Recognition of Phase Patterns in a Chemical Reactor Network

With a phase pattern in which all four reactors are in-phase, one of the two stored phase patterns ... network of four electrically coupled reactors (...
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J. Phys. Chem. 1996, 100, 19043-19048

19043

Recognition of Phase Patterns in a Chemical Reactor Network† G. Dechert, K.-P. Zeyer, D. Lebender, and F. W. Schneider* Institut fu¨ r Physikalische Chemie, UniVersita¨ t Wu¨ rzburg, Marcusstrasse 9/11, D-97070 Wu¨ rzburg, Germany ReceiVed: June 3, 1996; In Final Form: September 11, 1996X

Phase shifts between four Belousov-Zhabotinsky (BZ) oscillators are applied to encode phase patterns in an experimental network consisting of four reactors. Oscillations are established in a focus of the BZ reaction, which is sinusoidally driven by an applied electrical current. In addition to the global electrical coupling by the sinusoidal function the four reactors are electrically coupled by an optimized feedback function including time delay. Two of three possible phase patterns can be encoded in this Hopfield-type network of four reactors. With a phase pattern in which all four reactors are in-phase, one of the two stored phase patterns is recalled with a 50% probability. This indicates that the two encoded patterns have the same dynamic stability. It is possible to reversibly switch between the two patterns by adding Ce4+ solution. Higher order phase patterns permit a higher phase resolution. The phase method provides for a large amount of information to be stored and recalled in a multiunit network. Numerical calculations with the seven-variable Gyo¨rgyiField model of the BZ reaction are in good agreement with the experimental results. Generic similarities with the so-called binding problem in neurology are discussed.

1. Introduction The synchronization of the oscillatory responses of neurons is considered to be the solution of the so-called binding problem by which different features such as colors and shapes may be recognized in the cortex.1,2 We demonstrate that aspects of the binding problem can be modeled by a small chemical network consisting of only four chemical reactors that are interconnected by electrical potentials in such a way that they will recognize certain phase patterns. In principle, coupling between reactors can be performed by mass exchange,3 by flow rate,4 or by electrical current.5 Recently, the Boolean functions were calculated chemically in a feedforward chemical reactor network consisting of three reactors using several nonlinear chemical reactions: the bistable BZ reaction with flow rate coupling, the bistable minimal bromate system,6 and the monostable neutralization reaction7 where the last reaction was least efficient in switching between chemical binary states of high and low concentrations. Theoretical implementations of logic functions have been described by Hjelmfelt et al.8 and Lebender et al.9 Chemical pattern recognition has been demonstrated theoretically by the use of a Hopfield net.10 In a complex experimental feat, pattern recognition has been performed for the first time by Laplante, Pemberton, Hjelmfelt, and Ross (LPHR)11 with a network of eight open, bistable, mass-coupled chemical reactors where the bistable iodate-arseneous acid reaction was employed. In this work we choose electrical coupling instead of mass coupling for two reasons: electrical coupling is closer to its natural equivalent in neuronal networks and it readily enables the study of phase relations that have hitherto received only little attention. Owing to technical reasons, we use a small network of four electrically coupled reactors (Figure 1) that contain the BZ reaction12 in an excitable focal steady state near a Hopf bifurcation (Figure 2). All four focal steady states are * To whom correspondence should be addressed. † One of us (F.W.S.) feels particularly happy to dedicate this work to John Ross who has made so many important contributions to the emerging field of nonlinear dynamics in chemistry. A final remark may also be allowed, namely, that it is well nigh impossible to do one better than John. X Abstract published in AdVance ACS Abstracts, November 1, 1996.

S0022-3654(96)01606-1 CCC: $12.00

Figure 1. Experimental setup consisting of four CSTRs and a central chamber acting as a salt bridge (H2SO4). The redox potential in each CSTR is monitored by Ag/AgCl/Pt measuring electrodes. Each reactor also contains a Pt working electrode. Electrical perturbation and coupling are performed by applying electrical potentials to the working electrodes.

