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Controlling Neurological Disease at the Edge of Instability

  • John G. Milton
  • Jennifer Foss
  • John D. Hunter
  • Juan Luis Cabrera
Part of the Biocomputing book series (BCOM, volume 2)

Abstract

Rapid advances in technology are making the dream of treating human neurological diseases with implanted electronic devices a reality. The more such devices are able to exploit the properties of intrinsic neural control mechanisms, the more effective they will be in re-establishing control in the setting of disease. Noise and time delays are ubiquitous features of the nervous system. Three observations suggest that in order to understand control in noisy neural dynamical systems with retarded variables it will be necessary to change the focus from the identification and characterization of attractors to a study of phenomena that occur near stability boundaries (i.e., “critical phenomena”): 1) multistability has been identified in simple neural loops, the onset of epileptic seizures, and human postural sway; 2) on-off intermittency and 3) power laws arise in the nervous system. These observations support the possibility of developing strategies that treat neurological disease by the addition of appropriately designed stimuli, including noise.

Keywords

Epileptic Seizure Stochastic Resonance Parametric Noise Dynamic Disease Neural Synchrony 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [Ajmone Marsan, 1972]
    Ajmone Marsan, C. (1972). Focal electrical stimulation. In Purpura, D. P., Penry, J. K., Tower, D. B., Woodbury, D. M., and Walter, R. D., editors, Experimental models of epilepsy: A manual for the laboratory worker, pages 147–172, New York. Raven Press.Google Scholar
  2. [an der Heiden, 1979]
    an der Heiden, U. (1979). Delays in physiological systems. J. Math. Biol., 8: 345–364.MathSciNetMATHCrossRefGoogle Scholar
  3. [an der Heiden and Mackey, 1982]
    an der Heiden, U. and Mackey, M. C. (1982). The dynamics of production and destruction: Analytic insight into complex behavior. J. Math. Biol., 16: 75–101.MathSciNetMATHCrossRefGoogle Scholar
  4. [Arielle et al., 1996]
    Arielle, A., Sterkin, A.and Grinvald, A., and Aerstein, A. (1996). Dynamics of ongoing activity: Explanation of the large variability in evoked potential responses. Science, 273: 1868–1871.CrossRefGoogle Scholar
  5. [Arnold, 1998]
    Arnold, L. (1998). Random dynamical systems. Springer-Verlag, New York.MATHGoogle Scholar
  6. [Arnold et al., 1999]
    Arnold, L., Bleckert, G., and Schenk-Hoppé, K. (1999). The stochastic brusselator: Parametric noise destroys Hopf bifurcation. In Crauel, H. and Gundlach, M., editors, Stochastic Dynamics, pages 71–92. Springer-Verlag, New York.CrossRefGoogle Scholar
  7. [Baer et al., 1989]
    Baer, S. M., Ereux, T., and Rinzel, J. (1989). The slow passage through a Hopf bifurcation: Delay, memory effects, and resonance. SIAMJ. Appl. Math., 49: 55–71.MATHCrossRefGoogle Scholar
  8. [Bak et al., 1988]
    Bak, P., Tang, C., and Wiesenfeld, K. (1988). Self-organized criticality. Phys. Rev. A, 38: 364–374.MathSciNetCrossRefGoogle Scholar
  9. [Barabási, 2002]
    Barabási, A.-L. (2002). Linked: The new science of networks. Perseus Publishing, Cambridge, MA.Google Scholar
  10. [Bauer and Bertsch, 1990]
    Bauer, W. and Bertsch, G. F. (1990). Decay of ordered and chaotic systems. Phys. Rev. Lett., 65: 2213–2216.CrossRefGoogle Scholar
  11. [Bayer and an der Heiden, 1998]
