Dynamics of a neuron exposed to integer-and fractional-order discontinuous external magnetic flux
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摘要
We propose a modified Fitzhugh-Nagumo neuron(MFNN) model. Based on this model, an integerorder MFNN system(case A) and a fractional-order MFNN system(case B) were investigated. In the presence of electromagnetic induction and radiation, memductance and induction can show a variety of distributions. Fractionalorder magnetic flux can then be considered. Indeed, a fractional-order setting can be acceptable for non-uniform diffusion. In the case of an MFNN system with integer-order discontinuous magnetic flux, the system has chaotic and non-chaotic attractors. Dynamical analysis of the system shows the birth and death of period doubling, which is a sign of antimonotonicity. Such a behavior has not been studied previously in the dynamics of neurons. In an MFNN system with fractional-order discontinuous magnetic flux, different attractors such as chaotic and periodic attractors can be observed. However, there is no sign of antimonotonicity.
We propose a modified Fitzhugh-Nagumo neuron(MFNN) model. Based on this model, an integerorder MFNN system(case A) and a fractional-order MFNN system(case B) were investigated. In the presence of electromagnetic induction and radiation, memductance and induction can show a variety of distributions. Fractionalorder magnetic flux can then be considered. Indeed, a fractional-order setting can be acceptable for non-uniform diffusion. In the case of an MFNN system with integer-order discontinuous magnetic flux, the system has chaotic and non-chaotic attractors. Dynamical analysis of the system shows the birth and death of period doubling, which is a sign of antimonotonicity. Such a behavior has not been studied previously in the dynamics of neurons. In an MFNN system with fractional-order discontinuous magnetic flux, different attractors such as chaotic and periodic attractors can be observed. However, there is no sign of antimonotonicity.
We propose a modified Fitzhugh-Nagumo neuron(MFNN) model. Based on this model, an integerorder MFNN system(case A) and a fractional-order MFNN system(case B) were investigated. In the presence of electromagnetic induction and radiation, memductance and induction can show a variety of distributions. Fractionalorder magnetic flux can then be considered. Indeed, a fractional-order setting can be acceptable for non-uniform diffusion. In the case of an MFNN system with integer-order discontinuous magnetic flux, the system has chaotic and non-chaotic attractors. Dynamical analysis of the system shows the birth and death of period doubling, which is a sign of antimonotonicity. Such a behavior has not been studied previously in the dynamics of neurons. In an MFNN system with fractional-order discontinuous magnetic flux, different attractors such as chaotic and periodic attractors can be observed. However, there is no sign of antimonotonicity.
引文
Abdolmohammadi HR,Khalaf AJM,Panahi S,et al.,2018.A new 4D chaotic system with hidden attractor and its engineering applications:analog circuit design and field programmable gate array implementation.Pramana,90(6):70.https://doi.org/10.1007/s12043-018-1569-2
Baskonus HM,Bulut H,2015.On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method.Open Math,13(1):547-556.https://doi.org/10.1515/math-2015-0052
Bertsias P,Safari L,Minaei S,et al.,2018.Fractionalorder differentiators and integrators with reduced circuit complexity.Int Symp on Circuits and Systems,p.1-4.https://doi.org/10.1109/ISCAS.2018.8351452
Bla?ejczyk-Okolewska B,Kapitaniak T,1998.Co-existing attractors of impact oscillator.Chaos Sol Fract,9(8):1439-1443.https://doi.org/10.1016/S0960-0779(98)00164-7
Cafagna D,Grassi G,2013.Elegant chaos in fractional-order system without equilibria.Math Probl Eng,Article380436.https://doi.org/10.1155/2013/380436
Cafagna D,Grassi G,2015.Fractional-order systems without equilibria:the first example of hyperchaos and its application to synchronization.Chin Phys B,24(8):080502.https://doi.org/10.1088/1674-1056/24/8/080502
Chudzik A,Perlikowski P,Stefanski A,et al.,2011.Multistability and rare attractors in van der Pol-Duffing oscillator.Int J Bifurc Chaos,21(7):1907-1912.https://doi.org/10.1142/S0218127411029513
Dawson SP,Grebogi C,Yorke JA,et al.,1992.Antimonotonicity:inevitable reversals of period-doubling cascades.Phys Lett A,162(3):249-254.https://doi.org/10.1016/0375-9601(92)90442-O
Diethelm K,1997.An algorithm for the numerical solution of differential equations of fractional order.Electron Trans Numer Anal,5:1-6.
