Explicit Separation Of Time Scales Research Articles

Idealized multiple-timescale model of cortical spreading depolarization initiation and pre-epileptic hyperexcitability caused by NaV1.1/SCN1A mutations.

NaV1.1 (SCN1A) is a voltage-gated sodium channel mainly expressed in GABAergic neurons. Loss of function mutations of NaV1.1 lead to epileptic disorders, while gain of function mutations cause a migraine in which cortical spreading depolarizations (CSDs) are involved. It is still debated how these opposite effects initiate two different manifestations of neuronal hyperactivity: epileptic seizures and CSD. To investigate this question, we previously built a conductance-based model of two neurons (GABAergic and pyramidal), with dynamic ion concentrations (Lemaire et al. in PLoS Comput Biol 17(7):e1009239, 2021. https://doi.org/10.1371/journal.pcbi.1009239 ). When implementing either NaV1.1 migraine or epileptogenic mutations, ion concentration modifications acted as slow processes driving the system to the corresponding pathological firing regime. However, the large dimensionality of the model complicated the exploitation of its implicit multi-timescale structure. Here, we substantially simplify our biophysical model to a minimal version more suitable for bifurcation analysis. The explicit timescale separation allows us to apply slow-fast theory, where slow variables are treated as parameters in the fast singular limit. In this setting, we reproduce both pathological transitions as dynamic bifurcations in the full system. In the epilepsy condition, we shift the spike-terminating bifurcation to lower inputs for the GABAergic neuron, to model an increased susceptibility to depolarization block. The resulting failure of synaptic inhibition triggers hyperactivity of the pyramidal neuron. In the migraine scenario, spiking-induced release of potassium leads to the abrupt increase of the extracellular potassium concentration. This causes a dynamic spike-terminating bifurcation of both neurons, which we interpret as CSD initiation.

On Differential Geometric Formulations of Slow Invariant Manifold Computation: Geodesic Stretching and Flow Curvature

The theory of slow invariant manifolds (SIMs) is the foundation of various model-order reduction techniques for dissipative dynamical systems with multiple time-scales, e.g. in chemical kinetic models. The construction of SIMs and many approximation methods exploit the restrictive requirement of an explicit time-scale separation parameter. Most of those methods are also not formulated covariantly, i.e. in terms of tensorial constructions. We propose an intrinsically coordinate-free differential geometric approximation criterion approximating normally attracting invariant manifolds (NAIMs). We translate some ideas behind existing approximation approaches, the stretching based diagnostics (SBD) and the flow curvature method (FCM) to tensors of Riemannian geometry, specifically to spacetime curvature in extended phase space. For that purpose we derive from flow-generating smooth vector fields a metric tensor such that the original dynamical system is a geodesic flow on a Riemannian manifold. We apply the resulting method to test models.

Data-based reduced-order modeling of nonlinear two-time-scale processes

This work focuses on data-based reduced-order modeling of nonlinear processes that exhibit time-scale multiplicity. Using time-series data from all the state variables of a nonlinear process, an approach that involves nonlinear principal component analysis and neural network function approximators is employed to identify the fast and slow process state variables as well as compute a nonlinear slow manifold function approximation in which the fast variables are “slaved” in terms of the slow variables. Subsequently, a nonlinear sparse identification approach is employed to calculate a dynamic model of nonlinear first-order ordinary differential equations that describe the temporal evolution of the slow process states. The method is applied to two chemical process examples, and the advantages of the proposed method over using only sparse identification for the original two-time-scale process are discussed. The approach can be thought of as a data-based analogue of the classical singular perturbation modeling of two-time-scale processes where explicit time-scale separation is assumed (and expressed in terms of a small positive parameter ε multiplying the time derivative of the fast states), and the reduced-order slow subsystem is calculated analytically.

