Robust physiological mappings: from non-invasive to invasive
The goal of this paper is to highlight the challenges on the three methods of data analysis, namely: robust, component, and dynamical analysis with respect to the epilepsey. A forward and inverse mapping model for the human brain is presented. Research directions for obtaining robust inverse mapping...
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Інститут кібернетики ім. В.М. Глушкова НАН України
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irk-123456789-1247632017-10-05T03:02:47Z Robust physiological mappings: from non-invasive to invasive Syed, M.N. Georgiev, P.G. Pardalos, P.M. Системный анализ The goal of this paper is to highlight the challenges on the three methods of data analysis, namely: robust, component, and dynamical analysis with respect to the epilepsey. A forward and inverse mapping model for the human brain is presented. Research directions for obtaining robust inverse mapping, and conducting dynamical analysis of the epileptic brain are discussed. Проаналізовано проблеми, пов’язані з трьома методами аналізу даних щодо епілепсії головного мозку: робастним, покомпонентним і динамічним. Запропоновано пряму і обернену моделі відображення головного мозку. Також обговорюються напрями досліджень для отримання робастних обернених відображень і проведення динамічного аналізу епілептичного мозку 2015 Article Robust physiological mappings: from non-invasive to invasive / M.N. Syed, P.G. Georgiev, P.M. Pardalos // Кибернетика и системный анализ. — 2015. — Т. 51, № 1. — С. 111-120. — Бібліогр.: 44 назв. — англ. 0023-1274 http://dspace.nbuv.gov.ua/handle/123456789/124763 519.6 en Кибернетика и системный анализ Інститут кібернетики ім. В.М. Глушкова НАН України |
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Системный анализ Системный анализ Syed, M.N. Georgiev, P.G. Pardalos, P.M. Robust physiological mappings: from non-invasive to invasive Кибернетика и системный анализ |
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The goal of this paper is to highlight the challenges on the three methods of data analysis, namely: robust, component, and dynamical analysis with respect to the epilepsey. A forward and inverse mapping model for the human brain is presented. Research directions for obtaining robust inverse mapping, and conducting dynamical analysis of the epileptic brain are discussed. |
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Syed, M.N. Georgiev, P.G. Pardalos, P.M. |
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Syed, M.N. Georgiev, P.G. Pardalos, P.M. |
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Syed, M.N. |
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Robust physiological mappings: from non-invasive to invasive |
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Robust physiological mappings: from non-invasive to invasive |
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Robust physiological mappings: from non-invasive to invasive |
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Robust physiological mappings: from non-invasive to invasive |
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Robust physiological mappings: from non-invasive to invasive |
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robust physiological mappings: from non-invasive to invasive |
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Інститут кібернетики ім. В.М. Глушкова НАН України |
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Robust physiological mappings: from non-invasive to invasive / M.N. Syed, P.G. Georgiev, P.M. Pardalos // Кибернетика и системный анализ. — 2015. — Т. 51, № 1. — С. 111-120. — Бібліогр.: 44 назв. — англ. |
| series |
Кибернетика и системный анализ |
| work_keys_str_mv |
AT syedmn robustphysiologicalmappingsfromnoninvasivetoinvasive AT georgievpg robustphysiologicalmappingsfromnoninvasivetoinvasive AT pardalospm robustphysiologicalmappingsfromnoninvasivetoinvasive |
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2025-07-09T01:59:53Z |
| last_indexed |
2025-07-09T01:59:53Z |
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| fulltext |
UDC 519.6
M.N. SYED, P.G. GEORGIEV, and P.M. PARDALOS
ROBUST PHYSIOLOGICAL MAPPINGS: FROM NON-INVASIVE
TO INVASIVE
Abstract. The goal of this paper is to highlight the challenges on the three methods of data
analysis, namely: robust, component, and dynamical analysis with respect to the epilepsey.
A forward and inverse mapping model for the human brain is presented. Research directions
for obtaining robust inverse mapping, and conducting dynamical analysis of the epileptic brain
are discussed.
Keywords: epilepsy, electroencephalography (EEG), robust measures, blind signal separation,
independent component analysis, sparse component analysis, dynamical analysis.
