Appendix
Neuronal activity underlying the EEG is clearly nonlinear and important features of the
data might be missed if only linear techniques such as autocorrelation or fft are employed.
Indeed, it is clear that nonlinear techniques can provide useful characterizations of brain function
that complement the more traditional power analysis and variance measures, and contribute
significantly to our ability to discriminate between normal and pathological brain function
(Wackerman, 1999; Sarbadhikari and Chakrabarty, 2001; but see Palus, 1998).
The most commonly employed nonlinear methods for characterizing complex time series
are the largest Lyapunov exponent (L1) and the correlation dimension (D2). Correlation
dimension is thought to reflect complexity of cortical dynamics that underlie the EEG signal, and
a number of studies support its application to discrimination among brain functional ‘states’,
both normal and pathological (Pritchard & Duke, 1992). The neurophysiological significance of
L1 estimated from EEG tracings remains unclear, but it clearly is sensitive to different aspects of
time series dynamics, so we are convinced that we must investigate the utility of L1 as a measure
of therapeutic response.
It is important to note that we are not interested in assessing whether EEGs reflect
underlying low-dimensional chaotic dynamics. Our intent is to use nonlinear analysis as an
alternative characterization of cortical function in PTSD patients before, during, and after
treatment, and to incorporate those results into a biologic marker of neurophysiological response
to acupuncture.
The algorithms used by most workers have for calculating D2 (Grassberger and
Procaccia, 1984) and L1 (Wolf et al., 1985) are compromised by their requirement for infinitely
long time series of noise-free data, neither of which is realistic in an experimental setting. Of the
several available algorithms for computing L1 and D2 from small, noisy data sets, we selected
the one developed by Rosenstein et al. (1993), because it takes advantage of all available data,
it’s computationally tractable, and it yields intermediate results that can be used to compute D2
for the same time series. The Rosenstein algorithm is basically a statistical approach in which
L1 is calculated from the aggregate behavior of large number of points is used to ‘compensate
for’ the effects of noise. Like all algorithms used to compute L1 and D2, it is computationally
intense, but 5000 data points are more than sufficient to calculate both L1 and D2, which is well
within the capabilities of even contemporary laptop computers.
Since software implementing the Rosenstein et al. algorithm is not available, we
developed our own, employing the sliding window approach recommended by ***, first
extracting a 5000 data point section of the EEG and uses time-delay embedding to reconstruct
the attractor of the data, with values of time lag (J) and embedding dimension (m) optimized as
described in Pritchard and Duke (1992) and Sprott (2003). Euclidean distances for all points in
the resulting (5000 − (𝑚 − 1)𝐽)-by-𝑚 attractor matrix are calculated and the nearest neighbor
(in m-dimensional space) for each point is identified. The Theiler correction (Theiler, 1986) is
then applied as described in Sprott (2003) to avoid the impact of autocorrelation and the
temporal evolution of the distance between nearest neighbors is computed. The resulting data
are used to estimate L1 and D2 (see Preliminary Results for details). A total of five sections of
each electrode’s tracing were used in computation of L1 and D2.
The choice of time lag and embedding dimension is entirely arbitrary, and one must
balance the competing effects of demands on computer resources and susceptibility to noise in
the data. Small values for time lag can yield inappropriately low values of D2, while large
values for time lag increase the influence of noise in the data, leading to inappropriately high
values for D2 (Lo and Principe, 1989; Sprott, 2003). We used Rosenstein’s approximation of J
equal to the time required for the autocorrelation function to decline to 1 − 1/𝑒 (≈ 0.63), which
our simulation results indicate is appropriate. For our purposes, the significance of embedding
Mean Correlation Dimension, D2
dimension is that computation time increases with m2 . Embedding dimensions much larger than
6-8 result in undesirably long computation times, unless we utilize a supercomputer.
In general, the value of D2 increases with embedding dimension (Natarajan et al., ***;
Chae et al., ****). The dependence of D2 on m is simply the result of the fact that the minimum
nearest-neighbor distances necessarily increase as the position of points is specified in
increasingly higher-dimension hyperspace. Since the distances are lognormal distributed
(Toolson and Perry, unpubl.), no matter what the value of m, the effect of increasing m is to
increase the minimum nearest neighbor distances more rapidly than the maximum distances
which, in turn, leads to higher estimates for D2.