Analytical Description of Optimally Time-Dependent Modes for Reduced-Order Modeling of Transient Instabilities

The optimally time-dependent (OTD) modes form a time-evolving orthonormal basis that captures directions in phase space associated with transient and persistent instabilities. In the original formulation, the OTD modes are described by a set of coupled evolution equations that need to be solved along the trajectory of the system. For many applications where real-time estimation of the OTD modes is important, such as control or filtering, this is an expensive task. Here, we examine the low-dimensional structure of the OTD modes. In particular, we consider the case of slow-fast systems, and prove that OTD modes rapidly converge to a slow manifold, for which we derive an asymptotic expansion. The result is a parametric description of the OTD modes in terms of the system state in phase space. The analytical approximation of the OTD modes allows for their offline computation, making the whole framework suitable for real-time applications. In addition, we examine the accuracy of the slow-manifold approximation for systems in which there is no explicit time-scale separation. In this case, we show numerically that the asymptotic expansion of the OTD modes is still valid for regions of the phase space where strongly transient behavior is observed, and for which there is an implicit scale separation. We also find an analogy between the OTD modes and the Gram--Schmidt vectors (also known as orthogonal or backward Lyapunov vectors), and thereby establish new properties of the former. Several examples of low-dimensional systems are provided to illustrate the analytical formulation.

Relaxation oscillations and canards in the Jirsa\u2013Kelso excitator model: global flow perspective

Fenichel’s geometric singular perturbation theory and the blow-up method have been very successful in describing and explaining global non-linear phenomena in systems with multiple time-scales, such as relaxation oscillations and canards. Recently, the blow-up method has been extended to systems with flat, unbounded slow manifolds that lose normal hyperbolicity at infinity. Here, we show that transition between discrete and periodic movement captured by the Jirsa–Kelso excitator is a new example of such phenomena. We, first, derive equations of the Jirsa–Kelso excitator with explicit time scale separation and demonstrate existence of canards in the systems. Then, we combine the slow-fast analysis, blow-up method and projection onto the Poincaré sphere to understand the return mechanism of the periodic orbits in the singular case, ϵ = 0.

Ultrafast Reactive Quantum Dynamics Coupled to Classical Solvent Dynamics Using an Ehrenfest Approach.

The inclusion of solvent effects in the theoretical analysis of molecular processes becomes increasingly important. Currently, it is not feasible to directly include the solvent on the quantum level. We use an Ehrenfest approach to study the coupled time evolution of quantum dynamically treated solutes and classical solvents system. The classical dynamics of the solvent is coupled to the wavepacket dynamics of the solute and rotational and translational degrees of freedom of the solute are included classically. This allows quantum dynamics simulations for ultrafast processes that are decided by environment interactions without explicit separation of time scales. We show the application to the dissociation of ICN in liquid Ar as a proof of principal system and to the more applied example of uracil in water.

Acoustic streaming: an arbitrary Lagrangian-Eulerian perspective.

We analyse acoustic streaming flows using an arbitrary Lagrangian Eulerian (ALE) perspective. The formulation stems from an explicit separation of time scales resulting in two subproblems: a first-order problem, formulated in terms of the fluid displacement at the fast scale, and a second-order problem, formulated in terms of the Lagrangian flow velocity at the slow time scale. Following a rigorous time-averaging procedure, the second-order problem is shown to be intrinsically steady, and with exact boundary conditions at the oscillating walls. Also, as the second-order problem is solved directly for the Lagrangian velocity, the formulation does not need to employ the notion of Stokes drift, or any associated post-processing, thus facilitating a direct comparison with experiments. Because the first-order problem is formulated in terms of the displacement field, our formulation is directly applicable to more complex fluid-structure interaction problems in microacoustofluidic devices. After the formulation's exposition, we present numerical results that illustrate the advantages of the formulation with respect to current approaches.

Extending the zero-derivative principle for slow–fast dynamical systems

Slow–fast systems often possess slow manifolds, that is invariant or locally invariant sub-manifolds on which the dynamics evolves on the slow time scale. For systems with explicit timescale separation, the existence of slow manifolds is due to Fenichel theory, and asymptotic expansions of such manifolds are easily obtained. In this paper, we discuss methods of approximating slow manifolds using the so-called zero-derivative principle. We demonstrate several test functions that work for systems with explicit time scale separation including ones that can be generalized to systems without explicit timescale separation. We also discuss the possible spurious solutions, known as ghosts, as well as treat the Templator system as an example.

