ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


For citation:

Dmitrichev A. S., Kasatkin D. V., Klinshov V. V., Kirillov S. Y., Maslennikov O. V., Shchapin D. S., Nekorkin V. I. Nonlinear dynamical models of neurons: Review. Izvestiya VUZ. Applied Nonlinear Dynamics, 2018, vol. 26, iss. 4, pp. 5-58. DOI: 10.18500/0869-6632-2018-26-4-5-58

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text:
(downloads: 897)
Полный текст в формате PDF(En):
(downloads: 128)
Language: 
Russian
Article type: 
Review
UDC: 
621.373.1

Nonlinear dynamical models of neurons: Review

Autors: 
Dmitrichev Aleksej Sergeevich, Institute of Applied Physics of the Russian Academy of Sciences
Kasatkin Dmitry Vladimirovich, Institute of Applied Physics of the Russian Academy of Sciences
Klinshov V.  V., Federal state budgetary educational institution of higher professional education Nizhny Novgorod state University named N. And.Lobachevsky
Kirillov Sergej Yu., Institute of Applied Physics of the Russian Academy of Sciences
Maslennikov O.  V., Institute of Applied Physics of the Russian Academy of Sciences
Shchapin D. S., Institute of Applied Physics of the Russian Academy of Sciences
Nekorkin Vladimir Isaakovich, Institute of Applied Physics of the Russian Academy of Sciences
Abstract: 

Topic. A review of the basic dynamical models of neural activity is presented and individual features of their behavior are discussed, which can be used as a basis for the subsequent development and construction of various configurations of neural networks. The work contains both new original results and generalization of already known ones published earlier in different journals. Aim is to familiarize the reader with the basic dynamical properties of neurons, such as the existence of a rest state and the generation of the action potential; to outline the dynamical mechanisms underlying these properties which are used in the development of neural models with various levels of detailing. Investigated models. From the mathematical point of view, neuron models are divided into two classes. The first class is represented by models with continuous time described by ordinary differential equations. The section devoted to continuous-time models starts from the most detailed Hodgkin–Huxley model, which is a canonical model for neural activity in nonlinear dynamics. Further we describe simplified models, such as a two-dimensional model of Morris–Lecar for spiking and a three-dimensional model of Hindmarsh–Rose for bursting. The FitzHugh–Nagumo model is described in detail, and detailed bifurcation analysis is presented. We also present models for neurons with specific properties, namely a neuron with afterdepolarization and an inferior olives neuron. The last and the simplest model is the «integrate–fire» model. The second class of neural models are systems with discrete time represented by discrete maps. Such models have recently gained increasing popularity due to the richness of the demonstrated dynamics and the ease of numerical simulations. We describe such models as the Chialvo model, the Izhikevich model, the Rulkov model, and the Courbage–Nekorkin model. Results. The basic physical principles underlying the construction of mathematical models of neural activity, based on ion transport, are outlined. Using the FitzHugh–Nagumo model as an example, the main properties and mechanisms of the emergence of multithreshold excitation regimes in neurons are described. The mechanism of formation of burst oscillations in the Hindmarsh–Rose model is outlined. A dynamic mechanism for temporal decline of the excitation threshold and the emergence of periodic oscillations in a neuron with afterdepolarization are described. The formation of (Ca2+)- and (Na2+)-dependent spikes in inferior olive neurons is described. Dynamic mechanisms of formation of the major regular and chaotic regimes of neural activity in discrete models of Chialvo, Izhikevich, Rulkov and Courbage–Nekorkin are described. Discussion. In the Conclusion we briefly summarize the content of the survey.    

Reference: 
  1. McCulloch W., Pitts W. A logical calculus of the ideas immanent in nervous activity. Bulletin of Mathematical Biophysics, 1943, vol. 5, № 4, p. 115.
  2. Hodgkin A.L., Huxley A.F. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol., 1952, vol. 117, № 4, p. 500.
  3. Noble В. A modification of the Hodgkin–Huxley equations applicable to Purkinje fibre action and pacemaker potentials. J. Physiol., 1962, vol. 160, № 2, p. 317.
  4. Plant R.E., Kim M. Mathematical description of a bursting pacemaker neuron by a modification of the Hodgkin-Huxley equations. Biophys. J., 1976, Vol. 16, № 3. p. 227.
