===========For the Background section============
Analyzing EEG data
Nonlinear Analysis
Nonlinear analyses of EEG data stemmed from the explosive development of chaos theory in the
1970’s and the recognition that mathematically simple systems could exhibit extremely complex
temporal dynamics (i.e., low-dimensional chaos) such as those observed in an EEG.
Commencing in the mid-1980’s researchers began applying concepts derived from chaos theory
to the analysis of EEGs, in an effort to demonstrate the (presumed) existence of low-dimensional
chaos in the data and take advantage of chaos theory’s ability to elucidate simple mathematical
structure underlying complex time series (Natarajan et al., 2004). The de facto standard
nonlinear analyses involve computation of the largest Lyapunov exponent (L1) and/or the
correlation dimension (D2), both of which reflect the mathematical complexity underlying a
nonlinear time series.
EEG signals are clearly based on nonlinear processes and a number of studies over the years
have found values of L1 and D2 consistent with low-dimensional chaos (Kantz & Schreiber,
2004). However, the algorithms used to calculate L1 (Wolf et al, 1985) and D2 (Grassberger
and Procaccia, 1983) can yield misleading results when the data are of limited extent, corrupted
by noise, and nonstationary (Brown, 1993; Dammig et al., 1993; Dafilis et al., 2001; Kantz and
Schreiber, 2004; Schwilden, 2006). Under such conditions, heteroscedasticity, varying noise
levels, and even different data ranges can significantly bias estimates of L1 and D2 (Kantz &
Scheiber, 2004), and the autocorrelation structure of the data can be overriding (Paluš, 1999).
The purported advantages of nonlinear analyses of EEG data may thus be obviated by the now
well-documented high-dimensionality, nonstationarity (variation of parameters through time),
and noise that characterize EEG data (Garrett et al., 2003; Breakspear, 2006). Nonetheless,
Virtanen et al. (2007) found different patterns of variation in linear and nonlinear parameters (in
heart rate dynamics) during sleep in post-menopausal women, indicating that nonlinear analyses
are capable of detecting “features” in physiological time series that linear techniques do not. The
nature – and neurobiological significance – of these features, however, remains unclear.
Misgivings about the applicability of L1 and D2 to noisy data have led to the proposal of several
other analytical approaches, most notably entropy measures (Paluš, 1999; Kantz & Schreiber,
2004) and the recent multiscale entropy (MSE) approach of Costa et al. (2005). Linear
techniques are also enjoying a resurgence, with windowed variance (McSharry et al., 2003)
appearing to offer considerable promise. However, these have also been subject to criticism,
and to date have not been widely adopted by researchers.
Finally, a promising avenue for research is the utility of wavelet analysis, which overcomes the
stationarity assumption of FFT techniques. Sakkalis et al. (2006) analyzed frontal and central
EEG tracings during cognitive tasks and reported that wavelet analysis was superior as a tool for
assessing local cortical activation. Wavelet techniques have been applied with considerable
effectiveness to HRV data, as well (Yang and Liao, 1999), but have yet to be applied in the type
of research we are proposing.
The best way to garner information from EEG data is clearly still a contentious topic in the
research literature, and the discipline is still evolving. Especially in view of the serious demands
they place on computer resources, whether nonlinear measures really afford any useful insights
over those provided by traditional analyses such as FFT and simple variance measures remains
an open question (McSharry et al., 2003).
Literature Cited
Breakspear, M. (2006). The nonlinear theory of schizophrenia. Austr. New Zeal. J. Psych.
40:20-35.
Chae, J, Jeong, J, Peterson, BS, Kim, D, Bahk, W, Jun, T, Kim, S, Kim, K. (2004). Dimensional
complexity of the EEG in patients with posttraumatic stress disorder. Psychiatry Research:
Neuroimaging. 131:79-89.
Costa, M, Goldberger, ALPeng, C-K (2005). Multiscale entropy analysis of biological signals.
Phys. Rev. E 71:021906(1-18).
Dafilis,MP, Liley, DTJ, Cadusch, PJ, (2001). Robust chaos in a model of the
electroencephalogram: implications for brain dynamics. Chaos 11(3):474-478.