Identification of Criticality in Neuronal Avalanches: II. A Theoretical and Empirical Investigation of the Driven Case

The observation of apparent power laws in neuronal systems has led to the suggestion that the brain is at, or close to, a critical state and may be a self-organised critical system. Within the framework of self-organised criticality a separation of timescales is thought to be crucial for the observation of power-law dynamics and computational models are often constructed with this property. However, this is not necessarily a characteristic of physiological neural networks—external input does not only occur when the network is at rest/a steady state. In this paper we study a simple neuronal network model driven by a continuous external input (i.e. the model does not have an explicit separation of timescales from seeding the system only when in the quiescent state) and analytically tuned to operate in the region of a critical state (it reaches the critical regime exactly in the absence of input—the case studied in the companion paper to this article). The system displays avalanche dynamics in the form of cascades of neuronal firing separated by periods of silence. We observe partial scale-free behaviour in the distribution of avalanche size for low levels of external input. We analytically derive the distributions of waiting times and investigate their temporal behaviour in relation to different levels of external input, showing that the system’s dynamics can exhibit partial long-range temporal correlations. We further show that as the system approaches the critical state by two alternative ‘routes’, different markers of criticality (partial scale-free behaviour and long-range temporal correlations) are displayed. This suggests that signatures of criticality exhibited by a particular system in close proximity to a critical state are dependent on the region in parameter space at which the system (currently) resides.

Approximate linear response for slow variables of dynamics with explicit time scale separation

Many real-world numerical models are notorious for the time scale separation of different subsets of variables and the inclusion of random processes. The existing algorithms of linear response to external forcing are vulnerable to the time scale separation due to increased response errors at fast scales. Here we develop the approximate linear response algorithm for slow variables in a two-scale dynamical system with explicit separation of slow and fast variables, which has improved numerical stability and reduced computational expense.

Coarse-Graining Autonomous ODE Systems by Inducing a Separation of Scales: Practical Strategies and Mathematical Questions

This short note introduces a methodology for sequential multiscale modeling of autonomous, “microscopic”, systems of Ordinary Differential Equations through a redefinition of the original dynamics as an augmented system with an explicit separation of time scales arising merely from the definition of time-averaging. Associated mathematical questions are stated and discussed.

Discussion on: “Backstepping Controller Design Using Neural Networks for a Fighter Aircraft”

The paper by Sadati et al. presents a neural network(NN) based adaptive backstepping design for flightcontrol systems of fighter aircraft in high angle-of-attack (AoA) and/or high angular rate regimes. Theflight control system is designed for command fol-lowingoftheAoA,sideslipangle,androllangleaboutthe velocity vector. The design is an adaptive statefeedback design in which the NN is trained online tocompensate for uncertainties due to parametric var-iations in the aerodynamic coefficients and due to asudden change in aircraft dynamics in the event offailure or battle damage. The NN update laws arederived using Lyapunov-like stability analysis toprove uniform ultimate boundedness of the closed-loop system errors. The design of flight control sys-tems that are capable of delivering high performancein such regimes is very challenging due to the highlynonlinear and unsteady phenomena associated withhigh AoA flight and the modeling uncertainty. Thenonlinear adaptive control design provided in thepaper by Sadati et al. is thus an important contribu-tion to this problem. We first provide some commentsthat could have been included in the paper to clarifythe contributions and proceed to describe furtherdirections for research.The authors have commented in the introductionon the differences between feedback linearization andbackstepping approaches in nonlinear control design,and their motivation for choosing the backsteppingapproach. In our opinion, the authors should havementioned that the most significant reason why abackstepping design could be preferred over a feed-backlinearizationdesign isthat abacksteppingdesigncan account for unmatched uncertainties. However inproblems where both approaches are applicable, webelieve that backstepping designs can be more com-plex than feedback linearization designs. This can beseen in problems where there are multiple loops to bestabilized. Backstepping designs require choosinggains for stabilizing each individual loop and in gen-eral for the inner-loops the design could be high-gain.In addition, time derivatives of the intermediate con-trol commands appear in each step of the back-stepping procedure[e.g.,Eqs. (26),(29)and(30)ofthepaper]. These time derivatives cannot be directlycomputed when the plant model is unknown andapproximate techniques have to be developed.The authors have mentioned that one motivationfor pursuing the backstepping approach in flightcontrol systems design is that explicit time-scaleseparation assumptions are not required. While this istrue, we would like to bring to the authors’ attentionan adaptive NN-augmented feedback linearizationapproach in an almost identical problem [1]. Theapproachin[1]alsodoesnotrelyonexplicittime-scaleseparation assumption. The authors’ can carry out acomparison study of their approach with that in [1]that will illustrate the relative merits of the respectiveapproaches.

Singular perturbations and time scales in nonlinear models of power systems

Recent results on linear slow coherency and aggregation are extended to nonlinear electromechanical models of power systems and similar dynamic networks. Conservation and equilibrium properties are used for the derivation of nonlinear singular perturbation models with explicit separation of time scales.