Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics.

We address the question of whether or not childhood epidemics such as measles and chickenpox are chaotic, and argue that the best explanation of the observed unpredictability is that it is a manifestation of what we call chaotic stochasticity. Such chaos is driven and made permanent by the fluctuations from the mean field encountered in epidemics, or by extrinsic stochastic noise, and is dependent upon the existence of chaotic repellors in the mean field dynamics. Its existence is also a consequence of the near extinctions in the epidemic. For such systems, chaotic stochasticity is likely to be far more ubiquitous than the presence of deterministic chaotic attractors. It is likely to be a common phenomenon in biological dynamics.     

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。     

STOCHASTIC RESONANCE MEASURED BY THEMUTUAL INFORMATION IN THE NEURONSTHAT TRANSMIT CHAOTIC SPIKE TRAINS

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Food web complexity and chaotic population dynamics

In mathematical models, very simple communities consisting of three or more species frequently display chaotic dynamics which implies that long‐term predictions of the population trajectories in time are impossible. Communities in the wild tend to be more complex, but evidence for chaotic dynamics from such communities is scarce. We used supercomputing power to test the hypothesis that chaotic dynamics become less frequent in model ecosystems when their complexity increases. We determined the dynamical stability of a universe of mathematical, nonlinear food web models with varying degrees of organizational complexity. We found that the frequency of unpredictable, chaotic dynamics increases with the number of trophic levels in a food web but decreases with the degree of complexity. Our results suggest that natural food webs possess architectural properties that may intrinsically lower the likelihood of chaotic community dynamics.     

Effects of survival thresholds upon one-dimensional dynamics of single-species populations

A simple one-dimensional model of single-species populations is studied by means of computer simulations. Although the model has a rich spectrum of dynamics including chaotic behavior, the introduction of survival thresholds makes the chaotic region so small that it can be hardly observed. Stochastic fluctuations further reduce the chaotic region because they accidentally lead populations to extinction. The model thus naturally explains the observation that the majority of natural populations do not show chaotic behavior but a monotonic return to a stable equilibrium point following a disturbance.     

Nonlinear transient computation as a potential "kernel trick" in cortical processing

Evidence has been found for the presence of chaotic dynamics at all levels of the mammalian brain. This has led to some searching questions about the potential role that nonlinear dynamics may have in neural information processing. We propose that chaos equips the brain with the equivalent of a kernel trick for solving hard nonlinear problems. The approach presented, which is described as nonlinear transient computation, uses the dynamics of a well known chaotic attractor. The paper provides experimental results to show that this approach can be used to solve some challenging pattern recognition tasks. The paper also offers evidence to suggest that the efficacy of nonlinear transient computation for nonlinear pattern classification is dependent only on the generic properties of chaotic attractors and is not sensitive to the particular dynamics of specific sub-regions of chaotic phase space. If, as this work suggests, nonlinear transient computation is independent of the particulars of any given chaotic attractor, then it could be offered as a possible explanation of how the chaotic dynamics that have been observed in brain structures contribute to neural information processing tasks.     

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.     

STOCHASTIC RESONANCE MEASURED BY THEMUTUAL INFORMATION IN THE NEURONSTHAT TRANSMIT CHAOTIC SPIKE TRAINS

The spike trains generated by a neuron model are studied by the methods of nonlinear time series analysis. The results show that the spike trains are chaotic. To investigate effect of noise on transmission of chaotic spike trains, this chaotic spike trains are used as a discrete subthreshold input signal to the integrate-and-fire neuronal model and the FitzHugh-Nagumo(FHN) neuronal model working in noisy environment. The mutual information between the input spike trains and the output spike trains is calculated, the result shows that the transformation of information encoded by the chaotic spike trains is optimized by some level of noise, and stochastic resonance(SR) measured by mutual information is a property available for neurons to transmit chaotic spike trains.     

