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Information flow between weakly interacting lattices of coupled maps
Chaos, Solitons and Fractals 28 (2006) 755–767 www.elsevier.com/locate/chaos
Information flow between weakly interacting lattices of coupled maps York Dobyns a, Harald Atmanspacher b
b,*
a PEAR, Princeton University, Princeton, NJ 08544–5263, USA Institut fu¨r Grenzgebiete der Psychologie und Psychohygiene, Wilhelmstr. 3a, 79098 Freiburg, Germany
Accepted 25 July 2005
Abstract Weakly interacting lattices of coupled maps can be modeled as ordinary coupled map lattices separated from each other by boundary regions with small coupling parameters. We demonstrate that such weakly interacting lattices can nevertheless have unexpected and striking effects on each other. Under specific conditions, particular stability properties of the lattices are significantly influenced by their weak mutual interaction. This observation is tantamount to an efficacious information flow across the boundary. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Weak interactions between systems can produce unanticipated and dramatic effects on their behavior if their time evolution is nonlinear. The study of such effects (e.g., synchronization) has recently attracted much interest. Many investigations of weakly interacting systems focus on coupled nonlinear dynamical systems such as iterative maps or attractors. In these cases, remarkable synchronization phenomena have been observed experimentally and described theoretically [1]. In particular, phase synchronization has been found as an important indicator, much more sensitive than amplitude or frequency synchronization, for weak coupling. Beyond interactions between only a few individual systems it is interesting to study weak coupling for subsystems of spatiotemporally distributed systems consisting of many nonlinear elements. Such situations are generic whenever the evolution of a system as a whole leads to separately operating subsystems (domains) which themselves consist of many elements. Important examples are neuronal assemblies, subsystems of the cortex that are typically constituted by some thousands of neurons. In order to function as neural correlates of conscious states, the behavior of neuronal assemblies has to satisfy certain stability conditions (for more details see [2]). Coupled map lattices (CMLs) are much studied models of such systems [3,4]. Therefore they suggest themselves as suitable candidates to examine the possibility of a transfer of stability properties (i.e., information flow) due to weak interactions. Such interactions can be implemented in terms of boundary regions with weak internal coupling that separate regions with more strongly coupled elements undergoing a complex dynamical evolution. *
Corresponding author. Tel.: +49 761 207 2199. E-mail address: [email protected] (H. Atmanspacher).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.07.008
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In this contribution we describe corresponding effects of weak coupling between sublattices of CMLs separated by a boundary (or barrier). In the following Section 2, some basic notions concerning CMLs are briefly introduced. In Section 3 possible influences of the destabilizing behavior of one sublattice onto that of another one are investigated and quantified by appropriate measures. Two scenarios of isotropic and anisotropic coupling in the neighborhood of the boundary are explicitly distinguished. In contrast to the transfer of destabilizing behavior, Section 4 examines the transfer of stable patterns across boundaries. While the transfer of stability in Sections 3 and 4 is mainly explored for regions with a coupling parameter close to 0.5, Section 5 considers the case of strong coupling, ! 1. Section 6 gives a compact summary of the main results.
2. Coupled map lattices: some basic notions The basic evolution equation for a two-dimensional CML in discrete time is given by X uxy;tþ1 ¼ ð1 Þf ðuxy;t Þ þ gðuij;t Þ; N ði;jÞ2N
ð1Þ
where u is the state value at each cell in the lattice in the spatial dimensions (x, y), and t is the timestep of the iteration of the state values. The coupling constant determines how much input a cell gets from its neighbors in comparison to its own time evolution. The function f is usually taken to be the logistic map: x ! f(x) = rx(1 x) for 0 6 r 6 4. N is the set of cells that are neighbors of a given cell, and N is the number of cells in N. If the function g, applied to the values of neighboring cells, is given by g(x) = x, this implies that their input to u(x, y) is given by the values of the neighboring cells at the previous timestep. This condition is called causal coupling [5]. Alternatively, we may choose g(x) = f(x), in which case the input is given by the neighboring cell values after they have undergone the uncoupled evolution for the current timestep. This is called non-causal coupling [5], sometimes also future coupling [6]. The function g(x) may be parameterized as ax + (1 a)f(x), to consider mixtures of causal and noncausal coupling. A time scale important for the physical interpretation of Eq. (1) is the time interval Ds assumed for the updating mechanism, i.e., for the physical integration of signals from the neighborhood states with the state considered. If signals between cells are transmitted much slower than the time scale assumed for the updating mechanism, Ds Dt, the updating can be implemented (almost) instantaneously, or synchronously. If this is not the case, Ds J Dt, updating must be implemented asynchronously, implying that a proper updating sequence must be specified. For an inhomogeneous lattice the situation becomes more involved. In such a lattice, in general N, r and are not the same and may assume different values at or around each cell. Additional features of the CML, such as the function g or its interaction parameter a, can also be inhomogeneous. For the current investigation we restrict ourselves to lattices in which the coupling is variable within the simulation, = xy, but all other parameters are kept constant. Eq. (1) can thus be rewritten in the form xy X auij;t þ ð1 aÞruij;t ð1 uij;t Þ; ð2Þ uxy;tþ1 ¼ ð1 xy Þruxy;t ð1 uxy;t Þ þ N ði;jÞ2N Eq. (2) contains an implicit complication. Consider the case where two cells a and b with a 5 b occupy each othersÕ coupling neighborhoods. Cell a couples with strength a to every cell in its neighborhood, including b. Cell b couples with strength b to every cell in its neighborhood, including a. Thus, due to the inhomogeneity of , the coupling from a to b is not equal to the coupling from b to a. In addition to such cases of asymmetric coupling, we want to consider symmetric coupling as well. In order to achieve this, we specify the coupling strengths within a given neighborhood by additional indices i, j specifying the cell from which cell x, y receives coupling input and obtain P 1 X ði;jÞ2N xyij uxy;tþ1 ¼ 1 ruxy;t ð1 uxy;t Þ þ xyij ðauij;t þ ð1 aÞruij;t ð1 uij;t ÞÞ. ð3Þ N N ði;jÞ2N The condition for symmetric coupling between all pairs of points is then a symmetry condition for the four-dimensional -array: xyij = ijxy for all pairs (x, y) and (i, j). While Eq. (3) does not, in fact, require this symmetry condition, we will impose it whenever this form of the evolution equation is used. The asymmetric case of Eq. (2), where all cells in a given cellÕs neighborhood couple to it with the same strength, is referred to as an isotropic neighborhood. The case in which is given the additional degrees of freedom of Eq. (3) allows us to model anisotropic neighborhoods. When the coupling in an anisotropic neighborhood obeys the constraint xyij = ijxy, the coupling is nevertheless symmetric. All simulations in this paper that use anisotropic neighborhoods are designed for symmetric coupling.
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Most of the numerical results presented and discussed in the following are derived for von Neumann neighborhoods of order 1, defining a cellÕs neighbors as the orthogonally adjacent cells: the neighbors of (x, y) are the four cells (x ± 1, y) and (x, y ± 1). Some results are obtained using Moore neighborhoods; the Moore neighborhood of order 1 includes the von Neumann neighborhood of order 1 plus the four diagonally adjacent cells (x ± 1, y ± 1) and (x ± 1, y 1). Von Neumann and Moore neighborhoods of higher order are generalized in the obvious way. Although Eqs. (2) and (3) allow a to be varied continuously, all simulations used in the current analysis use either pure causal (a = 1) or pure non-causal (a = 0) coupling. In most cases the logistic parameter r is set to 4, so that uncoupled cells would engage in fully chaotic behavior that spans the entire range of state values from 0 to 1. Unless otherwise specified, simulations are started with random initial conditions and a 32 · 32 array with periodic boundary conditions in both directions is used.
3. Asynchronously updated CMLs with spontaneous stabilization In previous work [5,7,8] it has been found that coupled map lattices with large enough coupling exhibit spontaneous global stabilization at the locally unstable fixed point of individual maps. Specifically, for a causally coupled lattice with r = 4 and asynchronous update, the simulation stabilizes with all cell states at x = (r 1)/r = 0.75, the unstable fixed point, if P 0.5. This stabilization is completed within a few hundred timesteps in a 32 · 32 lattice. If < 0.5, the simulation arrives (equally rapidly) at a steady-state condition where the state values vary in some range around the fixed point, but continue to exhibit spatiotemporal chaos within that limited range. The size of the steady-state range shrinks with increasing . Fig. 1 shows state-value population distributions for the steady-state of CMLs just above and below, respectively, the critical value for stabilization. In the following we use this stabilization process to study the transfer of stability properties across boundaries between sublattices of a CML. The simulations are based on interactions between a stable sublattice with = 0.51 and an unstable sublattice with = 0.49. The full CML was split into two corresponding regions with the option of placing barriers between them. The two regions consisted of columns 0–15 and columns 16–31 of the 32 · 32 lattice, respectively, and barriers were placed in columns 16 and 31. (Due to periodic boundary conditions, column 0 neighbors column 31.) Subsequently the 512-cell region comprising columns 0–15, in which measurements are taken, is called the primary region. Barriers, if present, border the primary region, and the remainder of the CML beyond the barriers is called the secondary region. The values of the coupling constant within the primary, barrier, and secondary regions are labeled P, B, and S, respectively. (a)
(b) 0.06
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Fig. 1. Cell population histograms (bin width 1/128) averaged over 200 timesteps, after steady-state configuration achieved, in asynchronously updated and causally coupled CMLs: (a) = 0.51, full stabilization at the unstable fixed point x = 0.75, (b) = 0.49, ongoing spatiotemporal chaos within a limited range of state values.
