Attractor Reconstruction fromEvent-Related Multi-electrodeEEG-DataGottfried Mayer-KressCathleen BarczysWalter J. Freeman

SFI WORKING PAPER: 1991-01-007

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SANTA FE INSTITUTE

Attractor Reconstruction from Event-Related Multi-Electrode EEG-Data •

Gottfried Mayer-Kress

Mathematics Department, University of California at Santa Cruz

The Santa Fe Institute, 1120 Canyon Rd, Santa Fe, NM

Center for Nonlinear Studies, Los Alamos National Laboratory

and

Cathleen Barczys

Biophysics Graduate Group, University of California at Berkeley

and

Walter J. Freeman

Department of Molecular and Cell Biology, University of California at Berkeley

13 December 1990

Abstract

We combine methods from Karhunen-Loeve decomposition and from non-lineardynamical systems theory to describe the spatio-temporal chaos in signals from anarray of 8*8 electrodes implanted on the surface of the olfactory bulb of a rabbit.We propose several new approaches to analyse and quantify spatial and temporalmodes of complexity in arrays of irregular time series.

'Proceedings of the International Symposium MATHEMATICAL APPROACHES TO BRAIN FUNC­TIONING DIAGNOSTICS of the International Brain Research Organization, A.V. Holden (Ed.), WorldScientific, 1991

I

I. Introduction

The EEG literature contains increasing evidence that low dimensional deterministic dy­namical systems provide useful concepts to describe coherent, global behavior of brainevents (see e.g. [2],[5],[13]). This evidence has lead to numerous publications in whichsingle lead EEGs have been used for time delay reconstructions of vector data. If a subse­quent estimate of the correlation dimension of this vector data set appears to be low, thedynamics of the neural system, as manifested in the EEG data set, have been interpretedto approximate that of a chaotic attractor (see e.g. [10], [6]).

This interpretation has a theoretical justification if the EEG is indeed determinis­tic and globally coherent. In that case the measurement at a single lead would providecomplete information about the dynamics of the whole system (see e.g. [8], [7] for a dis­cussion ofthe spatio- temporal propagation of information). However, in typical cases thecoherence of the EEG is only partial, as evidenced by the differences among reconstructedattractors obtained from different leads (see e.g. [3], [12]). If in studies of specific sensoryand motor events one is interested in the stimulus- or response-dependent dynamics ofonly a limited area of the brain, single EEG traces suffice; but in general one is interestedin a larger neural region and thus, one has to take into account spatial as well as temporalaspects of EEG dynamics.

There appear to be basically three different approaches to the analysis of multi­channel EEG recordings. In one, the assumptions are made that the brain functions as anorganic whole, and that the various EEG time series recorded from the scalp in humansduring multi-channel recording, for example, derive in part from a global activity patternof finite dimension. The attractor reconstruction can then be based on vectors obtainedfrom the signals of an increasing number of leads [23]: Assume we observe signal Xi(t) attime t from lead i, i = 1, ... , N. Then for n < N an n-dimensional vector time series canbe given as: x(t) = (Xl (t), X2(t), . .. , xn(t)). The signal from each lead is inferred to havelocal components (stemming from the activity of "non-enslaved" neurons, i.e., neuronswhose firing is not synchronized with the global coherent modes) and a global componentwhich is associated with the deterministic brain attractor. Each of these componentscontributes significantly to the signal at each electrode, thereby contributing a large noisecomponent (from the "non-enslaved" neurons) as well as a large redundancy from theglobal, coherent component which is picked up at a number of electrodes simultaneously.To the extent that this approach succeeds, it will provide a univariate scale of complexityfor the whole human brain from the numerical values of attractor dimension. However, itoffers no insights into the underlying neural processes, just as the commonly-held divisionof the EEG spectrum into different frequency bands has been widely employed and yetnever convincingly demonstrated to be directly relevant to the underlying physiology.

The other extreme would be to assume that the dynamics at each lead are generatedby independent attractors, each of which may be associated with a functional area of thecortex. Thus we would obtain an array of dimension values d i from the reconstructedvector time-series Xi(t) = (Xi(t), Xi(t - T), . .. , Xi(t - (n -1 )T)) These dimensions di wouldbe smoothly changing as a function of electrode position indicated by the index i.

