Free Access
Issue
Math. Model. Nat. Phenom.
Volume 11, Number 3, 2016
Anomalous diffusion
Page(s) 142 - 156
DOI https://doi.org/10.1051/mmnp/20161139
Published online 21 June 2016
  1. M. Jucker, L. C. Walker, Pathogenic protein seeding in Alzheimer disease and other neurodegenerative disorders, Ann Neurol, vol. 70, no. 4, pp. 532–540, 2011. [CrossRef] [PubMed] [Google Scholar]
  2. S. B. Prusiner, Novel proteinaceous infectious particles cause scrapie, Science, vol. 216, no. 4542, pp. 136–144, 1982. [CrossRef] [PubMed] [Google Scholar]
  3. ——, Prion biology and diseases.Cold Spring Harbor Laboratory Press, 2004. [Google Scholar]
  4. K. Brown, J. A. Mastrianni, The prion diseases, J. Geriatr. Psychiatry Neurol., vol. 23, no. 4, pp. 277–298, 2010. [CrossRef] [PubMed] [Google Scholar]
  5. A. Aguzzi, L. Rajendran, The transcellular spread of cytosolic amyloids, prions, and prionoids, Neuron, vol. 64, no. 6, pp. 783–790, 2009. [CrossRef] [PubMed] [Google Scholar]
  6. M. Cushman, B. S. Johnson, O. D. King, A. D. Gitler, J. Shorter, Prion-like disorders: blurring the divide between transmissibility and infectivity, J. Cell Sci., vol. 123, no. 8, pp. 1191–1201, 2010. [CrossRef] [PubMed] [Google Scholar]
  7. C. Duyckaerts, B. Delatour, M.-C. Potier, Classification and basic pathology of Alzheimer disease, Acta Neuropathol., vol. 118, no. 1, pp. 5–36, 2009. [CrossRef] [PubMed] [Google Scholar]
  8. H. Braak, K. Del Tredici, Nervous system pathology in sporadic Parkinson disease, Neurol., vol. 70, no. 20, pp. 1916–1925, 2008. [CrossRef] [Google Scholar]
  9. L. C. Walker, H. LeVine, Corruption and spread of pathogenic proteins in neurodegenerative diseases, J. Bio. Chem., vol. 287, no. 40, pp. 33109–33115, 2012. [CrossRef] [Google Scholar]
  10. M. Jucker and L. C. Walker, Self-propagation of pathogenic protein aggregates in neurodegenerative diseases, Nature, vol. 501, no. 7465, pp. 45–51, 2013. [CrossRef] [PubMed] [Google Scholar]
  11. H. Baker, R. Ridley, L. Duchen, T. Crow, C. Bruton, Evidence for the experimental transmission of cerebral beta-amyloidosis to primates. Int. J Exp. Pathol., vol. 74, no. 5, p. 441, 1993. [PubMed] [Google Scholar]
  12. R. Ridley, H. Baker, C. Windle, R. Cummings, Very long term studies of the seeding of β-amyloidosis in primates, J Neural Transm., vol. 113, no. 9, pp. 1243–1251, 2006. [CrossRef] [PubMed] [Google Scholar]
  13. J. Goudsmit, C. H. Morrow, D. M. Asher, R. T. Yanagihara, C. L. Masters, C. J. Gibbs, D. C. Gajdusek, Evidence for and against the transmissibility of Alzheimer disease, Neurology, vol. 30, no. 9, pp. 945–945, 1980. [CrossRef] [PubMed] [Google Scholar]
  14. S. B. Prusiner, Prions, Proc. Nat. Aca. Sci., vol. 95, no. 23, pp. 13363–13383, 1998. [CrossRef] [Google Scholar]
  15. S.-J. Lee, P. Desplats, C. Sigurdson, I. Tsigelny, E. Masliah, Cell-to-cell transmission of non-prion protein aggregates, Nature Reviews Neurology, vol. 6, no. 12, pp. 702–706, 2010. [CrossRef] [PubMed] [Google Scholar]
  16. C. Saper, B. Wainer, D. German, Axonal and transneuronal transport in the transmission of neurological disease: potential role in system degenerations, including Alzheimer’s disease, Neuroscience, vol. 23, no. 2, pp. 389–398, 1987. [CrossRef] [PubMed] [Google Scholar]
  17. H. Braak, E. Braak, Neuropathological stageing of Alzheimer-related changes, Acta Neuropathol., vol. 82, no. 4, pp. 239–259, 1991. [CrossRef] [PubMed] [Google Scholar]
