Dynamical Invariants Research Articles

Automatized offline and online exploration to achieve a target dynamics in biohybrid neural circuits built with living and model neurons

Biohybrid circuits of interacting living and model neurons are an advantageous means to study neural dynamics and to assess the role of specific neuron and network properties in the nervous system. Hybrid networks are also a necessary step to build effective artificial intelligence and brain hybridization. In this work, we deal with the automatized online and offline adaptation, exploration and parameter mapping to achieve a target dynamics in hybrid circuits and, in particular, those that yield dynamical invariants between living and model neurons. We address dynamical invariants that form robust cycle-by-cycle relationships between the intervals that build neural sequences from such interaction. Our methodology first attains automated adaptation of model neurons to work in the same amplitude regime and time scale of living neurons. Then, we address the automatized exploration and mapping of the synapse parameter space that lead to a specific dynamical invariant target. Our approach uses multiple configurations and parallel computing from electrophysiological recordings of living neurons to build full mappings, and genetic algorithms to achieve an instance of the target dynamics for the hybrid circuit in a short time. We illustrate and validate such strategy in the context of the study of functional sequences in neural rhythms, which can be easily generalized for any variety of hybrid circuit configuration. This approach facilitates both the building of hybrid circuits and the accomplishment of their scientific goal.

Reduced Euler-Lagrange Equations of Floating-Base Robots: Computation, Properties, & Applications

At first glance, a floating-base robotic system is a kinematic chain, and its equations of motion are described by the inertia-coupled dynamics of its shape and movable base. However, the dynamics embody an additional structure due to the momentum evolution, which acts as a velocity constraint. In prior works of robot dynamics, matrix transformations of the dynamics revealed a block-diagonal inertia. However, the structure of the transformed matrix of Coriolis/Centrifugal (CC) terms was not examined, and is the primary contribution of this article. To this end, we simplify the CC terms from robot dynamics and derive the analogous terms from geometric mechanics. Using this interdisciplinary link, we derive a two-part structure of the CC matrix, in which each partition is iteratively computed using a self-evident velocity dependency. Through this CC matrix, we reveal a commutative property, the velocity dependencies of the skew-symmetry property, the invariance of the shape dynamics to the basis of momentum, and the curvature as a matrix operator. Finally, we show the application of the proposed CC matrix structure through controller design and locomotion analysis.

Three invariants of strange attractors derived through hypergeometric entropy

A new description of strange attractor systems through three geometrical and dynamical invariants is provided. They are the correlation dimension (D) and the correlation entropy (K), both having attracted attention over the past decades, and a new invariant called the correlation concentration (A) introduced in the present study. The correlation concentration is defined as the normalised mean distance between the reconstruction vectors, evaluated by the underlying probability measure on the infinite-dimensional embedding space. These three invariants determine the scaling behaviour of the system’s Rényi-type extended entropy, modelled by Kummer’s confluent hypergeometric function, with respect to the gauge parameter (ρ) coupled to the distance between the reconstruction vectors. The entropy function reproduces the known scaling behaviours of D and K in the ‘microscopic’ limit ρ→∞ while exhibiting a new scaling behaviour of A in the other, ‘macroscopic’ limit ρ→0. The three invariants are estimated simultaneously via nonlinear regression analysis without needing separate estimations for each invariant. The proposed method is verified through simulations in both discrete and continuous systems.