    Bayer, W. and an der Heiden, U. (1998). Oscillation types and bifurcations of a nonlinear second-order differential-difference equation. J. Dyn. Diff Eqns., 10: 303–326.MATHCrossRefGoogle Scholar
  12. [Beggs and Plenz, 2002]
    Beggs, J. M. and Plenz, D. (2002). Self-organized criticality of spontaneous activity in isolated cortical networks. Society of Neuroscience (Abstracts), page 28.Google Scholar
  13. [Bogdanoff, 1962]
    Bogdanoff, J. L. (1962). Influence on the behavior of a linear dynamical system of some imposed rapid movements of small amplitude. J. Acoust. Soc. Amer., 34: 1055–1062.MathSciNetCrossRefGoogle Scholar
  14. [Bogdanoff and Citron, 1965]
    Bogdanoff, J. L. and Citron, S. J. (1965). Experiments with an inverted pendulum subject to random parametric perturbations. J. Acoust. Soc. Amer., 38: 447–452.CrossRefGoogle Scholar
  15. [Borsellino et al., 1972]
    Borsellino, A., De Marco, A., Allazetta, A., Rinesi, S., and Bartolini, B. (1972). Reversal time distributions of visual ambiguous stimuli. Kybernetik, 10: 139–144.CrossRefGoogle Scholar
  16. [Brockman and Giesel, 2000]
    Brockman, D. and Giesel, T. (2000). The ecology of gaze shifts. Neurocomputing, 32–33: 643–650.CrossRefGoogle Scholar
  17. [Cabrera and Milton, 2002]
    Cabrera, J. L. and Milton, J. G. (2002). On–off intermittency in a human balancing task. Phys. Rev. Lett., 89:158702–1–4.Google Scholar
  18. [Cabrera and Milton, 2003]
    Cabrera, J. L. and Milton, J. G. (2003). Delays, scaling and the acquisition of motor skill. In Bezrukov, S., editor, Unsolved Problems of Noise and Fluctuations: UpoN 2002: Third International Conference on Unsolved Problems of Noise and Fluctuations in Physics, Biology and High Technology (AIP Proceedings Vol. 665 ), pages 250–256, Melville, NY. American Institute of Physics.Google Scholar
  19. [Campbell et al., 1995]
    Campbell, S. A., Bélair, J., Ohira, T., and Milton, J. G. (1995). Limit cycles, tori and complex dynamics in a second-order differential equation with delayed negative feedback. J. Diff Eqns., 7: 213–236.MATHCrossRefGoogle Scholar
  20. [Chialvo and Bak, 1999]
    Chialvo, D. R. and Bak, P. (1999). Learning from mistakes. Neuroscience, 90: 1137–1148.CrossRefGoogle Scholar
  21. [Chkhenkeli, 2002]
    Chkhenkeli, S. A. (2002). Direct deep brain stimulation: First steps towards the feedback control of seizures. In Milton, J. and Jung, P., editors, Epilepsy as a dynamic disease, pages 249–262, New York. Springer-Verlag.Google Scholar
  22. [Chkhenkeli and Milton, 2002]
    Chkhenkeli, S. A. and Milton, J. (2002). Dynamic epileptic systems versus static epileptic foci? In Milton, J. and Jung, P., editors, Epilepsy as a dynamic disease, pages 25–36, New York. Springer-Verlag.Google Scholar
  23. [Collins and De Luca, 1994]
    Collins, J J and De Luca, C. J. (1994). Random walking during quiet standing. Phys. Rev. Lett., 73: 764–767.CrossRefGoogle Scholar
  24. [Contreras et al., 1996]
    Contreras, D., Destexhe, A., Sejnowski, T. J., and Steraide, M. (1996). Control of spatiotemporal coherence of a thalamic oscillaton by corticothalamic feedback. Science, 274: 771–774.CrossRefGoogle Scholar
  25. [Coulter, 1997]
    Coulter, D. A. (1997). Thalamocortical anatomy and physiology. In Engel, Jr., J. and Pedley, T. A., editors, Epilepsy: A comprehensive textbook, pages 341–351, Philadelphia. Lippincott-Raven.Google Scholar
  26. [Ding and Yang, 1995]
    Ding, M. and Yang, W. (1995). Distribution of the first return time in fractional Brownian motion and its application to the study of on-off intermittency. Phys. Rev. E, 52: 207–213.MathSciNetCrossRefGoogle Scholar