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Finally in the third approach, which was first used in [18] and has also been suggestedin [24], the spatio-temporal signal is decomposed into orthogonal modes. There are severalbasic ways in which one could construct such optimal modes: one could use analyticbasis functions, which are selected according to a theory [14], [16] or based on symmetryconsiderations (see [26]), or one could use purely statistical criteria for the constructionof orthogonal modes [14].

A hybrid of this third approach was used in [18]. Principal components analysis(PCA), which is a type of factor analysis that extracts orthogonal components, was ap­plied to EEG data that were specifically linked to physiological structures and to definedbehavioral conditions. The EEG segments were then classified by stimulus type usingthe coefficients of the dominant PCA component. The hybrid aspect of their approachderives from the addition of several pre-processing steps which were incorporated to opti­mize the classification; these operations included segment selection optimization, spatialand temporal filtering, and normalization.

In this paper we want to focus on another type of hybrid approach involving anorthogonal decomposition - the Karhunen-Loeve decomposition and attractor reconstruc­tion. At the outset, however, we must distinguish this orthogonal decomposition first,from the singular value decomposition, as it is applied to reconstructed attractors fromsingle time series (via time delay) (see [28], [22]). Here the orthogonal (Karhunen-Loeve)modes are explicitly related to spatial structures and not to the temporal dynamics ofthe system. In that respect one selectively projects the spatio- temporal dynamics ontospatially (global) coherent modes. Incoherent local fluctuations are thereby averaged outand do not contribute to the deterministic dynamics of the reconstructed attractor.

In this paper we propose to combine the advantages of a spatial decomposition withthe power of modern chaos theory. The vector sequence which constitutes the recon­structed attractor has components which are neither given by time-delayed coordinatesnor are they given from the raw signals of spatially separated electrodes. Instead, they arecomposed of the time-dependent amplitudes of the Karhunen-Loeve modes. Thus we caninterpret the largest KL mode as that spatial pattern which has the largest contribution(averaged over the total observation time) to the spatial pattern of the observed signal.At any given instant of time its contribution to the observed pattern (at that specificmoment) would be given by the magnitude of its amplitude at that time.

In our approach we adhere to three conditions. First, the multiple channels of EEGdata are taken from an array of closely spaced electrodes that is placed onto one area ofcortex, in which short, locally dense anatomical connections provide the substrate for theemergence of cooperative neural activity with demonstrated widespread spatial coherence.Second, the EEG data for which attractors are to be derived have been demonstrated tocontain behavioral information. That is, the EEG segments can be classified reliablyby stimulus type using quantitative techniques. (The information-carrying component ofthe EEG upon which these classifications are based is the spatial patterns of amplitudemodulation of a common carrier waveform - that is, a time series that is a component ofall the channels.) Third, the EEG data are filtered to remove high frequency components(above 160 Hz) that are far above the spectral range of the dendritic generators, and toremove a low frequency component that reflects the external driving of the cortex by inputs

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extraneous to the behavioral task [15]. In this respect we attempt to work with thosecomponents of the data that reflect operation of the cortex in a quasi-autonomous manner,so as to meet an essential condition for the experimental delineation of an attractor fromsingle or multiple time series.

In the following we give a brief overview of previous work using principal componentanalysis of EEG data. Then we present results from the Karhunen-Loeve (KL) expansionon time series from a grid of 8*8 electrodes on the olfactory bulb of a rabbit under con­ditions of inhaling different odors. We present the spectrum and mode amplitudes of theKL expansion and reconstruct the attractor from the dynamics of the mode amplitudes.

The reconstruction of the attractor is based on the hypothesis that, because a spatialpattern is found to recur with the presentation of each odor, each stimulus puts thecortical dynamics into a condition which corresponds to a specific location on the attractormanifold. The time evolution of the EEG signal defines short line segments on thismanifold. Repetition of the identical experiment then corresponds to tracing the manifoldwith a short line segment starting from a different location within the same manifold.

In the case of a limit cycle attractor this would correspond to setting the phase of theoscillator. In a quasi-periodic attractor it would correspond to the selection of a point ona torus by fixing two phases. In both examples it is evident that the choice of a randomlyselected ensemble of phases and a subsequent short term evolution of the trajectories willrecover the structure of the limit cycle and torus. We expect that the same assumptionshould be true for chaotic attractors.

