Dynamic Manifold Research Articles

Dynamical manifold dimensionality as characterization measure of chimera states in bursting neuronal networks

Methods that distinguish dynamical regimes in networks of active elements make it possible to design the dynamics of models of realistic networks. A particularly salient example of such dynamics is partial synchronization, which may play a pivotal role in emergent behaviors of biological neural networks. Such emergent partial synchronization in structurally homogeneous networks is commonly denoted as chimera states. While several methods for detecting chimeras in networks of spiking neurons have been proposed, these are less effective when applied to networks of bursting neurons. In this study, we propose the correlation dimension as a novel approach that can be employed to identify dynamic network states. To assess the viability of this new method, we study networks of intrinsically bursting Hindmarsh–Rose neurons with non-local connections. In comparison to other measures of chimera states, the correlation dimension effectively characterizes chimeras in burst neurons, whether the incoherence arises in spikes or bursts. The generality of dimensionality measures inherent in the correlation dimension renders this approach applicable to a wide range of dynamic systems, thereby facilitating the comparison of simulated and experimental data. This methodology enhances our ability to tune and simulate intricate network processes, ultimately contributing to a deeper understanding of neural dynamics.

Transiently delocalized states enhance hole mobility in organic molecular semiconductors.

Evidence shows that charge carriers in organic semiconductors self-localize because of dynamic disorder. Nevertheless, some organic semiconductors feature reduced mobility at increasing temperature, a hallmark for delocalized band transport. Here we present the temperature-dependent mobility in two record-mobility organic semiconductors: dinaphtho[2,3-b:2',3'-f]thieno[3,2-b]-thiophene (DNTT) and its alkylated derivative, C8-DNTT-C8. By combining terahertz photoconductivity measurements with atomistic non-adiabatic molecular dynamics simulations, we show that while both crystals display a power-law decrease of the mobility (μ) with temperature (T) following μ ∝ T -n, the exponent n differs substantially. Modelling reveals that the differences between the two chemically similar semiconductors can be traced to the delocalization of the different states that are thermally accessible by charge carriers, which in turn depends on their specific electronic band structure. The emerging picture is that of holes surfing on a dynamic manifold of vibrationally dressed extended states with a temperature-dependent mobility that provides a sensitive fingerprint for the underlying density of states.

Dynamic functional connectivity MEG features of Alzheimer’s disease

Dynamic resting state functional connectivity (RSFC) characterizes time-varying fluctuations of functional brain network activity. While many studies have investigated static functional connectivity, it has been unclear whether features of dynamic functional connectivity are associated with neurodegenerative diseases. Popular sliding-window and clustering methods for extracting dynamic RSFC have various limitations that prevent extracting reliable features to address this question. Here, we use a novel and robust time-varying dynamic network (TVDN) approach to extract the dynamic RSFC features from high resolution magnetoencephalography (MEG) data of participants with Alzheimer’s disease (AD) and matched controls. The TVDN algorithm automatically and adaptively learns the low-dimensional spatiotemporal manifold of dynamic RSFC and detects dynamic state transitions in data. We show that amongst all the functional features we investigated, the dynamic manifold features are the most predictive of AD. These include: the temporal complexity of the brain network, given by the number of state transitions and their dwell times, and the spatial complexity of the brain network, given by the number of eigenmodes. These dynamic features have higher sensitivity and specificity in distinguishing AD from healthy subjects than the existing benchmarks do. Intriguingly, we found that AD patients generally have higher spatial complexity but lower temporal complexity compared with healthy controls. We also show that graph theoretic metrics of dynamic component of TVDN are significantly different in AD versus controls, while static graph metrics are not statistically different. These results indicate that dynamic RSFC features are impacted in neurodegenerative disease like Alzheimer’s disease, and may be crucial to understanding the pathophysiological trajectory of these diseases.

Joint Dynamic Manifold and Discriminant Information Learning for Feature Extraction.