  18. D. R. Thal, U. Rüb, M. Orantes, and H. Braak, Phases of Aβ-deposition in the human brain and its relevance for the development of ad, Neurology, vol. 58, no. 12, pp. 1791–1800, 2002. [CrossRef] [PubMed] [Google Scholar]
  19. H. Braak, K. Del Tredici, U. Rüb, R. A. de Vos, E. N. J. Steur, E. Braak, Staging of brain pathology related to sporadic Parkinson?s disease, Neurobiology of aging, vol. 24, no. 2, pp. 197–211, 2003. [Google Scholar]
  20. J. M. Ravits and A. R. LaSpada, ALS motor phenotype heterogeneity, focality, and spread deconstructing motor neuron degeneration, Neurology, vol. 73, no. 10, pp. 805–811, 2009. [CrossRef] [PubMed] [Google Scholar]
  21. J. Brettschneider, K. Del Tredici, J. B. Toledo, J. L. Robinson, D. J. Irwin, M. Grossman, E. Suh, V. M. Deerlin, E. M. Wood, Y. Baek et al., Stages of pTDP-43 pathology in amyotrophic lateral sclerosis, Annals of neurology, vol. 74, no. 1, pp. 20–38, 2013. [CrossRef] [PubMed] [Google Scholar]
  22. M. Goedert, M. G. Spillantini, K. Del Tredici, H. Braak, 100 years of Lewy pathology, Nature Reviews Neurology, vol. 9, no. 1, pp. 13–24, 2013. [CrossRef] [PubMed] [Google Scholar]
  23. J. Liu, L. Zhao, J. Nan, G. Li, S. Xiong, K. M. von Deneen, Q. Gong, F. Liang, W. Qin, J. Tian, The trade-off between wiring cost and network topology in white matter structural networks in health and migraine, Experimental neurology, vol. 248, pp. 196–204, 2013. [CrossRef] [PubMed] [Google Scholar]
  24. D. Le Bihan, J. F. Mangin, C. Poupon, C. Clark, S. Pappata, N. Molko, H. Chabriat, Diffusion tensor imaging: concepts and applications, J. Magn. Reson. Im., vol. 13, no. 4, pp. 534–546, 2001. [CrossRef] [Google Scholar]
  25. A. Raj, A. Kuceyeski, M. Weiner, A network diffusion model of disease progression in dementia, Neuron, vol. 73, no. 6, pp. 1204–1215, 2012. [CrossRef] [PubMed] [Google Scholar]
  26. E. Syková, C. Nicholson, Diffusion in brain extracellular space, Physio. Rev., vol. 88, no. 4, pp. 1277–1340, 2008. [CrossRef] [Google Scholar]
  27. E. Özarslan, P. J. Basser, T. M. Shepherd, P. E. Thelwall, B. C. Vemuri, S. J. Blackband, Observation of anomalous diffusion in excised tissue by characterizing the diffusion-time dependence of the MR signal, J. Magnet. Reson., vol. 183, no. 2, pp. 315–323, 2006. [CrossRef] [Google Scholar]
  28. E. Özarslan, T. M. Shepherd, C. G. Koay, S. J. Blackband, P. J. Basser, Temporal scaling characteristics of diffusion as a new MRI contrast: findings in rat hippocampus, Neuroimage, vol. 60, no. 2, pp. 1380–1393, 2012. [CrossRef] [PubMed] [Google Scholar]
  29. D. Le Bihan, E. Breton, D. Lallemand, P. Grenier, E. Cabanis, M. Laval-Jeantet et al., MR imaging of intravoxel incoherent motions: application to diffusion and perfusion in neurologic disorders, Radiology, vol. 161, no. 2, pp. 401–407, 1986. [CrossRef] [PubMed] [Google Scholar]
  30. R. Sener, Diffusion MRI: apparent diffusion coefficient (ADC) values in the normal brain and a classification of brain disorders based on ADC values, Comput. Med. Imag. Grap., vol. 25, no. 4, pp. 299–326, 2001. [CrossRef] [Google Scholar]
  31. R. V. Mulkern, H. Gudbjartsson, C.-F. Westin, H. P. Zengingonul, W. Gartner, C. R. Guttmann, R. L. Robertson, W. Kyriakos, R. Schwartz, D. Holtzman et al., Multi-component apparent diffusion coefficients in human brain, NMR Biomed., vol. 12, pp. 51–62, 1999. [CrossRef] [PubMed] [Google Scholar]
