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Energy level transition and mode transition in a neuron

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Abstract

During continuous diffusion and propagation of intracellular ions, energy transition between electric and magnetic field is proceeded to present appropriate firing patterns. For theoretical neuron models, an equivalent Hamilton energy is derived by Helmholtz theorem. For neural circuits, the Hamilton energy can also be obtained by applying scale transformation on the field energy function. External stimuli injects energy into the neuron, and the energy level transition is induced accompanying with mode transition in the neuronal activity. On the flip side, large external stimuli can induce shape deformation of the cell and possible parameter shift occurs to keep neuron on appropriate energy level in the deterministic neuron models. In this letter, energy function for Hindmarh–Rose neuron is estimated and a criterion for transition between energy levels and firing modes is defined and explained. It provides possible clues for understanding the dependencies of pattern selection in discharge mode on energy level and adaptive controllability in neurons, and thus the neural activities in neurons and nervous system can be controlled by regulating energy flow.

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The data in this study are available from the corresponding author upon right request.

References

  1. Bahramian, A., et al.: Collective behavior in a two-layer neuronal network with time-varying chemical connections that are controlled by a Petri net. Chaos 31, 033138 (2021)

    MathSciNet  Google Scholar 

  2. Chen, X., et al.: Searching for collective behavior in a small brain. Phys. Rev. E 99, 052418 (2019)

    Google Scholar 

  3. Diaz, M.M.S., et al.: Similar local neuronal dynamics may lead to different collective behavior. Phys. Rev. E 104, 064309 (2021)

    Google Scholar 

  4. Wang, Z., et al.: Chimeras in an adaptive neuronal network with burst-timing-dependent plasticity. Neurocomputing 406, 117–126 (2020)

    Google Scholar 

  5. Xu, Y., Ma, J.: Control of firing activities in thermosensitive neuron by activating excitatory autapse. Chin. Phys. B 30, 100501 (2021)

    Google Scholar 

  6. Xu, Y., Guo, Y., Ren, G., Ma, J.: Dynamics and stochastic resonance in a thermosensitive neuron. Appl. Math. Comput. 385, 125427 (2020)

    MathSciNet  Google Scholar 

  7. Zhang, X., Ma, J.: Wave filtering and firing modes in a light-sensitive neural circuit. J. Zhejiang Univ. Sci. A 22(9), 707–720 (2021)

    Google Scholar 

  8. Xie, Y., Ma, J.: How to discern external acoustic waves in a piezoelectric neuron under noise? J. Biol. Phys. 48, 339–353 (2022)

    Google Scholar 

  9. Zhou, P., Zhang, X., Ma, J.: How to wake up the electric synapse coupling between neurons? Nonlinear Dyn. 108, 1681–1695 (2022)

    Google Scholar 

  10. Njitacke, Z.T., et al.: Coexistence of firing patterns and its control in two neurons coupled through an asymmetric electrical synapse. Chaos 30, 023101 (2020)

    MathSciNet  Google Scholar 

  11. Mirzaei, S., et al.: Synchronization in repulsively coupled oscillators. Phys. Rev. E 107, 014201 (2023)

    MathSciNet  Google Scholar 

  12. Curti, S., O’Brien, J.: Characteristics and plasticity of electrical synaptic transmission. BMC Cell Biol. 17, 59–70 (2016)

    Google Scholar 

  13. Wang, R., et al.: Transition of spatiotemporal patterns in neuronal networks with chemical synapses. Commun. Nonlinear Sci. Numer. Simul. 40, 80–88 (2016)

    MathSciNet  Google Scholar 

  14. Wang, Z., et al.: Synchronization of the neurons coupled with sequential developing electrical and chemical synapses. Math. Biosci. Eng. 19, 1877–1890 (2022)

    Google Scholar 

  15. Hu, D., Cao, H.: Stability and synchronization of coupled Rulkov map-based neurons with chemical synapses. Commun. Nonlinear Sci. Numer. Simul. 35, 105–122 (2016)

    MathSciNet  Google Scholar 

  16. Shafiei, M., et al.: Time delayed chemical synapses and synchronization in multilayer neuronal networks with ephaptic inter-layer coupling. Commun. Nonlinear Sci. Numer. Simul. 84, 105175 (2020)

    MathSciNet  Google Scholar 

  17. Zhou, P., et al.: Energy balance between two thermosensitive circuits under field coupling. Nonlinear Dyn. 110, 1879–1895 (2022)

    Google Scholar 

  18. Yao, W., et al.: Exponential multistability of memristive Cohen–Grossberg neural networks with stochastic parameter perturbations. Appl. Math. Comput. 386, 125483 (2020)

    MathSciNet  Google Scholar 

  19. Rajagopal, K., et al.: Effect of magnetic induction on the synchronizability of coupled neuron network. Chaos 31, 083115 (2021)

