Restricted Subspace Research Articles

Role of the Na +-Ca 2+ Exchanger as an Alternative Trigger of CICR in Mammalian Cardiac Myocytes

Ca 2+ influx through the L-type Ca 2+ channels is the primary pathway for triggering the Ca 2+ release from the sarcoplasmic reticulum (SR). However, several observations have shown that Ca 2+ influx via the reverse mode of the Na +-Ca 2+ exchanger current ( I Na-Ca) could also trigger the Ca 2+ release. The aim of the present study was to quantitate the role of this alternative pathway of Ca 2+ influx using a mathematical model. In our model 20% of the fast sodium channels and the Na +-Ca 2+ exchanger molecules are located in the restricted subspace between the sarcolemma and the SR where triggering of the calcium-induced calcium release (CICR) takes place. After determining the strengths of the alternative triggers with simulated voltage-clamps in varied membrane voltages and resting [Na] i values, we studied the CICR in simulated action potentials, where fast sodium channel current contributes [Na] i of the subspace. In low initial [Na] i the Ca 2+ influx via the L-type Ca 2+ channels is the major trigger for Ca 2+ release from the SR, and the Ca 2+ influx via the reverse mode of the Na +-Ca 2+ exchanger cannot trigger the CICR. However, depending on the initial [Na] i, the contribution of the Ca 2+ entry via the exchanger may account for 25% (at [Na] i = 10 mM) to nearly 100% ([Na] i = 30 mM) of the trigger Ca 2+. The shift of the main trigger from L-type calcium channels to the exchanger reduced the delay between the action potential upstroke and the intracellular calcium transient. This may contribute to the function of the myocyte in physiological situations where [Na] i is elevated. These main results remain the same when using different estimates for the most crucial parameters in the modeling or different models for the exchanger.

Ground-state properties of a trapped few-boson system under rotation: Beyond the “lowest-Landau-level” approximation

We consider a harmonically trapped few-Boson system under rotation and investigate the ground state properties beyond the usual ``lowest Landau level'' approximation by using exact diagonalizations in a restricted Hilbert subspace. We find that both the effective interaction energy and density distribution are strongly affected by the two-body interaction strength.

A Geometrical Interpretation and Uniform Matrix Formulation of Multibody System Dynamics

The purpose of this paper is to associate the multibody dynamics procedures with a geometrical picture involving the concepts of configuration manifolds, tangent vector spaces, and orthogonality of constraint reactions to the constraint surfaces. An unconstrained mechanical system is assigned a free configuration manifold and is treated as a generalized particle on the manifold. The system dynamics is then formulated in the local tangent space to the manifold at the system representation point. Imposed constraints on the system, the tangent space splits into the velocity restricted and velocity admissible subspaces, while the system configuration manifold confines to the holonomic constraint manifold. Based on these geometrical concepts, a uniform vector-matrix formulation is developed. Both holonomic and nonholonomic systems are treated in a unified way, and the dynamic equations are expressible either in generalized velocities or in quasi-velocities. Using a geometrically grounded projection method, compact schemes for obtaining different types of equations of motion and for determination of constraint reactions are provided. Some fresh contributions to the theory of constrained systems are reported. A relationship between the present formulation and the other classical methods of analytical dynamics is shown.

On \(SU(2) \times \mathcal{S}_{n \geqslant 12} \) dual‐group tensorial sets and carrier spaces in the multiple invariant physics of multiquantum NMR. I. k>n/2 rank maximal \(\left\{ {\sum\nolimits_v {T^k } (v) \to \sum\nolimits_{\tilde \lambda '} {\Lambda _{ \cdot ,\tilde \lambda '} [\tilde \lambda ']} } \right\}\) Liouvillian \(\overline {\text{U}} \times \mathcal{P}\) mappings

