J . Phys. Chem. 1990, 94, 41 10-41 15
41 10
TABLE I: Characteristics in the Comparison of the Reactions of Oxalate and Formate Ions with Iodine
series I 2 3 4
5 6
oxalate reaction photoinduced reaction radical reaction inhibited by the radical scavengers reactive species are 12*and C , 0 , 2 rate is of half-order in I, and of first-order in C20d2rate is proportional to
formate reaction thermal reaction nonradical reaction no inhibition by the radical scavengers reactive species are I2 and HCOOrate is of first-order in both I, and HCOOrate is proportional to [I-]-‘
C. Comparison ofthe Reactions of Oxalate and Formate Ions with Iodine. For the sake of brevity, characteristics in the kinetics of the reaction of oxalate and formate ions with iodine are summarized in Table I. Formate is thermodynamically much harder than oxalate for the oxidation reaction. Generally, the oxidation of formate is much slower than that of oxalate. The results obtained in the present study are, however, completely of an opposite tendency: the oxalate ion was not thermally oxidized by iodine in the dark, but the formate ion was. In the thermal reaction of the formate ion
with iodine, the formation of an intermediate of reaction may be postulated in reaction 22, where a hydrogen-atom transfer occurs simultaneously with an electron transfer.
intermediatepostulated
21-
+
HC
+
COz
(22)
As far as the photoreactions are concerned, the outer-sphere type of reaction between HCOO- and I’ (and/or I**-) formed by irradiation with light would be much slower. than that for the oxalate ion, which is a general tendency for the oxidation of the oxalate and formate ions. Therefore, it is reasonable that the photoinduced reaction of formate by iodine does not occur, while the corresponding reaction for oxalate does.
Acknowledgment. This research was partly supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan No. 01470051. Registry No. I,, 7553-56-2; H02CC02H.Na2,62-76-0; HC02H.Na, 141-53-7; HCOZ-, 71-47-6; COZ-’, 14485-07-5; I, 14362-44-8.
Excitable Chemical Reaction Systems in a Continuous Stirred Tank Reactor J. Finkeovi, M. Dolnik, B. Hrudka, and M. Marek* Department of Chemical Engineering, Prague Institute of Chemical Technology, 166 28 Prague 6 , Czechoslocakia (Received: August 22, 1989: In Final Form: December 7 , 1989)
A methodology based on the use of CSTR to obtain experimental information on the threshold of excitation and response of excitable chemical systems to single and periodic forcing is described. The classical Belousov-Zhabotinskii (BZ) reaction mixture and a reaction mixture containing bromic acid were used to determine the excitation threshold with respect to AgN03 pulse stimulations. Both periodic and aperiodic behavior were observed in both reaction mixtures and are evaluated in the form of excitation diagrams showing the dependence of the firing number on the forcing period. Excitation diagrams for
both systems possess the form of the devil’s staircase. Sequences of excitation diagrams for increasing stimulation amplitudes were used to construct the dependence of the firing number on amplitude and frequency of stimulations. “Two-pulse” and “three-pulse” methods are proposed for the evaluation of the phase excitation curves (PEC’s). An iteration of the appropriate PEC then enables simulations of the full excitation diagram and thus prediction of the behavior of the continuously forced excitable system. Effects of noise are studied and generally satisfactory agreement between simulated and experimental excitation diagrams is demonstrated.
1. Introduction
Distributed excitable media are usually characterized by the presence of traveling waves of chemical, physical, or biological activity. The best known examples from biology are traveling waves of neural action potentials,’ contractions of heart tissue, waves of spreading depression in the cerebral cortex and the retina, waves of aggregation of slime mold amoebae on agar plates, and waves of infectious diseases traveling through biological populations; cf. collection of references in refs 2-4. Traveling waves of chemical activity in liquid reaction. mixtures were already studied in the beginning of the centurySand in the past 15 years were observed in a number of reacting mixtures.6-’0 Chemical excitable systems can serve in simpler situations as models of the underlying mechanism of excitation in more complicated biological systems. This is the case, for example, in the description of the function of chemical synapses. Excitable systems have certain characteristic common features: a stable rest (steady) state and characteristic behavior with respect to its stimulation. Small (low-amplitude) stimulations of the rest state are rapidly damped out. However, stimulations with an *To whom correspondence should be addressed.