  32. K. M. Bennett, K. M. Schmainda, D. B. Rowe, H. Lu, J. S. Hyde et al., Characterization of continuously distributed cortical water diffusion rates with a stretched-exponential model, Magnet. Reson. Med., vol. 50, no. 4, pp. 727–734, 2003. [CrossRef] [Google Scholar]
  33. M. G. Hall, T. R. Barrick, From diffusion-weighted MRI to anomalous diffusion imaging, Magnet. Reson. Med., vol. 59, no. 3, pp. 447–455, 2008. [CrossRef] [Google Scholar]
  34. S. De Santis, A. Gabrielli, M. Bozzali, B. Maraviglia, E. Macaluso, S. Capuani, Anisotropic anomalous diffusion assessed in the human brain by scalar invariant indices, Magnet. Reson. Med., vol. 65, no. 4, pp. 1043–1052, 2011. [CrossRef] [Google Scholar]
  35. M. G. Hall, T. R. Barrick, Two-step anomalous diffusion tensor imaging, NMR in Biomed., vol. 25, no. 2, pp. 286–294, 2012. [CrossRef] [Google Scholar]
  36. J. H. Jensen, J. A. Helpern, MRI quantification of non-Gaussian water diffusion by kurtosis analysis, NMR in Biomed., vol. 23, no. 7, pp. 698–710, 2010. [CrossRef] [Google Scholar]
  37. R. L. Magin, O. Abdullah, D. Baleanu, X. J. Zhou, Anomalous diffusion expressed through fractional order differential operators in the Bloch–Torrey equation, J. Magnet. Reson., vol. 190, no. 2, pp. 255–270, 2008. [Google Scholar]
  38. X. J. Zhou, Q. Gao, O. Abdullah, R. L. Magin, Studies of anomalous diffusion in the human brain using fractional order calculus, Magnet. Reson. Med., vol. 63, no. 3, pp. 562–569, 2010. [CrossRef] [Google Scholar]
  39. B. Fevrier, D. Vilette, F. Archer, D. Loew, W. Faigle, M. Vidal, H. Laude, G. Raposo, Cells release prions in association with exosomes, P. Natl. Acad. Sci. USA, vol. 101, no. 26, pp. 9683–9688, 2004. [CrossRef] [Google Scholar]
  40. A. C. Magalhães, G. S. Baron, K. S. Lee, O. Steele-Mortimer, D. Dorward, M. A. Prado, B. Caughey, Uptake and neuritic transport of scrapie prion protein coincident with infection of neuronal cells, J. Neurosci., vol. 25, no. 21, pp. 5207–5216, 2005. [CrossRef] [PubMed] [Google Scholar]
  41. R. Hilfer, L. Anton, Fractional master equations and fractal time random walks, Phys. Rev. E, vol. 51, p. R848, 1995. [Google Scholar]
  42. R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., vol. 339, no. 1, pp. 1–77, 2000. [NASA ADS] [CrossRef] [Google Scholar]
  43. E. Barkai, R. Metzler, J. Klafter, From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E, vol. 61, no. 1, p. 132, 2000. [CrossRef] [MathSciNet] [Google Scholar]
  44. I. M. Sokolov, J. Klafter, Field-induced dispersion in subdiffusion, Physical review letters, vol. 97, no. 14, p. 140602–1, 2006. [CrossRef] [PubMed] [Google Scholar]
  45. B. I. Henry, T. A. M. Langlands, P. Straka, Fractional Fokker-Planck equations for subdiffusion with space- and time-dependent forces, Phys. Rev. Lett., vol. 105, no. 17, p. 170602, 2010. [CrossRef] [PubMed] [Google Scholar]
  46. B. I. Henry, T. A. M. Langlands, S. L. Wearne, Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations, Phys. Rev. E, vol. 74, no. 3, p. 031116, 2006. [CrossRef] [MathSciNet] [Google Scholar]
  47. S. Fedotov, Non-Markovian random walks and nonlinear reactions: subdiffusion and propagating fronts, Physical Review E, vol. 81, no. 1, p. 011117, 2010. [CrossRef] [Google Scholar]
  48. C. N. Angstmann, I. C. Donnelly, B. I. Henry, Continuous time random walks with reactions, forcing and trapping, Math. Model. Nat. Phenom., vol. 8, pp. 17–27, 2013. [CrossRef] [EDP Sciences] [Google Scholar]
