Phase Diagram and Storage Capacity of Sequence Processing Neural Networks

We solve the dynamics of Hopfield-type neural networks which store sequences of patterns, close to saturation. The asymmetry of the interaction matrix in such models leads to violation of detailed balance, ruling out an equilibrium statistical mechanical analysis. Using generating functional methods we derive exact closed equations for dynamical order parameters, viz. the sequence overlap and correlation- and response functions, in the thermodynamic limit. We calculate the time translation invariant solutions of these equations, describing stationary limit-cycles, which leads to a phase diagram. The effective retarded self-interaction usually appearing in symmetric models is here found to vanish, which causes a significantly enlarged storage capacity of $\alpha_c\sim 0.269$, compared to \alpha_\c\sim 0.139 for Hopfield networks storing static patterns. Our results are tested against extensive computer simulations and excellent agreement is found.Comment: 17 pages Latex2e, 2 postscript figure

Dynamical Replica Theory for Disordered Spin Systems

We present a new method to solve the dynamics of disordered spin systems on finite time-scales. It involves a closed driven diffusion equation for the joint spin-field distribution, with time-dependent coefficients described by a dynamical replica theory which, in the case of detailed balance, incorporates equilibrium replica theory as a stationary state. The theory is exact in various limits. We apply our theory to both the symmetric- and the non-symmetric Sherrington-Kirkpatrick spin-glass, and show that it describes the (numerical) experiments very well.Comment: 7 pages RevTex, 4 figures, for PR

Magnon Localization in Mattis Glass

We study the spectral and transport properties of magnons in a model of a disordered magnet called Mattis glass, at vanishing average magnetization. We find that in two dimensional space, the magnons are localized with the localization length which diverges as a power of frequency at small frequencies. In three dimensional space, the long wavelength magnons are delocalized. In the delocalized regime in 3d (and also in 2d in a box whose size is smaller than the relevant localization length scale) the magnons move diffusively. The diffusion constant diverges at small frequencies. However, the divergence is slow enough so that the thermal conductivity of a Mattis glass is finite, and we evaluate it in this paper. This situation can be contrasted with that of phonons in structural glasses whose contribution to thermal conductivity is known to diverge (when inelastic scattering is neglected).Comment: 11 page

Quantum Annealing in the Transverse Ising Model

We introduce quantum fluctuations into the simulated annealing process of optimization problems, aiming at faster convergence to the optimal state. Quantum fluctuations cause transitions between states and thus play the same role as thermal fluctuations in the conventional approach. The idea is tested by the transverse Ising model, in which the transverse field is a function of time similar to the temperature in the conventional method. The goal is to find the ground state of the diagonal part of the Hamiltonian with high accuracy as quickly as possible. We have solved the time-dependent Schr\"odinger equation numerically for small size systems with various exchange interactions. Comparison with the results of the corresponding classical (thermal) method reveals that the quantum annealing leads to the ground state with much larger probability in almost all cases if we use the same annealing schedule.Comment: 15 pages, RevTeX, 8 figure

Order-Parameter Flow in the SK Spin-Glass II: Inclusion of Microscopic Memory Effects

We develop further a recent dynamical replica theory to describe the dynamics of the Sherrington-Kirkpatrick spin-glass in terms of closed evolution equations for macroscopic order parameters. We show how microscopic memory effects can be included in the formalism through the introduction of a dynamic order parameter function: the joint spin-field distribution. The resulting formalism describes very accurately the relaxation phenomena observed in numerical simulations, including the typical overall slowing down of the flow that was missed by the previous simple two-parameter theory. The advanced dynamical replica theory is either exact or a very good approximation.Comment: same as original, but this one is TeXabl

Expansion of the Gibbs potential for quantum many-body systems: General formalism with applications to the spin glass and the weakly non-ideal Bose gas

For general quantum systems the power expansion of the Gibbs potential and consequently the power expansion of the self energy is derived in terms of the interaction strength. Employing a generalization of the projector technique a compact representation of the general terms of the expansion results. The general aspects of the approach are discussed with special emphasis on the effects characteristic for quantum systems. The expansion is systematic and leads directly to contributions beyond mean-field of all thermodynamic quantities. These features are explicitly demonstrated and illustrated for two non-trivial systems, the infinite range quantum spin glass and the weakly interacting Bose gas. The Onsager terms of both systems are calculated, which represent the first beyond mean-field contributions. For the spin glass new TAP-like equations are presented and discussed in the paramagnetic region. The investigation of the Bose gas leads to a beyond mean-field thermodynamic description. At the Bose-Einstein condensation temperature complete agreement is found with the results presented recently by alternative techniques.Comment: 17 pages, 0 figures; revised version accepted by Phys Rev

The 3-SAT problem with large number of clauses in ∞\infty-replica symmetry breaking scheme

In this paper we analyze the structure of the UNSAT-phase of the overconstrained 3-SAT model by studying the low temperature phase of the associated disordered spin model. We derive the $\infty$ Replica Symmetry Broken equations for a general class of disordered spin models which includes the Sherrington - Kirkpatrick model, the Ising $p$-spin model as well as the overconstrained 3-SAT model as particular cases. We have numerically solved the $\infty$ Replica Symmetry Broken equations using a pseudo-spectral code down to and including zero temperature. We find that the UNSAT-phase of the overconstrained 3-SAT model is of the $\infty$-RSB kind: in order to get a stable solution the replica symmetry has to be broken in a continuous way, similarly to the SK model in external magnetic field.Comment: 19 pages, 7 figures; some section improved; iopart styl

Video and computer-based interactive exercises are safe and improve task-specific balance in geriatric and neurological rehabilitation: A randomised trial

© 2015. Question: Does adding video/computer-based interactive exercises to inpatient geriatric and neurological rehabilitation improve mobility outcomes? Is it feasible and safe? Design: Randomised trial. Participants: Fifty-eight rehabilitation inpatients. Intervention: Physiotherapist-prescribed, tailored, video/computer-based interactive exercises for 1 hour on weekdays, mainly involving stepping and weight-shifting exercises. Outcome measures: The primary outcome was the Short Physical Performance Battery (0 to 3) at 2 weeks. Secondary outcomes were: Maximal Balance Range (mm); Step Test (step count); Rivermead Mobility Index (0 to 15); activity levels; Activity Measure for Post Acute Care Basic Mobility (18 to 72) and Daily Activity (15 to 60); Falls Efficacy Scale (10 to 40), ED5D utility score (0 to 1); Reintegration to Normal Living Index (0 to 100); System Usability Scale (0 to 100) and Physical Activity Enjoyment Scale (0 to 126). Safety was determined from adverse events during intervention. Results: At 2 weeks the between-group difference in the primary outcome (0.1, 95% CI -0.2 to 0.3) was not statistically significant. The intervention group performed significantly better than usual care for Maximal Balance Range (38. mm difference after baseline adjustment, 95% CI 6 to 69). Other secondary outcomes were not statistically significant. Fifty-eight (55%) of the eligible patients agreed to participate, 25/29 (86%) completed the intervention and 10 (39%) attended > 70% of sessions, with a mean of 5.6 sessions (SD 3.3) attended and overall average duration of 4.5. hours (SD 3.1). Average scores were 62 (SD 21) for the System Usability Scale and 62 (SD 8) for the Physical Activity Enjoyment Scale. There were no adverse events. Conclusion: The addition of video/computer-based interactive exercises to usual rehabilitation is a safe and feasible way to increase exercise dose, but is not suitable for all. Adding the exercises to usual rehabilitation resulted in task-specific improvements in balance but not overall mobility. Registration: ACTRN12613000610730