Free Access
Issue
Math. Model. Nat. Phenom.
Volume 5, Number 2, 2010
Mathematics and neuroscience
Page(s) 185 - 207
DOI https://doi.org/10.1051/mmnp/20105207
Published online 10 March 2010
  1. C. Ajmone Marsan. Focal electrical stimulation. In: Experimental Models of Epilepsy: A manual for the laboratory worker. Eds D. P. Purpura, J. K. Penry, D. Tower, D. M. Woodbury and R. Walter, Raven Press, New York, 1972.
  2. S. Amari. Dynamics of pattern formation in lateral-inhibition type neural fields. Biol. Cybern., 27 (1977), No. 2, 77–87. [CrossRef] [PubMed]
  3. F. AtayA. Hutt. Stability and bifurcations in neural fields with finite propagation speed and general connectivity. SIAM J. Appl. Math., 65 (2005), No. 2, 644–666.
  4. U. B. Barnikol, O. V. Popovych, C. Hauptmann, V. Sturm, H. J. FreundP. A. Tass. Tremor entrainment by patterned low-frequency stimulation. Philos. Transact. A Math. Phys. Eng. Sci., 366 (2008), No. 1880, 3543–3573.
  5. R. Bartolow. Experimental investigations into the functions of the human brain. AM. J. Med. Sci., 1874, 305–313. [CrossRef]
  6. N. P. Bechtereva, A. N. BondarchukV. M. Smirnov. Therapeutic electrostimulations of deep brain structures. Vopr Neirokhir, 1 (1972), 115–120.
  7. A. L. Benabid, P. Pollak, A. Louveau, S. HenryJ. de Rougemont. Combined (thalamotomy and stimulation) stereotactic surgery of the Vim thalamic nucleus for bilateral Parkinson disease. Appl. Neurophysiol., 50 (1987), No. 1-6, 344–346. [PubMed]
  8. A. L. Benabid, W. Bradley, J. Mitrofanis, C. Xia, B. Piallat, V. Fraix, A. Batir, P. Krack, P. PollakF. Berger. Therapeutic electrical stimulation of the central nervous system. C. R. Biologies, 328 (2005), 177–186. [CrossRef]
  9. S. A. Chkhenkeli. Direct deep brain stimulation: first steps toward the feedback control of seizures. In: Epilepsy as a dynamical disease, p. 249-262. Eds J. Milton and P. Jung, Springer-Verlag, New York, 2003.
  10. J. Echauz, H. Firpi, G. Georgoulas. Intelligent control strategies for neurostimulation. In: Applications of intelligent control of engineering systems. Ed P. K. Valavanis, Springer, 2009.
  11. R. Edwards. Approximation of neural network dynamics by reaction-diffusion equations. Math. Meth. App. Sci., 19 (1996), 651–677. [CrossRef]
  12. G. B. ErmentroutJ. D. Cowan. A mathematical theory of visual hallucination patterns. Biol. Cybern., 34 (1979), No. 3, 137–150. [CrossRef] [PubMed]
  13. A. Eusebio, A. Pogosyan, S. Wang, B. Averbeck, L. D. Gaynor, S. Cantiniaux, T. Witjas, P. Limousin, J. P. AzulayP. Brown. Resonance in subthalamo-cortical circuits in Parkinson’s disease. Brain, 132 (2009), No. 8, 2139–2150. [CrossRef] [PubMed]
  14. W. Gerstner, R. Kempter, J. L. van HemmenH. Wagner. A neuronal learning rule for sub-millisecond temporal coding. Nature, 383 (1996), 76–81. [CrossRef] [PubMed]
  15. F. A. Gibbs, E. L. GibbsW. G. Lennox. The likeness of the cortical dysrhythmias of schizophrenia and psychomotor epilepsy. Am. J. Psychiatry, 95 (1938), 255–269.
  16. P. L. Gildenberg. History of electrical neuromodulation for chronic pain. Pain Medicine, 7 (2006), S7–S13. [CrossRef]
  17. B. J. Gluckman, E. J. Neel, T. I. Neto, W. L. Ditto, M. L. SpanoS. J. Schiff. Electric field suppression of epileptiform activity in hippocampal slices. J. Neurophysiol., 6 (1996), 4202–4205.
