Influence of timing jitter on nonlinear dynamics of a photonic crystal fiber ring cavity

We demonstrate that timing jitter has a strong influence on supercontinua generated in a photonic crystal fiber ring cavity synchronously pumped by 140?fs pulses. The global dynamics with respect to cavity detuning is analyzed both numerically and experimentally by tracking the cavity pulse energy. The results show that low-frequency timing jitter, induced by both the pump oscillator and the external cavity, masks the fine underlying bifurcation structure of the system. Numerical simulations in the absence of timing jitter reveal that the system dynamics fall into four qualitatively different regimes. The existence of these regimes is experimentally observed in first-return diagrams.     

Dynamics analysis on neural firing patterns by symbolic approach

Neural firing patterns are investigated by using symbolic dynamics. Bifurcation behaviour of the Hindmarsh--Rose (HR) neuronal model is simulated with the external stimuli gradually decreasing, and various firing activities with different topological structures are orderly numbered. Through constructing first-return maps of interspike intervals, all firing patterns are described and identified by symbolic expressions. On the basis of ordering rules of symbolic sequences, the corresponding relation between parameters and firing patterns is established, which will be helpful for encoding neural information. Moreover, using the operation rule of $\ast$ product, generation mechanisms and intrinsic configurations of periodic patterns can be distinguished in detail. Results show that the symbolic approach is a powerful tool to study neural firing activities. In particular, such a coarse-grained way can be generalized in neural electrophysiological experiments to extract much valuable information from complicated experimental data.     

Analysis of spontaneous activity in neuronal cultures through recurrence plots: impact of varying connectivity

We analyzed spontaneous activity of cortical neuronal networks in vitro using recurrence plots (RPs). Our data encompasses fluorescence traces of average network activity from two experimental explorations, namely the development of connections during the maturation of the network and the gradual weakening of connections through chemical action. The dynamical richness of the networks in these connectivity-evolving scenarios was examined through recurrence quantification analysis. Measures such as determinism and laminarity were used to portray the degree of uniformity and periodicity of the spontaneous activity patterns. The analysis shows that RPs are a powerful tool to visualize and interpret neuronal networks dynamics, and pinpoint its hallmarks.     

Neuronal avalanches and coherence potentials

The mammalian cortex consists of a vast network of weakly interacting excitable cells called neurons. Neurons must synchronize their activities in order to trigger activity in neighboring neurons. Moreover, interactions must be carefully regulated to remain weak (but not too weak) such that cascades of active neuronal groups avoid explosive growth yet allow for activity propagation over long-distances. Such a balance is robustly realized for neuronal avalanches, which are defined as cortical activity cascades that follow precise power laws. In experiments, scale-invariant neuronal avalanche dynamics have been observed during spontaneous cortical activity in isolated preparations in vitro as well as in the ongoing cortical activity of awake animals and in humans. Theory, models, and experiments suggest that neuronal avalanches are the signature of brain function near criticality at which the cortex optimally responds to inputs and maximizes its information capacity. Importantly, avalanche dynamics allow for the emergence of a subset of avalanches, the coherence potentials. They emerge when the synchronization of a local neuronal group exceeds a local threshold, at which the system spawns replicas of the local group activity at distant network sites. The functional importance of coherence potentials will be discussed in the context of propagating structures, such as gliders in balanced cellular automata. Gliders constitute local population dynamics that replicate in space after a finite number of generations and are thought to provide cellular automata with universal computation. Avalanches and coherence potentials are proposed to constitute a modern framework of cortical synchronization dynamics that underlies brain function.     

Identification of equivalent dynamics using ordinal pattern distributions

The frequency of occurrence of ordinal patterns in an observed (or measured) times series can be used to identify equivalent dynamical system. We demonstrate this approach for system identification and parameter estimation for dynamics that can (at least approximately) be described by one-dimensional iterated maps.     