Diethelm K,Ford NJ,2002.Analysis of fractional differential equations.J Math Anal Appl,265(2):229-248.https://doi.org/10.1006/jmaa.2000.7194
Diethelm K,Ford NJ,Freed AD,2004.Detailed error analysis for a fractional Adams method.Numer Algor,36(1):31-52.https://doi.org/10.1023/B:NUMA.0000027736.85078.be
Elwakil AS,2010.Fractional-order circuits and systems:an emerging interdisciplinary research area.IEEE Circ Syst Mag,10(4):40-50.https://doi.org/10.1109/MCAS.2010.938637
Fitzhugh R,1961.Impulses and physiological states in theoretical models of nerve membrane.Biophys J,1(6):445-466.https://doi.org/10.1016/S0006-3495(61)86902-6
Freeman WJ,1988.Strange attractors that govern mammalian brain dynamics shown by trajectories of electroencephalographic(EEG)potential.IEEE Trans Circ Syst,35(7):781-783.https://doi.org/10.1109/31.1822
Gu HG,Pan BB,2015.A four-dimensional neuronal model to describe the complex nonlinear dynamics observed in the firing patterns of a sciatic nerve chronic constriction injury model.Nonl Dynam,81(4):2107-2126.https://doi.org/10.1007/s11071-015-2129-7
Haghighi HS,Markazi AHD,2017.A new description of epileptic seizures based on dynamic analysis of a thalamocortical model.Sci Rep,7:13615.https://doi.org/10.1038/s41598-017-13126-4
Hodgkin AL,Huxley AF,1952.A quantitative description of membrane current and its application to conduction and excitation in nerve.J Physiol,117(4):500-544.https://doi.org/10.1113/jphysiol.1952.sp004764
Itoh M,Chua LO,2008.Memristor oscillators.Int J Bifurc Chaos,18(11):3183-3206.https://doi.org/10.1142/S0218127408022354
Izhikevich EM,2004.Which model to use for cortical spiking neurons?IEEE Trans Neur Netw,15(5):1063-1070.https://doi.org/10.1109/TNN.2004.832719
Jenson VG,Jeffreys GV,1977.Mathematical Methods in Chemical Engineering.Elsevier,Amsterdam,the Netherlands.
Kengne J,Negou AN,Tchiotsop D,2017.Antimonotonicity,chaos and multiple attractors in a novel autonomous memristor-based jerk circuit.Nonl Dynam,88(4):2589-2608.https://doi.org/10.1007/s11071-017-3397-1
Li CL,Zhang J,2016.Synchronisation of a fractional-order chaotic system using finite-time input-to-state stability.Int J Syst Sci,47(10):2440-2448.https://doi.org/10.1080/00207721.2014.998741
Lv M,Wang CN,Ren GD,et al.,2016.Model of electrical activity in a neuron under magnetic flow effect.Nonl Dynam,85(3):1479-1490.https://doi.org/10.1007/s11071-016-2773-6
Lv M,Ma J,Yao YG,et al.,2018.Synchronization and wave propagation in neuronal network under field coupling.Sci China Technol Sci,in press.https://doi.org/10.1007/s11431-018-9268-2
Ma J,Wang CN,Tang J,et al.,2010.Eliminate spiral wave in excitable media by using a new feasible scheme.Commun Nonl Sci Numer Simul,15(7):1768-1776.https://doi.org/10.1016/j.cnsns.2009.07.013
Ma J,Wu FQ,Wang CN,2016.Synchronization behaviors of coupled neurons under electromagnetic radiation.Int JMod Phys B,31(2):1650251.https://doi.org/10.1142/S0217979216502519