Figure 2. Experimental bifurcation diagram of the BZ reaction. The bifurcation diagram is in qualitative agreement with the bifurcation diagram of the seven-variable Gyo¨rgyi-Field model. A focus (steady state) (filled circle) is adjusted 11% above the Hopf bifurcation at 2.01 × 103 s-1. The black region marks small period-1 oscillations. Complex behavior emerges in the grey part. Below these regions period-1 oscillations with large amplitudes occur in a wide flow rate region.

electrically driven in-phase by an external sinusoidal potential that provides a weak global coupling of all four reactors at all © 1996 American Chemical Society

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Pot1(t) ) R sin(ωt) + w21(Pot2(t) - Pot2(t - 30 s)) and

Pot2(t) ) R sin(ωt) + w12(Pot1(t) - Pot1(t - 30 s)) (2)

Figure 3. Three phase patterns (1, 2, 3) for four coupled reactors (four circles). Open circles represent a phase angle of 0°, and filled circles represent a phase shift of 180°. The lines represent out-of-phase coupling. Patterns 1 and 2 are trained patterns; pattern 3 is untrained. With synchronous oscillations, patterns 1 and 2 were recalled with a 57-to-43 distribution in 15 independent experiments.

times. Electrical coupling via a working Pt electrode in each reactor is then activated, which leads to specific phase patterns depending on the coupling and the dynamics of the chemical reaction. Electrical coupling also leads to a doubling of the ratio between the response period and the perturbation period, which is necessary to recognize any phase shifts. At low coupling strengths each reactor is either in-phase or out-of-phase with any other reactor. The particular phase shifts constitute a phase pattern. Possible Phase Patterns. For a small network of four coupled oscillators there are altogether 8 ()24/2) possible phase patterns, since corresponding mirror images are equivalent in an oscillatory system. At optimal coupling strengths each individual oscillator displays either one of two states, in-phase (0°) or out-of-phase (180°). However, if any recognition process is to take place, the four-reactor network must be encoded with different trained patterns with which an untrained pattern may be associated. Permutations are not counted, since they are equivalent. Therefore, patterns containing a single (or triple) phase shift will not do, since there is no distinction between them. Thus, the number of distinguishable phase patterns is drastically reduced to three phase patterns containing two phase shifts each. They are shown schematically in Figure 3. For the recognition process two out of three patterns are encoded in the network, for example, patterns 1 and 2; pattern 3 is untrained in this case. Coupling Mode Producing a 180° Phase Shift. In order to store pattern 1 and pattern 2 (Figure 3) simultaneously in the four-reactor network, we use a coupling function similar to that of Pyragas,13 which has been successfully applied to control chemical chaos:14

Potj(t) ) R sin(ωt) + wij(Poti(t) - Poti(t - D))

(1)

where Poti(t) is the electrical potential imposed on reactor j by reactor i, R [V] denotes the amplitude of the perturbing external electrical potential, wij is the coupling strength between reactors i and j, and D is a delay time. When D is set equal to the perturbation period, this coupling function forces the two electrically coupled reactors i and j to oscillate exactly 180° out-of-phase as shown by experiment and model calculations. In order to establish the connectivities for storing patterns 1 and 2 simultaneously, the Pyragas coupling function is applied to reactors 1 and 2 as

where ω ) 0.209 rad/s corresponding to a perturbation period of 30 s. Thus, oscillators (reactors) 1 and 2 oscillate out-ofphase. It is important to note that their period has doubled to 60 s after coupling owing to the nonlinearity of the chemical reaction. In analogy, the coupling function is also applied to reactors 3 and 4, which also oscillate out-of-phase with an indentical period of 60 s. The phases between reactors 1 and 3, 1 and 4, 2 and 3, and 2 and 4 are unbiased, since coupling is equal to zero here as seen from the coupling matrix (eq 3). This is the reason why patterns 1 and 2 should possess equal stabilities. Therefore, they should be recalled with equal probability starting with a completely synchronous pattern or with pattern 3 (Figure 3). The present mode of coupling can also be expressed in terms of a Hopfield net.15 Hopfield nets are special types of artificial neural networks. All model neurons are arranged in one layer in contrast to feedforward nets, which contain input, hidden, and output layers. Here, a neuron is equivalent to a chemical reactor that may be connected with every other reactor. The mutual connectivities of each pair of reactors are symmetric. Therefore, n reactors result in a quadratic n × n connectivity matrix that is symmetric to its diagonal. In the present case each reactor may display one of two states, in-phase or out-ofphase oscillations, denoted by +1 and -1 in bipolar notation, respectively. For patterns 1 and 2 the connectivity matrix W is

