Towards dynamical system models of language-related brain potentials

Event-related brain potentials (ERP) are important neural correlates of cognitive processes. In the domain of language processing, the N400 and P600 reflect lexical-semantic integration and syntactic processing problems, respectively. We suggest an interpretation of these markers in terms of dynamical system theory and present two nonlinear dynamical models for syntactic computations where different processing strategies correspond to functionally different regions in the system's phase space.

Symmetry breaking, mixing, instability, and low frequency variability in a minimal Lorenz-like system

Starting from the classical Saltzman two-dimensional convection equations, we derive via a severe spectral truncation a minimal 10 ODE system which includes the thermal effect of viscous dissipation. Neglecting this process leads to a dynamical system which includes a decoupled generalized Lorenz system. The consideration of this process breaks an important symmetry and couples the dynamics of fast and slow variables, with the ensuing modifications to the structural properties of the attractor and of the spectral features. When the relevant nondimensional number (Eckert number Ec) is different from zero, an additional time scale of O(Ec−1) is introduced in the system, as shown with standard multiscale analysis and made clear by several numerical evidences. Moreover, the system is ergodic and hyperbolic, the slow variables feature long-term memory with 1/f3/2 power spectra, and the fast variables feature amplitude modulation. Increasing the strength of the thermal-viscous feedback has a stabilizing effect, as both the metric entropy and the Kaplan-Yorke attractor dimension decrease monotonically with Ec. The analyzed system features very rich dynamics: it overcomes some of the limitations of the Lorenz system and might have prototypical value in relevant processes in complex systems dynamics, such as the interaction between slow and fast variables, the presence of long-term memory, and the associated extreme value statistics. This analysis shows how neglecting the coupling of slow and fast variables only on the basis of scale analysis can be catastrophic. In fact, this leads to spurious invariances that affect essential dynamical properties (ergodicity, hyperbolicity) and that cause the model losing ability in describing intrinsically multiscale processes.

Stochastic perturbations to dynamical systems: a response theory approach

Using the formalism of the Ruelle response theory, we study how the invariant measure of an Axiom A dynamical system changes as a result of adding noise, and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics. We first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the additive and in the multiplicative case. We also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency-dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing. This provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings. Using Kramers-Kronig theory, it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the Green function of the system for any desired observable. We then extend our results to rather general patterns of random forcing, from the case of several white noise forcings, to noise terms with memory, up to the case of a space-time random field. Explicit formulas are provided for each relevant case analysed. As a general result, we find, using an argument of positive-definiteness, that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system. We provide an example of application of our results by considering the spatially extended chaotic Lorenz 96 model. These results clarify the property of stochastic stability of SRB measures in Axiom A flows, provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies, and introduce new ways to study the response of a system to external perturbations. Taking into account the chaotic hypothesis, we expect that our results have practical relevance for a more general class of system than those belonging to Axiom A.

A general method for finding extremal states of Hamiltonian dynamical systems, with applications to perfect fluids

In addition to the Hamiltonian functional itself, non-canonical Hamiltonian dynamical systems generally possess integral invariants known as ‘Casimir functionals’. In the case of the Euler equations for a perfect fluid, the Casimir functionals correspond to the vortex topology, whose invariance derives from the particle-relabelling symmetry of the underlying Lagrangian equations of motion. In a recent paper, Vallis, Carnevale & Young (1989) have presented algorithms for finding steady states of the Euler equations that represent extrema of energy subject to given vortex topology, and are therefore stable. The purpose of this note is to point out a very general method for modifying any Hamiltonian dynamical system into an algorithm that is analogous to those of Vallis etal. in that it will systematically increase or decrease the energy of the system while preserving all of the Casimir invariants. By incorporating momentum into the extremization procedure, the algorithm is able to find steadily-translating as well as steady stable states. The method is applied to a variety of perfect-fluid systems, including Euler flow as well as compressible and incompressible stratified flow.

Towards a general theory of extremes for observables of chaotic dynamical systems

In this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the cho- sen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan–Yorke dimension of the attractor. Preliminary numer- ical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.

A joint state and parameter estimation scheme for nonlinear dynamical systems

We present a novel algorithm for concurrent model state and parameter estimation in nonlinear dynamical systems. The new scheme uses ideas from three dimensional variational data assimilation (3D-Var) and the extended Kalman filter (EKF) together with the technique of state augmentation to estimate uncertain model parameters alongside the model state variables in a sequential filtering system. The method is relatively simple to implement and computationally inexpensive to run for large systems with relatively few parameters. We demonstrate the efficacy of the method via a series of identical twin experiments with three simple dynamical system models. The scheme is able to recover the parameter values to a good level of accuracy, even when observational data are noisy. We expect this new technique to be easily transferable to much larger models.