0022-3654/90/2094-4110$02.50/0
amplitude that exceeds a certain threshold value trigger an abrupt and large response-the occurrence of an excitation cycle before returning to a rest state (closed trajectory in an appropriate phase space). After the excitation, the excitable medium is refractory to further stimulation (“refractory period”) before it recovers full excitability. Waves in excitable media have been studied from experimental, numerical, and theoretical points of view. One chemical system, the Belousov-Zhabotinskii (BZ) reaction,* and one biological (1) Hcdgkin, A. L.; Huxley, A. F. J . Physiol. 1952, 117, 500. (2) Winfree, A. T. The Geometry of Biological Time; Springer: Berlin,
1980.
(3) Winfree, A. T. When Time Breaks Down; Princeton University Press: Princeton, NJ, 1987. (4) Tyson, J. J.; Keener, J . P. Physica 1988, D32, 327. ( 5 ) Luther, R. Z . Elektrochem. 1906, 12. 596. ( 6 ) Field, R. J.; Noyes, R. M. J . A m . Chem. SOC.1974, 96, 2001. (7) Hanna, A.; Saul, A.; Showalter, K. J . Am. Chem. Soc. 1982,104,3838. (8) Oscillations and Traveling Waves in Chemical Systems; Field, R. J., Burger, M, Eds.; Wiley: New York, 1985; pp 419-439. (9) (a) SevETkovl, H.; Marek, M. Physica 1983, 90,140. (b) Sevdkovi, H.; Marek, M. Physica, 1984, 130, 379. (c) SevEikovi, H.;Marek, M. Physica, 1986, 2ID, 61. (IO) Basza, G.; Epstein, I . R. J . Phys. Chem. 1985, 89, 3050.
0 1990 American Chemical Society
Excitable Chemical Reaction Systems in a CSTR system, heart t i s ~ u e ,have ~ , ~ been studied most often. A distributed excitable system can be viewed as consisting of locally excitable elements and the interaction of neighboring elements (eg., by a diffusive coupling) can produce traveling waves of excitation. I n one dimension the basic properties of wave trains are contained in the dispersion relation, which determines wave speed as a function of the wave train period. Dispersion relations for the BZ reaction systems were measured"J2 and calc~lated.'~The agreement between measured and theoretically predicted values has been discussed in a recent paper.14 On the other hand no summarizing experimental information on the local mechanism of generation of excitation and on the relations between the properties (amplitudes and frequencies) of stimulations causing wave excitations and properties of actually generated waves have been available until now for chemical systems. A large number of experimental studies on generation of waves have been performed in spatially distributed systems: in quasi-one-dimensional (narrow) tubular reactors, in quasi-two-dimensionalreaction media (Petri dish with a shallow layer of reaction mixture), and in the bulk of the reacting solution; cf. references in the review^.^,^ Only a few observations of excitability in well-mixed systems (batch or CSTR reactors) were r e p ~ r t e d . ' ~ - 'FieldIs ~ and Pacault et a1.I6 discussed excitability in an open Briggs-Rauscher oscillator and Ruoff" and Ruoff and SchwittersI8 have studied excitability qualitatively in a closed BZ system and tried to model the excitability reached by the addition of silver ions. Modeling of dynamics of the forced excitable BZ systems was also studied in ref 19. In this paper we describe the use of the continuous stirred tank reactor (CSTR) for the direct experimental characterization of excitability. First (section 2 ) , we describe the determination of the threshold of excitability with respect to stimulations by Ag+ ions used in our previous experiments, cf. ref 19. Then we use experiments with continuous forcing for the construction of the "excitation diagram", i.e., the dependence of the firing number (the ratio of the number of generated excitations to the number of applied stimulations) on the frequency of stimulations. A sequence of excitation diagrams for two excitable chemical systems, a system with the bromic acid and the classical BZ reaction system, will be given. We shall show that excitation diagrams possess the structure of the devil's staircase and that both periodic and aperiodic regimes of excitations can occur. A set of excitation diagrams is then used for the construction of the dependence of the firing number on the amplitude and on the period of stimulations. In the next section a methodology of the construction of the phase excitationarve, PEC (an analogy to the phase transition curve, PTC, used in the description of the dynamics of forced oscillatory systemsI9), based on the evaluation of the response to single-pulse stimulations applied at various phases of the excitation cycle will be described. In section 4 the use of the PEC for the construction of the excitation diagram will be discussed and the agreement between directly measured and modeled excitation diagrams will be demonstrated both for periodic and aperiodic regimes.
