Psychophysical laws as reflection of mental space properties.

The paper is devoted to the relationship between psychophysics and physics of mind. The basic trends in psychophysics development are briefly discussed with special attention focused on Teghtsoonian's hypotheses. These hypotheses pose the concept of the universality of inner psychophysics and enable us to speak about psychological space as an individual object with its own properties. Turning to the two-component description of human behavior I. Lubashevsky (2017) [9] the notion of mental space is formulated and human perception of external stimuli is treated as the emergence of the corresponding images in the mental space. On one hand, these images are caused by external stimuli and their magnitude bears the information about the intensity of the corresponding stimuli. On the other hand, the individual structure of such images as well as their persistence after emergence is determined only by the properties of mental space on its own. Finally, the mental operations of image comparison and their scaling are defined in a way allowing for the bounded capacity of human cognition. As demonstrated, the developed theory of stimulus perception is able to explain the basic regularities of psychophysics, e.g., (i) the regression and range effects leading to the overestimation of weak stimuli and the underestimation of strong stimuli, (ii) scalar variability (Weber's and Ekman' laws), and (iii) the sequential (memory) effects. As the final result, a solution to the Fechner-Stevens dilemma is proposed. This solution posits that Fechner's logarithmic law is not a consequences of Weber's law but stems from the interplay of uncertainty in evaluating stimulus intensities and the multi-step scaling required to overcome the stimulus incommensurability.

Double-well dynamics of noise-driven control activation in human intermittent control: the case of stick balancing.

When facing a task of balancing a dynamic system near an unstable equilibrium, humans often adopt intermittent control strategy: Instead of continuously controlling the system, they repeatedly switch the control on and off. Paradigmatic example of such a task is stick balancing. Despite the simplicity of the task itself, the complexity of human intermittent control dynamics in stick balancing still puzzles researchers in motor control. Here we attempt to model one of the key mechanisms of human intermittent control, control activation, using as an example the task of overdamped stick balancing. In doing so, we focus on the concept of noise-driven activation, a more general alternative to the conventional threshold-driven activation. We describe control activation as a random walk in an energy potential, which changes in response to the state of the controlled system. By way of numerical simulations, we show that the developed model captures the core properties of human control activation observed previously in the experiments on overdamped stick balancing. Our results demonstrate that the double-well potential model provides tractable mathematical description of human control activation at least in the considered task and suggest that the adopted approach can potentially aid in understanding human intermittent control in more complex processes.

To react or not to react? Intrinsic stochasticity of human control in virtual stick balancing.

Understanding how humans control unstable systems is central to many research problems, with applications ranging from quiet standing to aircraft landing. Increasingly, much evidence appears in favour of event-driven control hypothesis: human operators only start actively controlling the system when the discrepancy between the current and desired system states becomes large enough. The event-driven models based on the concept of threshold can explain many features of the experimentally observed dynamics. However, much still remains unclear about the dynamics of human-controlled systems, which likely indicates that humans use more intricate control mechanisms. This paper argues that control activation in humans may be not threshold-driven, but instead intrinsically stochastic, noise-driven. Specifically, we suggest that control activation stems from stochastic interplay between the operator's need to keep the controlled system near the goal state, on the one hand, and the tendency to postpone interrupting the system dynamics, on the other hand. We propose a model capturing this interplay and show that it matches the experimental data on human balancing of virtual overdamped stick. Our results illuminate that the noise-driven activation mechanism plays a crucial role at least in the considered task, and, hypothetically, in a broad range of human-controlled processes.

Model of oscillatory zoning in two dimensions: simulation and mode analysis.

Oscillatory zoning (OZ) occurs in all major classes of minerals and also in a wide range of geological environments. It is caused by self-organization and describes fluctuations of the spatial chemical composition profile of the crystal. We present here a two-dimensional model of OZ based on our previous one-dimensional (1D) analysis and investigate whether the results of the 1D stability analysis remain valid. With the additional second dimension we were able to study the origin of the spatially homogeneous layer formation by linear stability analysis. Numerical solutions of the model are presented and the results of a Fourier analysis delivers a detailed understanding of the crystal growth behavior as well as the limits of the model. Effects beyond linear stability analysis are important to finally understand the final structure formation.

Distributed self-regulation of living tissue: beyond the ideal limit.

