Optimization and Chaos Theory for Seizure Prediction

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Optimization and Chaos Theory for Seizure
Prediction

• Wanpracha Chaovalitwongse (Art)
• arty_at_epilepsy.health.ufl.edu
• Post Doctoral Associate
• Departments of Neuroscience and Industrial and
  Systems Engineering
• University of Florida

• Joint work with
• University of Florida
• P.M. Pardalos, Ph.D. (Industrial Systems
  Engineering)
• J.C. Sackellares, M.D. (Neurology)
• P.R. Carney, M.D. (Pediatrics)
• D.-S. Shiau, Ph.D. (Neuroscience)
• Arizona State University
• L.D. Iasemidis, Ph.D. (Electrical Computer
  Engineering)

This work was partially supported by the NIH,
NSF, and VA research grants.
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Outline

• Optimization Techniques in Seizure Prediction
• Introduction
• Tools to be used
• Chaos Theory
• Statistical Analysis
• Optimization
• Methods and Results
• Concluding Remarks and Future Works
• Appendix

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Introduction

• Epilepsy is among the most common disorders of
  the nervous system, affecting approximately 1 of
  the population.
• Epilepsy is characterized by recurrent,
  paroxysmal (short-term) electrical discharges of
  the cerebral cortex that result in intermittent
  disturbances of brain function.
• The hallmark of epilepsy is recurrent seizures.
• Seizures are due to sudden development of
  synchronous neuronal firing in the cerebral
  cortex.
• When neuronal networks are activated, they
  produce a change in voltage potential, which can
  be captured by an electroencephalogram (EEG).

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From Microscopic to Macroscopic Level
(Electroencephalogram - EEG)
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Machines
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Rationale

• This research is concerned with the problem of
• Predicting target episodes of events (onsets of
  seizures) from analyses based on multiple time
  series (multi-channel EEG time series).
• The motivation of this research is that
  traditional linear and nonlinear time series
  analyses
• have been routinely used
• did not successfully give insight into the
  characteristic and mechanism of time series.
• are limited by the stationarity assumption
  requirement of the time series and the normality
  and independence requirements of the residuals.

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Goals

• In this research we aim to
• Develop new techniques used to identify complex
  (nonperiodic, irregular, and chaotic)
  characteristics for the prediction of target
  events in time series arising in real world
  problems (e.g., electroencephalogram (EEG) time
  series, stock market time series).
• Identify spatiotemporal dynamical patterns in
  multiple EEG time series that characterize the
  epileptic brain, which can be used to predict
  seizures.
• Applying time series data mining concepts to
  multichannel EEG data for the prediction of
  epileptic seizures, we integrated methods based
  on
• Chaos Theory
• Statistical Analysis
• Optimization

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Why data mining?

• Data mining is fast emerging as a core component
  in many business and industries, especially those
  related to
• Finance and Banking
• Manufacturing
• Biosystems and Biotechnology
• Information Systems and Services
• Data mining
• Is the analysis of data with the goal of
  uncovering hidden information or patterns.
• Is defined as the search for valuable
  information in large volumes of data.
• Searches for very strong patterns in large data
  that can generalize to accurate future decisions.

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Data Mining in EEG Time Series

• Why chaos theory?
• Characterization and quantification of the
  dynamics of time series may enable us to predict
  the occurrences of some specific events which
  follow a change in the dynamics of the time
  series.
• The EEG is a multi-dimensional nonstationary time
  series of spatial extent.
• It has statistical properties that depend on both
  time (temporal) and space (spatial).
• Components of the brain (e.g., neurons, sets of
  neurons) are densely interconnected and the EEG
  recorded from one brain site is inherently
  related to the activity at other sites.
• These components may functionally interact at
  different time instants.

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Data Mining in EEG Time Series

• Why spatiotemporal patterns?
• There is only so much that we can do with a
  single variable. But with more than one variable,
  the analytic world opens up!
• Provide insight into the dynamical
  interrelationships (spatiotemporal patterns)
  between variables (time series).
• Why optimization?
• Identifying spatiotemporal patterns in multiple
  time series is combinatorial in nature, operating
  with the selection of critical components in the
  system of interest.
• Optimization techniques are developed to improve
  the performance of prediction in the time series
  by
• Identifying critical (optimal) spatiotemporal
  patterns related to the target events
• Determining the optimal parameters of the systems
  with one or more hidden variables

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Chaos Theory
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Measure of Chaoticity Lyapunov Exponents

• What are Lyapunov exponents?
• Lyapunov exponents are the average exponential
  rates of convergence or divergence of nearby
  trajectories, orbits, in phase space.
• Given a continuous dynamical system in an
  n-dimensional phase space, there are n Lyapunov
  exponents, which are always ordered so that ?1
  ?2 ?3 ?n.
• A system containing at least one positive
  Lyapunov exponents is defined to be chaotic.
• A positive Lyapunov exponent measures the
  chaoticity and stability of the system.

