Neuromorphic Engineering in Wetware: Discriminating Acoustic Frequencies through Their Effects on Chemical Waves.

Some features of the human nervous system can be mimicked not only through software or hardware but also through liquid solutions of chemical systems maintained under out-of-equilibrium conditions. We describe the possibility of exploiting a thin layer of the Belousov-Zhabotinsky (BZ) reaction as a surrogate for the cochlea for sensing acoustic frequencies. Experiments and simulations demonstrate that, as in the human ear where the cochlea transduces the mechanical energy of the acoustic frequencies into the electrochemical energy of neural action potentials and the basilar membrane originates topographic representations of sounds, our bioinspired chemoacoustic system, based on the BZ reaction, gives rise to spatiotemporal patterns as the representation of distinct acoustic bands through transduction of mechanical energy into chemical energy. Acoustic frequencies in the range 10-2000 Hz are partitioned into seven distinct bands based on three attributes of the emerging spatiotemporal patterns: (1) the types and frequencies of the chemical waves, (2) their velocities, and (3) the Faraday waves' wavelengths.

Effect of obstructions on growing Turing patterns.

We study how Turing pattern formation on a growing domain is affected by discrete domain discontinuities. We use the Lengyel-Epstein reaction-diffusion model to numerically simulate Turing pattern formation on radially expanding circular domains containing a variety of obstruction geometries, including obstructions spanning the length of the domain, such as walls and slits, and local obstructions, such as small blocks. The pattern formation is significantly affected by the obstructions, leading to novel pattern morphologies. We show that obstructions can induce growth mode switching and disrupt local pattern formation and that these effects depend on the shape and placement of the objects as well as the domain growth rate. This work provides a customizable framework to perform numerical simulations on different types of obstructions and other heterogeneous domains, which may guide future numerical and experimental studies. These results may also provide new insights into biological pattern growth and formation, especially in non-idealized domains containing noise or discontinuities.

Turing patterns on radially growing domains: experiments and simulations.

We study Turing pattern formation in a system undergoing radial growth in two dimensions. The photosensitive chlorine dioxide-iodine-malonic acid reaction is illuminated to inhibit patterning, with a growing non-illuminated circular domain in which the pattern develops. We examine the relationship between the linear radial growth rate and the resulting pattern morphology. Faster growth causes the pattern to form parallel to the growing boundary as concentric rings, while slower growth leads to pattern formation perpendicular to the growing boundary. We observe three distinct growth modes for the Turing patterns, which also depend on the radial growth rate. The experimental results are qualitatively reproduced in numerical simulations using the Lengyel-Epstein model with an additional term to account for the photosensitivity of the reaction. These results may provide new insight into how patterns form in growing biological systems.

Modulation of Turing Patterns in the CDIMA Reaction by Ultraviolet and Visible Light.

We have carried out the first systematic study of the effects of ultraviolet light, both alone and in combination with visible white light, on Turing patterns in the chlorine dioxide-iodine-malonic acid (CDIMA) reaction. The ultraviolet light used has a sharp peak at 368 nm and can perturb the system selectively. It primarily decomposes chlorine dioxide in a zeroth-order reaction, and when it is used to illuminate Turing patterns, shrunken spots are formed with an imperfect hexagonal arrangement. The ultraviolet light competes directly with the visible white light via the photoreaction with dissolved chlorine dioxide, which prevents the total suppression of patterns at intermediate intensities of white light. These results suggest that specific wavelengths of light in the ultraviolet spectrum selectively modify the chemistry behind the pattern formation and can be utilized to generate novel self-organized structures under forcing conditions. We propose a modified Lengyel-Epstein model to incorporate the effect of ultraviolet illumination and obtain good qualitative agreement between simulations and experiments. These results support the idea that chlorine dioxide photoreaction is a key step in modulating CDIMA patterns under ultraviolet illumination.

Multifold Increases in Turing Pattern Wavelength in the Chlorine Dioxide-Iodine-Malonic Acid Reaction-Diffusion System.

Turing patterns in the chlorine dioxide-iodine-malonic acid reaction were modified through additions of sodium halide salt solutions. The range of wavelengths obtained is several times larger than in the previously reported literature. Pattern wavelength was observed to significantly increase with sodium bromide or sodium chloride. A transition to a uniform state was found at high halide concentrations. The observed experimental results are qualitatively well reproduced in numerical simulations with the Lengyel-Epstein model with an additional chemically realistic kinetic term to account for the added halide and an adjustment of the activator diffusion rate to allow for interhalogen formation.

Analysis and prediction of aperiodic hydrodynamic oscillatory time series by feed-forward neural networks, fuzzy logic, and a local nonlinear predictor.

Forecasting of aperiodic time series is a compelling challenge for science. In this work, we analyze aperiodic spectrophotometric data, proportional to the concentrations of two forms of a thermoreversible photochromic spiro-oxazine, that are generated when a cuvette containing a solution of the spiro-oxazine undergoes photoreaction and convection due to localized ultraviolet illumination. We construct the phase space for the system using Takens' theorem and we calculate the Lyapunov exponents and the correlation dimensions to ascertain the chaotic character of the time series. Finally, we predict the time series using three distinct methods: a feed-forward neural network, fuzzy logic, and a local nonlinear predictor. We compare the performances of these three methods.