  5. Braun H.A., Huber M.T., Dewald M., Schafer K., and Voigt K. Computer simulations of neuronal signal transduction: The role of nonlinear dynamics and noise. Int. J. Bifurcation Chaos Appl. Sci. Eng., 1998, vol. 8, p. 881.
  6. Morris C., Lecar H. Voltage oscillations in the barnacle giant muscle fiber. Biophys. J., 1981, vol. 35, p. 193.
  7. Keynes R.D., Rojas E., Taylor R. E., Vergara J. Calcium and potassium systems of a giant barnacle muscle fibre under membrane potential control. J. Physiol. (Lond.), 1973, vol. 229, № 2, p. 409.
  8. Gutkin B.S., Ermetrout G.B. Dynamics of membrane excitability determine interspike interval variability: a link between spike generation mechanisms and cortical spike train statistics. Neural Computation, 1998, vol. 10, № 5, p. 1047.
  9. Rinzel J., Ermetrout G.B. Analysis of neural excitability and oscillations. Methods in Neuronal Modeling: From Ions to Networks (Eds. C. Koch, I. Segev). London: MIT Press. 1999. P. 251.
  10. Ermetrout G.B., Terman D.H. Mathematical Foundations of Neuroscience. New York: Springer. 2010. 422 p.
  11. Tsumoto K., Kitajima H., Yoshinaga T., Aihara K., Kawakami H. Bifurcations in Morris-Lecar neuron model. Neurocomputing, 2006, vol. 69, № 4–6, p. 293.
  12. Behdad R., Binczak S., Dmitrichev A.S., Nekorkin V.I., Bilbault J.M. Artificial electrical Morris–Lecar neuron. IEEE Trans. Neural Netw. Learn. Syst., 2015, vol. 26, № 9, p. 1875.
  13. Abbot L.F. A network of oscillators. J. Phys. A: Math. Gen., 1990, vol. 23, p. 3835.
  14. FitzHugh R. Thresholds and plateaus in the Hodgkin–Huxley nerve equations. J. Gen. Physiol., 1960, vol. 43, p. 867.
  15. FitzHugh R. Impulses and physiological states in theoretical models of nerve membranes. Biophysical Journal, 1961, vol. 1, p. 445.
  16. Nagumo J., Arimoto S., Yoshizawa S. An active pulse transmission line simulating nerve axon. Proc. IRE., 1962, vol. 50, p. 2061.
  17. Kepler T.B., Abbott L.F., Marder E. Reduction of conductance-based neuron models. Biol. Cybern., 1992, vol. 66, № 5, p. 381.
  18. Andronov A.A., Vitt A.A., Khaikin S.E. Theory of Oscillators. Oxford: Pergamon Press, 1966. 815 p.
  19. Mischenko E.F., Kolesov Yu.S., Kolesov A.Yu., Rozov N.Kh. Asymptotic Methods in Singularly Perturbed Systems, Monographs in Contemporary Mathematics. NY, Consultants Bureau, 1984. 294 p.
  20. Arnold V.I., Afrajmovich V.S., Il’yashenko Yu.S., Shil’nikov L.P. Bifurcation theory and catastrophe theory. Encyclopaedia of Mathematical Sciences: Dynamical Systems V., 1994, vol. 5, 274 p.
  21. Fenichel N. Geometric singular perturbation theory for ordinary differential equation. SIAM J. Diff. Eqns., 1979, Vol. 31, P. 53.
  22. Nekorkin V.I., Dmitrichev A.S., Shapin D.S., Kazantsev V.B. Dynamics if a neuron model with complex-threshold excitation. Mathematical Models and Computer Sim., 2005, vol. 17, № 6, p. 75. (In Russian).
  23. Binczak S., Kazantsev V.B., Nekorkin V.I., Bilbault J.M. Experimental study of bifurcations in a modified FitzHugh-Nagumo cell. Electron. Lett., 2003, vol. 39, p. 13.
  24. Shchapin D.S. Dynamics of two neuronlike elements with inhibitory feedback. Journal of Communications Technology and Electronics, 2009, vol. 54, № 2, p. 175.
  25. Hindmarsh J.L., Rose R.M. A model of neuronal bursting using three coupled first order differential equations. Proc. of the Royal Society London B., 1984, vol. 221, p. 87.
  26. Nekorkin V.I. Introduction to Nonlinear Oscillations. Wiley-VCH, 2015. 264 p.
  27. Wang X.-J. Genesis of bursting oscillations in the Hindmarsh–Rose model and homoclinicity to a chaotic saddle. Physica D., 1993, vol. 62, № 1–4, p. 263.
