Complexity of Brain Dynamics Inferred from the



Complexity of Brain Dynamics Inferred from the
Complexity of Brain Dynamics Inferred from the
Sample Entropy Analysis of Electroencephalogram
Pravitha Ramanand, R.Sreenivasan, and V. P. N. Nampoori
Abstract- Nonlinear time series analysis is a potential method
for gaining information regarding the dynamics of complex
physiological systems. The human brain is by far one of the
most complex systems in nature and the electrical signal from
the brain, the electroencephalogram (EEG) has been a subject of
nonlinear techniques to delve the dynamics of the brain. This
paper deals with the computation of a measure of complexity for
dense array EEG signals and the subsequent analysis of various
brain conditions. The complexity measure used in the study is
the Sample Entropy (SampEn), which quantifies the regularity
and ‘randomness’ in a time varying signal arising from a system
The present work computes the SampEn of the EEG signal
recorded for the three cases of passive, eyes closed condition;
mental arithmetic task and for epileptic spike data. The analysis
helps in gaining an insight into the nature of complexity
variations with brain state. It is observed that while during the
mental task only specific brain regions exhibit lowered
complexity, the dynamical complexity is very much lowered for
the pathological spike condition.
Keywords- Brain, EEG, Nonlinear time series analysis, Sample
Entropy (SampEn).
recent times the application of nonlinear methods to the
analysis of data obtained from complex systems in nature
has been of interest to the scientific world. The study of the
Electroencephalogram (EEG) signal generated by the human
brain is one of those complex signals, which is being strongly
pursued with the techniques based on nonlinear dynamics
and deterministic chaos [1]. This line of approach includes
the computation of invariant parameters such as generalized
dimensions, entropies, Lyapunov exponents and so on from
the EEG signal considered as a time series to characterize the
dynamics of particular brain states. Utmost care is however to
be exercised in accepting the quantitative results of the time
series analysis of the EEG signal which are mostly based on
Pravitha Ramanand is Research Fellow in the Prigogine Centre for
Complexity Studies, International School of Photonics, Cochin University of
Science and Technology, Kochi- 682022, Kerala
[email protected]).
R. Srenivasan is with the National Brain Research Centre, Near NSG
Campus, Nainwal Mode, Maneswar, Haryana 122050 (e-mail:
[email protected]).
V.P.N. Nampoori is with the Prigogine Centre for Complexity Studies,
International School of Photonics, Cochin University of Science and
Technology, Kochi- 682022 (e-mail: [email protected]).
the methods designed for infinitely long, stationary and noise
free data. Despite this, validity of such parameters with
statistical significance as comparative measures across
varying mental states has been more or less widely
established [2].
Complexity measures are another class of statistics
characterizing a time series that have arrived recently on the
scene. Measures characterizing the relative ‘randomness’ in
data sets have been thought of as useful markers
discriminating between the conditions generating them.
Though other measures of complexity exist [3,4], entropy
measures that quantify the rate of information generation in
the system are observed to be more widely useful in practical
applications. In the case of bio-signals, the need is for a
statistic that provides a meaningful measure of regularity
while remaining robust against noise. The Approximate
Entropy (ApEn) introduced by Pincus [5] is one such
measure. Lately, a new and related measure, the Sample
Entropy (SampEn) was devised as a measure that rectifies
some of the in-built drawbacks of the ApEn such as bias and
inconsistency in results. This novel measure has been
demonstrated to be useful in the discriminatory study of
cardio-vascular data [6], which is typical data occurring in
biological systems. The computation of complexity measures
such as SampEn scores above the more popular D2 estimation
due to the fact that these are algorithmically simpler and
hence easier to compute. The nonstationarity of the time
series, as in the EEG is also dealt with effectively since they
are designed to suit short data segments [7]. These and other
attractive features have made these measures popular in
physiological data analysis.
In this study, we apply the SampEn statistic to human EEG
data for a complexity characterization of three different states
(a) passive, eyes closed state (E.C.), (b) a mental task state
(M. A.) and (c) epileptic spike condition (Ep. Sp.). The
motivation in undertaking this analysis lies in assessing the
variations in complexity exhibited by the EEG due to the
underlying dynamics in these totally distinct conditions.
Our second interest lies in the complexity analysis of the
spike condition exhibited by epileptic subjects. Epilepsy is a
patterns in continuous EEG and seizure at times. A
complexity analysis of such a spike condition is carried out
and compared against the normal, eyes closed state. As
expected, the SampEn analysis shows more regular structure
in the spike data than in the normal state as reported earlier in
some other nonlinear measures [8]. Yet the clinical interest
that this statistic may hold as a computationally simpler tool
as compared to other quantities cannot be undermined.