(

)

0 -2 0 0 -2 0 0 0 (3) W) 0 0 0 -2 0 0 -2 0 The nonzero elements denote coupling according to Pyragas. All other connectivities are zero for patterns 1 and 2. 2. Experimental Section The experimental setup (Figure 1) consists of four Plexiglas continuous flow stirred tank reactors (CSTRs) of 4.21 mL volume each. These reactors are connected to a chamber (40 mL volume) via Teflon membranes (5 µm pore size) as cell partitions. The central chamber serves as a salt bridge through which sulfuric acid (1.125 mol/L) slowly flows with an average residence time of 7.1 h in order to avoid contamination with BZ species. Three feed lines are used for each reactor to avoid any chemical reactions in the feed streams. The reaction solutions are stored in three 50 mL glass syringes, which are driven by linear piston pumps. To achieve a focal steady state, the syringes contained the following solutions: syringe 1 contained 0.42 mol/L KBrO3 (Merck-Schuchardt; Nr. 4912); syringe 2 contained 1.5 × 10-3 mol/L Ce3+ from Ce2(SO4)3 (Fluka, Nr. 22420) and 0.9 mol/L malonic acid (MerckSchuchardt; Nr 800387); syringe 3 contained 1.125 mol/L sulfuric acid (Riedel-de Hae¨n, Nr. 30743). All reagents are of the highest available purity. The solutions were prepared using water purified by ion exchange (water purification system MilliQ, Millipore; specific resistance > 10 MΩ cm). The reactors, the feed lines, and the syringes were thermostated at 25 ( 0.2 °C. The three solutions enter each reactor through the bottom and are mixed with rotating Teflon stirrers (600 rpm). Each reactor contains a measuring Pt/Ag/AgCl electrode to monitor

Recognition of Phase Patterns

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Figure 5. At high coupling strength (w12 ) w21 ) w34 ) w43 ) 0.8) oscillations with periods (P ) 120 s) of 4 times the perturbation period (4:1) occur. In the present examples the pairs of coupled reactors (1-2 and 3-4) are 180° out-of-phase, whereas the phase shifts between reactors 1 and 3 are 0°, 90°, 180°, and 270° for patterns 1, 2, 3, and 4, respectively.

Figure 4. Individual time series of each reactor are shown separately. The coupled reactors 1 and 2 and the reactors 3 and 4 are 180° outof-phase; reactors 1 and 4 oscillate in-phase (pattern 2). The synchronous state and the start of the coupling are shown, whereas the transients (150-550 s) are not shown. Notice the doubling of the ratio between response period and perturbation period upon optimal coupling.

