Self-similar network model for fractional-order neuronal spiking: implications of dendritic spine functions


Fractional-order derivatives have been widely used to describe the spiking patterns of neurons, without considering their self-similar dendritic structures. In this study, a self-similar resistor–capacitor network is proposed to relate the spiny dendritic structure with fractional spiking properties. In order to achieve this goal, two types of networks comprising recursively staggered resistors and capacitors were developed to model the functional properties of smooth and spiny dendrites, respectively. Their overall electrotonic properties can be described by fractional order temporal operators derived by Heaviside operational calculus. According to this operator method, spiking patterns of spiny dendrites were controlled by the standard 0.5-order derivative, whereas an exponential modulation term was added in the governing fractional operator of the smooth dendrites. The application of these fractional operators in a leaky integrate-and-fire model reveals that the dendritic spine plays an important role in alternations of the spiking properties, including first-spike latency, firing rate adaptation, and afterhyperpolarization conductance. Further, the multilevel assembly of this network indicates that the fractional spiking behaviors of spiny neurons might originate from their hierarchical substructures, thereby highlighting possible functional consequences of alterations to dendritic self-similarity.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10


  1. 1.

    La Camera, G., Rauch, A., Thurbon, D., Lüscher, H.R., Senn, W., Fusi, S.: Multiple time scales of temporal response in pyramidal and fast spiking cortical neurons. J. Neurophysiol. 96(6), 3448–3464 (2006)

    Google Scholar 

  2. 2.

    Lundstrom, B.N., Higgs, M.H., Fairhall, A.L., Spain, W.J., Fairhall, A.L.: Fractional differentiation by neocortical pyramidal neurons. Nat. Neurosci. 11(11), 1335–1342 (2008)

    Google Scholar 

  3. 3.

    Chaudhuri, R., Knoblauch, K., Gariel, M.A., Kennedy, H., Wang, X.J.: A large-scale circuit mechanism for hierarchical dynamical processing in the primate cortex. Neuron 88(2), 419–431 (2015)

    Google Scholar 

  4. 4.

    Joglekar, M.R., Mejias, J.F., Yang, G.R., Wang, X.J.: Inter-areal balanced amplification enhances signal propagation in a large-scale circuit model of the primate cortex. Neuron 98(1), 222–234.e8 (2018)

    Google Scholar 

  5. 5.

    Higgs, M.H., Slee, S.J.: Diversity of gain modulation by noise in neocortical Neurons: regulation by the slow afterhyperpolarization conductance. J. Neurosci. 26(34), 8787–8799 (2006)

    Google Scholar 

  6. 6.

    Lundstrom, B.N., Fairhall, A.L., Maravall, M.: Multiple timescale encoding of slowly varying whisker stimulus envelope in cortical and thalamic neurons in vivo. J. Neurosci. 30(14), 5071–5077 (2010)

    Google Scholar 

  7. 7.

    Scutt, G., Allen, M., Kemenes, G., Yeoman, M.: A switch in the mode of the sodium/calcium exchanger underlies an age-related increase in the slow afterhyperpolarization. Neurobiol. Aging 36(10), 2838–2849 (2015)

    Google Scholar 

  8. 8.

    Disterhoft, J.F., Wu, W.W., Ohno, M.: Biophysical alterations of hippocampal pyramidal neurons in learning, ageing and Alzheimer’s disease. Ageing Res. Rev. 3(4), 383–406 (2004)

    Google Scholar 

  9. 9.

    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    Google Scholar 

  10. 10.

    Ionescu, C., Lopes, A., Copot, D., Machado, J.A.T., Bates, J.H.T.: The role of fractional calculus in modeling biological phenomena: a review. Commun. Nonlinear Sci. Numer. Simul. 51, 141–159 (2017)

    MathSciNet  Google Scholar 

  11. 11.

    Deseri, L., Di Paola, M., Zingales, M., Pollaci, P.: Power-law hereditariness of hierarchical fractal bones. Int. J. Numer. Methods Biomed. 29(12), 1338–1360 (2013)

    MathSciNet  Google Scholar 

  12. 12.

