具有时滞反馈控制的分数阶HR神经元模型的动力学分析.pdf

Vol.43(2023)No.5?J.of Math.(PRC)DYNAMICS ANALYSIS FOR THEFRACTIONAL-ORDER HINDMARSH-ROSENEURONAL MODEL WITH TIME-DELAY FEEDBACKCONTROLWANG Tian,WANG Hai-xia(College of Mathematics and Statistics,Nanjing University of Science and Technology,Nanjing 210094,China)Abstract:In this paper,we study the dynamics of the fractional-order Hindmarsh-Rose(HR)neuronal model with time-delay feedback control.Using stability theory and bifurcation analysismethod,the effects of fractional order and time-delay on equilibrium stability,bifurcation and dis-charge behavior are investigated in detail.At the same time,the analytical conditions for Hopfbifurcation in fractional-order model are obtained,and the relevant results of the integer-order HRneuronal model with time-delay feedback control are extended.Keywords:fractional-order;Hopf bifurcation;time-delay feedback control;stability;Hindmarsh-Rose neuronal model2010 MR Subject Classification:37J20;37J25Document code:AArticle ID:0255-7797(2023)05-0377-121 IntroductionIn recent years,the fractional differential equations have attracted more and more at-tention from researchers at home and abroad.Due to the wide application background,theyhave played an important role in mathematics 1,physics 2,biology 3,engineering 4and other fields 5.As a branch of calculus,fractional calculus is the extension of ordinaryinteger order differential and integral to arbitrary real order differential and integral.Withthe increasing maturity of fractional calculus theory and the rapid development of computerscience and technology,it is found that using fractional calculus to build a model can accu-rately describe the properties and morphology of matter,and the fractional calculus modelalso has a larger stable region compared with the integer model.The theoretical study offractional differential system not only has theoretical guiding significance,but also has a verywide application prospect.At present,the study of fractional neurodynamics has becomean important branch in the field of biological nervous system.Received date:2022-06-09Accepted date:2022-09-13Foundation item:Supported by National Natural Science Foundation of China(12272062).Biography:Wang Tian(1998),female,Tujia Nationality,born at Zhengzhou,Henan,master,major in chaotic dynamics.Corresponding author:Wang Haixia378Journal of MathematicsVol.43The neuronal system connected by a considerable number of neuron cells is a verycomplex nonlinear system.Integer order neuronal model can not well describe the neuronmovement process,since the fractional order neuronal model is established.We find that thefractional order neuronal model can more accurately describe some characteristics of neuronactivities 6.Hindmarsh-Rose(HR)neuronal model,as a classical neuronal model,was putforward by Hindmarsh and Rose in the 1980s 7.It plays an important role in describingthe physiological processes of neuronal system,neuron excitability and cardiac muscle fiber,and has a good effect in studying the firing behavior of neurons.The rich firing modes alsocontain abundant bifurcation structures.The study of bifurcation can not only reveal thedynamic mechanism of neuron firing and clarify the intrinsic nature of neural coding,butalso reveal the clustering nature of neural network.Hopf bifurcation is one of the important nonlinear characteristics.It describes thesudden situation of equilibrium stability and the dynamic behavior of nonlinear system suchas periodic solution and chaos.By Hopf bifurcation analysis,more detailed information ofperiodic solutions near the equilibrium point can be obtained.Shi M,et al 8 considered theinfluence of time delay on fractional HR neurons.A qualitative bifurcation condition is