Long food chains are in general chaotic

The question whether chaos exists in nature is much debated. In this paper we prove that chaotic parameter regions exist generically in food chains of length greater than three. While nonchaotic dynamics is also possible, the presence of chaotic parameter regions indicates that chaotic dynamics is likely. We show that the chaotic regions survive even at high exponents of closure. Our results have been obtained using a general food chain model that describes a large class of different food chains. The existence of chaos in models of such generality can be deduced from the presence of certain bifurcations of higher codimension.     

Rotifer Population Dynamics in Two Coupled Habitats: Invasion of Chaos

We study the role of interactions between habitats in rotifer dynamics. We use a simple discrete-time model to simulate the interactions between neighboring habitats with different intrinsic dynamics. Being uncoupled, one habitat shows periodical oscillations of the rotifer biomass while the other one demonstrates chaotic oscillations. As a result of the exchange of rotifer biomass, chaos replaces regular oscillations. As a result, the rotifer dynamics becomes chaotic in both habitats. We show that the invasion of chaos is followed by the synchronization of the chaotic regimes of both habitats, and this synchronization increases as coupling between the habitats is increased. We also demonstrate that the biological invasion of the rotifer species, which show chaotic dynamics, to a neighboring habitat with intrinsically regular plankton dynamics leads to the invasion of chaos and the synchronization of chaotic oscillations of the plankton biomass in both the habitats.     

Chaos and Ultradian Rhythms

Systems in a chaotic state have apparently random outputs despite a simple underlying kinetic mechanism. For instance, the interaction of two coupled oscillators (the mitotic oscillator and the ultradian clock) can produce chaotic behaviour over a limited range of parameter values. Mathematical modelling shows that physiologically realistic characteristics are thereby exhibited. Cell division cycles of lower eukaryotes (protozoa and yeasts) show both deterministic and stochastic properties. Both dispersion of cell cycle times and quantized values can be generated, as a deterministic chaotic consequence of oscillator interaction rather than from noisy limit cycles. Advantages may stem from chaotic operation; a controlled chaotic attractor could provide multifrequency outputs that determine rhythmic behaviour on different time scales (e.g. ultradian and circadian) with the facility for rapid state changes from one periodicity to another.     

Chaos and Ultradian Rhythms

Systems in a chaotic state have apparently random outputs despite a simple underlying kinetic mechanism. For instance, the interaction of two coupled oscillators (the mitotic oscillator and the ultradian clock) can produce chaotic behaviour over a limited range of parameter values. Mathematical modelling shows that physiologically realistic characteristics are thereby exhibited. Cell division cycles of lower eukaryotes (protozoa and yeasts) show both deterministic and stochastic properties. Both dispersion of cell cycle times and quantized values can be generated, as a deterministic chaotic consequence of oscillator interaction rather than from noisy limit cycles. Advantages may stem from chaotic operation; a controlled chaotic attractor could provide multifrequency outputs that determine rhythmic behaviour on different time scales ( e.g. ultradian and circadian) with the facility for rapid state changes from one periodicity to another.     

Detecting chaotic structures in noisy pulse trains based on interspike interval reconstruction

The nonlinear prediction method based on the interspike interval (ISI) reconstruction is applied to the ISI sequence of noisy pulse trains and the detection of the deterministic structure is performed. It is found that this method cannot discriminate between the noisy periodic pulse train and the noisy chaotic one when noise-induced pulses exist. When the noise-induced pulses are eliminated by the grouping of ISI sequence with the genetic algorithm, the chaotic structure of the chaotic firings becomes clear, and the noisy chaotic pulse train could be discriminated from the periodic one.     

Some implications of chaos theory for the genetic analysis of human development and variation.

Non-linear epigenetic processes are a potential underlying source of phenotypic differences in development. Simulation studies of twin pairs using simple non-linear development models characterised by chaotic or near-chaotic behavior are presented. The effect of chaotic processes on correlations is to lower them from their initial values, but high initial correlations are affected much less by chaotic and near-chaotic processes than intermediate correlations. Therefore, we would predict that traits affected by chaotic processes would have high MZ and low DZ twin correlations and this is reminiscent of certain traits such as EEG spectra. However the much more frequent observation of MZ correlations close to twice their DZ counterparts would suggest that the role of chaos in development is quite limited.     