  49. A. A. Nepomnyashchy, V. A. Volpert, An exactly solvable model of subdiffusion–reaction front propagation, Journal of Physics A: Mathematical and Theoretical, vol. 46, no. 6, p. 065101, 2013. [Google Scholar]
  50. V. A. Volpert, Y. Kanevsky, A. A. Nepomnyashchy, Propagation failure for a front between stable states in a system with subdiffusion, Phys. Rev. E, vol. 89, p. 012901, 2014. [CrossRef] [Google Scholar]
  51. B. Henry, T. Langlands, S. Wearne, Fractional cable models for spiny neuronal dendrites, Physical review letters, vol. 100, no. 12, p. 128103, 2008. [Google Scholar]
  52. T. A. M. Langlands, B. I. Henry, S. L. Wearne, Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions, Journal of mathematical biology, vol. 59, no. 6, pp. 761–808, 2009. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  53. T. Langlands, B. Henry, S. L. Wearne, Fractional cable equation models for anomalous electrodiffusion in nerve cells: finite domain solutions, SIAM Journal on Applied Mathematics, vol. 71, no. 4, pp. 1168–1203, 2011. [CrossRef] [Google Scholar]
  54. C. Ingo, R. L. Magin, L. Colon-Perez, W. Triplett, T. H. Mareci, On random walks and entropy in diffusion-weighted magnetic resonance imaging studies of neural tissue, Magnet. Reson. Med., vol. 71, no. 2, pp. 617–627, 2014. [CrossRef] [Google Scholar]
  55. C. N. Angstmann, I. C. Donnelly, B. I. Henry, Pattern formation on networks with reactions: A continuous-time random-walk approach, Phys. Rev. E, vol. 87, p. 032804, 2013. [CrossRef] [Google Scholar]
  56. C. N. Angstmann, I. C. Donnelly, B. I. Henry, T. A. M. Langlands, Continuous-time random walks on networks with vertex- and time-dependent forcing, Phys. Rev. E, vol. 88, p. 022811, 2013. [CrossRef] [Google Scholar]
  57. F. E. Cohen, K.-M. Pan, Z. Huang, M. Baldwin, R. J. Fletterick, S. B. Prusiner, Structural clues to prion replication, Science (New York, NY), vol. 264, no. 5158, p. 530, 1994. [CrossRef] [Google Scholar]
  58. T. A. M. Langlands, B. I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J Comp. Phys., vol. 205, no. 2, pp. 719–736, 2005. [CrossRef] [Google Scholar]
  59. A. L. Alexander, J. E. Lee, M. Lazar, A. S. Field, Diffusion tensor imaging of the brain, Neurotherapeutics, vol. 4, no. 3, pp. 316–329, 2007. [CrossRef] [Google Scholar]
  60. A. Erickson, B. Henry, P. Klaase, C. Angstmann, Predicting first traversal times for virions and nanoparticles in mucus with slowed diffusion, Biophysical Journal, vol. 109, pp. 164–172, 2015. [CrossRef] [PubMed] [Google Scholar]
  61. H. J. Haubold, A. M. Mathai, R. K. Saxena, Mittag-leffler functions and their applications, Journal of Applied Mathematics, vol. 2011, 2011. [CrossRef] [Google Scholar]
  62. J. A. Brown, J. D. Rudie, A. Bandrowski, J. D. VanHorn, S. Y. Bookheimer, The UCLA Multimodal Connectivity Database: A web-based platform for brain connectivity matrix sharing and analysis, Frontiers in Neuroinformatics, vol. 6, 28, 2012.no. [CrossRef] [Google Scholar]
  63. J.D. Rudie, J. Brown, D. Beck-Pancer, L. Hernandez, E. Dennis, P. Thompson, S. Bookheimer, M. Dapretto, Altered functional and structural brain network organization in autism, NeuroImage: Clinical, vol. 2, pp. 79–94, 2013. [CrossRef] [Google Scholar]
  64. K. B. Oldham and J. Spanier, The Fractional Calculus:Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, 1974. [Google Scholar]

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