Moreover its ability to distinguish the pathological epileptic
state even from data recorded in the absence of a seizure in
clinical terms makes it a potential tool in automated
A measure of the regularity of a time series will be
informative about the underlying complexities in the
processes giving rise to it. Pincus (1991) developed a
regularity measure related to the Kolmogorov entropy known
as the Approximate Entropy (ApEn), which assigns a
nonnegative number to a time series with larger values
corresponding to greater apparent randomness of the process
underlying the data while smaller values point to regular
features in the data. Despite being suitable for noisy data
usually encountered in biological systems, ApEn suffers from
the drawback of being a biased statistic. This and other
discrepancies are rectified in the newly developed Sample
entropy algorithm. A detailed description of the drawbacks in
the ApEn algorithm and the better results accorded by the
sample entropy statistic in the case of model as well as
clinical data is available in Richman and Moorman [6].
In order to evaluate the SampEn statistic for a given time
series of N points, the following method is adopted. Suppose
the time series is represented by {x(i )} where
i = 1,2, K , N ; then for a fixed window length m, vector
sequences of the form
y m (i ) = [x(i ), x(i + 1),K , x(i + (m − 1) )]
are created. On the basis of this embedding in the ℜ space,
we define the separation between two vectors y m (i ) and
y m ( j ) as
d [ y m (i ) , y m ( j )] = max
( x (i + k − 1) − x ( j + k − 1) )(2 )
where k =1,2…m and for each i, 1 ≤ i ≤ N − m + 1 .
Bim (r )
y m (i )
be ( N − m − 1) times the number of vectors
( j)
within r of m
with j ranging from 1 to (N-m) and
i ≠ j . Define the function,
N −m
B m (r ) =
∑ B (r )
i =1
N −m
(3a )
(r ) is defined as ( N − m − 1) −1 times
vectors y m +1 (i ) are within r of y m +1 ( j ) ;
In a like manner, Ai
the number of
again with j ranging from 1 to (N-m) and i ≠ j and the
function A
(r ) as,
N −m
A m (r ) =
Hence while B
∑ A (r )
i =1
N −m
(r ) gives
the probability that the two
sequences match for m points; A (r ) is the probability that
the match is for m +1 points. The sample entropy is defined
 Am
SampEn ( m, r ) = lim − ln m
N →∞
which is estimated by the statistic SampEn (m, r, N) defined
 Am
SampEn (m, r , N ) = − ln m
The SampEn (m,r,N) is a ‘family’ of statistics in the sense
that for a given application, comparison of the measure
between different systems holds only for fixed values of m
and r. Since SampEn varies in direct correlation with
changing background noise characteristics, it is suited for
comparison of data sets from a common experimental
protocol. This makes it a useful tool in the analysis of
multichannel EEG recordings pertaining to different mental
states. The choice of N, m and r usually depends on the
specific application of the measure just as in the ApEn case.
It has been reported that for numerous applications to
distinguish the data on the basis of regularity for m=1 and
m=2 for N varying from 50 to 5000 points, values of r
between 10% to 25% of the standard deviation (S.D) of the
data x(i) are seen to produce good statistical validity of ApEn
[7]. All these conditions appear to hold good for SampEn as
well, which is a measure related to ApEn.
In recent times, the limited spatial resolution of the
conventional EEG technology has been tackled by
introducing high resolution EEG (HR- EEG). In this study,
we resort to the use of data recorded using such a dense array
128-channel Electrical GeodesicTM system, with a sampling
frequency of 200Hz referenced to a vertex electrode. The
electrode configuration of the recording system is shown in
Fig (1). An online band pass filter from 0.1 to 70 Hz was
used and impedances were kept below 25kΩ. Vertical and
horizontal eye movements were monitored with a subset of
128 electrodes.
EEG’s were recorded from two different groups: (I)
normal subjects and (II) subjects suffering from the
pathological ailment of epilepsy. From the former group,
recording was carried out from 12 subjects with mean age 29
under two conditions (a) Passive relaxed eyes closed (E.C.)
(2) mental arithmetic of serial subtraction with eyes closed
(M.A.). The spike activity (Ep. Sp.) was recorded under eyes
closed state from a group of 5 subjects of mean age, 14
suffering from the pathological condition of epilepsy. The
resultant EEG records were transferred from the Net Station
acquisition system for further off-line analysis on a personal
Fig. 1. An illustration of the 128-electrode configuration of the dense
electrode array and the classification of the channels into 4 groups numbered
from I to IV representative of the major cortical regions of the brain.
