Energy estimation and coupling synchronization between biophysical neurons

  • FuQiang Wu
  • Jun MaEmail author
  • Ge Zhang


Static charges can induce spatial electric field while moving charges can induce magnetic field. As a result, continuous pumping and exchange of intercellular and extracellular Calcium, potassium and sodium of cells will generate time-varying magnetic field in the media. Therefore, the physical effect of electromagnetic induction in neural activities should be included in building biological neurons. On the other hand, the occurrence of action potential and propagation of ions require energy consumption and supply, so the estimation of physical energy becomes important. Based on our memristive biophysical neuron model, the Hamilton energy function is obtained by using the Helmholtz’s theorem, and this energy is contributed by the electric field and magnetic field described by magnetic flux. It is found that this improved neuron model can present the main dynamical properties in neural activities, and it characterizes the lower threshold behavior and subthreshold oscillation during refractory period. The external forcing current on an isolate is adjusted to calculate the firing patterns, energy function and mode transition, which shows the dependence of energy on electrical activities. Finally, magnetic coupling is triggered to modulate the phase synchronization between two identical neurons connected by electric synapse, respectively.


memristor energy magnetic field phase synchronization 


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  1. 1.
    Laughlin S B, Sejnowski T J. Communication in neuronal networks. Science, 2003, 301: 1870–1874CrossRefGoogle Scholar
  2. 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, 117: 500–544CrossRefGoogle Scholar
  3. 3.
    Izhikevich E M. Neural excitability, spiking and bursting. Int J Bifurcat Chaos, 2000, 10: 1171–1266MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Yeomans J S. Quantitative measurement of neural post-stimulation excitability with behavioral methods. Physiol Behav, 1975, 15: 593–602CrossRefGoogle Scholar
  5. 5.
    Rinzel J, Ermentrout B. Analysis of neural excitability and oscillations. Method Neuronal Model, 1989, 2: 251–292Google Scholar
  6. 6.
    de Vries G. Multiple bifurcations in a polynomial model of bursting oscillations. J Nonlinear Sci, 1998, 8: 281–316MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Izhikevich E M. Simple model of spiking neurons. IEEE Trans Neural Netw, 2003, 14: 1569–1572CrossRefGoogle Scholar
  8. 8.
    Izhikevich E M. Which model to use for cortical spiking neurons? IEEE Trans Neural Netw, 2004, 15: 1063–1070CrossRefGoogle Scholar
  9. 9.
    Grüsser O J, Grüsser-Cornehls U, Licker M D. Further studies on the velocity function of movement detecting class-2 neurons in the frog retina. Vision Res, 1968, 8: 1173–1185CrossRefGoogle Scholar
  10. 10.
    Izhikevich E M, Moehlis J. Dynamical systems in neuroscience: The geometry of excitability and bursting. SIAM Rev, 2008, 50: 397Google Scholar
  11. 11.
    Kim J I, Cho H Y, Han J H, et al. Which neurons will be the engram-activated neurons and/or more excitable neurons? Exp Neurobiol, 2016, 25: 55CrossRefGoogle Scholar
  12. 12.
    Roxin A, Riecke H, Solla S A. Self-sustained activity in a small-world network of excitable neurons. Phys Rev Lett, 2003, 92: 198101CrossRefGoogle Scholar
  13. 13.
    Liang X, Zhao L. Effect of nonidentical signal phases on signal amplification of two coupled excitable neurons. Neurocomputing, 2014, 127: 21–29CrossRefGoogle Scholar
  14. 14.
    Burić N, Ranković D, Todorović K, et al. Mean field approximation for noisy delay coupled excitable neurons. Physica A, 2010, 389: 3956–3964CrossRefGoogle Scholar
  15. 15.
    Tanabe S, Pakdaman K. Noise-induced transition in excitable neuron models. Biol Cybern, 2001, 85: 269–280zbMATHCrossRefGoogle Scholar
  16. 16.
    Xin B, Ma J, Chen T, et al. Delay-induced Hopf bifurcation in a noisedriven excitable neuron model. Int J Comput Math, 2011, 88: 3255–3270MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Lindner B, García-Ojalvo J, Neiman A, Schimansky-Geier L. Effects of noise in excitable systems. Phys Rep, 2004, 392: 321–424CrossRefGoogle Scholar
  18. 18.