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In addition to the width of the cell-population distribution as shown in Fig. 1, the domain transition rate (DTR) was primarily used to quantify the stabilization behavior in the sublattice under investigation. (We did not evaluate more elaborate measures, e.g., phase synchronization or conditional correlations.) A domain is a set of cells whose state values are all higher or all lower than the unstable fixed point at a given timestep. Since quasi-periodic oscillation with ‘‘period’’ 2 around the fixed point characterizes all cell values, a given cell was considered to have changed domains if it left this oscillatory behavior. The DTR was then defined as the average number of such occurrences per timestep in the 512 cells of the primary lattice. 3.1. Asymmetric coupling It turned out that the steady-state behavior of the unstable ( = 0.49) region was largely insensitive to contact with another, stable ( = 0.51) region. The distribution width did not change detectably, while the average DTR changed by an amount less than the timestep-to-timestep noise. In contrast, the stable sublattice proved sensitive to its surroundings. When the entire CML has = 0.51, it quickly (<200 timesteps) converges on a stable phase with no domain transitions and minimal width. In contrast, an = 0.51 sublattice in direct contact (B S) with an unstable = 0.49 sublattice yields a steady-state DTR of 7.26 ± 0.03 (1r statistical uncertainty). Fig. 2 shows the results for a one-cell-thick barrier in columns 16 and 31 with 0 6 B 6 0.49 (for B = 0.49 the ‘‘barrier’’ is simply part of the secondary region). The set of simulations used here maintained isotropic neighborhoods; therefore the coupling to cells in the barrier is asymmetric. Cells adjoining the barrier receive input from the barrier cells with the same coupling constant as from their other neighbors; the lowered coupling constant in the barrier only affects the cells of the barrier themselves. For vanishing barrier coupling, B = 0, the DTR of the region with P = 0.51 is 26.66, which is considerably higher than for vanishing barrier thickness. This is somewhat counterintuitive since the barrier cuts off all causal contact between the two main regions of the simulation. However, as a consequence of the vanishing coupling the cells of the barrier are simply free-running chaotic maps with r = 4, exhibiting the maximum possible variability from timestep to timestep. Thus, the barrier itself drives the increased instability of the stabilizing region with P = 0.51. As the barrier coupling B increases from 0 to 0.49, the DTR decreases non-monotonically. It is particularly remarkable that, even though B = 0 provides the maximum possible degree of chaotic variation, turning on even a very small coupling in the barrier increases, rather than decreases, the destabilizing activity in the primary region. Since the simulations use isotropic neighborhoods, the coupling strength with which cells at the boundary of the sublattices receive input from the barrier is given by B. Since the activity in the barrier decreases with increasing B, its influence on domain transitions in the primary region must likewise decrease. Nevertheless, the DTR is observed to increase.
Domain Transition Rate
25
20
15
10
0.0
0.1
0.2
0.3
0.4
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Barrier Coupling Fig. 2. Equilibrium domain transition rate in a sublattice with P = 0.51 separated from a sublattice with S = 0.49 by a one-cell-thick barrier of small B. At B = 0 the barrier is completely unaffected by the rest of the CML and produces a complete decoupling between regions. At B = 0.49 the barrier is part of the secondary region. Error bars are smaller than the plotting icons. Neighborhoods are isotropic first order von Neumann, hence the coupling between regions is asymmetric.
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Domain Transition Rate
27.2 27.1 27.0 26.9 26.8 Secondary C=0 Secondary C=.49 Secondary C=.8
26.7 26.6 0.0
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Barrier Coupling Fig. 3. Expanded view of the small-B region of Fig. 2, with additional data from simulations with S = 0 and S = 0.8 for the secondary region. Note the sharp increase in DTR at the transition from B = 0 to B = 0.001. Icons have been laterally displaced for clarity. Statistical errors are shown at 1r.
This demonstrates that even a minute change in the barrierÕs coupling properties can have disproportionate effects on the adjoining sublattice to which it couples. This does not, however, appear to be primarily due to information propagation across the barrier. Fig. 3 expands on Fig. 2 in two ways: first, it concentrates on the region 0 < B < 0.01, affording greater resolution; second, the simulations plotted in Fig. 2 have been supplemented by simulations in which the coupling constant in the secondary region has been lowered to S = 0 or raised to S = 0.8. In most cases any differences among the three values of DTR for different S at the same B are within the statistical resolution of the simulations. However, genuine differences are found for different B. Features such as the abrupt jump in DTR from B = 0 to 0.001 are driven primarily by the interaction between the primary region and the barrier, rather than by influences propagating from the secondary region across the barrier. A broader survey of simulation parameters in Fig. 4 shows similar behavior. As the coupling constant in the primary region changes from 0.5 to 0.7, the qualitative character of the response to the barrier remains unchanged: There is an abrupt increase in DTR between B = 0 and 0.001, and a continuous but non-monotonic progression as B = 0 is further increased. A number of values of primary and barrier coupling show an apparent sensitivity to the secondary coupling, but no single case stands out clearly from the simulation noise. The frequency with which such variations appear, however, does stand out above the simulation noise. For several simulations run with different values of the secondary coupling S and with the same values of primary coupling P and barrier coupling B, the mutual consistency of the measured values can be computed by a v2 statistic. Given a population N of measurements indexed by i, each of which has a value xi and an uncertainty ri, the quantity !, X X X 2 2 2 2 ðxi lÞ =r ; where l ¼ xi =ri 1=r2i ; ð4Þ v ¼ i
i
2
i
is distributed as a v with N 1 degrees of freedom. The distribution is exact if the observational errors are normally distributed, approximate otherwise. In the present case, the ‘‘observational error’’ consists of a statistical uncertainty from mutually independent simulations, such that the uncertainties can be treated as normal. All simulations were run at multiple values of S for 48 different combinations of P and B. Among these simulations, the variations due to S have v2 = 152.6 on 110 degrees of freedom (p = 0.004). This result can be checked for validity by noting that in those six cases with B = 0 the DTR of the primary region cannot, in principle, be affected by changes in S (and any such apparent dependence would denote a spurious effect in the analysis). Indeed it turned out that those cases produce v2 = 11.8 on 12 degrees of freedom as expected: the remaining 42 combinations have v2 = 140.8 on 98 degrees of freedom, p = 0.003. The conjecture that the behavior of the secondary region affects the primary region leads to the prediction that the DTR in the primary region should decrease as S increases (since then the stability of the secondary region increases). A statistical test for this prediction produces an interesting result shown in Fig. 5. All of the 42 cases where DTR measurements are available at multiple values of S for the same P, B with B 5 0 are shown as a scatterplot. Its horizontal
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Barrier Coupling Fig. 4. Small-B region as in Fig. 3 for different coupling constants P of the primary region between 0.5, the minimum for stabilization, and 0.7, as indicated on each subgraph. Icons are plotted for different barrier couplings and secondary-region couplings as in Fig. 3.
Residual in DTR
0.05
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0.10 0.0
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Coupling in exterior region Fig. 5. Survey of all simulations conducted to test DTR effects with isotropic coupling. The residual DTR is shown as a function of secondary-region coupling, after subtracting the mean DTR for all simulations run with the same primary and barrier coupling constants. Individual error bars are not shown due to the high degree of overlap: the horizontal dotted lines show the average 1r uncertainty of a typical simulation. The dashed line shows the slope of a linear regression fit: b = 0.041 ± 0.009 (statistical error).
