforeseen perturbations. Our simulations provide insight on the advantage of having a chaotic locomotive system and provide evidence as to how chaos can be used as an advantageous control scheme for the nervous system.     

Convergence Time towards Periodic Orbits in Discrete Dynamical Systems

We investigate the convergence towards periodic orbits in discrete dynamical systems. We examine the probability that a randomly chosen point converges to a particular neighborhood of a periodic orbit in a fixed number of iterations, and we use linearized equations to examine the evolution near that neighborhood. The underlying idea is that points of stable periodic orbit are associated with intervals. We state and prove a theorem that details what regions of phase space are mapped into these intervals (once they are known) and how many iterations are required to get there. We also construct algorithms that allow our theoretical results to be implemented successfully in practice.     

Multiple attractors and boundary crises in a tri-trophic food chain

The asymptotic behaviour of a model of a tri-trophic food chain in the chemostat is analysed in detail. The Monod growth model is used for all trophic levels, yielding a non-linear dynamical system of four ordinary differential equations. Mass conservation makes it possible to reduce the dimension by 1 for the study of the asymptotic dynamic behaviour. The intersections of the orbits with a Poincaré plane, after the transient has died out, yield a two-dimensional Poincaré next-return map. When chaotic behaviour occurs, all image points of this next-return map appear to lie close to a single curve in the intersection plane. This motivated the study of a one-dimensional bi-modal, non-invertible map of which the graph resembles this curve. We will show that the bifurcation structure of the food chain model can be understood in terms of the local and global bifurcations of this one-dimensional map. Homoclinic and heteroclinic connecting orbits and their global bifurcations are discussed also by relating them to their counterparts for a two-dimensional map which is invertible like the next-return map. In the global bifurcations two homoclinic or two heteroclinic orbits collide and disappear. In the food chain model two attractors coexist; a stable limit cycle where the top-predator is absent and an interior attractor. In addition there is a saddle cycle. The stable manifold of this limit cycle forms the basin boundary of the interior attractor. We will show that this boundary has a complicated structure when there are heteroclinic orbits from a saddle equilibrium to this saddle limit cycle. A homoclinic bifurcation to a saddle limit cycle will be associated with a boundary crisis where the chaotic attractor disappears suddenly when a bifurcation parameter is varied. Thus, similar to a tangent local bifurcation for equilibria or limit cycles, this homoclinic global bifurcation marks a region in the parameter space where the top-predator goes extinct. The 'Paradox of Enrichment' says that increasing the concentration of nutrient input can cause destabilization of the otherwise stable interior equilibrium of a bi-trophic food chain. For a tri-trophic food chain enrichment of the environment can even lead to extinction of the highest trophic level.     

Short-term depression and transient memory in sensory cortex

Persistent neuronal activity is usually studied in the context of short-term memory localized in central cortical areas. Recent studies show that early sensory areas also can have persistent representations of stimuli which emerge quickly (over tens of milliseconds) and decay slowly (over seconds). Traditional positive feedback models cannot explain sensory persistence for at least two reasons: (i) They show attractor dynamics, with transient perturbations resulting in a quasi-permanent change of system state, whereas sensory systems return to the original state after a transient. (ii) As we show, those positive feedback models which decay to baseline lose their persistence when their recurrent connections are subject to short-term depression, a common property of excitatory connections in early sensory areas. Dual time constant network behavior has also been implemented by nonlinear afferents producing a large transient input followed by much smaller steady state input. We show that such networks require unphysiologically large onset transients to produce the rise and decay observed in sensory areas. Our study explores how memory and persistence can be implemented in another model class, derivative feedback networks. We show that these networks can operate with two vastly different time courses, changing their state quickly when new information is coming in but retaining it for a long time, and that these capabilities are robust to short-term depression. Specifically, derivative feedback networks with short-term depression that acts differentially on positive and negative feedback projections are capable of dynamically changing their time constant, thus allowing fast onset and slow decay of responses without requiring unrealistically large input transients.     