The counter-propagating Rossby-wave perspective on baroclinic instability. I: Mathematical basis

It is shown that Bretherton's view of baroclinic instability as the interaction of two counter-propagating Rossby waves (CRWs) can be extended to a general zonal flow and to a general dynamical system based on material conservation of potential vorticity (PV). The two CRWs have zero tilt with both altitude and latitude and are constructed from a pair of growing and decaying normal modes. One CRW has generally large amplitude in regions of positive meridional PV gradient and propagates westwards relative to the flow in such regions. Conversely, the other CRW has large amplitude in regions of negative PV gradient and propagates eastward relative to the zonal flow there. Two methods of construction are described. In the first, more heuristic, method a ‘home-base’ is chosen for each CRW and the other CRW is defined to have zero PV there. Consideration of the PV equation at the two home-bases gives ‘CRW equations’ quantifying the evolution of the amplitudes and phases of both CRWs. They involve only three coefficients describing the mutual interaction of the waves and their self-propagation speeds. These coefficients relate to PV anomalies formed by meridional fluid displacements and the wind induced by these anomalies at the home-bases. In the second method, the CRWs are defined by orthogonality constraints with respect to wave activity and energy growth, avoiding the subjective choice of home-bases. Using these constraints, the same form of CRW equations are obtained from global integrals of the PV equation, but the three coefficients are global integrals that are not so readily described by ‘PV-thinking’ arguments. Each CRW could not continue to exist alone, but together they can describe the time development of any flow whose initial conditions can be described by the pair of growing and decaying normal modes, including the possibility of a super-modal growth rate for a short period. A phase-locking configuration (and normal-mode growth) is possible only if the PV gradient takes opposite signs and the mean zonal wind and the PV gradient are positively correlated in the two distinct regions where the wave activity of each CRW is concentrated. These are easily interpreted local versions of the integral conditions for instability given by Charney and Stern and by Fjørtoft.

The counter-propagating Rossby-wave perspective on baroclinic instability. Part III: Primitive-equation disturbances on the sphere

Baroclinic instability of perturbations described by the linearized primitive quations, growing on steady zonal jets on the sphere, can be understood in terms of the interaction of pairs of counter-propagating Rossby waves (CRWs). The CRWs can be viewed as the basic components of the dynamical system where the Hamiltonian is the pseudoenergy and each CRW has a zonal coordinate and pseudomomentum. The theory holds for adiabatic frictionless flow to the extent that truncated forms of pseudomomentum and pseudoenergy are globally conserved. These forms focus attention on Rossby wave activity. Normal mode (NM) dispersion relations for realistic jets are explained in terms of the two CRWs associated with each unstable NM pair. Although derived from the NMs, CRWs have the conceptual advantage that their structure is zonally untilted, and can be anticipated given only the basic state. Moreover, their zonal propagation, phase-locking and mutual interaction can all be understood by ‘PV-thinking’ applied at only two ‘home-bases’—potential vorticity (PV) anomalies at one home-base induce circulation anomalies, both locally and at the other home-base, which in turn can advect the PV gradient and modify PV anomalies there. At short wavelengths the upper CRW is focused in the mid-troposphere just above the steering level of the NM, but at longer wavelengths the upper CRW has a second wave-activity maximum at the tropopause. In the absence of meridional shear, CRW behaviour is very similar to that of Charney modes, while shear results in a meridional slant with height of the air-parcel displacement-structures of CRWs in sympathy with basic-state zonal angular-velocity surfaces. A consequence of this slant is that baroclinically growing eddies (on jets broader than the Rossby radius) must tilt downshear in the horizontal, giving rise to up-gradient momentum fluxes that tend to accelerate the barotropic component of the jet.

Decadal predictability of the Atlantic Ocean in a coupled GCM: forecast skill and optimal perturbations using Linear Inverse Modelling

The decadal predictability of three-dimensional Atlantic Ocean anomalies is examined in a coupled global climate model (HadCM3) using a Linear Inverse Modelling (LIM) approach. It is found that the evolution of temperature and salinity in the Atlantic, and the strength of the meridional overturning circulation (MOC), can be effectively described by a linear dynamical system forced by white noise. The forecasts produced using this linear model are more skillful than other reference forecasts for several decades. Furthermore, significant non-normal amplification is found under several different norms. The regions from which this growth occurs are found to be fairly shallow and located in the far North Atlantic. Initially, anomalies in the Nordic Seas impact the MOC, and the anomalies then grow to fill the entire Atlantic basin, especially at depth, over one to three decades. It is found that the structure of the optimal initial condition for amplification is sensitive to the norm employed, but the initial growth seems to be dominated by MOC-related basin scale changes, irrespective of the choice of norm. The consistent identification of the far North Atlantic as the most sensitive region for small perturbations suggests that additional observations in this region would be optimal for constraining decadal climate predictions.