axis is S, and the vertical axis is the residual change in DTR—that is, the difference between the DTR found for any particular values of (P, B, S) and the average DTR for all simulations with the same (P, B). The shift in the density of observed points as a function of S is evident. Much of the non-systematic noise in Fig. 5 is presumably due to simu-
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lation noise: the horizontal dotted lines show the average 1r uncertainty with respect to the residual DTR. The dashed line shows a linear regression on these data: it has a slope of 0.0408 ± 0.0093, some 4.4r away from 0 (p = 1.14 · 105, two-tailed). The overall trend shown in Fig. 5 can be confirmed by linear regressions in each of the 42 individual cases. The resulting population of slopes has a mean of 0.0410 ± 0.0093 that is in excellent agreement with the aggregate figure above. A goodness-of-fit test for the individual regression slopes against their population average shows a v2 of 66.1 on 41 degrees of freedom (p = 0.008), which is suggestive (though not conclusive) of non-random variation between cases, presumably driven by differential responses at different values of P and B. Since the baseline DTR for the performed simulations is in the range between 30 and 40 (cf. Fig. 4), the observed regression slopes correspond to a proportional change in DTR of approximately 1 part in 1000 per unit increment of S. 3.2. Symmetric coupling The results presented in the preceding subsection have been obtained for isotropic neighborhoods with asymmetric coupling between neighbors belonging to different regions. In the most extreme case, a barrier cell with B = 0 accepts no input from any neighbor and evolves purely according to the logistic equation, while it affects its neighbors at their full coupling constants. Fig. 6 is the analogue of Fig. 2 for the alternative case of symmetric coupling. As in Fig. 2, there is an increase in DTR as the coupling decreases. However, this increase is much smoother than with isotropic, asymmetric coupling. In particular, it joins smoothly onto B = 0 with no indication of a discontinuous jump as in the asymmetric case and without ever attaining a higher DTR than that for B = 0 (cf. Fig. 2). The increased DTR relative to the direct-contact situation must be caused by a mechanism different from the asymmetric case, since with symmetric coupling the increased noise of a low-coupling barrier cannot directly drive activity in the primary region. Due to the anisotropy of the symmetric coupling, each cell adjoining the barrier couples both to and from the barrier cell with B = 0. Both the coupling term and the suppression of the cellÕs internal chaotic evolution in Eq. (3) are therefore only 3/4 as large as they would be for isotropic coupling of the same strength. The same effect would be produced by lowering the primary regionÕs coupling constant to 3/4 of its nominal value, which in the case of P = 0.51 is insufficient to give rise to stabilization. As a consequence of this argument, stable behavior should reappear for B = 0 when P is increased to 4/3 of the stabilization-inducing value 0.5, i.e., to 0.666. Fig. 7 demonstrates that this is indeed the case. In addition, Fig. 7 shows the DTR as a function of P for two different barrier couplings, B = 0.001 and 0.01, and for the three values of S = 0, 0.4, and 0.8. Each set of simulations with the same B and S is plotted by a line (line type determined by B) connecting points (point type determined by S). The exact coincidence of the points indicates that the
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Fig. 6. Effect of low-coupling barrier when cell-wise symmetric coupling (anisotropic neighborhoods) is enforced. As in Fig. 2 (for asymmetric coupling), low coupling within the barrier increases the DTR in the stabilizing region. Unlike in the asymmetric case, the B-dependence joins smoothly onto B = 0. Error bars are smaller than plot icons.
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BC=0 BC=0.001 BC=0.01
Domain Transition Rate
25 20 15 BC=0 BC=.001, SC=0 BC=.001, SC=.4 BC=.001,SC=.8 BC=.01,SC=0 BC=.01,SC=.4 BC=.01,SC=.8
10 5 0 0.50
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0.70
Fig. 7. Effect of increasing coupling P in the stabilizing region for various barrier coupling constants, with anisotropic neighborhoods (symmetric coupling). Lines connect points generated with the same barrier coupling. For B = 0, full stabilization reappears at P = 2/ 3, as described in the text. For B 5 0, the exact superposition of points for different values of S indicates a lack of sensitivity to this parameter.
DTR does not depend on S. The v2 for S-driven variation is 140.7 on 158 degrees of freedom, well within the expected variation. The population of regression slopes has a mean of (7.5 ± 44.7) · 105. Thus, unlike in the case of isotropic neighborhoods, there is no evidence for any influence propagating from the secondary to the primary region with anisotropic neighborhoods.
4. Synchronously updated CMLs with stable spatial patterns Information propagation between sublattices cannot only be tested by stabilization processes in those sublattices but also by the transfer of stable spatial patterns resulting from such processes. As discussed in detail in [9], non-causally coupled lattices with synchronous update at parameter values such as r = 3.9, = 0.1 have a stable ‘‘checkerboard’’ phase in which each of a cellÕs orthogonal neighbors are in the domain opposite to that of the cell itself. Since each cell oscillates from a value below to a value above the unstable fixed point at each timestep, its domain does not change if it is viewed at period 2. Similar behavior can be seen in causally coupled lattices with synchronous updating [7]. For r = 4, = 0.8, the causally coupled lattice settles into a steady-state configuration with exactly two cell values, on opposite sides of the fixed point. This configuration is the spatial equivalent of the oscillatory feature addressed in Section 3. Corresponding behavior continues to hold for values of as low as 0.5, although the exact synchronization into only two states breaks down and each of the two domains expands into a continuous range of values. Spatial checkerboard patterns are interesting for the current investigation since they exist in two incompatible phases. Using L for the low-value domain and H for the high-value domain, any strip of, say, six cells in the simulation can hold the pattern LHLHLH or HLHLHL; in fact, each timestep will transform one of these patterns into the other. However, if one of them is established in one part of the lattice, and another one elsewhere, then the simultaneously existing checkerboard patterns are incompatible. They cannot be consistently extended into each other, and a defect in the regular pattern must appear at their zone of contact. If such a defect is a closed curve, the dynamics of the CML causes it to shrink and disappear within a few hundred timesteps. If a CML contains two defects that wind completely around it (recall that periodic boundary conditions are used), two incompatible checkerboards can persist for an almost arbitrarily long time. Typical lifetimes depend on the simulation parameters, which affect the mobility of defects and hence their potential to encounter each other, transform into closed curves, and disappear. This behavior can be complicated by the addition of inhomogeneous regions in the CML. A barrier of zero coupling that stretches across the CML, for example, provides an additional boundary, so that a single defect parallel to the bar-
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rier suffices to partition the simulation. Such a single defect has no partner with which it can annihilate, but it can be destroyed by contact with the zero-coupling barrier. Two parallel barriers of zero coupling, stretching completely across the CML, partition it into two causally disconnected regions, which may sustain opposite checkerboard patterns. As a side effect, this configuration increases the likelihood of partitioning by spontaneous defects. A linear defect perpendicular to the barriers can use them to anchor its endpoints and can, thus, be much shorter than a defect that winds around the entire CML. Such defects tend to be relatively immobile, leading fairly quickly to a steady-state configuration that shows no signs of further evolution despite the presence of defects. In order to utilize this behavior as a test for information propagation, 32 · 32 lattices with single-layer barriers at positions x = 0 and x = 16 were studied. Each row (set of cells with the same y) of the lattice is then cut by barriers at two positions. It is a straightforward task to compare the two sections of the row and determine whether or not they are in the same checkerboard pattern. Repeating this procedure for every row leads to a final figure between 0 and 32 giving the number of rows that are in the same state in both CML sections. If there were no commonality to the two regions separated by barriers, the expected average count of patternmatches would still be somewhat less than 16, because there are two ways of failing to match patterns. The two sides of a row may be in opposite checkerboard phases, or one or both of them may contain a defect and, hence, not be in any checkerboard phase at all. The extent of this bias depends on the defect survival rate and is hard to quantify analytically, so it is best to proceed numerically. In addition, even at B = 0, a barrier of thickness 1 exerts a common influence on both regions of the CML, even though it blocks information propagation between the regions. For both of these reasons, we conduct the analysis by (i) examining the pattern match rate for B = 0 as a baseline and (ii) comparing it with the pattern match rate when non-zero coupling is established in the barrier. Fig. 8 shows the result of simulations of synchronously updated non-causally coupled map lattices. The case of B = 0 barriers is used to establish a base distribution of match counts with which other barrier types can be compared. Barriers of thickness 2 were constructed by adding columns 1 and 17 to those at 0 and 16, respectively. Fig. 9 presents the same information for an alternative scenario of stable spatial patterning, a synchronously updated causally coupled map lattice with r = 4 and P = 0.6. Although there is no immediately obvious departure from the B = 0 distribution, there is a suggestion of bias in Fig. 8 for a barrier of thickness 1 with B = 0.01. This becomes clearer by considering the mean of each distribution rather than plotting its full density. These means are shown in Fig. 10. The barrier of thickness 1 with B = 0.01 in the non-causally coupled map lattice with r = 3.9 and P,S = 0.1 is clearly distinguished from the baseline case of B = 0. Even in the same class of simulations, this distinction does not survive when the barrier is thickened to two cells. Nevertheless, this provides a confirming example of information flow between weakly coupled sublattices across a barrier. It suggests that other cases in this paradigm may show smaller effects which would require much more extensive simulations to resolve.
Fraction of Simulations
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Fig. 8. Population of pattern-matches across barriers, where the base CML is non-causally coupled with r = 3.9 and P = 0.1. Stepfunction lines represent 1r limits of the causally decoupled case where walls have = 1. Icons represent other barrier configurations as marked on legend: BT is barrier thickness, C is coupling constant. Vertical strokes through icons are their 1r error bars.
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Fraction of Consistent Domains Fig. 9. Population of pattern-matches across barriers, where the base CML is causally coupled with r = 4 and P = 0.6. Significance of lines and icons is as in Fig. 8.
