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Bulletin of Mathematical Biology

, Volume 80, Issue 7, pp 1849–1870 | Cite as

A Simple Mathematical Model Inspired by the Purkinje Cells: From Delayed Travelling Waves to Fractional Diffusion

Original Article

Abstract

Recently, several experiments have demonstrated the existence of fractional diffusion in the neuronal transmission occurring in the Purkinje cells, whose malfunctioning is known to be related to the lack of voluntary coordination and the appearance of tremors. Also, a classical mathematical feature is that (fractional) parabolic equations possess smoothing effects, in contrast with the case of hyperbolic equations, which typically exhibit shocks and discontinuities. In this paper, we show how a simple toy-model of a highly ramified structure, somehow inspired by that of the Purkinje cells, may produce a fractional diffusion via the superposition of travelling waves that solve a hyperbolic equation. This could suggest that the high ramification of the Purkinje cells might have provided an evolutionary advantage of “smoothing” the transmission of signals and avoiding shock propagations (at the price of slowing a bit such transmission). Although an experimental confirmation of the possibility of such evolutionary advantage goes well beyond the goals of this paper, we think that it is intriguing, as a mathematical counterpart, to consider the time fractional diffusion as arising from the superposition of delayed travelling waves in highly ramified transmission media. The case of a travelling concave parabola with sufficiently small curvature is explicitly computed. The new link that we propose between time fractional diffusion and hyperbolic equation also provides a novelty with respect to the usual paradigm relating time fractional diffusion with parabolic equations in the limit. This paper is written in such a way as to be of interest to both biologists and mathematician alike. In order to accomplish this aim, both complete explanations of the objects considered and detailed lists of references are provided.

Keywords

Mathematical models Deduction from basic principles Time fractional equations Wave equations Purkinje cells Dendrites Neuronal arbours 

Mathematics Subject Classification

35Q92 92B05 92C20 

Notes

Acknowledgements

It is a pleasure to thank Elena Saftenku and Fidel Santamaria for very interesting discussions on neural transmissions. We also thank the Referees for their very valuable comments. This work has been supported by the Australian Research Council Discovery Project “N.E.W. Nonlocal Equations at Work”.

