Trajectories In State Space Research Articles

Turbulence from an Observer Perspective

Turbulence is often studied by tracking its spatiotemporal evolution and analyzing the dynamics of its different scales. The dual to this perspective is that of an observer who starts from measurements, or observations, of turbulence and attempts to identify their back-in-time origin, which is the foundation of data assimilation. This back-in-time search must contend with the action of chaos, which obfuscates the interpretation of the observations. When the available measurements satisfy a critical resolution threshold, the influence of chaos can be entirely mitigated and turbulence can be synchronized to the exact state–space trajectory that generated the observations. The critical threshold offers a new interpretation of the Taylor microscale, one that underscores its causal influence. Below the critical threshold, the origin of measurements becomes less definitive in regions where the flow is inconsequential to the observations. In contrast, flow events that influence the measurements, or are within their domain of dependence, are accurately captured. The implications for our understanding of wall turbulence are explored, starting with the highest density of measurements that entirely tame chaos and proceeding all the way to an isolated measurement of wall stress. The article concludes with a discussion of future opportunities and a call to action.

A Dynamic Entropy Approach Reveals Reduced Functional Network Connectivity Trajectory Complexity in Schizophrenia.

Over the past decade and a half, dynamic functional imaging has revealed low-dimensional brain connectivity measures, identified potential common human spatial connectivity states, tracked the transition patterns of these states, and demonstrated meaningful transition alterations in disorders and over the course of development. Recently, researchers have begun to analyze these data from the perspective of dynamic systems and information theory in the hopes of understanding how these dynamics support less easily quantified processes, such as information processing, cortical hierarchy, and consciousness. Little attention has been paid to the effects of psychiatric disease on these measures, however. We begin to rectify this by examining the complexity of subject trajectories in state space through the lens of information theory. Specifically, we identify a basis for the dynamic functional connectivity state space and track subject trajectories through this space over the course of the scan. The dynamic complexity of these trajectories is assessed along each dimension of the proposed basis space. Using these estimates, we demonstrate that schizophrenia patients display substantially simpler trajectories than demographically matched healthy controls and that this drop in complexity concentrates along specific dimensions. We also demonstrate that entropy generation in at least one of these dimensions is linked to cognitive performance. Overall, the results suggest great value in applying dynamic systems theory to problems of neuroimaging and reveal a substantial drop in the complexity of schizophrenia patients' brain function.

Spinal interneuron population dynamics underlying flexible pattern generation.

Features of spinal interneuron state space trajectories capture variations in the timing and magnitude of muscle activations across individual step cycles, with precision on the scales of milliseconds and microvolts respectively.

Abstract 7397: Application of state transition theory for predicting responses of acute myeloid leukemia to chemotherapy

Abstract Acute Myeloid Leukemia (AML) is aggressive cancer that initiates in the bone marrow (BM) and circulates in the peripheral blood (PB). The AML initiation and progressions are complex disease evolution associated with gene mutations and chromosomal abnormalities that change expression profiles of downstream genes characterizing functional networks of leukemic cells. We previously reported the application of the state-transition theory to characterize AML disease progression by modeling transcriptome changes during AML progression as a dynamic system that undergoes state-transition in an AML state-space characterized by an AML potential. Here, we investigate the impact of treatment on transcriptome state-transition and examine the hypothesis that treatment dynamically alters the AML potential. We use the Cbfb-MYH11 (CM) knock-in AML mouse model and a ”5+3” chemotherapy regimen to model the impact of current standard of care for newly diagnosed AML. CM leukemic mice were treated with a combination of cytarabine (Ara-C; 50mg/kg/day; 5 days) and Daunorubicin (DNR; 1.5mg/kg/day; 3 days) after detection of overt leukemia, defined by > 20% leukemia blast (cKit+) detected by flow cytometry in the PB. Time-series samples of peripheral blood mononuclear cell (PBMC) (n=174) were collected weekly before, during, and following “5+3” chemotherapy and subjected to RNA sequencing. All samples were mapped to the AML state-space constructed based on time-sequential RNA-seq collected over the course of CM AML development. Based on the state-transition model, all 10 treated AML mice achieved partial response to chemotherapy with a mean time to relapse (return to leukemia state) of 5 weeks confirming the anti-leukemia effects of treatment by the blast percentages (ckit%). We used the state-transition treatment model to study the dynamics of chemotherapy on the AML potential using state-space trajectories before, during, and after “5+3” chemotherapy. Dynamic changes of concentrations of Ara-C and DNR in the CM mouse model during ”5+3” chemotherapy were modeled by drug doses, half-life, and Heaviside step function. The drug effects were introduced to the AML potential as anti-leukemic treatment forces. Several treatment protocols were examined to find combinations of drug doses and timing that showed the longest survival rates. The model predicted that sequential doses reducing 50% of the original “5+3” chemotherapy dose would produce the longest survival rate. The model suggests that controlling the state-space trajectories in the healthy state with reduced doses will provide the most effective outcomes from “5+3” chemotherapy. Citation Format: Lisa Uechi, Yu-Hsuan Fu, David E. Frankhouser, Lianjun Zhang, Ying-Chieh Chen, Denis O'Meally, Sergio Branciamore, Jihyun Irizarry, Bin Zhang, Guido Marcucci, Ya-Huei Kuo, Russell C. Rockne. Application of state transition theory for predicting responses of acute myeloid leukemia to chemotherapy [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2024; Part 1 (Regular Abstracts); 2024 Apr 5-10; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2024;84(6_Suppl):Abstract nr 7397.

