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Journal of Combinatorial Optimization

, Volume 5, Issue 1, pp 9–26 | Cite as

Quadratic Binary Programming and Dynamical System Approach to Determine the Predictability of Epileptic Seizures

  • L.D. Iasemidis
  • P. Pardalos
  • J.C. Sackellares
  • D.-S. Shiau
Article

Abstract

Epilepsy is one of the most common disorders of the nervous system. The progressive entrainment between an epileptogenic focus and normal brain areas results to transitions of the brain from chaotic to less chaotic spatiotemporal states, the epileptic seizures. The entrainment between two brain sites can be quantified by the T-index from the measures of chaos (e.g., Lyapunov exponents) of the electrical activity (EEG) of the brain. By applying the optimization theory, in particular quadratic zero-one programming, we were able to select the most entrained brain sites 10 minutes before seizures and subsequently follow their entrainment over 2 hours before seizures. In five patients with 3–24 seizures, we found that over 90% of the seizures are predictable by the optimal selection of electrode sites. This procedure, which is applied to epilepsy research for the first time, shows the possibility of prediction of epileptic seizures well in advance (19.8 to 42.9 minutes) of their occurrence.

epileptic seizures Lyapunov exponents entrainment T-statistic optimization 

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References

  1. H.D.I. Abarbanel, Analysis of Observed Chaotic Data, Springer-Verlag: New York, 1996.Google Scholar
  2. B.W. Abou-Khalil, G.J. Seigel, J.C. Sackellares, S. Gilman, R. Hichwa, and R. Marshall, “Positron emission tomograghy studies of cerebral glucose metabolism in patients with chronic partial epilepsy,” Ann. Neurol., vol. 22, pp. 480-486, 1987.Google Scholar
  3. G.G. Athanasiou, C.P. Bachas, and W.F. Wolf, “Invariant geometry of spin-glass states,” Phy.Rev.B, vol. 35, pp. 1965-1968, 1987.Google Scholar
  4. F. Barahona, “On the computational complexity of spin glass models,” J. Phys. A: Math. Gen., vol. 15, pp. 3241-3253, 1982a.Google Scholar
  5. F. Barahona, “On the exact ground states of three-dimensional ising spin glasses,” J. Phys. A: Math. Gen., vol. 15, pp. L611-L615, 1982b.Google Scholar
  6. H. Berger, “Uber das elektroenkephalogramm des menchen,” Arch. Psychiatr. Nervenkr., vol. 87, pp. 527-570, 1929.Google Scholar
  7. D.E. Burdette, Sakuraisy, T.R. Henry, D.A. Ross, P.B. Pennell, K.A. Frey, J.C. Sackellares, and R. Albin, “Temporal lobe central benzodiazepine binding in unilateral mesial temporal lobe epilepsy,” Neurology, vol. 45, pp. 934-941, 1995.Google Scholar
  8. M. Casdagli, L.D. Iasemidis, R.L. Gilman, S.N. Roper, R.S. Savit, and J.C. Sackellares, “Nonlinearity in invasive EEG recordings from patients with temporal lobe epilepsy,” Electroenceph. Clin. Neurophysiol., vol. 102, pp. 98-105, 1997.Google Scholar
  9. M. Casdagli, L.D. Iasemidis, J.C. Sackellares, S.N. Roper, R.L. Gilman, and R.S. Savit, “Characterizing nonlin-earity in invasive EEG recordings from temporal lobe epilepsy,” Physica D, vol. 99, pp. 381-399, 1996.Google Scholar
  10. G. Casella and R.L. Berger, Statistical Inference, Duxbury Press: Belmont, CA, 1990.Google Scholar
  11. R. Caton, “The electric currents of the brain,” BMJ, vol. 2, p. 278, 1875.Google Scholar
  12. J.P. Eckmann, S.O. Kamphorst, D. Ruelle, and S. Ciliberto, “Lyapuunov exponents from time series,” Phys. Rev. A, vol. 34, pp. 4971-4972, 1986.Google Scholar
  13. C.E. Elger and K. Lehnertz, “Seizure prediction by non-linear time series analysis of brain electrical activity,” Europ. J. Neurosci., vol. 10, pp. 786-789, 1998.Google Scholar