神经元的确定性与随机性整数倍放电

在大鼠损伤背根节神经元的自发放电中发现了整数倍放电, 为了阐明这种放电所产生的原因, 首先研究神经元模型中确定性混沌所引起的整数倍放电与噪声所诱发的整数倍放电的峰峰间期（ＩＳＩ） 序列,通过分析得到前者的ＩＳＩ序列是非线性可预报的,具有确定的非线性特性,但由噪声所诱发的整数倍放电的ＩＳＩ序列是不可预报的, 这表明这两种机制所产生的整数倍放电具有不同的特点,存在着定性的差别,并且混沌运动所产生的整数倍放电是由混沌中各阶不稳定周期轨道决定的。从这种差别出发,分析了实验中整数倍放电的ＩＳＩ 序列,得到该ＩＳＩ 序列是可非线性预报的,这表明大鼠损伤背根节神经元自发放电中的整数倍放电更可能是由确定性机制所产生的     

Coexistence of Multiple Periodic and Chaotic Regimes in Biochemical Oscillations with Phase Shifts

The numerical study of a glycolytic model formed by a system of three delay differential equations reveals a multiplicity of stable coexisting states: birhythmicity, trirhythmicity, hard excitation and quasiperiodic with chaotic regimes. For different initial functions in the phase space one may observe the coexistence of two different quasiperiodic motions, the existence of a stable steady state with a stable torus, and the existence of a strange attractor with different stable regimes (chaos with torus, chaos with bursting motion, and chaos with different periodic regimes). For a single range of the control parameter values our system may exhibit different bifurcation diagrams: in one case a Feigenbaum route to chaos coexists with a finite number of successive periodic bifurcations, in other conditions it is possible to observe the coexistence of two quasiperiodicity routes to chaos. These studies were obtained both at constant input flux and under forcing conditions.     

New attractor states for synchronous activity in synfire chains with excitatory and inhibitory coupling

 In a feedforward network of integrate-and-fire neurons, where the firing of each layer is synchronous (synfire chain), the final firing state of the network converges to two attractor states: either a full activation or complete fading of the tailing layers. In this article, we analyze various modes of pattern propagation in a synfire chain with random connection weights and delta-type postsynaptic currents. We predict analytically that when the input is fully synchronized and the network is noise free, varying the characteristics of the weights distribution would result in modes of behavior that are different from those described in the literature. These are convergence to fixed points, limit cycles, multiple periodic, and possibly chaotic dynamics. We checked our analytic results by computer simulation of the network, and showed that the above results can be generalized when the input is asynchronous and neurons are spontaneously active at low rates. Received: 27 July 2001 / Accepted in revised form: 23 October 2001     

Chaotic attractor in two-prey one-predator system originates from interplay of limit cycles

We investigate the appearance of chaos in a microbial 3-species model motivated by a potentially chaotic real world system (as characterized by positive Lyapunov exponents (Becks et al., Nature 435, 2005). This is the first quantitative model that simulates characteristic population dynamics in the system. A striking feature of the experiment was three consecutive regimes of limit cycles, chaotic dynamics and a fixed point. Our model reproduces this pattern. Numerical simulations of the system reveal the presence of a chaotic attractor in the intermediate parameter window between two regimes of periodic coexistence (stable limit cycles). In particular, this intermediate structure can be explained by competition between the two distinct periodic dynamics. It provides the basis for stable coexistence of all three species: environmental perturbations may result in huge fluctuations in species abundances, however, the system at large tolerates those perturbations in the sense that the population abundances quickly fall back onto the chaotic attractor manifold and the system remains. This mechanism explains how chaos helps the system to persist and stabilize against migration. In discrete populations, fluctuations can push the system towards extinction of one or more species. The chaotic attractor protects the system and extinction times scale exponentially with system size in the same way as with limit cycles or in a stable situation.     

健康人心率变异性中的不稳定周期轨道

为刻划心脏节律存在的确定性动力学特征,运用不稳定周期轨道分析方法对健康青年人的RR间期时间序列数据进行分析。研究结果揭示健康人心脏节律中存在显著的不稳定周期轨道及不稳定周期轨道分级（周期1、周期2,周期3,周期4）现象,表明健康青年人心脏节律的动力学特性中包含着显著的确定性行为。通过跟踪不周期轨道随时间的演变,迹表明心脏节律的变化中存在着因有的非平稳性。     

Diversity of temporal self-organized behaviors in a biochemical system.