From the (automorphic) nature of nuclear permutational NMR spin symmetry established in the 1980s, the \(SU(2)\;{ \times }\;\mathcal{S}_n \) tensorial sets exhibit simple reducibility (SR) over their pattern algebraic‐based Liouvillian carrier space [F.P. Temme, Physica A 166 (1990) 676; 198 (1993) 245; Chem. Phys. 238 (1998) 245] for NMR. This underlies the quantum physical role of the \(\mathcal{S}_n \) group and its fundamental set of system invariants (SIs), i.e. beyond the Cartesian SIs of [P.L. Corio, J. Magn. Reson. 134 (1998) 131]. Here we examine the outer‐rank dual‐group tensorial subsets (over a \(\{ [\widetilde\lambda ]\} (\mathcal{S}_n )\) field) as standardised maximal forms for all \(\lambda | - n{\kern 1pt} :\) (λ<(n/2)) weakly‐branched partitions. Techniques based on Schur functions (SFs) “on a restricted subspace (RSS)”, classics in the Wybourne–Butler atomic physics tradition, serve to highlight the central role of combinatorics in physics. Recursive use of the SFs/RSS from the skew‐diagonals of \(\left\{ {\left| {I\overline M \rangle ,\;.\;.\;.\;,} \right|I\overline M \rangle } \right\}\) square generates the rank‐alone structure of Liouville space from SR \(\mathcal{S}_n \) decompositions of individual bipartite SF products. For high enough n‐indexed groups, the tensorial subsets exhibit maximal standard forms. Within a total \(\left( {_{\;n}^{2n} } \right)\)‐fold space, the \(\{ T^k (k_1 ,k_2 ,\;.\;.\;.\;,k_n :\mathcal{S}_n )\} \) (k=kmax−i)th subdimensionalities are governed by the related \(x_{1^{2n} }^{[2n - i,i]} (\mathcal{S}_{2n} )\) characters. The use of three separate combinatorial algorithms yields the $$\left\{ {\sum\limits_v {T^k } (v) \to \left\{ {\;\left[ {\widetilde\lambda } \right]\;} \right\}\left| {p \leqslant 2^2 } \right.{\text{part}}} \right\}$$ maps as the origins of subsequent studies (of part II) on democratic invariants. A strong case for the retention of \(\mathcal{S}_n \) group in NMR has been made in our above‐cited papers, in the context of spin dynamics [B.C. Sanctuary and T.K. Halstead, Adv. Opt. Magn. Reson. 15 (1991) 197; and references therein], and by [J.J. Sullivan and T.H. Sidall‐III, J. Phys. Chem. 96 (1992) 5789] for state‐space. The present work relates specifically to NMR evolution and to coherence transfer processes. Our understanding of spectral nuclear spin statistical weights of uniform fermionic (bosonic) cage‐isotopomers (as, e.g., in the ro‐vibrational CNP of 13Cn fullerenes) benefits from an appreciation of the dual group and its carrier space.

Criticality and Averaging in Cosmology

We propose comparing cosmological solutions in terms of their total spatial volumes $V(\tau)$ as functions of proper time $\tau$, assuming synchronous gauge, and with this intention evaluate the variations of $V(\tau)$ about the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) solutions for dust. This can be done successfully in a simple manner without solving perturbation equations. In particular, we find that first variations vanish with respect to all directions which do not possess homogeneity and isotropy preserving components; in other words, every FLRW solution is a {\it critical point} for $V(\tau)$ in the properly restricted subspace of the space of solutions. This property may support a validity of the interpretation of the FLRW solutions as constituting an averaged model. We also briefly investigate the second variations of $V(\tau)$.