Ma J,Wang Y,Wang CN,et al.,2017a.Mode selection in electrical activities of myocardial cell exposed to electromagnetic radiation.Chaos Sol Fract,99:219-225.https://doi.org/10.1016/j.chaos.2017.04.016
Ma J,Wu FQ,Ren GD,et al.,2017b.A class of initialsdependent dynamical systems.Appl Math Comput,298:65-76.https://doi.org/10.1016/j.amc.2016.11.004
Ma J,Wu FQ,Alsaedi A,et al.,2018.Crack synchronization of chaotic circuits under field coupling.Nonl Dynam,93(4):2057-2069.https://doi.org/10.1007/s11071-018-4307-x
McSharry PE,Smith LA,Tarassenko L,2003.Prediction of epileptic seizures:are nonlinear methods relevant?Nat Med,9(3):241-242.https://doi.org/10.1038/nm0303-241
Nagumo J,Arimoto S,Yoshizawa S,1962.An active pulse transmission line simulating nerve axon.Proc IRE,50(10):2061-2070.https://doi.org/10.1109/JRPROC.1962.288235
Panahi S,Aram Z,Jafari S,et al.,2017.Modeling of epilepsy based on chaotic artificial neural network.Chaos Sol Fract,105:150-156.https://doi.org/10.1016/j.chaos.2017.10.028
Perc M,Rupnik M,Gosak M,et al.,2009.Prevalence of stochasticity in experimentally observed responses of pancreatic acinar cells to acetylcholine.Chaos,19(3):037113.https://doi.org/10.1063/1.3160017
Pham VT,Kingni ST,Volos C,et al.,2017.A simple three-dimensional fractional-order chaotic system without equilibrium:dynamics,circuitry implementation,chaos control and synchronization.AEU Int J Electron Commun,78:220-227.https://doi.org/10.1016/j.aeue.2017.04.012
Pham VT,Volos C,Jafari S,et al.,2018.A novel cubicequilibrium chaotic system with coexisting hidden attractors:analysis,and circuit implementation.J Circ Syst Comput,27(4):1850066.https://doi.org/10.1142/S0218126618500664
Qian Y,Liu F,Yang KL,et al.,2017.Spatiotemporal dynamics in excitable homogeneous random networks composed of periodically self-sustained oscillation.Sci Rep,7(1):11885.https://doi.org/10.1038/s41598-017-12333-3
Ren GD,Zhou P,Ma J,et al.,2017.Dynamical response of electrical activities in digital neuron circuit driven by autapse.Int J Bifurc Chaos,27:1750187.https://doi.org/10.1142/S0218127417501875
Rostami Z,Pham VT,Jafari S,et al.,2018.Taking control of initiated propagating wave in a neuronal network using magnetic radiation.Appl Math Comput,338:141-151.https://doi.org/10.1016/j.amc.2018.06.004
Schmidt C,Grant P,Lowery M,et al.,2013.Influence of uncertainties in the material properties of brain tissue on the probabilistic volume of tissue activated.IEEETrans Biomed Eng,60(5):1378-1387.https://doi.org/10.1109/TBME.2012.2235835
Shah DK,Chaurasiya RB,Vyawahare VA,et al.,2017.FPGA implementation of fractional-order chaotic systems.AEU Int J Electron Commun,78:245-257.https://doi.org/10.1016/j.aeue.2017.05.005
Stamova I,Stamov G,2017.Functional and Impulsive Differential Equations of Fractional Order:Qualitative Analysis and Applications.CRC Press,Boca Raton,USA.
