A moment-based Kalman filtering approach for estimation in ensemble systems.

A persistent challenge in tasks involving large-scale dynamical systems, such as state estimation and error reduction, revolves around processing the collected measurements. Frequently, these data suffer from the curse of dimensionality, leading to increased computational demands in data processing methodologies. Recent scholarly investigations have underscored the utility of delineating collective states and dynamics via moment-based representations. These representations serve as a form of sufficient statistics for encapsulating collective characteristics, while simultaneously permitting the retrieval of individual data points. In this paper, we reshape the Kalman filter methodology, aiming its application in the moment domain of an ensemble system and developing the basis for moment ensemble noise filtering. The moment system is defined with respect to the normalized Legendre polynomials, and it is shown that its orthogonal basis structure introduces unique benefits for the application of Kalman filter for both i.i.d. and universal Gaussian disturbances. The proposed method thrives from the reduction in problem dimension, which is unbounded within the state-space representation, and can achieve significantly smaller values when converted to the truncated moment-space. Furthermore, the robustness of moment data toward outliers and localized inaccuracies is an additional positive aspect of this approach. The methodology is applied for an ensemble of harmonic oscillators and units following aircraft dynamics, with results showcasing a reduction in both cumulative absolute error and covariance with reduced calculation cost due to the realization of operations within the moment framework conceived.

Data-driven control of oscillator networks with population-level measurement.

Controlling complex networks of nonlinear limit-cycle oscillators is an important problem pertinent to various applications in engineering and natural sciences. While in recent years the control of oscillator populations with comprehensive biophysical models or simplified models, e.g., phase models, has seen notable advances, learning appropriate controls directly from data without prior model assumptions or pre-existing data remains a challenging and less developed area of research. In this paper, we address this problem by leveraging the network's current dynamics to iteratively learn an appropriate control online without constructing a global model of the system. We illustrate through a range of numerical simulations that the proposed technique can effectively regulate synchrony in various oscillator networks after a small number of trials using only one input and one noisy population-level output measurement. We provide a theoretical analysis of our approach, illustrate its robustness to system variations, and compare its performance with existing model-based and data-driven approaches.

Identification of network interactions from time series data: An iterative approach.

The first step toward advancing our understanding of complex networks involves determining their connectivity structures from the time series data. These networks are often high-dimensional, and in practice, only a limited amount of data can be collected. In this work, we formulate the network inference task as a bilinear optimization problem and propose an iterative algorithm with sequential initialization to solve this bilinear program. We demonstrate the scalability of our approach to network size and its robustness against measurement noise, hyper-parameter variation, and deviations from the network model. Results across experimental and simulated datasets, comprising oscillatory, non-oscillatory, and chaotic dynamics, showcase the superior inference accuracy of our technique compared to existing methods.

Optimal phase-selective entrainment of electrochemical oscillators with different phase response curves.

We investigate the entrainment of electrochemical oscillators with different phase response curves (PRCs) using a global signal: the goal is to achieve the desired phase configuration using a minimum-power waveform. Establishing the desired phase relationships in a highly nonlinear networked system exhibiting significant heterogeneities, such as different conditions or parameters for the oscillators, presents a considerable challenge because different units respond differently to the common global entraining signal. In this work, we apply an optimal phase-selective entrainment technique in both a kinetic model and experiments involving electrochemical oscillators in achieving phase synchronized states. We estimate the PRCs of the oscillators at different circuit potentials and external resistance, and entrain pairs and small sets of four oscillators in various phase configurations. We show that for small PRC variations, phase assignment can be achieved using an averaged PRC in the control design. However, when the PRCs are sufficiently different, individual PRCs are needed to entrain the system with the expected phase relationships. The results show that oscillator assemblies with heterogeneous PRCs can be effectively entrained to desired phase configurations in practical settings. These findings open new avenues to applications in biological and engineered oscillator systems where synchronization patterns are essential for system performance.