Functional holography analysis: simplifying the complexity of dynamical networks

We present a novel functional holography (FH) analysis devised to study the dynamics of task-performing dynamical networks. The latter term refers to networks composed of dynamical systems or elements, like gene networks or neural networks. The new approach is based on the realization that task-performing networks follow some underlying principles that are reflected in their activity. Therefore, the analysis is designed to decipher the existence of simple causal motives that are expected to be embedded in the observed complex activity of the networks under study. First we evaluate the matrix of similarities (correlations) between the activities of the network's components. We then perform collective normalization of the similarities (or affinity transformation) to construct a matrix of functional correlations. Using dimension reduction algorithms on the affinity matrix, the matrix is projected onto a principal three-dimensional space of the leading eigenvectors computed by the algorithm. To retrieve back information that is lost in the dimension reduction, we connect the nodes by colored lines that represent the level of the similarities to construct a holographic network in the principal space. Next we calculate the activity propagation in the network (temporal ordering) using different methods like temporal center of mass and cross correlations. The causal information is superimposed on the holographic network by coloring the nodes locations according to the temporal ordering of their activities. First, we illustrate the analysis for simple, artificially constructed examples. Then we demonstrate that by applying the FH analysis to modeled and real neural networks as well as recorded brain activity, hidden causal manifolds with simple yet characteristic geometrical and topological features are deciphered in the complex activity. The term "functional holography" is used to indicate that the goal of the analysis is to extract the maximum amount of functional information about the dynamical network as a whole unit.     

Turning point properties as a method for the characterization of the ergodic dynamics of one-dimensional iterative maps

Dynamical as well as statistical properties of the ergodic and fully developed chaotic dynamics of iterative maps are investigated by means of a turning point analysis. The turning points of a trajectory are hereby defined as the local maxima and minima of the trajectory. An examination of the turning point density directly provides us with the information of the position of the fixed point for the corresponding dynamical system. Dividing the ergodic dynamics into phases consisting of turning points and nonturning points, respectively, elucidates the understanding of the organization of the chaotic dynamics for maps. The turning point map contains information on any iteration of the dynamical law and is shown to possess an asymptotic scaling behaviour which is responsible for the assignment of dynamical structures to the environment of the two fixed points of the map. Universal statistical turning point properties are derived for doubly symmetric maps. Possible applications of the observed turning point properties for the analysis of time series are discussed in some detail. (c) 1997 American Institute of Physics.     

Ictal imaging using functional magnetic resonance

Magnetic resonance imaging (MRI) can now provide maps of human brain function with high spatial and temporal resolution. This noninvasive technique can also map the coritical activation that occurs during focal seizures, as demonstrated here by the results obtained using a conventional 1.5 T clinical MRI system for the investigation of a 4-year-old boy suffering from frequent partial motor seizures of his right side. FLASH images (TE = 60 ms) were acquired every 10 s over a period of 25 min, and activation images derived by subtracting baseline images from images obtained during clinical seizures. Functional MRI revealed sequential activation associated with specific gyri within the left hemisphere with each of five consecutive clinical seizures, and also during a period that was not associated with a detectable clinical seizure. The activated regions included gyri that were structurally abnormal. These results demonstrate (a) that functional MRI can potentially provide new insights into the dynamic events that occur in the epileptic brain and their relationship to brain structure; and (b) that there is the possibility of obtaining similar information in the absence of clinical seizures, suggesting the potential for studies in patients with interictal electrical disturbances.     

Unwrapping eigenfunctions to discover the geometry of almost-invariant sets in hyperbolic maps

The numerical approximation of Perron-Frobenius operators allows efficient determination of the physical invariant measure of chaotic dynamical systems as a fixed point of the operator. Eigenfunctions of the Perron-Frobenius operator corresponding to large subunit eigenvalues have been shown to describe “almost-invariant” dynamics in one-dimensional expanding maps. We extend these ideas to hyperbolic maps in higher dimensions. While the eigendistributions of the operator are relatively uninformative, applying a new procedure called “unwrapping” to regularised versions of the eigendistributions clearly reveals the geometric structures associated with almost-invariant dynamics. This unwrapping procedure is applied to a uniformly hyperbolic map of the unit square to discover this map’s dominant underlying dynamical structure, and to the standard map to pinpoint clusters of period 6 orbits.     