Vertical motion in the Galactic disc: unwinding the snail

ABSTRACTThe distribution of stars in the Milky Way disc shows a spiral structure − the snail − in the space of velocity and position normal to the Galactic mid-plane. The snail appears as straight lines in the vertical frequency−vertical phase plane when effects from sample selection are removed. Their slope has the dimension of inverse time, with the simplest interpretation being the inverse age of the snail. Here, we devise and fit a simple model in which the spiral starts as a lopsided perturbation from steady state, that winds up into the present-day morphology. The winding occurs because the vertical frequency decreases with vertical action. We use data from stars in Gaia EDR3 that have measured radial velocities, pruned by simple distance and photometric selection functions. We divide the data into boxels of dynamical invariants (radial action, angular momentum); our model fits the data well in many of the boxels. The model parameters have physical interpretations: one, A, is a perturbation amplitude, and one, t, is interpretable in the simplest models as the time since the event that caused the snail. We find trends relating the strength and age to angular momentum: (i) the amplitude A is small at low angular momentum (${\lt 1\, 600{\,}kpc\ km\ s}^{-1}$ or guiding-centre radius $\lt 7.3\,$kpc) and over a factor of three larger, with strong variations, in the outer disc; (ii) there is no single well-defined perturbation time, with t varying between 0.2 and 0.6 Gyr. Residuals between the data and the model display systematic trends, implying that the data call for more complex models.

Dynamic Transformation of Domain Walls in Chiral Ferrimagnets

The dynamics of domain walls in ferrimagnets in which spatial dynamics invariance is violated because of the presence of the chiral Dzyaloshinskii–Moriya interaction with energy linear in sublattice spin density gradients is investigated theoretically. Analysis is performed based on numerical integration of equations in the sigma model generalized to the case of a ferrimagnet near the sublattice spin compensation point. It is shown that in contrast to conventional or chiral ferromagnets, chiral ferrimagnets can exhibit effects of dynamic transformation of the domain wall structure with the formation of more complex walls with a nonmonotonic behavior of the spin density in a wall upon an increase in the wall velocity. These effects are possible in a quite narrow neighborhood of the compensation point, and the width of this region increases upon an increase in the Dzyaloshinskii–Moriya interaction constant.

Quantum control of tunable-coupling transmons using dynamical invariants of motion

We analyze the implementation of a fast nonadiabatic CZ gate between two transmon qubits with tunable coupling. The gate control method is based on a theory of dynamical invariants which leads to reduced leakage and robustness against decoherence. The gate is based on a description of the resonance between the and using an effective Hamiltonian with the six lowest energy states. A modification of the protocol allows us to take into account the higher-order perturbative corrections of this effective model. This enables a gate fidelity several orders of magnitude higher than other quasiadiabatic protocols, with gate times that approach the theoretical limit.

Dynamical-invariant-based holonomic quantum gates: Theory and experiment

Among existing approaches to holonomic quantum computing, the adiabatic holonomic quantum gates (HQGs) suffer errors due to decoherence, while the non-adiabatic HQGs either require additional Hilbert spaces or are difficult to scale. Here, we report a systematic, scalable approach based on dynamical invariants to realize HQGs without using additional Hilbert spaces. While presenting the theoretical framework of our approach, we design and experimentally evaluate single-qubit and two-qubits HQGs for the nuclear magnetic resonance system. The single-qubit gates acquire average fidelity 0.9972 by randomized benchmarking, and the controlled-NOT gate acquires fidelity 0.9782 by quantum process tomography. Our approach is also platform-independent, and thus may open a way to large-scale holonomic quantum computation.

Controlled nonautonomous matter–wave solitons in spinor Bose–Einstein condensates with spatiotemporal modulation

To study controlled evolution of nonautonomous matter–wave solitons in spinor Bose–Einstein condensates with spatiotemporal modulation, we focus on a system of three coupled Gross–Pitaevskii (GP) equations with space–time-dependent external potentials and temporally modulated gain/loss distributions. An integrability condition and a nonisospectral Lax pair for the coupled GP equations are obtained. Using it, we derive an infinite set of dynamical invariants, the first two of which are the mass and momentum. The Darboux transform is used to generate one- and two-soliton solutions. Under the action of different external potentials and gain/loss distributions, various solutions for controlled nonautonomous matter–wave solitons of both ferromagnetic and polar types are obtained, such as self-compressed, snake-like and stepwise solitons, and as well as breathers. In particular, the formation of states resembling rogue waves, under the action of a sign-reversible gain–loss distribution, is demonstrated too. Shape-preserving and changing interactions between two nonautonomous matter–wave solitons and bound states of solitons are addressed too. In this context, spin switching arises in the polar-ferromagnetic interaction. Stability of the nonautonomous matter–wave solitons is verified by means of systematic simulations of their perturbed evolution.