An alternative EMG normalization method: Heterogeneous recurrence quantification analysis of isometric maximum voluntary contraction movements

This study presents an alternative method of normalizing Surface Electromyography (EMG) signals, which provide valuable insights into the musculoskeletal properties of the human body. Traditional normalization through the peak maximum voluntary contraction (MVC) method may result in variable outcomes due to its dependence on pre-processing steps such as smoothing and window length. To address this issue, the study standardizes the EMG pre-processing steps by using non-linear techniques, namely recurrence quantification analysis (RQA) and heterogeneous RQA (HRQA). RQA, specifically HRQA, functions as EMG feature extractors to explore the nonlinear dynamical system and state space trajectories of chaotic EMG signals. Principal component analysis (PCA) and kernel PCA (KPCA) were further applied to the RQA and HRQA results to obtain data with more information. The results show that the first principal components (PC1 and KPC1) of HRQA have a very strong relationship with peak MVC (ρ > |0.90|, p < 0.001), indicating their potential compared to the traditional EMG normalization technique for lower limb muscle groups. Additionally, the study investigated the dynamic differences among isometric MVC movements in lower limb muscle groups and found that the RQA and HRQA methods indicated a greater separation for detecting these differences compared to peak MVC. This study offers a valuable method for assessing the musculoskeletal characteristics of individuals who may have difficulty with activities like walking or providing maximum force during EMG data collection, such as those suffering from Parkinson's disease or walking disabilities. This enhances our understanding of their musculoskeletal properties.

A cyclic perspective on transient gust encounters through the lens of persistent homology

Large-amplitude gust encounters exhibit a range of separated flow phenomena, making them difficult to characterize using the traditional tools of aerodynamics. In this work, we propose a dynamical systems approach to gust encounters, viewing the flow as a cycle (or a closed trajectory) in state space. We posit that the topology of this cycle, or its shape and structure, provides a compact description of the flow, and can be used to identify coordinates in which the dynamics evolve in a simple, intuitive way. To demonstrate this idea, we consider flowfield measurements of a transverse gust encounter. For each case in the dataset, we characterize the full-state dynamics of the flow using persistent homology, a tool that identifies holes in point cloud data, and transform the dynamics to a reduced-order space using a nonlinear autoencoder. Critically, we constrain the autoencoder such that it preserves topologically relevant features of the original dynamics, or those features identified by persistent homology. Using this approach, we are able to transform six separate gust encounters to a three-dimensional latent space, in which each gust encounter reduces to a simple circle, and from which the original flow can be reconstructed. This result shows that topology can guide the creation of low-dimensional state representations for strong transverse gust encounters, a crucial step towards the modelling and control of aerofoil–gust interactions.

Light evokes stereotyped global brain dynamics in Caenorhabditis elegans

Stereotyped oscillations in population neural activity recordings from immobilized Caenorhabditis elegans have garnered interest for their striking low dimensionality and their evocative state-space trajectories or manifolds. Previously these oscillations have been interpreted as intrinsically driven global motor commands. Here we test whether these oscillations are intrinsic. We show that similar oscillations are evoked by high-intensity blue light commonly used for calcium imaging. Oscillations are reduced or absent and have a lower frequency when a longer imaging wavelength is used. Under the original blue light illumination, oscillations are reduced or have a lower frequency in animals that lack GUR-3, an endogenous light- and hydrogen-peroxide-sensitive gustatory receptor. Additional experiments with hydrogen peroxide are consistent with GUR-3's involvement. We therefore propose that blue light evokes global oscillations in part through the creation of reactive oxygen species that activate the hydrogen-peroxide-sensing receptor GUR-3.