  28. Innocenti G., Morelli A., Genesio R., Torcini A. Dynamical phases of the Hindmarsh– Rose neuronal model: Studies of the transition from bursting to spiking chaos. Chaos, 2007, vol. 17, № 4, p. 043128.
  29. Shilnikov A., Kolomiets M. Methods of the qualitative theory for Hindmarsh–Rose model: A case study – A tutorial. Int. J. of Bifurcation and Chaos, 2008, vol. 18, № 8, p. 2141.
  30. Izhikevich E.M. Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. Cambridge: MIT Press, 2007. 441 p.
  31. Miura R.M. Analysis of excitable cell models. Journal of Computational and Applied Mathematics, 2002, vol. 144, № 1–2, p. 29.
  32. Yue C., Remy S., Su H., Beck H., Yaari Y. Proximal persistent Na+ channels drive spike afterdepolarizations and associated bursting in adult CA1 pyramidal cells. J. Neurosci., 2005, vol. 25, № 42, p. 9704.
  33. Lisman J.E., Idiart M.A. Storage of 7 +/-2 short-term memories in oscillatory subcycles. Science, 1995, vol. 267, № 5203, p. 1512.
  34. Jensen O., Idiart M.A.P. and Lisman J.E. Physiologically realistic formation of autoassociative memory in networks with theta/gamma oscillations: Role of fast NMDA channels. Learn. Mem., 1996, vol. 3, № 2–3, p. 243.
  35. Jensen O., Lisman J.E. Hippocampal sequence-encoding driven by a cortical multiitem working memory buffer. Trends in Neurosciences, 2005, vol. 28, № 2, p. 67.
  36. Haj-Dahmane S., Andrade К. Ionic mechanism of the slow afterdepolarization induced by muscarinic receptor activation in rat prefrontal cortex. J. Neurophysiol., 1998, vol. 80, № 3, p. 1197.
  37. Park J.-Y., Remy S., Varela О., Cooper D.C., Chung S., Kang H.-W., Lee J.-H., Spruston N. A post-burst after depolarization is mediated by group is metabotropic glutamate receptor-dependent upregulation of Ca(v)2.3 R-type calcium channels in CA1 pyramidal neurons. PLoS Biology, 2010, vol. 8, № 11, p. e1000534.
  38. Klin’shov V.V., Nekorkin V.I. Model of a neuron with afterdepolarization and shortterm memory. Radiophysics and Quantum Electronics, 2005, vol. 48, № 3, p. 203.
  39. Kepler T.B., Marder E. Spike initiation and propagation on axons with slow inward currents. Biol. Cybern., 1993, vol. 68, № 3, p. 209.
  40. Enns-Ruttan J., Miura R.M. Spontaneous secondary spiking in excitable cells. J. Theor. Biol., 2000, vol. 205, № 2, p. 181.
  41. Schweighofer N., Lang E.J., Kawato M. Role of the olivo-cerebellar complex in motor learning and control. Front. Neural Circuits, 2013, vol. 7, art. № 94, p. 1.
  42. Manor Y., Rinzel J., Segev I., Yarom Y. Low-amplitude oscillations in the inferior olive: A model based on electrical coupling of neurons with heterogeneous channel densities. J. Neurophysiol., 1997, vol. 77, № 5, p. 2736.
  43. Velarde M.G., Nekorkin V.I., Kazantsev V.B., Makarenko V.I., Llinas R. Modeling inferior olive neuron dynamics. Neural Netw., 2002, vol. 15, № 1, p. 5.
  44. Schweighofer N., Doya K., Kawato M. Electrophysiological properties of inferior olive neurons: A compartmental model. J. Neurophysiol., 1999, vol. 82, № 2, p. 804.
  45. Kazantsev V.B., Nekorkin V.I., Makarenko V.I., Llinas R. Olivo-cerebellar clusterbased universal control system. Proc. Natl. Acad. Sci. USA, 2003, vol. 100, № 22, p. 13064.
  46. Kazantsev V.B., Nekorkin V.I., Makarenko V.I., Llinas R. Self-referential phase reset based on inferior olive oscillator dynamics. Proc. Natl. Acad. Sci. USA, 2004, vol. 101, № 52, p. 18183.
  47. Llinas R., Yarom Y. Oscillatory properties of guinea-pig inferior olivary neurones and their pharmacological modulation: An in vitro study. J. Physiol., 1986, vol. 376, p. 163.
