Abstract
Most of nonlinear oscillators composed of capacitive and inductive variables can obtain the Hamilton energy by using the Helmholtz theorem when the models are rewritten in equivalent vector forms. The energy functions for biophysical neurons can be obtained by applying scale transformation on the physical field energy in their equivalent neural circuits. Realistic dynamical systems often have exact energy functions, while some mathematical models just suggest generic Lyapunov functions, and the energy function is effective to predict mode transition. In this paper, a memristive oscillator is approached by two kinds of memristor-based nonlinear circuits, and the energy functions are defined to predict the dependence of oscillatory modes on energy level. In absence of capacitive variable for capacitor, the physical timet and chargeq are converted into dimensionless variables by using combination of resistance and inductance (L,R), e.g.,τ=t×R/L. Discrete energy function for each memristive map is proposed by applying the similar weights as energy function for the memristive oscillator. For example, energy function for the map is obtained by replacing the variables and parameters of the memristive oscillator with corresponding variables and parameters for the memristive map. The memristive map prefers to keep lower average energy than the memristive oscillator, and chaos is generated in a discrete system with two variables. The scheme is helpful for energy definition in maps, and it provides possible guidance for verifying the reliability of maps by considering the energy characteristic.
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Heinrich M, Dahms T, Flunkert V, et al. Symmetry-breaking transitions in networks of nonlinear circuit elements. New J Phys, 2010, 12: 113030
Kenkel S W, Straley J P. Percolation theory of nonlinear circuit elements. Phys Rev Lett, 1982, 49: 767–770
Sivaganesh G, Srinivasan K, Arulgnanam A. Analytical studies on the dynamics of higher-dimensional nonlinear circuit systems. Pramana, 2022, 96: 185
Ardila V, Ramirez F, Suarez A. Analytical and numerical bifurcation analysis of circuits based on nonlinear resonators. IEEE Trans Microwave Theor Techn, 2021, 69: 4392–4405
Gao X, Mou J, Xiong L, et al. A fast and efficient multiple images encryption based on single-channel encryption and chaotic system. Nonlinear Dyn, 2022, 108: 613–636
Xin J, Hu H, Zheng J. 3D variable-structure chaotic system and its application in color image encryption with new Rubik’s Cube-like permutation. Nonlinear Dyn, 2023, 111: 7859–7882
Ning X, Dong Q, Zhou S, et al. Construction of new 5D Hamiltonian conservative hyperchaotic system and its application in image encryption. Nonlinear Dyn, 2023, 111: 20425–20446
Liu X, Tong X, Wang Z, et al. Construction of controlled multi-scroll conservative chaotic system and its application in color image encryption. Nonlinear Dyn, 2022, 110: 1897–1934
Wang R, Wang Y, Xu X, et al. Brain works principle followed by neural information processing: A review of novel brain theory. Artif Intell Rev, 2023, 56: 285–350
Quaranta G, Lacarbonara W, Masri S F. A review on computational intelligence for identification of nonlinear dynamical systems. Nonlinear Dyn, 2020, 99: 1709–1761
Groschner L N, Malis J G, Zuidinga B, et al. A biophysical account of multiplication by a single neuron. Nature, 2022, 603: 119–123
Tagluk M E, Isik I. Communication in nano devices: Electronic based biophysical model of a neuron. Nano Commun Netw, 2019, 19: 134–147
Wu F Q, Ma J, Zhang G. Energy estimation and coupling synchronization between biophysical neurons. Sci China Tech Sci, 2020, 63: 625–636
