Dynamics of Hodgkin–Huxley systems revisited

References

• Babin , AV and Vishik , MI . 1992 . Attractors of Evolution Equations , Amsterdam, London, New York, Tokyo : North-Holland .
   Google Scholar
• Hale , JK . 1988 . Asymptotic Behavior of Dissipative Systems , Providence, RI : American Mathematical Society .
   Google Scholar
• Miranville , A and Zelik , S . 2008 . “ Attractors for dissipative partial differential equations in bounded and unbounded domains ” . In Handbook of Differential Equations: Evolutionary Equations , Edited by: Dafermos , CM and Pokomy , M . Vol. IV , 103 – 200 . Amsterdam : Elsevier/North-Holland .
   Google Scholar
• Temam , R . 1997 . Infinite Dimensional Dynamical Systems in Mechanics and Physics , 2nd , Berlin, Heidelberg, New York : Springer-Verlag .
   Google Scholar
• Eden , A , Foias , C , Nicolaenko , B and Temam , R . 1994 . Exponential Attractors for Dissipative Evolution Equations , Paris : Masson .
   Google Scholar
• Dung , L and Nicolaenko , B . 2001 . Exponential attractors in Banach spaces . J. Dyn. Diff. Eqns , 13 : 791 – 806 .
   Google Scholar
• Efendiev , M , Miranville , A and Zelik , S . 2000 . Exponential attractors for a nonlinear reaction-diffusion system in ℝ3 . C. R. Math. Acad. Sci. Paris , 330 : 713 – 718 .
   Web of Science ®Google Scholar
• Efendiev , M , Miranville , A and Zelik , S . 2004 . Exponential attractors for a singularly perturbed Cahn–Hilliard system . Math. Nachr. , 272 : 11 – 31 .
   Web of Science ®Google Scholar
• Fabrie , P , Galusinski , C , Miranville , A and Zelik , S . 2004 . Uniform exponential attractors for a singularly perturbed damped wave equation . Discr. Contin. Dyn. Syst. , 10 : 211 – 238 .
   Web of Science ®Google Scholar
• Gatti , S , Grasselli , M , Miranville , A and Pata , V . 2006 . A construction of a robust family of exponential attractors . Proc. Amer. Math. Soc. , 134 : 117 – 127 .
   Google Scholar
• Babin , A and Nicolaenko , B . 1995 . Exponential attractors of reaction–diffusion systems in an unbounded domain . J. Dyn. Diff. Eqns. , 7 : 567 – 589 .
   Google Scholar
• Efendiev , M , Miranville , A and Zelik , S . 2005 . Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems . Proc. Roy. Soc. Edinburgh Sect. A , 13 : 703 – 730 .
   Google Scholar
• Miranville , A , Pata , V and Zelik , S . 2007 . Exponential attractors for singularly perturbed damped wave equations: A simple construction . Asymptotic Anal. , 53 : 1 – 12 .
   Web of Science ®Google Scholar
• Hodgkin , AL and Huxley , AF . 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
• Cavaterra , C and Grasselli , M . 2009 . Robust exponential attractors for singularly perturbed Hodgkin–Huxley equations . J. Diff. Eqns , 246 : 4670 – 4701 .
   Google Scholar
• Cronin , J . 1987 . Mathematical Aspects of Hodgkin–Huxley Neural Theory , Cambridge : Cambridge University Press .
   Google Scholar
• Fitzigibbon , WE and Parrott , M . 1994 . Convergence of singularly perturbed Hodgkin–Huxley systems . Nonlinear Anal. TMA , 22 : 363 – 379 .
   Google Scholar
• Fitzigibbon , WE , Parrott , M and You , Y . 1993 . “ Global dynamics of singularly perturbed Hodgkin–Huxley equations ” . In Semigroups of linear and nonlinear operations and applications , Edited by: Goldstein , GR and Goldstein , JA . 159 – 176 . Dordrecht : Kluwer .
   Google Scholar
• Fitzigibbon , WE , Parrott , M and You , Y . 1996 . Finite dimensionality and upper semicontinuity of the global attractor of singularly perturbed Hodgkin–Huxley systems . J. Diff. Eqns , 129 : 193 – 237 .
   Google Scholar
• Galusinski , C . 1998 . Existence and continuity of uniform exponential attractors of the singularly perturbed Hodgkin–Huxley system . J. Diff. Eqns , 144 : 99 – 169 .
   Google Scholar
• Lieberstein , H . 1967 . On the Hodgkin–Huxley partial differential equation . Math. Biosci. , 1 : 45 – 69 .
   Google Scholar
• Raugel , G . 2000 . “ Singularly perturbed hyperbolic equations revisited ” . In International Conference on Differential Equations , Vols 1 and 2 , 647 – 652 . River Edge, NJ : World Scientific Publishing .
   Google Scholar
• Marion , M . 1989 . Finite dimensional attractors associated with partially dissipative reaction diffusion systems . SIAM J. Math. Anal. , 20 : 815 – 844 .
   Google Scholar