Influence of dynamical noise on time series generated by nonlinear maps

We consider periodic and chaotic dynamics of discrete nonlinear maps in the presence of dynamical noise. We show that dynamical noise corrupting dynamics of a nonlinear map may be considered as a measurement “pseudonoise” with the distribution determined by the Jacobian of the map. The formula for the distribution of the measurement “pseudonoise” for one-dimensional quadratic maps has also been obtained in an explicit form. We expect that our results apply to an arbitrary distribution of low-level dynamical noise and hope that these results could help to find a universal method of discriminating dynamical from measurement noise.     

Spiking dynamics of interacting oscillatory neurons

Spiking sequences emerging from dynamical interaction in a pair of oscillatory neurons are investigated theoretically and experimentally. The model comprises two unidirectionally coupled FitzHugh-Nagumo units with modified excitability (MFHN). The first (master) unit exhibits a periodic spike sequence with a certain frequency. The second (slave) unit is in its excitable mode and responds on the input signal with a complex (chaotic) spike trains. We analyze the dynamic mechanisms underlying different response behavior depending on interaction strength. Spiking phase maps describing the response dynamics are obtained. Complex phase locking and chaotic sequences are investigated. We show how the response spike trains can be effectively controlled by the interaction parameter and discuss the problem of neuronal information encoding.     

Derivative temporal clustering analysis: detecting prolonged neuronal activity

Temporal clustering analysis (TCA) and independent component analysis (ICA) are promising data-driven techniques in functional magnetic resonance imaging (fMRI) experiments to obtain brain activation maps in conditions with unknown temporal information regarding the neuronal activity. Although comparable to ICA in detecting transient neuronal activities, TCA fails to detect prolonged plateau brain activations. To eliminate this pitfall, a novel derivative TCA (DTCA) method was introduced and its algorithms with different subtraction intervals were tested on simulated data with a pattern of prolonged plateau brain activation. It was found that the best performance of DTCA method in generating functional maps could be obtained if the subtraction interval is equal to or larger than the length of the rising time of the fMRI response. The DTCA method and its theoretical predication were further investigated and validated using in vivo fMRI data sets. By removing the limitations in the previous TCA, DTCA has shown its powerful capability in detecting prolonged plateau neuronal activities.     

Ergodic Theory of Generic Continuous Maps

We study the ergodic properties of generic continuous dynamical systems on compact manifolds. As a main result we prove that generic homeomorphisms have convergent Birkhoff averages under continuous observables at Lebesgue almost every point. In spite of this, when the underlying manifold has dimension greater than one, generic homeomorphisms have no physical measures—a somewhat strange result which stands in sharp contrast to current trends in generic differentiable dynamics. Similar results hold for generic continuous maps. To further explore the mysterious behaviour of C ⁰ generic dynamics, we also study the ergodic properties of continuous maps which are conjugated to expanding circle maps. In this context, generic maps have divergent Birkhoff averages along orbits starting from Lebesgue almost every point.     

Dynamical hysteresis and spatial synchronization in coupled non-identical chaotic oscillators

We identify a novel phenomenon in distinct (namely non-identical) coupled chaotic systems, which we term dynamical hysteresis. This behavior, which appears to be universal, is defined in terms of the system dynamics (quantified for example through the Lyapunov exponents), and arises from the presence of at least two coexisting stable attractors over a finite range of coupling, with a change of stability outside this range. Further characterization via mutual synchronization indices reveals that one attractor corresponds to spatially synchronized oscillators, while the other corresponds to desynchronized oscillators. Dynamical hysteresis may thus help to understand critical aspects of the dynamical behavior of complex biological systems, e.g. seizures in the epileptic brain can be viewed as transitions between different dynamical phases caused by time dependence in the brain’s internal coupling.     