givenwith time delay as a bifurcation parameter.In 9,based on the stability and bifurcationtheory of fractional systems,a two-dimensional improved fractional HR neuron model isanalyzed,and a state feedback method is proposed to control the Hopf bifurcation of themodel.Huang C,et al 10 studied the dynamics of a class of high-dimensional fractionalring structured neural networks with multiple delays.Taking time delay as bifurcationparameter,the delay-dependent stability and explicit conditions for Hopf bifurcation aregiven.The results show that the stability and bifurcation of the proposed network dependheavily on the sum of delays,and careful selection of the sum of delays can significantlyimprove the stability of the network.It is further proved that the order and number ofneurons have great influence on the stability and bifurcation of the network.We arrange the paper as follows:In section 2,some knowledge of fractional differentialis briefly introduced.The fractional-order HR neuronal model is given in Section 3.Stabilityand Hopf bifurcation analysis are investigated in Section 4.In Section 5,two examples aregiven to demonstrate the validity of the formula and to compare it with the integer-ordermodel.Finally,some useful conclusions are obtained.2 PreliminaryThere are several definitions of fractional-order calculus,of which the most commonare Gr unwald-Letnikov definition,Riemann-Liouville definition and Caputo definition.Inpractical application,Caputo definition is generally used to define the time fractional deriva-tive,while Gr unwald-Letnikov definition or Riemann-Liouville definition is generally usedto define the reciprocal of space fractional derivative.Whats more,the physical meaning ofthe initial condition of the Caputo definition is clear,so we will adopt it.No.5Dynamics analysis for the fractional-order hindmarsh-rose neuronal model 379Definition 2.1 11 The q-derivative of a function of one variable is defined as follows:aDqtf(t)=1(m q)Ztaf(m)(x)(t x)q+1mdx,(m 1 q m)(2.1)where m is the least integer that is not less than q(m 1 q m),a and t are the upperand lower limits of the integral,respectively.(m q)is the Gamma function.f(m)(x)isthe m-th derivative of the function f(x).Consider the fractional differential linear system as follows:(Dqx=Axx(0)=x0(2.2)where 0 q2;System(2.2)is stable if and only if|arg()|q2;System(2.2)is unstable if and only if|arg()|q2,where is the eigenvalue of the matrix A.Consider the fractional differential nonlinear system as follows:Dqx=g(x)(2.3)where 0 q2,is the eigenvalue of the Jacobian J=gxofsystem(2.3)at the equilibrium x0,that is,x0satisfies g(x0)=0,then system(2.3)is locallyasymptotically stable at the equilibrium point x0.3 Model descriptionHindmarsh-Rose neuronal model is a mathematical expression of neuron cluster behav-iors,which can imitate repeated peaks and irregular behaviors of mollusk neurons 14,15.The fractional Hindmarsh-Rose neuronal model used here has the following form:Dqtx(t)=y ax3+bx2 z+IDqty(t)=c dx2 yDqtz(t)=rs(x x)z(3.1)where x,y,z represent the membrane potential,the recovery variable and the slow adaptioncurrent,respectively.a,b,c,d,r,s,and x are system parameters,I is the external stimulus.Dqtis the differential operator by definition of Caputo,0 q 1 is the number of fractionalorder.380Journal of MathematicsVol.43We adopt the method of Pyragas 16,17,adding a time-delay difference feedbackkx(t )x(t)to the membrane potential of system(3.1),then the system is describedbyDqtx(t)=y ax3+bx2 z+I+kx(t )x(t)Dqty(t)=c dx2 yDqtz(t)=rs(x x)z(3.2)where k is the feedback gain and is the time delay.4 Dynamic analysis4.1 Stability analysisAs we know,the equilibrium of Eq.(3.1)is the same as Eq.(3.2).Assume that E(x,y,z)is the equilibrium of Eq.