    Deseri, L., Pollaci, P., Zingales, M., Dayal, K.: Fractional hereditariness of lipid membranes: instabilities and linearized evolution. J. Mech. Behav. Biomed. Mater. 58, 11–27 (2016)

    Google Scholar 

  13. 13.

    Guo, J., Yin, Y., Ren, G.: Abstraction and operator characterization of fractal ladder viscoelastic hyper-cell for ligaments and tendons. Appl. Math. Mech. Engl. Ed. 40(10), 1429–1448 (2019)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Perdikaris, P., Karniadakis, G.E.: Fractional-order viscoelasticity in one-dimensional blood flow models. Ann. Biomed. Eng. 42(5), 1012–1023 (2014)

    Google Scholar 

  15. 15.

    Mondal, A., Upadhyay, R.K.: Diverse neuronal responses of a fractional-order Izhikevich model: journey from chattering to fast spiking. Nonlinear Dyn. 91(2), 1275–1288 (2018)

    Google Scholar 

  16. 16.

    Meng, F., Zeng, X., Wang, Z.: Dynamical behavior and synchronization in time-delay fractional-order coupled neurons under electromagnetic radiation. Nonlinear Dyn. 95(2), 1615–1625 (2019)

    Google Scholar 

  17. 17.

    Paola, M.D., Zingales, M.: Exact mechanical models of fractional hereditary materials. J. Rheol. 56(5), 983–1004 (2012)

    Google Scholar 

  18. 18.

    Butera, S., Di Paola, M.: A physically based connection between fractional calculus and fractal geometry. Ann. Phys. 350, 146–158 (2014)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Di Paola, M., Pinnola, F.P., Zingales, M.: Fractional differential equations and related exact mechanical models. Comput. Math. Appl. 66(5), 608–620 (2013)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Teka, W., Marinov, T.M., Santamaria, F.: Neuronal spike timing adaptation described with a fractional leaky integrate-and-fire model. PLoS Comput. Biol. 10(3), 1–14 (2014)

    Google Scholar 

  21. 21.

    Shi, M., Wang, Z.: Abundant bursting patterns of a fractional-order Morris–Lecar neuron model. Commun. Nonlinear Sci. Numer. Simul. 19(6), 1956–1969 (2014)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Weinberg, S.H.: Membrane capacitive memory alters spiking in neurons described by the fractional-order Hodgkin–Huxley model. PLoS ONE 10(5), 1–27 (2015)

    MathSciNet  Google Scholar 

  23. 23.

    Teka, W., Stockton, D., Santamaria, F.: Power-law dynamics of membrane conductances increase spiking diversity in a Hodgkin–Huxley model. PLoS Comput. Biol. 12(3), 1–23 (2016)

    Google Scholar 

  24. 24.

    Upadhyay, R.K., Mondal, A., Teka, W.W.: Fractional-order excitable neural system with bidirectional coupling. Nonlinear Dyn. 87(4), 2219–2233 (2017)

    MathSciNet  Google Scholar 

  25. 25.

    Mankin, R., Paekivi, S.: Memory-induced resonancelike suppression of spike generation in a resonate-and-fire neuron model. Phys. Rev. E 97(1), 12125 (2018)

    MathSciNet  Google Scholar 

  26. 26.

    van Ooyen, A., Duijnhouwer, J., Remme, M.W.H., van Pelt, J.: The effect of dendritic topology on firing patterns in model neurons. Netw. Comput. Neural Syst. 13(3), 311–325 (2002)

    Google Scholar 

  27. 27.

    van Elburg, R.A.J., van Ooyen, A.: Impact of dendritic size and dendritic topology on burst firing in pyramidal cells. PLoS Comput. Biol. 6(5), e1000781 (2010)

    MathSciNet  Google Scholar 

  28. 28.

    de Sousa, G., Maex, R., Adams, R., Davey, N., Steuber, V.: Dendritic morphology predicts pattern recognition performance in multi-compartmental model neurons with and without active conductances. J. Comput. Neurosci. 38(2), 221–234 (2015)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Mainen, Z.F., Sejnowski, T.J.: Influence of dendritic structure on firing pattern in model neocortical neurons. Nature 382(6589), 363–366 (1996)

    Google Scholar 

  30. 30.

