Excitable neurons, firing threshold manifolds and Canards

Journal of Mathematical Neuroscience

Volumn 3, Issue 1, 2013, Pages 1-32

  Mitry, John ^a   McCarthy, Michelle ^b   Kopell, Nancy ^b   Wechselberger, Martin ^a  

^a UNIVERSITY OF SYDNEY   (Australia)

^b Boston University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

4 AMINOBUTYRIC ACID; PROPOFOL;

ARTICLE; BEHAVIOR; FIRING THRESHOLD; MATHEMATICAL MODEL; NERVE CELL EXCITABILITY; NERVOUS SYSTEM PARAMETERS; ORIENTATION; PRIORITY JOURNAL; REBOUND; SYNAPTIC INHIBITION; SYNAPTIC POTENTIAL; THEORY;

EID: 84887312141     PISSN: None     EISSN: 21908567     Source Type: Journal    
DOI: 10.1186/2190-8567-3-12     Document Type: Article

References (26)

1. A receptors. J Neurosci 1999, 19(24):10635-10646.
2. 3 et leur canards. Astérisque 1983, 109-110:159-191.
3. Benoît E, Callot J, Diener F, Diener M: Chasse au canard. Collect Math 1981, 31-32:37-119.
4. Brøns M, Kaper T, Rotstein H: Introduction to focus issue: mixed mode oscillations: experiment, computation, and analysis. Chaos 2008, 18:015101.
5. Desroches M, Guckenheimer J, Krauskopf B, Kuehn C, Osinga H, Wechselberger M: Mixed-mode oscillations with multiple time-scales. SIAM Rev 2012, 54:211-288.
6. Desroches M, Krupa M, Rodrigues S: Inflection, canards and excitability threshold in neuronal models. J Math Biol 2013, 67(4):989-1017.
7. Doedel EJ: AUTO: a program for the automatic bifurcation analysis of autonomous systems. Congr Numer 1981, 30:265-284.
8. Dumortier F, Roussarie R: Canard cycles and center manifolds. Mem Am Math Soc 1996, 121:577.
9. Fenichel N: Geometric singular perturbation theory. J Differ Equ 1979, 31:53-98.
10. FitzHugh R: Mathematical models of threshold phenomena in the nerve membrane. Bull Math Biophys 1955, 7:252-278.
11. FitzHugh R: Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1961, 1(6):445-466.
12. Hodgkin AL, Huxley AF: A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 1952, 117:500-544.
13. Izhikevich EM: Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. Cambridge: MIT Press; 2007.
14. Jones CKRT: Geometric singular perturbation theory. In Dynamical Systems; 1995:44-120. [Springer Lecture Notes Math., vol 1609.]
15. Krupa M, Szmolyan P: Relaxation oscillations and canard explosion. J Differ Equ 2001, 174:312-368.
16. Krupa M, Szmolyan P: Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions. SIAM J Math Anal 2001, 33(2):286-314.
17. McCarthy MM, Brown EN, Kopell N: Potential network mechanisms mediating electroencephalographic beta rhythm changes during propofol-induced paradoxical excitation. J Neurosci 2008, 28(50):13488-13504.
18. McCarthy MM, Kopell N: The effect of propofol anaesthesia on rebound spiking. SIAM J Appl Dyn Syst 2012, 11(4):1674-1697.
19. Rush ME, Rinzel J: The potassium A-current, low firing rates and rebound excitation in Hodgkin-Huxley models. Bull Math Biol 1995, 57:899-929.
20. Schwartz RS, Brown EN, Lydic R, Schiff ND: General anesthesia, sleep, and coma. N Engl J Med 2010, 363(27):2638-2650.
21. 3. J Differ Equ 2001, 177:419-453.
22. 3. J Differ Equ 2004, 200:69-104.
23. 3 in the case of a folded node. SIAM J Appl Dyn Syst 2005, 4:101-139.
24. Wechselberger M: À propos de canards (Apropos canards). Trans Am Math Soc 2012, 364:3289-3309.
25. Wechselberger M, Mitry J, Rinzel J: Canard theory and excitability. In Random and Nonautonomous Dynamical Systems in the Life Sciences; 2013, in press.
26. Wieczorek S, Ashwin P, Luke CM, Cox PM: Excitability in ramped systems: the compost-bomb instability. Philos Trans R Soc A, Math Phys Eng Sci 2011, 467(2129):1243-1269.