J. Phys. Chem. 1996, 100, 4437-4441
4437
Stochastic Resonance in Chemistry. 1. The Belousov-Zhabotinsky Reaction A. Guderian, G. Dechert, K.-P. Zeyer, and F. W. Schneider* Institute of Physical Chemistry, UniVersity of Wu¨ rzburg, Marcusstrasse 9/11, 97070 Wu¨ rzburg, Germany ReceiVed: August 4, 1995; In Final Form: October 11, 1995X
We demonstrate the phenomenon of stochastic resonance in a nonlinear chemical reaction. The term “stochastic resonance” (SR) denotes the detection of a weak periodic signal in a noisy system displaying a threshold. If the sum of the periodic signal and the noise amplitude crosses the threshold, an output pulse is triggered. At an optimal noise amplitude the distribution of pulse intervals as well as the signal-to-noise ratio will pass through a maximum. In the continuously stirred tank reactor (CSTR) experiments we superimpose a periodic flow rate signal on an excitable focal steady state located near a Hopf bifurcation in the Belousov-Zhabotinsky (BZ) reaction. In the Showalter-Noyes-Bar-Eli model of this reaction we vary the perturbation frequency and amplitude as well as the pulse length of the applied noise to elaborate the optimal conditions for stochastic resonance to occur in the model.
1. Introduction Stochastic resonance (SR) refers to the enhancement of a weak periodic signal by external noise in systems that display a threshold. When the sum of the subthreshold periodic signal and the noise amplitude crosses the threshold, the excitable system responds with a burst signal. The signal-to-noise ratio and the interspike distribution at the frequency of the weak periodic signal both pass through a maximum at an optimal noise amplitude. In a recent review1 stochastic resonance has been described to occur in several physical,2-11 biological,12,13 and technological14 systems. For the improvement of the signalto-noise ratio the term stochastic resonance was used in NMR experiments already in 1970 where broad-band excitation occurred simultaneously with detection.15 In recent work periodic and random perturbations of focal steady states have been investigated in the BZ, the peroxidaseoxidase (PO), and the methylene blue-sulfide-oxygen systems via the flow rate and the electrical current.16-22 In the present study we investigate SR in the Belousov-Zhabotinsky (BZ)23 reaction in a focus where a sinusoidal signal together with noise is imposed on the flow rate of reactants into the reactor. The (excitable) focus is located close to a primary Hopf bifurcation beyond which small-amplitude period-1 oscillations are observed. Subsequently a secondary Hopf bifurcation occurs which leads to large-amplitude mixed-mode oscillations called bursts. The latter bifurcation is called a torus bifurcation.17 The threshold of the system is represented by a complex function of several parameters involving the modulation frequency, the noise amplitude, the duration of the noise, and the distance of the focus from the primary and the secondary Hopf bifurcation. The analysis of the time series of bursts demonstrates that an optimal noise amplitude exists for the detection of the imposed sinusoidal signal. In concurrent work (part 2 of this series) SR has been demonstrated also in a nonlinear enzymatic reaction, the PO reaction which displays similar threshold properties.24 * To whom correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, February 15, 1996.
0022-3654/96/20100-4437$12.00/0
Figure 1. CSTR (3.35 mL volume): The flow rate of the three feed lines is varied by a sinusoidal modulation and by stochastic noise. The redox potential is measured via a Pt/Ag/AgCl redox electrode.
2. Experimental Setup Materials. Malonic acid was purchased from Merck and recrystallized twice from acetone to remove trace impurities.25-27 Sulfuric acid, cerous sulfate (Riedel-de Hae¨n), and potassium bromate (Merck) were of analytical grade and used without further purification. The water was purified by ion exchange (water purification system Milli-Q, Millipore; specific resistance g10 MΩ cm). All solutions were equilibrated with air. Reactor. The experimental setup is shown in Figure 1. The CSTR (3.35 mL volume) contains a Pt/Ag/AgCl reference electrode (Ingold) to monitor the change in the redox potential. Due to the variations in the sensitivity of the redox electrode, we present the output in arbitrary units. The solutions are mixed effectively by a Teflon stirrer at a high stirring rate of 1100 rpm. Previous work has shown that stirring rates between ∼600 and ∼1100 rpm produce identical results in the BZ reaction.17 In contrast, the minimal bromate oscillator is sensitive to the stirring rate in the same interval due to the effect of fast reaction steps in its mechanism.28,29 The reactor is fed with three syringes containing the reactant solutions which enter through © 1996 American Chemical Society
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Guderian et al.