(3.1)which satisfies the followingequations:y ax3+bx2 z+I=0c dx2 y=0rs(x x)z=0(4.1)The Jacobian matrix of Eq.(3.1)near equilibrium Eis:J=3ax2+2bx112dx10rs0r(4.2)In this paper,we always choose a=1,b=3,c=1,d=5,s=4,x=1.6,r=0.005,and I=1.7,so the single HR neuron without time-delay feedback control exhibits periodicfiring with period-1.We can get the equilibrium(1.2147,6.3772,1.5413)with eigenvalues:1=12.7468,2=0.0137+0.0348i,3=0.0137 0.0348i.From Lemma 2.2,we calculate that|arg(2,3)|=q02,then q0=0.7612,it means thecritical order of periodic firing in HR neuronal model.Thus,when 0 q q0=0.7612,theequilibrium(1.2147,6.3772,1.5413)is stable,that means the neuron is in a rest state;when q0=0.7612 0,then it satisfies the characteristic equation(5.4),substitute i into Eq.(5.4)andseparate the real part and imaginary part,so we have(A+Q12qcos(q )+Q2qcos(q2)+Q3cos()=0B+Q12qsin(q )+Q2qsin(q2)Q3sin()=0(5.6)where(A=3qcos(3q2)+P12qcos(q)+P2qcos(q2)+P3B=3qsin(3q2)+P12qsin(q)+P2qsin(q2)(5.7)Using the trigonometric function formula,Eq.(5.7)can be rewritten as(M1cos()+M2sin()=AM2cos()M1sin()=B(5.8)where M1=Q12qcos(q)+Q2qcos(q2)+Q3,M2=Q12qsin(q)+Q2qsin(q2).Then,we can getcos()=AM1+BM2M21+M22sin()=AM2BM1M21+M22(5.9)No.5Dynamics analysis for the fractional-order hindmarsh-rose neuronal model 383According to cos2()+sin2()=1,we can obtain thatA2+B2=M21+M22.SupposethatF()=A2+B2(M21+M22)=0(5.10)substituteA,B,M1and M2into Eq.(5.10),we obtain the following expression:F()=6q+15q+24q+33q+42q+5q+6=0,(5.11)where1=2P1cos(q2),2=P21 Q21+2P2cos(q),3=2P3cos(3q2)+2(P1P2 Q1Q2)cos(q2),4=P22 Q22+2(P1P3 Q1Q3)cos(q),5=2(P2P3 Q2Q3)cos(q2),6=P23 Q23.(5.12)Lemma 5.1 8 Assume that Eq.(5.11)has at least one positive real root 0,then when=j,Eq.(5.4)exists a pair of purely imaginary roots i0,wherej=10arctan(AM2BM1AM1+BM2)+j,j=0,1,2,.(5.13)Next,we use time delay as a variable parameter to investigate the Hopf bifurcationbehavior of Eq.(3.2).Assume that the roots of Eq.(5.4)are s()=()+i()whichis time-delay depended.We substitute s()into Eq.(5.4),and according to the implicitfunction theorem,we getdsd=s(Q1s2q+Q2sq+Q3)(3qs3q1+2qP1s2q1+qP2sq1)es+(2qQ1s2q1+qQ2sq1)(Q1s2q+Q2sq+Q3)(5.14)From Lemma 5.1,we know that when =j,(j)=0,(j)=0.Define the bifurcationpoint 0=minj,j=0,1,2,we haveRe?dsd?|=0,=0=N1N3+N2N4N23+N24(5.15)384Journal of MathematicsVol.43whereN1=Q12q+10sin(q)+Q2q+10sin(q2)cos(00)+Q12q+10cos(q)+Q2q+10cos(q2)+Q30sin(00)N2=Q12q+10cos(q)+Q2q+10cos(q2)+Q30cos(00)+Q12q+10sin(q)+Q2q+10sin(q2)sin(00)N3=3q3q10sin(3q2)+2qP12q10sin(q)+qP2q10sin(q2)+2qQ12q10sin(q)+qQ2q10sin(q2)Q102q0cos(q)Q20q0cos(q2)Q30cos(00)2qQ12q10cos(q)+qQ2q10cos(q2)+Q102q0sin(q)+Q20q0sin(q2)sin(00)N4=3q3q10cos(3q2)2qP12q10cos(q)qP2q10cos(q2)2qQ12q10cos(q)+qQ2q10cos(q2)+Q102q0sin(q)+Q20q0sin(q2)cos(00)2qQ12q10sin(q)+qQ2q10sin(q2)Q102q0cos(q)Q20q0cos(q2)sin(00)(5.16)In order to prove the occurrence of Hopf bifurcation in Eq.(3.2),we give the following lemma.Lemma 5.2 8 Assume the roots s()=()+i()of Eq.(5.4),it satisfies(0)=0,(0)=0,it means that when =0,Eq.(5.4)has a pair of purely imaginary roots.Then the following transversality conditionRe?dsd?|=0,=0=N1N3+N2N4N23+N246=0(5.17)ensures the occurrence of Hopf bifurcation near the bifurcation.By Lemma 5.1 and Lemma 5.2,we can get the following theorem.Theorem 5.1When the feedback gain k 0,if there is a critical time-delay 0,Eq.(5.11)has at least one positive real root 0,and the transversality condition Eq.(5.17)holds,the equilibrium of Eq.(3.2)is asymptotically stable as 0,Hopf bifurcation occurs at the equilibrium of Eq.(3.2).ProofWhen Eq.(5.11)has at least one positive real root 0,it means Eq.(5.4)has atleast a pair of pure imaginary roots,and the transversality condition Eq.(5.17)holds,thenthe Hopf bifurcation will occur at the equilibrium of Eq.