2. Experiments with Periodic .Forcing 2.1. Experimental Arrangement. Most experiments were performed with the modified BZ reaction mixture in which the ( I 1) Marek. M.; SevEikovl, H. In From Chemical to Biological Organization; Markus, M., Muller, S. c.,Nicolis, G., Springer Verlag: Berlin, 1988; p 103. (12) Pagola, A.; Ross, J.; Vidal, C. J . Phys. Chem. 1988, 92, 163. (13) E)ockery,-J. D.: Keener, J. P.; Tyson, J. J . Physica 1988, 3 0 0 , 177. (14) SevEikova, H.; Marek, M. Physica 1989, 3 9 0 , 15. ( I 5) Field, R. J. In Theoretical Chemistry; Vol. 4, Eyring, H., Henderson, D., Eds.; Academic Press: New York, 1978. (16) Pacault, A.; Hanusse, P.; DeKepper, P.; Vidal, C.; Boissonade, J. Acc. Chew. Res. 1976, 9, 438. (17) Ruoff, P. Chem. Phys. Lett. 1982, 90,76. (18) Ruoff, P.; Schwitters, B. J . Phys. Chem. 1984, 88., 6434. (19) Dolnik, M.; Finkeovl, J.; Schreiber, I.; Marek, M. J . Phys. Chem. 1989, 93, 2765.
The Journal of Physical Chemistry, Vol. 94, No. 10, 1990 4111
------
Figure 1. Experimental arrangement: (A) solution of HBrO,, (B) KBr, (C) malonic acid, (D) ferroin, (E) AgNO,, (CE) calomel electrode, (PE) platinum electrode, (BE) bromide selective electrode, (S) stirrer, (SV) solenoid valve, (0)overflow of reaction mixture.
/
a2
.
150 I 1
,
0.4
0.6
08
1.0 vlmll
Figure 2. Dependence of the system response on the volume of 0.01 M AgNOl pulse for the modified BZ reaction mixture. The determination of the excitability threshold. The full circles denote the maximum values of potential measured by the bromide selective electrode, the empty circles the responses of the platinum electrode.
solutions of HISO, and NaBrO, were replaced by HBr0,. The same initial concentrations of reactants, 0.2 M HBr03 (synthesized as described in ref 20), 0.05 M CH2(COOH)2,0.007 M KBr, and 0.00375 M ferroin were used both in batch and CSTR experiments. These reaction conditions are comparable to those used in our group in the studies of wave initiation and propagation on a Petri dish2' and in a capillary tube.I9 In the CSTR system the reactants were pumped into the reactor separately by four input channels. The chosen residence time in the experiments reported here was 39 min. The temperature was kept constant at 20 "C. The experiments were carried out in a Plexiglas reactor; see Figure 1. The reaction volume was 45 mL. A specially designed stirrer of the Rushton turbine type was used, the stirring intensity (20) Gmelins Handbuch der Anorganischen Chemie. 8. Auflage System Number 7; Brom. bearbeitet von R. J. Meyer; Verlag Chemie: Berlin, 1931; p 305. (21) Marek, M.; Schreiber, 1.; Vroblovi, L. In Structure, Coherence and Chaos in Dynamical Systems; Christiansen and Parmentiers, Eds.; Man-
chester University Press: Manchester, U.K., 1988.