Figure 2. Experimental time series of 14 000 s (upper part) and a section (for better viewing) of the flow rate modulation of the CSTR inflow (lower part). The sinusoidal frequency was 8.38 × 10-3 rad/s (T ) 750 s). Notice that the focus occurs at a higher flow rate than the threshold. The effective threshold in the flow rate is approximated by a dashed line. (a, top left) Time series with a noise amplitude of β ) 0.11 slightly below the excitation threshold. (b, bottom left) Time series with β ) 0.13 at threshold showing some noise-induced bursts. (c, top right) Time series with β ) 0.15 close to the optimal signal-to-noise ratio. (d, bottom right) Noise amplitude above the optimal signal-to-noise ratio (β ) 0.25).
the bottom of the CSTR:
syringe 1:
0.42 M KBrO3
syringe 2:
1.5 × 10-3 M Ce3+ (from Ce2(SO4)3) 0.9 M malonic acid
syringe 3:
1.125 M H2SO4
To obtain reactor concentrations, divide by three. The reactor, the feed lines, and the syringes are thermostated at 25.0 ( 0.2 °C. Continuous Flow Conditions. The feed lines of the reactor are supplied by a precise linear pump, which is driven by a computer via a DA converter. At a flow rate of kfH ) 2.01 × 10-3 s-1, corresponding to a residence time of τ ) 8.3 min, the BZ reaction displays a supercritical Hopf bifurcation with the parameters used. When the flow rate is decreased a small region of period-1 oscillations with low amplitudes is observed. Decreasing the flow rate further to 1.89 × 10-3 s-1 (τ ) 8.8 min) leads to mixed-mode oscillations with high amplitudes which occur beyond a secondary Hopf bifurcation.
Periodic and Stochastic Modulation of the Flow Rate. A sinusoidal modulation is imposed on the flow rate with a given frequency and amplitude via a DA converter. Equation 1 shows the total variation imposed on the flow rate, where kf° is the constant flow rate producing the focus, ω is the frequency of the sinusoidal flow rate signal, R is the amplitude, δ the length of a noise pulse, and β the amplitude of the equally distributed noise. The noise R(δ) is generated externally by a random number generator.
kf ) kf°(1 + R sin(ωt) + βR(δ))
(1)
Data Evaluation. The Fourier spectra of each time series were calculated. The signal-to-noise ratio was determined from the Fourier spectra as the ratio of the fundamental peak to the background noise at the signal frequency. Alternatively, the distribution of the intervals between the bursts, the so called interspike histograms, were calculated.30,31 In the experiments 14 000 data points were recorded in intervals of 1 s. The Fourier transformation was carried out with 4.48 × 10-4 rad/s frequency resolution.
Stochastic Resonance in Chemistry. 1
Figure 3. (a, top) Stochastic resonance: signal-to-noise ratio (SNR) as evaluated from the Fourier spectra versus the noise amplitude. The signal-to-noise ratio passes through a maximum at approximately β ) 0.15. (b, bottom) Interspike histogram curve: Number of burst intervals occurring from 700 to 799 s versus the noise amplitude (maximum at approximately β ) 0.16).