Synchronization, clustering, and weak chimeras in a densely coupled transcription-based oscillator model for split circadian rhythms.

The synchronization dynamics for the circadian gene expression in the suprachiasmatic nucleus is investigated using a transcriptional circadian clock gene oscillator model. With global coupling in constant dark (DD) conditions, the model exhibits a one-cluster phase synchronized state, in dim light (dim LL), bistability between one- and two-cluster states and in bright LL, a two-cluster state. The two-cluster phase synchronized state, where some oscillator pairs synchronize in-phase, and some anti-phase, can explain the splitting of the circadian clock, i.e., generation of two bouts of daily activities with certain species, e.g., with hamsters. The one- and two-cluster states can be reached by transferring the animal from DD or bright LL to dim LL, i.e., the circadian synchrony has a memory effect. The stability of the one- and two-cluster states was interpreted analytically by extracting phase models from the ordinary differential equation models. In a modular network with two strongly coupled oscillator populations with weak intragroup coupling, with appropriate initial conditions, one group is synchronized to the one-cluster state and the other group to the two-cluster state, resulting in a weak-chimera state. Computational modeling suggests that the daily rhythms in sleep-wake depend on light intensity acting on bilateral networks of suprachiasmatic nucleus (SCN) oscillators. Addition of a network heterogeneity (coupling between the left and right SCN) allowed the system to exhibit chimera states. The simulations can guide experiments in the circadian rhythm research to explore the effect of light intensity on the complexities of circadian desynchronization.

Finding influential nodes in networks using pinning control: Centrality measures confirmed with electrochemical oscillators.

The spatiotemporal organization of networks of dynamical units can break down resulting in diseases (e.g., in the brain) or large-scale malfunctions (e.g., power grid blackouts). Re-establishment of function then requires identification of the optimal intervention site from which the network behavior is most efficiently re-stabilized. Here, we consider one such scenario with a network of units with oscillatory dynamics, which can be suppressed by sufficiently strong coupling and stabilizing a single unit, i.e., pinning control. We analyze the stability of the network with hyperbolas in the control gain vs coupling strength state space and identify the most influential node (MIN) as the node that requires the weakest coupling to stabilize the network in the limit of very strong control gain. A computationally efficient method, based on the Moore-Penrose pseudoinverse of the network Laplacian matrix, was found to be efficient in identifying the MIN. In addition, we have found that in some networks, the MIN relocates when the control gain is changed, and thus, different nodes are the most influential ones for weakly and strongly coupled networks. A control theoretic measure is proposed to identify networks with unique or relocating MINs. We have identified real-world networks with relocating MINs, such as social and power grid networks. The results were confirmed in experiments with networks of chemical reactions, where oscillations in the networks were effectively suppressed through the pinning of a single reaction site determined by the computational method.

Optimal phase-selective entrainment of heterogeneous oscillator ensembles.

We develop a framework to design optimal entrainment signals that entrain an ensemble of heterogeneous nonlinear oscillators, described by phase models, at desired phases. We explicitly take into account heterogeneity in both oscillation frequency and the type of oscillators characterized by different Phase Response Curves. The central idea is to leverage the Fourier series representation of periodic functions to decode a phase-selective entrainment task into a quadratic program. We demonstrate our approach using a variety of phase models, where we entrain the oscillators into distinct phase patterns. Also, we show how the generalizability gained from our formulation enables us to meet a wide range of design objectives and constraints, such as minimum-power, fast entrainment, and charge-balanced controls.

Inferring dynamic topology for decoding spatiotemporal structures in complex heterogeneous networks.