  14. T. Elbert, W.J. Ray, J. Kowalik, J.E. Skinner, K.E. Graf, and N. Birbaumer, “Chaos and physiology: Deterministic chaos in excitable cell assemblies,” Physiol. Rev., vol. 74, pp. 1-47, 1994.Google Scholar
  15. J. Engel Jr., D.E. Kuhl, M.E. Phelps, and J.C. Mazziota, “Interictal cerebral glucose metabolism in partial epilepsy and its relation to EEG changes,” Ann. Neurol., vol. 12, pp. 510-517, 1982.Google Scholar
  16. M.A. Falconer, E.A. Serefetinides, and J.A.N. Corsellis, “Aetiology and pathogenesis of temporal lobe epilepsy,” Arch. Neurol., vol. 19, pp. 233-240, 1964.Google Scholar
  17. M. Feucht, U. Moller, H. Witte, F. Benninger, S. Asenbaum, D. Prayer, and M.H. Friedrich, “Application of correlation dimension and pointwise dimension for non-linear topographical analysis of focal onset seizures,” Med. Biol. Eng. Comp., vol. 37, pp. 208-217, 1999.Google Scholar
  18. A.M. Fraser and H.L. Swinney, “Independent coordinates for strange attractors from mutual information,” Phys. Rev. A, vol. 33, pp. 1134-1140, 1986.Google Scholar
  19. P. Gloor, Hans Berger on the Electroencephalogram of Man, Elsevier: Amsterdam, 1969.Google Scholar
  20. P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D, vol. 9, pp. 189-208, 1983a.Google Scholar
  21. P. Grassberger and I. Procaccia, “Characterization of strange attractors,” Phys. Rev. Lett., vol. 50, pp. 346-349, 1983b.Google Scholar
  22. P. Grassberger, T. Schreiber, and C. Schaffrath, “Nonlinear time sequence analysis,” Int. J. Bifurc. Chaos, vol. 1, pp. 521-547, 1991.Google Scholar
  23. A.V. Holden, Chaos-Nonlinear Science: Theory and Applications, University Press: Manchester, 1986.Google Scholar
  24. H. Horst, P.M. Pardalos, and V. Thoai, Introduction to Global Optimization, Series on Nonconvex Optimization and its Applications, 3, Kluwer Academic Publishers: Dordrecht, 1995.Google Scholar
  25. L.D. Iasemidis, H.P. Zaveri, J.C. Sackellares, and W.J. Williams, “Linear and nonlinear modeling of ECoG in temporal lobe epilepsy,” 25th Annual Rocky Mountain Bioengineering Symposium, vol. 24, pp. 187-193, 1988.Google Scholar
  26. L.D. Iasemidis, J.C. Sackellares, H.P. Zaveri, and W.J. Williams, “Phase space topography of the electrocorticogram and the Lyapunov exponent in partial seizures,” Brain Topogr., vol. 2, pp. 187-201, 1990.Google Scholar
  27. L.D. Iasemidis, “On the dynamics of the human brain in temporal lobe epilepsy,” PhD Thesis, University of Michigan, Ann Arbor, 1991.Google Scholar
  28. L.D. Iasemidis and J.C. Sackellares, “The temporal evolution of the largest Lyapunov exponent on the human epileptic cortex,” in Measuring Chaos in the Human Brain, D.W. Duke and W.S. Pritchard (Eds.), World Scientific: Singapore, 1991.Google Scholar
  29. L.D. Iasemidis, J.C. Sackellares, and R.S. Savit, “Quantification of hidden time dependencies in the EEG within the framework of nonlinear dynamics,” in Nonlinear Dynamical Analysis of the EEg, B.H. Jansen and M.E. Brandt (Eds.), World Scientific: Singapore, 1993.Google Scholar
  30. L.D. Iasemidis, L.D. Olson, J.C. Sackellares, and R. Savit, “Time dependencies in the occurrences of epileptic seizures: A nonlinear approach,” Epilepsy Research, vol. 17, pp. 81-94, 1994.Google Scholar
  31. L.D. Iasemidis, J.C. Principe, and J.C. Sackellares, “Spatiotemporal dynamics of human epileptic seizures,” in 3 rd Experimental Chaos Conference, R.G. Harrison, W. Lu, W. Ditto, L. Pecora, M. Spano, and S. Vohra (Eds.), World Scientific: Singapore, 1996.Google Scholar
  32. L.D. Iasemidis and J.C. Sackellares, “Chaos theory and epilepsy,” The Neuroscientist, vol. 2, pp. 118-126, 1996.Google Scholar