Clark R, Fuller L, Platt J A, et al. Reduced-dimension, biophysical neuron models constructed from observed data. Neural Comput, 2022, 34: 1545–1587
Sotero R C, Trujillo-Barreto N J. Biophysical model for integrating neuronal activity, EEG, fMRI and metabolism. NeuroImage, 2008, 39: 290–309
Ma J. Biophysical neurons, energy, and synapse controllability: A review. J Zhejiang Univ Sci A, 2023, 24: 109–129
Wu F Q, Guo Y T, Ma J. Energy flow accounts for the adaptive property of functional synapses. Sci China Tech Sci, 2023, 66: 3139–3152
Yang F, Xu Y, Ma J. A memristive neuron and its adaptability to external electric field. Chaos-An Interdisciplinary J NOnlinear Sci, 2023, 33: 023110
Xie Y, Yao Z, Ma J. Formation of local heterogeneity under energy collection in neural networks. Sci China Tech Sci, 2023, 66: 439–455
Yang F, Wang Y, Ma J. Creation of heterogeneity or defects in a memristive neural network under energy flow. Commun NOnlinear Sci Numer Simul, 2023, 119: 107127
Real E, Asari H, Gollisch T, et al. Neural circuit inference from function to structure. Curr Biol, 2017, 27: 189–198
Pan Y, Monje M. Activity shapes neural circuit form and function: A historical perspective. J Neurosci, 2020, 40: 944–954
Davis F P, Nern A, Picard S, et al. A genetic, genomic, and computational resource for exploring neural circuit function. eLife, 2020, 9: e50901
Sussillo D. Neural circuits as computational dynamical systems. Curr Opin Neurobiol, 2014, 25: 156–163
Wu F, Yao Z. Dynamics of neuron-like excitable Josephson junctions coupled by a metal oxide memristive synapse. Nonlinear Dyn, 2023, 111: 13481–13497
Xie Y, Yao Z, Hu X, et al. Enhance sensitivity to illumination and synchronization in light-dependent neurons. Chin Phys B, 2021, 30: 120510
Zhou P, Yao Z, Ma J, et al. A piezoelectric sensing neuron and resonance synchronization between auditory neurons under stimulus. Chaos Solitons Fractals, 2021, 145: 110751
Xu Y, Liu M, Zhu Z, et al. Dynamics and coherence resonance in a thermosensitive neuron driven by photocurrent. Chin Phys B, 2020, 29: 098704
Tagne J F, Edima H C, Njitacke Z T, et al. Bifurcations analysis and experimental study of the dynamics of a thermosensitive neuron conducted simultaneously by photocurrent and thermistance. Eur Phys J Spec Top, 2022, 231: 993–1004
Zhu Z, Ren G, Zhang X, et al. Effects of multiplicative-noise and coupling on synchronization in thermosensitive neural circuits. Chaos Solitons Fractals, 2021, 151: 111203
Shen H, Yu F, Wang C, et al. Firing mechanism based on single memristive neuron and double memristive coupled neurons. Nonlinear Dyn, 2022, 110: 3807–3822
Wu F, Hu X, Ma J. Estimation of the effect of magnetic field on a memristive neuron. Appl Math Comput, 2022, 432: 127366
Wen Z, Wang C, Deng Q, et al. Regulating memristive neuronal dynamical properties via excitatory or inhibitory magnetic field coupling. Nonlinear Dyn, 2022, 110: 3823–3835
Xu Q, Ju Z, Ding S, et al. Electromagnetic induction effects on electrical activity within a memristive Wilson neuron model. Cogn Neurodyn, 2022, 16: 1221–1231
Kafraj M S, Parastesh F, Jafari S. Firing patterns of an improved Izhikevich neuron model under the effect of electromagnetic induction and noise. Chaos Solitons Fractals, 2020, 137: 109782
Narayanan R, Johnston D. Functional maps within a single neuron. J Neurophysiol, 2012, 108: 2343–2351
Ibarz B, Casado J M, Sanjuan M A F. Map-based models in neuronal dynamics. Phys Rep, 2011, 501: 1–74
Muni S S, Fatoyinbo H O, Ghosh I. Dynamical effects of electromagnetic flux on chialvo neuron map: Nodal and network behaviors. Int J Bifurcation Chaos, 2022, 32: 2230020
Ramakrishnan B, Mehrabbeik M, Parastesh F, et al. A new memristive neuron map model and its network’s dynamics under electrochemical coupling. Electronics, 2022, 11: 153