Thus we are able to replace the time average, typically done with continuous longtime-series, with an ensemble average (assuming a form of ergodicity), that allows us toapproach the problem of data limitation from a different angle.

Finally we introduce the notion of vector recurrence plot based on the vectors con­structed from the mode amplitudes with respect to the KL base. From the attractorsreconstructed from the mode amplitudes we can give a coarse estimation of the (time­dependent) point-wise dimensions [1] .

The KL spectra for the three different data sets (" odor A" , "odor B", and" control")provide a tool for ordering the data sets according to their spatial complexity. On theother hand, the values of the approximate point-wise dimension values indicate a differentordering of the data sets according to their temporal complexity.

II. Dynamics of Coherent Modes

From the KL expansion we can identify a number of dominant spatial modes as well as thevariation of their amplitudes. A rather naive conclusion would be that each spatial modecan be described by just one first order ordinary differential equation. Only in that casewould the number of dominant KL modes be an estimate of the dimension of the globaldynamical system. (Hereby the dimension would have been determined based on attractorreconstruction from a single time series.) This assumption might be valid near the onset

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of pattern formation in the EEG - as it is, for example, in fluid convection. There, modesof different spatial wave lengths become successively unstable and start to oscillate. Lowdimensional temporal chaos can only occur when the proliferation of oscillating spatialmodes is constrained (in many cases geometrically through small aspect ratios). ThusKL analysis in combination with dimension analysis of the time-series can provide newtools to determine if the increased complexity in the signal stems from a larger numberof spatial modes or from a higher degree of chaos in the already existing modes.

Coherent dynamical modes are defined experimentally through the measured dy­namics of observed quantities such as the electrical potential in the case of EEG andthe velocity field in the case of fluid convection. In electro-convection the flow field isinfluenced by the ion-distribution which typically is not observed. Thus, non-observabledynamical modes can selectively drive specific flow patterns with changing temporal com­plexity but with the same spatial pattern.

Similarly it is thinkable that different electro-chemical processes in the brain are as­sociated with global and coherent structures which can coexist and are only weakly cou­pled such that they can exhibit different temporal complexity. Non-observable dynamicalmodes arise in studies of cortical neuro-dynamics when the limitations of experimentaltechniques allow us to observe only one or a small number of interactive populations ofneurons in the brain of a behaving animal or human subject. For example, we may seeonly the forward limb (excitatory neurons) or the feedback limb (inhibitory neurons) ofcortical negative feedback loops that generate oscillations in the gamma range, but notboth simultaneously. However, our models allow us to predict the wave forms of these"non-observables" in special experimental circumstances where they can be made acces­sible for verification.

III. Definitions

Let us consider data from a 2-dimensional array of observation points (leads): qj =

(Xi, Yi), i = 1, ... , N, where N = p2 denotes the number of array points (i.e. Y(~-l)P+M =Y(~-l)p+v, and also: X(~-l)P+M = X(v-l)P+M' for all />', fl, v = 1, ... ,p. (These conditions justmake sure that we relabel the two-dimensional grid as one-dimensional array.) A spatio­temporal signal is given by a three-dimensional array u(qj, t 19 ), where the observationinterval is sampled at points t19, {) = 1, ... ,T. The covariance matrix C of the data set(normalized to have zero mean) is given by:

(1)

The n - th Karhunen-Loeve mode ,pn E Rn is determined through the eigen-valueequation:

(2)

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where A1 :2: A2 :2: ... :2: AN :2: a determine the "energy" of the modes, - that is,the average contribution of each mode to the overall pattern of the signal during theobservation time. In explicit form the eigen-value equation is given as:

The constant c is determined from the ortho normalization equation:

N

I>Pn(<Jj),pm(<Jj) = cOn,mj=l

The eigenvalues An are normalized: 2:::;=1 An = 1.

Now we can decompose the signal U(qi, tll) into the KL modes according to:

N

u(<Ii, tll) = L An (tll ),pn (<u)n=l

(3)

(4)

(5)

where we have for the time dependent amplitudes An (tll ), Am (tll) , n,m = 1, ... ,N :

(6)

Thus the amplitude An (tll) indicates how much each KL mode,pn contributes to thepattern U(qi' t ll ) at the time instant t ll .

IV. Some Test Examples

The interpretation of the dynamics of KL modes is not completely obvious and widelyknown; therefore, we want to discuss some simple test examples, before we proceed withthe experimental data.