Rate-induced tipping in ecosystems and climate: the role of unstable states, basin boundaries and transient dynamics

Abstract. The climate system as well as ecosystems might undergo relatively sudden qualitative changes in the dynamics when environmental parameters or external forcings vary due to anthropogenic influences. The study of these qualitative changes, called tipping phenomena, requires the development of new methodological approaches that allow phenomena observed in nature to be modeled, analyzed and predicted, especially concerning the climate crisis and its consequences. Here we briefly review the mechanisms of classical tipping phenomena and investigate rate-dependent tipping phenomena which occur in non-autonomous systems characterized by multiple timescales in more detail. We focus on the mechanism of rate-induced tipping caused by basin boundary crossings. We unravel the mechanism of this transition and analyze, in particular, the role of such basin boundary crossings in non-autonomous systems when a parameter drift induces a saddle-node bifurcation in which new attractors and saddle points emerge, including their basins of attraction. Furthermore, we study the detectability of those bifurcations by monitoring single trajectories in state space and find that depending on the rate of environmental parameter drift, such saddle-node bifurcations might be masked or hidden, and they can only be detected if a critical rate of environmental drift is crossed. This analysis reveals that unstable states of saddle type are the organizing centers of the global dynamics in non-autonomous multistable systems and as such need much more attention in future studies.

Polynomial Attitude Trajectory Planning for Spacecraft with Movable Parts Using Decoupled Strategy

The problem of attitude planning for spacecraft with movable parts under complex pointing constraints is addressed in this paper. To avoid tackling the complex pointing constraints in the high-dimensional system, a novel decoupled strategy is proposed to map the pointing space to the attitude space. Path-planning algorithm Rapidly-exploring Random Tree (RRT)*-Smart is employed in pointing space to generate a collision-free pointing path, and piecewise quintic polynomials are used to represent the attitude path. By proving the differential flatness of spacecraft with movable parts, the inverse dynamics is used to acquire attitude state trajectory in state space. Benefiting from the decoupled strategy, the attitude state trajectory is optimized by a new heuristic time allocation algorithm using binary search with no gradient information needed. The simulation results are discussed, and the performance during each scenario is analyzed.

Instability of the optimal edge trajectory in the Blasius boundary layer

In the context of linear stability analysis, considering unsteady base flows is notoriously difficult. A generalisation of modal linear stability analysis, allowing for arbitrarily unsteady base flows over a finite time, is therefore required. The recently developed optimally time-dependent (OTD) modes form a projection basis for the tangent space. They capture the leading amplification directions in state space under the constraint that they form an orthonormal basis at all times. The present numerical study illustrates the possibility to describe a complex flow case using the leading OTD modes. The flow under investigation is an unsteady case of the Blasius boundary layer, featuring streamwise streaks of finite length and relevant to bypass transition. It corresponds to the state space trajectory initiated by the minimal seed; such a trajectory is unsteady, free from any spatial symmetry and shadows the laminar–turbulent separatrix for a finite time only. The finite-time instability of this unsteady base flow is investigated using the 8 leading OTD modes. The analysis includes the computation of finite-time Lyapunov exponents as well as instantaneous eigenvalues, and of the associated flow structures. The reconstructed instantaneous eigenmodes are all of outer type. They map unambiguously the spatial regions of largest instantaneous growth. Other flow structures, previously reported as secondary, are identified with this method as relevant to streak switching and to streamwise vortical ejections. The dynamics inside the tangent space features both modal and non-modal amplification. Non-normality within the reduced tangent subspace, quantified by the instantaneous numerical abscissa, emerges only as the unsteadiness of the base flow is reduced.

Learning stability guarantees for constrained switching linear systems from noisy observations

We present a data-driven framework based on Lyapunov theory to provide stability guarantees for a family of hybrid systems. In particular, we are interested in the asymptotic stability of switching linear systems whose switching sequence is constrained by labeled graphs, namely constrained switching linear systems. In order to do so, we provide chance-constrained bounds on stability guarantees, that can be obtained from a finite number of observations with bounded noise. We first present a method providing stability guarantees from sampled trajectories in the hybrid state space of the system. We then study the harder situation where one only observes the continuous part of the hybrid states. We show that in this case, one may still obtain formal chance-constrained stability guarantees. For this latter result we provide a new upper bound of general interest, also for model-based stability analysis.

Single-neuron and population measures of neuronal activity in working memory tasks.