INTRODUCTION
Robustness has been unanimously identified as one of the critical factors in developing
successful analysis methods for experimental data. Typically, statistical inference
methods are sensitive to the outliers, and they dictated the conventional theory of
analyzing the data. In 1962, Tukey [1] in his awe inspiring paper kindled the
importance of robust methods. He differentiated the term “Data Analysis” from
“Statistical Analysis” by stating that the former can be considered as science, but
the later is subjective upon the statistician’s approach. Supporting Tukey’s
ideology, Huber [2] encouraged the usage of term data analysis, as the other term
is often misinterpreted in an overly narrow sense (restricted to mathematics and
probability). Thus, the seminal work of Tukey [1, 3] enlarged the scope of data
analysis from mere statistical inference to something more.
During the shift from the traditional statistical analysis to the contemporary data
analysis, one of the key analysis elements that remained valid is the usage of
optimization based approaches in extracting knowledge from the data. Typically, the
efficiency of an optimization based approach depends upon the type of objective
function, feasible space, and the data quality. Traditionally, the data analysis methods
were based on the impractical assumptions that provided mathematical advantage in
proposing solution algorithms to the optimization problem. However, the traditional
approaches neglected the criticality pertaining to the practicability and data quality.
Existence of outliers (or noise) often taint the solution space, hence, practical data
analysis calls for robust methods.
The importance of robust methods in data analysis has been early recognized, and
can be traced back to the old famous dispute between Fisher and Eddington. Based on
practical observations, Eddington [4] proposed the suitability of the absolute error as
an appropriate measure. Fisher [5] countered the idea of Eddington by theoretically
showing that under ideal circumstances (errors are normally distributed, and outliers
free data) the mean square error is better than the absolute error. The dispute between
Eddington and Fisher actually played a prominent role in shaping the theory of
statistical analysis. After Fisher’s illustration, many researchers incorporated mean
square error as a default similarity measure in their analysis. Later, Tukey [1] reasoned
that occurrence of the ideal circumstances for the practical scenarios is very rare. Noise
as less as 02. %, which is ideal for many practical data, will favor the usage of the
absolute error instead of the mean square error [6, 7]. Presently, the prevalence of
sensitive measure in data analysis can be solely attributed to their mathematical
ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2015, òîì 51, ¹ 1 111
© M.N. Syed, P.G. Georgiev and P.M. Pardalos, 2015
advantage in proposing the solution algorithms. Nevertheless, the practitioners, under
the preamble of robust statistics, have been conducting research in the directions of
robust methods, which have led to the insightful explicit studies [6–8].
Obviously relaxing all the distribution or statistical assumptions in a data analysis
method is the most appropriate case for analyzing experimental data. However, the
distribution assumptions cannot be discarded in most of the scenarios, mainly due to
the loss of mathematical convenience in the analysis approach. Thus, most of the
research in robust methods is based on incorporating ideas into the traditional methods
that will result in insensitivity to the outliers and uncertainty. The major approaches of
robust methods in data analysis can be divided into the following categories:
� Robust measure based approaches: In these approaches, a measure which is
insensitive to outliers is used as an objective function [7, 8].
� Robust algorithm based approaches: In these approaches, subsamples from the
given data sample is analyzed separately, and an average of all the subsample analysis
results is considered as representative result [9–11].
� Robust optimization based approaches: In these approaches, an uncertainty
based domain is considered around each data sample, and stochastic optimization based
algorithms are used to conduct the analysis [12, 13].
In physiological data analysis, specifically in computational neuroscience, the
invasive data recordings enhance the analysis and predictability than compared to
non-invasive data recordings [14]. On the other hand, invasive data recordings are not
easily available, and are recorded only in certain specific cases. The critical difference
between the data, collected form invasive and non-invasive approaches, is the mixing
of the sources, apart from the noise interference. For example, in the case of
ElectroEncephaloGraphy (EEG) data recordings, the source signals generated at the
active neurons are smeared through the surrounding brain matter by the volume
conduction. Although the volume conduction is a passive resistive process in nature,
signals at each scalp electrode is influenced by a local set of underlying active
neurons [15]. In addition to that, the recordings collected on scalp not only involves
mixture of the true source signals, but also involves the mixture of source signals and the
influential artifacts. The typical artifacts may include ocular activity (eye movements,
eye blinks), myographic activity (muscle, jaw tightening), cardiac cycle activity,
electrical activity (50 Hz or 60 Hz noise). Moreover, the volume conduction from the
active neurons to the electrodes involves no time delays, which is attributed to the
effectively instantaneous mixing within the minuscular intra-cortical distances [16].
