Slow Stochastic Switching by Collective Chaos of Fast Elements
aa r X i v : . [ n li n . C D ] M a y Slow Stochastic Switching by Collective Chaos of Fast Elements
Hidetoshi Aoki and Kunihiko Kaneko
Research Center for Complex Systems Biology, Graduate School of Arts and Sciences,The University of Tokyo 3-8-1 Komaba, Meguro-ku Tokyo 153-8902, Japan (Dated: February 25, 2018)Coupled dynamical systems with one slow element and many fast elements are analyzed. Byaveraging over the dynamics of the fast variables, the adiabatic kinetic branch is introduced for thedynamics of the slow variable in the adiabatic limit. The dynamics without the limit are foundto be represented by stochastic switching over these branches mediated by the collective chaos ofthe fast elements, while the switching frequency shows a complicated dependence on the ratio ofthe two timescales with some resonance structure. The ubiquity of the phenomena in the slow–fastdynamics is also discussed.
Dynamics with distributed timescales are ubiquitousin nature, not only in physicochemical and geophysicalsystems but also in biological, neural, and social sys-tems. In biological rhythms, for example, dynamics withtimescales as long as a day coexist and interfere with thedynamics of much faster biochemical reactions occurringon subsecond timescales [1]. A similar hierarchy existseven within protein dynamics [2]. Electroencephalogra-phy (EEG) of the brain is known to involve a broad rangeof frequencies, and the functional significance of multi-ple timescales has been extensively discussed [3–5]: Neu-ral dynamics in higher cortical areas alter our attentionon a slower timescale and switch the neural activities offaster timescales in lower cortical areas. Faster sensorydynamics are stored successively in short-term to long-term memory. Unveiling the salient intriguing behaviorthat is a result of the interplay of dynamics with differenttimescales is thus of general importance.To treat dynamics with fast and slow timescales,several theoretical tools have been developed sincethe proposition of Born–Oppenheimer approximation inquantum physics. Consider dynamical systems of theform dy i /dt = F i ( { x j , y j } ); ǫdx i /dt = G i ( { x j , y j } ) , (1)where ǫ is small so that { x i } are faster variables than { y j } . According to adiabatic elimination or Haken’s slav-ing principle [6–9], fast variables are eliminated by solv-ing dx i /dt = 0 for a given { y j } , and by using this solu-tion of { x i } as a function of { y j } , closed equations forthe slow variables are obtained. This is a powerful tech-nique when the fast variables are relaxing to fixed pointsfor the given slow variables, whereas to include a case forwhich the fast variables have oscillatory dynamics, theaveraging method is useful [7, 10]. That is, the long-termaverage of the fast variables < x i > is taken for a given { y j } , and by inserting the average into the equation for { y j } , a set of closed equations for the slow variables is ob-tained. When the number of variables involved is small,additional techniques developed with the use of a slowmanifold can be beneficial [11]. Dynamical systems withmutual interference between the fast and slow variables have also been investigated [12–18].In this Letter, we study a case that involves a largenumber of fast variables which show chaotic dynamics.We introduce the adiabatic kinetic plot (AKP) to ac-count for the kinetics of the slow variables under the adi-abatic limit ǫ → ǫ for which stochastic transitivedynamics over different modes are observed and are ex-plained as switches over the adiabatic kinetic branches(AKB) obtained from the AKP. This stochasticity in theswitches is shown to originate from the collective chaosof an ensemble of fast variables. y
005 to obtain the plot. (b) The timeseries of y for ǫ = 0 . As a specific example, we consider the case for a singleslow variable y , where F ≡ h ( { x j } , y ) − y and G ( { x j } , y )are chosen from the threshold dynamics as dydt = h ( { x j } , y ) − y ≡ tanh ( β √ N N − X j =1 ( J j x j + J y )) − y, (2) ǫ dx i dt = G ( { x j } , y ) ≡ tanh ( β √ N N − X j =1 ( J ij x j + J i y )) − x i , (3)where β is taken to be 10. Here, J ij is chosen as a ho-mogeneous random number in the interval [ − x = 1) or “off” ( x = − J ij >
0) or inhibited ( J ij <
0) by otherelements. Note, however, that the method and findingsdiscussed here are not restricted to the specific choices ofthe functions F and G ; they are valid for any choice.The dynamics of the slow variable y is represented us-ing the averaging method as dy/dt = < h ( { x j } ( y )) > − y, where < h ( { x j } ( y )) > is the temporal average of h for agiven y , i.e., the average input that y receives from { x j } ,in the adiabatic limit. To compute the average < · > , wefirst fix the y value, obtain the attractors for x j , and thencompute the temporal average for each attractor. Bychanging the value of y , < h ( { x j } ( y ) > is obtained, andthis forms a continuously changing branch. At this point,it is useful to introduce the plot ( y, < h ( { x j } ( y )) > )(Fig. 1) If there are multiple attractors that depend onthe initial condition of x j , then there are several branchesin the AKP. Starting from a given y and initial condition x j , the dynamical system falls on a specific branch. Ac-cording to the equation for y , if < h > is larger (smaller)than y , then dy/dt > dy/dt < y along each branch. When a branchcrosses the line y = < h > , then y falls on a fixed point.If the slope of the branch at the fixed point is less thanunity, then the system is attracted to the point so thatthe slow variable falls on a fixed point attractor (at least)in the limit of ǫ → ǫ .The periodic motion of y can also be explained by theAKP. For example, see the top and bottom branches inFig. 