Dammig, M, Mitschke, F, (1993). Estimation of Lyzpunov exponents from time series: the
stochastic case. Physics Letters. A. 178:385-394.
Garrett, D, Peterson, DA, Anderson, CW, Thaut, MH (2003). Comparison of linear, nonlinear,
and feature selection methods for EEG signal classification. IEEE Transactions on Neural
Systems and Rehabilitation Engineering. 11(2):141-144.
Grassberger, P., Procaccia, I. (1983). Measuring the strangeness of strange attractors. Physica
D. 9:189-208.
Kantz, H, Schreiber, T (2004) Nonlinear Time Series Analysis Second Ed. New York:
Cambridge Press. 369 pp.
McSharry, PE, Smith, LA, Tarassenko, L (2003). Prediction of epileptic seizures: are nonlinear
methods relevant? Nat. Med. 9(2):241-242.
Natarajan, K, U, RA, Alias, F, Tiboleng, T, Puthusserypady, SK (2004). Nonlinear analysis of
EEG signals at different mental states. BioMedical Engineering OnLine. 2004:3:7.
Paluš, M (1999). Nonlinear Dynamics in the EEG analysis: disappointments and perspectives.
In: Rapp, P.E., Pradhan, N., Sreenivasan, R. (eds.) Nonlinear Dynamics and Brain Functioning,
Nova Science Publishers.
Sakkalis, V, Zervakis, M, Micheloyannis, S. (2006). Significant EEG Features Involved in
Mathematical Reasoning: Evidence from Wavelet Analysis. Brain Topography. 19(1-2):53-60.
Schwilden, H. (2006). Concepts of EEG processing: from power spectrum to bispectrum,
fractals, entropies and all that. Best Practice & Research Clinical Anaesthesiology. 20(1):31-48.
Virtanen, I, Ekholm, E, Polo-Kantola, P, Huikuri, H. (2007). Sleep stage dependent patterns of
nonlinear heart rate dynamics in postmenopausal women. Autonomic Neuroscience: Basic and
Clinical. 134:74-80.
Wolf, A, Swift, J, Swinney, H, Vastano, J. (1983). Determining Lyapunov exponents from a
time series. Physica D. 16:285-293.
===========end of For the Background section============
===========For the Design & Methods section============
***Add to Linear Analyses***
Finally, we will incorporate a relatively recent development in the linear analysis of EEG data,
the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model of Engle
(1982), as implemented by Galka et al. (2004) and Wong et al. (2006). The GARCH technique
works very well for analyzing non-stationary data, because it allows for good resolution of the
transients and short time-scale changes in spectral properties that characterize EEG data.
***end Add to Linear Analyses section***
From Bell’s proposal:
 Temporal dynamics of physiological subsystems are nonlinear
o Something from the book “Nonlinear analysis of physiological data”
 Complexity of EEG and HRV data decreases in disease states
 Entropy measures discriminate between OCD and borderline personality disorder
(Mandell and Seiz, 1995)
 Bell et al., 2002, 2004c,e –
 Fell et al. (1996) – combined linear/nonlinear approach best discriminates among
nighttime sleep stages.
 ***
Data Analysis
We will assume that both EEG and HRV data form a complex, nonlinear time series, but that
chaotic dynamics of a low-dimensional attractor is not the primary source of the variation. Our
analytical philosophy will be based on the observations of Schreiber and Schmidt (1997), who
reported that nonlinear analysis of EEG data worked well in some cases but failed completely in
others; their most consistent results came from a combination of linear and nonlinear techniques.
Similarly, Fell et al. (1996) found that a combined linear/nonlinear analysis is the more insightful
approach, and Shen et al. (2003) found that the dimensional complexity of the sleep EEG is
influenced by both linear and nonlinear features,. Thus, while the applicability of nonlinear
measures, especially to EEG data, is fraught with pitfalls as discussed above, our opinion is that
our purposes will best be served by a combination of linear and nonlinear analyses. Clearly,
however, this is a dynamic field. Therefore, we will stay current with theoretical and practical
developments and incorporate them into our research protocol as appropriate.