Let us consider the case of two spatial modes 1>1 (<u), 1>2(<U) which have temporalamplitudes B 1(tll),B2 (tll), for a ::::: tll ::::: 100. That means we consider a spatio-temporalsignal:

(7)

We make the assumption here that we do not know the origin of u(<u, tll ) and wantto recover the modes from KL decomposition. In all examples below we have chosen the

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trigonometricfunctions rP1(X,y) = sin(1rx)cos(41rY), and rP2(X,y) = sin(41rx)cos(1rY) thatmeans we want to analyze simple two-dimensional spatial patterns that evolve in time.

We now want to understand the results of the KL decomposition for different typesof temporal evolutions of the original mode amplitudes Bi(t~).

Example 1:

We first chose a linear time-dependence of the modes, i.e. we have: B1 (t~) =t~ , B2(t~) = 1 - t~. In Fig. 1 we see the corresponding numerical eigen modes,pm, m = 1,2,3. From the KL decomposition we observe that for ,pI the eigenvaluesare A1 = 0.77104 and A2 = 0.22872. Thus the first two modes already account for morethan 99.9% of the variance of the signal. The third mode has less than A3 = .01% ofthe energy and a noisy spatial structure. That means we have an agreement between thetheoretical result ( A3 = 0.0) and the discretized computational result which is better than99.9 %. The amplitude of the first mode is large at all times with a maximum at t = 0.5whereas the second mode shows a nonlinear monotonic increase. That means we cannotexpect that the details of the dynamics of the original modes will be reproduced by theKL modes.

Example 2:

Each of the original amplitudes is a sinusoidal oscillator:

Bl(t~) = cos(51rt~) , B2(t~) = cos(c't~)

, where c' = 26.83873. This means in the temporal domain we have a quasi-periodicmotion of dimension d = 2. In the case of a single time series we would reconstructa 2 dimensional attractor, but this dynamics could come from a single coherent modewith quasi-periodic time dependence or it could stem from the super-position of twocoherent periodic modes. In Fig.2 we see the amplitudes of the first (solid) and second(dashed) KL modes for the case u(q;, t~) = (B1(t~) + B2(t~))rP2(q;). As in the firstcase the discretization in the numerical code produces a second mode with an energy ofE2 = 1.001%. All the higher modes are exactly zero. In Fig.3 we see the first threeKL modes for: u(qj, t~) = B1(t~)rPl(qj) + B2(t~)rP2(qj). The energies are: E1 = 59.76%,E 2 = 39.59%, E 3 = 0.651%. Again we see the spurious mode at about 1% energy. Whatwe also note is the correlation between the slope of one mode and the amplitude of thesucceeding mode. This indicates how the KL algorithm selects modes which are maximallyuncorrelated. (It can be seen easily that the correlation function of a dynamical variableof a bounded attractor and its time derivative vanish.) Thus we can verify for theseexamples that the KL algorithm separates spatial from temporal modes in the sense thatthe spectrum and the dynamics of the amplitudes reflect this separation.

Example 3:

We have also tested several discontinuous time evolutions of the original amplitudesBl(t~), B2(t~). In the case of step function dynamics of the amplitudes (i.e. B1(t~) = 0

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and B 2(t{}) = 1 for t < .5 and B 1(t{}) = 1 and B 1(t{}) = 0 for t > .5 we obtain basicallythe same results with a spurious amplitude which is zero except at the discontinuityt = 0.5. If we replace the sine functions in example 2 by discrete chaotic maps (i.e.B i (t{}+l) = 1 - aBi (t{})2, a = 2 - E, E > 0 small.) we obtain spurious contributions ofenergies of ::; 4%, but we can still clearly distinguish from the KL-spectrum the case of twodimensional dynamics in one mode or the super-position of one dimensional dynamics oftwo modes. We also did several runs with additional noise terms added where the noise wasof the same magnitude as the deterministic signal. In the case of spatially and temporallyuncorrelated noise we observe one large energy eigenvalue (E1 ~ 75%) whereas all theother energies are in the 1% range. If the noise terms are only in the amplitudes of thecoherent modes (i.e. spatially completely correlated), then we can distinguish again thetwo different cases from the spectrum qualitatively similar to the quasi-periodic case.