1. As y increases along the top branch, it eventuallyreaches the endpoint of the branch and then switches tothe bottom branch, which corresponds to an alternativeattractor of x . The process then repeats itself as y de-creases along the bottom branch to the endpoint before switching to the top branch. Indeed, this periodic oscil-lation exists as an attractor, as shown in Fig. 1(b). Inthis example, the { x j } attractor is a fixed point at eachbranch, but in many other examples, the attractor maybe a limit cycle or chaos. However, the present analysisof the y dynamics is still valid in such cases. In fact, theperiodic oscillation of y as analyzed from the AKP existsup to a certain value of ǫ (e.g., ∼ ǫ is added to the slow y oscillation, if { x j } exhibits oscillation.In general, AKP has much more branches that makethe oscillatory dynamics complex. A rather more com-plicated example is shown in Fig. 2. In this case, in thelimit of ǫ → y switches between two branches. In theexample in Fig. 3 (a) for ǫ = 2 × − , y periodicallyswitches between the branch a + and a section of branch j − (“4”). For a larger ǫ value, however, complex oscilla-tions of y are seen, as shown in Fig. 2. This is describedas the switching over all 2 ×
10 branches, a ± , b ± , c ± , ...,j ± (the first 12 are labeled explicitly in the figure), where ± denotes the symmetric branches of y > y < a + → b + (“1”) or a + → i − (“3” → “4”) are both possible, as are d + → j − (“1” → “2”) and d + → e + (“5” → “6”). As ǫ is decreased,a larger number of branches is visited by the stochasticswitches (see Fig. 3(a) to 3(b) and to 3 c)] until only acycle between two branches remains in the limit of ǫ → x i switchamong (at least) 2 ×
10 types of attractors including fixedpoints, limit cycles, and chaos. This type of switching isreminiscent of chaotic itinerancy [23–26] where the orbititinerates over “attractor ruins”. Here, in contrast, thestochastic switches progress among attractors for a givenvalue of the slow variable y , while the chaotic dynamicsof the fast variables provides a source for the stochas-tic switching. Indeed, at the boundary of the branches a, d, f, · · · , the fast variables { x i } show chaotic oscilla-tion.For a detailed analysis of the stochastic switching dueto the chaotic dynamics, we consider the simpler examplegiven in Fig. 4 with a different matrix J ij . In this case,as ǫ → y shows periodic oscillation between the twobranches a + and a − , whereas for ǫ > ǫ c ≈ . × − ,the branches b + and b − are also available, and stochasticswitching a + → a − , b + and its symmetric counterpartappear. The choice between a + → a − and a + → b + isstochastic, and indeed, we have computed the Shannonentropy of the n -tuple symbol sequence of the branches a, b visited by the slow y variable and confirmed that itincreases linearly with n ( ∼ . n )[27].To examine if the origin of the stochasticity lies in thechaos of the fast variables, we measured the maximalLyapunov exponent for the ( N − { x i } for a given y at each branch. As shown a + c + b + e + d + h + g + f + j + i + a - b - c - d - e - g - f - i-j- h - ① ⑦⑥⑤④③ ② ⑪⑩⑨ ⑧ ⑫⑦ʼ⑪ʼ⑨ʼ
9) for the branchesb, c, and h. in Fig. 4 (c), the exponent is positive around the end-points of the branches where stochastic switching occurs.Several other examples also show stochastic transition-ing beyond a critical value of ǫ , a Poisson switching-timedistribution, and a positive Lyapunov exponent at thebranch endpoint (for example see Supplementary Figs.2). The stochastic switching from the branches a and fin Fig. 2 is also explained by the chaotic dynamics of thefast { x i } variables at the branches. -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 50 60 70 80 90 100 y time (a) -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 50 60 70 80 90 100 y time (b) -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 50 60 70 80 90 100 y time (c)Fig. 3. (Color online): Time series of the slow variables y corresponding to Fig,2, except for ǫ = 2 × − (a), ǫ =1 × − (b), and ǫ = 1 × − (c). The time series of (c)corresponds to Fig.2(b) but is plotted for a longer time span. a + b - a - b + y
02) is colored as blue.(See Supplementary Fig.1 for the Lyapunov exponents at eachbranch). (b) The time series of y for ǫ = 0 . When { x i } shows chaotic dynamics, one might expectthat the variable h can be regarded as just the noise rep-resented by the sum of random { x i } variables. If thiswere the case, then the amplitude of this noise woulddecrease as the number of fast elements N is increased.The variable h would then approach a constant in the N → ∞ limit, and the frequency of the stochastic switch-ings would decrease accordingly. However, this is doesnot occur. We simulated the present model by increas- ε fr e qu e n c y o f t h e s w it c h i ng Fig. 5. (Color online) Frequency of the switching in thebranches corresponding to the dynamics in Fig. 4 as a func-tion of ǫ . For each ǫ value, the number of a+ to b+ switchingevents divided by those from a+ to a- are computed as thefraction of the number of times that y goes beyond 0 . ing the number of fast elements by 2 k ( k = 1 , , · · · ,
6) bycloning the matrix J ij and confirmed that the frequencydoes not decrease with an increase in N . This suggeststhat there is still some correlation among the fast vari-ables x , so that h shows collective chaotic motion, as hasbeen studied extensively [28–31]. In fact, the oscillationof the collective variable h has a larger amplitude thanthe typical mean-field oscillation in the collective chaos incoupled chaotic systems studied thus far [28, 30]. Indeed,some of the fast elements x i undergo a large-amplitudechange between the on and off (1 and −