The present paper is devoted to mathematical description of the vascular network response to local perturbations in the cellular tissue state, being one of the basic mechanisms controlling the inner environment of human body. Keeping in mind individual organs we propose a model for distributed self-regulation of living tissue, which is regarded as an active hierarchical system without any controlling center. This model is based on the self-processing of information about the cellular tissue state and cooperative interaction of blood vessels governing redistribution of blood flow over the vascular network. The information self-processing is implemented via mass conservation, i.e., conservation of blood flow as well as special biochemical compounds called activators transported by blood. The cooperative interaction of blood vessels stems from the response of individual vessels to activators in blood flowing through them. The general regularities are used to specify the vessel behavior. The arterial and venous beds are considered to be individually of the tree form. The constructed governing equations are analyzed numerically. In particular, first, we show that the blood perfusion rate approximately (in the analyzed case within 10% accuracy) depends only on the local concentration of activators in the cellular tissue. It is due to the hierarchical structure of the vascular network rather than the ideal behavior of individual vessels accepted previously. Second, we demonstrate the distinction between the reaction thresholds of individual vessels and that of the vascular network as a whole. The latter effect is the cause for introducing the notion of activators instead of using such quantities as temperature in describing the living tissue self-regulation.

Continuous-time multidimensional Markovian description of Lévy walks.

The paper presents a multidimensional model for nonlinear Markovian random walks that generalizes the one we developed previously [I. Lubashevsky, R. Friedrich, and A. Heuer, Phys. Rev. E 79, 011110 (2009)] in order to describe the Lévy-type stochastic processes in terms of continuous trajectories of walker motion. This approach may open a way to treat Lévy flights or Lévy random walks in inhomogeneous media or systems with boundaries in the future. The proposed model assumes the velocity of a wandering particle to be affected by a linear friction and a nonlinear Langevin force whose intensity is proportional to the magnitude of the velocity for its large values. Based on the singular perturbation technique, the corresponding Fokker-Planck equation is analyzed and the relationship between the system parameters and the Lévy exponent is found. Following actually the previous paper we demonstrate also that anomalously long displacements of the wandering particle are caused by extremely large fluctuations in the particle velocity whose duration is determined by the system parameters rather than the duration of the observation interval. In this way we overcome the problem of ascribing to Lévy random-walk non-Markov properties.

Realization of Lévy walks as Markovian stochastic processes.

Based on multivariate Langevin processes we present a realization of Lévy flights as a continuous process. For the simple case of a particle moving under the influence of friction and a velocity-dependent stochastic force we explicitly derive the generalized Langevin equation and the corresponding generalized Fokker-Planck equation describing Lévy flights. Our procedure is similar to the treatment of the Kramers-Fokker-Planck equation in the Smoluchowski limit. The proposed approach may open a way to treat Lévy flights in inhomogeneous media or systems with boundaries in the future.

Different routes towards oscillatory zoning in the growth of solid solutions.

Oscillatory zoning, i.e., self-formation of spatial quasiperiodic oscillations in the composition of solid growing from aqueous solution, is analyzed theoretically. Keeping in mind systems like (Ba,Sr)SO4 , we propose a one-dimensional model that takes into account the nonideality of the solid solution and the system asymmetry, in particular, reflecting itself in different solubilities for such systems. Based on a linear stability analysis, different parameter regions can be identified. Even an ideal solution with a sufficiently large asymmetry can display oscillatory zoning. Numerical simulations complement the linear stability analysis as well as the qualitative consideration of the instability development and reveal the nature of the limit cycles.

Boundary-reaction-diffusion model for oscillatory zoning in binary crystals grown from solution.

Oscillatory Zoning (OZ) is a phenomenon exhibited by many geologically formed crystals. It is characterized by quasiperiodic oscillations in the composition of a solid solution, caused by self-organization. We present a model for OZ. The growth mechanism applied includes species diffusion through the solution bulk, particle adsorption, surface diffusion, and subsequently desorption or incorporation into the crystal. This mechanism, in particular, can provide the synchronization effects necessary to reproduce the layered structure of experimentally obtained crystals, lacking in other models. We conduct a linear stability analysis combined with numerical simulations. Our results reproduce the experimental findings with respect to the patterns formed and a critical supersaturation necessary for OZ to occur.

Rational-driver approximation in car-following theory.

