Delay-induced Hopf bifurcation in a noise-driven excitable neuron model

References

• Chian , A. , Borotto , F. , Rempel , E. and Rogers , C. 2005 . Attractor merging crisis in chaotic business cycles . Chaos Solitons Fractals , 24 : 869 – 875 .
   Web of Science ®Google Scholar
• Engelborghs , K. , Luzyanina , T. and Roose , D. 2002 . Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL . ACM Trans. Math. Software , 28 : 1 – 21 .
   Web of Science ®Google Scholar
• Fitzhugh , R. 1961 . Impulses and physiological states in theoretical models of nerve membrane . Biophys. J , 1 : 445 – 466 .
   PubMed Web of Science ®Google Scholar
• Gong , P. , Xu , J. and Hu , S. 2002 . Resonance in a noise-driven excitable neuron model . Chaos Solitons Fractals , 13 : 885 – 895 .
   Google Scholar
• Hassard , B. , Kazarinoff , N. and Wan , Y. 1981 . Theory and Application of Hopf Bifurcation , Cambridge, , UK : Cambridge University Press .
   Google Scholar
• He , J. 2006 . A modified Hodgkin–Huxley model . Chaos Solitons Fractals , 29 : 303 – 306 .
   Google Scholar
• Hodgkin , A. and Huxley , A. 1952 . A quantitative description of membrane current and its application to conduction and excitation in nerve . J. Physiol , 117 : 500 – 544 .
   PubMed Web of Science ®Google Scholar
• Li , Z. , Li , X. and Zhuang , K. 2009 . Hopf bifurcation analysis in a delayed human respiratory system . Nonlinear Anal. Real World Appl , 10 : 2104 – 2116 .
   Google Scholar
• Lu , Q. , Gu , H. , Yang , Z. , Shi , X. , Duan , L. and Zheng , Y. 2008 . Dynamics of firing patterns, synchronization and resonances in neuronal electrical activities: Experiments and analysis . Acta Mech. Sin , 24 : 593 – 628 .
   Web of Science ®Google Scholar
• Lu , Q. , Yang , Z. , Duan , L. , Gu , H. and Ren , W. 2009 . Dynamics and transitions of firing patterns in deterministic and stochastic neuronal systems . Chaos Solitons Fractals , 40 : 577 – 597 .
   Google Scholar
• Ma , J. and Gao , Q. 2009 . Stability and Hopf bifurcations in a business cycle model with delay . Appl. Math. Comput , 215 : 829 – 834 .
   Google Scholar
• Mao , X. and Hu , H. 2009 . Dynamics of a delayed four-neuron network with a short-cut connection: Analytical, numerical and experimental studies . Int. J. Nonlinear Sci. Numer. Simul , 10 : 523 – 538 .
   Google Scholar
• Ma , S. and Lu , Q. 2005 . Dynamical bifurcation for a predator–prey metapopulation model with delay . Int. J. Nonlinear Sci. Numer. Simul , 6 : 13 – 17 .
   Google Scholar
• Ma , Y. , Xiu , Z. , Sun , L. and Feng , E. 2006 . Hopf bifurcation and chaos analysis of a microbial continuous culture model with time delay . Int. J. Nonlinear Sci. Numer. Simul , 7 : 305 – 308 .
   Google Scholar
• Ma , S. , Feng , Z. and Lu , Q. 2009 . Dynamics and double Hopf bifurcations of the Rose–Hindmarsh model with time delay . Internat. J. Bifur. Chaos , 19 : 3733 – 3751 .
   Web of Science ®Google Scholar
• Ma , S. , Lu , Q. and Feng , Z. 2010 . Synchrony and lag synchrony on a neuron model coupling with time delay . Internat. J. Non-Linear Mech , 45 : 659 – 665 .
   Google Scholar
• Song , X. , Zhou , X. and Zhao , X. 2010 . Properties of stability and Hopf bifurcation for a HIV infection model with time delay . Appl. Math. Model , 34 : 1511 – 1523 .
   Web of Science ®Google Scholar
• Sun , C. , Han , M. , Lin , Y. and Chen , Y. 2007 . Global qualitative analysis for a predator–prey system with delay . Chaos Solitons Fractals , 32 : 1582 – 1596 .
   Google Scholar
• Wan , A. and Wei , J. 2010 . Hopf bifurcation analysis of a food-limited population model with delay . Nonlinear Anal. Real World Appl , 11 : 1087 – 1095 .
   Web of Science ®Google Scholar
• Wang , L. and Wu , X. 2010 . Bifurcation analysis of a Kaldor–Kalecki model of business cycle with time delay . Electron. J. Qual. Theory Differ. Equ , 2010 : 1 – 20 .
   Google Scholar
• Wang , Q. , Lu , Q. , Chen , G. , Feng , Z. and Duan , L. 2009 . Bifurcation and synchronization of synaptically coupled FHN models with time delay . Chaos Solitons Fractals , 39 : 918 – 925 .
   Google Scholar
• Wei , J. and Zou , X. 2006 . Bifurcation analysis of a population model and epidemic model with delay . J. Comput. Appl. Math , 197 : 169 – 187 .
   Google Scholar
• Wu , J. 1998 . Symmetric functional differential equations and neural networks with memory . Trans. Amer. Math. Soc , 350 : 4799 – 4838 .
   Web of Science ®Google Scholar
• Wu , X. and Eshete , M. 2011 . Bifurcation analysis for a model of gene expression with delays . Commun. Nonlinear Sci. Numer. Simul , 16 : 1073 – 1088 .
   Google Scholar
• Wu , W. , Chen , Z. and Yuan , Z. 2009 . The evolution of a novel four-dimensional autonomous system: Among 3-torus, limit cycle, 2-torus, chaos and hyperchaos . Chaos Solitons Fractals , 39 : 2340 – 2356 .
   Google Scholar
• Xin , B. , Chen , T. and Liu , Y. 2011 . Complexity evolvement of a chaotic fractional-order finance system . Acta Phys. Sin , 60 : 040000-6
   Google Scholar
• Xu , J. and Pei , L. 2008 . The nonresonant double Hopf bifurcation in delayed neural network . Int. J. Comput. Math , 85 : 925 – 935 .
   Web of Science ®Google Scholar