V. EEG from the Olfactory Bulb

The EEG signals are recorded from the olfactory bulb of a rabbit under three differentexperimental conditions corresponding to three different odor stimuli: odor A, odor B,and no odor (the control). We thus denote the signals as test odor" AT", test odor "BT",and controls" AC" and "BC" depending on whether the data are taken with respect to ATor BT (the no- odor control segments are taken from the pre-stimulus period; see [15] fordetails). Each signal consists of a spatial array of 8*8 time-series. For each experimentaltrial, 6 time segments of data are selected: 3 preceding the stimulus presentation and3 following the stimulus. Earn time series segment covers approximately 76 msec at asampling rate of 512 Hz, which is equivalent to 38 data points per electrode, or 64 * 38= 2432 data points for each segment. In the preliminary, exploratory data set describedhere there are 10 segments for each of the 3 experimental conditions. In Fig. 4 we showa sequence of snapshots of one AT data segment as it evolves over time (the index 1 inAT2.1 denotes that it was the first out ofthe ten runs). The x and y directions correspondto the electrode location; the z-axis indicates the electrical potential; the time index, t,can range from 1 to 38, although only 6 of the 38 snapshots are shown here.

The strongest KL modes (with eigen value A1 > An' n = 1, ... , N) for the 10 differentruns of the experiment are given in Fig. 5. We note that all modes have the same formexcept AT2.1,AT2.2,AT2.7 AT2.8 but AT2.2 and AT2.8 are similar and appear to be areflection (z ---t -z) of the standard modes. Mode AT2.1 and AT2.7 have a differentpattern but also seem to be symmetric to each other. This suggests that there exist onlya very few classes (up to symmetry) of spatial patterns in the observed bulb activity. Theamplitudes of the strongest KL mode across 10 repetitions of the odor A experimentalcondition are presented in Fig. 6. The great variability among the different runs in themode amplitudes is apparent and also expected from our simple test examples. This isalso to be expected if the state of the system is associated with a strange attractor forwhich individual trajectories can show a large variation.

Since for a discrete sequence of time-series data points the time ordering of the KLprocedure does not enter explicitly into the averaging procedure, we concatenated all 10runs of the experiment into one sequence and computed the KL modes of the total data

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set. The first six KL modes are presented in Fig. 7. We see that there are two dominantmodes with 77 % and 15 %of the energy, whereas the other modes have only 2 %of theenergy or less. Thus, the first two modes cover a total of 92 % of the energy. On onehand, this percentage could indicate that the modes beyond the first 2 are relatively weak;in the context of turbulent flow, for example, a typical cutoff for deterministic modes is90 %. On the other hand, it takes a total of six modes to cover 98 % of the energy,which suggests that perhaps the other, less coherent (possibly noisy) modes contain somerelevant information [21].

The mode-amplitudes as a function of time are given in Figs. 8. We note the strongperiodic component of the first two modes and also some correlation between the slope ofthe first mode with the amplitude of the second mode. This correlation in the time seriesmight result from a spatial gradient in phase at the maximal temporal frequency of timeseries that was observed experimentally in these data [19] up to a maximal phase differencenear 1/4 cycle. An effect on the spatial KL modes is unlikely because the direction ofthe gradient varied randomly over successive segments. Mode 3 has small amplitude andirregular time dependence. The spectral properties of all sequences of "odor A", "odorB", and "control" is summarized in Fig. 9 on a logarithmic scale.