  27. [Durand, 1993]
    Durand, D. M. (1993). Ictal patterns in experimental models of epilepsy. Clin. Neurophysiol., 10: 281–297.CrossRefGoogle Scholar
  28. [Ermentrout and Kopell, 1994]
    Ermentrout, B. B. and Kopell, N. (1994). Learning of phase lags in coupled neural oscillators. Neural Corp., 6: 225–241.CrossRefGoogle Scholar
  29. [Eurich and Milton, 1996]
    Eurich, C. W. and Milton, J. G. (1996). Noise- induced transitions in human postural sway. Phys. Rev. E, 54: 6681–6684.CrossRefGoogle Scholar
  30. [Fitts and Posner, 1973]
    Fitts, P. M. and Posner, M. I. (1973). Human performance. Prentice-Hall, London.Google Scholar
  31. [Foss et al., 1997a]
    Foss, J., Eurich, C. W., Milton, J., and Ohira, T. (1997a). Noise, multistability and long-tailed interspike interval (ISI) histograms. Bull. Amer. Phys. Soc., 42: 781.Google Scholar
  32. [Foss et al., 1996]
    Foss, J., Longtin, A., Mensour, B., and Milton, J. (1996). Multistability and delayed recurrent loops. Phys. Rev. Lett., 76: 708–711.CrossRefGoogle Scholar
  33. [Foss and Milton, 2000]
    Foss, J. and Milton, J. (2000). Multistability in recurrent neural loops arising from delay. J Neurophysiology, 84: 975–985.Google Scholar
  34. [Foss and Milton, 2002]
    Foss, J. and Milton, J. (2002). Aborting seizures with a single stimulus: The case for multistability. In Milton, J. and Jung, P., editors, Epilepsy as a dynamic disease, pages 283–295, New York. Springer-Verlag.Google Scholar
  35. [Foss et al., 1997b]
    Foss, J., Moss, F., and Milton, J. (1997b). Noise, multistability, and delayed recurrent loops. Phys. Rev. E, 55: 4536–4543.CrossRefGoogle Scholar
  36. [Gammaitoni et al., 1998]
    Gammaitoni, L., Hänggi, P., Jung, P., and Marchesoni, F. (1998). Stochastic resonance. Rev. Mod. Phys., 70: 223–288.CrossRefGoogle Scholar
  37. [Glanz, 1997]
    Glanz, J. (1997). Mastering the nonlinear brain. Science, 277: 1758–1760.CrossRefGoogle Scholar
  38. [Glass and Mackey, 1979]
    Glass, L. and Mackey, M. C. (1979). Pathological conditions resulting from instabilities in physiological control systems. Ann. N. Y. Acad. Sci., 620: 22–44.Google Scholar
  39. [Gotman, 1983]
    Gotman, J. (1983). Measurement of small time differences between EEG channels method and application to epileptic seizure propagation. Electroenceph. Clin. Neurophysiol, 56: 501–514.CrossRefGoogle Scholar
  40. [Grotta-Ragazzo et al., 1999]
    Grotta-Ragazzo, C., Pakdaman, K., and Malta, C. P. (1999). Metastability for delayed differential equations. Phys. Rev. E, 60: 6230–6233.MathSciNetCrossRefGoogle Scholar
  41. [Guillouzic et al., 1999]
    Guillouzic, S., L’Heureux, I., and Longtin, A. (1999). Small delay approximation of stochastic delay differential equations. Phys. Rev. E, 59: 3970–3982.CrossRefGoogle Scholar
  42. [Guttman et al., 1980]
    Guttman, R., S., L., and Rinzel, J. (1980). Control of repetitive firing in squid axon membrane as a model for a neuron oscillator. J. Physiol., 305: 377–395.Google Scholar
  43. [Haken et al., 1985]
    Haken, H., Kelso, J. A. S., and Bunz, H. (1985). A theoretical model of phase transitions in human movement. Biol. Cybern., 53: 247–257.MathSciNetGoogle Scholar
  44. [Harris and Wolpert, 1998]
    Harris, C. M. and Wolpert, D. M. (1998). State-dependent noise determines motor planning. Nature (London), 394: 780–784.CrossRefGoogle Scholar
  45. [Heagy et al., 1994]
    Heagy, J. F., Platt, N., and Hammel, S. M. (1994). Characterization of on-off intermittency. Phys. Rev. Lett., 49: 1140–1150.Google Scholar
  46. [Hetling, 2002]
