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Model of electrical activity in a neuron under magnetic flow effect

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Abstract

The electric activities of neurons are dependent on the complex electrophysiological condition in neuronal system, and it indicates that the complex distribution of electromagnetic field could be detected in the neuronal system. According to the Maxwell electromagnetic induction theorem, the dynamical behavior in electric activity in each neuron can be changed due to the effect of internal bioelectricity of nervous system (e.g., fluctuation of ion concentration inside and outside of cell). As a result, internal fluctuation of electromagnetic field is established and the effect of magnetic flux across the membrane should be considered during the emergence of collective electrical activities and signals propagation among a large set of neurons. In this paper, the variable for magnetic flow is proposed to improve the previous Hindmarsh–Rose neuron model; thus, a four-variable neuron model is designed to describe the effect of electromagnetic induction on neuronal activities. Within the new neuron model, the effect of magnetic flow on membrane potential is described by imposing additive memristive current on the membrane variable, and the memristive current is dependent on the variation of magnetic flow. The dynamics of this modified model is discussed, and multiple modes of electric activities can be observed by changing the initial state, which indicates memory effect of neuronal system. Furthermore, a practical circuit is designed for this improved neuron model, and this model could be suitable for further investigation of electromagnetic radiation on biological neuronal system.

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References

  1. Buschman, T.L., Denovellis, E.L., Diogo, C., et al.: Synchronous oscillatory neural ensembles for rules in the prefrontal cortex. Neuron 76, 838–846 (2012)

    Article  Google Scholar 

  2. Wig, G.S., Schlaggar, B.L., Petersen, S.E.: Concepts and principles in the analysis of brain networks. Ann. N.Y. Acad. Sci. 1224, 126–146 (2011)

    Article  Google Scholar 

  3. Seely, J., Crotty, P.: Optimization of the leak conductance in the squid giant axon. Phys. Rev. E 82, 021906 (2010)

    Article  Google Scholar 

  4. Postnov, D.E., Koreshkov, R.N., Brazhe, N.A., et al.: Dynamical patterns of calcium signaling in a functional model of neuron–astrocyte networks. J. Biol. Phys. 35, 425–445 (2009)

    Article  Google Scholar 

  5. Volman, V., Bazhenov, M., Sejnowski, T.J.: Computational models of neuron–astrocyte interaction in epilepsy. Front. Comput. Neurosci. 6, 58 (2012)

    Article  Google Scholar 

  6. Volman, V., Perc, M., Bazhenov, M.: Gap junctions and epileptic seizures-two sides of the same coin? PLoS One 6, e20572 (2011)

    Article  Google Scholar 

  7. Ozer, M., Ekmekci, N.H.: Effect of channel noise on the time-course of recovery from inactivation of sodium channels. Phys. Lett. A 338, 150–154 (2005)

    Article  MATH  Google Scholar 

  8. Barthélemy, M.: Spatial networks. Phys. Rep. 499, 1–101 (2011)

    Article  MathSciNet  Google Scholar 

  9. Herz, A.V.M., Gollisch, T., Machens, C.K., et al.: Modeling single-neuron dynamics and computations: a balance of detail and abstraction. Science 314, 80–85 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gerstner, W., Kistler, W.M.: Spiking Neuron Models Single Neurons, Populations, Plasticity. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  11. Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)

    Article  Google Scholar 

  12. Morris, C., Lecar, H.: Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35, 193–213 (1981)

    Article  Google Scholar 

  13. Izhikevich, E.M.: Which model to use for cortical spiking neurons? IEEE Trans. Neural Netw. 15, 1063–1070 (2004)

    Article  Google Scholar 

  14. Ibarz, B., Casado, J.M., Sanjuán, M.A.F.: Map-based models in neuronal dynamics. Phys. Rep. 501, 1–74 (2011)

    Article  Google Scholar 

  15. Perc, M., Marhl, M.: Amplification of information transfer in excitable systems that reside in a steady state near a bifurcation point to complex oscillatory behavior. Phys. Rev. E 71, 026229 (2005)

    Article  Google Scholar 

  16. Hindmarsh, J.L., Rose, R.M.: A model of the nerve impulse using two first-order differential equations. Nature (London) 296, 162–164 (1982)