Extracting complex interactions (i.e., dynamic topologies) has been an essential, but difficult, step toward understanding large, complex, and diverse systems including biological, financial, and electrical networks. However, reliable and efficient methods for the recovery or estimation of network topology remain a challenge due to the tremendous scale of emerging systems (e.g., brain and social networks) and the inherent nonlinearity within and between individual units. We develop a unified, data-driven approach to efficiently infer connections of networks (ICON). We apply ICON to determine topology of networks of oscillators with different periodicities, degree nodes, coupling functions, and time scales, arising in silico, and in electrochemistry, neuronal networks, and groups of mice. This method enables the formulation of these large-scale, nonlinear estimation problems as a linear inverse problem that can be solved using parallel computing. Working with data from networks, ICON is robust and versatile enough to reliably reveal full and partial resonance among fast chemical oscillators, coherent circadian rhythms among hundreds of cells, and functional connectivity mediating social synchronization of circadian rhythmicity among mice over weeks.

Biophysically interpretable inference of single neuron dynamics.

Identification of key ionic channel contributors to the overall dynamics of a neuron is an important problem in experimental neuroscience. Such a problem is challenging since even in the best cases, identification relies on noisy recordings of membrane potential only, and strict inversion to the constituent channel dynamics is mathematically ill-posed. In this work, we develop a biophysically interpretable, learning-based strategy for data-driven inference of neuronal dynamics. In particular, we propose two optimization frameworks to learn and approximate neural dynamics from an observed voltage trajectory. In both the proposed strategies, the membrane potential dynamics are approximated as a weighted sum of ionic currents. In the first strategy, the ionic currents are represented using voltage dependent channel conductances and membrane potential in a parametric form, while in the second strategy, the currents are represented as a linear combination of generic basis functions. A library of channel activation/inactivation and time-constant curves describing prototypical channel kinetics are used to provide estimates of the channel variables to approximate the ionic currents. Finally, a linear optimization problem is solved to infer the weights/scaling variables in the membrane-potential dynamics. In the first strategy, the weights can be used to recover the channel conductances, and the reversal potentials while in the second strategy, using the estimated weights, active channels can be inferred and the trajectory of the gating variables are recovered, allowing for biophysically salient inference. Our results suggest that the complex nonlinear behavior of the neural dynamics over a range of temporal scales can be efficiently inferred in a data-driven manner from noisy membrane potential recordings.

Design of charge-balanced time-optimal stimuli for spiking neuron oscillators.

In this letter, we investigate the fundamental limits on how the interspike time of a neuron oscillator can be perturbed by the application of a bounded external control input (a current stimulus) with zero net electric charge accumulation. We use phase models to study the dynamics of neurons and derive charge-balanced controls that achieve the minimum and maximum interspike times for a given bound on the control amplitude. Our derivation is valid for any arbitrary shape of the phase response curve and for any value of the given control amplitude bound. In addition, we characterize the change in the structures of the charge-balanced time-optimal controls with the allowable control amplitude. We demonstrate the applicability of the derived optimal control laws by applying them to mathematically ideal and experimentally observed neuron phase models, including the widely studied Hodgkin-Huxley phase model, and by verifying them with the corresponding original full state-space models. This work addresses a fundamental problem in the field of neural control and provides a theoretical investigation to the optimal control of oscillatory systems.

Optimal pulse design in quantum control: a unified computational method.

Many key aspects of control of quantum systems involve manipulating a large quantum ensemble exhibiting variation in the value of parameters characterizing the system dynamics. Developing electromagnetic pulses to produce a desired evolution in the presence of such variation is a fundamental and challenging problem in this research area. We present such robust pulse designs as an optimal control problem of a continuum of bilinear systems with a common control function. We map this control problem of infinite dimension to a problem of polynomial approximation employing tools from geometric control theory. We then adopt this new notion and develop a unified computational method for optimal pulse design using ideas from pseudospectral approximations, by which a continuous-time optimal control problem of pulse design can be discretized to a constrained optimization problem with spectral accuracy. Furthermore, this is a highly flexible and efficient numerical method that requires low order of discretization and yields inherently smooth solutions. We demonstrate this method by designing effective broadband π/2 and π pulses with reduced rf energy and pulse duration, which show significant sensitivity enhancement at the edge of the spectrum over conventional pulses in 1D and 2D NMR spectroscopy experiments.