    Jia B, Gu H. Dynamics and physiological roles of stochastic firing patterns near bifurcation points. Int J Bifurcat Chaos, 2017, 27: 1750113MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Desai N S, Rutherford L C, Turrigiano G G. Plasticity in the intrinsic excitability of cortical pyramidal neurons. Nat Neurosci, 1999, 2: 515–520CrossRefGoogle Scholar
  20. 20.
    Tsumoto K, Kitajima H, Yoshinaga T, et al. Bifurcations in Morris-Lecar neuron model. Neurocomputing, 2006, 69: 293–316CrossRefGoogle Scholar
  21. 21.
    Canavier C C, Clark J W, Byrne J H. Routes to chaos in a model of a bursting neuron. Biophys J, 1990, 57: 1245–1251CrossRefGoogle Scholar
  22. 22.
    Silva L R, Amitai Y, Connors B W. Intrinsic oscillations of neocortex generated by layer 5 pyramidal neurons. Science, 1991, 251: 432–435CrossRefGoogle Scholar
  23. 23.
    Connor J A, Stevens C F. Prediction of repetitive firing behaviour from voltage clamp data on an isolated neurone soma. J Physiol, 1971, 213: 31–53CrossRefGoogle Scholar
  24. 24.
    Song X L, Wang C N, Ma J, et al. Transition of electric activity of neurons induced by chemical and electric autapses. Sci China Tech Sci, 2015, 58: 1007–1014CrossRefGoogle Scholar
  25. 25.
    Marder E, Goaillard J M. Variability, compensation and homeostasis in neuron and network function. Nat Rev Neurosci, 2006, 7: 563–574CrossRefGoogle Scholar
  26. 26.
    Lu Q, Gu H, Yang Z, et al. Dynamics of firing patterns, synchronization and resonances in neuronal electrical activities: Experiments and analysis. Acta Mech Sin, 2008, 24: 593–628zbMATHCrossRefGoogle Scholar
  27. 27.
    Ma J, Wu F, Ren G, et al. A class of initials-dependent dynamical systems. Appl Math Comput, 2017, 298: 65–76MathSciNetzbMATHGoogle Scholar
  28. 28.
    Ge M, Jia Y, Kirunda J B, et al. Propagation of firing rate by synchronization in a feed-forward multilayer Hindmarsh-Rose neural network. Neurocomputing, 2018, 320: 60–68CrossRefGoogle Scholar
  29. 29.
    Lu L, Jia Y, Kirunda J B, et al. Effects of noise and synaptic weight on propagation of subthreshold excitatory postsynaptic current signal in a feed-forward neural network. Nonlinear Dyn, 2019, 95: 1673–1686CrossRefGoogle Scholar
  30. 30.
    Gu H, Pan B, Chen G, et al. Biological experimental demonstration of bifurcations from bursting to spiking predicted by theoretical models. Nonlinear Dyn, 2014, 78: 391–407MathSciNetCrossRefGoogle Scholar
  31. 31.
    Ruiz L G B, Rueda R, Cuéllar M P, et al. Energy consumption forecasting based on Elman neural networks with evolutive optimization. Expert Syst Appl, 2018, 92: 380–389CrossRefGoogle Scholar
  32. 32.
    Wang Y, Wang C, Ren G, et al. Energy dependence on modes of electric activities of neuron driven by multi-channel signals. Nonlinear Dyn, 2017, 89: 1967–1987CrossRefGoogle Scholar
  33. 33.
    Yue Y, Liu L, Liu Y, et al. Dynamical response, information transition and energy dependence in a neuron model driven by autapse. Nonlinear Dyn, 2017, 90: 2893–2902CrossRefGoogle Scholar
  34. 34.
    Magistretti P. Neuron-glia metabolic coupling: Role in plasticity and neuroprotection. J Neurol Sci, 2017, 381: 24CrossRefGoogle Scholar
  35. 35.
    Zhang D, Chen Z, Ren J, et al. Energy-harvesting-aided spectrum sensing and data transmission in heterogeneous cognitive radio sensor network. IEEE Trans Veh Technol, 2017, 66: 831–843CrossRefGoogle Scholar
  36. 36.
    Seitzman B A, Abell M, Bartley S C, et al. Cognitive manipulation of brain electric microstates. Neuroimage, 2017, 146: 533–543CrossRefGoogle Scholar
  37. 37.