Sun H,Abdelwahab A,Onaral B,1984.Linear approximation of transfer function with a pole of fractional power.IEEE Trans Autom Contr,29(5):441-444.https://doi.org/10.1109/TAC.1984.1103551
Vastarouchas C,Psychalinos C,Elwakil AS,2018.Fractionalorder model of a commercial ear simulator.IEEE Int Symp on in Circuits and Systems,p.1-4.https://doi.org/10.1109/ISCAS.2018.8351400
Wang CN,Lv M,Alsaedi A,et al.,2017.Synchronization stability and pattern selection in a memristive neuronal network.Chaos,27(11):113108.https://doi.org/10.1063/1.5004234
Wu FQ,Wang CN,Xu Y,et al.,2016.Model of electrical activity in cardiac tissue under electromagnetic induction.Sci Rep,6:28.https://doi.org/10.1038/s41598-016-0031-2
Wu FQ,Wang Y,Ma J,et al.,2018.Multi-channels couplinginduced pattern transition in a tri-layer neuronal network.Phys A,493:54-68.https://doi.org/10.1016/j.physa.2017.10.041
Wu HG,Bao BC,Liu Z,et al.,2016.Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator.Nonl Dynam,83(1-2):893-903.https://doi.org/10.1007/s11071-015-2375-8
Xu Y,Jia Y,Ma J,et al.,2018.Collective responses in electrical activities of neurons under field coupling.Sci Rep,8(1):1349.https://doi.org/10.1038/s41598-018-19858-1
Yang SM,Wei XL,Deng B,et al.,2018.Efficient digital implementation of a conductance-based globus pallidus neuron and the dynamics analysis.Phys A,494:484-502.https://doi.org/10.1016/j.physa.2017.11.155
Zambrano-Serrano E,Mu?oz-Pacheco J,Campos-Cantón E,2017.Chaos generation in fractional-order switched systems and its digital implementation.AEU Int JElectron Commun,79:43-52.https://doi.org/10.1016/j.aeue.2017.05.032
Baskonus HM,Bulut H,2015.On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method.Open Math,13(1):547-556.https://doi.org/10.1515/math-2015-0052
Bertsias P,Safari L,Minaei S,et al.,2018.Fractionalorder differentiators and integrators with reduced circuit complexity.Int Symp on Circuits and Systems,p.1-4.https://doi.org/10.1109/ISCAS.2018.8351452
Bla?ejczyk-Okolewska B,Kapitaniak T,1998.Co-existing attractors of impact oscillator.Chaos Sol Fract,9(8):1439-1443.https://doi.org/10.1016/S0960-0779(98)00164-7
Cafagna D,Grassi G,2013.Elegant chaos in fractional-order system without equilibria.Math Probl Eng,Article380436.https://doi.org/10.1155/2013/380436
Cafagna D,Grassi G,2015.Fractional-order systems without equilibria:the first example of hyperchaos and its application to synchronization.Chin Phys B,24(8):080502.https://doi.org/10.1088/1674-1056/24/8/080502
Chudzik A,Perlikowski P,Stefanski A,et al.,2011.Multistability and rare attractors in van der Pol-Duffing oscillator.Int J Bifurc Chaos,21(7):1907-1912.https://doi.org/10.1142/S0218127411029513
Dawson SP,Grebogi C,Yorke JA,et al.,1992.Antimonotonicity:inevitable reversals of period-doubling cascades.Phys Lett A,162(3):249-254.https://doi.org/10.1016/0375-9601(92)90442-O
Diethelm K,1997.An algorithm for the numerical solution of differential equations of fractional order.Electron Trans Numer Anal,5:1-6.