    Hetling, J. R. (2002). Prospects for building a therapeutic cortical stimulator. In Milton, J. and Jung, P., editors, Epilepsy as a dynamic disease, pages 323–339, New York. Springer-Verlag.Google Scholar
  47. [Horsthemenke and Lefever, 1984]
    Horsthemenke, J. and Lefever, R. (1984). Noise-induced transitions: Theory and applications in physics, chemistry and biology. Springer-Verlag, New York.Google Scholar
  48. [Hunter and Milton, 2001]
    Hunter, J. D. and Milton, J. G. (2001). Synaptic heterogeneity and stimulus induced modulation of depression in central synapses. J. Neurosci., 21: 1427–1438.Google Scholar
  49. [Hunter and Milton, 2002]
    Hunter, J. D. and Milton, J. G. (2002). Using inhibitory interneurons to control neural synchrony. In Milton, J. G. and Jung, P., editors, Epilepsy as a Dynamic Disease, pages 115–130, New York, Springer-Verlag.Google Scholar
  50. [Hunter and Milton, 2003]
    Hunter, J. D. and Milton, J. G. (2003). Amplitude and frequency dependence of spike timing: Implications for dynamic regulation. J. Neurophysiology 90: 387–394.CrossRefGoogle Scholar
  51. [Hunter et al., 1998]
    Hunter, J. D., Milton, J. G., Thomas, P.J., and Cowan, J. D. (1998). Resonance effect for neural spike time reliability. J. Neurophysiol, 80 (3): 1427–38.Google Scholar
  52. [Izhikevich, 2000]
    Izhikevich, E. M. (2000). Neural excitability, spiking, and bursting. Int. J. Bifurc. Chaos, 10: 1171–1266.MathSciNetMATHCrossRefGoogle Scholar
  53. [Jackson, 1931]
    Jackson, J. H. (1931). Selected Writings. Hodder & Soughton, London.Google Scholar
  54. [Jasper, 1969]
    Jasper, H. H. (1969). Mechanisms of propagation: Extracellular studies. In Jasper, H. H., Ward, A. A., and Pope, A., editors, Basic mechanisms of the epilepsies, pages 421–440, Boston. Little Brown.Google Scholar
  55. [Kadanoff, 1993]
    Kadanoff, L. P. (1993). From order to chaos, essays: Critical, chaotic and otherwise. World Scientific, Singapore.MATHCrossRefGoogle Scholar
  56. [Kelso, 1984]
    Kelso, J. A. S. (1984). Phase transitions and critical behavior in human bimanual coordination. Amer. J. Physiology: Regulation, integrative and comparative physiology, 15: R1000–R1004.Google Scholar
  57. [Kelso, 1999]
    Kelso, J. A. S. (1999). Dynamical patterns: The self-organization of brain and behavior. The MIT Press, Cambridge, MA.Google Scholar
  58. [Kelso et al., 1992]
    Kelso, J. A. S., Bressler, S. L., Buchanan, S., DeGuzman, G. C., Ding, M., Fuchs, A., and Holroyd, T. (1992). A phase transition in human brain and behavior. Phys. Lett. A, 169: 134–144.CrossRefGoogle Scholar
  59. [Knight, 1972]
    Knight, B. K. (1972). Dynamics of encoding in a population of neurons. J Gen. Physiol., 59: 734–766.CrossRefGoogle Scholar
  60. [Kopell, 1995]
    Kopell, N. (1995). Chains of coupled oscillators. In Arbib, M. A., editor, Brain theory and neural networks, pages 178–183, Cambridge, MA. MIT Press.Google Scholar
  61. [Kramers, 1940]
    Kramers, H. A. (1940). Brownian motion in a field of force and the diffusion model of chemical reactions. Physica, 7: 284–304.MathSciNetMATHCrossRefGoogle Scholar
  62. [Kruse and Stadler, 1995]
    Kruse, P. and Stadler, M., editors (1995). Ambiguity in Mind and Nature. Springer-Verlag, New York.MATHGoogle Scholar
  63. [Lechner et al., 1996]
    Lechner, H. A., Baxter, D. A., Clark, J. W., and Byrne, J. H. (1996). Bistability and its regulation by serotonin in the endogenously bursting neuron R15 of Aplysia. J. Neurophysiology, 75: 957–962.Google Scholar
  64. [Legrand and Sornette, 1990]