  48. Klinshov V., Franovic I. Slow rate fluctuations in a network of noisy neurons with coupling delay. EPL (Europhysics Letters), 2016, vol. 116, № 4, p. 48002.
  49. Klinshov V., Franovic I. Mean field dynamics of a random neural network with noise. Physical Review E., 2015, vol. 92, № 6, p. 62813.
  50. Franovic I., Klinshov V. Clustering promotes switching dynamics in networks of noisy neurons. Chaos, 2018, vol. 28, p. 23111.
  51. Brunel N. Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. Journal of Computational Neuroscience, 2000, vol. 8, № 3, p. 183.
  52. Olmi S., Politi A., Torcini A. Collective chaos in pulse-coupled neural networks. EPL (Europhysics Letters), 2010, vol. 92, № 6, p. 60007.
  53. Ullner E., Politi A. Self-sustained irregular activity in an ensemble of neural oscillators. Physical Review X., 2016, vol. 6, № 1, p. 1.
  54. Hasegawa H. Population rate codes carried by mean, fluctuation and synchrony of neuronal firings. Physica A: Statistical Mechanics and its Applications, 2009, vol. 388, № 4, p. 499.
  55. Hasegawa H. Synchrony and variability induced by spatially correlated additive and multiplicative noise in the coupled Langevin model. Physical Review E., 2008, vol. 78, № 3, p. 31110.
  56. Nykamp D.Q., Friedman D., Shaker S., Shinn M., Vella M., Compte A., Roxin A. Mean-field equations for neuronal networks with arbitrary degree distributions. Physical Review E., 2017, vol. 95, № 4, p. 1.
  57. Montbrio E., Pazo D., Roxin A. Macroscopic description for networks of spiking neurons. Physical Review X., 2015, vol. 5, № 2, p. 1.
  58. Lapicque L. Recherches quantitatives sur l’excitation electrique des nerfs traitee comme une polarization. J. Physiol. Pathol. Generale, 1907, 9, p. 620.
  59. Lazar A.A. Time encoding with an integrate-and-fire neuron with a refractory period. Neurocomputing, 2004, vol. 58, p. 53.
  60. Liu Y.-H., Wang X.-J. Spike-frequency adaptation of a generalized leaky integrateand-fire model neuron. Journal of Computational Neuroscience, 2001, vol. 10, № 1, p. 25.
  61. Brette R., Gerstner W. Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. Journal of neurophysiology, 2005, vol. 94, № 5, p. 3637.
  62. Abbot L.F., van Vreeswijk C. Asynchronous states in networks of pulse-coupled oscillators. Physical Review E., 1993, vol. 48, p. 1483.
  63. Ermentrout B. Type I membranes, phase resetting curves, and synchrony. Neural Computation, 1996, vol. 8, № 5, p. 979.
  64. Latham P.E., Richmond B.J., Nelson P., Nirenberg S. Intrinsic dynamics in neuronal networks. I. Theory. J. Neurophysiology, 2000, vol. 83, № 2, p. 808.
  65. Hansel D., Mato G. Existence and stability of persistent states in large neuronal networks. Phys. Rev. Letters, 2001, vol. 86, p. 4175.
  66. Izhikevich E.M. Neural excitability, spiking and bursting. International Journal of Bifurcation and Chaos, 2000, vol. 10, № 6, p. 1171.
  67. Courbage M., Nekorkin V.I. Map based models in neurodynamics. International Journal of Bifurcation and Chaos, 2010, vol. 20, № 6, p. 1631.
  68. Ibarz B., Casado J.M., Sanjuan M.A.F. Map-based models in neuronal dynamics. Physics Reports, 2011, vol. 501, № 1–2, p. 1.
  69. Girardi-Schappo M., Tragtenberg M.H.R., Kinouchi O. A brief history of excitable map-based neurons and neural networks. Journal of Neuroscience Methods, 2013, vol. 220, № 2, p. 116.
  70. Chialvo D.R. Generic excitable dynamics on a two-dimensional map. Chaos, Solitons & Fractals, 1995, vol. , № 3–4, p. 461.
  71. Rulkov N.F. Modeling of spiking-bursting neural behavior using two-dimensional map. Physical Review E., 2002, vol. 65, № 4, p. 041922.
  72. Shilnikov A.L., Rulkov N.F. Subthreshold oscillations in a map-based neuron model. Physics Letters A., 2004, vol. 328, № 2–3, p. 177.