Neighborhood reconstruction is a good recipe to learn the local manifold structure. Representation-based discriminant analysis methods normally learn the reconstruction relationship between each sample and all the other samples. However, reconstruction graphs constructed in these methods have three limitations: 1) they cannot guarantee the local sparsity of reconstruction coefficients; 2) heterogeneous samples may own nonzero coefficients; and 3) they learn the manifold information prior to the process of dimensionality reduction. Due to the existence of noise and redundant features in the original space, the prelearned manifold structure may be inaccurate. Accordingly, the performance of dimensionality reduction would be affected. In this article, we propose a joint model to simultaneously learn the affinity relationship, reconstruction relationship, and projection matrix. In this model, we actively assign neighbors for each sample and learn the inter-reconstruction coefficients between each sample and their neighbors with the same label information in the process of dimensionality reduction. Specifically, a sparse constraint is employed to ensure the sparsity of neighbors and reconstruction coefficients. The whitening constraint is imposed on the projection matrix to remove the relevance between features. An iterative algorithm is proposed to solve this method. Extensive experiments on toy data and public datasets show the superiority of the proposed method.

Multi-band oscillations emerge from a simple spiking network.

In the brain, coherent neuronal activities often appear simultaneously in multiple frequency bands, e.g., as combinations of alpha (8-12 Hz), beta (12.5-30 Hz), and gamma (30-120 Hz) oscillations, among others. These rhythms are believed to underlie information processing and cognitive functions and have been subjected to intense experimental and theoretical scrutiny. Computational modeling has provided a framework for the emergence of network-level oscillatory behavior from the interaction of spiking neurons. However, due to the strong nonlinear interactions between highly recurrent spiking populations, the interplay between cortical rhythms in multiple frequency bands has rarely been theoretically investigated. Many studies invoke multiple physiological timescales (e.g., various ion channels or multiple types of inhibitory neurons) or oscillatory inputs to produce rhythms in multi-bands. Here, we demonstrate the emergence of multi-band oscillations in a simple network consisting of one excitatory and one inhibitory neuronal population driven by constant input. First, we construct a data-driven, Poincaré section theory for robust numerical observations of single-frequency oscillations bifurcating into multiple bands. Then, we develop model reductions of the stochastic, nonlinear, high-dimensional neuronal network to capture the appearance of multi-band dynamics and the underlying bifurcations theoretically. Furthermore, when viewed within the reduced state space, our analysis reveals conserved geometrical features of the bifurcations on low-dimensional dynamical manifolds. These results suggest a simple geometric mechanism behind the emergence of multi-band oscillations without appealing to oscillatory inputs or multiple synaptic or neuronal timescales. Thus, our work points to unexplored regimes of stochastic competition between excitation and inhibition behind the generation of dynamic, patterned neuronal activities.

Stem girth changes in response to soil water potential in lowland dipterocarp forest in Borneo: An individualistic time-series analysis.

Time-series data offer a way of investigating the causes driving ecological processes as phenomena. To test for possible differences in water relations between species of different forest structural guilds at Danum (Sabah, NE Borneo), daily stem girth increments (gthi), of 18 trees across six species were regressed individually on soil moisture potential (SMP) and temperature (TEMP), accounting for temporal autocorrelation (in GLS-arima models), and compared between a wet and a dry period. The best-fitting significant variables were SMP the day before and TEMP the same day. The first resulted in a mix of positive and negative coefficients, the second largely positive ones. An adjustment for dry-period showers was applied. Interactions were stronger in dry than wet period. Negative relationships for overstorey trees can be interpreted in a reversed causal sense: fast transporting stems depleted soil water and lowered SMP. Positive relationships for understorey trees meant they took up most water at high SMP. The unexpected negative relationships for these small trees may have been due to their roots accessing deeper water supplies (if SMP was inversely related to that of the surface layer), and this was influenced by competition with larger neighbour trees. A tree-soil flux dynamics manifold may have been operating. Patterns of mean diurnal girth variation were more consistent among species, and time-series coefficients were negatively related to their maxima. Expected differences in response to SMP in the wet and dry periods did not clearly support a previous hypothesis differentiating drought and non-drought tolerant understorey guilds. Trees within species showed highly individual responses when tree size was standardized. Data on individual root systems and SMP at several depths are needed to get closer to the mechanisms that underlie the tree-soil water phenomena in these tropical forests. Neighborhood stochasticity importantly creates varying local environments experienced by individual trees.