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Barrier thickness and coupling constant Fig. 10. Overall mean pattern-match rates for the CMLs of Figs. 8 and 9. Baselines show the match rate for B = 0 barriers; parallel dotted lines show the 95% confidence interval. Icons show match rates with nonzero coupling. Barrier thickness and coupling are given at the bottom of the graph; error bars show 95% confidence limits. (a) Non-causally coupled map lattices with r = 3.9, P = 0.1. (b) Causally coupled map lattices with r = 4, P = 0.6.
5. CMLs with strong coupling It was recently found that, under fairly general conditions, neurons display non-monotonic transfer functions, in which the response to a nominally excitatory stimulus reaches a peak at a certain level of input and then declines with further increase of the stimulus [10]. Such a non-monotonic transfer function is qualitatively similar to the logistic function. Since neurons generally have no self-excitation, eliminating the self-term in the CML evolution by setting = 1 seems a reasonable choice to represent such situations. As has been noted earlier [11] two-dimensional lattices with strong coupling generally tend to display ongoing spatiotemporal chaos as their final steady-state behavior. CMLs under ‘‘neuronal’’ conditions as mentioned above exhibit a long-range spatial structure which varies irregularly. Domains tend to persist over several timesteps, longer than for uncoupled maps with r = 4, but they are never truly stable.
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The possibility of information propagation in this case was investigated by examining the influence of spatial patterns across boundaries much as in Section 4. The CML was partitioned by two barriers, which had the effect of cutting across the visible patterns. To determine the effect of the partitioning, a correlation coefficient between state values in the columns of cells adjoining the barrier was computed; e.g., for a barrier at x = 16, the correlation would be computed between the cell values at x = 15 and those at x = 17. Since the adjoining cells are strongly perturbed by the presence of the barrier, correlations between the next columns outward (x = 14 and x = 18 in this example) were calculated as well. Fig. 11 shows results for isotropic neighborhoods (asymmetric coupling), for synchronous updating, for = 1, for first order von Neumann and Moore neighborhoods, and for different separations (2 columns and 4 columns). The coupling constant B of the barrier is plotted horizontally. The case marked NA means that there is no barrier—the sim-
Von Neumann
Correlations across barrier position
S=2
Moore S=4
S=2
S=4
NA0.001.01.1
NA0.001.01.1
0.6
0.4
0.2
0.0 NA0.001.01.1
NA0.001.01.1
Coupling in barrier Fig. 11. Effect of barriers cutting short-range structure in case of strong coupling ( ! 1). The figure shows the correlation coefficient between columns of cells with the given separation with a one-cell-thick barrier as a function of B. Error bars show a 95% confidence interval. NA indicates no barrier (homogeneous CML). S = 2 is the smallest possible separation, representing the two columns of cells immediately adjacent to the barrier on each side. Separation S = 4 shows the correlation between the next adjoining columns, where the immediate perturbing effects of the barrier are not visible to inspection.
Correlations across barrier position
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0
0.2
0.4 0.6 Barrier coupling
0.8
1.0
Fig. 12. Correlation adjacent to the barrier for the full range of barrier coupling (barrier thickness 1, first order von Neumann neighborhoods). (Fig. 11 amplifies the low-coupling regime of this plot.) Density of plotted points is increased in some sections to resolve regions of particular interest.
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Coupling in barrier Fig. 13. Effect of extending the barrier from 1 to 2 cells, so that the minimum separation is 3 and the next pair of adjoining columns has separation 5. Other details are as in Fig. 11.
ulation is homogeneous. This case was included as a measure of the normal degree of influence to be compared with the influence across barriers with small B. From Fig. 11 it is clear that a considerable degree of spatial structure remains between columns even after they are separated by a barrier of B = 0. This does not, however, denote a true causal connection so much as a shared causal influence: Both regions of cells are influenced by the same barrier layer. There is no sign of an abrupt change when small coupling is restored to the barrier. There is, however, a somewhat counterintuitive continuing decrease in the correlation as the barrier coupling increases. Fig. 12 plots the correlation for columns adjacent to a 1-cell-thick barrier with von Neumann neighborhoods for the full range of barrier coupling constants. The case of no barrier is given by B = 1. The correlation decreases in a smooth curve as B is decreased, reaching a minimum slightly below B = 0.2. For smaller barrier coupling, the correlation increases again and reaches a local maximum at B = 0. The behavior shown in Figs. 11 and 12 shows a surprising phenomenological richness, but does not fully address the question of information transfer across a barrier. To ensure that the barrier with = 0 not only cuts off contact but eliminates all shared causal influences, the barrier was thickened from a one-cell to a two-cell layer in Fig. 13. This increases the separation between adjoining cell layers from 2 to 3 units, with the next layers out now 5 units apart. Somewhat unexpectedly, the von Neumann lattice loses all correlations at these ranges, even when no barrier is interposed, despite the fact that the case of separation 4 in Fig. 11 showed strong correlation. On the other hand, Moore neighborhoods retain strong correlations if there is no barrier. When the two-layer barrier not only disconnects but isolates the two regions, the correlation vanishes completely, and is not restored by moderate levels of increased barrier coupling (cf. right half of Fig. 13).
6. Summary Information propagating between weakly coupled systems is often regarded as inconsequential relative to environmental noise. However, since nonlinear dynamical systems can drastically amplify specific components of their input, such assessments may be premature for such systems. In fact, there is increasing evidence that weak interactions between complex nonlinear systems can cause unexpected but significant consequences for their behavior. In this contribution, we used sublattices of coupled map lattices separated by a boundary with low-coupling parameter as a tool to examine the possibility of efficacious information flow between the sublattices. Our attention was focused on the influence of the stability properties of one sublattice on those of the other. In detail, we investigated particular features of both the stabilizing process and the spatial patterns resulting from it. The results depend on particular lattice parameters such as causal versus non-causal coupling, isotropic versus anisotropic neighborhoods, synchronous versus asynchronous updating, and on the coupling strength of the sublattices and the barrier. We found significant indications for information flow across the barrier for two scenarios:
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• Propagation of destabilizing behavior from sublattices with low coupling to sublattices with high coupling occurs in causally coupled, asynchronously updated lattices with isotropic neighborhoods (asymmetric coupling). • Propagation of stable pattern information occurs in non-causally coupled, synchronously updated lattices. On the other hand, no evidence for significant information flow was found in the following scenarios: • No propagation of destabilizing or stabilizing behavior occurs in causally coupled, asynchronously updated lattices with anisotropic neighborhoods (symmetric coupling). • No propagation of stable pattern information occurs in causally coupled, synchronously updated lattices. • No propagation of long-range structure occurs in lattices with strong coupling. As a fairly obvious area of applications for the obtained results we refer to investigations of stability properties of neuronal assemblies, subsystems of cortical networks. In this case, issues of stability are particularly important since stable neuronal behavior is required for neural correlates of conscious states. Another class of interesting but more speculative applications might be found in the study of mind-matter systems in the more general sense addressed, e.g., in [12–14]. The basic observation in these studies indicates correlations between material and mental states without any known interaction. No plausible understanding of these correlations is currently available. The possibility of an efficacious and detectable information exchange between systems with extremely weak interaction may be relevant for this situation.
References [1] Pikovsky A, Rosenblum M, Kurths J. Synchronization. A universal concept in nonlinear sciences. Berlin: Springer; 2001. [2] Atmanspacher H, Filk T, Scheingraber H. The significance of causally coupled, stable neuronal assemblies for the psychological time arrow. In: Buccheri R et al., editors. Endophysics, time, quantum, and the subjective. Singapore: World Scientific; 2005. p. 149–62. [3] Kaneko K, editor. Theory and applications of coupled map lattices. New York: Wiley; 1993. [4] Kaneko K, Tsuda I. Complex systems: chaos and beyond. Berlin: Springer; 2000. [5] Atmanspacher H, Scheingraber H. Stabilization of causally and non-causally coupled map lattices. Physica A 2005;345:435–47. [6] Mehta M, Sinha S. Asynchronous updating of coupled maps leads to synchronization. Chaos 2000;10:350–8. [7] Atmanspacher H, Scheingraber H. Inherent global stabilization of unstable local behavior in coupled map lattices. Int J Bifurcat Chaos 2005;5:1665–76. [8] Atmanspacher H, Filk T, Scheingraber H. Stability analysis of coupled map lattices at locally unstable fixed points. Eur Phys J B 2005;44:229–39. [9] Dobyns Y, Atmanspacher H. Characterizing spontaneous irregular behavior in coupled map lattices. Chaos, Solitons, & Fractals 2005;24:313–27. [10] Kuhn A, Aertsen A, Rotter S. Neuronal integration of synaptic input in the fluctuation-driven regime. J Neurosci 2004;24:2345–56. [11] Kaneko K. Spatiotemporal chaos in one- and two-dimensional coupled map lattices. Physica D 1989;37:60–82. [12] Radin DI, Nelson RD. Evidence for consciousness-related anomalies in random physical systems. Found Phys 1989;19:1499–514. [13] Jahn R, Dunne B, Bradish G, Dobyns Y, Lettieri A, Nelson R, et al. Mind/machine interaction consortium: PortREG replication experiments. J Scient Explor 2000;14:499–555. [14] Atmanspacher H, Jahn RG. Problems of reproducibility in complex mind-matter systems. J Scient Explor 2003;17:243–70.