    MathSciNet  Google Scholar 

  20. Yang, Z., Zhang, Y., Wu, F.: Memristive magnetic coupling feedback induces wave-pattern transition. Nonlinear Dyn. 100, 647–658 (2020)

    Google Scholar 

  21. Wang, Y., et al.: Passivity and passification of memristive recurrent neural networks with multi-proportional delays and impulse. Appl. Math. Comput. 369, 124838 (2020)

    MathSciNet  Google Scholar 

  22. Bao, H., et al.: Memristor synapse-coupled memristive neuron network: synchronization transition and occurrence of chimera. Nonlinear Dyn. 100, 937–950 (2020)

    Google Scholar 

  23. Liu, F., Song, Q., Cao, J.: Improvements and applications of entrainment control for nonlinear dynamical systems. Chaos 18, 043120 (2008)

    MathSciNet  Google Scholar 

  24. Wu, F., Guo, Y., Ma, J.: Reproduce the biophysical function of chemical synapse by using a memristive synapse. Nonlinear Dyn. 109, 2063–2084 (2022)

    Google Scholar 

  25. Lin, H., et al.: Firing multistability in a locally active memristive neuron model. Nonlinear Dyn. 100, 3667–3683 (2020)

    Google Scholar 

  26. Lin, H., et al.: Review on chaotic dynamics of memristive neuron and neural network. Nonlinear Dyn. 106, 959–973 (2021)

    Google Scholar 

  27. Shilnikov, A., Calabrese, R.L., Cymbalyuk, G.: Mechanism of bistability: tonic spiking and bursting in a neuron model. Phys. Rev. E 71, 056214 (2005)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  29. Channell, P., Cymbalyuk, G., Shilnikov, A.: Origin of bursting through homoclinic spike adding in a neuron model. Phys. Rev. Lett. 98, 134101 (2007)

    Google Scholar 

  30. Innocenti, G., et al.: Dynamical phases of the Hindmarsh–Rose neuronal model: studies of the transition from bursting to spiking chaos. Chaos 17, 043128 (2007)

    MathSciNet  Google Scholar 

  31. Yang, F., Xu, Y., Ma, J.: A memristive neuron and its adaptability to external electric field. Chaos 33, 023110 (2023)

    MathSciNet  Google Scholar 

  32. Xie, Y., et al.: Enhance sensitivity to illumination and synchronization in light-dependent neurons. Chin. Phys. B 30, 120510 (2021)

    Google Scholar 

  33. Guo, Y., et al.: Biophysical mechanism of signal encoding in an auditory neuron. Nonlinear Dyn. 105, 3603–3614 (2021)

    Google Scholar 

  34. Khasabov, S.G., et al.: Responses of neurons in the primary somatosensory cortex to itch-and pain-producing stimuli in rats. J. Neurophysiol. 123, 1944–1954 (2020)

    Google Scholar 

  35. Wan, Q., et al.: Complex dynamics in a Hopfield neural network under electromagnetic induction and electromagnetic radiation. Chaos 32, 073107 (2022)

    MathSciNet  Google Scholar 

  36. Zhang, S., et al.: Multi-scroll hidden attractor in memristive HR neuron model under electromagnetic radiation and its applications. Chaos 31, 011101 (2021)

    MathSciNet  Google Scholar 

  37. Takembo, C.N., et al.: Effect of electromagnetic radiation on the dynamics of spatiotemporal patterns in memristor-based neuronal network. Nonlinear Dyn. 95, 1067–1078 (2019)

    Google Scholar 

  38. Wu, J., Ma, S.: Coherence resonance of the spiking regularity in a neuron under electromagnetic radiation. Nonlinear Dyn. 96, 1895–1908 (2019)

    Google Scholar 

  39. Mitaim, S., Kosko, B.: Adaptive stochastic resonance in noisy neurons based on mutual information. IEEE Trans. Neural Netw. 15, 1526–1540 (2004)

    Google Scholar 

  40. Lee, S.G., Neiman, A., Kim, S.: Coherence resonance in a Hodgkin–Huxley neuron. Phys. Rev. E 57, 3292 (1998)

    Google Scholar 

  41. Shinohara, Y., et al.: Array-enhanced coherence resonance and forced dynamics in coupled FitzHugh–Nagumo neurons with noise. Phys. Rev. E 65, 051906 (2002)

    Google Scholar 

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

    Google Scholar 

  43. Yao, Y., Ma, J.: Logical stochastic and vibrational resonances induced by periodic force in the FitzHugh–Nagumo neuron. Eur. Phys. J. Plus. 137, 1214 (2022)