  33. L.D. Iasemidis, J.C. Principe, J.M. Czaplewski, R.L. Gilman, S.N. Roper, and J.C. Sackellares, “Spatiotemporal transition to epileptic seizures: A nonlinear dynamical analysis of scalp and intracranial EEG recordings,” in Spatiotemporal Models in Biological and Artifical Systems, F. Lopes da Silva, J.C. Principe, and L.B. Almeida (Eds.), IOS Press: Amsterdam, 1997.Google Scholar
  34. L.D. Iasemidis, J.C. Principe, and J.C. Sackellares, “Measurement and quantification of spatiotemporal dynamics of human epileptic seizures,” in Nonlinear Biomedical Signal Processing, 2, M. Akay (Ed.), IEEE Press, pp. 294-298, 2000.Google Scholar
  35. L.D. Iasemidis, D.S. Shidu, P. Pardalos, and J.C. Sackellares, “Transition to epileptic seizures-an optimization approach into its dynamics,” in Discrete Problems with Medical Applications, D.Z. Du, P.M. Pardalos, and J. Wang, DIMACS Series American Mathematical Society Publishing Co., vol. 55, pp. 55-74, 2000.Google Scholar
  36. B.H. Jansen, “Is it and so what? A critical review of EEG-chaos,” in Measuring Chaos in the Human Brain, D.W. Duke and W.S. Pritchard (Eds.), World Scientific: Singapore, 1991.Google Scholar
  37. A.N. Kolmogorov, “The general theory of dynamical systems and classical mechanics,” in Foundations of Mechanics, R. Abraham and J.E. Marsden (Eds.), 1954.Google Scholar
  38. E.J. Kostelich, “Problems in estimating dynamics from data,” Physica D, vol. 58, pp. 138-152, 1992.Google Scholar
  39. K. Lehnertz and C.E. Elger, “Spatio-temporal dynamics of the primary epileptogenic area in temporal lobe epilepsy characterized by neuronal complexity loss,” Electroenceph. Clin. Neurophysiol., vol. 95, pp. 108-117, 1995.Google Scholar
  40. K. Lehnertz and C.E. Elger, “Can epileptic seizures be predicted? Evidence from nonlinear time series analysis of brain electrical activity,” Phys. Rev. Lett., vol. 80, pp. 5019-5022, 1998.Google Scholar
  41. M. Le Van Quyen, J. Martinerie, C. Adam, and F.J. Varela, “Nonlinear spatio-temporal interdependences of interictal intracranial EEG recordings from patients with temporal lobe epilepsy: Localizing of epileptogenic foci,” Physica D, vol. 127, pp. 250-266, 1999a.Google Scholar
  42. M. Le Van Quyen, J. Martinerie, M. Baulac, and F. Varela, “Anticipating epileptic seizures in real time by a non-linear analysis of similarity between EEG recordings,” NeuroReport, vol. 10, pp. 2149-2155, 1999b.Google Scholar
  43. F. Lopes da Silva, “EEG analysis: Theory and practice; computer-assisted EEG diagnosis: Pattern recognition tech-niques,” in Electroencephalography: Basic Principles, Clinical Applications and Related Field, E. Niedermeyer and F. Lopes da Silva (Eds.), Urban and Schwarzenberg: Baltimore, 1987.Google Scholar
  44. J. Martinerie, C. Adam, M. Le Van Quyen, M. Baulac, S. Clemenceau, B. Renault, and F.J. Varela, “Epileptic seizures can be anticipated by non-linear analysis,” Nature Medicine, vol. 4, pp. 1173-1176, 1998.Google Scholar
  45. J.H. Margerison and J.A.N. Corsellis, “Epilepsy and the temporal lobes,” Brain, vol. 89, pp. 499-530, 1966.Google Scholar
  46. G. Mayer-Kress, Dimension and Entropies in Chaotic Systems, Springer-Verlag: Berlin, 1986.Google Scholar
  47. J.W. McDonald, E.A. Garofalo, T. Hood, J.C. Sackellares, S. Gilman, P.E. McKeever, J.C. Troncaso, and M.V. John-ston, “Altered excitatory and inhibitory aminoacid receptor binding in hippocampus of patients with temporal lobe epilepsy,” Annals of Neurology, vol. 29, pp. 529-541, 1991.Google Scholar
  48. M. Mezard, G. Parisi, and M.A. Virasoro, Spin Glass Theory and Beyond, World Scientific: Singapore, pp. 112-133, 1987.Google Scholar
  49. J.J. Moré and S.J. Wright, Optimization Software Guide, SIAM, Philadelphia, 1994.Google Scholar