Challenges involved in extracting original source signals form EEG recordings of
an epileptic brain is the theme of this paper. In addition to that, usage of synchronization
based dynamical analysis methods on the source signals is highlighted. A forward
mixing system, that mathematically describes the mixing process in human brain is
presented in section 1. In section 2, series of assumptions that are used in the
traditional and novel class of algorithms will be discussed. Section 3 presents the
assumptions and limitations of the unmixing methods. Ultimately, the challenges in
extracting the sources from an epileptic brain are presented in section 4. Finally, we
conclude the paper by presenting few research directions.
1. THE FORWARD MIXING SYSTEM
The majority of the non-invasive vitals are instantaneous mixtures of their sources.
The severity or triviality of the mixing problem, in physiological data analysis,
depends upon the inter-source distances and artifact-source interference. Thus,
a challenging task is to rightly identify the underlying function that maps
non-invasive data to the invasive data. The task mathematically reshapes to finding
112 ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2015, òîì 51, ¹ 1
the solution of the inverse system, posed by the following forward system:
X S M N� ��( , ) , (1)
where � is assumed to be continuous bijective (homeomorphism) unknown
mapping, X is any representation of the observed non-invasive data, typically can be
transformed into flat representation, i.e. X � �
�
m P . Similarly, S � �
�
s P may denote
the source signals inside the human body, M � �
�
a P may represent the mixed
artifacts, and N � �
�
m P represents unmixed artifacts or external outliers. It is critical
to highlight the difference between M and N. Artifacts of type N can be filtered out
using digital filters, whereas, artifacts of type M cannot be easily removed. P is always
finite, representing the finite amount of data. Typically, P represents any acquisition
variable, over which a sample of mixture (a column of X matrix) is collected. The
most common types of acquisition variables are time and frequency. However,
position, wave number, and other indices can be used depending on the nature of the
physical process under investigation. Lastly, s and m may represent quantitative
information of observed and actual data. Typically, m corresponds to the total number
of observations, and s corresponds to the actual number of active sources.
Mixing in Epileptic Brain. For the epileptic brain, X can be seen as the EEG
data recordings where each row is a channel and each column is a time point, N can be
seen as those noise elements, that can be filtered using a bandpass filter. Let X f be the
filtered data represented as:
X X Nf � � , (2)
where X f � �
�
m P represents a filtered form of given mixture data X. Typically,
exact information of N is unknown, and classical methods to solve Equation (2)
involves designing a filter using frequency, amplitude, smoothness or geometry
based information of N [17]. The preciseness and the solution quality of the
inverse system are the two key factors that determines the validity and success of
any physiological data analysis. Most of the experienced clinicians and surgeons
merely observe X f , and are able to accurately identify pathological condition for
certain cases. For example, asymmetrical slowing of EEG data (which is typically
observed from X f ) can indicate existence of pathological conditions [18].
Typically, existence and location of background asymmetry between left and right
brain hemispheres is sufficient to identify focal slowing. Thus, mixing is not
a critical issue when the focal slowing is dominant. Successful results for the focal
slowing have been reported for these cases [19, 20]. Similarly, mixing does not
play a critical issue when diffused slowing is diagnosed [21] (in the diffused
slowing case, the total slowing of brain is considered).
Contrary to the above average source analysis scenario, epilepsy analysis does
demand analysis of S rather than analyzing X f . For instance, consider the difficult
problem of epilepsy prediction. Linear, nonlinear, dynamical methods have been
rigorously applied to X f , and ambiguous results from different groups have been
reported prior to formation of International Seizure Prediction Group (ISPG). After
standardizations proposed by ISPG, a new hope arose in the field of epilepsy
prediction. However, the results of the ISPG conferences were inconsistent and
contradictory (see [22] and the references therein). But the worth noting summary from
these conferences is that analysis from the invasive data techniques performed better
than the scalp data techniques in early identification of the epilepsy [23]. This
unequivocal result from decades of research directly points towards importance of
analyzing S rather than analyzing X f .
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2. UNMIXING APPROACHES
The representation of the inverse system for the system defined in Equation (1) is
non-trivial. However, a traditional approach is to represent the inverse system as:
S XM f� �� 1 ( ) , (3)
where
S
S
M
SM M�
�
�
� � � �, ( )
�
s a P . (4)
Similar to any typical notorious mathematical problem, the simple representation
of inverse system illustrated in Equation (3) is veiling its complexity. Obviously, when
the governing dynamics of a system in the analytical form is well known and
invertible, solution to Equation (3) might be trivial. Unfortunately, such underlying
knowledge of many physiological systems is intangible. Identifying ��1, is principally
impossible without additional assumptions on the sources. Thus, a specific function
class from � is pre-selected for identifying ��1. In fact, solutions for Equation (3) are
available only when ��1 is taken as a linear mapping. Currently, there are no
successful results reported for any other class of mappings. Assuming ��1 is linear,
Equation (3) can be rewritten as:
S W XM f� T , (5)
where W � � �
�
m s a( ) . The system represented in Equation (5) is a variant representation
of the linear matrix factorization. Since, the goal is not to find any factor matrices, but
specific matrices SM and W, the problem forms a specific case of linear matrix
factorization. A well known name of this problem is Blind Source Separation (BSS),
where the term “Blind” is used for emphasizing unknowness of the factor matrices.
