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

Statistical Mechanics of Dilute Batch Minority Games with Random External Information

We study the dynamics and statics of a dilute batch minority game with random external information. We focus on the case in which the number of connections per agent is infinite in the thermodynamic limit. The dynamical scenario of ergodicity breaking in this model is different from the phase transition in the standard minority game and is characterised by the onset of long-term memory at finite integrated response. We demonstrate that finite memory appears at the AT-line obtained from the corresponding replica calculation, and compare the behaviour of the dilute model with the minority game with market impact correction, which is known to exhibit similar features.Comment: 22 pages, 6 figures, text modified, references updated and added, figure added, typos correcte

News and price returns from threshold behaviour and vice-versa: exact solution of a simple agent-based market model

Starting from an exact relationship between news, threshold and price return distributions in the stationary state, I discuss the ability of the Ghoulmie-Cont-Nadal model of traders to produce fat-tailed price returns. Under normal conditions, this model is not able to transform Gaussian news into fat-tailed price returns. When the variance of the news so small that only the players with zero threshold can possibly react to news, this model produces Levy-distributed price returns with a -1 exponent. In the special case of super-linear price impact functions, fat-tailed returns are obtained from well-behaved news.Comment: 4 pages, 3 figures. This is quite possibly the final version. To appear in J. Phys

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

Closure of Macroscopic Laws in Disordered Spin Systems: A Toy Model

We use a linear system of Langevin spins with disordered interactions as an exactly solvable toy model to investigate a procedure, recently proposed by Coolen and Sherrington, for closing the hierarchy of macroscopic order parameter equations in disordered spin systems. The closure procedure, based on the removal of microscopic memory effects, is shown to reproduce the correct equations for short times and in equilibrium. For intermediate time-scales the procedure does not lead to the exact equations, yet for homogeneous initial conditions succeeds at capturing the main characteristics of the flow in the order parameter plane. The procedure fails in terms of the long-term temporal dependence of the order parameters. For low energy inhomogeneous initial conditions and near criticality (where zero modes appear) deviations in temporal behaviour are most apparent. For homogeneous initial conditions the impact of microscopic memory effects on the evolution of macroscopic order parameters in disordered spin systems appears to be mainly an overall slowing down.Comment: 14 pages, LateX, OUTP-94-24

Stochastic learning in a neural network with adapting synapses

We consider a neural network with adapting synapses whose dynamics can be analitically computed. The model is made of $N$ neurons and each of them is connected to $K$ input neurons chosen at random in the network. The synapses are $n$-states variables which evolve in time according to Stochastic Learning rules; a parallel stochastic dynamics is assumed for neurons. Since the network maintains the same dynamics whether it is engaged in computation or in learning new memories, a very low probability of synaptic transitions is assumed. In the limit $N\to\infty$ with $K$ large and finite, the correlations of neurons and synapses can be neglected and the dynamics can be analitically calculated by flow equations for the macroscopic parameters of the system.Comment: 25 pages, LaTeX fil

Attractor Modulation and Proliferation in 1+∞\infty Dimensional Neural Networks

We extend a recently introduced class of exactly solvable models for recurrent neural networks with competition between 1D nearest neighbour and infinite range information processing. We increase the potential for further frustration and competition in these models, as well as their biological relevance, by adding next-nearest neighbour couplings, and we allow for modulation of the attractors so that we can interpolate continuously between situations with different numbers of stored patterns. Our models are solved by combining mean field and random field techniques. They exhibit increasingly complex phase diagrams with novel phases, separated by multiple first- and second order transitions (dynamical and thermodynamic ones), and, upon modulating the attractor strengths, non-trivial scenarios of phase diagram deformation. Our predictions are in excellent agreement with numerical simulations.Comment: 16 pages, 15 postscript figures, Late

Cluster derivation of Parisi's RSB solution for disordered systems

We propose a general scheme in which disordered systems are allowed to sacrifice energy equi-partitioning and separate into a hierarchy of ergodic sub-systems (clusters) with different characteristic time-scales and temperatures. The details of the break-up follow from the requirement of stationarity of the entropy of the slower cluster, at every level in the hierarchy. We apply our ideas to the Sherrington-Kirkpatrick model, and show how the Parisi solution can be {\it derived} quantitatively from plausible physical principles. Our approach gives new insight into the physics behind Parisi's solution and its relations with other theories, numerical experiments, and short range models.Comment: 7 pages 5 figure

Electrical stimulation in the treatment of bladder dysfunction: Technology update

The urinary bladder has two functions: urine storage and voiding. Clinically, two major categories of lower urinary tract symptoms can be defined: storage symptoms such as incontinence and urgency, and voiding symptoms such as feeling of incomplete bladder emptying and slow urinary stream. Urgency to void with or without incontinence is called overactive bladder (OAB). Slow urinary stream, hesitancy, and straining to void with the feeling of incomplete bladder emptying are often called underactive bladder (UAB). The underlying causes of OAB or UAB can be either non-neurogenic (also referred to as idiopathic) and neurogenic, for example due to spinal cord injury or multiple sclerosis. OAB and UAB can be treated conservatively by lifestyle intervention or medication. In the case that conservative treatment does not provide sufficient benefit, electrical stimulation can be used. Sacral neurostimulation or neuromodulation (SNM) is offered as a third-line therapy to patients with non-neurogenic OAB or UAB. In SNM, the third or fourth sacral nerve root is stimulated and after a test period, a neuromodulator is implanted in the buttock. Until recently only a non-rechargeable neuromodulator was approved for clinical use. However, nowadays, a rechargeable sacral neuromodulator is also on the market, with similar safety and effectiveness to the non-rechargeable SNM system. The rechargeable device was approved for full body 1.5T and 3T MRI in Europe in February 2019. Regarding neurogenic lower urinary tract dysfunction, electrical stimulation only seems to benefit a selected group of patients

Slowly evolving geometry in recurrent neural networks I: extreme dilution regime

We study extremely diluted spin models of neural networks in which the connectivity evolves in time, although adiabatically slowly compared to the neurons, according to stochastic equations which on average aim to reduce frustration. The (fast) neurons and (slow) connectivity variables equilibrate separately, but at different temperatures. Our model is exactly solvable in equilibrium. We obtain phase diagrams upon making the condensed ansatz (i.e. recall of one pattern). These show that, as the connectivity temperature is lowered, the volume of the retrieval phase diverges and the fraction of mis-aligned spins is reduced. Still one always retains a region in the retrieval phase where recall states other than the one corresponding to the `condensed' pattern are locally stable, so the associative memory character of our model is preserved.Comment: 18 pages, 6 figure