the actual redox potential. Owing to unavoidable variations in the sensitivities of the redox electrodes, the amplified and normalized output is presented in arbitrary units. At the start of each experiment the response amplitudes are arbitrarily adjusted to 1000 units each before the Pyragas coupling (eqs 1 and 2) is initiated. The redox potentials are digitally recorded each second. The coupling function is reevaluated every second. Each reactor and the salt bridge contain platinum working electrodes (3.0 cm2 surface area each). In order to drive each focus sinusoidally, an external periodic electrical potential is imposed on the working electrodes. When the central chamber (Figure 1) contains the cathode, all BZ reactors contain the anode and vice versa. In order to avoid crosstalk between the reactors, we used separate current supplies for each reactor. Furthermore, the detecting and coupling circuits are separated by optocouplers according to Crowley and Field.5 Noise with frequencies >20 Hz was filtered. 3. Results 3.1. Experiments. All experiments were performed at a fixed flow rate of kf ) 2.23 × 10-3 s-1 (residence time 450 s) to generate the same focal steady state in each reactor. This focus has been placed 11% above the supercritical Hopf bifurcation as shown in the experimental bifurcation diagram (Figure 2). Initial Conditions. A synchronous starting pattern is established in all four BZ reactors in all experiments, i.e., all four reactors oscillate in-phase at the start. These synchronous (inphase) starting oscillations are obtained by imposing sinusoidal electrical perturbations by an electrical current generator on the working Pt electrodes with an amplitude of R ) 1.0 V (1.0 mA) and a frequency of ω ) 0.2094 rad/s (period ) 30 s). The synchronous pattern contains two errors with respect to each of the stored patterns 1 and 2 (Figure 3). Therefore, the recognition process consists of recalling patterns 1 and 2 with equal probabilities in a large number of experiments. Electrical Coupling Strength. When the electrical coupling is activated according to eq 2, the computing process begins.

At low coupling strengths (wij ) 0.50) the original synchronous (1:1) pattern (Figure 4) is only slightly perturbed (not shown) when coupling is turned on and pattern selection has not taken place. For an optimal coupling strength wij ) 0.60 the results of a typical experiment are shown in Figure 4. From t ) 0 s to t ) 150 s all four reactors oscillate in-phase; then electrical coupling is turned on at t ) 150 s. After about 400 s coupling the network has selected pattern 2 in this case. On account of the electrical coupling and the nonlinearity of the BZ reaction, the ratio between the response period and the perturbation period has doubled from 1:1 to 2:1. When the experiment was repeated 15 times under identical conditions, a nearly 50% distribution of recalled pattern 1 (eight times) and pattern 2 (seven times) (Figure 3) was observed. These results indicate that both phase patterns are of comparable stability within experimental error. At high coupling strengths (wij ) 0.80) the response period turned out to be 4 times as large as the perturbation period. This 4:1 ratio enables the stabilization of several phase patterns using the above coupling function (eq 2) whose possible phases also contain 90° shifts as shown in Figure 5. One of the experimental phase patterns is displayed in Figure 6. Transition between Patterns. It is easily possible to switch back and forth between patterns 1 and 2 by single perturbations at the optimal coupling strength of wij ) 0.60. Figure 7 shows the results of a switching experiment starting arbitrarily with pattern 1 and adding a small amount of Ce4+ solution at the minimum of the redox potential (at t ) 100 s) in reactor 1. This perturbation caused a transition to pattern 2 requiring only ∼150 s. According to the phase-resetting curve for the BZ reaction,16 a maximum sensitivity is expected when the perturbation is applied at a phase (φ ) 270°) where the phase shift is maximal. This is the case approximately at the minimum in the redox potential. 3.2. Model Calculations. Model calculations were performed with the seven-variable Gyo¨rgyi-Field model17 for the BZ reaction. This model has already been used to simulate the experimental phase-resetting curves as obtained by electrical perturbations.16 The initial concentrations of the variables and the parameters are given in Tables 1 and 2. As in the experiments, a focal steady state was used at a flow rate of kf ) 1.20 × 10-3 s-1 (τ ) 13.9 min) located 11% above the supercritical Hopf bifurcation (kfc ) 1.08 × 10-3 s-1). The effects of the electrical perturbation and coupling are modeled by the addition of eq 4 to the differential equations of the Ce4+ variables. Recent experiments support the assumption18 that the electrical current mainly influences the cerium redox couple.16

∆n1(t) ) R′ sin(ωt) + w21([Ce4+]2(t) - [Ce4+]2(t - D)) ∆n2(t) ) R′ sin(ωt) + w12([Ce4+]1(t) - [Ce4+]1(t - D)) ∆n3(t) ) R′ sin(ωt) + w43([Ce4+]4(t) - [Ce4+]4(t - D)) ∆n4(t) ) R′ sin(ωt) + w34([Ce4+]3(t) - [Ce4+]3(t - D)) (4)

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Figure 6. Pattern 4 in Figure 5 showing four time series at a high coupling strength of w12 ) w21 ) w34 ) w43 ) 0.8. Oscillations with periods (P ) 120 s) of 4 times the perturbation period are observed. Possible phase angles are drawn for one oscillation.