  18. B. J. Gluckman, H. Nguyen, S. L. WeinsteinS. J. Schiff. Adaptive electric field control of epileptic seizures. J. Neurosci., 21 (2001), No. 2, 290–600.
  19. S. Grillner, A. KozlovJ. H. Kotaleski. Integrative neuroscience: linking levels of analyses. Curr. Opin. Neurobiol., 15 (2005), No. 5, 614–621. [CrossRef] [PubMed]
  20. R. Hassler, F. MundigerT. Riechert. Correlations between clinical and autoptic findings in stereotaxic operations in parkinsonism. Confin. Neurol., 26 (1965), 282–290. [PubMed]
  21. A. L. HodgkinA. F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol., 117 (1952), No. 4, 500–544. [CrossRef] [PubMed]
  22. J. C. Horton, D. L. Adams. The cortical column: a structure without a function. Phil. Trans. of the Royal Soc. B, 360 (2005), No. 1456, 837–862. [CrossRef]
  23. X. Huang, W. C. Troy, Q. Yang, H. Ma, C. R. Laing, S. J. SchiffJ. Y. Wu. Spiral waves in disinhibited mammalian neocortex. J. Neurosci., 24 (2004), 9897–9902. [CrossRef] [PubMed]
  24. E. M. Izhikevich. Simple model of spiking neurons. Transactions on Neural Networks, 14 (2003), 1569–1572. [CrossRef] [PubMed]
  25. E. M. Izhikevich. Polychronization: computation with spikes. Neural Computation, 18 (2006), 245–282. [CrossRef] [MathSciNet] [PubMed]
  26. H. H. Jasper. Recording from microelectrodes in stereotactic surgery for Parkinson’s disease. J. Neurosurg., 24 (1966), 219–221.
  27. E. I. Kandel. Functional and stereotactic neurosurgery. Plenum Medical Book Co, New York, 1966.
  28. R. R. Llinas, U. Ribary, D. Jeanmonod, E. Kronberg, P. P. Mitra. Thalamocortical dysrhythmia: a neurological and neuropsychiatric syndrome characterized by magnetoencephalography. Proc. Natl. Acad. Sci. USA, 96 (1999), No 26, 15222–15227. [CrossRef] [PubMed]
  29. H. O. Lüders. Deep brain stimulation and epilepsy. Martin Dunitz, New York, 2004.
  30. C. C. McIntyre, S. Mori, D. L. Sherman, N. V. ThakorJ. L. Vitek. Electric field and stimulating influence generated by deep brain stimulation of the subthalamic nucleus. Clin. Neurophysiol., 115 (2004), No. 3, 589–595. [CrossRef] [PubMed]
  31. W. Meissner, A. Leblois, D. Hansel, B. Bioulac, C. E. Gross, A. BenazzouzT. Boraud. Subthalamic high frequency stimulation resets subthalamic firing and reduces abnormal oscillations. Brain, 128 (2005), No. 10, 2372–2382. [CrossRef] [PubMed]
  32. JMilton, P. Jung. Epilepsy as a dynamical disease. Springer-Verlag, New York, 2003.
  33. J. Modolo, J. HenryA. Beuter. Dynamics of the subthalamo-pallidal complex in Parkinson’s disease during deep brain stimulation. J. Biol. Phys., 34 (2008), No. 3-4, 351–366.
  34. J. Modolo, A. Beuter. Contribution of cortical inputs to subthalamic activity during deep brain stimulation. Proceedings of the Neurocomp 2008 conference, Marseille, France (2008).
  35. J. ModoloA. Beuter. Linking brain dynamics, neural mechanisms and deep brain stimulation in Parkinson’s disease: an integrated perspective. Med. Eng. Phys., 31 (2009), 615–623. [CrossRef] [PubMed]
  36. D. Q. NykampD. Tranchina. A population density approach that facilitates largescale modeling of neural networks : analysis and an application to orientation tuning. J. Comput. Neurosci., 8 (2000), No. 1, 19–50. [CrossRef] [PubMed]
  37. J. Olszewski. The thalamus of the Macaca Mulatta. An atlas for use with the stereotactic instrument. Basel Karger, 1952.