The study is undertaken with the objective of looking at
the variations in the complexity exhibited by a global EEG
recording as a result of the varying dynamics of the system
with the change in brain state. The complexity of the EEG
time trace emerging from each recording channel is
quantified by the statistic of SampEn. In order to take into
account the nonstationary nature of the EEG signal, the data
is divided into non-overlapping windows of 1000 data points
and the SampEn measure evaluated for each of these.
A. Eyes closed state (E.C.)Vs. Mental task state(M.A.)
The SampEn measure evaluated for the passive state of
eyes closed condition when compared with mental arithmetic
shows that the complexity was lowered in the mental task
state. Fig. (2) gives the plot for the evaluated complexity
measure under the two conditions for the 128 channel data
grouped into four, representative of the major cortical
regions. A complexity drop is seen in regions I and IV during
M.A. as compared to the E.C. state. Over the rest of the
brain, there is not much complexity difference in the two
states. The reduction in SampEn for the prefrontal region in
region I may be looked upon as the reduced neuronal
complexity of this region during mental arithmetic as
compared with the others. Moreover the decrease in
complexity in the parieto-occipital regions covered by region
IV points to the involvement of this region in the subtraction
It is concluded on the basis of appropriate repeated
measures analysis of the computed values that the effect due
to the change in state from E.C. to M.A. gives a significant
decrease in the SampEn measure especially in regions I and
Fig. 2. A detailed comparison of SampEn measure for the mental arithmetic
(M.A.) state with that of eyes closed (E.C.) case for each of the four groups
numbered I to IV. Significant variation in the complexity in the electrodes of
the frontal electrodes in region I and of the occipito-parietal electrodes in
region IV is visible.
IV (p<0.001). On the other hand, the channel effect is not
significant though the interaction effect (state*channel) is
observed to be significant in region IV covering the occipitoparietal regions. This analysis also throws light on the
advantage of using a dense array recording paradigm. With a
limited number of electrodes as in the more popular 10-20
system, the gross analysis may fail reveal the significant
difference in the effect caused by the mental arithmetic state
on the electrode sites in region IV.
Mental arithmetic is a complex task that requires
integration of multiple processes carried out by a large-scale
network of distributed yet interconnected local networks [9].
The complexity analysis supports this idea and is also in
accordance with imaging evidence that suggests the
participation of parietal areas in addition to the prefrontal
areas in mental calculation [10]. The lowered complexity in
the occipito-parietal channels suggests the activation of
visuo-verbal regions while the frontal activation has been
always been linked with working memory and attention [11].
Hence, the observations point to the advantage of noninvasive techniques like the EEG, in comparison with
relatively expensive imaging methods as primary tools in
studying the human brain under myriad conditions.
B. Epileptic Spike (Ep.Sp.) Condition
The data recorded during the epileptic spike activity was
found to have much lowered complexity when compared
with the eyes closed state. Fig. (3) shows that the SampEn
measure during the epileptic spike is smaller than the
corresponding values over all the channels distributed over
the cortex. Bhattacharya (2000) [12] recently reported that
there is indeed a lowering of ApEn measure for a 16-channel
data set for epileptic subjects. However, the dense array EEG
gives a closer view of the generalized statement made in that
condition. The sharp reduction in the complexity measure of
sample entropy for the epileptic spike data over the whole
skull space is in agreement with the finding that epileptic
behaviour in general, is reflected in the lowering of various
parameters such as correlation dimension [8].
Nonlinear parameters such as the SampEn are potential
markers of particular brain states and can be used judiciously
in gaining more information of the brain functioning in both
normal states and pathological conditions. The complexity
analysis can be considered as a first tool to classify mental
processes especially with regard to abnormal conditions such
as schizophrenia, mental depression and so on. Such studies
have potential in diagnostic as well as research purposes.
Fig. 3. The complexity analysis for the epileptic spike data (Ep. Sp.) in
comparison with the relaxed eyes closed (E.C.) case is presented.
From a mechanistic point of view, the strong reduction in
the SampEn measure points to the greater component or
subsystem autonomy. Thus for normal subjects in
comparison with the subjects suffering from a pathological
illness, there are more independent, parallel, functional
processes active. The complexity loss during the spike
activity associated with epileptic disorder points to the longrange order or enhanced synchrony reported earlier in
epileptic subjects [13].