3. Results 3.1. Experimental Results with the BZ Reaction. The BZ reaction was run in a focal steady state at a constant flow rate of kf° ) 2.48 × 10-3 s-1 (τ ) 6.7 min) which is 23% above the supercritical Hopf bifurcation.17,21,22 After at least three residence times have elapsed, a sinusoidal flow rate modulation with ω ) 8.38 × 10-3 rad/s (T ) 750 s) was imposed on the focus. At a signal amplitude of R ) 0.62 simple response oscillations of the same period were observed without external noise. When small amplitude external flow rate noise (β ) 0.11) was added to the subthreshold signal, the response consisted of simple oscillations without bursts (Figure 2a). However, when a slightly higher level of external noise (β ) 0.13) was added to the subthreshold signal (ω ) 8.38 × 10-3 rad/s; R ) 0.62) bursting started (Figure 2b). Notice that the present Hopf bifurcation from the focus to the periodic regime occurs in the direction of decreasing flow rates. This is why the lower end values of the flow rate modulation with varying noise amplitudes are shown in Figure 2a-d. The pulse length of the noise was held constant at δ ) 5 s for all experiments. Shortly above threshold the bursts consist of one or two spikes per oscillation (Figure 2b). The number of spikes per oscillation increases with increasing noise (Figure 2b,c). Figure 2c shows a time series obtained near the maximum of the stochastic resonance effect. At this noise strength (β ) 0.15) the system shows several bursts consisting of closely spaced spikes. Figure 2d shows a time series at β ) 0.25 above the optimal value of the noise where the number of bursts has decreased. We determined the signal-to-noise ratios at various noise amplitudes from the Fourier spectra of the individual time series (Figure 3a). At a noise amplitude of β ≈ 0.15 the maximum signalto-noise ratio was observed. From the individual histograms the number of burst intervals occuring between 700 and 799 s as a function of the noise level is shown in Figure 3b. A maximum value is found at a noise amplitude of β ≈ 0.16 which
J. Phys. Chem., Vol. 100, No. 11, 1996 4439 shows good agreement with the value obtained from the maximum in the signal-to-noise ratio (Figure 3a). All points display a relatively large scatter particularly at high noise amplitudes. Stochastic resonance was also measured for a sinusoidal signal of a shorter period ω ) 4.5 × 10-3 rad/s (T ) 400 s) and R ) 0.52 imposed on a focus (kf° ) 2.11 × 10-3 s-1; τ ) 7.9 min) located 5% above the Hopf bifurcation. The flow rate in this experiment was placed closer to the Hopf bifurcation. In this case the sinusoidally perturbed focus displays bursting for R > 0.54 without external noise, which is a lower value of R than in the case above (R > 0.62). Adding flow rate noise to the subthreshold signal (ω ) 4.5 × 10-3 rad/s; R ) 0.52) causes bursting already at a low noise level of β ≈ 0.03. Here the pulse length was also held constant at δ ) 5 s. This experiment shows the same qualitative behavior to increasing noise amplitudes as the one at higher T and higher R. The number of spikes per burst increases with increasing noise amplitude. The interspike histograms at T ) 400 s pass through a maximum at a noise amplitude of β ≈ 0.12. The signal-tonoise ratio reaches a maximum at a noise amplitude of β ≈ 0.13 verifying the presence of SR. The time series for these experiments (T ) 400 s) are not shown. 3.2. Model Calculations with the SNB Model. The SNB Model. The SNB model32 was used with the rate constants given by Field and Fo¨rsterling33 except for k2, which was set to 200.0 M-1 s-1 (Table 1).17 A supercritical Hopf bifurcation occurs at kfH ) 3.662 × 10-2 -1 s and period-1 oscillations of low amplitudes emerge at lower flow rates as in the case of the experiments. Lowering the flow rate further leads to complex periodic oscillations with high amplitudes. We chose a focal steady state at kf° ) 3.7719 × 10-2 s-1, which is located 3% above the Hopf bifurcation described above. This focal steady state was perturbed sinusoidally according to eq 1. Its resonance frequency was determined by the scan method16 to be 23.5 s. A total of 60 000 points were recorded at intervals of 1 s to determine the signal-to-noise ratio from the corresponding Fourier spectra. In the simulations the Fourier transformation was done with 2.0 × 10-4 rad/s frequency resolution. The interspike histogram was obtained by evaluating 1000 interval times between bursts. The first 20 000 s transients were always discarded. For a sinusoidal signal of frequency ω ) 0.062 83 rad/s (T ) 100 s) and perturbation amplitude of R ) 0.21 bursts are observed only if noise is added. The noise leads to bursting when it drives the system across the excitation threshold for a sufficiently long time. Figure 4a shows the number of burst intervals occuring between 95 and 104 s around T ) 100 s. To show the scatter due to a different choice of random numbers, five different runs are depicted in Figures 4a,b and 5. The number of burst intervals between 95 to 104 s passes through a maximum at a noise amplitude of β ≈ 0.27 as typical for the phenomenon of stochastic resonance. However, when noise of a longer pulse duration of 10 s is chosen the maximum effect of stochastic resonance is shifted to a lower noise amplitude of β ≈ 0.17 (Figure 4b), whereas the general shape of the resonance curve is similar to the case of a pulse length of 2 s. The signalto-noise ratios from the Fourier spectra (Figure 4c) pass through a maximum at the same noise amplitude as the interspike histograms. We further studied a periodic signal of a lower frequency of ω ) 0.020 94 rad/s (T ) 300 s) with an amplitude of R ) 0.24.