TABLE 1: Seven-Variable Model (Nonstoichiometric Steps)a HBrO2 + Br- + H+ f 2BrMA BrO3- + Br- + 2H+ f HBrO2 + BrMA 2HBrO2 f BrO3- + BrMA + H+ BrO3- + HBrO2 + H+ h 2BrO2• + H2O Ce3+ + BrO2• + H+ h HBrO2 + Ce4+ MA + Ce4+ f MA• + Ce3+ + H+ BrMA + Ce4+ f Ce3+ + BrMA• + BrMA f MA + Br2MA• f MA

S1 S2 S3 S4/S5 S6/S7 S8 S9 S10 S11

a MA ) malonic acid; MA• ) malonic acid radical; BrMA ) bromomalonic acid.

∆ni(t) [mol/(L s)] is the number of electrons per time and volume, which is turned over in reactor i. R′ [mol/(L s)] is the amplitude of the sinusoidal perturbation, and ω is the frequency. All other parameters have the same significance as in eq 1. R′ was set to 5.0 × 10-7 mol/(L s) for all calculations. The perturbation frequency was chosen to be ω ) 0.0314 rad/s (period ) 200.0 s). As in the experiments, the delay time D was fixed at the same value as the perturbation period (D ) 200.0 s). All coupling strengths were set equal to w12 ) w21 ) w34 ) w43 ) 0.019 s-1. Without coupling the sinusoidal perturbation of the focus produced oscillations with periods equal to the perturbation period (P ) 200.0 s). To enable pattern recognition to take place, equally distributed noise (interval 0.002) was superimposed on the system variables. When the coupling according to the Pyragas equation was turned on at 1000 s, oscillations with periods of twice the perturbation period emerged as also observed in the experiments. The calculations were performed 100 times; pattern 1 was obtained 52 times while pattern 2 occurred 48 times. Therefore, both patterns display nearly the same stability as also observed in the experiments. Figure 8 shows pattern 2. Increasing the sinusoidal perturbation frequencies to ω ) 0.0628 rad/s (period ) 100.0 s) and keeping the perturbation

Figure 7. Transition between patterns 1 and 2 by perturbation with the addition of 0.1 mL Ce4+ solution (concentration 5.0 × 10-3 mol/ L) to CSTR 1. The transience time (gray area) is relatively short (∼150 s).

TABLE 2: Rate Constants of the Seven-Variable Model and Reactor Concentrations (Experiment and Model) kS1 kS3 kS5 kS7 kS9 kS11 [MA] [Ce3+]

2.0 × 106 s-1 M-2 3.0 × 103 s-1 M-1 7.6 × 105 s-1 M-2 7.0 × 103 s-1 M-1 3.0 × 101 s-1 M-1 3.0 × 109 s-1 M-1 0.3 M 5.0 × 10-4 M

kS2 kS4 kS6 kS8 kS10

2.0 s-1 M-3 3.3 × 101 s-1 M-2 6.2 × 104 s-1 M-2 3.0 × 10-1 s-1 M-1 2.4 × 104 s-1 M-1

[H+] [BrO3-]

0.375 M 0.14 M

amplitudes constant, one obtains period-2 oscillations. The periods of these oscillations (200.0 s) are now twice the perturbation period. All coupling strengths were set equal to w12 ) w21 ) w34 ) w43 ) 0.010 s-1. The delay time was set to 100.0 s. The 2:1 ratio of the oscillation period and the perturbation period is conserved upon coupling. We applied Gaussian distributed white noise (variance 0.002) to all variables of the model. If the trained pattern 1(2) was chosen as an initial condition, the network recognized pattern 1(2) by retaining pattern 1(2) in the calculations. If the untrained pattern 3 or the synchronous state was chosen as an initial condition, the network answered with either pattern 1 or pattern 2 with 50% probability as required by the chosen mode of coupling. 4. Discussion The phase method proves to be a relatively rapid method for pattern recognition. However, owing to experimental limitations, only one given set of initial conditions could be tested using focal steady states, namely, all four reactors being in synchrony (in-phase) with a 1:1 initial ratio of the response period to perturbation period. The sine perturbation in eqs 1, 2, and 4 is always active in all experiments. It represents weak global coupling of all four reactors. It also prevents a

