J. Phys. Chem. 1994, 98, 3999-4002
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Oscillation Phase Dynamics in the Belousov-Zhabotinsky Reaction. Implementation to Image Processing Rubin R. Alievf Institute for Theoretical and Experimental Biophysics, Puschino, Moscow Region, 142292 Russia Received: May 26, 1993; In Final Form: December 6. 1993’
The phase jump dynamics in a photosensitive Belousov-Zhabotinsky reaction has been studied. It was found that an initially inhomogeneous phase distribution evolved to a homogeneous one, so that the phase a t phase-lag points rises to the value a t phase-lead points. A steep phase distribution was shown to become smoother in the course of time, inducing a dephasing wave which restores the system to a uniform phase. Applications of photosensitive chemical reactions to image processing (image storage, inversing, contouring) are discussed.
Introduction A number of processes in solid-state physics, theory of lasers, chemistry, and biology can be described in terms of the theory of coupled oscillators.’ Autooscillating chemical reactions, such as the Belousov-Zhabotinsky (BZ) reaction, may be regarded from the same viewpoint: each microvolume of the medium presents a nonlinear oscillator diffusively coupled to its neighbors. Study of two and several connected cells containing BZ reagents shows the existence of antiphase oscillations, chaos, and other kinds of nontrivial behavior.**’ In a continuous array of coupled oscillators, realized by the BZ reaction, traveling waves were observed.M Further investigations show that there are two types of waves having different characteristic^:^-'^ (i) Waves of constant shape and low speed (1-4 mm/min) are called trigger waves. Such waves are emitted by spiral or concentric wave sources. (ii) The second type are rapid waves whose velocity is many times superior to the velocity of trigger ones, with no upper limit for the velocity. These waves can penetrate through any barrier impermeable for diffusion. These are phase waves or pseudowaves that are similar to phaseshifted oscillations in a chain of weakly coupled pendula. Another wave classification was introduced by Bodet and Ross‘s and in refs 16 and 17: (i) trigger, (ii) diffusion phase waves, and (iii) kinematic waves were separated. To avoid the confusion, consider the equation for wave velocity u:
Here &/at and &/ax are partial derivatives of phase 4 with respect to time t and space x (gradient 4). At high phase gradient, we obtain slow trigger waves, and at low phase gradient, “phase” = “kinematic” waves appear. In the intermediate regi0n“diffusion phase waves” occur. Hereafter the term “phase waves” stands for both diffusion phase and kinematic waves. It should be noted that there is no strict boundary between different types of waves. Trigger and kinematic waves in an oscillatory system denote the two limiting cases of traveling waves: kinematic waves assume no mass transfer; Le., the dynamics of a cell is not affected by its neighbors. In contrast, trigger waves are regarded as a result of the reaction and diffusion, with saying nothing about the influence of system autooscillations on wave propagation. As observed, trigger waves in the BZreaction have been studied in detail (see ref 14 for references), while less attention was paid to phase waves. The phase dynamics in distributed systems was studied mainly theoretically.1”26 In these papers rapid accelerating phase waves and chaotic behavior have been described. t Internet: aliev8venus.iteb.serpukhov.su,
[email protected]. a Abstract published in Advance ACS Abstracts, January 15, 1994.