Information flow between weakly interacting lattices of coupled maps York Dobyns a, Harald Atmanspacher b
b,*
a PEAR, Princeton University, Princeton, NJ 08544–5263, USA Institut fu¨r Grenzgebiete der Psychologie und Psychohygiene, Wilhelmstr. 3a, 79098 Freiburg, Germany
Accepted 25 July 2005
Abstract Weakly interacting lattices of coupled maps can be modeled as ordinary coupled map lattices separated from each other by boundary regions with small coupling parameters. We demonstrate that such weakly interacting lattices can nevertheless have unexpected and striking effects on each other. Under specific conditions, particular stability properties of the lattices are significantly influenced by their weak mutual interaction. This observation is tantamount to an efficacious information flow across the boundary. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Weak interactions between systems can produce unanticipated and dramatic effects on their behavior if their time evolution is nonlinear. The study of such effects (e.g., synchronization) has recently attracted much interest. Many investigations of weakly interacting systems focus on coupled nonlinear dynamical systems such as iterative maps or attractors. In these cases, remarkable synchronization phenomena have been observed experimentally and described theoretically [1]. In particular, phase synchronization has been found as an important indicator, much more sensitive than amplitude or frequency synchronization, for weak coupling. Beyond interactions between only a few individual systems it is interesting to study weak coupling for subsystems of spatiotemporally distributed systems consisting of many nonlinear elements. Such situations are generic whenever the evolution of a system as a whole leads to separately operating subsystems (domains) which themselves consist of many elements. Important examples are neuronal assemblies, subsystems of the cortex that are typically constituted by some thousands of neurons. In order to function as neural correlates of conscious states, the behavior of neuronal assemblies has to satisfy certain stability conditions (for more details see [2]). Coupled map lattices (CMLs) are much studied models of such systems [3,4]. Therefore they suggest themselves as suitable candidates to examine the possibility of a transfer of stability properties (i.e., information flow) due to weak interactions. Such interactions can be implemented in terms of boundary regions with weak internal coupling that separate regions with more strongly coupled elements undergoing a complex dynamical evolution. *
Corresponding author. Tel.: +49 761 207 2199. E-mail address: [email protected] (H. Atmanspacher).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.07.008
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In this contribution we describe corresponding effects of weak coupling between sublattices of CMLs separated by a boundary (or barrier). In the following Section 2, some basic notions concerning CMLs are briefly introduced. In Section 3 possible influences of the destabilizing behavior of one sublattice onto that of another one are investigated and quantified by appropriate measures. Two scenarios of isotropic and anisotropic coupling in the neighborhood of the boundary are explicitly distinguished. In contrast to the transfer of destabilizing behavior, Section 4 examines the transfer of stable patterns across boundaries. While the transfer of stability in Sections 3 and 4 is mainly explored for regions with a coupling parameter close to 0.5, Section 5 considers the case of strong coupling, ! 1. Section 6 gives a compact summary of the main results.
2. Coupled map lattices: some basic notions The basic evolution equation for a two-dimensional CML in discrete time is given by X uxy;tþ1 ¼ ð1 Þf ðuxy;t Þ þ gðuij;t Þ; N ði;jÞ2N
ð1Þ
where u is the state value at each cell in the lattice in the spatial dimensions (x, y), and t is the timestep of the iteration of the state values. The coupling constant determines how much input a cell gets from its neighbors in comparison to its own time evolution. The function f is usually taken to be the logistic map: x ! f(x) = rx(1 x) for 0 6 r 6 4. N is the set of cells that are neighbors of a given cell, and N is the number of cells in N. If the function g, applied to the values of neighboring cells, is given by g(x) = x, this implies that their input to u(x, y) is given by the values of the neighboring cells at the previous timestep. This condition is called causal coupling [5]. Alternatively, we may choose g(x) = f(x), in which case the input is given by the neighboring cell values after they have undergone the uncoupled evolution for the current timestep. This is called non-causal coupling [5], sometimes also future coupling [6]. The function g(x) may be parameterized as ax + (1 a)f(x), to consider mixtures of causal and noncausal coupling. A time scale important for the physical interpretation of Eq. (1) is the time interval Ds assumed for the updating mechanism, i.e., for the physical integration of signals from the neighborhood states with the state considered. If signals between cells are transmitted much slower than the time scale assumed for the updating mechanism, Ds Dt, the updating can be implemented (almost) instantaneously, or synchronously. If this is not the case, Ds J Dt, updating must be implemented asynchronously, implying that a proper updating sequence must be specified. For an inhomogeneous lattice the situation becomes more involved. In such a lattice, in general N, r and are not the same and may assume different values at or around each cell. Additional features of the CML, such as the function g or its interaction parameter a, can also be inhomogeneous. For the current investigation we restrict ourselves to lattices in which the coupling is variable within the simulation, = xy, but all other parameters are kept constant. Eq. (1) can thus be rewritten in the form xy X auij;t þ ð1 aÞruij;t ð1 uij;t Þ; ð2Þ uxy;tþ1 ¼ ð1 xy Þruxy;t ð1 uxy;t Þ þ N ði;jÞ2N Eq. (2) contains an implicit complication. Consider the case where two cells a and b with a 5 b occupy each othersÕ coupling neighborhoods. Cell a couples with strength a to every cell in its neighborhood, including b. Cell b couples with strength b to every cell in its neighborhood, including a. Thus, due to the inhomogeneity of , the coupling from a to b is not equal to the coupling from b to a. In addition to such cases of asymmetric coupling, we want to consider symmetric coupling as well. In order to achieve this, we specify the coupling strengths within a given neighborhood by additional indices i, j specifying the cell from which cell x, y receives coupling input and obtain P 1 X ði;jÞ2N xyij uxy;tþ1 ¼ 1 ruxy;t ð1 uxy;t Þ þ xyij ðauij;t þ ð1 aÞruij;t ð1 uij;t ÞÞ. ð3Þ N N ði;jÞ2N The condition for symmetric coupling between all pairs of points is then a symmetry condition for the four-dimensional -array: xyij = ijxy for all pairs (x, y) and (i, j). While Eq. (3) does not, in fact, require this symmetry condition, we will impose it whenever this form of the evolution equation is used. The asymmetric case of Eq. (2), where all cells in a given cellÕs neighborhood couple to it with the same strength, is referred to as an isotropic neighborhood. The case in which is given the additional degrees of freedom of Eq. (3) allows us to model anisotropic neighborhoods. When the coupling in an anisotropic neighborhood obeys the constraint xyij = ijxy, the coupling is nevertheless symmetric. All simulations in this paper that use anisotropic neighborhoods are designed for symmetric coupling.
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Most of the numerical results presented and discussed in the following are derived for von Neumann neighborhoods of order 1, defining a cellÕs neighbors as the orthogonally adjacent cells: the neighbors of (x, y) are the four cells (x ± 1, y) and (x, y ± 1). Some results are obtained using Moore neighborhoods; the Moore neighborhood of order 1 includes the von Neumann neighborhood of order 1 plus the four diagonally adjacent cells (x ± 1, y ± 1) and (x ± 1, y 1). Von Neumann and Moore neighborhoods of higher order are generalized in the obvious way. Although Eqs. (2) and (3) allow a to be varied continuously, all simulations used in the current analysis use either pure causal (a = 1) or pure non-causal (a = 0) coupling. In most cases the logistic parameter r is set to 4, so that uncoupled cells would engage in fully chaotic behavior that spans the entire range of state values from 0 to 1. Unless otherwise specified, simulations are started with random initial conditions and a 32 · 32 array with periodic boundary conditions in both directions is used.
3. Asynchronously updated CMLs with spontaneous stabilization In previous work [5,7,8] it has been found that coupled map lattices with large enough coupling exhibit spontaneous global stabilization at the locally unstable fixed point of individual maps. Specifically, for a causally coupled lattice with r = 4 and asynchronous update, the simulation stabilizes with all cell states at x = (r 1)/r = 0.75, the unstable fixed point, if P 0.5. This stabilization is completed within a few hundred timesteps in a 32 · 32 lattice. If < 0.5, the simulation arrives (equally rapidly) at a steady-state condition where the state values vary in some range around the fixed point, but continue to exhibit spatiotemporal chaos within that limited range. The size of the steady-state range shrinks with increasing . Fig. 1 shows state-value population distributions for the steady-state of CMLs just above and below, respectively, the critical value for stabilization. In the following we use this stabilization process to study the transfer of stability properties across boundaries between sublattices of a CML. The simulations are based on interactions between a stable sublattice with = 0.51 and an unstable sublattice with = 0.49. The full CML was split into two corresponding regions with the option of placing barriers between them. The two regions consisted of columns 0–15 and columns 16–31 of the 32 · 32 lattice, respectively, and barriers were placed in columns 16 and 31. (Due to periodic boundary conditions, column 0 neighbors column 31.) Subsequently the 512-cell region comprising columns 0–15, in which measurements are taken, is called the primary region. Barriers, if present, border the primary region, and the remainder of the CML beyond the barriers is called the secondary region. The values of the coupling constant within the primary, barrier, and secondary regions are labeled P, B, and S, respectively. (a)
(b) 0.06
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Fig. 1. Cell population histograms (bin width 1/128) averaged over 200 timesteps, after steady-state configuration achieved, in asynchronously updated and causally coupled CMLs: (a) = 0.51, full stabilization at the unstable fixed point x = 0.75, (b) = 0.49, ongoing spatiotemporal chaos within a limited range of state values.