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Figure 3 2-D Example Circle of initial
conditions evolves into an ellipse.
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• Embed the data set (EEG). Xi (x(ti),x(tit),,x(
  ti(p-1)t))T where t is the selected time lag
  between the components of each vector in the
  phase space, p is the selected dimension of the
  embedding phase space, and ti ? 1,T-(p-1) t.
• Pick a point x(t0) somewhere in the middle of the
  trajectory. Find that point's nearest neighbor.
  Call that point z0 (t0).
• Compute z0 (t0) - x(t0) L0.
• Follow the difference trajectory" -- the dashed
  line -- forwards in time, computing z0 (ti) -
  x(ti) L0(i) and incrementing i, until L0(i) gt
  e. Call that value L0' and that time t1.
• Find z1 (t1), the nearest neighbor of x(t1),
  and go to step 3. Repeat the procedure to the end
  of the fiduciary trajectory t tn, keeping track
  of the Li and Li' .

where M is the number of times we went through
the loop above, and N is the number of time-steps
in the fiduciary. N?t tn - t0
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Measure of Chaoticity in EEG Short-Term Maximum
Lyapunov Exponents (STLmax)
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STLmax Profiles from Multiple EEG Time Series
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Statistical Analysis
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Statistics to quantify the convergence of STLmax

• By paired-T statistic
• For electrode sites i and j, suppose their STLmax
  values in a window Wt (of length 60 points, 10
  minutes) are

and
The T-index at time window Wt between electrode
sites i and j is defined as
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Drop in T-index
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Entrainment Convergence of STLmax
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Resetting of Epileptic Brain

• This figure shows the existence of resetting of
  the brain after seizures' onset, that is,
  divergence of STLmax profiles after seizures.
• Resetting of the epileptic brain occurs after
  seizure onset and involves a subset of cortical
  electrode sites.

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Models
Homoclinic Chaos (Silnikovs Theorem) Rössler
systems, Lorentz systems, population dynamical
systems
(1)
(2)
(3)

• w, a, b and g are intrinsic parameters.
• e and e are directional coupling strengths.
• N number of oscillators

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Spatio-Temporal Chaos

• w1 gt w2

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Spatial Synchronization and Hysteresis(Oscillator
s 1 and 2)
Similarity Function between 2 Oscillators

• For non-identical signals mintS(t)1
• For identical signals mintS(t)0 for t0
• For identical signals with finite time shift t
• mintS(t) 0 for some nonzero t.

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Lmax from the 12-dimensional Model versus coupling

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Lmax versus time and coupling

Iasemidis, et. al (in press)
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Optimization
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Why Optimization? Not every electrode site shows
the convergence
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Optimization Problems

• Optimization
• We apply optimization techniques to find a group
  of electrode sites such that
• They are the most converged (in STLmax) electrode
  sites during 10-min window before the seizure
• They show the dynamical resetting (diverged in
  STLmax) during 10-min window after the seizure.
• Such electrode sites are defined as critical
  electrode sites.
• Hypothesis
• The critical electrode sites should be most
  likely to show the convergence in STLmax again
  before the next seizure.

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Why MQIP?

• To select critical electrode sites, we formulated
  this problem as a multi-quadratic integer (0-1)
  programming (MQIP) problem with
• objective function to minimize the average
  T-index among electrode sites
• a linear constraint to identify the number of
  critical electrode sites
• a quadratic constraint to ensure that the
  selected electrode sites show the dynamical
  resetting

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Notation and Modeling

• x is an n-dimensional column vector (decision
  variables), where each xi represents the
  electrode site i.
• xi 1 if electrode i is selected to be one of
  the critical electrode sites.
• xi 0 otherwise.
• Q is an (n?n) matrix, whose each element qij
  represents the T-index between electrode i and j
  during 10-minute window before a seizure.
• b is an integer constant. (the number of critical
  electrode sites)
• D is an (n?n) matrix, whose each element dij
  represents the T-index between electrode i and j
  during 10-minute window after a seizure.
• a 2.662k(k-1), an integer constant. 2.662 is
  the critical value of T-index, as previously
  defined, to reject H0 two brain sites acquire
  identical STLmax values within 10-minute window

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KKT Conditions Approach

• Consider the quadratic 0-1 programming problem
• eT (1,1,,1)
• Relax x 0, we then have the following KKT
  conditions

Q is an (n?n) matrix. b is an integer constant x
is an n-dimensional column vector
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KKT Conditions Approach

• Add slack variables a and define s u.e a
• Minimizing slack variables, we can formulate this
  problem as
• Note that this problem formulation is an
  efficient approach in our application, as n
  increases, because it has the SAME number of 0-1
  variables (n) however, this technique is
  heuristic in nature (it does not guarantee the
  optimality to the original Linearly Constrained
  Quadratic 0-1 Problem).

Relax x?0,1
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Connections Between QIP problems and MLIP problems
Equivalent
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MLIP formulation for MQIP problem

• Consider the MQIP problem
• We proved that the MQIP problem is EQUIVALENT to
  a MLIP problem with the SAME number of integer
  variables.