Non-linear altimetric geoid inversion for lithospheric elastic thickness and crustal density

SUMMARY The inversion of high-resolution geoid anomaly maps derived from satellite altimetry should allow one to retrieve the lithospheric elastic thickness, T e , and crustal density, r c . Indeed, the bending of a lithospheric plate under the load of a seamount depends on both parameters, and the associated geoid anomaly is correspondingly dependent on the two parameters. The diVerence between the observed and modelled geoid signatures is estimated by a cost function, J, of the two variables, T e and r c . We show that this cost function forms a valley structure along which many local minima appear, the global minimum of J corresponding to the true values of the lithospheric parameters. Classical gradient methods fail to find this global minimum because they converge to the first local minimum of J encountered, so that the final parameter estimate strongly depends on the starting pair of values (T e , r c ). We here implement a non-linear optimization algorithm to recover these two parameters from altimetry data. We demonstrate from the inversion of synthetic data that this approach ensures robust estimates of T e and r c by activating two search phases alternately: a gradient phase to find a local minimum of J, and a tunnelling phase through high values of the cost function. The accuracy of the solution can be improved by a search in an iteratively restricted parameter subspace. Applying our non-linear inversion to the Great Meteor Seamount geoid data, we further show that the inverse problem is intrinsically illposed. As a consequence, minute geoid (or gravity) data errors can induce large changes in any recovery of lithospheric elastic thickness and crustal density.

Cardiac Ca 2+ Dynamics: The Roles of Ryanodine Receptor Adaptation and Sarcoplasmic Reticulum Load

We construct a detailed mathematical model for Ca 2+ regulation in the ventricular myocyte that includes novel descriptions of subcellular mechanisms based on recent experimental findings: 1) the Keizer-Levine model for the ryanodine receptor (RyR), which displays adaptation at elevated Ca 2+; 2) a model for the L-type Ca 2+ channel that inactivates by mode switching; and 3) a restricted subspace into which the RyRs and L-type Ca 2+ channels empty and interact via Ca 2+. We add membrane currents from the Luo-Rudy Phase II ventricular cell model to our description of Ca 2+ handling to formulate a new model for ventricular action potentials and Ca 2+ regulation. The model can simulate Ca 2+ transients during an action potential similar to those seen experimentally. The subspace [Ca 2+] rises more rapidly and reaches a higher level (10–30 μM) than the bulk myoplasmic Ca 2+ (peak [Ca 2+] i ≈ 1 μM). Termination of sarcoplasmic reticulum (SR) Ca 2+ release is predominately due to emptying of the SR, but is influenced by RyR adaptation. Because force generation is roughly proportional to peak myoplasmic Ca 2+, we use [Ca 2+] i in the model to explore the effects of pacing rate on force generation. The model reproduces transitions seen in force generation due to changes in pacing that cannot be simulated by previous models. Simulation of such complex phenomena requires an interplay of both RyR adaptation and the degree of SR Ca 2+ loading. This model, therefore, shows improved behavior over existing models that lack detailed descriptions of subcellular Ca 2+ regulatory mechanisms.

Supersymmetry and localization in the quantum Hall effect

We study the localization transition in the integer quantum Hall effect as described by the network model of quantum percolation. Starting from a path integral representation of transport Green functions for the network model, which employs both complex (bosonic) and Grassmann (fermionic) fields, we map the problem of localization to the problem of diagonalizing a one-dimensional non-hermitian Hamiltonian of interacting bosons and fermions. An exact solution is obtained in a restricted subspace of the Hilbert space which preserves boson-fermion supersymmetry. The physically relevant regime is investigated using the density matrix renormalization group (DMRG) method, and critical behavior is found at the plateau transition.

On Wannier Excitons

We rederive the exciton concept from scratch i.e. from N electrons in the valence and conduction bands. In a first part, we diagonalize the semiconductor Hamiltonian in a restricted N electron subspace having valence electrons and one conduction electron. We show that the introduction of Wannier states is unnecessary and that the main steps of this diagonalization are quite clear in terms of valence and conduction Bloch states only. We also show why the dielectric constant is irremediably lost by the diagonalization inside this restricted subspace, its appearance being then phenomenologically forces as a "reasonable screening of the Coulomb interaction". In a second part, we look for the response function of a semiconductor to a photon field using the standard Green's function procedure. The main advantage of this second approach is to generate in a natural way the appearance of the semiconductor dielectric constant.