    Legrand, O. and Sornette, D. (1990). Coarse-grained properties of the chaotic trajectories in the stadium. Physica D, 44: 229–247.MathSciNetMATHCrossRefGoogle Scholar
  65. [Lesser et al., 1999]
    Lesser, R. P., Kim, S. H., Beyderman, L., Miglioretti, D. L., Webber, W. R. S., Bare, M., Cysyk, B., Krauss, G., and Gordon, B. (1999). Brief bursts of pulse stimulation terminate afterdischarges caused by cortical stimulation. Neurology, 53: 2073–2081.Google Scholar
  66. [Longtin, 1991]
    Longtin, A. (1991). Noise-induced transitions at a Hopf bifurcation in a first-order delay-differential equation. Phys. Rev. A, 44: 4801–4813.CrossRefGoogle Scholar
  67. [Longtin and Hinzer, 1996]
    Longtin, A. and Hinzer, K. (1996). Encoding with bursting, subthreshold oscillations, and noise in mammalian cold receptors. Neural Computation, 8: 215–255.CrossRefGoogle Scholar
  68. [Longtin et al., 1990]
    Longtin, A., Milton, J. G., Bos, J. E., and Mackey, M. C. (1990). Noise and critical behavior of the pupil light reflex at oscillation onset. Phys. Rev. A, 41: 6992–7005.CrossRefGoogle Scholar
  69. [Losson et al., 1993]
    Losson, J., Mackey, M. C., and Longtin, A. (1993). Solution multistability in first-order nonlinear differential delay equations. Chaos, 3: 167–176.MathSciNetMATHCrossRefGoogle Scholar
  70. [Mackey and an der Heiden, 1984]
    Mackey, M. C. and an der Heiden, U. (1984). The dynamics of recurrent inhibition. J. Math. Biol., 19: 211–225.MathSciNetMATHCrossRefGoogle Scholar
  71. [Mackey and Glass, 1977]
    Mackey, M. C. and Glass, L. (1977). Oscillations and chaos in physiological control systems. Science, 197: 287–289.CrossRefGoogle Scholar
  72. [Mackey and Milton, 1987]
    Mackey, M. C. and Milton, J. G. (1987). Dynamical diseases. Ann. N. E. Acad. Sci., 504: 16–32.CrossRefGoogle Scholar
  73. [Mackey and Milton, 1990]
    Mackey, M. C. and Milton, J. G. (1990). A deterministic approach to survival statistics. J. Math. Biol., 28: 33–48.MathSciNetMATHCrossRefGoogle Scholar
  74. [Manuca et al., 1998]
    Manuca, R., Casdagli, M., and Savit, R. (1998). Nonstationarity in epileptic EEG and implications for neural dynamics. Math. Biosci., 147: 1–22.MATHCrossRefGoogle Scholar
  75. [Matsumoto and Kunisawa, 1978]
    Matsumoto, G. and Kunisawa, T. (1978). Critical slowing-down near the transition region from the resting to time-ordered states in squid giant axons. J. Phys. Soc. Japan, 44: 1047–1048.CrossRefGoogle Scholar
  76. [Mehta and Schaal, 2002]
    Mehta, B. and Schaal, S. (2002). Forward models in visuomotor control. J Neurophysiology, 88: 942–953.Google Scholar
  77. [Meyer-Lindenberg et al., 2002]
    Meyer-Lindenberg, A., Ziemann, U., Hajak, G., Cohen, L., and Berman, K. (2002). Transitions between dynamical states of differing stability in the human brain. Proc Natl Acad Sci USA, 99: 10948–53.CrossRefGoogle Scholar
  78. [Miller, 1994]
    Miller, R. (1994). What is the contribution of axonal conduction delay to temporal structure in brain dynamics? In Pantev, C., editor, Oscillatory event-related brain dynamics, pages 53–57, New York. Plenum Press.Google Scholar
  79. [Milton, 2002]
    Milton, J. (2002). Insights into seizure propagation from axonal conduction times. In Milton, J. and Jung, P., editors, Epilepsy as a dynamic disease,pages 15–23, New York. springer-Verlag.Google Scholar
  80. [Milton and Black, 1995]
    Milton, J. and Black, D. (1995). Dynamic diseases in neurology and psychiatry. CHAOS, 5: 8–13.CrossRefGoogle Scholar
  81. [Milton and Foss, 1997]
