Rotating spiral waves are a form of self-organization observed in spatially extended systems of physical, chemical, and biological natures. A small perturbation causes gradual change in spatial location of spiral's rotation center and frequency, i.e., drift. The response functions ͑RFs͒ of a spiral wave are the eigenfunctions of the adjoint linearized operator corresponding to the critical eigenvalues =0, Ϯ i. The RFs describe the spiral's sensitivity to small perturbations in the way that a spiral is insensitive to small perturbations where its RFs are close to zero. The velocity of a spiral's drift is proportional to the convolution of RFs with the perturbation. Here we develop a regular and generic method of computing the RFs of stationary rotating spirals in reaction-diffusion equations. We demonstrate the method on the FitzHugh-Nagumo system and also show convergence of the method with respect to the computational parameters, i.e., discretization steps and size of the medium. The obtained RFs are localized at the spiral's core.

Rotating spiral waves are a form of self-organization observed in spatially extended systems of physical, chemical, and biological nature. In the presence of a small perturbation, the spiral wave's center of rotation and fiducial phase may change over time, i.e., the spiral wave drifts. In linear approximation, the velocity of the drift is proportional to the convolution of the perturbation with the spiral's response functions, which are the eigenfunctions of the adjoint linearized operator corresponding to the critical eigenvalues λ=0,±iω . Here, we demonstrate that the response functions give quantitatively accurate prediction of the drift velocities due to a variety of perturbations: a time dependent, periodic perturbation (inducing resonant drift); a rotational symmetry-breaking perturbation (inducing electrophoretic drift); and a translational symmetry-breaking perturbation (inhomogeneity induced drift) including drift due to a gradient, stepwise, and localized inhomogeneity. We predict the drift velocities using the response functions in FitzHugh-Nagumo and Barkley models, and compare them with the velocities obtained in direct numerical simulations. In all cases good quantitative agreement is demonstrated.

We have recently suggested that neural flow parsing mechanisms act to subtract global optic flow consistent with observer movement to aid in detecting and assessing scene-relative object movement. Here, we examine whether flow parsing can occur independently from heading estimation. To address this question we used stimuli comprising two superimposed optic flow fields comprising limited lifetime dots (one planar and one radial). This stimulus gives rise to the so-called optic flow illusion (OFI) in which perceived heading is biased in the direction of the planar flow field. Observers were asked to report the perceived direction of motion of a probe object placed in the OFI stimulus. If flow parsing depends upon a prior estimate of heading then the perceived trajectory should reflect global subtraction of a field consistent with the heading experienced under the OFI. In Experiment 1 we tested this prediction directly, finding instead that the perceived trajectory was biased markedly in the direction opposite to that predicted under the OFI. In Experiment 2 we demonstrate that the results of Experiment 1 are consistent with a positively weighted vector sum of the effects seen when viewing the probe together with individual radial and planar flow fields. These results suggest that flow parsing is not necessarily dependent on prior estimation of heading direction. We discuss the implications of this finding for our understanding of the mechanisms of flow parsing.

We describe an approach to numerical simulation of spiral waves dynamics of large spatial extent, using small computational grids.

Here we examine the relationship between the perception of heading and flow parsing. In a companion study we have investigated the pattern of dependence of human heading estimation on the quantity (amount of dots per frame) and quality (amount of directional noise) of motion information in an optic flow field. In the present study we investigated whether the flow parsing mechanism, which is thought to aid in the assessment of scene-relative object movement during observer movement, exhibits a similar pattern of dependence on these stimulus manipulations. Finding that the pattern of flow parsing effects was similar to that observed for heading thresholds would provide some evidence that these two complementary roles for optic flow processing are reliant on the same, or similar, neural computation. We found that the pattern of flow parsing effects observed does indeed display a striking similarity to the heading thresholds. As with judgements of heading, there is a critical value of around 25 dots per frame; below this value flow parsing effects rapidly deteriorate and above this value flow parsing effects are stable [see Warren et al. (1988) for similar results for heading]. Also, as with judgements of heading, when there were 50 or more dots there was a systematic effect of noise on the magnitude of the flow parsing effect. These results are discussed in the context of different possible schemes of flow processing to support both heading and flow parsing mechanisms.