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TABLE 1 the SNB model A + Y + 2H h X + P k1 ) 2.0 M-3 s-1; k2 ) 200 M-1 s-1 X + Y + H h 2P k3 ) 3 × 106 M-2 s-1; k4 ) 2 × 10-5 M-1 s-1 A + X + H h 2W k5 ) 42 M-2 s-1; k6 ) 4.2 × 107 M-1 s-1 W + C + H h X + Z′ k7 ) 8 × 104 M-2 s-1; k8 ) 8.9 × 103 M-1 s-1 2X h A + P + H k9 ) 3 × 103 M-1 s-1; k10 ) 1 × 10-8 M-2 s-1 Z′ f gY + C k11 ) 0.2 s-1 A ) BrO3-, Z′ ) Ce4+, C ) Ce3+, P ) HOBr, H ) H+, W ) BrO2•, X ) HBrO2, Y ) Br-, g ) 0.45 (stoichiometric factor) initial values A(0) ) 0.08731 M W(0) ) 5.931 × 10-7 M Y(0) ) 2.6 × 10-6 M C(0) ) 9.0122 × 10-4 M X(0) ) 9.687 × 10-6 M Z′(0) ) 9.87 × 10-5 M P(0) ) 5.129 × 10-4 M inflow concentrations A0 ) 0.0875 M C0 ) 1 × 10-3 M H+ ) 0.7466 M (constant; not included in the rate constants)
Figure 5. (a, top) SNB model: number of burst intervals occurring between 295 and 304 s using a perturbation period of 300 s and a noise pulse length of 2 s. 1000 burst intervals were evaluated. (b, bottom) Same as Figure 5a but with a noise pulse length of 10 s.
4. Discussion
Figure 4. (a, top) SNB model: Number of burst intervals occurring between 95 and 104 s using a perturbation period of 100 s and a pulse length of 2 s. 1000 burst intervals were evaluated. (b, middle) Same as Figure 4a but with a noise pulse length of 10 s. (c, bottom) Signalto-noise ratio for the constraints of Figure 4b.
This amplitude was chosen to be 18% below the bursting threshold as in the case above. When we additionally applied noise of a pulse length of 2 s, we observed stochastic resonance at a very low noise amplitude of β ≈ 6.6 × 10-4 (Figure 5a). For a longer pulse duration of 10 s SR occurs at β ≈ 3.3 × 10-4 (Figure 5b). Periodic signals of higher frequencies than ω ) 0.157 rad/s (T ) 40 s) did not show SR. It is assumed that a signal period of 40 s is about equal to the recovery time (refractory time) for a burst.
We have demonstrated the phenomenon of stochastic resonance with a nonlinear chemical reaction in an excitable focal steady state close to a supercritical Hopf bifurcation.17,21,22 Our experiments with the BZ reaction are in qualitative agreement with the SNB model.32 The calculations as well as the experiments show that optimal noise levels for the enhancement of periodic signals also exist in nonlinear chemical reactions when certain conditions are fulfilled. For example, the chosen focus and the amplitude of the periodic signal must be sufficiently close to the excitation threshold. The period of the signal must not be smaller than the refractory time, which denotes the shortest possible time interval between two subsequent excitations (bursts). For even higher sinusoidal signal frequencies only multiples of the sinusoidal period are observed in the model system. An increase in the noise frequency and an increase in the sinusoidal signal frequency both require higher optimal noise levels for SR to occur. Thus, in a chemical reaction the effective threshold is not a constant quantity but it is a complex function of the described parameters. SR has also been demonstrated in other nonlinear chemical reactions with