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In addition to the width of the cell-population distribution as shown in Fig. 1, the domain transition rate (DTR) was primarily used to quantify the stabilization behavior in the sublattice under investigation. (We did not evaluate more elaborate measures, e.g., phase synchronization or conditional correlations.) A domain is a set of cells whose state values are all higher or all lower than the unstable fixed point at a given timestep. Since quasi-periodic oscillation with ‘‘period’’ 2 around the fixed point characterizes all cell values, a given cell was considered to have changed domains if it left this oscillatory behavior. The DTR was then defined as the average number of such occurrences per timestep in the 512 cells of the primary lattice. 3.1. Asymmetric coupling It turned out that the steady-state behavior of the unstable ( = 0.49) region was largely insensitive to contact with another, stable ( = 0.51) region. The distribution width did not change detectably, while the average DTR changed by an amount less than the timestep-to-timestep noise. In contrast, the stable sublattice proved sensitive to its surroundings. When the entire CML has = 0.51, it quickly (<200 timesteps) converges on a stable phase with no domain transitions and minimal width. In contrast, an = 0.51 sublattice in direct contact (B S) with an unstable = 0.49 sublattice yields a steady-state DTR of 7.26 ± 0.03 (1r statistical uncertainty). Fig. 2 shows the results for a one-cell-thick barrier in columns 16 and 31 with 0 6 B 6 0.49 (for B = 0.49 the ‘‘barrier’’ is simply part of the secondary region). The set of simulations used here maintained isotropic neighborhoods; therefore the coupling to cells in the barrier is asymmetric. Cells adjoining the barrier receive input from the barrier cells with the same coupling constant as from their other neighbors; the lowered coupling constant in the barrier only affects the cells of the barrier themselves. For vanishing barrier coupling, B = 0, the DTR of the region with P = 0.51 is 26.66, which is considerably higher than for vanishing barrier thickness. This is somewhat counterintuitive since the barrier cuts off all causal contact between the two main regions of the simulation. However, as a consequence of the vanishing coupling the cells of the barrier are simply free-running chaotic maps with r = 4, exhibiting the maximum possible variability from timestep to timestep. Thus, the barrier itself drives the increased instability of the stabilizing region with P = 0.51. As the barrier coupling B increases from 0 to 0.49, the DTR decreases non-monotonically. It is particularly remarkable that, even though B = 0 provides the maximum possible degree of chaotic variation, turning on even a very small coupling in the barrier increases, rather than decreases, the destabilizing activity in the primary region. Since the simulations use isotropic neighborhoods, the coupling strength with which cells at the boundary of the sublattices receive input from the barrier is given by B. Since the activity in the barrier decreases with increasing B, its influence on domain transitions in the primary region must likewise decrease. Nevertheless, the DTR is observed to increase.
Domain Transition Rate
25
20
15
10
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0.1
0.2
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Barrier Coupling Fig. 2. Equilibrium domain transition rate in a sublattice with P = 0.51 separated from a sublattice with S = 0.49 by a one-cell-thick barrier of small B. At B = 0 the barrier is completely unaffected by the rest of the CML and produces a complete decoupling between regions. At B = 0.49 the barrier is part of the secondary region. Error bars are smaller than the plotting icons. Neighborhoods are isotropic first order von Neumann, hence the coupling between regions is asymmetric.
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Domain Transition Rate
27.2 27.1 27.0 26.9 26.8 Secondary C=0 Secondary C=.49 Secondary C=.8
26.7 26.6 0.0
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Barrier Coupling Fig. 3. Expanded view of the small-B region of Fig. 2, with additional data from simulations with S = 0 and S = 0.8 for the secondary region. Note the sharp increase in DTR at the transition from B = 0 to B = 0.001. Icons have been laterally displaced for clarity. Statistical errors are shown at 1r.
This demonstrates that even a minute change in the barrierÕs coupling properties can have disproportionate effects on the adjoining sublattice to which it couples. This does not, however, appear to be primarily due to information propagation across the barrier. Fig. 3 expands on Fig. 2 in two ways: first, it concentrates on the region 0 < B < 0.01, affording greater resolution; second, the simulations plotted in Fig. 2 have been supplemented by simulations in which the coupling constant in the secondary region has been lowered to S = 0 or raised to S = 0.8. In most cases any differences among the three values of DTR for different S at the same B are within the statistical resolution of the simulations. However, genuine differences are found for different B. Features such as the abrupt jump in DTR from B = 0 to 0.001 are driven primarily by the interaction between the primary region and the barrier, rather than by influences propagating from the secondary region across the barrier. A broader survey of simulation parameters in Fig. 4 shows similar behavior. As the coupling constant in the primary region changes from 0.5 to 0.7, the qualitative character of the response to the barrier remains unchanged: There is an abrupt increase in DTR between B = 0 and 0.001, and a continuous but non-monotonic progression as B = 0 is further increased. A number of values of primary and barrier coupling show an apparent sensitivity to the secondary coupling, but no single case stands out clearly from the simulation noise. The frequency with which such variations appear, however, does stand out above the simulation noise. For several simulations run with different values of the secondary coupling S and with the same values of primary coupling P and barrier coupling B, the mutual consistency of the measured values can be computed by a v2 statistic. Given a population N of measurements indexed by i, each of which has a value xi and an uncertainty ri, the quantity !, X X X 2 2 2 2 ðxi lÞ =r ; where l ¼ xi =ri 1=r2i ; ð4Þ v ¼ i
i
2
i
is distributed as a v with N 1 degrees of freedom. The distribution is exact if the observational errors are normally distributed, approximate otherwise. In the present case, the ‘‘observational error’’ consists of a statistical uncertainty from mutually independent simulations, such that the uncertainties can be treated as normal. All simulations were run at multiple values of S for 48 different combinations of P and B. Among these simulations, the variations due to S have v2 = 152.6 on 110 degrees of freedom (p = 0.004). This result can be checked for validity by noting that in those six cases with B = 0 the DTR of the primary region cannot, in principle, be affected by changes in S (and any such apparent dependence would denote a spurious effect in the analysis). Indeed it turned out that those cases produce v2 = 11.8 on 12 degrees of freedom as expected: the remaining 42 combinations have v2 = 140.8 on 98 degrees of freedom, p = 0.003. The conjecture that the behavior of the secondary region affects the primary region leads to the prediction that the DTR in the primary region should decrease as S increases (since then the stability of the secondary region increases). A statistical test for this prediction produces an interesting result shown in Fig. 5. All of the 42 cases where DTR measurements are available at multiple values of S for the same P, B with B 5 0 are shown as a scatterplot. Its horizontal
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Barrier Coupling Fig. 4. Small-B region as in Fig. 3 for different coupling constants P of the primary region between 0.5, the minimum for stabilization, and 0.7, as indicated on each subgraph. Icons are plotted for different barrier couplings and secondary-region couplings as in Fig. 3.
Residual in DTR
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Coupling in exterior region Fig. 5. Survey of all simulations conducted to test DTR effects with isotropic coupling. The residual DTR is shown as a function of secondary-region coupling, after subtracting the mean DTR for all simulations run with the same primary and barrier coupling constants. Individual error bars are not shown due to the high degree of overlap: the horizontal dotted lines show the average 1r uncertainty of a typical simulation. The dashed line shows the slope of a linear regression fit: b = 0.041 ± 0.009 (statistical error).