References

  1. Abatangelo N, Valdinoci E (2019) Getting acquainted with the fractional Laplacian. To appear in Springer INdAM SerGoogle Scholar
  2. Allen M (2018) A nondivergence parabolic problem with a fractional time derivative. Differ Integral Equ 31(3–4):215–230MathSciNetMATHGoogle Scholar
  3. Allen M, Caffarelli L, Vasseur A (2016) A parabolic problem with a fractional time derivative. Arch Ration Mech Anal 221(2):603–630MathSciNetCrossRefMATHGoogle Scholar
  4. Anastasio TJ (1998) Nonuniformity in the linear network model of the oculomotor integrator produces approximately fractional-order dynamics and more realistic neuron behavior. Biol Cybern 79:377–391CrossRefMATHGoogle Scholar
  5. Appali R, van Rienen U, Heimburg T (2012) A comparison of the Hodgkin-Huxley model the soliton theory for the action potential in nerves. Adv Planar Lipid Bilayers Liposomes 16:275–298CrossRefMATHGoogle Scholar
  6. Bagley R (2007) On the equivalence of the Riemann–Liouville and the Caputo fractional order derivatives in modeling of linear viscoelastic materials. Fract Calc Appl Anal 10(2):123–126MathSciNetMATHGoogle Scholar
  7. Balanis CA (2012) Advanced engineering electromagnetics. Wiley, HobokenGoogle Scholar
  8. ben-Avraham D, Havlin S (2000) Diffusion and reactions in fractals and disordered systems. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  9. Blanco A, Moyano R, Vivo J, Flores-Acuña R, Molina A, Blanco C, Monterde JG (2006) Purkinje cell apoptosis in arabian horses with cerebellar abiotrophy. J Vet Med Physiol Pathol Clin Med 53(6):286–287CrossRefGoogle Scholar
  10. Bucur C (2017) Local density of Caputo-stationary functions in the space of smooth functions. ESAIM Control Optim Calc Var 23(4):1361–1380MathSciNetCrossRefMATHGoogle Scholar
  11. Bucur C, Valdinoci E (2016) Nonlocal diffusion and applications, vol 20. Lecture Notes of the Unione Matematica Italiana. Springer, BolognaMATHGoogle Scholar
  12. Caputo M (1967) Linear model of dissipation whose \(Q\) is almost frequency independent-II. Geophys J R Astron Soc 13(5):529–539CrossRefGoogle Scholar
  13. Coombes S (2006) Neural fields. Scholarpedia 1(6):1373CrossRefGoogle Scholar
  14. Dáger R, Zuazua E (2006) Wave propagation, observation and control in 1-d flexible multi-structures, vol 50. Mathématiques & applications. Springer-Verlag, BerlinCrossRefMATHGoogle Scholar
  15. Diethelm K (2004) The analysis of fractional differential equations. An application-oriented exposition using differential operators of Caputo type. Lecture Notes in Mathematics. Springer, BerlinMATHGoogle Scholar
  16. Dipierro S, Savin O, Valdinoci E (2017) All functions are locally \(s\)-harmonic up to a small error. J Eur Math Soc (JEMS) 19(4):957–966MathSciNetCrossRefMATHGoogle Scholar
  17. Dipierro S, Savin O, Valdinoci E (2016) Local approximation of arbitrary functions by solutions of nonlocal equations. arXiv:1609.04438
  18. Du M, Wang Z, Hu H (2013) Measuring memory with the order of fractional derivative. Sci Rep 3:3431CrossRefGoogle Scholar
  19. El Hady A, Machta BB (2015) Mechanical surface waves accompany action potential propagation. Nat Commun 6:6697 EPCrossRefGoogle Scholar
  20. Ermentrout GB, Kleinfeld D (2001) Traveling electrical waves in cortex: insights from phase dynamics and speculation on a computational role. Neuron 29:33–44CrossRefGoogle Scholar
  21. Ermentrout GB, McLeod JB (1993) Existence and uniqueness of travelling waves for a neural network. Proc R Soc Edinb 123A:461–478MathSciNetCrossRefMATHGoogle Scholar
  22. Evans LC (1998) Partial differential equations, vol 19. Graduate studies in mathematics. American Mathematical Society, ProvidenceMATHGoogle Scholar
  23. Fiala JC, Harris KM (1999) Dendrite structure. In: Stuart G, Nelson S, Häusser M (eds) Dendrites. Oxford Scholarship Online. Oxford University Press, OxfordGoogle Scholar
  24. Gonzalez-Perez A, Mosgaard LD, Budvytyte R, Villagran-Vargas E, Jackson AD, Heimburg T (2016) Solitary electromechanical pulses in lobster neurons. Biophys Chem 216:51–59CrossRefGoogle Scholar
  25. Heimburg T, Jackson AD (2005) On soliton propagation in biomembranes and nerves. Proc Natl Acad Sci 102(28):9790–9795CrossRefGoogle Scholar
  26. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117(4):500–544CrossRefGoogle Scholar
  27. Ionescu C, Lopes A, Copot D, Machado JAT, Bates JHT (2017) The role of fractional calculus in modelling biological phenomena: a review. Commun Nonlinear Sci Numer Simul 51:141–159MathSciNetCrossRefGoogle Scholar
  28. Ivancevic VG, Ivancevic TT (2010) Quantum neural computation. Springer, DordrechtCrossRefMATHGoogle Scholar
  29. Karakash JJ (1950) Transmission lines and filter networks. Macmillan, New YorkGoogle Scholar