The BSS problem suffers from uniqueness and identifiability:
� Uniqueness
Let � �, ( ) ( )� � � �
�
s a s a be a diagonal matrix and permutation matrix
respectively. Consider the following:
S W XM f� T ,
( ) ( )� � � �S W XM f� T ,
S W X f� �� T .
There can be infinite equivalent solutions of the form S� and W� . The goal of
a good BSS algorithm should be to find at least one of the equivalent solutions. Due to
the inability of finding the unique solution, we not only loose the information
regarding the order of sources and mixing artifacts, but also loose the information of
energy contained in the sources. Generally, normalization of rows of S� may be used
to tackle scalability. Also, relative or normalized form of energy can be used in the
further analysis. Theoretically any information pertaining to the order is impossible to
recover. However, practically, problem specific knowledge will be helpful in identifying
correct order for the further analysis.
� Identifiability
Let � � � � �
�
( ) ( )s a s a be any arbitrary matrix, consider the following:
S W XM f� T ,
� �S W XM f� T ,
S W X f� �� T .
114 ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2015, òîì 51, ¹ 1
The scenario depicted by the above equations is the critical identifiability issue.
Clearly, the BSS problem without any further assumptions is intractable. The key idea
of rightly identifying both the matrices (of course with unavoidable scaling and
permutation ambiguity) is to impose structural restrictions on SM , while solving the
BSS problem.
The BSS problem can be mathematically stated as: Let X � �
�
m N be generated
by a linear mixing of sources S � �
�
s N . Given X, the BSS problem is to find two
matrices A � �
�
m s and S, such that the three matrices are related as X AS� . Matrix A
is called as mixing matrix. In order to solve this problem up to certain level of
uniqueness, following identifiability conditions are imposed on A and S matrices.
� Statistical Independence Assumptions:
One of the earliest approaches to solve the BSS problem was to assume statistical
independence of the source signals. The widely known method that is dedicated to
the above assumption is the Independent Component Analysis (ICA) method. The
fundamental assumption in ICA is that the rows of matrix S are statistically independent
and non-gaussian [24, 25].
� Sparse Assumptions:
Apart from ICA, the other type of algorithms, which provides sufficient
identifiability conditions are based on the notion of sparsity in the S matrix. There are
two major categories in the sparse assumptions:
— Nonnegative sources:
In this category, along with certain level of sparsity, the elements of S are
assumed to be nonnegative. Ideas of this type of approach can be traced back to the
Nonnegative Matrix Factorization (NMF) method. The basic assumption in NMF is
that the sources (and dictionary) are assumed to be nonnegative [26]. However, in
certain cases for the BSS problem the nonnegativity assumption on the elements of
matrix A can be relaxed [27] without damaging the identifiability of A and S.
— Real sources:
In this category, no sign restrictions are assumed on the elements of S, i.e.
si j, �� . The only assumption used to define the identifiability conditions is the
sparsity of S. The methods using only sparsity assumptions are called as Sparse
Component Analysis (SCA) [28].
At present, these are the only two (statistical and sparsity assumptions) available BSS
approaches that can provide sufficient identifiability conditions (uniqueness upto
permutation and scalability). In fact, the sparsity based methods (see [27, 29]) are
relatively new in the area of BSS when compared to the traditional statistical independence
methods (see [25]). For neurological data, recent studies have shown relevance of sparsity
assumption when compared to statistical independence assumption [30].
3. ASSUMPTIONS AND LIMITATIONS
Makeig et al. [16] were pioneers in explaining the usage of BSS for EEG signals.
They identified four critical properties that should be satisfied by source signals so
as to successfully implement BSS approach. However, their focus was limited to
ICA view of BSS. In the following we extend the properties in terms of general
BSS, which is more broader area than ICA.
� The source signals should have at least one of the following properties:
— The source signals are statistically independent, and not more than one source
signal follows Gaussian distribution.