We now want to proceed to reconstruct attractors from the ensemble of electrodesignals for the estimation of its dimensional complexity. A direct approach would be to re­construct an attractor via time-delays for each electrode and then estimate a dimensionalparameter for each of them. This could lead to interesting information about the localiza­tion of complex activity if enough data points were available. State related changes can bereflected in a change in the dimensional complexity of the activity only in localized areasof the cortex [12] . In the case of this data set, however, time series of sufficient lengthwere not available. Thus, the attractors were reconstructed from a set of non- contiguousdata segments arranged into a single time series. For this type of reconstruction one has totake precautions to choose the reconstructed vector components appropriately; otherwiseartificial outliers will be introduced (see [1]). This problem can be circumvented by using,for the reconstruction, the amplitudes of the (spatial) KL modes. Here the embeddingdimension is determined by the number of KL-modes used for the reconstruction. 1 InFig. 10 we have the projection of a time series of amplitude vectors of the two modeswith the largest KL eigenvalues from the AT2 sequence above. In Fig. 11 we have thecorresponding projection on the KL-modes 3 and 4. We observe an increased noisinessof the attractor together with a decreased amplitude (see Fig. 8). In general one wouldexpect the extension of the reconstructed attractor to fall off with the KL eigenvalues.This is because of the decreasing amplitudes of the KL-modes as a function of eigenvalues.In Fig. 12 we have generalized the concept of recurrence plots (see e.g. [29]) to vectortime- series: The grey levels indicate the distance in a five- dimensional mode- amplitudespace between one of the ten samples of the AT2 and each of the other nine samples.The distance in the recurrence increases from black to white. The complexity of theserecurrences in amplitude space can be quantified with the help of a dimensional com­plexity estimate from the reconstructed attractor (taking the five most dominant modes).Preliminary results indicate that the temporal (dimensional) complexity (see e.g. [29],[1])

1Higher dimensional embeddings can be achieved through a combination of spatial vectors and time­delay embedding (see e.g. [27]).

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of the response to both odors (AT2,BT2) is reduced compared to a control state (Ac2),the difference being more pronounced for BT2 (Fig. 13).

VI. Conclusions

We present some preliminary evidence that spatio-temporal complexity can be analysedwith the help of spatial Karhunen-Loeve mode decomposition in combination with tem­poral dimension analysis of the vector of the mode amplitudes. It appears that thosedata sets with a small number of relevant spatial modes have relative high dimensionalcomplexity and vice versa. These preliminary results show that with a synthesis of spatialKL decomposition and temporal dimension estimates we might be able to describe datafrom spatio-temporal chaos in the brain more efficiently.

AcknowledgementsWe would like to thank J. Ringland for his KL code and helpful e-mail exchanges and M.Koebbe for providing us with the vector recurrence data and dimension estimates. One ofus (GMK) gratefully acknowledges fruitful discussions with N. Birbaumer, M. Castellanos,T. Elbert, R. Friedrich, J. Holzfuss, M. Kirby, T. Kurz, W. Lutzenberger, and O. Rossler.

References

[IJ G. Mayer-Kress, F. E. Yates, L. Benton, M. Keidel, W. Tirsch, S.J. PappI, K.Geist, Dimensional Analysis of Nonlinear Oscillations in Brain, Heart and Muscle,Mathematical Biosciences, 90, 155-182,1988

[2] "Perspectives in Biological Dynamics and Theoretical Medicine", A.S. Mandell, S.Koslow, M.F. Shlesinger (eds.) Annals of the New York Academy of Sciences, Vol.504, 62-86, New York, 1987

[3J G. Mayer-Kress, S.P. Layne, Dimensionality of the Human Electroencephalogram,62-86, in [2J

[4] B. Rockstroh, T. Elbert, A. Canavan, W. Lutzenberger, N. Birbaumer "Slow Corti­cal Potentials and Behaviour", Urban & Schwarzenberg, Baltimore, Munich, Vienna,1989

[5] Springer Series in Brain Dynamics, E. Basar(Ed.), 1, Springer-Verlag Berlin, Hei­delberg 1988

[6] S.P. Layne, G. Mayer-Kress, J. Holzfuss, Problems Associated with DimensionalAnalysis of Electro-Encephalogram Data, in [9]

[7] G. Mayer-Kress, K. Kaneko Spatiotemporal Chaos and Noise, J. Stat. Phys.,54,1489-1508, (1989)

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[8] J.A. Vastano, H.L. Swinney, Information Transport in Spatio Temporal Systems,Phys. Rev. Let. 60, 1773-1776, (1989)

[9] "Dimensions and Entropies in Chaotic Systems", G. Mayer- Kress, (ed.), SpringerSeries in Synergetics, 32, Springer-Verlag Berlin, Heidelberg 1986

[10] A. Babloyantz, A. Destexhe, Proc. Nat!. Acad. Sci. 83, 3513, 1986

[11] Proc. of "Temporal Disorder in Human Oscillatory Systems", 1. Rensing, W. ander Heyden, M.C. Mackey (Eds.), Springer Series in Synergetics, Vo!. 36, Springer­Verlag Berlin, Heidelberg 1987