0022-3654/94/2098-3999$04.50/0
BZ reaction simulations performed in the Oregonator model by Reusser and Field27 show that depending on the value of the phase gradient either slow trigger or rapid phase waves occur. The influence of the phase gradient on the type of waves was also studied experimentally by Bodet and Ross.15 The present paper describes experimental investigations of wave processes occurring in the inhomogeneous in phase medium. There is one more point of interest in studying phase distributions in thelight-sensitive BZ reaction. This is the possible utilization of such a system as an image processor.28 The idea looks attractive because a processor unit created on the basis of the BZ reaction would be an extremely parallel one, enabling one to solve some problems of image treatment.29 Such a system can store images, make negatives, and perform an edge detection.2**30 This paper reports the evaluations for storage time available in the reaction and discusses possible structure of image filters. Experimental Setup The photosensitive Belousov-Zhabotinsky reaction was used for experimentation. It consisted of malonic acid (0.167 M), sodium bromate (0.33 M),sulfuric acid (0.33 M), and ruthenium bipyridyl (1.7 mM) at 20 O C . A 4.8-mL sample of the mixture was stirred, poured into a Petri dish (8 cm in diameter), and covered with a lid. After about 10-min delay (time responsible for the bromomalonic acid synthesis and the equilibrium establishing), the medium was ready for experimentation. A slide projecting device (LETI-60M) equipped with a 500-W lamp, optical focusing system, and I R cut filters was used as a light source for image projecting and for background lighting. The desired light field was created by projecting a slide mask. All the experiments were preceded by a careful mixture stirring and strong light exposure during 3-5 min. This procedure suppresses any activity and resets the phase of oscillation which causes the medium to be homogeneous in phase. The image registration system involved a RGB CCD camera with a blue filter (passing light of wavelengths below 480 nm), VCR, and AT386 computer with frame grabber and frame processor cards (DT2851-58, Data Translation). An ordinary technique was used for image processing: patterns arising in the BZ reaction were stored on a video cassette, and then spatial and temporal characteristics were retrieved by means of a computer image processing. The figures representing X-t diagrams were obtained from a series of 2D frames stored on a video cassette. On each frame a cut along the straight line was taken and stored on disk. A number of such cuts were linked together to form an x-t diagram. From this diagram equiphase lines were extracted. These are the lines of equal phase measured a t the front edge at level of 0.7 amplitude of excitation at every point (see Figure 1). The time 0 1994 American Chemical Society
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Figure 2. x-t diagrams of the phase jump soon after mask removal (a, bottom) and in 20 oscillations(b, top). The decrease of the phase gradient with time is seen. The x axis is scaled as in Figure 1; time ranged from -2 to 126 s (a) and from 510 to 638 s (b).
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Figure 1. (a, top) Space-time (x-t) diagram of phase-shifted oscillations. The x axis stands for space in the range 0-38.3 mm; they axis represents time in the range -2 to 638 s. Zero time corresponds to the removal of the mask used to shadow a part of the medium and induce the initial phasejump. Black regions denote large light transmission. Initial phase jump was 35 s; the period of bulk oscillationswas 21.8 s. In the upper-left corner parasitic oscillationsappear because of an outer spiral wave source. (b, bottom) Equiphase lines show the phase evolution in space and time.
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Experimental Results Figure 1 shows phase-shifted oscillations obtained in the BZ reaction after the left-hand part of the medium was subjected to the light irradiation during 50 s, while the right-hand part was shadowed with a mask; the cut shown in the figure was parallel to the light-field gradient. As a result, oscillations in the left part of the medium are behind in phase with respect to those in the right part (phase count started at the moment of removing the mask). It is seen that the medium can be separated into three distinct areas: left- and right-hand parts where synchronized oscillations occur and an intermediate area of a phase jump, connecting the said parts together. Such a jumpon an x-t diagram stands for a phase wave propagating from right to left. The jump becomes smoother with time (Figure 2); Le., a phase alignment occurs in the system. Comparison of parts a and b of Figure 2 shows that, over the course of time, more and more points of the phase-lagged left part of the medium adjust their phase toward that occurring in the phase-leading right part. This process for several points inside a phase jump is clearly seen in Figure 3.
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b Figure 3. Phase trajectories in the BZ reaction. (a) Dotted lines represent the phase of unperturbed bulk oscillationsat both sides of a phase jump. Solid lines between them are taken at points initially placed at a phase jump. Such points adjust their phase to that of phase-lead points (lower dotted line). The lines from up to down measured at points a distance of 12.9,19.0,20.5, and 23.6 mm from the left wall in Figure 1. (b) Phase shift with respect to phase-lead points. Phase shift decreases and tends to zero as a dephasing wave passes. The uppermost line measured at 12.9 mm from the left wall of Figure 1. The lower lines spaced with an increment of 1.53 mm.
The phase gradient decrease results in an increase of the absolute value of the phase wave velocity (eq 1). The values measured in the experiment show the increase of propagating velocity, u+, from 0.123 to 0.186 mm/s during 25 oscillations.