4112
The Journal of Physical Chemistry, Vol. 94, No. 10, 1990
200’
I
1
E,
ImV I ,
9501
I
9001 8501
8004
’ b
0
200
400
600
750’
50
t[sl
1
100
150 E,,[mVl
Figure 3. Dynamic regime 1 / 1 . Modified BZ reaction mixture. (a) Time course of the potential of the bromide selective electrode. (b) Recording of the redox potential. (c) Phase portrait. Forcing, 7 . 5 X IO” mol of A g N 0 3 ; T = 285 s.
950 -
50
900-
950 900
850 -
850
800-
I
6
750’
OM@”
50
100
t I51
150 gImVl
Figure 4. Dynamic regime 1/4. Modified BZ reaction mixture. (a) Time course of the potential of the bromide selective electrode. (b) Recording of the redox potential. (c) Phase portrait. Forcing, 1.0 X lo-’ mol of AgNO,; T = 60 s.
was 700 rpm, and it was checked that further increase of the stirring intensity does not significantly affect the course of measured concentrations. The concentration of Br- and the redox potential (proportional to the Fe3+/Fe2+)were measured by a bromide selective electrode and a platinum wire electrode with a standard calomel electrode with a salt bridge as a reference electrode, respectively. The perturbation was realized by a fast addition of a controlled constant volume of AgNO, solution to a reacting medium. Calibrated pumps were used for feeding separately the reactants into the CSTR. 2.2. Experiments in CSTR. A steady state characterized by a constant course of the Br-concentration and of the redox potential is established in the CSTR after the transient period. Single-pulse experiments can be then used both for the determination of the threshold of excitability with respect to the chosen parameter (the concentration of particular component, temperature) and for the evaluation of the phase excitation curve. The threshold of excitability was determined by comparing the response of the system to the single-pulse stimulation of an increasing amplitude; see Figure 2 . The full circles denote the maximum values of potential measured by the bromide electrode and t h e empty circles the responses of the platinum electrode.
Finkeovi et al. Low-amplitude pulses of added AgNO, changed only the course of the concentration of Br- ions but they had almost insignificant influence on the course of the redox potential. When the threshold of excitability was reached, high-amplitude peaks were observed on the recorded potentials of both electrodes. 2.2.1. Periodic Excitation of the Modvied BZ Reaction Mixture. Generally, excitable systems (“excitators”) have no apparent natural rhythm similar to the autonomously oscillating systems (“oscillators”); however, when the excitator is periodically forced it is possible to find dynamical regimes similar to those, which are observed in forced oscillators. Similarly as in the case of autonomous oscillators we are then interested in the classification and characterization of dynamical regimes resulting from variations of the forcing amplitude and period. In our experimental study with the bromic acid reaction mixture responses to variations of both the amount of added Ag+ ions (the forcing amplitude) and the frequency of AgN0, addition (the forcing frequency) were investigated. The volume of pulse additions was held constant at 0.5 mL and the concentration of added AgNO, was varied from 0.01 to 0.0225 M; hence the amount of AgNO, added changed from 5 X to 1.125 X mol. The observed dynamical regimes were classified by evaluating the firing number v = m/n, where m denotes the number of excitation events and n is the number of applied stimulations. Hence if q stimulations repeatedly elicit p excitations, the regime is denoted as a p / q periodic regime. Most often the regimes of the 1/ q type were observed. A typical dynamical behavior showing periodic regimes with firing numbers 1/ 1 and 1 /4, respectively, is depicted in Figures 3 and 4. The time dependence of both measured electrode potentials is given in Figures 3a,b and 4a,b, and the corresponding two-dimensional phase portraits (constructed from the measured Br- and Pt electrode signals) are depicted in Figures 3c and 4c. The 1 / 1 regime (Figure 3) is fully synchronized with external stimulations, and the