Statistical mechanics of training in neural networks

A statistical mechanical approach for neural networks in which couplings are the slow dynamical variables is considered. The couplings are assumed to be confined in a restricted subspace near the Hebb-rule structure corresponding to quenched patterns. We study the situation when the couplings thermalize at a temperature different from that of the spin degrees of freedom, which makes it possible to treat the system in terms of the traditional replica approach with a finite number of replicas. The structure of the model is such that the effective evolution of the couplings tends to deepen the free energy minima corresponding to the learnt patterns. The phase diagram obtained exhibits a substantial increase of the retrieval region.

Optimum image restoration filters and generalized inverses of operators

AbstractAn optimal restoration filter is a restoration filter which meets some optimization criterion. It is known that a close relationship exists between optimal image restoration filters and generalized inverses of operators. By clarifying this relationship, the understanding of both optimal image restoration filters and generalized inverse is deepened.This paper clarifies the following points concerning the aforementioned relationship. First, the projection filter and averaged projection filter satisfy an optimal criterion in some restricted subspace. Furthermore, it is shown that whether that optimal criterion is evaluated in the original image space or the observed image space, the same result is achieved. Finally, it is shown clearly how the Wiener filter behaves as a generalized inverse.

Lattice dynamics, structural stability, and phase transitions in incommensurate and commensurate A2BX4 materials.

Numerous materials with the general formula ${\mathit{A}}_{2}$${\mathit{BX}}_{4}$, where A and B are cations and X is the anion, are isomorphous to \ensuremath{\beta}-${\mathrm{K}}_{2}$${\mathrm{SO}}_{4}$ (space group Pnam) at high temperatures. A considerable number of them exhibit a structural instability leading to an incommensurately modulated phase with identical superspace symmetry. ${\mathrm{K}}_{2}$${\mathrm{SeO}}_{4}$ is the archetypical example. From the analysis of the incommensurate structures, the polarization vector of the unstable frozen mode can be determined, being similar in all investigated compounds. We report here a comparative energetic study and lattice-dynamics analysis of a set of compounds in this family. The set of materials ${\mathrm{K}}_{2}$${\mathrm{SO}}_{4}$, ${\mathrm{Rb}}_{2}$${\mathrm{SeO}}_{4}$, ${\mathrm{Cs}}_{2}$${\mathrm{SeO}}_{4}$, ${\mathrm{Cs}}_{2}$${\mathrm{ZnCl}}_{4}$, ${\mathrm{Cs}}_{2}$${\mathrm{ZnBr}}_{4}$, ${\mathrm{K}}_{2}$${\mathrm{CrO}}_{4}$, ${\mathrm{K}}_{2}$${\mathrm{SeO}}_{4}$, ${\mathrm{K}}_{2}$${\mathrm{ZnCl}}_{4}$, ${\mathrm{Rb}}_{2}$${\mathrm{ZnCl}}_{4}$, and ${\mathrm{Rb}}_{2}$${\mathrm{ZnBr}}_{4}$, includes compounds with and without an incommensurate phase.An empirical rigid-ion force model has been used with only three adjustable parameters, the tetrahedral ${\mathit{BX}}_{4}$ groups being reduced to rigid bodies. The adjusted force model, optimized for each compound with use of only static structural data, is sufficient to explain the eventual presence of an incommensurate lattice instability at lower temperatures. The calculated phonon dispersion curves of those compounds with an incommensurate phase include an unstable ${\mathrm{\ensuremath{\Sigma}}}_{2}$ phonon branch with a minimum close to 1/3${\mathbf{a}}^{\mathrm{*}}$. In the simulations, the unstable or soft-mode branch is always an optical branch in the extended zone scheme or the consequence of an anticrossing of an optical branch with the ${\mathrm{\ensuremath{\Sigma}}}_{3\mathrm{\ensuremath{-}}}$${\mathrm{\ensuremath{\Sigma}}}_{2}$ acoustic branch; this result discredits any attempt to explain the soft-mode mechanism in terms of a one-dimensional model with an acoustic soft branch. The polarization vectors of the soft or unstable modes obtained in the simulations fairly agree with the experimental ones. They are rather insensitive to the details of the interactions, explaining their strong similarities.On the other hand, the form of the soft branch depends strongly on the material, and clearly distinguishes those materials having the ${\mathit{BX}}_{4}$ groups disordered in the normal phase, from those having a soft-mode mechanism. The simulations indicate that the static and dynamic features of potassium chromate are similar to those of potassium selenate, raising the possibility that potassium chromate could exhibit a similar mode softening at low temperatures. The existence of an incommensurate lattice instability in these compounds depends basically on the effective volume of the A cations compared with the size of the ${\mathit{BX}}_{4}$ tetrahedra. The charge distribution within the tetrahedral anion groups also plays a significant secondary role; smaller values of the charge of B tend to stabilize the Pnam structure. The static energy of some of the compounds has been investigated in a restricted configuration subspace, which includes the order-parameter distortion. The energy maps obtained show a clear ``multiple-well'' structure, that can be quantitatively related with the transition temperatures by means of a local-mode model.