Diethelm K,Ford NJ,2002.Analysis of fractional differential equations.J Math Anal Appl,265(2):229-248.https://doi.org/10.1006/jmaa.2000.7194
Diethelm K,Ford NJ,Freed AD,2004.Detailed error analysis for a fractional Adams method.Numer Algor,36(1):31-52.https://doi.org/10.1023/B:NUMA.0000027736.85078.be
Elwakil AS,2010.Fractional-order circuits and systems:an emerging interdisciplinary research area.IEEE Circ Syst Mag,10(4):40-50.https://doi.org/10.1109/MCAS.2010.938637
Fitzhugh R,1961.Impulses and physiological states in theoretical models of nerve membrane.Biophys J,1(6):445-466.https://doi.org/10.1016/S0006-3495(61)86902-6
Freeman WJ,1988.Strange attractors that govern mammalian brain dynamics shown by trajectories of electroencephalographic(EEG)potential.IEEE Trans Circ Syst,35(7):781-783.https://doi.org/10.1109/31.1822
Gu HG,Pan BB,2015.A four-dimensional neuronal model to describe the complex nonlinear dynamics observed in the firing patterns of a sciatic nerve chronic constriction injury model.Nonl Dynam,81(4):2107-2126.https://doi.org/10.1007/s11071-015-2129-7
Haghighi HS,Markazi AHD,2017.A new description of epileptic seizures based on dynamic analysis of a thalamocortical model.Sci Rep,7:13615.https://doi.org/10.1038/s41598-017-13126-4
Hodgkin AL,Huxley AF,1952.A quantitative description of membrane current and its application to conduction and excitation in nerve.J Physiol,117(4):500-544.https://doi.org/10.1113/jphysiol.1952.sp004764
Itoh M,Chua LO,2008.Memristor oscillators.Int J Bifurc Chaos,18(11):3183-3206.https://doi.org/10.1142/S0218127408022354
Izhikevich EM,2004.Which model to use for cortical spiking neurons?IEEE Trans Neur Netw,15(5):1063-1070.https://doi.org/10.1109/TNN.2004.832719
Jenson VG,Jeffreys GV,1977.Mathematical Methods in Chemical Engineering.Elsevier,Amsterdam,the Netherlands.
Kengne J,Negou AN,Tchiotsop D,2017.Antimonotonicity,chaos and multiple attractors in a novel autonomous memristor-based jerk circuit.Nonl Dynam,88(4):2589-2608.https://doi.org/10.1007/s11071-017-3397-1
Li CL,Zhang J,2016.Synchronisation of a fractional-order chaotic system using finite-time input-to-state stability.Int J Syst Sci,47(10):2440-2448.https://doi.org/10.1080/00207721.2014.998741
Lv M,Wang CN,Ren GD,et al.,2016.Model of electrical activity in a neuron under magnetic flow effect.Nonl Dynam,85(3):1479-1490.https://doi.org/10.1007/s11071-016-2773-6
Lv M,Ma J,Yao YG,et al.,2018.Synchronization and wave propagation in neuronal network under field coupling.Sci China Technol Sci,in press.https://doi.org/10.1007/s11431-018-9268-2
Ma J,Wang CN,Tang J,et al.,2010.Eliminate spiral wave in excitable media by using a new feasible scheme.Commun Nonl Sci Numer Simul,15(7):1768-1776.https://doi.org/10.1016/j.cnsns.2009.07.013
Ma J,Wu FQ,Wang CN,2016.Synchronization behaviors of coupled neurons under electromagnetic radiation.Int JMod Phys B,31(2):1650251.https://doi.org/10.1142/S0217979216502519
Ma J,Wang Y,Wang CN,et al.,2017a.Mode selection in electrical activities of myocardial cell exposed to electromagnetic radiation.Chaos Sol Fract,99:219-225.https://doi.org/10.1016/j.chaos.2017.04.016
Ma J,Wu FQ,Ren GD,et al.,2017b.A class of initialsdependent dynamical systems.Appl Math Comput,298:65-76.https://doi.org/10.1016/j.amc.2016.11.004
Ma J,Wu FQ,Alsaedi A,et al.,2018.Crack synchronization of chaotic circuits under field coupling.Nonl Dynam,93(4):2057-2069.https://doi.org/10.1007/s11071-018-4307-x
McSharry PE,Smith LA,Tarassenko L,2003.Prediction of epileptic seizures:are nonlinear methods relevant?Nat Med,9(3):241-242.https://doi.org/10.1038/nm0303-241
Nagumo J,Arimoto S,Yoshizawa S,1962.An active pulse transmission line simulating nerve axon.Proc IRE,50(10):2061-2070.https://doi.org/10.1109/JRPROC.1962.288235
