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Time series analysis – ‘standard’ techniques Mean, variance – entire series, or windowed Autocorrelation ARMA – autoregressive moving average Spectral analysis – Fast Fourier Transform EEGs as Time Series

Chaos theory Formulated in 1960s Lorenz model (1961) Mathematically simple… …but behaviorally (dynamically) complex EEGs as Chaotic Time Series

Chaos theory Formulated in 1960s Multiple definitions – all include some form of Aperiodicity – output never returns exactly to starting conditions Boundedness EEGs as Chaotic Time Series

Chaos theory, cont. Termed low-dimensional, deterministic chaos 3 equations sufficed to produce “chaotic dynamics” No stochastic input (“noise”) was required to produce complicated behavior Extreme sensitivity to initial conditions (starting point)  Can’t predict outcome, no matter how accurately initial conditions are specified. EEGs as Chaotic Time Series

Chaos adherents noted that EEGs were Complex time series Bounded Seemingly aperiodic  Hypothesis that EEGs represented low-dimensional chaotic dynamics EEGs as Chaotic Time Series

Historically, two parameters from chaos theory have dominated the EEG/HRV literature: Lyapunov exponent ( 1 or L1) Correlation Dimension (D2) Nonlinear Analysis of Time Series

Lyapunov exponent ( 1 or L1) Measures rate of divergence of two nearby points = ‘unpredictability’ Said to measure “degree of chaos” Meaning of that is unclear… Positive L1 is necessary condition for chaotic dynamics L1 for Lorenz system is 1.5 – 2+, depending on values for , , and  Nonlinear Analysis of Time Series

Correlation Dimension (D2) Measures number of independent variables required to produce dynamics of time series > 2  6-ish for low-dimensional chaotic dynamics Infinite for a pure noise time series Nonlinear Analysis of Time Series

Algorithms for computing L1 & D2 from time series developed in early 1980s Developed for time series based on solutions of chaos-producing systems of differential equations (e.g., Lorenz model) Applied to EEG and HRV data soon after Early authors overlooked or neglected to mention that their algorithms require time series of (almost) infinite length don’t work when noise is present in time series Nonlinear Analysis of Time Series

Algorithms for computing L1 & D2 from time series developed in early 1980s Developed for time series based on solutions of chaos-producing systems of differential equations (e.g., Lorenz model) Applied to EEG and HRV data soon after Early authors overlooked or neglected to mention that their algorithms require time series of (almost) infinite length don’t work when noise is present in time series Nonlinear Analysis of Time Series

Suitable(?) algorithm developed by Rosenstein et al. in 1993 Relatively easy implementation, in contrast with earlier algorithms Averages behavior of a sample of points in a time series Divergence through time of two nearby points is the datum N = 5000 appears sufficient for estimation of L1 and D2 Not yet applied as extensively to EEG & HRV analysis as it should be Nonlinear Analysis of Time Series

We developed software that allowed us to ‘section out’ eye-blink and other artifacts: Nonlinear Analysis of EEGs

We then modified the Rosenstein et al. algorithm slightly Select points from time series Locate nearest neighbor in m–dimensional phase space Allow distance between nearest neighbors to evolve through time Apply algorithm to successive point chunks of the sectioned time series Calculate L1 from L1 = ln(divergence)/  t Nonlinear Analysis of Time Series

Calculate L1 from L1 = ln(divergence)/  t Lyapunov Exponent (L1) Calculation

Calculate D2 from distribution of nearest neighbor distances Correlation Dimension (D2) Calculation

Nearest neighbor distances ‘partitioned’ into “shells” of increasing (m–dimensional) size Cumulative proportion of nearest neighbor distances computed as function of shell size Termed Correlation Sum, or Cmr Log-log plot of Cmr vs. shell size yields D2 Correlation Dimension (D2) Calculation

Calculate D2 from proportion of nearest neighbor distances that are within successively larger distances Correlation Dimension (D2) Calculation

Calculate D2 from proportion of nearest neighbor distances that are within successively larger distances Correlation Dimension (D2) Calculation 

The time (lag) when the autocorrelation function drops below (1 – 1/e) is important Estimates the period of the m–dimensional time series. Autocorrelation Function

Results of ANOVA L1D2Period F P0.027n.s.<0.001 Control0.80 a a PTSD0.62 a a PS & F1.56 b b L1 and Period significantly increased by therapy Increase persists until follow-up EEG May be decreased by PTSD; possible sample size problem. D2 unaffected by PTSD or therapy

Nonlinear Analysis of EEGs Next step is to apply L1,D2(?), the autocorrelation function, and other nonlinear time series analysis techniques Goal is an algorithm that allows early discrimination of responders from non-responders Our index must therefore exhibit a high degree of sensitivity specificity

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