Bao H, Li K X, Ma J, et al. Memristive effects on an improved discrete Rulkov neuron model. Sci China Tech Sci, 2023, 66: 3153–3163
Li Y, Wang Z, Midya R, et al. Review of memristor devices in neuromorphic computing: Materials sciences and device challenges. J Phys D-Appl Phys, 2018, 51: 503002
Khalid M. Review on various memristor models, characteristics, potential applications, and future works. Trans Electr Electron Mater, 2019, 20: 289–298
Thakkar P, Gosai J, Gogoi H J, et al. From fundamentals to frontiers: A review of memristor mechanisms, modeling and emerging applications. J Mater Chem C, 2024, 12: 1583–1608
Lin H, Wang C, Deng Q, et al. Review on chaotic dynamics of memristive neuron and neural network. Nonlinear Dyn, 2021, 106: 959–973
Lai Q, Lai C. Design and implementation of a new hyperchaotic memristive map. IEEE Trans Circuits Syst II, 2022, 69: 2331–2335
Liu X, Sun K, Wang H, et al. A class of novel discrete memristive chaotic map. Chaos Solitons Fractals, 2023, 174: 113791
Ramadoss J, Almatroud O A, Momani S, et al. Discrete memristance and nonlinear term for designing memristive maps. Symmetry, 2022, 14: 2110
Rong K, Bao H, Li H, et al. Memristive Hénon map with hidden Neimark–Sacker bifurcations. Nonlinear Dyn, 2022, 108: 4459–4470
Bao B, Zhao Q, Yu X, et al. Complex dynamics and initial state effects in a two-dimensional sine-bounded memristive map. Chaos Solitons Fractals, 2023, 173: 113748
Fox R F, Gatland I R, Roy R, et al. Fast, accurate algorithm for numerical simulation of exponentially correlated colored noise. Phys Rev A, 1988, 38: 5938–5940
Li X, Xu Y. Energy level transition and mode transition in a neuron. Nonlinear Dyn, 2024, 112: 2253–2263
Mantegna R N. Fast, accurate algorithm for numerical simulation of Lévy stable stochastic processes. Phys Rev E, 1994, 49: 4677–4683
Guo Y, Xie Y, Ma J. How to define energy function for memristive oscillator and map. Nonlinear Dyn, 2023, 111: 21903–21915
Ma J. Energy function for some maps and nonlinear oscillators. Appl Math Comput, 2024, 463: 128379
Isah A, Bilbault J M. Review on the basic circuit elements and memristor interpretation: Analysis, technology and applications. J Low Power Electron Appl, 2022, 12: 44
Abraham I. The case for rejecting the memristor as a fundamental circuit element. Sci Rep, 2018, 8: 10972
Ramakrishnan B, Durdu A, Rajagopal K, et al. Infinite attractors in a chaotic circuit with exponential memristor and Josephson junction resonator. AEU-Int J Electron Commun, 2020, 123: 153319
Isah A, Nguetcho AST, Binczak S, et al. Dynamics of a charge-controlled memristor in master–slave coupling. Electron Lett, 2020, 56: 211–213
Sun J, Yang J, Liu P, et al. Design of general flux-controlled and charge-controlled memristor emulators based on hyperbolic functions. IEEE Trans Comput-Aided Des Integr Circuits Syst, 2022, 42: 956–967
Sharma P K, Ranjan R K, Khateb F, et al. Charged controlled mem-element emulator and its application in a chaotic system. IEEE Access, 2020, 8: 171397–171407
Wang C, Tang J, Ma J. Minireview on signal exchange between nonlinear circuits and neurons via field coupling. Eur Phys J Spec Top, 2019, 228: 1907–1924
Wang X, Yu D, Wu Y, et al. Effects of potassium channel blockage on inverse stochastic resonance in Hodgkin-Huxley neural systems. J Zhejiang Univ Sci A, 2023, 24: 735–748
Huang W, Yang L, Zhan X, et al. Synchronization transition of a modular neural network containing subnetworks of different scales. Front Inform Technol Electron Eng, 2023, 24: 1458–1470
Xie Y, Yao Z, Ma J. Phase synchronization and energy balance between neurons. Front Inform Technol Electron Eng, 2022, 23: 1407–1420