Recognition of Phase Patterns

Figure 8. For the model simulations the individual time series of each reactor are shown separately. The synchronous state, the start of coupling, and the transients are also shown. As in the experiments the ratio of the response to the perturbation period doubles from 1:1 to 2:1 owing to optimal coupling. In this case pattern 2 emerged (compare with experimental results in Figure 4).

randomization of the phases between reactors 1 and 3, 2 and 4. Initial conditions of the known pattern 1 (or pattern 2) could not be achieved experimentally, since these patterns are generated only after the coupling interaction has been turned on. Thus, the optimal coupling process generates a doubled ratio of the response to the perturbation period. The numerical simulations were in good agreement with experimental results. Numerically, arbitrary initial conditions were chosen and recognition by the phase method could be verified directly. For example, starting with trained pattern 1(2) retained pattern 1(2) whereas starting with untrained pattern 3 (or with a synchronous pattern) led to a 50% distribution of the recalled patterns 1 and 2 in the presence of noise just like in the experiments. This means that recognition by the four-reactor network does not change a “wrong” state into its “correct” state (as in recognition using bistability) but that the two different patterns 1 and 2 will be recalled with equal probability starting with all four oscillators in-phase. This equal probability criterion corresponds to the recognition process. If patterns 1 and 2 would be recalled with vastly different probabilities, the recognition process by the chemical network would have failed. At higher coupling strengths a nonlinear oscillator permits operation at a higher phase resolution with a concomitant increase in information storage and transfer as demonstrated by the 4:1 ratio of the response period to the perturbation period (Figure 5). However, the region of coupling strengths to achieve higher ratios becomes narrower and more difficult to achieve experimentally. At still higher coupling strengths the phases start to fluctuate and the phase method becomes unreliable. Comparison with LPHR. In their recent experiments LPHR11 used (nonoscillatory) bistable steady states and mass

J. Phys. Chem., Vol. 100, No. 49, 1996 19047 exchange to test a chemical network (iodate-arseneous acid reaction) consisting of 8 reactors and 24 connections with three different sets of patterns and several initial conditions. These authors found that the recall of the stored patterns depended on the position of the system within the bistability limits. At the equistability point where both states (high and low iodine) are equally stable, the transition from one state to the other occurred with equal facility. If the experiments were done at a distance from the bistability point, the stored pattern was recalled only as a transient and, finally, nonrecognition occurred. The mass exchange rate between the reactors was another important factor where relatively high rates were favorable for the recall of a stable pattern. This finding is somewhat analogous to the present phase method, which gives optimal results at a relatively high coupling strength. In the LPHR experiments the relative stability of a stored pattern, while depending on the above factors, was important for the recall of the pattern: the most stable pattern was recalled most of the time as demonstrated by their pattern 1 (their Figure 1). Thus, an eight-reactor network was able to recognize patterns efficiently. On the other hand, the present four-reactor network using phase relations was limited experimentally to a single synchronous initial state from which only equal distributions of the two stored patterns were indeed obtained as a criterion for recognition. In numerical simulations, however, any initial state may be chosen and recognition of phase patterns is very efficient and more rapid than recognition of bistable states coupled by mass exchange. In principle, the phase method is a very sensitive and rapid method that is able to distinguish between phase patterns differing not only by 180° but by only a few degrees in numerical simulations. Other Dynamic States. Numerical simulations were also carried out with four coupled autonomous P1 oscillators (as in Figure 2) instead of four periodically driven foci. Upon coupling, the P1 oscillators lapsed into a 2-to-1 ratio between the response period and the perturbation period. Storing two known patterns similar to those in Figure 3 according to eq 2 and starting with completely synchronous initial conditions led to a ∼50% distribution of the two stored patterns. In all cases an association with one of the two stored patterns occurred in the presence of small levels of noise imposed on all variables. Similar to the sinusoidally driven foci, pattern 1(2) was conserved when it was chosen as the initial condition. The experimental use of autonomous P1 oscillations instead of sinusoidally driven focal steady states is difficult to achieve experimentally (albeit easy numerically) even with additional periodic driving as our preliminary experiments have shown. It is also difficult to achieve identical periods in four coupled experimental chemical oscillators. The extension from four reactors to a larger number of reactors may be possible by employing microtechnology, for example. However, a similar problem regarding initial conditions would remain; experimentally, only initial conditions corresponding to all oscillators in synchrony would be possible and the recognition process would consist of the criterion of equal probabilities. Binding Problem. In the aforementioned binding problem1,2 initial phase patterns would be provided by the input from the receptors and the recognition process would be carried out in the cortex. For example, if the neurons due to the color red are in a defined and trained phase relation with the neurons due to a triangle, the cortex “recognizes” a red triangle and not a green one. In analogy, a similar phase pattern situation would exist with other objects. Recognition in the cortex would thus be carried out not only for completely synchronized neurons (phase angle 0°) but also for other defined phase angles where