Stochastic Resonance in Chemistry. 1 threshold properties, namely in the enzymatic peroxidaseoxidase reaction24 (part 2), and in the minimal bromate oscillator34 (part 3). Further work is in progress. Acknowledgment. We thank the Deutsche Forschungsgemeinschaft and the Volkswagenstiftung for partial financial support. We thank A. Fo¨rster for valuable discussions. References and Notes (1) Wiesenfeld, K.; Moss, F. Nature 1995, 373, 33. (2) Benzi, R.; Sutera, A.; Vulpiani, A. J. Phys. 1981, 14A, L453. (3) Nicolis, C. Tellus 1982, 34, 1. (4) Benzi, R.; Parisi, G.; Sutera, A.; Vulpiani, A. Tellus, 1982, 34, 10. (5) Fauve, S.; Heslot, F. Phys. Lett. 1983, 97A, 5. (6) McNamara, B.; Wiesenfeld, K.; Roy, R. Phys. ReV. Lett. 1988, 60, 2626. (7) Simon, A.; Libchaber, A. Phys. ReV. Lett. 1992, 68, 3375. (8) Gammaitoni, L.; Martinelli, M.; Pardi, L.; Santucci, S. Phys. ReV. Lett. 1991, 67, 1799; J. Statist. Phys. 1993, 70, 425. (9) Dykman, M. I.; Velikovich, A. L.; Golubev, G. P.; Luchinskii, D. G.; Tsuprikov, S. V. Pis’ma Zh. Eksp. Teor. Fiz. 1991, 53, 182. (10) Grohs, J.; Apanasevich, S.; Jung, P.; Issler, H.; Burak, D.; Klingshirn, C. Phys. ReV. 1994, 49a, 2199. (11) Spano, M. L.; Wun-Fogle, M.; Ditto, W. L. Phys. ReV. 1992, 46a, 5253. (12) Douglass, J. K.; Wilkens, L.; Pantazelou, E.; Moss, F. Nature 1993, 365, 337. (13) Moss, F.; Douglass, J. K.; Wilkens, L.; Pierson, D.; Pantazelou, E. Ann. N.Y. Acad. Sci. 1993, 706, 26. (14) Hibbs, A. D.; Singsaas, A. L.; Jacobs, E. W.; Bulsara, A. R.; Bekkedahl, J. J.; Moss, F. J. Appl. Phys. 1995, 77, 2582.
J. Phys. Chem., Vol. 100, No. 11, 1996 4441 (15) Ernst, R. R. J. Magn. Reson. 1970, 3, 10. (16) Buchholtz, F.; Schneider, F. W. J. Am. Chem. Soc. 1983, 105, 7450. (17) Schneider, F. W.; Mu¨nster, A. F. J. Phys. Chem. 1991, 95, 2130. (18) Resch, P.; Mu¨nster, A. F.; Schneider, F. W. J. Phys. Chem. 1991, 95, 6270. (19) Mu¨nster, A. F.; Schneider, F. W. Ber. Bunsen-Ges. Phys. Chem. 1992, 96, 32. (20) Fo¨rster, A.; Hauck, T.; Schneider, F. W. J. Phys. Chem. 1994, 98, 184. (21) Fo¨rster, A.; Zeyer, K.-P.; Schneider, F. W. J. Phys. Chem. 1995, 99, 11889. (22) Zeyer, K.-P.; Mu¨nster, A. F.; Schneider, F. W. J. Phys. Chem. 1995, 99, 13173. (23) Belousov, B. P. Sb. Ref. Radiat. Med. 1959, 145. (24) Fo¨rster, A.; Merget, M.; Schneider, F. W. J. Phys. Chem., following paper in this issue. (25) Noszticzius, Z.; McCormick, W. D.; Swinney, H. L. J. Phys. Chem. 1987, 91, 5129. (26) Coffman, K. G.; McCormick, W. D.; Noszticzius, Z.; Simoyi, R. H.; Swinney, H. L. J. Chem. Phys. 1987, 86, 119. (27) Gyo¨rgyi, L.; Field, R. J.; Noszticzius, Z.; McCormick, W. D.; Swinney, H. L. J. Phys. Chem. 1992, 96, 1228. (28) Hauser, M. J. B.; Lebender, D.; Schneider, F. W. J. Chem. Phys. 1992, 97, 2163. (29) Hauser, M. J. B.; Lebender, D.; Schneider, F. W. J. Phys. Chem. 1992, 96, 9332. (30) Zhou, T.; Moss, F.; Jung, P. Phys. ReV. 1990, 42A, 3161. (31) Longtin, A.; Bulsara, A.; Moss, F. Phys. ReV. Lett. 1991, 67, 656. (32) Showalter, K.; Noyes, R. M.; Bar-Eli, K. J. Chem. Phys. 1978, 69, 2514. (33) Field, R. J.; Fo¨rsterling, H.-D. J. Phys. Chem. 1986, 90, 5400. (34) Hohmann, W.; Mu¨ller, J.; Schneider, F. W. J. Phys. Chem., in press.
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