Системный подход в оценке динамических состояний технических объектов: элементы концепции динамических инвариантов

The scientific and methodological base is being developed within the framework of systems theory and system analysis in application to the tasks of assessment, control and formation of dynamic states of technical objects, the design schemes of which are mechanical oscillatory systems with concentrated parameters. In the context of the fundamental problems of the theory of system studies of forced movements under deterministic disturbances there are considered the problems, whose solutions are aimed at developing a methodological approach to assessing the dynamic states of technical objects operating under coherent vibration loads. The purpose of the study is to develop a method for evaluating, controlling and forming dynamic states of the objects under harmonic force loads. It is assumed that mechanical oscillatory systems formed by a solid body performing small steady-state oscillations relative to the position of the static equilibrium can be used as calculation schemes of the technical objects. To construct mathematical models, Lagrange formalism, Laplace integral transformations taking into account zero conditions and structural methods are used, which make it possible to consider mechanical oscillatory systems as dynamically equivalent structural schemes of automatic control systems. The dynamic state of an object is assessed by using a transfer function, which in the physical sense represents the interpretation of a dynamic compliance. The amplitude-frequency characteristics used to assess the dynamic state of the object present a parametric family, for which the first parameter is the coefficient of connectivity of external force disturbances, and the second parameter is the coordinate of the point for which the dynamic accuracy is defined. An infinite family of amplitude-frequency characteristics is regularized due to the introduction of dynamic invariants, which can be considered as generalized dynamic states of mechanical oscillatory systems. To determine the dynamic invariants of a family of amplitude-frequency characteristics, taking into account two parameters, a map of dynamic invariants is constructed, which divides the plane of parameters into disjoint sets using boundaries. A theorem is proved to determine the explicit analytic expressions of these boundaries.

Invariant-Parameterized Exact Evolution Operator for SU(2) Systems with Time-Dependent Hamiltonian.

We report the step-by-step construction of the exact, closed and explicit expression for the evolution operator U(t) of a localized and isolated qubit in an arbitrary time-dependent field, which for concreteness we assume to be a magnetic field. Our approach is based on the existence of two independent dynamical invariants that enter the expression of SU(2) by means of two strictly related time-dependent, real or complex, parameters. The usefulness of our approach is demonstrated by exactly solving the quantum dynamics of a qubit subject to a controllable time-dependent field that can be realized in the laboratory. We further discuss possible applications to any SU(2) model, as well as the applicability of our method to realistic physical scenarios with different symmetry properties.

Dynamic event-triggered practical stabilization of random suspension system based on immersion and invariance

This article investigates the practical stabilization problem of random quarter-car active suspension systems. An adaptive dynamic event-trigger strategy is proposed to stabilize the states of vehicle suspension in response to system uncertainty and controller area network resource constraints. Moreover, the model of random active suspension systems is extended to the general random robot systems; the controller is developed with the aid of a double dynamic surface filter, immersion and invariance (I&I) techniques, and event-triggered mechanisms. The results show that the semi-global stability of error systems is achieved, and there are some improvements in triggering times and adaptive estimation performance under the control framework. Finally, simulation comparison results are provided to prove the advantages of the proposed scheme.

Maps of dynamic invariants in the estimation of modes of motion of mechanical oscillatory systems

Maps of dynamic invariants in the estimation of modes of motion of mechanical oscillatory systems

Exact Solution of a Time-Dependent Quantum Harmonic Oscillator with Two Frequency Jumps via the Lewis-Riesenfeld Dynamical Invariant Method.