The general analytical approach detailed in Fell et al. (1996) and Sprott (2003) will be employed.
The embedding dimension will be optimized by the delay reconstruction technique, using the first
zero or, if necessary, first minimum of the autocorrelation function as our choice of time lag
(Kantz and Schreiber, 2004). Maximum Lyapunov exponents (MLE, or L1), the correlation
dimension (D2), and entropy measures (Mandell and Seiz, 1995) will then be calculated and
analyzed as detailed in Kantz and Schreiber (2004) and Costa et al. (2005). Although chaotic
time series are characterized by a spectrum of Lyapunov exponents, knowledge of L1 alone will
suffice for our purposes since it establishes a quantitative limit to predictability. Finally, even
though L1, D2, and entropy measures are mathematically interrelated, they represent different
views of the dynamics underlying the time series data. It is therefore important to compute each
of them because we cannot predict a priori which will be the most important in discriminating
between responders and non-responders.
Nonlinear analyses of our time series data will be conducted using the TISEAN freeware package
(Hegger et al., 1999; Kantz and Schreiber, 2004), the program published by Costa et al. (2005),
and other programs that Toolson will write as required. The important procedural caveats
delineated in Hegger et al. (1999) and Kantz and Schreiber (2004) will be rigorously heeded
when conducting the computations and when interpreting the results.
Statistical analysis of the results of our linear and nonlinear analyses will be conducted using R v.
2.7.2 (The R Foundation). In particular, stepwise discriminant analysis of the results will be
employed to develop the best multivariate index of response versus non-response.
Literature Cited
Engle, RF (1982). Autoregressive conditional heteroskedasticity with estimates of the variance
of United Kingdom inflation, Econometrica 50:987–1008.
Fell, J. Roschke, J., Mann, K., Schaffner, C. (1996). Discrimination of sleep stages: a
comparison between spectral and nonlinear EEG measures. EEG Clin. Neurophysiol. 98:401410.
Galka, A, Yamashita, O, Ozaki, T. (2004). GARCH modelling of covariance in dynamical
estimation of inverse solutions, Phys. Lett. A 333:261–268.
Hegger, R, Kantz, H, Schreiber, T. (1999). Practical implementation of nonlinear time series
methods: the TISEAN package. Chaos. 9(2):413-435.
Mandell, AJ, Seiz, KA. (1995). Nonlinear dynamical patterns as personality theory for
neurobiology and psychiatry. Psychiatry. 58:371-390.
Schreiber, T, Schmitz, A. (1997). Discrimination power of measures for nonlinearity in a time
series. Phys. Rev. E. 55(5):5443-5447.
Shen, Y, Olbrich, E, Achermann, P, Meier, PF. (2003). Dimensional complexity and spectral
properties of the human sleep EEG. Clinical Neurophysiology 114:199–209.
Sprott, JC. (2003). Chaos and Time-Series Analysis. Oxford: Oxford Press. 507 pp.
Takahashi, M, Matsukawa, K, Nakamoto, T, Tsuchimochi, H, Sakaguchi, A, Kawaguchi, K,
Onari , K (2007). Cardiovascular control during voluntary static exercise in humans with
tetraplegia. J. Appl. Physiol, 103(5): 1669-1677.
Vincent, P, Gaspoz, J-M, Molliex, S, Antoniadis, A, Busso, T, Roche, F, Costes, F, Quintin, L,
Lacour, J-R, Barthélémy, J-C (1999). Wavelet transform to quantify heart rate variability and to
assess its instantaneous changes. J. Appl. Physiol. 86(3):1081–1091.
Wong, K, Galka, A, Yamashita, O, Ozaki, T (2006). Modelling non-stationary variance in EEG
time series by state space GARCH model. Computers in Biology and Medicine 36:1327-1335.
Yang, F, Wangcai, L. (1997). Modeling and decomposition of HRV signals with wavelet
transforms. Engineering in Medicine and Biology Magazine, IEEE. 16(4): 17 - 22
===========end of For the Design & Methods section============