  30. Kim I, Kim K-H, Lim S (2017) An \(L_q (L_p )\)-theory for the time fractional evolution equations with variable coefficients. Adv Math 306:123–176MathSciNetCrossRefMATHGoogle Scholar
  31. Larkum ME, Watanabe S, Nakamura T, Lasser-Ross N, Ross WN (2003) Synaptically activated Ca2+ waves in layer 2/3 and layer 5 rat neocortical pyramidal neurons. J Physiol 549:471–488CrossRefGoogle Scholar
  32. Lautrup B, Appali R, Jackson AD, Heimburg T (2011) The stability of solitons in biomembranes and nerves. Eur Phys J 34(57):1–9Google Scholar
  33. Magin RL (2010) Fractional calculus models of complex dynamics in biological tissues. Comput Math Appl 59:1586–1593MathSciNetCrossRefMATHGoogle Scholar
  34. Marinov T, Santamaria F (2013) Modeling the effects of anomalous diffusion on synaptic plasticity. BMC Neurosci 14(Suppl. 1):P343CrossRefGoogle Scholar
  35. Marinov T, Santamaria F (2014) Computational modeling of diffusion in the cerebellum. Prog Mol Biol Transl Sci 123:169–89CrossRefGoogle Scholar
  36. Mavroudis IA, Fotiou DF, Adipepe LF, Manani MG, Njau SD, Psaroulis D, Costa VG, Baloyannis SJ (2010) Morphological changes of the human Purkinje cells and deposition of neuritic plaques and neurofibrillary tangles on the cerebellar cortex of Alzheimer’s disease. Am J Alzheimer’s Dis Other Demen 25(7):585–591CrossRefGoogle Scholar
  37. Metzler R, Klafter J (2000) The random walks guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339(1):1–77MathSciNetCrossRefMATHGoogle Scholar
  38. Miranker WL (2006) A neural network wave formalism. Adv Appl Math 37:19–30MathSciNetCrossRefMATHGoogle Scholar
  39. Nakamura T, Lasser-Ross N, Nakamura K, Ross WN (2002) Spatial segregation and interaction of calcium signalling mechanisms in rat hippocampal CA1 pyramidal neurons. J Physiol 543:465–480CrossRefGoogle Scholar
  40. Neymotin SA, McDougal RA, Sherif MA, Fall CP, Hines ML, Lytton WW (2015) Neuronal Calcium wave propagation varies with changes in endoplasmic reticulum parameters: a computer model. Neural Comput 27(4):898–924CrossRefGoogle Scholar
  41. Nimchinsky EA, Sabatini BL, Svoboda K (2002) Structure and function of dendritic spines. Annu Rev Physiol 64:313–353CrossRefGoogle Scholar
  42. Pinto DJ, Ermentrout GB (2001) Spatially structured activity in synaptically coupled neuronal networks: I. Travelling fronts and pulses. SIAM J Appl Math 62:206–225MathSciNetCrossRefMATHGoogle Scholar
  43. Rigatos GG (2015) Advanced models of neural networks. Nonlinear dynamics and stochasticity in biological neurons. Springer, BerlinCrossRefMATHGoogle Scholar
  44. Ross WN (2002) Understanding calcium waves and sparks in central neurons. Nat Rev Neurosci 13:157–168CrossRefGoogle Scholar
  45. Saftenku EÈ (2005) Modeling of slow glutamate diffusion and AMPA receptor activation in the cerebellar glomerulus. J Theor Biol 234:363–382MathSciNetCrossRefGoogle Scholar
  46. Santamaria F, Wils S, De Schutter E, Augustine GJ (2006) Anomalous diffusion in Purkinje cell dendrites caused by spines. Neuron 52:635–648CrossRefGoogle Scholar
  47. Santamaria F, Wils S, De Schutter E, Augustine GJ (2011) The diffusional properties of dendrites depend on the density of dendritic spines. Eur J Neurosci 34(4):561–568CrossRefGoogle Scholar
  48. Saxton MJ (1996) Anomalous diffusion due to binding: a Monte Carlo study. Biophys J 70:1250–1262CrossRefGoogle Scholar
  49. Thorson J, Biederman-Thorson M (1974) Distributed relaxation processes in sensory adaptation: spatial nonuniformity in receptors can explain both the curious dynamics and logarithmic statics of adaptation. Science 183(4121):161–172CrossRefGoogle Scholar
  50. Trommershäuser J, Marienhagen J, Zippelius A (1999) Stochastic model of central synapses: slow diffusion of transmitter interacting with spatially distributed receptors and transporters. J Theor Biol 198:101–120CrossRefGoogle Scholar
  51. von Schweidler ER (1907) Studien über die Anomalien im Verhalten der Dielectrika. Ann Phys 24:711–770CrossRefMATHGoogle Scholar
  52. Wikipedia: Drawing of Purkinje cells (A) and granule cells (B) from pigeon cerebellum by Santiago Ramón y Cajal, 1899; Instituto Cajal, Madrid, Spain. File:PurkinjeCell.jpg https://en.wikipedia.org/wiki/Purkinje_cell#/media/File:PurkinjeCell.jpg
  53. Zacher R (2005) Maximal regularity of type \(L_p\) for abstract parabolic Volterra equations. J Evol Equ 5(1):79–103MathSciNetCrossRefMATHGoogle Scholar
  54. Zacher R (2013) A De Giorgi–Nash type theorem for time fractional diffusion equations. Math Ann 356(1):99–146MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli studi di MilanoMilanItaly
  2. 2.School University of Western AustraliaCrawleyAustralia
  3. 3.School of Mathematics and StatisticsUniversity of MelbourneParkvilleAustralia
  4. 4.Istituto di Matematica Applicata e Tecnologie InformaticheConsiglio Nazionale delle RicerchePaviaItaly

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