    Milton, J. and Foss, J. (1997). Oscillations and multi-stability in delayed feedback control. In Othmer, H. G., Adler, F. R., Lewis, M. A., and Dallon, J. C., editors, Case studies in mathematical modeling: Ecology, physiology, and cell biology, pages 179–198, Upper Saddle River, New Jersey. Prentice Hall.Google Scholar
  82. [Milton and Jung, 2002]
    Milton, J. and Jung, P., editors (2002). Epilepsy as a dynamic disease. Springer-Verlag, New York.Google Scholar
  83. [Milton, 1996]
    Milton, J. G. (1996). Dynamics of small neural populations. American Mathematical Society, Providence, Rhode Island.MATHGoogle Scholar
  84. [Milton, 2000]
    Milton, J. G. (2000). Epilepsy: multistability in a dynamic disease. In Walleczek, J., editor, Self-organized biological dynamics and nonlinear control, pages 374–386, New York. Cambridge University Press.CrossRefGoogle Scholar
  85. [Milton et al., 1990]
    Milton, J. G., an der Heiden, U., Longtin, A., and Mackey, M. C. (1990). Complex dynamics and noise in simple neural networks with delayed mixed feedback. Biomed. Biochim. Acta, 49: 697–707.Google Scholar
  86. [Milton et al., 1993]
    Milton, J. G., Chu, P. H., and Cowan, J. D. (1993). Spiral waves in integrate-and-fire neural networks. In Hanson, S. J., Cowan, J. D., and Giles, C. L., editors, Advances in Neural Information Processing Systems 5, pages 1001–1007, San Mateo, CA. Morgan Kaufmann.Google Scholar
  87. [Milton et al., 1987]
    Milton, J. G., Gotman, J., Remillard, G. M., and Andermann, F. (1987). Timing of seizure recurrence in adult epileptic patients: a statistical analysis. Epilepsia, 28: 471–478.CrossRefGoogle Scholar
  88. [Milton et al., 1989]
    Milton, J. G., Longtin, A., Reuter, A., Mackey, M. C., and Glass, L. (1989). Complex dynamics and bifurcations in neurology. J. theoret. Biol., 138: 129–147.MathSciNetCrossRefGoogle Scholar
  89. [Milton and Mackey, 2000]
    Milton, J. G. and Mackey, M. C. (2000). Neural ensemble coding and statistical periodicity: Speculations on the operation of the mind’s eye. J. Physiol. (Paris), 94: 489–503.CrossRefGoogle Scholar
  90. [Mоеll, 1985]
    Morrell, F. (1985). Secondary epileptogenesis in man. Arch. Neurol., 42: 318–335.Google Scholar
  91. [Moss et al., 1994]
    Moss, F., Pierson, D., and O’Gorman, D. (1994). Stochastic resonance: tutorial and update. Int. J Bifurc. Chaos, 4: 1383–1397.MathSciNetMATHCrossRefGoogle Scholar
  92. [Motamedi et al., 2002]
    Motamedi, G. K., Lesser, R. P., Miglioretti, D. L., Mizuno-Matsumo, Y., Gordon, B., Webber, W. R. S., Jackson, D. C., Sepkuty, J. P., and Crone, N. E. (2002). Optimizing parameters for terminating cortical afterdischarges with pulse stimulation. Epilepsia, 43: 836–846.CrossRefGoogle Scholar
  93. [Ohira and Yamane, 2000]
    Ohira, T. and Yamane, T. (2000). Delayed stochastic systems. Phys. Rev. E, 61: 1247–1257.CrossRefGoogle Scholar
  94. [Pakdaman et al., 1998]
    Pakdaman, K., Grotta-Ragazzo, C., Malta, C. P., Arino, O., and Vibert, J.-F. (1998). Effect of delay on the boundary of the basin of attraction in a system of two neurons. Neural Networks, 11: 509–519.CrossRefGoogle Scholar
  95. [Penfield and Jasper, 1954]
    Penfield, W. and Jasper, H. (1954). Epilepsy and the functional anatomy of the human brain. Churchill, London.Google Scholar
  96. [Platt et al., 1993]
    Platt, N., Spiegel, E. A., and Tresser, C. (1993). On-off intermittency: A mechanism for bursting. Phys. Rev. Lett., 70: 279–282.CrossRefGoogle Scholar
  97. [Pomeau and Manneville, 1980]