    Article  Google Scholar 

  17. Gu, H.G., Pan, B.B., Chen, G.R.: Biological experimental demonstration of bifurcations from bursting to spiking predicted by theoretical models. Nonlinear Dyn. 78, 391–407 (2014)

    Article  MathSciNet  Google Scholar 

  18. Storace, M., Linaro, D., de Lange, E.: The Hindmarsh–Rose neuron model: bifurcation analysis and piecewiselinear approximations. Chaos 18, 033128 (2008)

    Article  MathSciNet  Google Scholar 

  19. Pinto, R.D., Varona, P., Volkovskii, A., et al.: Synchronous behavior of two coupled electronic neurons. Phys. Rev. E 62, 2644 (2000)

    Article  Google Scholar 

  20. Moujahid, A., d’Anjou, A., Torrealdea, F.J., et al.: Efficient synchronization of structurally adaptive coupled Hindmarsh–Rose neurons. Chaos Solitons Fractals 44, 929–933 (2011)

    Article  Google Scholar 

  21. Selverston, A., Rabinovich, M., Abarbanel, H.D., et al.: Reliable circuits for irregular neurons: a dynamical approach to understanding central pattern generators. J. Physiol. 94, 357 (2000)

    Google Scholar 

  22. Rech, P.C.: Dynamics in the parameter space of a neuron model. Chin. Phys. Lett. 29, 060506 (2012)

    Article  Google Scholar 

  23. Gu, H.G., Pan, B.B.: A four-dimensional neuronal model to describe the complex nonlinear dynamics observed in the firing patterns of a sciatic nerve chronic constriction injury model. Nonlinear Dyn. 81, 2107–2126 (2015)

    Article  MathSciNet  Google Scholar 

  24. Uzun, R., Ozer, M., Perc, M.: Can scale-freeness offset delayed signal detection in neuronal networks? EPL 105, 60002 (2014)

    Article  Google Scholar 

  25. Ozer, M., Uzuntarla, M., Perc, M., et al.: Spike latency and jitter of neuronal membrane patches with stochastic Hodgkin–Huxley channels. J. Theor. Biol. 261, 83–92 (2009)

    Article  Google Scholar 

  26. Aggarwal, A., Kumar, M., Rawat, T.K., et al.: Optimal design of 2-D FIR filters with quadrantally symmetric properties using fractional derivative constraints. Circ. Syst. Signal Process. (2016). doi:10.1007/s00034-016-0283-x

    Google Scholar 

  27. Kumar, M., Rawat, T.K.: Fractional order digital differentiator design based on power function and least-squares. Int. J. Electron. (2016). doi:10.1080/00207217.2016.1138520

    Google Scholar 

  28. Wang, S.T., Wang, W., Liu, F.: Propagation of firing rate in a feed-forward neuronal network. Phys. Rev. Lett. 96, 018103 (2006)

    Article  Google Scholar 

  29. Suffczynski, p, Kalitzina, S., Lopes Da Silva, F.H.: Dynamics of non-convulsive epileptic phenomena modeled by a bistable neuronal network. Neuroscience 126, 467–484 (2004)

    Article  Google Scholar 

  30. Cullheim, S., Thams, S.: The microglial networks of the brain and their role in neuronal network plasticity after lesion. Brain Res. Rev. 55, 89–96 (2007)

    Article  Google Scholar 

  31. Wang, R., Zhang, Z.Z., Ma, J., et al.: Spectral properties of the temporal evolution of brain network structure. Chaos 25, 123112 (2015)

    Article  MathSciNet  Google Scholar 

  32. Bao, B.C., Liu, Z., Xu, J.P.: Steady periodic memristor oscillator with transient chaotic behaviors. Electron. Lett. 46, 228–230 (2010)

    Article  Google Scholar 

  33. Muthuswamy, B.: Implementing memristor based chaotic circuits. Int. J. Bifurc. Chaos 20, 1335–1350 (2010)

    Article  MATH  Google Scholar 

  34. Li, Q.D., Zeng, H.Z., Li, J.: Hyperchaos in a 4D memristive circuit with infinitely many stable equilibria. Nonlinear Dyn. 79, 2295–2308 (2015)

    Article  MathSciNet  Google Scholar 

  35. Ma, J., Tang, J.: A review for dynamics of collective behaviors of network of neurons. Sci. China Technol. Sci. 58, 2038–2045 (2015)