Harmonic oscillators with multiple abrupt jumps in their frequencies have been investigated by several authors during the last decades. We investigate the dynamics of a quantum harmonic oscillator with initial frequency ω0, which undergoes a sudden jump to a frequency ω1 and, after a certain time interval, suddenly returns to its initial frequency. Using the Lewis−Riesenfeld method of dynamical invariants, we present expressions for the mean energy value, the mean number of excitations, and the transition probabilities, considering the initial state different from the fundamental. We show that the mean energy of the oscillator, after the jumps, is equal or greater than the one before the jumps, even when ω1<ω0. We also show that, for particular values of the time interval between the jumps, the oscillator returns to the same initial state.

Quantum electrodynamics in the null-plane causal perturbation theory. II.

We develop a complete formulation of quantum gauge invariance in light-front dynamics for interacting theories with massless vector gauge fields in the framework of null-plane causal perturbation theory. We apply the general results to quantum electrodynamics, showing that the so-called "gauge terms" present in the photon commutation distribution when quantized under the null-plane gauge condition have no contribution in the calculation of the physical S-operator matrix elements at any order. We use this result to prove the normalizability of the theory, and to calculate the electron's self-energy at second order.

Non-adiabatic quantum control of quantum dot arrays with fixed exchange using Cartan decomposition.

In semiconductor spin qubits which typically interact through short-range exchange coupling, shuttling of spin is a practical way to generate quantum operations between distant qubits. Although the exchange is often tunable through voltages applied to gate electrodes, its minimal value can be significantly large, which hinders the applicability of existing shuttling protocols to such devices, requiring a different approach. In this work, we extend our previous results for double- and triple-dot systems, and describe a method for implementing spin state transfer in long chains of singly occupied quantum dots in a non-adiabatic manner. We make use of Cartan decomposition to break down the interacting problem into simpler problems in a systematic way, and use dynamical invariants to design smooth non-adiabatic pulses that can be implemented in devices with modest control bandwidth. Finally, we discuss the extensibility of our results to directed shuttling of spin states on two-dimensional lattices of quantum dots with fixed coupling. This article is part of the theme issue 'Shortcuts to adiabaticity: theoretical, experimental and interdisciplinary perspectives'.

A technology of structural mathematical modelling for engineering objects under vibrational loading: Interaction forms and dynamic invariants

We present a system approach to evaluating dynamic forms of interactions between elements of mechanical oscillatory systems, which can be applied as design diagrams of engineering objects used for transport or technology purposes. The approach of structural mathematical modelling was used, where a mechanical oscillatory system (with the finite number of freedom degrees) is compared to the block diagram of a dynamically equivalent automatic control system. The research object was represented by a family of mechanical oscillatory systems with the finite number of freedom degrees under the action of connected force harmonic disturbances. The main research focus was on a set of dynamic states of an engineering object, which are caused by the application of common-phase external force disturbances at various frequencies and characterised by a connection coefficient. For a family of chain mechanical oscillatory systems with two degrees of freedom, a method for interpreting a set of dynamic states in the form of oriented graphs was developed. A method for constructing a population graph of dynamic states based on the frequency response of system transfer functions was developed. Within the framework of the developed interpretation, the graphs of dynamic states can be considered as peculiar invariants that persist on sets of mechanical oscillatory system parameters. Using the methodology of structural mathematical modelling, a concept of dynamic invariants was developed, according to which the totality of dynamic invariants can be used to provide a general evaluation of the variety of dynamic states and forms of dynamic interactions between the elements of mechanical oscillatory systems. The proposed concept of dynamic invariants extends the methodology of structural mathematical modelling in relation to the problems of system analysis for ensuring the safety of engineering objects for transport and technology purposes under the conditions of connected vibrational loading.

Reverse engineering of one-qubit filter functions with dynamical invariants

We derive an integral expression for the filter-transfer function of an arbitrary one-qubit gate through the use of dynamical invariant theory and Hamiltonian reverse engineering. We use this result to define a cost function which can be efficiently optimized to produce one-qubit control pulses that are robust against specified frequency bands of the noise power spectral density. We demonstrate the utility of our result by generating optimal control pulses that are designed to suppress broadband detuning and pulse amplitude noise. We report an order of magnitude improvement in gate fidelity in comparison with known composite pulse sequences. More broadly, we also use the same theoretical framework to prove the robustness of nonadiabatic geometric quantum gates under specific error models and control constraints.