phase portrait characterizes local dynamics of the excitable system. The comparison of parts a and b of Figure 4 for the firing number v = 1/ 4 illustrates well the all or none character of the response of the excitable system. The signal corresponding to the redox potential (Figure 4b) fires only at each fourth stimulation. This can be also clearly seen in the corresponding phase portrait (Figure 4c). An aperiodic behavior resulting from a periodic forcing is depicted in Figure 5a-c. Here the response of the system alternates between sections with the firing numbers l / l and l /2; the length of these sections varies apparently in a random manner; cf. the course of potentials in Figure 5, a and b. The stroboscopic map shown in Figure 5c contains two groups of points corresponding to 1 / 1 and 1/ 2 regimes; the trajectory jumps randomly between them. The dependences of the firing number on the forcing period T (“excitation diagrams”) are shown in Figures 6 and 7 . The devil’s staircase-like structure is similar to the structures wellknown from the forced oscillatory systems.19 Periodic regimes in the excitation diagrams are denoted by full lines and the aperiodic ones by dotted lines. Only periodic regimes were observed for the amplitude of perturbation 5 X 10” mol, as is shown in Figure 6. Both periodic and aperiodic regimes occur for the perturbation amplitude 1.125 X 10-5 mol. The aperiodic regimes are formed by random combination of l / q and l / ( q + 1) periodic c ‘
0
15
30
45
75
E,,ImVl
Figure 5. Aperiodic regime Y = 0.63. Modified BZ reaction mixture. (a) Time course of the potential of the bromide selective electrode. (b) Recording of the redox potential (c) Stroboscopic map. Forcing, 1.125 X mol of AgNO,; T = 255 s
Excitable Chemical Reaction Systems in a CSTR
+I
The Journal of Physical Chemistry, Vol. 94, No. 10, 1990 4113 3 1'1
1.0
1.0
o,*I 0.6
1:2
i 0.21
0
TB
0'1 I
0
MO
100
300
Tlsl
Figure 6. Excitation diagram for the modified BZ reaction mixture.
Dependence of the firing number v on the forcing period T for the constant amplitude of pulse. Forcing, 5.0 X lod mol of AgNO,.
1 I
I
0.8
,p
0.2
-... 13
I
300 Tis1 Figure 7. Excitation diagram for the modified BZ reaction mixture. Dependence of the firing number v on the forcing period T for the constant amplitude of pulse. Forcing, 1.0 X 10" mol of AgN0,. 0
200
100
-
0 025020-
0.015-
1
s
14
1 3
0.010-
~~~~
0.005
0
100 200 300 T i s ] Figure 8. Diagram of forcing amplitude A-forcing period T for the modified BZ reaction mixture. Point A corresponds to the dynamical regime depicted in Figure 5 .
regimes. For higher amplitudes of forcing pulses (1 X mol AgN03) the overlap of neighboring regimes is observed suggesting multistable behavior. The results of the above experiments for several amplitudes were then used for the construction of the diagram forcing amplitude-forcing period, cf. Figure 8. The approximate boundaries of resonance tongues are drawn in the diagram to indicate the range of the parameters where periodic regimes with the specified firing number exist. Aperiodic regimes are located inside the regions where the neighboring tongues are overlapping. The inevitable presence of experimental noise is probably responsible for the experimental observation that both l / q and l / ( q + 1) regimes can be observed between the resonance boundaries. 2.2.2. Periodic Excitation of the Classical BZ Reaction Mixture. Experiments with the classical BZ reaction mixture (malonic acid (MA), NaBr03, KBr, and Ce4+ catalyst) were performed in the same experimental arrangement (CSTR) as in the above reported case of the modified BZ reaction. Initial concentrations of the reactants were set at the following values: 0.3 M NaBr03, 0.1 M MA, 0.41 M H2S04, 0.006 M Ce(S04),,
I
LO 80 120 160 200 ns1 Figure 9. Excitation diagram for the classical BZ reaction mixture. Dependence of the firing number v on the forcing period T for the constant amplitude of pulse. The regime 0/1 denotes no pulse generation after the stimulation. 0