    Zhu Z, Wang R, Zhu F. The energy coding of a structural neural network based on the Hodgkin-Huxley model. Front Neurosci, 2018, 12: 122CrossRefGoogle Scholar
  38. 38.
    Moujahid A, D’Anjou A, Torrealdea F J, et al. Energy and information in Hodgkin-Huxley neurons. Phys Rev E, 2011, 83: 031912MathSciNetCrossRefGoogle Scholar
  39. 39.
    Wang Z, Wang R. Energy distribution property and energy coding of a structural neural network. Front Comput Neurosci, 2014, 8: 14CrossRefGoogle Scholar
  40. 40.
    Ma J, Tang J. A review for dynamics of collective behaviors of network of neurons. Sci China Tech Sci, 2015, 58: 2038–2045CrossRefGoogle Scholar
  41. 41.
    Ma J, Tang J. A review for dynamics in neuron and neuronal network. Nonlinear Dyn, 2017, 89: 1569–1578MathSciNetCrossRefGoogle Scholar
  42. 42.
    Barry J F, Turner M J, Schloss J M, et al. Optical magnetic detection of single-neuron action potentials using quantum defects in diamond. Proc Natl Acad Sci USA, 2016, 113: 14133–14138CrossRefGoogle Scholar
  43. 43.
    Reilly J P. peripheral nerve stimulation by induced electric currents: Exposure to time-varying magnetic fields. Med Biol Eng Comput, 1989, 27: 101–110CrossRefGoogle Scholar
  44. 44.
    Ueno S, Lövsund P, Oberg P A. Effect of time-varying magnetic fields on the action potential in lobster giant axon. Med Biol Eng Comput, 1986, 24: 521–526CrossRefGoogle Scholar
  45. 45.
    Wikswo J P, Barach J P, Freeman J A. Magnetic field of a nerve impulse: First measurements. Science, 1980, 208: 53–55CrossRefGoogle Scholar
  46. 46.
    Kwong K K, Belliveau J W, Chesler D A, et al. Dynamic magnetic resonance imaging of human brain activity during primary sensory stimulation. Proc Natl Acad Sci USA, 1992, 89: 5675–5679CrossRefGoogle Scholar
  47. 47.
    Fox M D, Raichle M E. Spontaneous fluctuations in brain activity observed with functional magnetic resonance imaging. Nat Rev Neurosci, 2007, 8: 700–711CrossRefGoogle Scholar
  48. 48.
    Barbieri F, Trauchessec V, Caruso L, et al. Local recording of biological magnetic fields using Giant Magneto Resistance-based microprobes. Sci Rep, 2016, 6: 39330CrossRefGoogle Scholar
  49. 49.
    Lv M, Wang C, Ren G, et al. Model of electrical activity in a neuron under magnetic flow effect. Nonlinear Dyn, 2016, 85: 1479–1490CrossRefGoogle Scholar
  50. 50.
    Wu F, Wang C, Xu Y, et al. Model of electrical activity in cardiac tissue under electromagnetic induction. Sci Rep, 2016, 6: 28CrossRefGoogle Scholar
  51. 51.
    Ma J, Wu F, Hayat T, et al. Electromagnetic induction and radiation-induced abnormality of wave propagation in excitable media. Physica A, 2017, 486: 508–516MathSciNetCrossRefGoogle Scholar
  52. 52.
    Tian C, Cao L, Bi H, et al. Chimera states in neuronal networks with time delay and electromagnetic induction. Nonlinear Dyn, 2018, 93: 1695–1704CrossRefGoogle Scholar
  53. 53.
    Lu L L, Jia Y, Xu Y, et al. Energy dependence on modes of electric activities of neuron driven by different external mixed signals under electromagnetic induction. Sci China Tech Sci, 2019, 62: 427–440CrossRefGoogle Scholar
  54. 54.
    Xu Y, Jia Y, Wang H, et al. Spiking activities in chain neural network driven by channel noise with field coupling. Nonlinear Dyn, 2019, 95: 3237–3247CrossRefGoogle Scholar
  55. 55.
    Ge M, Jia Y, Xu Y, et al. Wave propagation and synchronization induced by chemical autapse in chain Hindmarsh-Rose neural network. Appl Math Comput, 2019, 352: 136–145MathSciNetGoogle Scholar
  56. 56.
