Non-specular reflections in a macroscopic system with wave-particle duality: Spiral waves in bounded media

A growing number of dynamical situations involve the coupling of particles or singularities with physical waves. In principle these situations are very far from the wave particle duality at quantum scale where the wave is probabilistic by nature. Yet some dual characteristics were observed in a system where a macroscopic droplet is guided by a pilot wave it generates. Here we investigate the behaviour of these entities when confined in a two-dimensional harmonic potential well. A discrete set of stable orbits is observed, in the shape of successive generalized Cassinian-like curves (circles, ovals, lemniscates, trefoils and so on). Along these specific trajectories, the droplet motion is characterized by a double quantization of the orbit spatial extent and of the angular momentum. We show that these trajectories are intertwined with the dynamical build-up of central wave-field modes. These dual self-organized modes form a basis of eigenstates on which more complex motions are naturally decomposed.

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“…and its mean non-dimensional angular momentum z L : 7 where rk is the position of the k th bounce and N is the total number of bounces.…”

Section: A Double Quantizationmentioning

confidence: 99%

Self-organization into quantized eigenstates of a classical wave-driven particle

et al. 2014

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“…This invites comparison with a growing number of macroscopic systems in which waves propagating in a nonlinear medium are associated with some degree of spatial localization [6], including liquid 'walker' droplets bouncing on a vibrated bath [7,8], various optical solitons [9,10] and chemical wave segments [11]. Among other common properties, each of these examples exhibits nonspecular reflections from obstacles or medium perturbations [12][13][14][15][16] and the dynamics involved in the reflection process can be quite complex [17]. It is within this context that we have undertaken the present investigation.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic dynamics of reflecting spiral waves

2014

Self Cite

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Resonantly forced spiral waves in excitable media drift in straight-line paths, their rotation centers behaving as pointlike objects moving along trajectories with a constant velocity. Interaction with medium boundaries alters this velocity and may often result in a reflection of the drift trajectory. Such reflections have diverse characteristics and are known to be highly nonspecular in general. In this context we apply the theory of response functions, which via numerically computable integrals, reduces the reaction-diffusion equations governing the whole excitable medium to the dynamics of just the rotation center and rotation phase of a spiral wave. Spiral reflection trajectories are computed by this method for both small- and large-core spiral waves in the Barkley model. Such calculations provide insight into the process of reflection as well as explanations for differences in trajectories across parameters, including the effects of incidence angle and forcing amplitude. Qualitative aspects of these results are preserved far beyond the asymptotic limit of weak boundary effects and slow resonant drift.

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“…In these models, traveling waves encode motor patterns defining movement , whilst synchrony constitutes the resting state . The current model, in which spiral waves encode for the active holding of posture, sits well in this framework, because spirals, unlike other traveling waves, have and (almost) hold a location in a specific sense (Biktasheva and Biktashev, 2003; Langham and Barkley, 2013). This makes them interesting for the kind of active almost-stillness characterizing postural control.…”

Section: Transient Neurodynamics and Spiral Wavesmentioning

confidence: 79%

“…The multiscale ubiquity of spiral waves in nature and biology (Toomre, 1969; Lechleiter et al, 1991; Winfree, 2001), and their interesting dynamical properties (Boerlijst and Hogeweg, 1991; Biktashev and Holden, 1993, 1995; Langham and Barkley, 2013), have motivated many physical, chemical, and mathematical studies. Arthur Winfree pioneered computational and empirical investigations of toroidal dynamics in chemical and biological systems (Winfree, 1967, 1972).…”

Section: Transient Neurodynamics and Spiral Wavesmentioning

confidence: 99%

Capture of fixation by rotational flow; a deterministic hypothesis regarding scaling and stochasticity in fixational eye movements

Wilkinson

Metta

2014

Front. Syst. Neurosci.

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Visual scan paths exhibit complex, stochastic dynamics. Even during visual fixation, the eye is in constant motion. Fixational drift and tremor are thought to reflect fluctuations in the persistent neural activity of neural integrators in the oculomotor brainstem, which integrate sequences of transient saccadic velocity signals into a short term memory of eye position. Despite intensive research and much progress, the precise mechanisms by which oculomotor posture is maintained remain elusive. Drift exhibits a stochastic statistical profile which has been modeled using random walk formalisms. Tremor is widely dismissed as noise. Here we focus on the dynamical profile of fixational tremor, and argue that tremor may be a signal which usefully reflects the workings of oculomotor postural control. We identify signatures reminiscent of a certain flavor of transient neurodynamics; toric traveling waves which rotate around a central phase singularity. Spiral waves play an organizational role in dynamical systems at many scales throughout nature, though their potential functional role in brain activity remains a matter of educated speculation. Spiral waves have a repertoire of functionally interesting dynamical properties, including persistence, which suggest that they could in theory contribute to persistent neural activity in the oculomotor postural control system. Whilst speculative, the singularity hypothesis of oculomotor postural control implies testable predictions, and could provide the beginnings of an integrated dynamical framework for eye movements across scales.

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Non-specular reflections in a macroscopic system with wave-particle duality: Spiral waves in bounded media

Cited by 17 publications

References 42 publications

Self-organization into quantized eigenstates of a classical wave-driven particle

Self-organization into quantized eigenstates of a classical wave-driven particle

Asymptotic dynamics of reflecting spiral waves

Capture of fixation by rotational flow; a deterministic hypothesis regarding scaling and stochasticity in fixational eye movements

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