Optimal waveform for fast entrainment of weakly forced nonlinear oscillators.

For many biological and engineered systems, a central function or design goal is to abbreviate the time required to synchronize a rhythmic process to an external forcing signal. We present a theory for deriving the input that effectively minimizes the average transient time required to entrain a phase model, which enables a practical technique for constructing fast entrainment waveforms for general nonlinear oscillators. This result is verified in numerical simulations using the Hodgkin-Huxley neuron model, and in experiments on an oscillatory electrochemical system.

Interpretable Design of Reservoir Computing Networks Using Realization Theory.

The reservoir computing networks (RCNs) have been successfully employed as a tool in learning and complex decision-making tasks. Despite their efficiency and low training cost, practical applications of RCNs rely heavily on empirical design. In this article, we develop an algorithm to design RCNs using the realization theory of linear dynamical systems. In particular, we introduce the notion of α -stable realization and provide an efficient approach to prune the size of a linear RCN without deteriorating the training accuracy. Furthermore, we derive a necessary and sufficient condition on the irreducibility of the number of hidden nodes in linear RCNs based on the concepts of controllability and observability from systems theory. Leveraging the linear RCN design, we provide a tractable procedure to realize RCNs with nonlinear activation functions. We present numerical experiments on forecasting time-delay systems and chaotic systems to validate the proposed RCN design methods and demonstrate their efficacy.

Engineering spatiotemporal patterns: information encoding, processing, and controllability in oscillator ensembles.

The ability to finely manipulate spatiotemporal patterns displayed in neuronal populations is critical for understanding and influencing brain functions, sleep cycles, and neurological pathologies. However, such control tasks are challenged not only by the immense scale but also by the lack of real-time state measurements of neurons in the population, which deteriorates the control performance. In this paper, we formulate the control of dynamic structures in an ensemble of neuron oscillators as a tracking problem and propose a principled control technique for designing optimal stimuli that produce desired spatiotemporal patterns in a network of interacting neurons without requiring feedback information. We further reveal an interesting presentation of information encoding and processing in a neuron ensemble in terms of its controllability property. The performance of the presented technique in creating complex spatiotemporal spiking patterns is demonstrated on neural populations described by mathematically ideal and biophysical models, including the Kuramoto and Hodgkin-Huxley models, as well as real-time experiments on Wein bridge oscillators.

Data-Driven Control of Neuronal Networks with Population-Level Measurement.

Controlling complex networks of nonlinear neurons is an important problem pertinent to various applications in engineering and natural sciences. While in recent years the control of neural populations with comprehensive biophysical models or simplified models, e.g., phase models, has seen notable advances, learning appropriate controls directly from data without any model assumptions remains a challenging and less developed area of research. In this paper, we address this problem by leveraging the network's local dynamics to iteratively learn an appropriate control without constructing a global model of the system. The proposed technique can effectively regulate synchrony in a neuronal network using only one input and one noisy population-level output measurement. We provide a theoretical analysis of our approach and illustrate its robustness to system variations and its generalizability to accommodate various physical constraints, such as charge-balanced inputs.

Moment-Based Reinforcement Learning for Ensemble Control.

Problems involving controlling the collective behavior of a population of structurally similar dynamical systems, the so-called ensemble control, arise in diverse emerging applications and pose a grand challenge in systems science and control engineering. Owing to the severely under-actuated nature and the difficulty of placing large-scale sensor networks, ensemble systems are limited to being actuated and monitored at the population level. Moreover, mathematical models describing the dynamics of ensemble systems are often elusive. Therefore, it is essential to design broadcast controls that excite the entire population in such a way that the heterogeneity in system dynamics is robustly compensated. In this article, we propose a reinforcement learning (RL)-based data-driven control framework incorporating population-level aggregated measurement data to learn a global control signal for steering a dynamic population in the desired manner. In particular, we introduce the notion of ensemble moments induced by aggregated measurements and derive the associated moment system to the original ensemble system. Then, using the moment system, we learn an approximation of optimal value functions and the associated policies in terms of ensemble moments through RL. We illustrate the feasibility and scalability of the proposed moment-based approach via numerical experiments using a population of linear, bilinear, and nonlinear dynamic ensemble systems. We report that the proposed method achieves the desired control objectives of various ensemble control tasks and obtains significantly better averaged-reward when compared with three existing methods.