Dynamic Manifold Learning for Land Deformation Forecasting

Landslides refer to occurrences of massive ground movements due to geological (and meteorological) factors, and can have disastrous impact on property, economy, and even lead to loss of life. The advances of remote sensing provide accurate and continuous terrain monitoring, enabling the study and analysis of land deformation which, in turn, can be used for possible landslides forecast. Prior studies either rely on independent observations for displacement prediction or model static land characteristics without considering the subtle interactions between different locations and the dynamic changes of the surface conditions. We present DyLand -- Dynamic Manifold Learning with Normalizing Flows for Land deformation prediction -- a novel framework for learning dynamic structures of terrain surface and improving the performance of land deformation prediction. DyLand models the spatial connections of InSAR measurements and estimates conditional distributions of deformations on the terrain manifold with a novel normalizing flow-based method. Instead of modeling the stable terrains, it incorporates surface permutations and captures the innate dynamics of the land surface while allowing for tractable likelihood estimates on the manifold. Our extensive evaluations on curated InSAR datasets from continuous monitoring of slopes prone to landslides show that DyLand outperforms existing bechmarking models.

Hybrid Zero Dynamics Control for Gait Guidance of a Novel Adjustable Pediatric Lower-Limb Exoskeleton.

Exoskeleton technology has undergone significant developments for the adult population but is still lacking for the pediatric population. This paper presents the design of a hip–knee exoskeleton for children 6 to 11 years old with gait abnormalities. The actuators are housed in an adjustable exoskeleton frame where the thigh part can adjust in length and the hip cradle can adjust in the medial-lateral and posterior-anterior directions concurrently. Proper control of exoskeletons to follow nominal healthy gait patterns in a time-invariant manner is important for ease of use and user acceptance. In this paper, a hybrid zero dynamics (HZD) controller was designed for gait guidance by defining the zero dynamics manifold to resemble healthy gait patterns. HZD control utilizes a time-invariant feedback controller to create dynamically stable gaits in robotic systems with hybrid models containing both discrete and continuous dynamics. The effectiveness of the controller on the novel pediatric exoskeleton was demonstrated via simulation. The presented preliminary results suggest that HZD control provides a viable method to control the pediatric exoskeleton for gait guidance.

Dynamic manifold Boltzmann optimization based on self‐supervised learning for human motion estimation

It is a challenge work to estimate the 3D human motion from image sequence. There are some problems, such as unsatisfactory estimation error, ambiguous matching and transient occlusion. Although the prior information of learning large-scale samples exists, these problems are still difficult to be solved. How to extract the feature of the high-dimensional (HD) sample of 3D human motion and find the desired one will become the key to solve these problems above. Some dimension reduction methods can extract the sample features and build the low-dimensional (LD) space to view their LD features, but how to search the relevant valid and desired LD samples remains the bottleneck problem, which can be used to reconstruct the 3D human motions denoted by the corresponding high-dimensional samples. Thus, a new method called dynamic manifold Boltzmann optimization (DMBO) is proposed to estimate the 3D human motion from multi-view images. DMBO can find the best matching 3D human motion model by the help of the self-supervised learning from Gaussian incremental dimension reduction model (GIDRM). DMBO can avoid the local optimum during searching and solve the problems above, so that the generation of the accurate 3D human motion corresponding to multi-view images can be achieved.

Oscillatory Resonance and Dynamic Manifolds in Cortical Networks With Time Delay and Multiple External Stimuli.

Rhythmic oscillation is crucial for information transmission and neural communication among different brain areas. Stochastic resonance (SR) can evoke different patterns of neural oscillation. However, the characteristics of network resonance and underlying dynamical mechanisms are still unclear. In this paper, a biological model of cortical network is established and its dynamical response to external periodic stimulation is investigated. We explore the oscillatory resonance of excitatory and inhibitory populations in cortical network. It is found that the intrinsic parameters of neural populations determine the extent of resonant activity, indicating that the firing rate exhibits coherent oscillation when the frequency of external stimulation is close to intrinsic frequency of neural population. In addition, the nonlinear dynamics of cortical network in oscillatory resonance can be represented by helical manifolds in low-dimensional state space. The geometry of neural manifolds reveals the periodic dynamics and state transition in oscillatory resonance. Moreover, time delay in chemical synapses can induce multiple resonances, which appear intermittently at integer multiples of the period of input signal. The dynamical response of neural population achieves maximal periodically, due to the transition of network states induced by time delay. Furthermore, mean-field theory is applied to analyze theoretical dynamic of cortical networks with time delay and demonstrate the effective transmission of stimulation information via oscillatory resonance in the brain. Consequently, the obtained results contribute to the improvement of neuromodulation for neurological disease from the viewpoint of the neural basis.