Cingulate seizure-like activity reveals neuronal avalanche regulated by network excitability and thalamic inputs

Background

Cortical neurons display network-level dynamics with unique spatiotemporal patterns that construct the backbone of processing information signals and contribute to higher functions. Recent years have seen a wealth of research on the characteristics of neuronal networks that are sufficient conditions to activate or cease network functions. Local field potentials (LFPs) exhibit a scale-free and unique event size distribution (i.e., a neuronal avalanche) that has been proven in the cortex across species, including mice, rats, and humans, and may be used as an index of cortical excitability. In the present study, we induced seizure activity in the anterior cingulate cortex (ACC) with medial thalamic inputs and evaluated the impact of cortical excitability and thalamic inputs on network-level dynamics. We measured LFPs from multi-electrode recordings in mouse cortical slices and isoflurane-anesthetized rats.

Results

The ACC activity exhibited a neuronal avalanche with regard to avalanche size distribution, and the slope of the power-law distribution of the neuronal avalanche reflected network excitability in vitro and in vivo. We found that the slope of the neuronal avalanche in seizure-like activity significantly correlated with cortical excitability induced by γ-aminobutyric acid system manipulation. The thalamic inputs desynchronized cingulate seizures and affected the level of cortical excitability, the modulation of which could be determined by the slope of the avalanche size.

Conclusions

We propose that the neuronal avalanche may be a tool for analyzing cortical activity through LFPs to determine alterations in network dynamics.     

Temporal and spatial persistence of combustion fronts in paper

The spatial and temporal persistence, or first-return distributions are measured for slow-combustion fronts in paper. The stationary temporal and (perhaps less convincingly) spatial persistence exponents agree with the predictions based on the front dynamics, which asymptotically belongs to the Kardar-Parisi-Zhang universality class. The stationary short-range and the transient behavior of the fronts are non-Markovian, and the observed persistence properties thus do not agree with the predictions based on Markovian theory. This deviation is a consequence of additional time and length scales, related to the crossovers to the asymptotic coarse-grained behavior.     

Differentiability implies continuity in neuronal dynamics

Recent work has identified nonlinear deterministic structure in neuronal dynamics using periodic orbit theory. Troublesome in this work were the significant periods of time where no periodic orbits were extracted — “dynamically dark” regions. Tests for periodic orbit structure typically require that the underlying dynamics are differentiable. Since continuity of a mathematical function is a necessary but insufficient condition for differentiability, regions of observed differentiability should be fully contained within regions of continuity. We here verify that this fundamental mathematical principle is reflected in observations from mammalian neuronal activity. First, we introduce a null Jacobian transformation to verify the observation of differentiable dynamics when periodic orbits are extracted. Second, we show that a less restrictive test for deterministic structure requiring only continuity demonstrates widespread nonlinear deterministic structure only partially appreciated with previous approaches.     

人脑默认模式网络的动力学行为

大脑具有自适应、自组织、多稳态等重要特征,是典型的复杂系统.人脑在静息态下的关键功能子网络--默认模式网络(DMN)的激活处于多状态间持续跳转的非平衡过程,揭示该过程背后的动力学机制具有重要的科学意义和临床应用前景.本文基于功能磁共振获得的血氧水平依赖(BOLD)信号,建立了DMN吸引子跳转非平衡过程的能量图景、吸引子非联通图、跳转关系网络等;以高级视觉皮层和听觉等皮层活动为例,通过对应激活DMN状态空间的分布,以及XGBoost、深度神经网络等算法验证了DMN状态变化与外部脑区状态的密切依赖关系;通过偏相关、收敛交叉映射等方法分析了DMN内各个脑区之间的相互作用.本文结果有助于理解静息态下大脑内在非平衡过程的动力学机制,以及从动力学的角度探索具有临床意义的脑功能障碍生物标志物.     

From the Perron-Frobenius equation to the Fokker-Planck equation

We show that for certain classes of deterministic dynamical systems the Perron-Frobenius equation reduces to the Fokker-Planck equation in an appropriate scaling limit. By perturbative expansion in a small time scale parameter, we also derive the equations that are obeyed by the first- and second-order correction terms to the Fokker-Planck limit case. In general, these equations describe non-Gaussian corrections to a Langevin dynamics due to an underlying deterministic chaotic dynamics. For double-symmetric maps, the first-order correction term turns out to satisfy a kind of inhomogeneous Fokker-Planck equation with a source term. For a special example, we are able solve the first- and second-order equations explicitly.