    Liu Z, Wang C, Zhang G, et al. Synchronization between neural circuits connected by hybrid synapse. Int J Mod Phys B, 2019, 33: 1950170MathSciNetCrossRefGoogle Scholar
  57. 57.
    Lv M, Ma J, Yao Y G, et al. Synchronization and wave propagation in neuronal network under field coupling. Sci China Tech Sci, 2019, 62: 448–457CrossRefGoogle Scholar
  58. 58.
    Xu Y, Jia Y, Ma J, et al. Collective responses in electrical activities of neurons under field coupling. Sci Rep, 2018, 8: 1349CrossRefGoogle Scholar
  59. 59.
    Ma J, Zhang G, Hayat T, et al. Model electrical activity of neuron under electric field. Nonlinear Dyn, 2019, 95: 1585–1598CrossRefGoogle Scholar
  60. 60.
    Sussillo D. Neural circuits as computational dynamical systems. Curr Opin Neurobiol, 2014, 25: 156–163CrossRefGoogle Scholar
  61. 61.
    Zhang J, Liao X. Effects of initial conditions on the synchronization of the coupled memristor neural circuits. Nonlinear Dyn, 2019, 95: 1269–1282CrossRefGoogle Scholar
  62. 62.
    Bao H, Liu W, Chen M. Hidden extreme multistability and dimensionality reduction analysis for an improved non-autonomous memristive FitzHugh-Nagumo circuit. Nonlinear Dyn, 2019, 96: 1879–1894CrossRefGoogle Scholar
  63. 63.
    Hu X, Liu C, Liu L, et al. Chaotic dynamics in a neural network under electromagnetic radiation. Nonlinear Dyn, 2018, 91: 1541–1554CrossRefGoogle Scholar
  64. 64.
    Innocenti G, Di Marco M, Forti M, et al. Prediction of period doubling bifurcations in harmonically forced memristor circuits. Nonlinear Dyn, 2019, 96: 1169–1190CrossRefGoogle Scholar
  65. 65.
    Zhang G, Ma J, Alsaedi A, et al. Dynamical behavior and application in Josephson Junction coupled by memristor. Appl Math Comput, 2018, 321: 290–299MathSciNetzbMATHGoogle Scholar
  66. 66.
    Yuan F, Deng Y, Li Y, et al. The amplitude, frequency and parameter space boosting in a memristor-meminductor-based circuit. Nonlinear Dyn, 2019, 96: 389–405CrossRefGoogle Scholar
  67. 67.
    Hodgkin A L, Huxley A F, Katz B. Measurement of current-voltage relations in the membrane of the giant axon of Loligo. J Physiol, 1952, 116: 424–448CrossRefGoogle Scholar
  68. 68.
    Chua L. Memristor: The missing circuit element. IEEE Trans Circuit Theor, 1971, 18: 507–519CrossRefGoogle Scholar
  69. 69.
    Strukov D B, Snider G S, Stewart D R, et al. The missing memristor found. Nature, 2008, 453: 80–83CrossRefGoogle Scholar
  70. 70.
    Sarasola C, Torrealdea F J, D’Anjou A, et al. Energy balance in feedback synchronization of chaotic systems. Phys Rev E, 2004, 69: 011606CrossRefGoogle Scholar
  71. 71.
    Guo S, Xu Y, Wang C, et al. Collective response, synapse coupling and field coupling in neuronal network. Chaos Solitons Fractals, 2017, 105: 120–127MathSciNetCrossRefGoogle Scholar
  72. 72.
    Grossman N, Bono D, Dedic N, et al. Noninvasive deep brain stimulation via temporally interfering electric fields. Cell, 2017, 169: 1029–1041.e16CrossRefGoogle Scholar
  73. 73.
    Ge M, Jia Y, Xu Y, et al. Mode transition in electrical activities of neuron driven by high and low frequency stimulus in the presence of electromagnetic induction and radiation. Nonlinear Dyn, 2018, 91: 515–523CrossRefGoogle Scholar
  74. 74.
    Wu F, Ma J, Ren G. Synchronization stability between initial-dependent oscillators with periodical and chaotic oscillation. J Zhejiang Univ Sci A, 2018, 19: 889–903CrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of PhysicsLanzhou University of TechnologyLanzhouChina
  2. 2.NAAM-Research GroupKing Abdulaziz UniversityJeddahSaudi Arabia

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