[12] T. Elbert, W. Lutzenberger, N. Birbaumer, private communication, 1990

[13] Proceedings of the International Symposium MATHEMATICAL APPROACHESTO BRAIN FUNCTIONING DIAGNOSTICS of the International Brain ResearchOrganisation, A.V. Holden (Ed.), World Scientific, 1991

[14] W.J. Freeman, "Mass Action in the Nervous System", New York, Academic Press,(1975)

[15] Viana Di Prisco, G. and W.J. Freeman, Odor-related bulbar EEG spatial patternanalysis during appetitive conditioning in rabbits, Behavioral Neuroscience 99, 962­978, (1985)

[16] W.J. Freeman, G. Viana Di Prisco, Relation of olfactory EEG to behavior: Timeseries analysis. Behavioral Neuroscience 100,753-763 (1986)

[17] W.J. Freeman, Techniques used in the search for the physiological basis of the EEG,In: A. S. Gevins, and A. Remond,(eds.) Handbook of Electroencephalography andclinical Neurophysiology.Vol 3A, Part 2, Ch. 18. Amsterdam, Elsevier, (1987)

[18] W.J. Freeman, B. Van Dijk, Spatial patterns of visual cortical fast EEG duringconditioned reflex in a rhesus monkey, Brain Research 422, 267-276, (1987)

[19] W.J. Freeman, B. Baird. Relation of olfactory EEG to behavior: Spatial analysis.,Behavioral Neuroscience 101, 393-408 (1987)

[20J J.L. Lumley, Stochastic Tools in Turbulence, Academic Press, NY, New York 1970

[21] L. Sirovich, M. Kirby, An eigenfunction approach to large scale transitional struc­tures in jet flow, Phys. Fuids., A2, 127-136, 1990

[22] M. Palus, 1. Dvorak, On the reliability of singular value decomposition in attractorreconstruction, submitted to Physica D

[23] M. Palus, 1. Dvorak, in [13]

[24] R. Friedrich, A. Fuchs, H. Haken, D. Lehmann, Spatio-temporal analysis of multichannel Ci - EEG maps, in: "Computational Systems - Natural and Artificial",H.Haken (ed.), Springer 1987

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[25] R. Friedrich, A. Fuchs, H. Haken, Synergetic analysis of spatio-temporal EEG­pattern, Proc. "Nonlinear Phenomena in Excitable Media", A. Holden (Ed.), Leeds,1989

[26] R. Friedrich, A. Fuchs, H. Haken, Modeling of spatio- temporal EEG-patterns, in[13]

[27] G. Mayer-Kress, T. Kurz, Dimension Density of Coupled Map Lattices, J. ComplexSystems 1 (1987), 821-829

[28] D.S. Broomhead, G.P. King, Physica 20D, 217, 1986

[29] G. Mayer-Kress, A. Hiibler, Time evolution of local complexity measures and aperi­odic perturbations of nonlinear dynamical systems, LA-UR-89-2202, in "Measures ofComplexity and Chaos", N.B. Abraham, A.M. Albano, A. Passamante, P.E. Rapp(Eds.), Plenum, New York, 1990

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Figure 1: First three eigen modes of a linear superposition of trigonometric spatial modes.The third IIlode vanishes up to numerical accuracy.

Figure 2: Amplitudes of the first (solid) and second (dashed) KL modes for the caseu(q;, t~) = (B1(t19) + B2(t19))<P2(Qi).

Figure 3: Amplitudes of the first (solid),second (short-dashed), and third (long-dashed)KL modes for the case u(q;, t19) = B1(t19)<Pl(q;) +B2(t19 )<P2(q;).

Figure 4: Snap-shots of the time-series AT2.1 of the potential (u(x,y,t), arbitrary units)of the grid of 8*8 electrodes (x-y- plane)

Figure 5: Strongest KL mode for repetitions AT2.1 - 10 of the experiment for which odorA was present. Note the sirn.ilarity of AT2.3, AT2.4, AT2.5, AT2.9, AT2.10, as well asthat of AT2.2 and AT2.8 which appears to be symmetrical to the modes of the first group.

Figure 6: The amplitudes of the strongest KL mode across 10 repetitions of the odor Aexperimental condition AT2.1 - 10. Upper figure: surface representation, lower figure:greyscale repesentation ( increasing from black to white).