Fronts and patterns in a spatially forced CDIMA reaction.

We use the CDIMA chemical reaction and the Lengyel-Epstein model of this reaction to study resonant responses of a pattern-forming system to time-independent spatial periodic forcing. We focus on the 2 : 1 resonance, where the wavenumber of a one-dimensional periodic forcing is about twice the wavenumber of the natural stripe pattern that the unforced system tends to form. Within this resonance, we study transverse fronts that shift the phase of resonant stripe patterns by π. We identify phase fronts that shift the phase discontinuously, and pairs of phase fronts that shift the phase continuously, clockwise and anti-clockwise. We further identify a front bifurcation that destabilizes the discontinuous front and leads to a pair of continuous fronts. This bifurcation is the spatial counterpart of the nonequilibrium Ising-Bloch (NIB) bifurcation in temporally forced oscillatory systems. The spatial NIB bifurcation that we find occurs as the forcing strength is increased, unlike earlier studies of the NIB bifurcation. Furthermore, the bifurcation is subcritical, implying a range of forcing strength where both discontinuous Ising fronts and continuous Bloch fronts are stable. Finally, we find that both Ising fronts and Bloch fronts can form discrete families of bound pairs, and we relate arrays of these front pairs to extended rectangular and oblique patterns.

Target Turing patterns and growth dynamics in the chlorine dioxide-iodine-malonic acid reaction.

We study the growth dynamics of Turing patterns in the chlorine dioxide-iodine-malonic acid reaction-diffusion system in response to perturbations with visible light. We describe several mechanisms by which Turing patterns reappear after they are suppressed by illumination with a disc-shaped geometry. We observe that under specific conditions the patterns reorganize from a random configuration of spots and stripes to a set of ordered, concentric rings, which we refer to as target Turing patterns. These patterns closely resemble the unit cells of the Turing hexagonal superlattices known as black eye patterns. However, these target Turing patterns are not part of a larger superlattice structure, and they usually have a larger number of concentric rings. Numerical simulations support the experimental findings.

Forcing of Turing patterns in the chlorine dioxide-iodine-malonic acid reaction with strong visible light.

We investigate the sensitivity of Turing patterns in the chlorine dioxide-iodine-malonic acid reaction to illumination by strong white light. Intense illumination results in an increase of [I(-)], in contrast to previous studies, which found only decreased [I(-)] for weak and intermediate intensities of illumination. We propose an expanded mechanism to explain the experimental observations. Both experimental and numerical results suggest that [ClO2] is the key parameter that determines whether the high iodide state is obtained under strong illumination. When strong illumination is applied through a spatially periodic mask with black and white stripes, a dark state with high [I(-)] is produced in the illuminated domain and a light state with low [I(-)] forms in the nonilluminated domain. Depending on the black:white ratio of the mask and its wavelength, Turing patterns can coexist with either the light or the dark state in the nonilluminated domain.

Turing patterns in the chlorine dioxide-iodine-malonic acid reaction with square spatial periodic forcing.

We use the photosensitive chlorine dioxide-iodine-malonic acid reaction-diffusion system to study wavenumber locking of Turing patterns to two-dimensional "square" spatial forcing, implemented as orthogonal sets of bright bands projected onto the reaction medium. Various resonant structures emerge in a broad range of forcing wavelengths and amplitudes, including square lattices and superlattices, one-dimensional stripe patterns and oblique rectangular patterns. Numerical simulations using a model that incorporates additive two-dimensional spatially periodic forcing reproduce well the experimental observations.

Interaction of chemical patterns in coupled layers.

We investigate the interaction between reaction-diffusion systems coupled by diffusion. The photosensitive CDIMA (chorine dioxide-iodine-malonic acid) reaction allows us to study experimentally the mutual influence of two layers of Turing patterns coupled via a diffusive interaction. By illuminating each of the layers with different intensities of homogeneous external light, the chemical conditions in each layer can be shifted, allowing us to study the result of diffusive interaction between Turing patterns with different spatial configurations. Our experiments suggest a complex scenario for the interaction between different patterns, strongly dependent on the spatial characteristics of the interacting patterns. Numerical simulations are also reported in full agreement with experimental observations.

Oligomerization oscillations of L-lactic acid in solution.

We employ high-performance liquid chromatography with diode array, evaporative light scattering, and mass spectrometric detection to monitor the oligomerization of L-lactic acid in pure acetonitrile and in 70% aqueous ethanol. The production of higher oligomers appears to proceed in an oscillatory fashion. A model is presented that involves the formation of aggregates (micelles), which catalyze the oligomerization.

Locking of Turing patterns in the chlorine dioxide-iodine-malonic acid reaction with one-dimensional spatial periodic forcing.