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CY* Figure 4. Phase wave velocity measured close to a phase jump. Each curve presents velocity at a single point and has a bell-shaped form: the velocity is infinity (i.e., bulk oscillations) far from the phase jump; the velocity is minimum at a place of maximum phase gradient. Because of the dephasing wave, the phase jump travels in space. Note also the minimum phase velocity increases with time. Dotted line above shows the velocity of a trigger wave emitted by a spiral wave source. Curves 1-7 spaced by 1.53 mm starting at 12.9 mm from the left wall of Figure 1.
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Figure 6. Light transmission profiles at different moments. (a) t = 13.75 s, right part of the medium is excited; (b) t = 27.5 s, negative to (a); (c) t = 32.0 s, “edge detection”: excitation exists only at the place of the phase gradient location. Note that in (a) and (b) phase waves also emphasize edges. These occur because the initial phase shift exceeded the period of bulk oscillations.
region (dark in Figure 5 ) the points advance their phase to the value of phase-leading points. Note that this region of short period moves from right to left, having a distinct front edge and a more gradual tail. This phenomenon can be referred to as a dephasing wave. This wave moves from the region where phaselead points occur to phase-lag regions and synchronizes oscillations, establishing the uniform phase in the medium. The shift of the dephasing wave is seen in Figure 2 as well. Note that the points of collision of phase waves and bulk oscillations (seen as angularities in Figures 1 and 2) move from right to left. These points are situated on the front of the dephasing wave, as seen in combining Figures 5 and 1. The velocity of the dephasing wave, measured from Figure 5, is constant at about 0.0087 mm/s. This value is significantly lower than the velocity of both phase and trigger waves and determines the rate with which the system loses information about its initial phase distribution. Implementation to Image Processing
Figure 5. Dephasing wave on an x-t plane. Darkness represents the period of oscillations: dark region correspondsto faster oscillations.Points of the fastest oscillations (period 10-1596 smaller than that of bulk oscillations) form the front of the dephasing wave. Axes scaled as in Figure 1.
These are the minimum values occurring at the point of maximum gradient inside a phase jump (Figure 4). As discussed above, there is no upper limit for the phase wave velocity; the value tends to infinity at the places of zero phase gradient. These are the areas ahead and back of a phase jump where synchronized bulk oscillations occur (Figure 4). The dotted line in the upper part of this figure designates the velocity uo of trigger waves radiated by a spiral wave source. uo = 0.07 mm/s, measured 10 min after beginning the experiment, varies by less than 10% during recording. (A train of trigger waves, emitted by an external spiral wave source, is seen in the upper-left corner of Figure 1.35) The value of uo is much smaller than ub (the velocity of waves initiated on the phase jump). This allows us to refer to waves initiated on the phase jump as phase waves. The phase synchronization implies that different points of the medium have either higher or lower frequency of oscillations, depending on whether the phase rises or drops at these points. Figure 5 presents the period of oscillations vs space and time. (The period at a point (x0,to) was assumed to be the time interval between successive excitations at xo.) One can see that the period is shorter in the region where phase gradient is steeper. In this
Recently, a series of papers appeared discussing a possible structure of an image processor on the basis of the photosensitive BZ r e a c t i ~ n . ~Projecting ~ . ~ ~ an image to the reaction creates a nonuniform oscillation phase distribution. It is the nonlinear phase interactionsthat are responsible for the observed phenomena of image processing in the system. In the current work, the simplest element of an image, an edge, was applied to the BZ processor. While studying the phase dynamics induced by projecting such pictures, it is easy to comprehend the main features of image processor operations described by Kuhnert, Agladze, and Krinsky.28 The image storage observed in the system28 implies retaining the initial phase distribution. The main parameter for possible implementations is the time of reliable storage. This time is limited because of nonideal conditions in the system, which lead to spontaneous appearance of parasitic wave sources disturbing the stored phase distribution (see Figure 1, upper left). In the absence of such disturbing wave sources, the limitation is imposed by the dephasing wave described above. The existence of this wave means that the system “forgets” objects of size L in time L/u,u being the dephasing wavevelocity. The smaller the picture details that must be retained, the faster the memory regeneration that must be applied. Under the used conditions, the system stores imagesof size0.2 mm during about onecycle, which restricts spatial resolution of the system. On the other hand, the dephasing wave results in image smoothing, so it can be implemented for low-pass filtering. Figure 6 illustrates other image processing operations. Neg-
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ative and positive image switching is illustrated in parts a and b. Another, the most nontrivial operation maintained by such an image processor is the “edge detection”. This implies that, after projecting an image onto the system, an excitation periodically appears at the places of picture edge locations (Figure 6c). The reason underlying the phenomenon is the phase wave which propagates at the place of phase gradient. This wave excites the points of the medium at the moment when equiphased medium regions are at rest (Figure 6c), emphasizing the phase gradient location. It should be noted that in the case where the phase shift is larger than the period of bulk oscillations, the phase wave a t the place of edge position is seen in Figure 6a,b as well. One more interesting phenomenon described in the previous section is that the final phase established in the system is not the mean phase, but the phase of the most phase-leading area (see Figure 3). This results in a nonequivalence of phase-leading and phase-lagging points; i.e., black and white parts of an image induce the appearance of phase-leading and phase-lagging areas in the BZ medium after projecting and therefore evolve in a different manner. Such nonsymmetric behavior enables one to construct intensity-dependent filters that could, for example, filter out noise consisting of dark spots on a light background.