Relativistic potential scattering and phase shift analysis

It is shown that two-body relativistic scattering cross sections can be represented in terms of phase shift analysis in essentially the same form as that of nonrelativistic scattering theory. The representation is covariant; the variable that corresponds to the orbital quantum number (and approaches it in the nonrelativistic limit) is relativistically invariant. The Levinson theorem is valid, and provides a link between the bound states, which have support in an O(2,1) invariant subspace of the full spacelike region of relative coordinates, and the scattering states that contain resonant behavior. These scattering states consequently have support in the same restricted subspace, and the same procedure may be used for the separation of variables. As an example, the resonances of an O(3,1) invariant ‘‘square well’’ are discussed.

Fluctuations and Dissipation in a Barotropic Flow Field

Abstract Herein we present the first systematic derivation of stochastic mode rate equations for a geophysical hydrodynamic system. Coarse graining concepts from nonequilibrium statistical mechanics are applied to the vorticity equations for barotropic motion on a β-plane. The projection of the initial field equations onto a restricted subspace yields nonlinear stochastic mode rate equations for the physical observables with completely determined statistical properties. It is shown that the usual assumptions of ergodicity and Markovicity are valid only under some very restrictive conditions.

The RPA description of the quantum damped harmonic oscillator

We have developed a microscopic, time reversible model which behaves like a damped harmonic oscillator for times short compared to the system's Poincaré time. The model assumes an RPA description of initial collective states within a restricted subspace, thentraces their time evolution when an additional subspace (the reservoir) is coupled to the restricted subspace. The assumption that this time evolution is also governed by RPA dynamics is utilized. The evolution of packets initialized as the shifted eigenstates of an undamped oscillator of arbitrary frequency (different from the frequency of the collective phonon) allows the explicit study of the expectation values and the fluctuations in the collective coordinates. In all cases, the fluctuations decay to their zero point values. Due to the decay of the fluctuations, the energy loss rate deviates from Rayleigh's result for classical mechanics. The RPA description contrasts with both the Albrecht and the Kostin descriptions.

Statistical properties of quantum systems: The linear oscillator

Statistical fluctuations in linear quantum-mechanical systems are shown to result from a projection of the total quantum system onto a restricted subspace. The resulting equations of motion are of the generalized Langevin form, with fluctuating and dissipative terms. These terms are related by a quantum-mechanical fluctuation-dissipation relation that ensures thermal equilibration. We analyze the dynamical behavior of the subsystem and elucidate the meaning and interrelation of several ubiquitous concepts in the following context: weak-coupling limit, Markovian limit, rotating-wave approximation (RWA), and low-temperature behavior. The three most salient consequences of our analysis are as follows: (1) The time scale for the correlation of fluctuations and the dissipation can be quite distinct, (2) the traditional implementation of the RWA only gives valid results in the strict weak-coupling limit, and (3) a reformulation of the RWA valid at arbitrary coupling strengths, and hence at arbitrarily low temperatures, is possible.