We use the photosensitive chlorine dioxide-iodine-malonic acid reaction-diffusion system to study wavenumber locking of Turing patterns with spatial periodic forcing. Wavenumber-locked stripe patterns are the typical resonant structures that labyrinthine patterns exhibit in response to one-dimensional forcing by illumination when images of stripes are projected on a working medium. Our experimental results reveal that segmented oblique, hexagonal and rectangular patterns can also be obtained. However, these two-dimensional resonant structures only develop in a relatively narrow range of forcing parameters, where the unforced stripe pattern is in close proximity to the domain of hexagonal patterns. Numerical simulations based on a model that incorporates the forcing by illumination using an additive term reproduce well the experimental observations. These findings confirm that additive one-dimensional forcing can generate a two-dimensional resonant response. However, such a response is considerably less robust than the effect of multiplicative forcing.

Rearrangement dynamics of fishbonelike Turing patterns generated by spatial periodic forcing.

External forcing can greatly affect the evolution of Turing patterns in reaction-diffusion (RD) media. Here, we employ spatially periodic illumination in a photosensitive RD system to create Turing structures that are repetitive in one direction. We then study their relaxation in the absence of light. These unforced, fishbonelike configurations undergo self-reorganization and establish stationary arrangements, some of which fully exhibit, while others partially or completely lose, the symmetry of the initially imposed pattern.

Breathing spiral waves in the chlorine dioxide-iodine-malonic acid reaction-diffusion system.

Breathing spiral waves are observed in the oscillatory chlorine dioxide-iodine-malonic acid reaction-diffusion system. The breathing develops within established patterns of multiple spiral waves after the concentration of polyvinyl alcohol in the feeding chamber of a continuously fed, unstirred reactor is increased. The breathing period is determined by the period of bulk oscillations in the feeding chamber. Similar behavior is obtained in the Lengyel-Epstein model of this system, where small amplitude parametric forcing of spiral waves near the spiral wave frequency leads to the formation of breathing spiral waves in which the period of breathing is equal to the period of forcing.

Coupled and forced patterns in reaction-diffusion systems.

Several reaction-diffusion systems that exhibit temporal periodicity when well mixed also display spatio-temporal pattern formation in a spatially distributed, unstirred configuration. These patterns can be travelling (e.g. spirals, concentric circles, plane waves) or stationary in space (Turing structures, standing waves). The behaviour of coupled and forced temporal oscillators has been well studied, but much less is known about the phenomenology of forced and coupled patterns. We present experimental results focusing primarily on coupled patterns in two chemical systems, the chlorine dioxide-iodine-malonic acid reaction and the Belousov-Zhabotinsky reaction. The observed behaviour can be simulated with simple chemically plausible models.

Turing patterns beyond hexagons and stripes.

The best known Turing patterns are composed of stripes or simple hexagonal arrangements of spots. Until recently, Turing patterns with other geometries have been observed only rarely. Here we present experimental studies and mathematical modeling of the formation and stability of hexagonal and square Turing superlattice patterns in a photosensitive reaction-diffusion system. The superlattices develop from initial conditions created by illuminating the system through a mask consisting of a simple hexagonal or square lattice with a wavelength close to a multiple of the intrinsic Turing pattern's wavelength. We show that interaction of the photochemical periodic forcing with the Turing instability generates multiple spatial harmonics of the forcing patterns. The harmonics situated within the Turing instability band survive after the illumination is switched off and form superlattices. The square superlattices are the first examples of time-independent square Turing patterns. We also demonstrate that in a system where the Turing band is slightly below criticality, spatially uniform internal or external oscillations can create oscillating square patterns.

Effect of axial growth on Turing pattern formation.

We have performed one-dimensional and two-dimensional experiments and simulations to study the formation of patterns in a system that grows continuously in one direction. Depending on the growth velocity, three basic spatial configurations can be obtained: stripes that are parallel, oblique, or perpendicular to the growth direction. The dependence of the wavelength on the growth velocity has also been observed. Our results illustrate the importance of these growth mechanisms in determining the final configuration of chemical and biological pattern-forming processes.

Dynamic mechanism of photochemical induction of turing superlattices in the chlorine dioxide-iodine-malonic acid reaction-diffusion system.

We study the mechanism of development of superlattice Turing structures from photochemically generated hexagonal patterns of spots with wavelengths several times larger than the characteristic wavelength of the Turing patterns that spontaneously develop in the nonilluminated system. Comparison of the experiment with numerical simulations shows that interaction of the photochemical periodic forcing with the Turing instability results in generation of multiple resonant triplets of wave vectors, which are harmonics of the external forcing. Some of these harmonics are situated within the Turing instability band and are therefore able to maintain their amplitude as the system evolves and after illumination ceases, while photochemically generated harmonics outside the Turing band tend to decay.

Turing pattern formation in a two-layer system: superposition and superlattice patterns.

Turing patterns in the chlorine dioxide-iodine-malonic acid reaction are studied in a system consisting of two coupled gel layers. Patterns with two wavelengths are observed. Changing the strength of the interlayer coupling causes a transition between a superposition of Turing patterns and a superlattice pattern. The effects of the reactant concentrations on the pattern wavelengths are delineated.