  50. E. Niedermeyer, “Depth electroencephalography,” in Electroencephalography: Basic Principles, Clinical Appli-cations and Related Fields, E. Niedermeyer and F. Lopes da Silva (Eds.), Urban and Schwarzenberg: Baltimore, 1987.Google Scholar
  51. A. Oseledec, “A multiplicative ergodic theorum-Lyapunov characteristic numbers for dynamical systems (English translation),” IEEE Int. Conf. ASSP, vol. 19, pp. 179-210, 1968.Google Scholar
  52. N.H. Packard, J.P. Crutchfield, J.D. Farmer, and R.S. Shaw, “Geometry from time series,” Phys. Rev. Lett., vol. 45, pp. 712-716, 1980.Google Scholar
  53. M. Palus, V. Albrecht, and I. Dvorak, “Information theoretic test for nonlinearity in time series,” Phys. Lett. A, vol. 175, pp. 203-209, 1993.Google Scholar
  54. P.M. Pardalos and G. Rodgers, “Parallel branch and bound algorithms for unconstrained quadratic zero-one programming,” in Impact of Recent Computer Advances on Operations Research, R. Sharda et al. (Eds.), North-Holland, 1989.Google Scholar
  55. P.M. Pardalos and G. Rodgers, “Computational aspects of a branch and bound algorithm for quadratic zero-one programming,” Computing, vol. 45, pp. 131-144, 1990.Google Scholar
  56. J. Pesin, “Characteristic Lyapunov exponents and smooth ergodic theory,” Russian Math. Survey, vol. 4, pp. 55-114, 1977.Google Scholar
  57. A. Renyi, Probability Theory, Elsevier: Amsterdam, 1970.Google Scholar
  58. J.C. Sackellares, L.D. Iasemidis, R.L. Gilman, and S.N. Roper, “Epilepsy-When chaos fails,” in Chaos in the Brain? P. Grassberger, C.E. Elger, and K. Lehnertz (Eds.), World Scientific: Singapore, pp. 112-133, 2000.Google Scholar
  59. J.C. Sackellares, L.D. Iasemidis, H.P. Zaveri, and W.J. Williams, “Measurement of chaos to localize seizure onset,” Epilepsia, vol. 30, p. 663, 1989a.Google Scholar
  60. J.C. Sackellares, L.D. Iasemidis, H.P. Zaveri, W.J. Williams, and T.W. Hood, “Inference on the chaotic behavior of the epileptogenic focus,” Epilepsia, vol. 29, p. 682, 1989b.Google Scholar
  61. J.C. Sackellares, G.J. Siegel, B.W. Abou-Khalil, T.W. Hood, S. Gilman, P. McKeever, R.D. Hichwa, and G.D. Hutchins, “Differences between lateral and mesial temporal metabolism interictally in epilepsy of mesial temporal origin,” Neurology, vol. 40, p. 1420-1426, 1990.Google Scholar
  62. F. Takens, “Detecting strange attractors in turbulence,” in Dynamical Systems and Turbulence, Lecture Notes in Mathematics, D.A. Rand and L.S. Young (Eds.), Springer-Verlag: Heidelburg, 1981.Google Scholar
  63. M.J. Van der Heyden, D.N. Velis, B.P.T. Hoekstra, J.P.M. Pijn, W. Van Emde Boas, C.W.M. Van Veelen, P.C. Van Rijen, F.H. Lopes da Silva, and J. DeGoede, “Non-linear analysis of intracranial human EEG in temporal lobe epilepsy,” Clinical Neurophysiology, vol. 110, pp. 1726-1740, 1999.Google Scholar
  64. J.A. Vastano and E.J. Kostelich, “Comparison of algorithms for determining Lyapunov exponents from experimental data,” in Dimensions and Entropies in Chaotic Systems: Quantification of Complex Behavior, G. Mayer-Kress (Ed.), Springer-Verlag: Berlin, 1986.Google Scholar
  65. P. Walters, An Introduction to Ergodic Theory, Springer-Verlag: Berlin, 1982.Google Scholar
  66. Weisberg, Applied Linear Regression, Wiley: New York, 1990.Google Scholar
  67. A. Wolf, J.B. Swift, H.L. Swinney, and J.A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D, vol. 16, pp. 285-317, 1985.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • L.D. Iasemidis
    • 1
  • P. Pardalos
    • 2
  • J.C. Sackellares
    • 3
  • D.-S. Shiau
    • 4
  1. 1.BioengineeringArizona State UniversityUSA
  2. 2.Industrial and Systems EngineeringUniversity of FloridaUSA
  3. 3.Neurology; Bioengineering; NeuroscienceUniversity of FloridaUSA
  4. 4.StatisticsUniversity of FloridaUSA

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