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MLIP formulation for MQIP problem

• Consider the MQIP problem
• We proved that the MQIP program is EQUIVALENT to
  a MLIP problem with the SAME number of integer
  variables.

Equivalent
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Linearization Approach
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Linearization Approach for Multi-Quadratic 0-1
Problem
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Table 1. Performance characteristics of two
proposed approaches compared with complete
enumerations.
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Figure 3. Performance characteristics of two
proposed approaches compared with complete
enumerations.
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Predictability of Epileptic Seizures
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Methods and Results
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Hypothesis Testing - Simulation

• Hypothesis
• The critical electrode sites should be most
  likely to show the convergence in STLmax (drop in
  T-index below the critical value) again before
  the next seizure.
• The critical electrode sites are electrode sites
  that
• are the most converged (in STLmax ) electrode
  sites during 10-min window before the seizure
• show the dynamical resetting (diverged in STLmax
  ) during 10-min window after the seizure
• Simulation
• Based on 3 patients with 20 seizures, we compare
  the probability of showing the convergence in
  STLmax (drop in T-index below the critical value)
  before the next seizure between the electrode
  sites, which are
• Critical electrode sites
• Randomly selected (5,000 times)

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Simulation - Results
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EEG Characteristics of Seizures

• Traditional view (Interictal ? Ictal ? Postictal)
• The occurrences of seizures are random.
• The transition from the interictal to ictal state
  is very abrupt(seconds).
• Existence of a recovery period? (postictal state)
• Emerging view (Interictal ? Preictal ? Ictal ?
  Postictal)
• The occurrences of seizures are deterministic.
• Existence of a preictal state. The transition
  from the interictal to preictal to ictal state is
  progressive (minutes to hours).
• Postictal disentrainment of the epileptogenic
  focus from normal brain sites.

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Findings

• Chaos Theory
• Quantify the dynamics (degree of stability) of
  EEG
• Convergence in STLmax before the seizure onset
• Divergence in STLmax after the seizure onset
• Statistical Analysis
• Quantify the degree of convergence or divergence
  of the STLmax profiles.
• Optimization
• Select critical electrode sites such that they
  show the spatio-temporal changes in dynamics of
  EEG

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Automated Seizure Warning Algorithm
Continuously calculate STLmax from sequential
10.24 sec epoch of EEG from each electrode site.
Select critical electrode sites after every
subsequent seizure
EEG Signals
Give a warning when T-index value is
greater than 5, then drops to a value of 2.662
or less
Monitor the average T-index curve of Critical
electrodes
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Evaluation of ASWA

• To test this algorithm, a warning was considered
  to be true if a seizure occurred within 3 hours
  after the warning.
• Sensitivity
• False Prediction Rate average number of false
  warnings per hour

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Figure 6 Performance characteristics of
automated seizure warning algorithm with optimal
parameter-settings of training data set.
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Figure 7 ROC curve analysis for optimal
parameter settings of 5 patients
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Figure 8 Performance characteristics of
automated seizure warning algorithm on testing
data set.
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Concluding Remarks and Future Works
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Concluding Remarks

• Because the algorithm analyzes continuous EEG
  recordings of several days duration, the
  computational approach to calculate STLmax and to
  solve the optimization problem has to be very
  efficient.
• The linearization technique, based on KKT
  conditions, allow us to solve multi-quadratic
  integer programming very efficiently.
• Based on the KKT linearization technique, the
  mixed linear 0-1 programming formulation can be
  applied in other applications.

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Maximum Clique Problem
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Maximum Clique Problem
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Maximum Independent Set Problem
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Concluding Remarks

• The results of this study confirm our hypothesis
  that it is possible to predict an impending
  seizure based on optimization and nonlinear
  dynamics.
• Allow the patients to take measures against
  potential hazards to themselves or others (e.g.,
  stop driving, call for assistance).
• Evaluate effectiveness of medication
• Allow intervention schemes for early abolition of
  an upcoming seizure.
• Development of a seizure prediction algorithm was
  complicated by three factors
• the cortical sites participating in the preictal
  transition varied from seizure to seizure
• the length of the preictal transition varied from
  seizure to seizure
• it was not known whether or not this type of
  spatiotemporal transition was unique to the
  preictal period.

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Future Works Applications to other dynamical
disorders

• Early warning for transitions in other brain
  dynamical disorders (e.g., sleep disorders,
  migraine attacks, Parkinsonian tremors, memory
  lapses, cognition and learning disorders)
• Early warning for transitions in biomedical data
  from other modalities like EKG (e.g., heart
  attacks and fibrillation), EEG changes due to
  level of anesthesia in the operating room,
  diabetic attacks
• Early warning for synchronization in coupled
  physical systems (e.g., laser arrays, phase
  locked loops, artificial neural networks)

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Future Works Applications in FinanceAA ALCOA
INC. Stock Index
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CAT CATERPILLAR INC. Stock Index
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DD E.I. DUPONT DE NEMOURS Stock Index
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Appendix
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References

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