Embedding dimension has generally been optimized by choosing a dimension high
enough to ‘saturate’ the estimate of D2, i.e., to reach a point where estimates of D2 do not
increase with embedding dimension (Sprott, 2003). However, this method was not effective for
us, because our estimates of D2 did not exhibit the well-defined plateau reported by other
workers (Natarajan et al., ***; Chae et al., ****):
14
12
10
8
6
4
2
0
2
4
6
8
10
12
14
16
18
20
Embedding Dimension
This failure of D2 to exhibit a definite saturation plateau has been reported for other types of
physiological data, including EEG and HRV data, and appears to be a typical result when the
data are corrupted by the noise that characterizes most physiological data (Kantz et al., 1998).
Most previous workers have reported values for D2 in the range of 3-6, without reporting
numeric values for m, which, based on our work and the results reported by Natarajan et al.
(****) must have been in the range of 3  6. For the exploratory analyses we report here, we
chose to use an embedding dimension of 4.
Computation of L1
The Rosenstein et al. algorithm records the temporal evolution of the distance between
nearest neighbors, yielding necessary for computing L1. Since L1 measures the exponential rate
of divergence of nearest neighbors, a plot of ln < 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒 > versus time yields a graph with
a section where the plot is more-or-less linear (Rosenstein et al., 1993). The slope of a leastsquares linear regression line fit to those data yields the estimate of L1. We wrote an algorithm
that optimizes fit of a straight line to the widest possible time span, selecting the line that
maximized the r2 value for the least-squares regression line. Typical results are illustrated in the
following figure:
ln (D)
3
2.5
2
1.5
0
0.1
0.2
0.3
0.4
0.5
Divergence Time
Computation of Correlation Dimension
Correlation dimension is computed by first calculating all nearest neighbor distances (in
m-dimensional space), then computing the the correlation sum, 𝐶(𝑚, 𝜖) for a series of
increasingly larger-radius ‘shells’ as described in Sprott (2003). The dependence of 𝐶(𝑚, 𝜖) on
𝜖 is a power curve (since the volume of the shell is proportional to 𝜖 𝑚 ), and the correlation
dimension is then defined as:
𝐷2 =
𝜕 log 𝐶(𝑚, 𝜖)
𝜖→0,𝑁→∞
𝜕 log 𝜖
lim
which is simply the slope of the plot of log 𝐶(𝑚, 𝜖) versus log 𝜖, taken as 𝜖 → 0. A sample plot
of log 𝐶(𝑚, 𝜖) versus 𝜖 is as follows:
0
-1
log ( Cm, )
-2
-3
-4
-5
-6
-7
-8
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
log (  )
The ‘jumps’ in the line for large negative values of log 𝜖 result from statistical fluctuations in the
necessarily small number of nearest neighbors found within very small distances of each other.
Most workers omit these data when computing D2, and fit a linear model to the most linear
portion of the curve by least-squares regression. The slope of this line is taken as the estimate of
D2. However, since the graph is nowhere truly linear, and since D2 is defined at the limit 𝜖 → 0,
we wrote a program that optimizes the fit of a power curve to the raw 𝐶(𝑚, 𝜖) versus 𝜖 data. The
model has the form:
𝐶(𝑚, 𝜖) ∝ 𝜖 𝑘
and k is taken as the estimate of D2. Our algorithm returns values for D2 that are consistently
10-15% greater than those computed by the linear fit method (Toolson and Perry, unpubl.).