Projected gradient approach to the numerical solution of the SCoTLASS

The SCoTLASS problem-principal component analysis modified so that the components satisfy the Least Absolute Shrinkage and Selection Operator (LASSO) constraint-is reformulated as a dynamical system on the unit sphere. The LASSO inequality constraint is tackled by exterior penalty function. A globally convergent algorithm is developed based on the projected gradient approach. The algorithm is illustrated numerically and discussed on a well-known data set. (c) 2004 Elsevier B.V. All rights reserved.

Unifying syntactic theory and sentence processing difficulty through a connectionist minimalist parser

Syntactic theory provides a rich array of representational assumptions about linguistic knowledge and processes. Such detailed and independently motivated constraints on grammatical knowledge ought to play a role in sentence comprehension. However most grammar-based explanations of processing difficulty in the literature have attempted to use grammatical representations and processes per se to explain processing difficulty. They did not take into account that the description of higher cognition in mind and brain encompasses two levels: on the one hand, at the macrolevel, symbolic computation is performed, and on the other hand, at the microlevel, computation is achieved through processes within a dynamical system. One critical question is therefore how linguistic theory and dynamical systems can be unified to provide an explanation for processing effects. Here, we present such a unification for a particular account to syntactic theory: namely a parser for Stabler's Minimalist Grammars, in the framework of Smolensky's Integrated Connectionist/Symbolic architectures. In simulations we demonstrate that the connectionist minimalist parser produces predictions which mirror global empirical findings from psycholinguistic research.

Inverse problems in dynamic cognitive modeling

Inverse problems for dynamical system models of cognitive processes comprise the determination of synaptic weight matrices or kernel functions for neural networks or neural/dynamic field models, respectively. We introduce dynamic cognitive modeling as a three tier top-down approach where cognitive processes are first described as algorithms that operate on complex symbolic data structures. Second, symbolic expressions and operations are represented by states and transformations in abstract vector spaces. Third, prescribed trajectories through representation space are implemented in neurodynamical systems. We discuss the Amari equation for a neural/dynamic field theory as a special case and show that the kernel construction problem is particularly ill-posed. We suggest a Tikhonov-Hebbian learning method as regularization technique and demonstrate its validity and robustness for basic examples of cognitive computations.

Enhancing dominant modes in nonstationary time series by means of the symbolic resonance analysis

We present the symbolic resonance analysis (SRA) as a viable method for addressing the problem of enhancing a weakly dominant mode in a mixture of impulse responses obtained from a nonlinear dynamical system. We demonstrate this using results from a numerical simulation with Duffing oscillators in different domains of their parameter space, and by analyzing event-related brain potentials (ERPs) from a language processing experiment in German as a representative application. In this paradigm, the averaged ERPs exhibit an N400 followed by a sentence final negativity. Contemporary sentence processing models predict a late positivity (P600) as well. We show that the SRA is able to unveil the P600 evoked by the critical stimuli as a weakly dominant mode from the covering sentence final negativity. (c) 2007 American Institute of Physics. (c) 2007 American Institute of Physics.

Finite departure from convective quasi-equilibrium: periodic cycle and discharge-recharge mechanism

A simple self–contained theory is proposed for describing life cycles of convective systems as a discharge–recharge process. A closed description is derived for the dynamics of an ensemble of convective plumes based on an energy cycle. The system consists of prognostic equations for the cloud work function and the convective kinetic energy. The system can be closed by intro ducing a functional relationship between the convective kinetic energy and the cloud–base mass flux. The behaviour of this system is considered under a bulk simplification. Previous cloud–resolving mo delling as well as bulk statistical theories for ensemble convective systems suggest that a plausible relationship would be to assume that the convective kinetic energy is linearly proportional to the cloud–base mass flux. As a result, the system reduces to a nonlinear dynamical system with two dependent variables, the cloud–base mass flux and the cloud work function. The fully nonlinear solution of this system always represents a periodic cycle regardless of the initial condition under constant large–scale forcing. Importantly, the inclusion of energy dissipation in this model does not in itself lead the system to an equilibrium.

State estimation using model order reduction for unstable systems

The problem of state estimation occurs in many applications of fluid flow. For example, to produce a reliable weather forecast it is essential to find the best possible estimate of the true state of the atmosphere. To find this best estimate a nonlinear least squares problem has to be solved subject to dynamical system constraints. Usually this is solved iteratively by an approximate Gauss–Newton method where the underlying discrete linear system is in general unstable. In this paper we propose a new method for deriving low order approximations to the problem based on a recently developed model reduction method for unstable systems. To illustrate the theoretical results, numerical experiments are performed using a two-dimensional Eady model – a simple model of baroclinic instability, which is the dominant mechanism for the growth of storms at mid-latitudes. It is a suitable test model to show the benefit that may be obtained by using model reduction techniques to approximate unstable systems within the state estimation problem.