    Google Scholar 

  44. Yao, Y., et al.: Chaos-induced Set–Reset latch operation. Chaos Soliton. Fractal. 152, 111339 (2021)

    MathSciNet  Google Scholar 

  45. Nobukawa, S., et al.: Analysis of chaotic resonance in Izhikevich neuron model. PLoS ONE 10, e0138919 (2015)

    Google Scholar 

  46. Nobukawa, S., Shibata, N.: Controlling chaotic resonance using external feedback signals in neural systems. Sci. Rep. 9, 4990 (2019)

    Google Scholar 

  47. Akgul, A., et al.: Complex bio rhythms. Eur. Phys. J Spec. Top. 231, 815–818 (2022)

    Google Scholar 

  48. Parastesh, F., et al.: Synchronizability of two neurons with switching in the coupling. Appl. Math. Comput. 350, 217–223 (2019)

    MathSciNet  Google Scholar 

  49. Upadhyay, R.K., Mondal, A., Teka, W.W.: Mixed mode oscillations and synchronous activity in noise induced modified Morris–Lecar neural system. Int. Journal Bifurcat. Chaos 27, 1730019 (2017)

    MathSciNet  Google Scholar 

  50. Zhang, S., et al.: Multi-scroll hidden attractor in memristive HR neuron model under electromagnetic radiation and its applications. Chaos 31, 011101 (2021)

    MathSciNet  Google Scholar 

  51. Zhao, Z., Li, L., Gu, H.: Excitatory autapse induces different cases of reduced neuronal firing activities near Hopf bifurcation. Commun. Nonlinear Sci. Numer. Simul. 85, 105250 (2020)

    MathSciNet  Google Scholar 

  52. Xu, Y., Wu, Y.: Analytical predictions of stable and unstable firings to chaos in a Hindmarsh–Rose neuron system. Chaos 32, 113113 (2022)

    MathSciNet  Google Scholar 

  53. Wu, F., et al.: Can Hamilton energy feedback suppress the chameleon chaotic flow? Nonlinear Dyn. 94, 669–677 (2018)

    Google Scholar 

  54. Njitacke, Z.T., et al.: Hamiltonian energy and coexistence of hidden firing patterns from bidirectional coupling between two different neurons. Cogn. Neurodyn. 16, 899–916 (2022)

    Google Scholar 

  55. Lu, L.L., et al.: Energy dependence on modes of electric activities of neuron driven by different external mixed signals under electromagnetic induction. Sci. China Technol. Sci. 62, 427–440 (2019)

    Google Scholar 

  56. Usha, K., Subha, P.A.: Energy feedback and synchronous dynamics of Hindmarsh–Rose neuron model with memristor. Chin. Phys. B 28, 020502 (2019)

    Google Scholar 

  57. Kobe, D.H.: Helmholtz’s theorem revisited. Am. J. Phys. 54, 552–554 (1986)

    Google Scholar 

  58. Xie, Y., Yao, Z., Ma, J.: Phase synchronization and energy balance between neurons. Front. Inf. Technol. Electron. Eng. 23, 1407–1420 (2022)

    Google Scholar 

  59. Ma, X., Xu, Y.: Taming the hybrid synapse under energy balance between neurons. Chaos Soliton. Fract. 159, 112149 (2022)

    MathSciNet  Google Scholar 

  60. Xie, Y., Zhou, P., Ma, J.: Energy balance and synchronization via inductive-coupling in functional neural circuits. Appl. Math. Modell. 113, 175–187 (2023)

    MathSciNet  Google Scholar 

  61. Torrealdea, F.J., et al.: Energy aspects of the synchronization of model neurons. Phys. Rev. E 74, 011905 (2006)

    Google Scholar 

  62. Yao, Z., et al.: Energy flow-guided synchronization between chaotic circuits. Appl. Math. Comput. 374, 124998 (2020)

    MathSciNet  Google Scholar 

  63. Xie, Y., et al.: Estimate physical reliability in Hindmarsh–Rose neuron. Phys. Lett. A 464, 128693 (2023)

    MathSciNet  Google Scholar 

  64. Gu, H.: Biological experimental observations of an unnoticed chaos as simulated by the Hindmarsh–Rose model. PLoS ONE 8, e81759 (2013)

    Google Scholar 

  65. Yang, J., et al.: Responsiveness of a neural pacemaker near the bifurcation point. Neurosci. Lett. 392, 105–109 (2006)

    Google Scholar 

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Funding

This project is partially supported by the Postdoctoral Research Foundation of China Nos. 2021M702036.

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  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China

    Xiaodi Li & Ying Xu

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  1. Xiaodi Li
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  2. Ying Xu
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Correspondence to Ying Xu.

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Li, X., Xu, Y. Energy level transition and mode transition in a neuron. Nonlinear Dyn 112, 2253–2263 (2024). https://doi.org/10.1007/s11071-023-09147-6

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