    Henry, B.I., Langlands, T.A.M., Wearne, S.L.: Fractional cable models for spiny neuronal dendrites. Phys. Rev. Lett. 100(12), 128103 (2008)

    Google Scholar 

  31. 31.

    Nimchinsky, E.A., Sabatini, B.L., Svoboda, K.: Structure and function of dendritic spines. Annu. Rev. Physiol. 64(1), 313–353 (2002)

    Google Scholar 

  32. 32.

    Martin, K.A.C., Douglas, R.J., Binzegger, T.: Axons in cat visual cortex are topologically self-similar. Cereb. Cortex 15(2), 152–165 (2004)

    Google Scholar 

  33. 33.

    Smith, T.G., Lange, G.D., Marks, W.B.: Fractal methods and results in cellular morphology—dimensions, lacunarity and multifractals. J. Neurosci. Methods 69(2), 123–136 (1996)

    Google Scholar 

  34. 34.

    Fernández, E., Jelinek, H.F.: Use of fractal theory in neuroscience: methods, advantages, and potential problems. Methods 24(4), 309–321 (2001)

    Google Scholar 

  35. 35.

    Rothnie, P., Kabaso, D., Hof, P.R., Henry, B.I., Wearne, S.L.: Functionally relevant measures of spatial complexity in neuronal dendritic arbors. J. Theor. Biol. 238(3), 505–526 (2006)

    MathSciNet  Google Scholar 

  36. 36.

    Mandelbrot, B.B., Vicsek, T.: Directed recursion models for fractal growth. J. Phys. A Math. Theor. 22(9), L377 (1989)

    MATH  Google Scholar 

  37. 37.

    Puškaš, N., Zaletel, I., Stefanović, B.D., Ristanović, D.: Fractal dimension of apical dendritic arborization differs in the superficial and the deep pyramidal neurons of the rat cerebral neocortex. Neurosci. Lett. 589(88), 88–91 (2015)

    Google Scholar 

  38. 38.

    Baer, S.M., Rinzel, J.: Propagation of dendritic spikes mediated by excitable spines: a continuum theory. J. Neurophysiol. 65(4), 874–890 (1991)

    Google Scholar 

  39. 39.

    Mikusinski, J.: Operational Calculus, 2nd edn. Pergamon Press, Oxford (1983)

    Google Scholar 

  40. 40.

    Cartea, Á., Del-Castillo-Negrete, D.: Fluid limit of the continuous-time random walk with general Lévy jump distribution functions. Phys. Rev. E 76(4), 41105 (2007)

    Google Scholar 

  41. 41.

    Cao, J., Li, C., Chen, Y.: On tempered and substantial fractional calculus. In: 2014 IEEE/ASME 10th International Conference on Mechatronic and Embedded Systems and Applications (MESA), pp. 1–6 (2014)

  42. 42.

    Chen, M., Deng, W.: Discretized fractional substantial calculus. ESAIM Math. Model. Numer. Anal. 49(2), 373–394 (2015)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Li, C., Deng, W.: High order schemes for the tempered fractional diffusion equations. Adv. Comput. Math. 42(3), 543–572 (2016)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Friedrich, R., Jenko, F., Baule, A., Eule, S.: Anomalous diffusion of inertial, weakly damped particles. Phys. Rev. Lett. 96(23), 230601 (2006)

    Google Scholar 

  45. 45.

    Sabzikar, F., Meerschaert, M.M., Chen, J.: Tempered fractional calculus. J. Comput. Phys. 293, 14–28 (2015)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Ding, H., Li, C.: A high-order algorithm for time-Caputo-tempered partial differential equation with Riesz derivatives in two spatial dimensions. J. Sci. Comput. 80(1), 81–109 (2019)

    MathSciNet  MATH  Google Scholar 

  47. 47.