Panahi S,Aram Z,Jafari S,et al.,2017.Modeling of epilepsy based on chaotic artificial neural network.Chaos Sol Fract,105:150-156.https://doi.org/10.1016/j.chaos.2017.10.028
Perc M,Rupnik M,Gosak M,et al.,2009.Prevalence of stochasticity in experimentally observed responses of pancreatic acinar cells to acetylcholine.Chaos,19(3):037113.https://doi.org/10.1063/1.3160017
Pham VT,Kingni ST,Volos C,et al.,2017.A simple three-dimensional fractional-order chaotic system without equilibrium:dynamics,circuitry implementation,chaos control and synchronization.AEU Int J Electron Commun,78:220-227.https://doi.org/10.1016/j.aeue.2017.04.012
Pham VT,Volos C,Jafari S,et al.,2018.A novel cubicequilibrium chaotic system with coexisting hidden attractors:analysis,and circuit implementation.J Circ Syst Comput,27(4):1850066.https://doi.org/10.1142/S0218126618500664
Qian Y,Liu F,Yang KL,et al.,2017.Spatiotemporal dynamics in excitable homogeneous random networks composed of periodically self-sustained oscillation.Sci Rep,7(1):11885.https://doi.org/10.1038/s41598-017-12333-3
Ren GD,Zhou P,Ma J,et al.,2017.Dynamical response of electrical activities in digital neuron circuit driven by autapse.Int J Bifurc Chaos,27:1750187.https://doi.org/10.1142/S0218127417501875
Rostami Z,Pham VT,Jafari S,et al.,2018.Taking control of initiated propagating wave in a neuronal network using magnetic radiation.Appl Math Comput,338:141-151.https://doi.org/10.1016/j.amc.2018.06.004
Schmidt C,Grant P,Lowery M,et al.,2013.Influence of uncertainties in the material properties of brain tissue on the probabilistic volume of tissue activated.IEEETrans Biomed Eng,60(5):1378-1387.https://doi.org/10.1109/TBME.2012.2235835
Shah DK,Chaurasiya RB,Vyawahare VA,et al.,2017.FPGA implementation of fractional-order chaotic systems.AEU Int J Electron Commun,78:245-257.https://doi.org/10.1016/j.aeue.2017.05.005
Stamova I,Stamov G,2017.Functional and Impulsive Differential Equations of Fractional Order:Qualitative Analysis and Applications.CRC Press,Boca Raton,USA.
Sun H,Abdelwahab A,Onaral B,1984.Linear approximation of transfer function with a pole of fractional power.IEEE Trans Autom Contr,29(5):441-444.https://doi.org/10.1109/TAC.1984.1103551
Vastarouchas C,Psychalinos C,Elwakil AS,2018.Fractionalorder model of a commercial ear simulator.IEEE Int Symp on in Circuits and Systems,p.1-4.https://doi.org/10.1109/ISCAS.2018.8351400
Wang CN,Lv M,Alsaedi A,et al.,2017.Synchronization stability and pattern selection in a memristive neuronal network.Chaos,27(11):113108.https://doi.org/10.1063/1.5004234
Wu FQ,Wang CN,Xu Y,et al.,2016.Model of electrical activity in cardiac tissue under electromagnetic induction.Sci Rep,6:28.https://doi.org/10.1038/s41598-016-0031-2
Wu FQ,Wang Y,Ma J,et al.,2018.Multi-channels couplinginduced pattern transition in a tri-layer neuronal network.Phys A,493:54-68.https://doi.org/10.1016/j.physa.2017.10.041
Wu HG,Bao BC,Liu Z,et al.,2016.Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator.Nonl Dynam,83(1-2):893-903.https://doi.org/10.1007/s11071-015-2375-8
Xu Y,Jia Y,Ma J,et al.,2018.Collective responses in electrical activities of neurons under field coupling.Sci Rep,8(1):1349.https://doi.org/10.1038/s41598-018-19858-1
Yang SM,Wei XL,Deng B,et al.,2018.Efficient digital implementation of a conductance-based globus pallidus neuron and the dynamics analysis.Phys A,494:484-502.https://doi.org/10.1016/j.physa.2017.11.155
Zambrano-Serrano E,Mu?oz-Pacheco J,Campos-Cantón E,2017.Chaos generation in fractional-order switched systems and its digital implementation.AEU Int JElectron Commun,79:43-52.https://doi.org/10.1016/j.aeue.2017.05.032