and 0.01 M KBr. The same reaction temperature (20 "C) and the same intensity of stirring were used as in the ferroin-catalyzed bromic acid reaction system. However, the inlet flow rate into the CSTR was higher (the residence time was 30 min). The threshold of excitability was found to be lower than in the case of the bromic acid mixture and equal to 5 X lo-' mol of AgN03. The concentration pulses of the same amplitude were then used for the periodic perturbation and for the evaluation of phase excitation curves. The experimentally determined dependence of the firing number on the forcing period is depicted in Figure 9. Contrary to excitation diagrams observed for the modified BZ reaction mixture (cf. Figures 6 and 7), only periodic regimes 1/1, 1/2, 1/3, and 0/1 and no higher resonances were observed in this case. The sharp boundaries were found to occur between the regimes 1/1 and 1/2, 1 / 2 and 1/3, respectively. Aperiodic regime (empty circle in Figure 9) was observed to occur between the regimes O / 1 and 1/3. Here randomly four, five, or six pulses were necessary to elicit the excitation. 3. Methods of Evaluation of Phase Excitation Curves The periodically oscillating BZ reaction in a CSTR exhibits positive or negative phase shifts of oscillations when a single-pulse perturbation by the bromide or the silver ion is used.22 By plotting old (p) and new (p') phases we obtain the phase transition curve (PTC). In an excitable system the length of the excitation cycle (cf. Figure 3c) determined by the refractory period corresponds formally to the length of the period Qf oscillations. Similarly as in the oscillatory system we can then define an old phase (corresponding to the position at the original excitation cycle) and a new phase (corresponding to the position along the shifted cycle after an applied stimulation). The phase excitation curve (PEC) then describes the change in the length of the excitation cycle after the application of the stimulation. It is defined by means of the old phase determined as p = To/TBand the new phase cp' = 1 (a - TR/TB,cf. Figure 10. We have chosen as a basic refractory period TB,the lowest period for which an excitation with the firing number 1 / 1 is still possible (using the corresponding excitation diagram). Two methods of evaluation of the PEC were tested. Both methods are schematically represented in Figure IOa,b. They are based on the application of two subsequent stimulations as depicted in Figure loa. The first pulse elicits an excitation cycle; the second pulse applied with a varying time delay (at different values of the phase cp of the excitation cycle) then causes a phase shift of the excitation cycle ( i t . , a lengthening or a shortening of the actual refractory period, Le., of the time when the system regains excitable properties). The methods then differ in the way of determination of the actual refractory period TR. The first method, a direct one (cf. Figure loa), can be used in situations when the refractory state of the system can be determined directly, for example, by the known value of the measured
+
(22) (a) Dolnik, M.; Schreiber, I.; Marek, M. Phys. Left.A 1984, 100, 316. (b) Dolnik, M.; Schreiber, I.; Marek, M. Physica 1986, 21D, 78.
4114
The Journal of Physical Chemistry, Vol. 94, No. 10, 1990
Finkeovi et al.
0.8
I
1
b 1000 0
0.6
0.1
0.2
0.8
y
1.0
Figure 12. Phase excitation curve (PEC) for classical BZ reaction mixture. The dependence of the “new phase” p’ on the “old phase” p.
Jk
800
Figure 10. Experimental evaluation of PEC. ( 1 ) Superthreshold stimulation starting an excitation cycle; (2) stimulation applied at the time To. (a) Direct PEC evaluation. E;: The level of the redox potential below which the system is excitable; (---) course of the original excitation cycle corresponding to the single stimulation I ; (-) course of the shifted excitation cycle after stimulation 2. (b) Iterative method of PEC evaluation. 3 1.32.33: Sequence of testing stimulations applied at times Ti, T2, T, ... Stimulation 32 elicits excitation, the lowest value of T, E ( T I %T 2 ) is determined iteratively.
I
0
I
I
1
0.5
1.0
lf
Figure 13. One-dimensional mapping constructed with the use of PEC. 1/4-periodicregime is denoted by dashed line; T = 98.5 s.