Temporal Dynamics of Neural Responses in Human Visual Cortex.

Neural responses to visual stimuli exhibit complex temporal dynamics, including subadditive temporal summation, response reduction with repeated or sustained stimuli (adaptation), and slower dynamics at low contrast. These phenomena are often studied independently. Here, we demonstrate these phenomena within the same experiment and model the underlying neural computations with a single computational model. We extracted time-varying responses from electrocorticographic recordings from patients presented with stimuli that varied in duration, interstimulus interval (ISI) and contrast. Aggregating data across patients from both sexes yielded 98 electrodes with robust visual responses, covering both earlier (V1-V3) and higher-order (V3a/b, LO, TO, IPS) retinotopic maps. In all regions, the temporal dynamics of neural responses exhibit several nonlinear features. Peak response amplitude saturates with high contrast and longer stimulus durations, the response to a second stimulus is suppressed for short ISIs and recovers for longer ISIs, and response latency decreases with increasing contrast. These features are accurately captured by a computational model composed of a small set of canonical neuronal operations, that is, linear filtering, rectification, exponentiation, and a delayed divisive normalization. We find that an increased normalization term captures both contrast- and adaptation-related response reductions, suggesting potentially shared underlying mechanisms. We additionally demonstrate both changes and invariance in temporal response dynamics between earlier and higher-order visual areas. Together, our results reveal the presence of a wide range of temporal and contrast-dependent neuronal dynamics in the human visual cortex and demonstrate that a simple model captures these dynamics at millisecond resolution.SIGNIFICANCE STATEMENT Sensory inputs and neural responses change continuously over time. It is especially challenging to understand a system that has both dynamic inputs and outputs. Here, we use a computational modeling approach that specifies computations to convert a time-varying input stimulus to a neural response time course, and we use this to predict neural activity measured in the human visual cortex. We show that this computational model predicts a wide variety of complex neural response shapes, which we induced experimentally by manipulating the duration, repetition, and contrast of visual stimuli. By comparing data and model predictions, we uncover systematic properties of temporal dynamics of neural signals, allowing us to better understand how the brain processes dynamic sensory information.

ON SOME NUMERICAL MODEL TO SOLVING DYNAMICAL EQUATIONS OF NONRELATIVISTIC AND RELATIVISTIC BACKWARD-WAVE TUBE

It is developed an effective computational approach to solution the master corresponding system of differential equations, which describe the nonlinear stationary and non-stationary electromagnetic processes in the nonrelativistic and relativistic backward-wave tubes (carcinotrons) with maximal accounting for the different physical factors such as the relativistic effects, effects of dissipation, the presence of a space charge, wave reflections at the ends of the slowing system, stochastic factors by means including the special elements in a whole system etc as well as the detailed investigation of characteristics (dynamical and topological invariants) of dynamics of a carcinotron in automodulation and chaotic regimes with construction the corresponding bi-furcation diagrams. Below in order to further solve the master system of dynamical equations for carcinotron it is presented in brief the realizing numerical scheme, based on the use of the conservative finite-difference schemes of the "predictor-corrector" type and the sweep algorithm.

NON-LINEAR ANALYSIS OF CHAOTIC SELF-OSCILLATIONS IN BACKWARD-WAVE TUBE

The paper presents the results of the analysis and modelling of topological and dynamic invariants for the regimes of chaotic self-oscillations in the backward-wave oscillator, in particular, the analysis of chaotic time series for the amplitude of the output signal, which is the solution of the equations of the non-stationary nonlinear theory for the O-type backward-wave oscillator (without taking into account space charge, relativistic effects, energy losses, etc.). The main attention is paid to the calculation and analysis of the spectrum of Lyapunov exponents based on the Sano-Savada algorithm. Numerical data of the Lyapunov backward-wave oscillator calculated for the time series of the amplitudes of the output signal are given, which definitely indicate the presence of elements of advanced chaos in the dynamics of the system.