Dissecting transition cells from single-cell transcriptome data through multiscale stochastic dynamics

Advances in single-cell technologies allow scrutinizing of heterogeneous cell states, however, detecting cell-state transitions from snap-shot single-cell transcriptome data remains challenging. To investigate cells with transient properties or mixed identities, we present MuTrans, a method based on multiscale reduction technique to identify the underlying stochastic dynamics that prescribes cell-fate transitions. By iteratively unifying transition dynamics across multiple scales, MuTrans constructs the cell-fate dynamical manifold that depicts progression of cell-state transitions, and distinguishes stable and transition cells. In addition, MuTrans quantifies the likelihood of all possible transition trajectories between cell states using coarse-grained transition path theory. Downstream analysis identifies distinct genes that mark the transient states or drive the transitions. The method is consistent with the well-established Langevin equation and transition rate theory. Applying MuTrans to datasets collected from five different single-cell experimental platforms, we show its capability and scalability to robustly unravel complex cell fate dynamics induced by transition cells in systems such as tumor EMT, iPSC differentiation and blood cell differentiation. Overall, our method bridges data-driven and model-based approaches on cell-fate transitions at single-cell resolution.

On the Topological Dynamics of Dynamical Manifolds and Their Fundamental Group

The paper aims to deduce the relation between the category of topology and algebra from viewpoint of geometry and dynamical system. We introduce and define a dynamical manifold as a manifold associated with a time parameter. We obtain the induced chain of topological dynamics on the fundamental group from the chain of dynamical maps on a dynamical manifold. For many adjunctions in this context, we deduce the limit topological dynamics and conditional topological dynamics on the fundamental group. We use the category of commutative diagrams as chains of dynamical manifolds to deduce the chains on fundamental groups. Also, we describe how the manifold changes in a dynamical system from the view of the fundamental group.

Center manifolds for nonautonomous dynamics via evolution semigroups

Using nonlinear evolution semigroups, we construct center-stable, center-unstable and center invariant manifolds for a nonautonomous dynamics in a simple manner. Our approach consists essentially of two steps: after introducing a nonlinear evolution semigroup for the original dynamics, we show that the problem of finding center-stable invariant manifolds is equivalent to a corresponding problem for the nonlinear semigroup. As a nontrivial consequence, we show that the regularity properties of the invariant manifolds for the evolution family and for the evolution semigroup are the same. The main novelty of our work is the method of proof, which leads to quite simple arguments for important results.

Sparse representation preserving embedding based on extreme learning machine for process monitoring

Extreme learning machine (ELM) is a fast learning mechanism used in many domains. Unsupervised ELM has improved to extract nonlinear features. A nonlinear dynamic process monitoring method named sparse representation preserving embedding based on ELM (SRPE-ELM) is proposed in this paper. First, the noise is removed by sparse representation and the sparse coefficient is applied to construct the adjacency graph. The adjacency graph with a data-adaptive neighborhood can extract dynamic manifold structure better than a specified neighborhood parameter. Secondly, a new objection function considered the sparse reconstruction and output weights is established to extract nonlinear dynamic manifold structure. Thirdly, the statistic SPE and T2 based on SRPE-ELM are built to monitor the whole process. Finally, SRPE-ELM is applied in the IRIS data classification example, a numerical case and Tennessee Eastman benchmark process to verify the effectiveness of process monitoring.

Walking With Confidence: Safety Regulation for Full Order Biped Models

Safety guarantees are valuable in the control of walking robots, as falling can be both dangerous and costly. Unfortunately, set-based tools for generating safety guarantees (such as sums-of-squares optimization) are typically restricted to simplified, low-dimensional models of walking robots. For more complex models, methods based on hybrid zero dynamics can ensure the local stability of a pre-specified limit cycle, but provide limited guarantees. This letter combines the benefits of both approaches by using sums-of-squares optimization on a hybrid zero dynamics manifold to generate a guaranteed safe set for a ten-dimensional walking robot model. Along with this set, this letter describes how to generate a controller that maintains safety by modifying the manifold parameters when on the edge of the safe set. The proposed approach, which is applied to a bipedal RABBIT model, provides a roadmap for applying sums-of-squares techniques to high-dimensional systems. This opens the door for a broad set of tools that can generate flexible and safe controllers for complex walking robot models.