axis is S, and the vertical axis is the residual change in DTR—that is, the difference between the DTR found for any particular values of (P, B, S) and the average DTR for all simulations with the same (P, B). The shift in the density of observed points as a function of S is evident. Much of the non-systematic noise in Fig. 5 is presumably due to simu-
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lation noise: the horizontal dotted lines show the average 1r uncertainty with respect to the residual DTR. The dashed line shows a linear regression on these data: it has a slope of 0.0408 ± 0.0093, some 4.4r away from 0 (p = 1.14 · 105, two-tailed). The overall trend shown in Fig. 5 can be confirmed by linear regressions in each of the 42 individual cases. The resulting population of slopes has a mean of 0.0410 ± 0.0093 that is in excellent agreement with the aggregate figure above. A goodness-of-fit test for the individual regression slopes against their population average shows a v2 of 66.1 on 41 degrees of freedom (p = 0.008), which is suggestive (though not conclusive) of non-random variation between cases, presumably driven by differential responses at different values of P and B. Since the baseline DTR for the performed simulations is in the range between 30 and 40 (cf. Fig. 4), the observed regression slopes correspond to a proportional change in DTR of approximately 1 part in 1000 per unit increment of S. 3.2. Symmetric coupling The results presented in the preceding subsection have been obtained for isotropic neighborhoods with asymmetric coupling between neighbors belonging to different regions. In the most extreme case, a barrier cell with B = 0 accepts no input from any neighbor and evolves purely according to the logistic equation, while it affects its neighbors at their full coupling constants. Fig. 6 is the analogue of Fig. 2 for the alternative case of symmetric coupling. As in Fig. 2, there is an increase in DTR as the coupling decreases. However, this increase is much smoother than with isotropic, asymmetric coupling. In particular, it joins smoothly onto B = 0 with no indication of a discontinuous jump as in the asymmetric case and without ever attaining a higher DTR than that for B = 0 (cf. Fig. 2). The increased DTR relative to the direct-contact situation must be caused by a mechanism different from the asymmetric case, since with symmetric coupling the increased noise of a low-coupling barrier cannot directly drive activity in the primary region. Due to the anisotropy of the symmetric coupling, each cell adjoining the barrier couples both to and from the barrier cell with B = 0. Both the coupling term and the suppression of the cellÕs internal chaotic evolution in Eq. (3) are therefore only 3/4 as large as they would be for isotropic coupling of the same strength. The same effect would be produced by lowering the primary regionÕs coupling constant to 3/4 of its nominal value, which in the case of P = 0.51 is insufficient to give rise to stabilization. As a consequence of this argument, stable behavior should reappear for B = 0 when P is increased to 4/3 of the stabilization-inducing value 0.5, i.e., to 0.666. Fig. 7 demonstrates that this is indeed the case. In addition, Fig. 7 shows the DTR as a function of P for two different barrier couplings, B = 0.001 and 0.01, and for the three values of S = 0, 0.4, and 0.8. Each set of simulations with the same B and S is plotted by a line (line type determined by B) connecting points (point type determined by S). The exact coincidence of the points indicates that the
22
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20 18 16 14 12 10 8 0.0
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0.2 0.3 Barrier Coupling
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Fig. 6. Effect of low-coupling barrier when cell-wise symmetric coupling (anisotropic neighborhoods) is enforced. As in Fig. 2 (for asymmetric coupling), low coupling within the barrier increases the DTR in the stabilizing region. Unlike in the asymmetric case, the B-dependence joins smoothly onto B = 0. Error bars are smaller than plot icons.
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BC=0 BC=0.001 BC=0.01
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Fig. 7. Effect of increasing coupling P in the stabilizing region for various barrier coupling constants, with anisotropic neighborhoods (symmetric coupling). Lines connect points generated with the same barrier coupling. For B = 0, full stabilization reappears at P = 2/ 3, as described in the text. For B 5 0, the exact superposition of points for different values of S indicates a lack of sensitivity to this parameter.
DTR does not depend on S. The v2 for S-driven variation is 140.7 on 158 degrees of freedom, well within the expected variation. The population of regression slopes has a mean of (7.5 ± 44.7) · 105. Thus, unlike in the case of isotropic neighborhoods, there is no evidence for any influence propagating from the secondary to the primary region with anisotropic neighborhoods.
4. Synchronously updated CMLs with stable spatial patterns Information propagation between sublattices cannot only be tested by stabilization processes in those sublattices but also by the transfer of stable spatial patterns resulting from such processes. As discussed in detail in [9], non-causally coupled lattices with synchronous update at parameter values such as r = 3.9, = 0.1 have a stable ‘‘checkerboard’’ phase in which each of a cellÕs orthogonal neighbors are in the domain opposite to that of the cell itself. Since each cell oscillates from a value below to a value above the unstable fixed point at each timestep, its domain does not change if it is viewed at period 2. Similar behavior can be seen in causally coupled lattices with synchronous updating [7]. For r = 4, = 0.8, the causally coupled lattice settles into a steady-state configuration with exactly two cell values, on opposite sides of the fixed point. This configuration is the spatial equivalent of the oscillatory feature addressed in Section 3. Corresponding behavior continues to hold for values of as low as 0.5, although the exact synchronization into only two states breaks down and each of the two domains expands into a continuous range of values. Spatial checkerboard patterns are interesting for the current investigation since they exist in two incompatible phases. Using L for the low-value domain and H for the high-value domain, any strip of, say, six cells in the simulation can hold the pattern LHLHLH or HLHLHL; in fact, each timestep will transform one of these patterns into the other. However, if one of them is established in one part of the lattice, and another one elsewhere, then the simultaneously existing checkerboard patterns are incompatible. They cannot be consistently extended into each other, and a defect in the regular pattern must appear at their zone of contact. If such a defect is a closed curve, the dynamics of the CML causes it to shrink and disappear within a few hundred timesteps. If a CML contains two defects that wind completely around it (recall that periodic boundary conditions are used), two incompatible checkerboards can persist for an almost arbitrarily long time. Typical lifetimes depend on the simulation parameters, which affect the mobility of defects and hence their potential to encounter each other, transform into closed curves, and disappear. This behavior can be complicated by the addition of inhomogeneous regions in the CML. A barrier of zero coupling that stretches across the CML, for example, provides an additional boundary, so that a single defect parallel to the bar-
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rier suffices to partition the simulation. Such a single defect has no partner with which it can annihilate, but it can be destroyed by contact with the zero-coupling barrier. Two parallel barriers of zero coupling, stretching completely across the CML, partition it into two causally disconnected regions, which may sustain opposite checkerboard patterns. As a side effect, this configuration increases the likelihood of partitioning by spontaneous defects. A linear defect perpendicular to the barriers can use them to anchor its endpoints and can, thus, be much shorter than a defect that winds around the entire CML. Such defects tend to be relatively immobile, leading fairly quickly to a steady-state configuration that shows no signs of further evolution despite the presence of defects. In order to utilize this behavior as a test for information propagation, 32 · 32 lattices with single-layer barriers at positions x = 0 and x = 16 were studied. Each row (set of cells with the same y) of the lattice is then cut by barriers at two positions. It is a straightforward task to compare the two sections of the row and determine whether or not they are in the same checkerboard pattern. Repeating this procedure for every row leads to a final figure between 0 and 32 giving the number of rows that are in the same state in both CML sections. If there were no commonality to the two regions separated by barriers, the expected average count of patternmatches would still be somewhat less than 16, because there are two ways of failing to match patterns. The two sides of a row may be in opposite checkerboard phases, or one or both of them may contain a defect and, hence, not be in any checkerboard phase at all. The extent of this bias depends on the defect survival rate and is hard to quantify analytically, so it is best to proceed numerically. In addition, even at B = 0, a barrier of thickness 1 exerts a common influence on both regions of the CML, even though it blocks information propagation between the regions. For both of these reasons, we conduct the analysis by (i) examining the pattern match rate for B = 0 as a baseline and (ii) comparing it with the pattern match rate when non-zero coupling is established in the barrier. Fig. 8 shows the result of simulations of synchronously updated non-causally coupled map lattices. The case of B = 0 barriers is used to establish a base distribution of match counts with which other barrier types can be compared. Barriers of thickness 2 were constructed by adding columns 1 and 17 to those at 0 and 16, respectively. Fig. 9 presents the same information for an alternative scenario of stable spatial patterning, a synchronously updated causally coupled map lattice with r = 4 and P = 0.6. Although there is no immediately obvious departure from the B = 0 distribution, there is a suggestion of bias in Fig. 8 for a barrier of thickness 1 with B = 0.01. This becomes clearer by considering the mean of each distribution rather than plotting its full density. These means are shown in Fig. 10. The barrier of thickness 1 with B = 0.01 in the non-causally coupled map lattice with r = 3.9 and P,S = 0.1 is clearly distinguished from the baseline case of B = 0. Even in the same class of simulations, this distinction does not survive when the barrier is thickened to two cells. Nevertheless, this provides a confirming example of information flow between weakly coupled sublattices across a barrier. It suggests that other cases in this paradigm may show smaller effects which would require much more extensive simulations to resolve.
Fraction of Simulations
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BT 1, C=.001 BT 1, C=.01 BT 2, C=.001 BT 2, C=.01
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Fig. 8. Population of pattern-matches across barriers, where the base CML is non-causally coupled with r = 3.9 and P = 0.1. Stepfunction lines represent 1r limits of the causally decoupled case where walls have = 1. Icons represent other barrier configurations as marked on legend: BT is barrier thickness, C is coupling constant. Vertical strokes through icons are their 1r error bars.
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Fraction of Consistent Domains Fig. 9. Population of pattern-matches across barriers, where the base CML is causally coupled with r = 4 and P = 0.6. Significance of lines and icons is as in Fig. 8.