The human brain is a highly complex system exhibiting
nonlinear and nonequilibrium behaviour. The study of the
electric signals generated in the brain using techniques from
nonlinear time series analysis has been advantageous in
gaining information about the dynamics of the system. The
noninvasive nature and computational viability of these
methods score above the more expensive imaging techniques
such as the Positron emission and magnetic resonance
imaging in studying the brain.
In the present report, we have looked at the complexity
aspect of EEG data recorded during the three conditions of
(a) normal eyes closed, (b) mental arithmetic and (c) epileptic
spike activity. The dense array configuration provides
simultaneous records of the EEG signal from a large number
of channels that is useful to gain a more complete picture of
the activity distributed over the cortex. The normal eyes
closed state is considered as a base line state and the
complexity during the other two states are compared with
this. It is found that even during a cognitive task such as
mental arithmetic that consists of a series of processes such
as perception of information, processing of arithmetical
signs, retrieval and memorization mechanisms; the
complexity in most of the brain regions remains the same as
that during the relaxed, passive state. The prefrontal region
and some parts of the occipito-parietal region exhibit lowered
complexity during serial subtraction task in comparison with
the corresponding regions during the passive eyes closed
The first author acknowledges C.S.I.R. India for a research
fellowship during the period of this work. R.S acknowledges
The Wellcome Trust, UK for setting up the Dense-array EEG
laboratory at the University of Cape Town. V.P.N.N.
expresses gratitude to the UGC, India for a research award
and acknowldeges Science, Technology & Environment
Dept. Govt. of Kerala for research grants.
T. Elbert, W. J. Ray, Z. J. Kowalik, J. E. Skinner, and N. Birbaumer,
“Chaos and Physiology: Deterministic chaos in excitable cell
assemblies,” Physiological Reviews, 1994, vol. 74, no. 1, pp. 1-97.
P.E. Rapp, “Chaos in the neurosciences - Cautionary tales from the
frontier”, Biologist, 1993, vol.40, pp. 89-94.
N. Radhakrishnan and B. N. Gangadhar, “Estimating regularity in
epileptic seizure time-series data”, IEEE Engineering in Medicine and
Biology, May/June1998, vol.17, pp. 89-94.
R. Wackerbauer, A. Witt, H. Atmanspacher, J. Kurths, and H.
Scheingraber, “A comparative classification of complexity measures,”
Chaos, Sol. & Frac., 1994, vol. 4, pp. 133-173.
S. M. Pincus, “Approximate entropy as a measure of system
complexity”, Proc. Natl. Acad. (USA), 1991, vol. 88,:3017-3026.
J. S. Richman, and J. R. Moorman, “Physiological time-series analysis
using approximate entropy and sample entropy”, Am. J. Physiol Hear
Circ Physiol, 2000, vol. 278, pp. H2039-H2049.
P. Indic, R. Pratap, V. P. N. Nampoori, and N. Pradhan, “Significance
of time scales in nonlinear dynamical analysis of Electroencephalogram
signals, Int. J. Neurosci., 1999, vol. 99, pp. 181-194.
K. Lehnertz, and C. E. Elger, “Spatio-temporal dynamics of the primary
epileptogenic area in temporal lobe epilepsy characterized by neuronal
complexity loss,” Electroencephalogr. Clin. Neurophysiol., 1995, vol.
95, pp. 108-117.
M. McCloskey, “Cognitive mechanisms in numerical processing:
evidence from acquired Dyscalculia”, Cognition, 1992, vol. 44, pp.
H. Kazui, H. Kitagaki, and E. Mori, “Cortical activation during
retrieval of arithmetic facts and actual calculation: a functional
magnetic imaging study,” Psychiat. Clin. Neurosci., 2000, vol. 54, pp.
T. Harmony, T. Fernández, J. Silva, J. Bosch, P. Valdés, A. F.- Bouzas,
L. Galán, E. Aubert, and D. Rodríguez, “Do specific EEG frequencies
indicate different processes during mental calculation?” Neuroscience
Letters, 1999, vol. 268, pp. 25-28.
J. Bhattacharya, “Complexity analysis of spontaneous EEG,” Acta
Neurobiol. Exp., 2000, vol. 60, pp. 495-501.
R. Pravitha, and V. P. N. Nampoori, “Coordination in the dynamics of
the Brain as inferred from EEG analysis”, Int. J. Neurosci., 2002, vol.
112(10), pp. 1245-1262.