G OL
00
02
06
Figure 11. Phase excitation curve (PEC) for modified BZ system. The dependence of the “new phase” on the “old phase”. (a) Forcing, 5.0 X IO” mol of AgNO,; (b) forcing, 1.125 X mol of AgNO,.
variable (in our case the value of the redox potential E&). In the iterative (indirect) method (cf. Figure lob) the length of the new refractory period is determined iteratively by means of repeated experiments with an application of the third testing pulse, cf. Figure lob. In the following we describe an application of the two discussed methods in two excitable media: (a) the bromic acid reaction mixture (direct method), (b) the classical BZ reaction mixture (indirect method). ( a ) Direct Method of Evaluation of PEC: Bromic Acid Mixture. The experimental evaluation of the PEC was in this case based on the “method of two subsequent pulses”, Le., on the method based on two additions of AgNO, of a chosen amplitude and with varying time delay between the pulses To. The excitation diagrams, firing number-forcing frequency, showing the location of the 1 / 1 firing number were used to determine the basic refractory period T B ( T Bwas equal to 285 s). The obtained PEC’s for two amplitudes of stimulation are depicted in Figure 1 la,b. We can observe that the PEC’s cross the line q = p’, Le., the perturbation by AgNO, causes for low values of cp an increase of the refractory period and for high values of 9 a decrease of the refractory period. As could be expected, differences between old and new phases increase with increasing amplitude of stimulation. ( b ) lteratice Method of Evaluation of PEC: Classical BZ Mixture. Phase excitation curves were determined on the basis
of an iteratively estimated refractory period TR by the method of “three pulses” (cf. Figure lob). In this method we determine the length of the excitation cycle (the refractory time) iteratively, by means of a subsequent application of the third perturbation pulse at varying time intervals after the application of the second perturbation. The proper length of the excitation cycle then corresponds to the shortest time interval between the first stimulation and the third testing stimulation causing an excitation. The accuracy of this method is limited by the time interval required for the discrimination between the successful and unsuccessful excitations resulting from the application of the third pulse. Accuracy of the estimation of TR was in our case equal to 2.5% relative to the value of TB( T Bwas equal to 190 s). The phase excitation curve for the above-mentioned amplitude of the stimulation is shown in Figure 12. The entire curve is located below the straight line cp = cp’, Le., only negative phase shifts in the excitation cycle were observed.
4. Comparison between Simulated and Measured Excitation Diagrams The experimentally determined PEC’s were used for testing the possibility of predictions of dynamic regimes arising by periodic forcing with corresponding amplitudes. To find the relation between the subsequent two phases cpk and pk+l after repeated stimulations we add the value of the dimensionless forcing period T I T B to the mapping cp’(cp). If the resulting phase ‘pk+l at the next stimulation exceeds 1 then the pulse elicits the excitation event and in the first approximation the value of 9’(~,0~+~) can be set to zero. Therefore the mapping which relates qk and pk+]can be written as
This mapping is discontinuous at (o = 1 and, in addition, independent of cp for cp L 1; see Figure 13. Thus the asymptotic dynamical behavior produced by the successive iterations of the
The Journal of Physical Chemistry, Vol. 94, No. 10, 1990 4115
Excitable Chemical Reaction Systems in a CSTR
diagram for the classical BZ reaction mixture is shown in Figure 15. This diagram can be compared with the experimental results of periodic forcing presented in Figure 9. Again a good agreement between simulated and measured excitation diagrams can be observed. However, it is evident that for lower values of the forcing period T the iteration computation reveals finer details of the resonant structure than was observed in the experiments.