Data-Efficient Inference of Nonlinear Oscillator Networks.

Decoding the connectivity structure of a network of nonlinear oscillators from measurement data is a difficult yet essential task for understanding and controlling network functionality. Several data-driven network inference algorithms have been presented, but the commonly considered premise of ample measurement data is often difficult to satisfy in practice. In this paper, we propose a data-efficient network inference technique by combining correlation statistics with the model-fitting procedure. The proposed approach can identify the network structure reliably in the case of limited measurement data. We compare the proposed method with existing techniques on a network of Stuart-Landau oscillators, oscillators describing circadian gene expression, and noisy experimental data obtained from Rössler Electronic Oscillator network.

A moment kernel machine for clinical data mining to inform medical decision making.

Machine learning-aided medical decision making presents three major challenges: achieving model parsimony, ensuring credible predictions, and providing real-time recommendations with high computational efficiency. In this paper, we formulate medical decision making as a classification problem and develop a moment kernel machine (MKM) to tackle these challenges. The main idea of our approach is to treat the clinical data of each patient as a probability distribution and leverage moment representations of these distributions to build the MKM, which transforms the high-dimensional clinical data to low-dimensional representations while retaining essential information. We then apply this machine to various pre-surgical clinical datasets to predict surgical outcomes and inform medical decision making, which requires significantly less computational power and time for classification while yielding favorable performance compared to existing methods. Moreover, we utilize synthetic datasets to demonstrate that the developed moment-based data mining framework is robust to noise and missing data, and achieves model parsimony giving an efficient way to generate satisfactory predictions to aid personalized medical decision making.

Optimal control of gene mutation in DNA replication.

We propose a molecular-level control system view of the gene mutations in DNA replication from the finite field concept. By treating DNA sequences as state variables, chemical mutagens and radiation as control inputs, one cell cycle as a step increment, and the measurements of the resulting DNA sequence as outputs, we derive system equations for both deterministic and stochastic discrete-time, finite-state systems of different scales. Defining the cost function as a summation of the costs of applying mutagens and the off-trajectory penalty, we solve the deterministic and stochastic optimal control problems by dynamic programming algorithm. In addition, given that the system is completely controllable, we find that the global optimum of both base-to-base and codon-to-codon deterministic mutations can always be achieved within a finite number of steps.

Deep multi-modal learning for joint linear representation of nonlinear dynamical systems.

Dynamical systems pervasively seen in most real-life applications are complex and behave by following certain evolution rules or dynamical patterns, which are linear, non-linear, or stochastic. The underlying dynamics (or evolution rule) of such complex systems, if found, can be used for understanding the system behavior, and furthermore for system prediction and control. It is common to analyze the system's dynamics through observations in different modality approaches. For instance, to recognize patient deterioration in acute care, it usually relies on monitoring and analyzing vital signs and other observations, such as blood pressure, heart rate, respiration, and electroencephalography. These observations convey the information describing the same target system, but the dynamics is not able to be directly characterized due to high complexity of individual modality and maybe time-delay interactions among modalities. In this work, we suppose that the state behavior of a dynamical system follows an intrinsic dynamics shared among these modalities. We specifically propose a new deep auto-encoder framework using the Koopman operator theory to derive the joint linear dynamics for a target system in a space spanned by the intrinsic coordinates. The proposed method aims to reconstruct the original system states by learning the information provided among multiple modalities. Furthermore, with the derived intrinsic dynamics, our method is capable of restoring the missing observations within and across modalities, and used for predicting the future states of the system that follows the same evolution rule.