— The source signals are non-negative, and partial spatial sparsity exists among
the sources.
ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2015, òîì 51, ¹ 1 115
— The source signals are always spatially sparse.
� The mixing mechanism should have all of the following properties:
— Mixing is linear.
— Mixing is not illconditioned.
— The propagation delays occurs in the mixing medium are negligible.
� The number of source signals and observed signals are nearly (not necessarily
exact) equal in number. Although, for ICA number of observed signals should be
greater than or equal to the number of source signals.
Currently, in EEG analysis usage of ICA is prevalent. However, ICA in EEG is
merely used for separating the mixing artifacts. The basic idea is to perform ICA on
X f , and get as statistically independent as possible sources and artifacts. Let S ICA
denote the sources obtained after applying ICA. Since the mixing artifacts M are
anatomically, fundamentally and functionally independent from the sources S, the ICA
solution S ICA can be decomposed into S
�
and M blocks, i.e.
S
S
M
ICA �
�
�
�
� . (6)
The enthusiasm among researchers to use ICA is mostly based on the notion of
identifying and separating M form S
�
. However, the relation between S
�
and S
(the true sources) has been ignored in the research. Most of the time, authors were able
to provide satisfactory arguments (based on their experimentation) that the overall
information content in S
�
is sufficient for analysis than identifying the precise
mapping between the rows of S
�
and X [31].
However, assumption of statistical independent sources S, in human brain is hard
to verify (due to the connectivity among the nodes). Thus, even after ignoring the
ordering among rows, the remaining relation between S
�
and S is of critical
importance. In fact, if statistical independent sources is invalid for epileptic brain, then
S
�
is nothing but another mixture of rows of S, where S
�
has as independent rows as
possible. This can be seen as another way of representing S, and such representation do
have a practical advantage is physiological data analysis. Nevertheless, the curiosity to
identify and extract S from the observed X should not be extinguished.
In the following section, we present the challenges that remains unanswered in
EEG analysis, specifically in the diagnosis of epileptic seizures.
4. CHALLENGES
In this section, several challenges pertaining to the development of robust mappings
in epileptic brain is systematical enlisted. We begin with the development of the
tractable robust mapping formulation, proceed towards explaining the challenges
that underlie in the development of such mappings.
Developing Tractable Robust Formulation for Seizure Analysis
The general Robust Physiological Mapping (RPM) for EEG data can be
mathematically illustrated as:
find
such that
:
:
( )
�
�S X�
(7)
where � is the robust inverse physiological mapping. X is the EEG data, where
each row represents a channel. Similarly, S represents the source signals, where each
row is a source. The abstract formulation needs precise definition of robustness and
116 ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2015, òîì 51, ¹ 1
mapping to obtain a tractable solution method. Since the superposition of signals is
typical mixing phenomenon in the human brain, linear mapping assumption can be
used without loss of generality. Furthermore, any known information pertaining to M
and N is an additional advantage. Thus, a suitable modification of formulation (7),
which is tangible for optimization will be:
minimize
such that:
T
:
( )�
�
�
S W X
S
W
f�
�
�
(8)
where � is a set of structural assumptions placed on S based on the epileptic brain,
and � is a set of assumptions that must be placed on W for successful unmixing.
The function � is the robust function which is the key element for analyzing the
experimental data.
Selecting Suitable Robust Measures �
Robustness can be defined as insensitivity to the outliers in the context of data
analysis. Traditional functions used in ICA like kutrosis, cross-cumilants are sensitive
to outliers [25, 32]. Thus, a robust measure that ignores outliers while extracting
sources is the key element of the EEG source analysis. However, adding a robust
objective function � raises issue of finding optimal solutions, i.e. robustness,
convexity and smoothness very rarely exists in a single real function. Thus, robustness
comes with a trade off in either convexity or smoothness. Although in the literature, no
specific criterion is available to choose from convexity or smoothness, but from
optimization perspective, a robust smooth function with pseudodonvexity is
auspicious. Thus, any function � that satisfies above criterion may be taken as the
objective function. Under mild boundedness assumption, the correntropic loss function
�
�
� has been identified recently as one of the robust smooth invex function [33]:
�
�
�
�
�
�
( ) exp expv � �
��
�
��
�
�
��
�
�
� �
��
�
�
�
��
1
1
2
1
22
1 2
2
i
�
�
�
�
�
�
�
�
� �
�
�
i
n
n
1
v � (9)
where � � 0 is the kernel width parameter. Moreover, specific robust functions
towards a similar direction of research (robust, pseudoconvex and smooth) can be
designed based on the theoretical and experimental knowledge of the epileptic brain.
