Canard Theory and Excitability

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• 作者：Martin Wechselberger (4)
  John Mitry (4)
  John Rinzel (5)
  
• 关键词：Canards ; Geometric singular perturbation theory ; Excitability ; Neural dynamics ; Firing threshold manifold ; Separatrix ; Transient attractor
• 刊名：Lecture Notes in Mathematics
• 出版年：2013
• 出版时间：2013
• 年：2013
• 卷：1
• 期：1
• 页码：89-132
• 全文大小：2,827 KB
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• 作者单位：Martin Wechselberger (4)
  John Mitry (4)
  John Rinzel (5)
  
  4. School of Mathematics and Statistics, University of Sydney, Sydney, Australia
  5. Courant Institute, New York University, New York, USA
  
• ISSN：1617-9692

文摘

An important feature of many physiological systems is that they evolve on multiple scales. From a mathematical point of view, these systems are modeled as singular perturbation problems. It is the interplay of the dynamics on different temporal and spatial scales that creates complicated patterns and rhythms. Many important physiological functions are linked to time-dependent changes in the forcing which leads to nonautonomous behaviour of the cells under consideration. Transient dynamics observed in models of excitability are a prime example.Recent developments in canard theory have provided a new direction for understanding these transient dynamics. The key observation is that canards are still well defined in nonautonomous multiple scales dynamical systems, while equilibria of an autonomous system do, in general, not persist in the corresponding driven, nonautonomous system. Thus canards have the potential to significantly shape the nature of solutions in nonautonomous multiple scales systems. In the context of neuronal excitability, we identify canards of folded saddle type as firing threshold manifolds. It is remarkable that dynamic information such as the temporal evolution of an external drive is encoded in the location of an invariant manifold—the canard.