Dynamical Behaviors As Well As The Bifurcation Mechanism Of A Coupled Hodgkin-Huxley Model With Two Scales In Frequency Domain

The complex behaviors and mechanism analysis of dynamical systems with different scales coupling have always been a hot topic among nonlinear dynamics.Based on the bifurcation theory as well as the slow-fast analysis method,the dynamical behaviors as well as the bifurcation mechanism of a coupled Hodgkin-Huxley model with two scales in frequency domain under periodic excitation are emphatically investigated.The main contents of this paper are as follows.Firstly,for a coupled system with external periodic excitation,when the excitation frequency is far less than the natural frequency,the whole excitation term can be regarded as a slow-varying parameter.The equilibrium branches and the corresponding bifurcations of the generalized autonomous system are obtained.By introducing the transformed phase portraits and the slow-varying parameter is further considered as generalized state variable,the influence on the bursting oscillations with the excitation amplitude varying is explored.As the excitation amplitude increases,two co-existed asymmetric bursting attractors which are respectively located in different regions evolve continuously in phase space.It can be found out that one pair of fold bifurcations occurs,leading the trajectory to jump from one stable focus to a stable limit cycle,which causes the symmetric bursting attractors.Further investigation shows that the characteristics of the spiking state and quiescent state in the two asymmetric bursting attractors remain in the symmetric bursting attractors,and the oscillation regions of the two kinds of attractors are almost periodic type,whose oscillation frequency can be evaluated by the natural frequency of the system and the frequency of the relevant Hopf bifurcation.The emergence of a pair of supercritical Hopf bifurcations leads to different types of bursting oscillations.The slow passage effect causes the trajectory to pass through the supercritical Hopf bifurcation point along the stable equilibrium branches,almost moves along the unstable equilibrium branches,and delaying the oscillations that tend to the stable limit cycle.Secondly,we consider the coupled system with periodic parametric excitation.The coupling effect of two scales occurs since there is an order gap between the natural frequency and the exciting frequency.Taking suitable parameters,the bifurcation sets of the slow-varying parameter as well as the bifurcation parameter are acquired.By analyzing bifurcating features in different regions,the mode of bifurcation and thecorresponding dynamical behaviors in fixed slow-varying parameter regions are given under the two typical bifurcation parameters.Furthermore,using the transformed phase portraits,the effect of magnitude change of excitation frequency on the bursting oscillations is discussed.Through analysis,we found that the magnitude change of the excitation frequency not only results in the magnitude difference in the periodic time of corresponding bursting oscillation,but also the trajectory tends to different attractors,which leads to the different transition behaviors between the spiking state and quiescent state,even between the spiking state and the spiking state,special transition phenomenon occurs,and then inducing multiple bursting oscillations.Thirdly,we found that the order gap causes the effect of the fast-slow coupling.Different from the circumstances above,we here introduce parametric and external excitation both,which will cause not only the complexity of the equilibrium branches as well as the bifurcations but also some unique nonlinear phenomena in certain.Taking the same coupled system as an example,the fast and slow subsystems,respectively,are established via introducing the parametric and the external excitation.Various bifurcation modes and corresponding bifurcation behaviors of the fast subsystem are revealed with the slow-varying parameter changing.By overlapping the transformed phase portraits,the mechanism and those dynamical behaviors of the coupling system are uncovered.For the situation that the amplitude is relatively small,the almost periodic oscillations occur,which can finally evolve to the bursting oscillations with the exciting amplitude increasing.The main reason for these bursting oscillations is that there exists a unique stable equilibrium branch in the part of space with slow-varying parameter changing,the trajectory of which may oscillate down to the equilibrium branch,appearing in quiescent state,which may bifurcate to repetitive spiking oscillations and change with slow-varying parameter.Meanwhile,the attractors in the fast subsystems will change along with the exciting amplitude,leading to different forms of bursting oscillations.Besides,unlike the situation with sole excitation,when there coexist the parametric and external excitation,some of the stable attractors may have no influence on the oscillations since they are buried in other stable attractors.Finally,the work of this thesis is briefly summarized,some deficiencies are pointed out,and the prospect is given.