  38. A. Omurtag, B. W. Knight, L. Sirovich. On the simulation of large populations of neurons. J. Comput. Neurosci., 8 (2000), No. 5, 51–63. [CrossRef] [PubMed]
  39. A. Pascual, J. Modolo, A. Beuter. Is a computational model useful to understand the effect of deep brain stimulation in Parkinson’s disease? J. Integr. Neurosci., 5 (2006), No. 4, 541–559. [CrossRef] [PubMed]
  40. J. Richmond. The 3Rs-Past, present and future. Scand. J. Lab. Anim. Sci., 27 (2000), 84–92.
  41. J. E. RubinD. Terman. High frequency stimulation of the subthalamic nucleus eliminates pathological thalamic rhythmicity in a computational model. J. Comput. Neurosci., 16 (2004), No. 3, 211–235. [CrossRef] [PubMed]
  42. D. Rubino, K. A. RobbinsN. G. Hatsopoulos. Propagating waves mediate information transfer in the motor cortex. Nature Neurosci., 9 (2006), No. 12, 1549–1557. [CrossRef]
  43. J. D. SpeelmanD. A. Bosch. Resurgence of functional neurosurgery for Parkinson’s disease: a historical perspective. Mov. Disord., 13 (1998), No. 3, 582–588. [CrossRef] [PubMed]
  44. E. A. Spiegel, H. T. Wycis, M. MarksA. S. Lee. Stereotaxic apparatus for operations on the human brain. Science, 106 (1947), 349–350. [CrossRef] [PubMed]
  45. A. A. Spiegel, H. T. Wycis. Stereoencephalotomy (thalamic related procedures) part 1: Methods and atlas for the human brain. Grune and Stratton, New York, 1952.
  46. P. A. Tass. Phase Resetting in Medicine and Biology. Stochastic Modelling and Data Analysis. Series: Springer Series in Synergetics, 1999.
  47. D. Terman, J. E. Rubin, A. C. YewC. J. Wilson. Activity patterns in a model for the subthalamopallidal network of the basal ganglia. J. Neurosci., 22 (2002), No. 7, 2963–2976. [PubMed]
  48. L. Timmermann, J. Gross, M. Dirks, J. Volkmann, H. J. FreundA. Schnitzler. The cerebral oscillatory network of parkinsonian resting tremor. Brain, 126 (2003), No. 1, 199–212. [CrossRef] [PubMed]
  49. L. Timmermann, E. Florin, C. Reck. Pathological cerebral oscillatory activity in Parkinson’s disease: a critical review on methods, data and hypotheses. Expert Rev. Med. Dev., 4 (2007), No 5, 651–61. [CrossRef]
  50. M. S. Titcombe, L. Glass, D. GuehlA. Beuter. Dynamics of Parkinsonian tremor during deep brain stimulation. Chaos, 11 (2001), No. 4, 766–773. [CrossRef] [PubMed]
  51. J. L. P. Velazquez. Brain, behaviour and mathematics: Are we using the right approaches? Physica D, 212 (2005), 161–182. [CrossRef] [MathSciNet]
  52. J. A. VilenskyS. Gilman. Horsley was the first to use electrical stimulation of the human cerebral cortex intraoperatively. Surg. Neurol., 58 (2002), 425–426. [CrossRef] [PubMed]
  53. H. R. WilsonJ. D. Cowan. A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik, 13 (1973), No. 2, 55–80. [CrossRef] [PubMed]
  54. T. WichmannM. R. Delong. Deep brain stimulation for neurologic and neuropsychiatric disorders. Neuron, 52 (2006), No. 1, 197–204. [CrossRef] [PubMed]
  55. A. Winfree. Are cardiac waves relevant to epileptic waves propagation? In: Epilepsy as a dynamical disease, p. 165-188. Eds J. Milton and P. Jung, Springer-Verlag, New York, 2003.
  56. J. S. Yeomans. Principles of Brain Stimulation. Oxford University Press, New York, 1990.

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