(3.2).5 Numerical simulationsIn this part,we will test the valid of the above theoretical results.No.5Dynamics analysis for the fractional-order hindmarsh-rose neuronal model 38501020304050607080t-4-20246x(a)01020304050607080t-4-3-2-1012345x(b)01020304050607080t-4-20246x(c)01020304050607080t-4-3-2-1012345x(d)Fig.2.Time evolution of membrane potential when(a)=1.0,(b)=1.5,(c)=1.6,(d)=2.3.Example 1We take q=0.83 q0=0.7612,k=5.From Eq.(5.11),we can obtainthree positive real roots,they are 1=0.0226,2=0.1169 and 3=0.6182,respectively.Then from Eq.(5.13),the critical delays of the Eq.(3.2)producing Hopf bifurcation at theequilibrium(1.2147,6.3772,1.5413)are 1=1.0128,2=3.2398,3=2.2122.Afterverification,the transversality condition(5.17)is satisfied.Given =1.0,=1.5,=1.6,=2.3,the time evolution of the membrane potential is shown in Fig.2,and we can find thatwhen =1.0 1=1.0128,the equilibrium is unstable,and the neuron is chaotic.When =1.6 3=2.2122,the neuron is inperiodic firing with period-1.According to Theorem 5.1,this confirms the occurrence ofHopf bifurcation near the critical time-delays.01020304050607080t-3-2-10123456x(a)01020304050607080t-4-3-2-1012345x(b)Fig.3.Time evolution of membrane potential when(a)=0.1,(b)=0.3Example 2We take q=0.86 q0=0.7612,k=8.From Eq.(5.11),we can obtainthree positive real roots,they are 1=0.0204,2=0.1224 and 3=5.0037,respectively.386Journal of MathematicsVol.43Then from Eq.(5.13),the critical delays of the Eq.(3.2)producing Hopf bifurcation at theequilibrium(1.2147,6.3772,1.5413)are 1=61.8556,2=1.8982,3=0.1779.Afterverification,the transversality condition(5.17)is satisfied.Given =0.1,=0.3,thetime evolution of the membrane potential is shown in Fig.3.When =0.1 3=0.1779,the equilibrium isunstable,and the neuron is in periodic firing with period-1.According to Theorem5.1,thisconfirms the occurrence of Hopf bifurcation near the critical time-delays.01020304050607080t-101234567x(a)01020304050607080t-4-3-2-1012345x(b)Fig.4.Time evolution of membrane potential when(a)=7.0,(b)=8.75RemarksWhen the feedback gain k 0,the phenomenon at the equilibrium of thesystem is different from that when the feedback gain k q0=0.7612,k=12.From Eq.(5.11),we can obtain twopositive real roots,they are 1=0.0159 and 2=0.1439,respectively.Then fromEq.(5.13),the critical delays of the Eq.(3.2)producing Hopf bifurcation at the equilibrium(1.2147,6.3772,1.5413)are 1=79.8026,2=8.6746.After verification,the transver-sality condition(5.17)is satisfied.Given =7.0,=8.75,the time evolution of themembrane potential is shown in Fig.4.When =7.0 2=8.6746,the equilibrium is asymptotically stable.From Fig.4b,we can see that the equilibrium is in a resting state,but the lines in the time evolutiondiagram are intermittent and not smooth.Obviously,the effect of Fig.4(b)is not good.Through the numerical simulation,it can be shown that Hopf bifurcation can occur at theequilibrium of the system when feedback gain k is positive value or negative value.6 ConclusionIn this paper,the stability and Hopf bifurcation of the fractional-order HR neuronalmodel with time-delay feedback control are studied.In the uncontrolled model,we get thecritical order.When the order of the system is smaller than the critical order,the equilibriumis stable;otherwise,the equilibrium is unstable.In the controlled model,we prove theexistence of the critical time-delay which generates Hopf bifurcation at the equilibrium ofthe system,and prove the reliability of the theory through numerical simulations.FromFig.2 and Fig.3,the feedback gain is negative,when the time-delay is less than the criticalNo.5Dynamics analysis for the fractional-order hindmarsh-rose neuronal model 387delay,the equilibrium of the system is in a rest