A Novel Hybrid Control Strategy for an Underactuated 3-D Biped with Asymmetric Structure

In reality, due to the manufacturing error or the component loss in the service process, the structural parameters of bipedal robots may exhibit asymmetry. In this work, we consider the stable walking of an underactuated 3-D bipedal robot with asymmetric structure, and a novel hybrid control strategy is proposed. The control strategy consists of a continous heuristic motion controller, which asymptotically drive the state of the robot to the zero dynamics manifold, and an event-based feedback controller that renders the hybrid zero dynamics locally asymptotically stable. The heuristic motion controller uses heuristic state variables as controlled variables rather than simply the actuated variables, and the controller parameters of the event-based feedback controller are designed in an analytical method rather than relying on the left–right symmetry property. The effectiveness of the presented control strategy is illustrated by a numerical simulation example.

Transitioning Between Underactuated Periodic Orbits: An Optimal Control Approach for Settling Time Reduction

In underactuated systems, a transition between two periodic orbits is generally characterized by slow convergence. This is due to the fact that the unactuated degree of freedom (DoF) hinders the state of the system to enter the domain of attraction of the target orbit close to the fixed point of the Poincaré Map. In this paper, we introduce an optimal control algorithm to reduce the settling time of transitions between periodic orbits of underactuated walking robots. This is achieved by utilizing the hybrid zero dynamics (HZD) framework to express the feasibility condition of the transition which can be imposed as an inequality constraint in the proposed optimal control problem. In addition, the cost function penalizes deviations from the fixed point of the target periodic orbit in the zero dynamics manifold while at the same time all dynamic and kinematic assumptions are treated as constraints. Furthermore, high magnitude torques are also penalized. The numerical results show that the proposed methodology can indeed improve the settling time compared to the transition methodology usually found in the bibliography and at the same time provide a feasible and smooth motion.

Implications of causality for quantum biology – I: topology change

ABSTRACTA framework for describing the causal, topology changing, evolution of interacting biomolecules is developed. The quantum dynamical manifold equations (QDMEs) derived from this framework can be related to the causality restrictions implied by a finite speed of light and to Planck's constant to set a transition frequency scale. The QDMEs imply conserved stress-energy, angular-momentum and Noether currents. The functional whose extremisation leads to this result provides a causal, time-dependent, non-equilibrium generalisation of the Hohenberg–Kohn theorem. The system of dynamical equations derived from this functional and the currents J derived from the QDMEs are shown to be causal and consistent with the first and second laws of thermodynamics. This has the potential of allowing living systems to be quantum mechanically distinguished from non-living ones.

Model reaching adaptive-robust control law for vibration isolation systems with parametric uncertainty

Adaptive control has been used for active vibration isolation and vehicle suspensions systems. A model reference adaptive control law is used for the plant to track the ideal reference model. In a model reaching adaptive control approach, the ideal of a skyhook target without using a reference model is achieved. In this paper, a novel approach, a model reaching adaptive-robust control law is studied for active vibration isolation systems. A dynamic manifold for ideal system is defined using the ideal of a skyhook target model system parameters. First, a new Lyapunov function is defined. Based on the Lyapunov stability theory, a model reaching adaptive and a robust control laws are derived for the uncertain system to reach the ideal manifold. Parameters and upper bounding functions are estimated as a trigonometric function depending on the relative displacements, velocities and the defined manifold. The developed adaptive and the robust compensators are combined and this combination is proposed as an adaptive-robust control law. After that, the controller is applied to a vehicle suspension system and the ideal of a skyhook target without using a reference model is achieved. The results also show that the proposed robust control law can increase the comfort of the vehicle active suspension systems and the ride comfort is remarkably increased.

Extensions to a manifold learning framework for time-series analysis on dynamic manifolds in bioelectric signals.

This paper addresses the challenge of extracting meaningful information from measured bioelectric signals generated by complex, large scale physiological systems such as the brain or the heart. We focus on a combination of the well-known Laplacian eigenmaps machine learning approach with dynamical systems ideas to analyze emergent dynamic behaviors. The method reconstructs the abstract dynamical system phase-space geometry of the embedded measurements and tracks changes in physiological conditions or activities through changes in that geometry. It is geared to extract information from the joint behavior of time traces obtained from large sensor arrays, such as those used in multiple-electrode ECG and EEG, and explore the geometrical structure of the low dimensional embedding of moving time windows of those joint snapshots. Our main contribution is a method for mapping vectors from the phase space to the data domain. We present cases to evaluate the methods, including a synthetic example using the chaotic Lorenz system, several sets of cardiac measurements from both canine and human hearts, and measurements from a human brain.