frequency as compared to those in the nonilluminated region. That is why during the illumination and a short period after switching it off there is a gradient of frequency in addition to the phase gradient. The measurements reported here were carried out after such transients have died out. (To check this, compare hardly distinguishable periods of oscillation in the left, “exirradiated” and right parts of Figures 1 and 2.) The reasons for the use of the liquid layer BZ reaction for the experimentation lie in its simplicity. Figures 1 and 5 show that the period of bulk oscillations was approximately constant during the registration time (21.8 s at the beginning and 24.4 s in 30 oscillations). Such small period drift allows us to neglect reaction aging and refrain from using complicated continuously fed gel reactors.
Discussion Propagation of the dephasing wave described above is clearly seen in Figure 5 during 25 cycles (10 min). Existence of such a wave was not predicted theoretically, because the majority of theoretical studied geheric cases and were restricted to terms of second order in function expansions. These are diffusion-like terms that in general case predict the phase blurring, as seen in the tail of the dephasing wave (Figure 5 ) . It is the account of peculiar BZ reaction characteristics that allows us to correctly describe the sharp front of the dephasing wave experimentally ~ b t a i n e d . ~ ] One of the referees has rightfully noticed that, at the place of phase discontinuity, trigger waves could be initiated; meanwhile, the waves are referred to as phase waves everywhere in the paper. The formal answer is that in our experiments initial phase gradient was not large enough to initiate a trigger wave. On the other hand, this question concerns the problem of wave classification in an oscillatory system. The term “trigger waves” has been introduced while studying excitable media, where such waves are described as waves of constant shape with sharp fronts and low velocity.14J7 Direct implementation of thedefinition to oscillatory media is impossible. Consider, for instance, a propagating solitary trigger wave in an unbounded oscillatory system. Spontaneous bulk oscillations occurring far ahead and back of such a wave would interfere with the wave propagation, raising its velocity and smoothing the shape. (The phenomenon was noticed by Zeldovich.34) In an oscillatory medium, solitary trigger waves or finite trains of such waves should evolve into phase waves. Thus, neat wave classification applicable for oscillatory systems is the problem for a future work. Phase waves initiated at the places of phase gradient most likely correspond to so-called “big waves”32.33 recently observed in the BZ reaction.36 Common features of these phenomena are a very high propagating velocity and self-acceleration. As described above, this is the result of dephasing wave propagation, which smooths phase gradients and according to eq 1 speeds up wave propagation. The data of a single experiment are presented in the figures, but a dozen have been carried out. The presented data are from the long-term experiment. The great difficulty in studying the photosensitive BZ reaction is the spontaneous appearance of highfrequency wave sources, spiral waves, and target patterns, which destroy the phase distribution studied. Short-time observations confirm the results presented. It should be noted that the illumination used to create inhomogeneous phase distribution results in change of local
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Acknowledgment. We thank Drs. Vadim N. Biktashev and Mikhail R. Stepanov for helpful discussionsof thesubject. Special acknowledgement is given to Professor Arthur T. Winfree, who read the manuscript carefully and helped in its structure improvement. References and Notes