1'
t
1 2
0 2L
I
0
1 2
5. Discussion and Conclusions
1M)
200
100
300
203
300 T Is1
Figure 14. Simulated excitation diagram for the modified BZ reaction mixture. (a) Noise level 0%. (b) Noise level 5%. Forcing, 1.125 X mol of AgNO, 31
1
1.01
0.31 1 2
OOX' I
13 -
0.21
,
01
,
0 LO 80 1M 160 BO Tlsl Figure 15. Simulated excitation diagram for the classical BZ reaction mixture.
+
mapping is always periodic because of the presence of the constant part of the map. Two successive iterates Vk, Pk+l such that (Pk < 1 5 (ok+l correspond to the occurrence of the excitation event and hence the firing number can be easily computed. The nondecreasing character of p'(cp) in the interval 0 Icp I1 implies that every q-periodic sequence {pk}has the firing number u = l / q . We have used cubic splines to interpolate the experimental data in Figure 1 1 b, and applied eq 1 to obtain the u versus T plot; see Figure 14. Obviously, this devil's staircase is discontinuous as it possesses only values of v of the form u = l / q but it seems that all q's are present in this structure. The effects of experimental noise can be included into simulations by means of the consideration of a random change of phase with the chosen amplitude. Such a case is for the bromic acid reaction mixture and 5% variance with respect to the mean depicted in Figure 14b. Comparing Figure 14 with the excitation diagrams determined directly in the periodic forcing experiments (cf. Figures 6 and 7) we can observe that the character of both excitation diagrams is similar. Periodic and aperiodic regimes (denoted by circles) of similar character occur in Figure 14. Higher resonances (regimes with values of q > 5) can be also observed in Figure 14. The introduction of the effects of noise into simulations (Figure 14b) causes the disappearance of some of the higher resonances and the increase of the regions of aperiodic behavior. This chaotic behavior is of the same type as that observed in periodic forcing experiments (including random transitions between neighboring resonances). The simulated excitation
Excitation diagrams represent a summary of responses of the system to periodic perturbations of varying amplitudes and frequencies. They give basic information on the ability of the excitable media to transfer frequency-coded in for ma ti or^.^^^^^ The arrangement of the observed periodic responses with the sequence of firing numbers correspond evidently to Farey sequence; however, the inevitable presence of experimental noise precludes the possibility of determining higher resonance regimes. The diagram, depicting the boundaries of regimes with differing firing numbers dependent on the amplitude and the frequency of stimulation, can be constructed from the excitation diagrams, as it was illustrated in the case of bromic acid mixture. Such a diagram then can be used for predictions of responses of the excitable system to signals of varying amplitudes and frequencies. Periodically forced oscillatory and excitable systems reveal striking similarities in the arrangement of response regions and in the parameter dependences of the firing numbers. The behavior of strongly relaxational periodically forced oscillatory systems can be simulated in certain situations by means of the iteration of the corresponding phase transition curve.'g This conclusion also holds for excitable media as was demonstrated in this paper. The PEC's can be determined relatively precisely, as the experiments at the chosen phase cp can be repeated at will. The same methodology of the construction of phase excitation diagrams and phase excitation curves can be used for other chemical systems. Phase excitation curves of different character will be then most probably determined for different reaction conditions for the above-studied systems and/or completely different chemical systems as follows from experiments that are in progress in our research group. The flow rate into the CSTR can be adjusted to simulate processes of pulse generation in a particular distributed We propose to use the phase excitation curves for the characterization of excitable media. The phase excitation curve can be either determined experimentally as was illustrated above or evaluated from the model similarly as was demonstrated in the case of the PTC for oscillatory systems in ref 19. Biological systems consisting, e.g., from immobilized enzymes, organelles, membranes, or entire cells can be studied in a similar way. This could be of importance for example in studies of the function of chemical synapses or of isolated heart ~ e l l s . ~ ~ , ~ ~
Acknowledgment. We express our sincere thanks to H. SevEikovi and I. Schreiber for useful discussions. Registry No. BrO,, 15541-45-4; AgNO,, 7761-88-8; malonic acid, 141-82-2; ferroin, 14708-99-7. ~~
~
~~
~
~
(23) Eccles, J. The Understanding ofthe Brain, McGraw-Hill, New York, 1973. (24) Bublock, T.; Grinnebl, A,; Orkland, R. Introducrion to Neruous Systems; Freeman: San Francisco, 1977.