(a)
(b)
Mean match count
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Barrier thickness and coupling constant Fig. 10. Overall mean pattern-match rates for the CMLs of Figs. 8 and 9. Baselines show the match rate for B = 0 barriers; parallel dotted lines show the 95% confidence interval. Icons show match rates with nonzero coupling. Barrier thickness and coupling are given at the bottom of the graph; error bars show 95% confidence limits. (a) Non-causally coupled map lattices with r = 3.9, P = 0.1. (b) Causally coupled map lattices with r = 4, P = 0.6.
5. CMLs with strong coupling It was recently found that, under fairly general conditions, neurons display non-monotonic transfer functions, in which the response to a nominally excitatory stimulus reaches a peak at a certain level of input and then declines with further increase of the stimulus [10]. Such a non-monotonic transfer function is qualitatively similar to the logistic function. Since neurons generally have no self-excitation, eliminating the self-term in the CML evolution by setting = 1 seems a reasonable choice to represent such situations. As has been noted earlier [11] two-dimensional lattices with strong coupling generally tend to display ongoing spatiotemporal chaos as their final steady-state behavior. CMLs under ‘‘neuronal’’ conditions as mentioned above exhibit a long-range spatial structure which varies irregularly. Domains tend to persist over several timesteps, longer than for uncoupled maps with r = 4, but they are never truly stable.
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The possibility of information propagation in this case was investigated by examining the influence of spatial patterns across boundaries much as in Section 4. The CML was partitioned by two barriers, which had the effect of cutting across the visible patterns. To determine the effect of the partitioning, a correlation coefficient between state values in the columns of cells adjoining the barrier was computed; e.g., for a barrier at x = 16, the correlation would be computed between the cell values at x = 15 and those at x = 17. Since the adjoining cells are strongly perturbed by the presence of the barrier, correlations between the next columns outward (x = 14 and x = 18 in this example) were calculated as well. Fig. 11 shows results for isotropic neighborhoods (asymmetric coupling), for synchronous updating, for = 1, for first order von Neumann and Moore neighborhoods, and for different separations (2 columns and 4 columns). The coupling constant B of the barrier is plotted horizontally. The case marked NA means that there is no barrier—the sim-
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Correlations across barrier position
S=2
Moore S=4
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NA0.001.01.1
NA0.001.01.1
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Coupling in barrier Fig. 11. Effect of barriers cutting short-range structure in case of strong coupling ( ! 1). The figure shows the correlation coefficient between columns of cells with the given separation with a one-cell-thick barrier as a function of B. Error bars show a 95% confidence interval. NA indicates no barrier (homogeneous CML). S = 2 is the smallest possible separation, representing the two columns of cells immediately adjacent to the barrier on each side. Separation S = 4 shows the correlation between the next adjoining columns, where the immediate perturbing effects of the barrier are not visible to inspection.
Correlations across barrier position
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0
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0.4 0.6 Barrier coupling
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1.0
Fig. 12. Correlation adjacent to the barrier for the full range of barrier coupling (barrier thickness 1, first order von Neumann neighborhoods). (Fig. 11 amplifies the low-coupling regime of this plot.) Density of plotted points is increased in some sections to resolve regions of particular interest.
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Coupling in barrier Fig. 13. Effect of extending the barrier from 1 to 2 cells, so that the minimum separation is 3 and the next pair of adjoining columns has separation 5. Other details are as in Fig. 11.
ulation is homogeneous. This case was included as a measure of the normal degree of influence to be compared with the influence across barriers with small B. From Fig. 11 it is clear that a considerable degree of spatial structure remains between columns even after they are separated by a barrier of B = 0. This does not, however, denote a true causal connection so much as a shared causal influence: Both regions of cells are influenced by the same barrier layer. There is no sign of an abrupt change when small coupling is restored to the barrier. There is, however, a somewhat counterintuitive continuing decrease in the correlation as the barrier coupling increases. Fig. 12 plots the correlation for columns adjacent to a 1-cell-thick barrier with von Neumann neighborhoods for the full range of barrier coupling constants. The case of no barrier is given by B = 1. The correlation decreases in a smooth curve as B is decreased, reaching a minimum slightly below B = 0.2. For smaller barrier coupling, the correlation increases again and reaches a local maximum at B = 0. The behavior shown in Figs. 11 and 12 shows a surprising phenomenological richness, but does not fully address the question of information transfer across a barrier. To ensure that the barrier with = 0 not only cuts off contact but eliminates all shared causal influences, the barrier was thickened from a one-cell to a two-cell layer in Fig. 13. This increases the separation between adjoining cell layers from 2 to 3 units, with the next layers out now 5 units apart. Somewhat unexpectedly, the von Neumann lattice loses all correlations at these ranges, even when no barrier is interposed, despite the fact that the case of separation 4 in Fig. 11 showed strong correlation. On the other hand, Moore neighborhoods retain strong correlations if there is no barrier. When the two-layer barrier not only disconnects but isolates the two regions, the correlation vanishes completely, and is not restored by moderate levels of increased barrier coupling (cf. right half of Fig. 13).
6. Summary Information propagating between weakly coupled systems is often regarded as inconsequential relative to environmental noise. However, since nonlinear dynamical systems can drastically amplify specific components of their input, such assessments may be premature for such systems. In fact, there is increasing evidence that weak interactions between complex nonlinear systems can cause unexpected but significant consequences for their behavior. In this contribution, we used sublattices of coupled map lattices separated by a boundary with low-coupling parameter as a tool to examine the possibility of efficacious information flow between the sublattices. Our attention was focused on the influence of the stability properties of one sublattice on those of the other. In detail, we investigated particular features of both the stabilizing process and the spatial patterns resulting from it. The results depend on particular lattice parameters such as causal versus non-causal coupling, isotropic versus anisotropic neighborhoods, synchronous versus asynchronous updating, and on the coupling strength of the sublattices and the barrier. We found significant indications for information flow across the barrier for two scenarios:
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• Propagation of destabilizing behavior from sublattices with low coupling to sublattices with high coupling occurs in causally coupled, asynchronously updated lattices with isotropic neighborhoods (asymmetric coupling). • Propagation of stable pattern information occurs in non-causally coupled, synchronously updated lattices. On the other hand, no evidence for significant information flow was found in the following scenarios: • No propagation of destabilizing or stabilizing behavior occurs in causally coupled, asynchronously updated lattices with anisotropic neighborhoods (symmetric coupling). • No propagation of stable pattern information occurs in causally coupled, synchronously updated lattices. • No propagation of long-range structure occurs in lattices with strong coupling. As a fairly obvious area of applications for the obtained results we refer to investigations of stability properties of neuronal assemblies, subsystems of cortical networks. In this case, issues of stability are particularly important since stable neuronal behavior is required for neural correlates of conscious states. Another class of interesting but more speculative applications might be found in the study of mind-matter systems in the more general sense addressed, e.g., in [12–14]. The basic observation in these studies indicates correlations between material and mental states without any known interaction. No plausible understanding of these correlations is currently available. The possibility of an efficacious and detectable information exchange between systems with extremely weak interaction may be relevant for this situation.
References [1] Pikovsky A, Rosenblum M, Kurths J. Synchronization. A universal concept in nonlinear sciences. Berlin: Springer; 2001. [2] Atmanspacher H, Filk T, Scheingraber H. The significance of causally coupled, stable neuronal assemblies for the psychological time arrow. In: Buccheri R et al., editors. Endophysics, time, quantum, and the subjective. Singapore: World Scientific; 2005. p. 149–62. [3] Kaneko K, editor. Theory and applications of coupled map lattices. New York: Wiley; 1993. [4] Kaneko K, Tsuda I. Complex systems: chaos and beyond. Berlin: Springer; 2000. [5] Atmanspacher H, Scheingraber H. Stabilization of causally and non-causally coupled map lattices. Physica A 2005;345:435–47. [6] Mehta M, Sinha S. Asynchronous updating of coupled maps leads to synchronization. Chaos 2000;10:350–8. [7] Atmanspacher H, Scheingraber H. Inherent global stabilization of unstable local behavior in coupled map lattices. Int J Bifurcat Chaos 2005;5:1665–76. [8] Atmanspacher H, Filk T, Scheingraber H. Stability analysis of coupled map lattices at locally unstable fixed points. Eur Phys J B 2005;44:229–39. [9] Dobyns Y, Atmanspacher H. Characterizing spontaneous irregular behavior in coupled map lattices. Chaos, Solitons, & Fractals 2005;24:313–27. [10] Kuhn A, Aertsen A, Rotter S. Neuronal integration of synaptic input in the fluctuation-driven regime. J Neurosci 2004;24:2345–56. [11] Kaneko K. Spatiotemporal chaos in one- and two-dimensional coupled map lattices. Physica D 1989;37:60–82. [12] Radin DI, Nelson RD. Evidence for consciousness-related anomalies in random physical systems. Found Phys 1989;19:1499–514. [13] Jahn R, Dunne B, Bradish G, Dobyns Y, Lettieri A, Nelson R, et al. Mind/machine interaction consortium: PortREG replication experiments. J Scient Explor 2000;14:499–555. [14] Atmanspacher H, Jahn RG. Problems of reproducibility in complex mind-matter systems. J Scient Explor 2003;17:243–70.