    Article  Google Scholar 

  36. Song, X.L., Jin, W.Y., Ma, J.: Energy dependence on the electric activities of a neuron. Chin. Phys. B 24, 128710 (2015)

    Article  Google Scholar 

  37. Song, X.L., Wang, C.N., Ma, J., et al.: Transition of electric activity of neurons induced by chemical and electric autapses. Sci. China Technol. Sci. 58, 1007–1014 (2015)

    Article  Google Scholar 

  38. Qin, H.X., Ma, J., Jin, W.Y., et al.: Dynamics of electrical activities in neuron and neurons of network induced by autapses. Sci. China Technol. Sci. 57, 936–946 (2014)

    Article  Google Scholar 

  39. Yılmaz, E., Baysal, V., Perc, M., et al.: Enhancement of pacemaker induced stochastic resonance by an autapse in a scale-free neuronal network. Sci. China Technol. Sci. 59, 364–370 (2016)

    Article  Google Scholar 

  40. Qin, H.X., Ma, J., Wang, C.N., et al.: Autapse-induced target wave, spiral wave in regular network of neurons. Sci. China Phys. Mech. Astron. 57, 1918–1926 (2014)

    Article  Google Scholar 

  41. Ren, G.D., Wu, G., Ma, J.: Simulation of electric activity of neuron by setting up a reliable neuronal circuit driven by electric autapse. Acta Phys. Sin. 64, 058702 (2015). In Chinese

    Google Scholar 

  42. Ma, J., Chen, Z.Q., Wang, Z.L., et al.: A four-wing hyper-chaotic attractor generated from a 4-D memristive system with a line equilibrium. Nonlinear Dyn. 81, 1275–1288 (2015)

    Article  Google Scholar 

  43. Chen, M., Li, M.Y., Yu, Q., et al.: Dynamics of self-excited attractors and hidden attractors in generalized memristor-based Chua’s circuit. Nonlinear Dyn. 81, 215–226 (2015)

    Article  MathSciNet  Google Scholar 

  44. Pei, J.S., Wright, J.P., Todd, M.D., et al.: Understanding memristors and memcapacitors in engineering mechanics applications. Nonlinear Dyn. 80, 457–489 (2015)

    Article  Google Scholar 

  45. Li, Q.D., Tang, S., Zeng, H.Z., et al.: On hyperchaos in a small memristive neural network. Nonlinear Dyn. 78, 1087–1099 (2014)

    Article  MATH  Google Scholar 

  46. Pham, V.T., Jafari, S., Vaidyanathan, S., et al.: A novel memristive neural network with hidden attractors and its circuitry implementation. Sci. China Technol. Sci. 59, 358–363 (2016)

    Article  Google Scholar 

  47. Wu, H.G., Bao, B.C., Liu, Z., et al.: Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator. Nonlinear Dyn. 83, 893–903 (2016)

    Article  MathSciNet  Google Scholar 

  48. Ma, J., Tang, J., Zhang, A.H., et al.: Robustness and breakup of the spiral wave in a two-dimensional lattice network of neurons. Sci. China Phys. Mech. Astron. 53, 672–679 (2010)

    Article  Google Scholar 

  49. Ma, J., Wu, Y., Wu, N.J., et al.: Detection of ordered wave in the networks of neurons with changeable connection. Sci. China Phys. Mech. Astron. 56, 952–959 (2013)

    Article  Google Scholar 

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11265008 (MJ) and 11365014 (WCN).

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Authors and Affiliations

  1. Department of Physics, Lanzhou University of Technology, Lanzhou, 730050, China

    Mi Lv, Chunni Wang, Guodong Ren, Jun Ma & Xinlin Song

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  1. Mi Lv
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  2. Chunni Wang
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  3. Guodong Ren
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  4. Jun Ma
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  5. Xinlin Song
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Correspondence to Jun Ma.

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Lv, M., Wang, C., Ren, G. et al. Model of electrical activity in a neuron under magnetic flow effect. Nonlinear Dyn 85, 1479–1490 (2016). https://doi.org/10.1007/s11071-016-2773-6

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  • DOI: https://doi.org/10.1007/s11071-016-2773-6

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