Microscopic nuclear dissipation: (II). Damping of collective states in subspaces which include 2p-2h states

We have formulated a microscopic, nonperturbative, time reversible model which exhibits a dissipative decay of collective motion for times short compared to the system's Poincaré time. The model assumes an RPA approximate description of the initial collective state within a restricted subspace, then traces its time evolution when an additional subspace is coupled to the restricted subspace by certain simplified matrix elements. It invokes no statistical assumptions. The damping of the collective motion occurs via real tansitions from the collective state to other more complicated nuclear states of the same energy. It corresponds therefore to the so called “one-body” long mean free path limit of nuclear dissipation when the collective state describes a surface vibration. When the simplest RPA approximation is used, this process associates the dissipation with the escape width for direct particle emission to the continuum. When the more detailed second RPA is used, it associates the dissipation with the spreading width for transitions to the 2p-2h components of the nuclear compound states as well. The energy loss rate for sharp n-phonon initial states is proportional to the total collective energy, unlike the dissipation of a classical damped oscillator, where it is proportional to the kinetic energy only. However, for coherent, multi-phonon wave packets, which explicitly describe the time-dependent oscillations of the mean field, dissipation proportional only to the kinetic energy is obtained. Canonical coordinates for the collective degree of freedom are explicitly introduced and a nonlinear frictional hamiltonian to describe such systems is specified by the requirement that it yield the same time dependence for the collective motion as the microscopic model. Thus, for the first time a descriptive nonlinear hamiltonian is derived explicitly from the underlying microscopic model of a nuclear system.

Order and disorder lines in systems with competing interactions: II. The IRF model

Systems with competing interactions can be often exactly solved on a restricted subspace of the parameter space, called an order or disorder trajectory. A simple method introduced within the transfer matrix formalism allows for the calculation of the free energy and spin-spin correlation functions along the order and disorder lines of the Ising model with all possible interactions around a face of the square lattice (IRF model). The general eight-vertex model is thoroughly examined and shows full analogy with the quantum spin chain results of the previous paper.

Nuclear dissipation as damping of collective motion in the time-dependent RPA: (I). The microscopic model

A microscopic model for describing nuclear dissipation as the damping of collective motion in the time-dependent RPA is considered. The collective state is defined as a solution of the RPA equations of motion in a restricted subspace, S 1, of discrete 1p-1h states. The damping of the collective motion is described by the time evolution of the wave packet which solves the time-dependent Schrödinger equation in an extended subspace, S, when initialized with the collective state. In the present paper we consider additional state of the 1p-1h structure only. The RPA solution for the wave packet describing the damping enables one to calculate — as a function of time — both the probability amplitude for finding the system in the collective state and the collective energy, and to extract their time decay under appropriate conditions. The rate of decrease of the collective energy is interpreted as the energy dissipation rate.

Converging SCF calculations on excited states

AbstractA practical scheme for the calculation of excited states of the same symmetry as a given reference state is outlined in the context of the Hartree‐Fock method. In order to prevent the excited state from “collapsing” into a lower‐lying state, the prediagonalized Fock matrix is diagonalized in a restricted subspace, deleting the component associated with the orbital which participates in the excitation. Computationally, the deletion is accomplished by means of a “big shift” of the associated diagonal element of the prediagonalized Fock matrix. The resulting wave function will not be fully relaxed, but can be shown to be orthogonal to the reference state. The method has been implemented in a molecular UHF program. Applications to the 4σ hole state of CO and to an excited state of the CuCl ion are reported.