Information represented in working memory is reflected in the firing rate of neurons in the prefrontal cortex and brain areas connected to it. In recent years, there has been an increased realization that population measures capture more accurately neural correlates of cognitive functions. We examined how single neuron firing in the prefrontal and posterior parietal cortex of two male monkeys compared with population measures in spatial working memory tasks. Persistent activity was observed in the dorsolateral prefrontal and posterior parietal cortex and firing rate predicted working memory behavior, particularly in the prefrontal cortex. These findings had equivalents in population measures, including trajectories in state space that became less separated in error trials. We additionally observed rotations of stimulus representations in the neuronal state space for different task conditions, which were not obvious in firing rate measures. These results suggest that population measures provide a richer view of how neuronal activity is associated with behavior, largely confirming that persistent activity is the core phenomenon that maintains visual-spatial information in working memory.NEW & NOTEWORTHY Recordings from large numbers of neurons led to a reevaluation of neural correlates of cognitive functions, which traditionally were defined based on responses of single neurons or averages of firing rates. Analysis of neuronal recordings from the dorsolateral prefrontal and posterior parietal cortex revealed that properties of neuronal firing captured in classical studies of persistent activity can account for population representations, though some population characteristics did not have clear correlates in single neuron activity.

LLC Resonant Converter as a Current Source Using Simple Trajectory Control

This paper discusses a simple method for controlling an LLC resonant DC-DC converter using a state space trajectory based on a linear combination of the voltage across the capacitor and the current through the inductance in the series resonant tank. An analysis of the converter using the state plane technique for the continuous current operation mode is proposed. The output, control and load characteristics of the converter, required for its design and application, are built for the different inductance ratios. They specify the limitations in the control range of the LLC resonant converter which result from the use of the control method. It is shown that under this control, the converter possesses linear control characteristics and has current source behavior. Therefore, the converter can successfully be used as a battery-charging device. Simulations of the converter’s operation, both in steady state and when the control parameter is changed significantly, are implemented. Studies using the model confirm that the considered simple trajectory control provides a fast dynamic response and stable operation of the LLC resonant converter, even in cases of a significant change in the control.

Temporary disruption in language processing reflected as multiscale temporal discoordination in a recurrent network

By juxtaposing time series analyses of activity measured from a fully recurrent network undergoing disrupted processing and of activity measured from a continuous meta-cognitive report of disruption in real-time language comprehension, we present an opportunity to compare the temporal statistics of the state-space trajectories inherent to both systems. Both the recurrent network and the human language comprehension process appear to exhibit long-range temporal correlations and low entropy when processing is undisrupted and coordinated. However, when processing is disrupted and discoordinated, they both exhibit more short-range temporal correlations and higher entropy. We conclude that by measuring human language comprehension in a dense-sampling manner similar to how we analyze the networks, and analyzing the resulting data stream with nonlinear time series analysis techniques, we can obtain more insight into the temporal character of these discoordination phases than by simply marking the points in time at which they peak.

Temporary disruption in language processing reflected as multiscale temporal discoordination in a recurrent network

By juxtaposing time series analyses of activity measured from a fully recurrent network undergoing disrupted processing and of activity measured from a continuous meta-cognitive report of disruption in real-time language comprehension, we present an opportunity to compare the temporal statistics of the state-space trajectories inherent to both systems. Both the recurrent network and the human language comprehension process appear to exhibit long-range temporal correlations and low entropy when processing is undisrupted and coordinated. However, when processing is disrupted and discoordinated, they both exhibit more short-range temporal correlations and higher entropy. We conclude that by measuring human language comprehension in a dense-sampling manner similar to how we analyze the networks, and analyzing the resulting data stream with nonlinear time series analysis techniques, we can obtain more insight into the temporal character of these discoordination phases than by simply marking the points in time at which they peak.

REFINEMENT OF OPTIMAL CONTROL PROBLEM FOR PRACTICAL IMPLEMENTATION OF ITS SOLUTION

The solution of the optimal control problem in the classical formulation is control in the form of a function of time. The implementation of such a solution leads to an open control system and therefore cannot be applied directly in practice. It is believed that solving the classical optimal control problem leads to an optimal control program and program trajectory in state space. To implement the movement of the control object along the program trajectory, it is necessary to build an additional movement stabilization system. The problem of synthesizing a system for stabilizing movement along a program trajectory and the requirements that this system should meet do not arise from the classical setting of the optimal control problem. An updated statement of the optimal control problem is given, which includes an additional requirement for an optimal trajectory, and the solution of which can be directly applied in practice in a real control object.