    Pomeau, Y. and Manneville, P. (1980). Intermittent transition to turbulence in dissipative systems. Commun. Math. Phys., 74: 189–197.MathSciNetCrossRefGoogle Scholar
  98. [Rinzel and Baer, 1988]
    Rinzel, J. and Baer, S. M. (1988). Threshold for repetitive activity for a low stimulus RAMP: A memory effect and its dependence on fluctuations. Biophys. J, 54: 551–555.CrossRefGoogle Scholar
  99. [S. A. Campbell et al., 1995]
    S. A. Campbell, S. A., Bélair, J., Ohira, T., and Milton, J. G. (1995). Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback. CHAOS, 5: 640–645.MATHGoogle Scholar
  100. [Schiff et al., 1994]
    Schiff, S. J., Jerger, K., Duong, D. H., Chay, T., Spano, M. L., and Ditto, W. L. (1994). Controlling chaos in the brain. Nature (London), 370: 615–620.CrossRefGoogle Scholar
  101. [Schöner et al., 1986]
    Schöner, G., Haken, H., and Kelso, J. A. S. (1986). A stochastic theory of phase transitions in human movement. Biol. Cyner., 53: 247–257.MATHGoogle Scholar
  102. [Schwartzkroin and McIntyre, 1997]
    Schwartzkroin, P. A. and McIntyre, D. C. (1997). Limbic anatomy and physiology. In Engel, Jr., J. and Pedley, T. A., editors, Epilepsy: A comprehensive textbook, pages 323–340, Philadelphia. Lippincott-Raven.Google Scholar
  103. [Segev et al., 2002]
    Segev, R., Benveniste, M., Hulata, E., Cohen, N., Palevski, A., Kapon, E., Shapira, Y., and Ben–Jacob, E. (2002). Long term behavior of lithographically prepared in vitro neuronal networks. Phys. Rev. Lett., 88:118102–1–4.Google Scholar
  104. [Spano et al., 2002]
    Spano, M. L., Ditto, W. L., Moss, F., and Dolan, K. (2002). Unstable periodic orbits (UPOs) and chaos control in neural systems. In Milton, J. and Jung, P., editors, Epilepsy as a dynamic disease, pages 297–322, New York. Springer-Verlag.Google Scholar
  105. [Stanley et al., 1998]
    Stanley, H. E., Amaral, L. A. N., Andrade, J. S., Buldyrev, S. V., Havlin, S., Makse, H. A., Peng, C.-K., Suki, B., and Viswanathan, G. (1998). Scale-invariant correlations in the biological and social sciences. Phil. Mag. B, 77: 1373–1388.Google Scholar
  106. [Stark et al., 1958]
    Stark, L., Campbell, F. W., and Atwood, J. (1958). Pupil-lary unrest: An example of noise in a biological servo-mechanism. Nature (London), 182: 857–858.CrossRefGoogle Scholar
  107. [Stépàn, 1989]
    Stépàn, G. (1989). Retarded dynamical systems: Stability and characteristic functions, volume 210 of Pitman Research Notes in Mathematics Series. Wiley & Sons, New York.Google Scholar
  108. [Venkataramani et al., 1996]
    Venkataramani, S. C., Antonsen, T. M., ott, E., and Sommerer, J. C. (1996). On-off intermittency: Power spectra and fractal properties of time series. Physica D, 96: 66–99.MathSciNetMATHCrossRefGoogle Scholar
  109. [Verveen and DeFelice, 1974]
    Verveen, A. A. and DeFelice, L. J. (1974). Membrane noise. Prog. Biophys. Mol. Biol., 28: 253–264.CrossRefGoogle Scholar
  110. [Viswanathan et al., 1996]
    Viswanathan, G. M., Afanasyev, V., Buldyrev, S. V., Murphy, E. J., and Stanley, H. E. (1996). Lévy search patterns of wandering albatrosses. Nature (London), 381: 413–415.CrossRefGoogle Scholar
  111. [Winfree, 1980]
    Winfree, A. T. (1980). The geometry of biological time. Springer-Verlag, New York.MATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • John G. Milton
    • 1
  • Jennifer Foss
    • 2
  • John D. Hunter
    • 1
  • Juan Luis Cabrera
    • 3
  1. 1.Department of NeurologyUniversity of ChicagoUSA
  2. 2.Department of PsychologyUniversity of New OrleansUSA
  3. 3.Centro de FísicaIVIC VenezuelaVenezuela

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