19048 J. Phys. Chem., Vol. 100, No. 49, 1996 subsets of patterns are still in synchrony with each other. These subsets of patterns arise as the result of the coupling strengths between the neurons. “Helper cells” are not required here. The use of phase patterns for recognition, as in this work, would dramatically increase the number of representative states leading to increased information storage and transfer in the cortex. The interesting question remains, however, of how the connectivities are adjusted in a real neural system. Acknowledgment. We thank the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie for financial support of this work. References and Notes (1) von der Malsburg, C.; Singer, W. Neurobiology of Neocortex, Proceedings of the Dahlem Conference; Rakic¸ , P., Singer, W., Eds.; Wiley: Chichester, 1988; p 69. (2) Gray, C. M.; Ko¨nig, P.; Engel, A. K.; Singer, W. Nature 1989, 338, 334. Kunstmann, N.; Hillermeier, C.; Rabus, B.; Tavan, P. Biol. Cybern. 1994, 72, 119. (3) Marek, M.; Stuchl, I. Biophys. Chem. 1975, 3, 241. Nakajima, K.; Sawada, Y. J. Chem. Phys. 1980, 72, 2231. Stuchl, I.; Marek, M. J. Chem. Phys. 1982, 77, 1607. Bar-Eli, K. J. Phys. Chem. 1984, 88, 3616. Bar-Eli, K.; Reuveni, S. J. Phys. Chem. 1985, 89, 1329. Boukalouch, M.; Elezgaray, J.; Arneodo, A.; Boissonade, J.; De Kepper, P. J. Phys. Chem. 1987, 91, 5843. Crowley, M. F.; Epstein, I. R. J. Phys. Chem. 1989, 93, 2496. Yoshimoto, M.; Yoshikawa, K.; Mori, Y.; Hanazaki, I. Chem. Phys. Lett. 1992, 189, 18. Laplante, J.-P.; Erneux, T. J. Phys. Chem. 1992, 96, 4931. Doumbouya, S. I.; Mu¨nster, A. F.; Doona, C. J.; Schneider, F. W. J. Phys. Chem. 1993, 97, 1025. Doumbouya, S. I.; Schneider, F. W. J. Phys. Chem. 1993, 97, 6945. Hauser, M. J. B.; Schneider, F. W. J. Chem. Phys. 1994, 100, 1058. Booth, V.; Erneux, T.; Laplante, J.-P. J. Phys. Chem. 1994, 98, 6537. (4) Zhabotinskii, A. M.; Zaikin, A. N.; Rovinskii, A. B. React. Kinet. Catal. Lett. 1982, 20, 29. Weiner, J.; Schneider, F. W.; Bar-Eli, K. J. Phys. Chem. 1989, 93, 2704. Chevalier, T.; Freund, A.; Ross, J. J. Chem. Phys. 1991, 95, 308. Roesky, P. W.; Doumbouya, S. I.; Schneider, F. W. J. Phys.

