Models for saccadic motion and postsaccadic oscillations.

In a recent letter [S. Bouzat et al., Phys. Rev. Lett. 120, 178101 (2018)10.1103/PhysRevLett.120.178101], a mathematical model for eyeball and pupil motion was developed allowing for the understanding of the postsaccadic oscillations (PSO) as inertial effects. The model assumes that the inner part of the iris, which defines the pupil, moves driven by inertial forces induced by the eyeball rotation, in addition to viscous and elastic forces. Among other achievements, the model correctly reproduces eye-tracking experiments concerning PSO profiles and their dependence on the saccade size. In this paper we propose various extensions of the mentioned model, we provide analytical solutions, and we perform an exhaustive analysis of the dynamics. In particular, we consider a more general time dependence for the eyeball velocity enabling the description of saccades with vanishing initial acceleration. Moreover, we give the analytical solution in terms of hypergeometric functions for the constant parameter version of the model and we provide particular expressions for some cases of interest. We also introduce a new version of the model with inhomogeneous viscosity that can improve the fitting of the experimental results. Our analysis of the solutions explores the dependence of the PSO profiles on the system parameters for varying saccade sizes. We show that the PSO emerge in critical-like ways when parameters such as the elasticity of the iris, the global eyeball velocity, or the saccade size vary. Moreover, we find that the PSO profiles with the first overshoot smaller than the second one, which are usually observed in experiments, can be associated to parameter regions close to criticality.

Inertial Movements of the Iris as the Origin of Postsaccadic Oscillations.

Recent studies on the human eye indicate that the pupil moves inside the eyeball due to deformations of the iris. Here we show that this phenomenon can be originated by inertial forces undergone by the iris during the rotation of the eyeball. Moreover, these forces affect the iris in such a way that the pupil behaves effectively as a massive particle. To show this, we develop a model based on the Newton equation on the noninertial reference frame of the eyeball. The model allows us to reproduce and interpret several important findings of recent eye-tracking experiments on saccadic movements. In particular, we get correct results for the dependence of the amplitude and period of the postsaccadic oscillations on the saccade size and also for the peak velocity. The model developed may serve as a tool for characterizing eye properties of individuals.

Game theory in models of pedestrian room evacuation.

We analyze the pedestrian evacuation of a rectangular room with a single door considering a lattice gas scheme with the addition of behavioral aspects of the pedestrians. The movement of the individuals is based on random and rational choices and is affected by conflicts between two or more agents that want to advance to the same position. Such conflicts are solved according to certain rules closely related to the concept of strategies in game theory, cooperation and defection. We consider game rules analogous to those from the Prisoner's Dilemma and Stag Hunt games, with payoffs associated to the probabilities of the individuals to advance to the selected site. We find that, even when defecting is the rational choice for any agent, under certain conditions, cooperators can take advantage from mutual cooperation and leave the room more rapidly than defectors.

Directed transport induced by α-stable Lévy noises in weakly asymmetric periodic potentials.

We study the motion of a particle in spatially periodic potentials with broken mirror symmetry under the influence of white α-stable Lévy noises. We consider both time-independent and fluctuating potentials. We focus on cases in which the spatial asymmetry of the potential is due not to a difference between the distances from an absolute minimum to the absolute maximum on its left and to the absolute maximum on its right but only to the curvatures of the potential profiles. The analysis is performed using the fractional Fokker-Planck formalism. Consistent results from Langevin simulations are also presented. We analyze the influence of the symmetry properties of the potentials in combination with the fluctuating characteristics of the system in the determination of the current. We find different situations in which both the absolute value and the direction of the current depend not only on the properties of the potential but also on the parameters characterizing the α-stable Lévy noise. Among other features, we analyze the case of supersymmetric potentials. In particular, we show that a static supersymmetric potential produces no current, and we analyze the conditions for observing a nonvanishing current when the potential fluctuates between different supersymmetric profiles.

Dynamics of learning in coupled oscillators tutored with delayed reinforcements.

In this work we analyze the solutions of a simple system of coupled phase oscillators in which the connectivity is learned dynamically. The model is inspired by the process of learning of birdsongs by oscine birds. An oscillator acts as the generator of a basic rhythm and drives slave oscillators which are responsible for different motor actions. The driving signal arrives at each driven oscillator through two different pathways. One of them is a direct pathway. The other one is a reinforcement pathway, through which the signal arrives delayed. The coupling coefficients between the driving oscillator and the slave ones evolve in time following a Hebbian-like rule. We discuss the conditions under which a driven oscillator is capable of learning to lock to the driver. The resulting phase difference and connectivity are a function of the delay of the reinforcement. Around some specific delays, the system is capable of generating dramatic changes in the phase difference between the driver and the driven systems. We discuss the dynamical mechanism responsible for this effect and possible applications of this learning scheme.

Pattern dynamics in bidimensional oscillatory media with bistable inhomogeneities.

By means of numerical simulations, we study pattern dynamics in selected examples of inhomogeneous active media described by a reaction diffusion model of the activator-inhibitor type. We consider inhomogeneities corresponding to a variation in space of the (nonlinear) reaction characteristics of the system or the diffusion constants. Three different bidimensional systems are analyzed: an oscillatory medium in a square reactor with a circular central bistable domain, and cases of a bistable stripe immersed in an oscillatory medium in a trapezoidal reactor and in a rectangular reactor with inhomogeneous diffusion. The different types of complex behavior that arise in these systems are analyzed.

Stochastic resonance in extended bistable systems: the role of potential symmetry.

We study the role of potential symmetry in a three-field reaction-diffusion system presenting bistability by means of a two-state theory for stochastic resonance in general asymmetric systems. By analyzing the influence of different parameters in the optimization of the signal-to-noise ratio, we observe that this magnitude always increases with the symmetry of the system's potential, indicating that it is this feature which governs the optimization of the system's response to periodic signals.

Reaction kinetics of diffusing particles injected into a reactive substrate.

We analyze the kinetics of trapping (A+B-->B) and annihilation (A+B-->0) processes on a one-dimensional substrate with homogeneous distribution of immobile B particles while the A particles are supplied by a localized source. For the imperfect reaction case, we analyze both problems by means of a stochastic model and compare the results with numerical simulations. In addition, we present the exact analytical results of the stochastic model for the case of perfect trapping.