    Santamaria, F., Wils, S., De Schutter, E., Augustine, G.J.: Anomalous diffusion in Purkinje cell dendrites caused by spines. Neuron 52(4), 635–648 (2006)

    Google Scholar 

  48. 48.

    Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1–77 (2000)

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Oldham, K.B.: The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York, NY (1974)

    Google Scholar 

  50. 50.

    Marinov, T., Nelson, R., Fidel, S.: Fractional integration toolbox. Fract. Calc. Appl. Anal. 16, 670–681 (2013)

    MathSciNet  MATH  Google Scholar 

  51. 51.

    Heil, P.: First-spike latency of auditory neurons revisited. Curr. Opin. Neurobiol. 14(4), 461–467 (2004)

    Google Scholar 

  52. 52.

    Ikeda, M., Hojo, Y., Komatsuzaki, Y., Okamoto, M., Kato, A., Takeda, T., Kawato, S.: Hippocampal spine changes across the sleep-wake cycle: corticosterone and kinases. J. Endocrinol. 226(2), M13–M27 (2015)

    Google Scholar 

  53. 53.

    Maravall, M., Petersen, R.S., Fairhall, A.L., Arabzadeh, E., Diamond, M.E.: Shifts in coding properties and maintenance of information transmission during adaptation in barrel cortex. PLoS Biol. 5(2), e19 (2007)

    Google Scholar 

  54. 54.

    Segev, I., London, M.: Untangling dendrites with quantitative models. Science 290(5492), 744–750 (2000)

    Google Scholar 

  55. 55.

    Suetsugu, M., Mehraein, P.: Spine distribution along the apical dendrites of the pyramidal neurons in Down’s syndrome. Acta Neuropathol. 50(3), 207–210 (1980)

    Google Scholar 

  56. 56.

    Jacobs, B., Driscoll, L., Schall, M.: Life-span dendritic and spine changes in areas 10 and 18 of human cortex: a quantitative Golgi study. J. Comp. Neurol. 386(4), 661–680 (1997)

    Google Scholar 

  57. 57.

    Duan, H., Wearne, S.L., Rocher, A.B., Macedo, A., Morrison, J.H., Hof, P.R.: Age-related dendritic and spine changes in corticocortically projecting neurons in Macaque monkeys. Cereb. Cortex 13(9), 950–961 (2003)

    Google Scholar 

  58. 58.

    Zador, M., Diego, S., Zador, A.M., Agmon-Snir, H., Segev, I.: The morphoelectrotonic transform: a graphical approach to dendritic function. J. Neurosci. 15(3), 1669–1682 (1995)

    Google Scholar 

  59. 59.

    Cafagna, D., Grassi, G.: On the simplest fractional-order memristor-based chaotic system. Nonlinear Dyn. 70(2), 1185–1197 (2012)

    MathSciNet  Google Scholar 

  60. 60.

    Si, G., Diao, L., Zhu, J.: Fractional-order charge-controlled memristor: theoretical analysis and simulation. Nonlinear Dyn. 87(4), 2625–2634 (2017)

    Google Scholar 

  61. 61.

    Yang, N., Xu, C., Wu, C., Jia, R., Liu, C.: Fractional-order cubic nonlinear flux-controlled memristor: theoretical analysis, numerical calculation and circuit simulation. Nonlinear Dyn. 97(1), 33–44 (2019)

    MATH  Google Scholar 

Download references


We thank Yiheng Han and Yuhao Sun for the helpful discussion on this topic. This research was partially supported by the National Natural Science Foundation of China (Grant Nos. 61836014, 11672150).

Author information



Corresponding author

Correspondence to Gexue Ren.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest with regard to the publication of this manuscript.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Guo, J., Yin, Y., Hu, X. et al. Self-similar network model for fractional-order neuronal spiking: implications of dendritic spine functions. Nonlinear Dyn 100, 921–935 (2020).

Download citation


  • Fractional-order model
  • Dendritic morphology
  • Dendritic spine
  • Self-similar
  • Operator method
  • Leaky integrate-and-fire model