The problem of a car following a lead car driven with constant velocity is considered. To derive the governing equations for the following car dynamics a cost functional is constructed. This functional ranks the outcomes of different driving strategies, which applies to fairly general properties of the driver behavior. Assuming rational-driver behavior, the existence of the Nash equilibrium is proved. Rational driving is defined by supposing that a driver corrects continuously the car motion to follow the optimal path minimizing the cost functional. The corresponding car-following dynamics is described quite generally by a boundary value problem based on the obtained extremal equations. Linearization of these equations around the stationary state results in a generalization of the widely used optimal velocity model. Under certain conditions (the "dense traffic" limit) the rational car dynamics comprises two stages, fast and slow. During the fast stage a driver eliminates the velocity difference between the cars, the subsequent slow stage optimizes the headway. In the dense traffic limit an effective Hamiltonian description is constructed. This allows a more detailed nonlinear analysis. Finally, the differences between rational and bounded rational driver behavior are discussed. The latter, in particular, justifies some basic assumptions used recently by the authors to construct a car-following model lying beyond the frameworks of rationality.

Long-lived states in synchronized traffic flow: empirical prompt and dynamical trap model.

The present paper proposes an interpretation of the widely scattered states (called synchronized traffic) stimulated by Kerner's hypothesis about the existence of a multitude of metastable states in the fundamental diagram. Using single-vehicle data collected at the German highway A1, temporal velocity patterns have been analyzed to show a collection of certain fragments with approximately constant velocities and sharp jumps between them. The particular velocity values in these fragments vary in a wide range. In contrast, the flow rate is more or less constant because its fluctuations are mainly due to the discreteness of traffic flow. Subsequently, we develop a model for synchronized traffic that can explain these characteristics. Following previous work [I. A. Lubashevsky and R. Mahnke, Phys. Rev. E 62, 6082 (2000)] the vehicle flow is specified by car density, mean velocity, and additional order parameters h and a that are due to the many-particle effects of the vehicle interaction. The parameter h describes the multilane correlations in the vehicle motion. Together with the car density it determines directly the mean velocity. The parameter a, in contrast, controls the evolution of h only. The model assumes that a fluctuates randomly around the value corresponding to the car configuration optimal for lane changing. When it deviates from this value the lane change is depressed for all cars forming a local cluster. Since exactly the overtaking maneuvers of these cars cause the order parameter a to vary, the evolution of the car arrangement becomes frozen for a certain time. In other words, the evolution equations form certain dynamical traps responsible for the long-time correlations in the synchronized mode.

Probabilistic description of traffic breakdowns.

We analyze the characteristic features of traffic breakdown. To describe this phenomenon we apply the probabilistic model regarding the jam emergence as the formation of a large car cluster on a highway. In these terms, the breakdown occurs through the formation of a certain critical nucleus in the metastable vehicle flow, which enables us to confine ourselves to one cluster model. We assume that, first, the growth of the car cluster is governed by attachment of cars to the cluster whose rate is mainly determined by the mean headway distance between the car in the vehicle flow and, maybe, also by the headway distance in the cluster. Second, the cluster dissolution is determined by the car escape from the cluster whose rate depends on the cluster size directly. The latter is justified using the available experimental data for the correlation properties of the synchronized mode. We write the appropriate master equation converted then into the Fokker-Planck equation for the cluster distribution function and analyze the formation of the critical car cluster due to the climb over a certain potential barrier. The further cluster growth irreversibly causes jam formation. Numerical estimates of the obtained characteristics and the experimental data of the traffic breakdown are compared. In particular, we draw a conclusion that the characteristic intrinsic time scale of the breakdown phenomenon should be about 1 min and explain the case why the traffic volume interval inside which traffic breakdown is observed is sufficiently wide.

Towards a variational principle for motivated vehicle motion.

We deal with the problem of deriving the microscopic equations governing individual car motion based on assumptions about the strategy of driver behavior. We presume the driver behavior to be a result of a certain compromise between the will to move at a speed that is comfortable for him under the surrounding external conditions, comprising the physical state of the road, the weather conditions, etc., and the necessity to keep a safe headway distance between the cars in front of him. Such a strategy implies that a driver can compare the possible ways of further motion and so choose the best one. To describe the driver preferences, we introduce the priority functional whose extremals specify the driver choice. For simplicity we consider a single-lane road. In this case solving the corresponding equations for the extremals we find the relationship between the current acceleration, velocity, and position of the car. As a special case we get a certain generalization of the optimal velocity model similar to the "intelligent driver model" proposed by Treiber and Helbing.