Analysis of Chaotic Resonance in Izhikevich Neuron Model

In stochastic resonance (SR), the presence of noise helps a nonlinear system amplify a weak (sub-threshold) signal. Chaotic resonance (CR) is a phenomenon similar to SR but without stochastic noise, which has been observed in neural systems. However, no study to date has investigated and compared the characteristics and performance of the signal responses of a spiking neural system in some chaotic states in CR. In this paper, we focus on the Izhikevich neuron model, which can reproduce major spike patterns that have been experimentally observed. We examine and classify the chaotic characteristics of this model by using Lyapunov exponents with a saltation matrix and Poincaré section methods in order to address the measurement challenge posed by the state-dependent jump in the resetting process. We found the existence of two distinctive states, a chaotic state involving primarily turbulent movement and an intermittent chaotic state. In order to assess the signal responses of CR in these classified states, we introduced an extended Izhikevich neuron model by considering weak periodic signals, and defined the cycle histogram of neuron spikes as well as the corresponding mutual correlation and information. Through computer simulations, we confirmed that both chaotic states in CR can sensitively respond to weak signals. Moreover, we found that the intermittent chaotic state exhibited a prompter response than the chaotic state with primarily turbulent movement.     

Hindmarsh-Rose 神经网络的混沌同步

研究了通过特殊构造的非线性函数耦合连接的神经网络的混沌同步问题。在发展基于稳定性准则的混沌同步方法的基础上,给出了计算同步稳定性的误差发展方程,当耦合强度取参考值时,可实现稳定的混沌同步而不需要计算最大条件Lyapunov指数去判定是否稳定。通过对按照完全连接形式构成的Hindmarsh-Rose神经网络的数值模拟,显示可仅从两个耦合神经的耦合强度的稳定性范围预期到许多耦合神经实现同步的稳定性范围。该方法在噪声影响下,对实现神经元的混沌同步仍具有较强的鲁棒性。此外发现随着耦合神经数的增加,满足同步稳定性的耦合强度减小,与耦合神经的数量成反比。     

时滞援助的两抑制性突触耦合的Chay 神经元的同步

一对抑制性突触耦合的混沌Chay神经元的同步模式被研究。结果表明当耦合强度超过临界值时,两抑制耦合的混沌Chay神经元能达到反相的同步。与此同时,两混沌的神经元变为周期而不是原来的混沌运动。然而,如果考虑耦合神经元信息的传导时滞,在有效的时滞下,两个耦合神经元的在相簇同步能增加。在相簇同步窗口的大小随着耦合强度的增加而增加。此结果对于我们理解神经元集群的运动是一个指导。     

A chaotic neural network mimicking an olfactory system and its application on image recognition

1 Introduction A biological neural system is complicated and ef-ficient. People have tried for years to simulate it to per-form complex signal processing functions. For example,the artificial neural network is a kind of model derivedfrom a biological neural system. Most artificial neuralnetworks simulate some important features such as thethreshold behaviour and plasticity of synapses. However,they are primary simulations and still much simpler incomparison with specific biological neural…     

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。     

A flexible annealing chaotic neural network to maximum clique problem

Based on the analysis and comparison of several annealing strategies, we present a flexible annealing chaotic neural network which has flexible controlling ability and quick convergence rate to optimization problem. The proposed network has rich and adjustable chaotic dynamics at the beginning, and then can converge quickly to stable states. We test the network on the maximum clique problem by some graphs of the DIMACS clique instances, p-random and k random graphs. The simulations show that the flexible annealing chaotic neural network can get satisfactory solutions at very little time and few steps. The comparison between our proposed network and other chaotic neural networks denotes that the proposed network has superior executive efficiency and better ability to get optimal or near-optimal solution.