Figure 7: KL-modes with the six largest eigenvalues of the concatenation of the sequenceAT2.1-10. The value of the (n-th) KL mode ,pn(lli) is plotted at the locations lli on the8*8 grid in the x-y-plane (see eq. (3)).

Figure 8: Amplitudes of the first three KL modes for the data AT2.1- 10 of fig 7.

Figure 9: Summary of the KL spectra of sequences ACl.1-10 - AC3.1-10, BCl.1-10 ­BC3.1-10, ATU-10 - AT3.1-10, and BTU-10 - BT3.1-10 on a logarithmic scale. All ofthe spectra have been computed from the 380 points of the concatenated files of the 10repetioins of each condition. A and B refer to Odor A and Odor B. "C" denotes controlsegments; "T" denotes test (experimental) segments. The numbers 1, 2, and 3 denotechronological order with respect to the onset time of the stimulus; thus, a chronologicalordering of the segments with a trial would be: C3, C2, C1, Stimulus, T1, T2, T3.

Figure 10: Projection of a time series of amplitude vectors of the two modes with thelargest KL eigenvalues from the concatenated AT2 sequence above. We observe a regularstructure in the reconstructed "attractor" with a high concentration of points in the 4thquadrant.

13

Figure 11: Same as in fig. 10 for dimensions (amplitudes) 3 and 4.

Figure 12: Vector recurrence plot of the first 5 KL-modes of data files AC2, AT2, BT2.Horizontal axis has trial number, vertical axis is time and the grayscale corresponds to thedistance between the vectors in 5-dimensional amplitude space (distances increase fromblack to white).

Figure 13: Very rough estimation of the qualitative change in the dimensional complexityvariation for vector timeseries of experiments AC2, AT2, BT2. The data are the same asthe ones used for the vector- recurrence plots of fig. 12

14

·

surfaces.rna

0.02

o

1/LI

r c, i h

Surfaces.rna

0.02

a

1

1000

surfaces.rna

100 0

r

1

\\\\

(t2.2.1amp.)

1(t2.1.1amp.)

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J

1II

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\

,I

I/

/1

/I

\...'/12 = cos(fivepi * /) *fl +cos(26.83873 * /) *12

(E1 =0.59761, E2 =0.39588, E3 =0.00651)

112 = (cos(fivepi * /) +cos(26.83873 * I)) *12(E1 = 0.98999, E2 =0.01001)

\\..__J L-.J

l

0.75

0.5

0.25

-0.25 KL-3

-0.5

-0.75

-l

t =21

2~4Y

8

8

2

4

x

8

8

4 Y

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200

,r., /,'

At2.1

2

At2. 3

At2. 5

4

6

Largest Karhunen-Loeve Modes

At2. 2

8

At2. 4

2

8

8

8

8

200

oo

At2.7

At2.9

,'I

/

5 lO

8

8

l5 20 25 30

8

8

35time

...-./( -1.--1. " /.'

E =0.77444

E =0.02347

Largest Karhunen-Loeve Modes AT2 (380 pts)

E =0.15131

8

E =0.01465

8

E =0.00655

8

8

8

8

K·L Mode amplitudes AT2 (]80 pts)

t

1- M

rv ~rv'~

5

t- IC 1~ 00 300 - 5

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KL·2 1!

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KL·] 1

0.75

0.5

0.25

-0.25

-0.5

-0.75

-1 (al2amp.o.)

Karhunen-Loeve Eigen values (380 frames)

B

A

t1

-c3

c2 J

c1

oEn

-1

-2

-3

t2

t3

t2

t1

c3

c2cl

t3

10

Karhunen Loeve Attractor, AT2

A2

1

A41

A1

.. ". ,

I ,!

-.,(

-0.75

-1

A3

BnAC2

Vector Recurrences (5 KL modes)

Dimensional Complexity (S KL-Modes)

I I I I It:L5 20 25 30 35

1\( \

1\1 \

1 \ .... 1 \\

1 \/ .......

1 .....

1 \ "- 1

1....... / \ /- - -/

D5 AC2

I

4

3

2

:L

I

5 :La

D5 I

AT24

3

2

:L - - - -,

5 :La 20 25 30 35 t

D5

BT24

3

2

:L

,/--- /--- ,... /---- ",--......-..,..--------" '-

5 :La :L5 20 25 30 35 t