Developing Extendable Identifiability Conditions on A and S w.r.t Epileptic
Brain
Another important challenge is to develop suitable identifiability conditions on A
and S that imitate the brain mixing and brain sources respectively. Currently,
independent source assumption prevail the literature of EEG analysis, due to the early
development and widespread of ICA. Moreover, recent sparsity based conditions are
still at the early stages, needs further exploration. Apart from independence and
sparsity, other assumptions specific to epileptic brain can be exploited in developing
sufficient identifiability conditions on A and S. In fact, for epileptic brain, the
hypothesized synchronization phenomena among the source brain signals can be
exploited in developing the conditions. For example:
� Rows of S can be assumed to be as synchronous as possible at the time of
seizure. Methods using this information can be developed to extract S from X.
� Since epilepsy is a dynamical phenomenon, rank of the Henkel matrix between
pair of signals can be used to identify total number of dynamically independent signals.
ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2015, òîì 51, ¹ 1 117
Understanding Temporal Changes in A and S
Typically, the impulsive synchronous neuronal activity in the cerebral cortex is
considered as the main reason behind the occurrence of seizures. Furthermore, the
synchronous activity may be kindled locally (in specific portions of cerebral
hemispheres) or globally (in both cerebral hemispheres) in the brain. The seizures that
are initiated from local activity and remained confined to the region are called as
partial or focal seizures. Whereas, the seizures that are initiated from global activity
involving almost the entire brain are termed as generalized seizures. However, if the
mixing model of the brain (see Equation (1)) is relevant, then answering the following
key point will open the doors of understanding dynamics of epilepsy.
� Is it the sources that are getting synchronized or is it the mixing that is getting
singular?
� How does the rank of matrix A changes from pre-ictal to ictal periods?
Dynamical Inter-manifold Analysis of S
After resolving all the issues pertaining to the extraction of S form X, the next
challenging task is to properly study the dynamics embedded in S. Typical methods like
time delay embeddings [34] can be used to reconstruct manifolds. Furthermore, robust
nonlinear embeddings [35] can be used to enhance the understanding of the core
differences between pre-ictal and ictal periods of the epileptic brain. A manifold
constructed from S can be used to calculate the traditional measures (correlation
dimensions [36], Lyapunov exponents [37] and Kolmogorov entropy [38]) or the novel
measures (see [39, 40]). Moreover, different multivariate synchronization measures can
be used to identify the development of seizure [41]. Finally, for any traditional or novel
dynamical analysis, surrogate test [42] should be conducted to validate the results.
See [43] for the current literature on dynamical methods in analyzing the EEG data.
Currently, the multi channel analysis in dynamical methods is mere a concatenation of
the single channel methods. Alternatively, dynamical analysis of every row of S can be
conducted to create several manifolds. Then the critical issues in the dynamical
analysis of EEG source data will be to:
� Develop a robust inter-manifold measures that can detect dynamically
equivalent manifolds, and/or synchronous manifolds.
� Develop a multivariate method which results in the elimination of redundant
information from S, during the manifold construction.
� Develop the corresponding efficient surrogate test for the proposed measures.
CONCLUSIONS
The noise factor in the success of dynamical analysis methods in the prediction of
epilepsy can be extended via robust approaches. Furthermore, the advantage
of invasive data over the scalp EEG data can be incorporated by developing
a mapping between the two data sets. Since, typical EEG recording involve
artifacts, robust mappings are the key approaches to tackle noise and invasive data
factors. The reasons behind the patient specific (and for a given patient seizure
specific) nature of dynamical measures can be studied by analyzing the source
signals. Furthermore, the localization issue [44] among the sources can be
eliminated by developing suitable methods that leaves markers in the sources, i.e.
suitable experiments can be designed that indicates the linking of a particular
source to a particular time span.
While applying the new sparsity based source separation methods, existence of
sparsity in the actual EEG sources may appear dubious. However, similar to ICA
techniques where independent noise is induced in the data via experimentation,
sparsity can be induced by conducting experiments that results in the sparse activation
of the neurons. Furthermore, transformations like discrete wavelet transform can be
applied on X hoping the wavelet coefficients of sources S are sparse. Nevertheless, the
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key issue is to identify structures in � and � which are both relevant and tractable to
the practical physiological conditions in the epileptic brain. Thus, developing RPM
that maps non-invasive data to invasive data will open the doors of the unidentified
physiological phenomena, the epilepsy.
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