State estimation in minimal turbulent channel flow: A comparative study of 4DVar and PINN

The state of turbulent, minimal-channel flow is estimated from spatio-temporal sparse observations of the velocity, using both a physics-informed neural network (PINN) and adjoint-variational data assimilation (4DVar). The performance of PINN is assessed against the benchmark results from 4DVar. The PINN is efficient to implement, takes advantage of automatic differentiation to evaluate the governing equations, and does not require the development of an adjoint model. In addition, the flow evolution is expressed in terms of the network parameters which have a far smaller dimension than the predicted trajectory in state space or even just the initial condition of the flow. Provided adequate observations, network architecture and training, the PINN can yield satisfactory estimates of the flow field, both for the missing velocity data and the entirely unobserved pressure field. However, accuracy depends on the network architecture, and the dependence is not known a priori. In comparison to 4DVar estimation which becomes progressively more accurate over the observation horizon, the PINN predictions are generally less accurate and maintain the same level of errors throughout the assimilation time window. Another notable distinction is the capacity to accurately forecast the flow evolution: while the 4DVar prediction depart from the true flow state gradually and according to the Lyapunov exponent, the PINN is entirely inaccurate immediately beyond the training time horizon unless re-trained. Most importantly, while 4DVar satisfies the discrete form of the governing equations point-wise to machine precision, in PINN the equations are only satisfied in an L2 sense.

Altered basal ganglia output during self-restraint.

Suppressing actions is essential for flexible behavior. Multiple neural circuits involved in behavioral inhibition converge upon a key basal ganglia output nucleus, the substantia nigra pars reticulata (SNr). To examine how changes in basal ganglia output contribute to self-restraint, we recorded SNr neurons during a proactive behavioral inhibition task. Rats responded to <i>Go!</i> cues with rapid leftward or rightward movements, but also prepared to cancel one of these movement directions on trials when a <i>Stop!</i> cue might occur. This action restraint - visible as direction-selective slowing of reaction times - altered both rates and patterns of SNr spiking. Overall firing rate was elevated before the <i>Go!</i> cue, and this effect was driven by a subpopulation of direction-selective SNr neurons. In neural state space, this corresponded to a shift away from the restrained movement. SNr neurons also showed more variable inter-spike intervals during proactive inhibition. This corresponded to more variable state-space trajectories, which may slow reaction times via reduced preparation to move. These findings open new perspectives on how basal ganglia dynamics contribute to movement preparation and cognitive control.

Regimes and mechanisms of transient amplification in abstract and biological neural networks.

Neuronal networks encode information through patterns of activity that define the networks’ function. The neurons’ activity relies on specific connectivity structures, yet the link between structure and function is not fully understood. Here, we tackle this structure-function problem with a new conceptual approach. Instead of manipulating the connectivity directly, we focus on upper triangular matrices, which represent the network dynamics in a given orthonormal basis obtained by the Schur decomposition. This abstraction allows us to independently manipulate the eigenspectrum and feedforward structures of a connectivity matrix. Using this method, we describe a diverse repertoire of non-normal transient amplification, and to complement the analysis of the dynamical regimes, we quantify the geometry of output trajectories through the effective rank of both the eigenvector and the dynamics matrices. Counter-intuitively, we find that shrinking the eigenspectrum’s imaginary distribution leads to highly amplifying regimes in linear and long-lasting dynamics in nonlinear networks. We also find a trade-off between amplification and dimensionality of neuronal dynamics, i.e., trajectories in neuronal state-space. Networks that can amplify a large number of orthogonal initial conditions produce neuronal trajectories that lie in the same subspace of the neuronal state-space. Finally, we examine networks of excitatory and inhibitory neurons. We find that the strength of global inhibition is directly linked with the amplitude of amplification, such that weakening inhibitory weights also decreases amplification, and that the eigenspectrum’s imaginary distribution grows with an increase in the ratio between excitatory-to-inhibitory and excitatory-to-excitatory connectivity strengths. Consequently, the strength of global inhibition reveals itself as a strong signature for amplification and a potential control mechanism to switch dynamical regimes. Our results shed a light on how biological networks, i.e., networks constrained by Dale’s law, may be optimised for specific dynamical regimes.

Hilbert space as a computational resource in reservoir computing

Accelerating computation with quantum resources is limited by the challenges of high-fidelity control of quantum systems. Reservoir computing presents an attractive alternative, as precise control and full calibration of system dynamics are not required. Instead, complex internal trajectories in a large state space are leveraged as a computational resource. Quantum systems offer a unique venue for reservoir computing, given the presence of interactions unavailable in classical systems and a potentially exponentially-larger computational space. With a reservoir comprised of a single $d$-dimensional quantum system, we demonstrate clear performance improvement with Hilbert space dimension at two benchmark tasks and advantage over the physically analogous classical reservoir. Quantum reservoirs as realized by current-era quantum hardware offer immediate practical implementation and a promising outlook for increased performance in larger systems.