Wu F, Gu H, Jia Y. Bifurcations underlying different excitability transitions modulated by excitatory and inhibitory memristor and chemical autapses. Chaos Solitons Fractals, 2021, 153: 111611
Majhi S, Perc M, Ghosh D. Dynamics on higher-order networks: A review. J R Soc Interface, 2022, 19: 20220043
Li X, Ghosh D, Lei Y. Chimera states in coupled pendulum with higher-order interaction. Chaos Solitons Fractals, 2023, 170: 113325
Parastesh F, Mehrabbeik M, Rajagopal K, et al. Synchronization in Hindmarsh-Rose neurons subject to higher-order interactions. Chaos-An Interdiscipl J Nonlinear Sci, 2022, 32: 013125
Ramasamy M, Devarajan S, Kumarasamy S, et al. Effect of higher-order interactions on synchronization of neuron models with electromagnetic induction. Appl Math Comput, 2022, 434: 127447
Kundu S, Ghosh D. Higher-order interactions promote chimera states. Phys Rev E, 2022, 105: L042202
Atay F M, Jost J, Wende A. Delays, connection topology, and synchronization of coupled chaotic maps. Phys Rev Lett, 2004, 92: 144101
Koronovskii A A, Moskalenko O I, Shurygina S A, et al. Generalized synchronization in discrete maps. New point of view on weak and strong synchronization. Chaos Solitons Fractals, 2013, 46: 12–18
Winkler M, Sawicki J, Omelchenko I, et al. Relay synchronization in multiplex networks of discrete maps. Europhys Lett, 2019, 126: 50004
Muni S S, Rajagopal K, Karthikeyan A, et al. Discrete hybrid Izhikevich neuron model: Nodal and network behaviours considering electromagnetic flux coupling. Chaos Solitons Fractals, 2022, 155: 111759
Ma M, Yang Y, Qiu Z, et al. A locally active discrete memristor model and its application in a hyperchaotic map. Nonlinear Dyn, 2022, 107: 2935–2949
Peng Y, He S, Sun K. A higher dimensional chaotic map with discrete memristor. AEU-Int J Electron Commun, 2021, 129: 153539
Zhong H, Li G, Xu X. A generic voltage-controlled discrete memristor model and its application in chaotic map. Chaos Solitons Fractals, 2022, 161: 112389
Ren L, Mou J, Banerjee S, et al. A hyperchaotic map with a new discrete memristor model: Design, dynamical analysis, implementation and application. Chaos Solitons Fractals, 2023, 167: 113024
Bao H, Hua Z Y, Liu W B, et al. Discrete memristive neuron model and its interspike interval-encoded application in image encryption. Sci China Tech Sci, 2021, 64: 2281–2291
Li C, Yang Y, Yang X, et al. Application of discrete memristors in logistic map and Hindmarsh-Rose neuron. Eur Phys J Spec Top, 2022, 231: 3209–3224
Bao H, Chen Z G, Cai J M, et al. Memristive cyclic three-neuron-based neural network with chaos and global coexisting attractors. Sci China Tech Sci, 2022, 65: 2582–2592
Lu L L, Yi M, Liu X Q. Energy-efficient firing modes of chay neuron model in different bursting kinetics. Sci China Tech Sci, 2022, 65: 1661–1674
Yuan Y Y, Yang H, Han F, et al. Traveling chimera states in locally coupled memristive Hindmarsh-Rose neuronal networks and circuit simulation. Sci China Tech Sci, 2022, 65: 1445–1455
Yu Y, Fan Y B, Han F, et al. Transcranial direct current stimulation inhibits epileptic activity propagation in a large-scale brain network model. Sci China Tech Sci, 2023, 66: 3628–3638
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This work was supported by the National Natural Science Foundation of China (Grant No. 12072139).
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Guo, Y., Ma, J., Zhang, X. et al. Memristive oscillator to memristive map, energy characteristic. Sci. China Technol. Sci. 67, 1567–1578 (2024). https://doi.org/10.1007/s11431-023-2637-1
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DOI: https://doi.org/10.1007/s11431-023-2637-1