Dechert et al. Chem. 1993, 97, 398. Weiner, J.; Holz, R.; Schneider, F. W.; Bar-Eli, K. J. Phys. Chem. 1992, 96, 8915. Holz, R.; Schneider, F. W. J. Phys. Chem. 1993, 97, 12239. Zeyer, K.-P.; Holz, R.; Schneider, F. W. Ber. BunsenGes. Phys. Chem. 1993, 97, 1112. (5) Crowley, M. F.; Field, R. J. In Nonlinear Phenomena in Chemical Dynamics; Vidal, C., Pacault, A., Eds.; Springer: Berlin, 1981; p 147. Crowley, M. F.; Field, R. J. In Lecture Notes in Biomathematics, Nonlinear Oscillations in Biology and Chemistry; Othmer, H., Ed.; Springer: Berlin; 1986; p 68. Crowley, M. F.; Field, R. J. J. Phys. Chem. 1986, 90, 1907. Botre´, C.; Lucarini, C.; Memoli, A.; D’Ascenzo, E. Bioelectrochem. Bioenerg. 1981, 8, 201. Schneider, F. W.; Hauser, M. J. B.; Reising, J. Ber. Bunsen-Ges. Phys. Chem. 1993, 97, 55. Zeyer, K.-P.; Mu¨nster, A. F.; Hauser, M. J. B.; Schneider, F. W. J. Chem. Phys. 1994, 101, 5126. (6) Zeyer, K.-P.; Dechert, G.; Hohmann, W.; Blittersdorf, R.; Schneider, F. W. Z. Naturforsch. 1994, 49A, 953. (7) Blittersdorf, R.; Mu¨ller, J.; Schneider, F. W. J. Chem. Educ. 1995, 72, 760. (8) Hjelmfelt, A.; Weinberger, E. D.; Ross, J. Proc. Natl. Acad. Sci. U.S.A. 1991, 88, 10983; Proc. Natl. Acad. Sci. U.S.A. 1992, 89, 383. (9) Lebender, D.; Schneider, F. W. J. Phys. Chem. 1994, 98, 7533. (10) Hjelmfelt, A.; Schneider, F. W.; Ross, J. Science 1993, 260, 335. Hjelmfelt, A.; Ross, J. J. Phys. Chem. 1993, 97, 7988. (11) Laplante, J.-P.; Pemberton, M.; Hjelmfelt, A.; Ross, J. J. Phys. Chem. 1995, 99, 10063. (12) Belousov, B. P. Sb. Ref. Radiat. Med. 1959, 145. Zhabotinsky, A. M. Biofizika 1964, 9, 306. Field, R. J.; Burger, M. In Oscillations and TraVelling WaVes in Chemical Systems; J. Wiley & Sons: New York, 1985. (13) Pyragas, K. Phys. Lett. 1992, 170A, 421; Z. Naturforsch. 1993, 48A, 629. (14) Schneider, F. W.; Blittersdorf, R.; Fo¨rster, A.; Hauck, T.; Lebender, D.; Mu¨ller, J. J. Phys. Chem. 1993, 97, 12244. Lekebusch, A.; Fo¨rster, A.; Schneider, F. W. J. Phys. Chem. 1995, 99, 681. (15) Hopfield, J. J. Proc. Natl. Acad. Sci. U.S.A. 1982, 79, 2554; Proc. Natl. Acad. Sci. U.S.A. 1984, 81, 3088. (16) Dechert, G.; Lebender, D.; Schneider, F. W. J. Phys. Chem. 1995, 99, 11432. (17) Gyo¨rgyi, L.; Field, R. J. J. Phys. Chem. 1991, 95, 6594. (18) Crowley, M. F.; Field, R. J. J. Phys. Chem. 1986, 90, 1907.

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