A significant advantage of the Rosenstein et al. algorithm is that its results are quite
robust across a broad range of values for m and J; values for m considerably smaller than the
Takens criterion (Takens, 19**) don’t adversely impact the estimates of L1 (Rosenstein et al.,
1993; Toolson and Perry, unpubl.). However, we will have to wait until we actually start
gathering data before we make final decisions about the best way to proceed with optimizing
time lag and embedding dimension. Therefore, for the proposed research, we intend to reduce
computation time and, especially, the impact of noise in the data on our estimates of D2 and L1
by using the smallest embedding dimension that gives us consistent results. Our preliminary
work so far suggests that a value of m in the range of 4-6 will be appropriate, but we will
investigate this – and the optimization of time lags – thoroughly once we start gathering our own
data, and will adjust the value of m and J as necessary to obtain valid results.
Preliminary results and discussion
We have applied the algorithms described above to EEG tracings obtained from 9
Control and 9 PTSD subjects. The results are as follows:
Largest Lyapunov Exponent, L1
Controls
Electrode
F3
F4
C3
C4
T7
T8
P3
P4
O1
O2
Mean
0.90
1.15
0.92
0.99
0.92
1.22
0.91
1.27
1.25
1.19
PTSD
S. D.
0.17
0.43
0.21
0.35
0.53
0.55
0.14
0.42
0.36
0.42
Mean
0.95
1.02
1.02
1.04
1.18
0.90
0.87
0.89
1.01
1.06
S. D.
0.37
0.31
0.36
0.44
0.21
0.33
0.32
0.22
0.37
0.28
t
0.33
0.70
0.73
0.24
1.34
1.48
0.27
2.30
1.33
0.70
P
(2-tailed)
0.75
0.49
0.48
0.81
0.20
0.16
0.79
0.04*
0.20
0.50
t
0.51
0.22
0.61
2.76
1.09
0.57
0.12
0.12
1.96
P
(2-tailed)
0.62
0.83
0.55
0.01**
0.29
0.58
0.91
0.90
0.07
Correlation Dimension, D1
Controls
Electrode
F3
F4
C3
C4
T7
T8
P3
P4
O1
Mean
3.69
3.63
3.53
3.51
3.72
3.68
3.49
3.70
3.85
PTSD
S. D.
0.24
0.32
0.14
0.29
0.26
0.31
0.23
0.27
0.24
Mean
3.76
3.67
3.59
3.87
3.85
3.75
3.50
3.69
3.64
S. D.
0.32
0.33
0.24
0.24
0.26
0.19
0.20
0.29
0.18
O2
3.68
0.23
3.63
0.26
0.36
0.73
***Should we say anything about small sample sizes, comorbidity, etc.?***
Conclusions
The modified algorithms and associated software we developed work well for computing
L1 and D2 values from EEG tracings. The values we obtained are comparable to those reported
by earlier workers (except those who explicitly used much larger values for the embedding
dimension), and the standard deviations for our estimates are acceptably low, with coefficients of
variation for L1 of 33% and for D2 of only 7%. (***accurate estimates of L1 are significantly
more difficult to obtain; Palus, 1998). With reasonable sample sizes, statistical power will be
more than sufficient for our purposes. The same algorithms will also be directly applicable to
the heart rate variability (HRV) data that we will collect along with the EEG data.
Finally, nonlinear time series analysis is a dynamic field, and new algorithms appear in
the research literature on a regular basis, each claimed to be immune to the problems associated
with L1 and D2. We will carefully and thoroughly evaluate the merits of each of these in the
context of our data, and incorporate them as appropriate into our biological marker of therapeutic
response. The multiscale approaches developed by Gao et al. (200*) and Costa and her
colleagues (Costa et al., 200*) seem particularly promising, and we are in the process of
producing and validating the computer code to incorporate those algorithms into our analyses of
EEG and HRV data.
Literature Cited
Chae et al., ****
Costa et al. (****)
Gao et al., (****)
Grassberger and Procaccia, 1984
Kantz, H., Kurths, J., Mayer-Kress, G. 1998. Nonlinear Analysis of Physiological Data.
Springer, Berlin. 344 pp.
Lo and Principe, 1989
Natarajan et al., ****
Palus, 1998
Pritchard & Duke, 1992
Rosenstein et al. (1993)
Sarbadhikari and Chakrabarty, 2001
Sprott, 2003
Theiler, 1986
Wackerman, 1999
Wolf et al., 1985