The numerical study of a glycolytic model formed by a system of three delay-differential equations revealed a notable richness of temporal structures which included the three main routes to chaos, as well as a multiplicity of stable coexisting states. The Feigenbaum, intermitency and quasiperiodicity routes to chaos can emerge in the biochemical oscillator. Moreover, different types of birhythmicity, trirhythmicity and hard excitation emerge in the phase space. For a single range of the control parameter it can be observed the coexistence of two quasiperiodicity routes to chaos, the coexistence of a stable steady state with a stable torus, and the coexistence of a strange attractor with different stable regimes such as chaos with different periodic regimes, chaos with bursting behavior, and chaos with torus. In most of the numerical studies, the biochemical oscillator has been considered under periodic input flux being the mean input flux rate 6 mM/h. On the other hand, several investigators have observed quasiperiodic time patterns and chaotic oscillations by monitoring the fluorescence of NADH in glycolyzing yeast under sinusoidal glucose input flux. Our numerical results match well with these experimental studies.     

Global stability and periodic orbits for two-patch predator-prey diffusion-delay models

We consider a predator-prey system where the prey can diffuse between one patch with a low level of food and without predation and one patch with a higher level of food but with predation. We assume a Volterra within-patch dynamics, and we assume further that the benefit for the predator comes also from predation in the past through an exponential-delay memory function. By homotopy techniques we prove that, if the prey diffusion is weak enough, then a nonzero globally stable equilibrium exists. This result essentially depends upon the self-regulating coefficient of the predator. If we put this coefficient equal to zero, assuming that the predator density is regulated only by predation, then we can prove the existence of a Hopf bifurcating orbit from the positive equilibrium. The main cause of periodic orbits is the time delay in the predator response functional. We prove that diffusion, lack of delay in the predator response, and increase in the rate of the exponential decay of the memory play stabilizing roles.     

Low-Dimensional Dynamics in Sensory Biology 2: Facial Cold Receptors of the Rat

We report the results of a search for evidence of unstable periodic orbits in the sensory afferents of the facial cold receptors of the rat. Cold receptors are unique in that they exhibit a diversity of action potential firing patterns as well as pronounced transients in firing rate following rapid temperature changes. These characteristics are the result of an internal oscillator operating at the level of the membrane potential. If such oscillators have three or more degree of freedom, and at least one of which also exhibits a nonlinearity, they are potentially capable of complex activity. By detecting the existence of unstable periodic orbits, we demonstrate low-dimensional dynamical behavior whose characteristics depend on the temperature range, impulse pattern, and temperature transients.     

Accuracy and response-time distributions for decision-making: linear perfect integrators versus nonlinear attractor-based neural circuits

Animals choose actions based on imperfect, ambiguous data. “Noise” inherent in neural processing adds further variability to this already-noisy input signal. Mathematical analysis has suggested that the optimal apparatus (in terms of the speed/accuracy trade-off) for reaching decisions about such noisy inputs is perfect accumulation of the inputs by a temporal integrator. Thus, most highly cited models of neural circuitry underlying decision-making have been instantiations of a perfect integrator. Here, in accordance with a growing mathematical and empirical literature, we describe circumstances in which perfect integration is rendered suboptimal. In particular we highlight the impact of three biological constraints: (1) significant noise arising within the decision-making circuitry itself; (2) bounding of integration by maximal neural firing rates; and (3) time limitations on making a decision. Under conditions (1) and (2), an attractor system with stable attractor states can easily best an integrator when accuracy is more important than speed. Moreover, under conditions in which such stable attractor networks do not best the perfect integrator, a system with unstable initial states can do so if readout of the system’s final state is imperfect. Ubiquitously, an attractor system with a nonselective time-dependent input current is both more accurate and more robust to imprecise tuning of parameters than an integrator with such input. Given that neural responses that switch stochastically between discrete states can “masquerade” as integration in single-neuron and trial-averaged data, our results suggest that such networks should be considered as plausible alternatives to the integrator model.