  73. Rulkov N.F. Regularization of synchronized chaotic bursts. Physical Review Letters, 2001, vol. 86, № 1, p. 183.
  74. Nekorkin V.I., Vdovin L.V. Map-based model of the neural activity. Izvestiya VUZ. Applied Nonlinear Dynamics, 2007, vol. 15, № 5, p. 36. (In Russian).
  75. Courbage M., Nekorkin V.I., Vdovin L.V. Chaotic oscillations in a map-based model of neural activity. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2007, vol. 17, № 4, p. 043109.
  76. Izhikevich E.M., Hoppensteadt F. Classification of bursting mappings. International Journal of Bifurcation and Chaos, 2004, vol. 14, № 11, p. 3847.
  77. Maslennikov O.V., Nekorkin V.I. // Nonlinear Dynamics and Complexity (Eds. V. Afraimovich, A.C.J. Luo, X. Fu). Springer, 2014. P. 143.
  78. Hess A., Yu L., Klein I., De Mazancourt M., Jebrak G., Mal H., Brugiere O., Fournier M., Courbage M., Dauriat G. Neural mechanisms underlying breathing complexity. PloS ONE, 2013, vol. 8, № 10, p. e75740.
  79. Courbage M., Maslennikov O.V., Nekorkin V.I. Synchronization in time-discrete model of two electrically coupled spike-bursting neurons. Chaos, Solitons & Fractals, 2012, vol. 45, № 5, p. 645.
  80. Maslennikov O.V., Nekorkin V.I. Modular networks with delayed coupling: Synchronization and frequency control. Physical Review E., 2014, vol. 90, № 1, p. 012901.
  81. Maslennikov O.V., Nekorkin V.I. Discrete model of the olivo-cerebellar system: structure and dynamics. Radiophysics and Quantum Electronics, 2012, vol. 55, № 3, p. 198.
  82. Nekorkin V.I., Maslennikov O.V. Spike-burst synchronization in an ensemble of electrically coupled discrete model neurons. Radiophysics and Quantum Electronics, 2011, vol. 54, № 1, p. 56.
  83. Maslennikov O.V., Nekorkin V.I., Kurths J. Basin stability for burst synchronization in small-world networks of chaotic slow-fast oscillators. Physical Review E., 2015, vol. 92, № 4, p. 042803.
  84. Maslennikov O.V., Nekorkin V.I. Evolving dynamical networks with transient cluster activity. Communications in Nonlinear Science and Numerical Simulation, 2015, vol. 23, № 1-3, p. 10.
  85. Yu L., De Mazancourt M., Hess A., Ashadi F.R., Klein I., Mal H., Courbage M., Mangin L. Functional connectivity and information flow of the respiratory neural network in chronic obstructive pulmonary disease. Human Brain Mapping, 2016, vol. 37, № 8, p. 2736.
  86. Maslennikov O.V., Kasatkin D.V., Rulkov N.F., Nekorkin V.I. Emergence of antiphase bursting in two populations of randomly spiking elements. Physical Review E., 2013, vol. 88, № 4, p. 042907.
  87. Maslennikov O.V., Shchapin D.S., Nekorkin V.I. Transient sequences in a hypernetwork generated by an adaptive network of spiking neurons. Phil. Trans. R. Soc. A., 2017, vol. 375, № 2096, p. 20160288.
  88. Yue Y., Liu Y.-J., Song Y.-L., Chen Y., Yu L.-C. Information capacity and transmission in a Courbage–Nekorkin–Vdovin map-based neuron model. Chinese Physics Letters, 2017, vol. 34, № 4, p. 048701.
  89. Franovic I., Maslennikov O.V., Bacic I., Nekorkin V.I. Mean-field dynamics of a population of stochastic map neurons. Physical Review E., 2017, vol. 96, № 1, p. 012226.
  90. Mangin L., Courbage M. Respiratory neural network: Activity and connectivity. Advances in dynamics, patterns, cognition (Eds. Aronson, I.S., Pikovsky, A., Rulkov, N.F., Tsimring, L.S.). Nonlinear Systems and Complexity. Springer Int. 2017. vol. 20. p. 227.
  91. Yang X., Wang M. The evolution to global burst synchronization in a modular neuronal network. Modern Physics Letters B., 2016, vol. 30, № 14, p. 1650210. 
Received: 
01.06.2018
Accepted: 
28.06.2018
Published: 
31.08.2018
Short text (in English):
(downloads: 64)