Phase locking in integrate-and-fire models with refractory periods and modulation

Journal of Mathematical Biology

Volumn 49, Issue 6, 2004, Pages 577-603

  Gedeon, Tomáš ^a   Holzer, Matt ^a  

^a MONTANA STATE UNIVERSITY   (United States)

Author keywords

Absolute and relative refractory period; Adaptation; Integrate and fire neuron; Phase locking; Spike modulation

Indexed keywords

ACTION POTENTIAL; ARTICLE; BIOLOGICAL MODEL; NERVE CELL; PHYSIOLOGY;

ACTION POTENTIALS; MODELS, NEUROLOGICAL; NEURONS;

EID: 11244352150     PISSN: 03036812     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00285-004-0268-4     Document Type: Article

References (39)

1. Abbott, L., Kepler, T.: Model Neurons from Hodgkin-Huxley to Hopfield. Lecture Notes in Phys. Springer Verlag, Garrido ed. 368, 5-18 (1990)
2. Adrian, E., Zotterman, Y.: The impulse produced by sensory nerve endings, Part 2: the response of a single end-organ. J. of Physiology (London) 61, 151-171 (1926)
3. Aihara, K., Matsumoto, G., Ikegaya, Y.: Periodic and non-periodic responses of periodically forced Hodgkin-Huxley oscillator. J. of Theoret. Biol. 109, 249-269 (1984)
4. Azouz, R., Gray, C.: Dynamic Spike Threshold Reveals a Mechanism for Synaptic Coincidence Detection in Cortical Neurons In Vivo. Proc. Natl. Acad. Sci., 97, 8110-8115 (2000)
5. Barge, M., Mathison, T.: A Poincaré-Birkhoff theorem on invariant plane continua. Ergod. Th. & Dynam. Sys. 18, 41-52 (1998)
6. Barge, M., Gillette, R. M.: Rotation and periodicity in plane separating continua. Ergod. Th. & Dynam. Sys. 11, 619-631 (1991)
7. Birkhoff, G.D.: Sur quelques courbes fermés remarquables. Bull. Soc. Math. France 60, 1-26 (1932)
8. Brette, R.: Dynamics of one-dimensional spiking neuron models. J. Math. Biol. (2003), DOI: 10.1007/s00285-003-0223-9 Available from Springer Online First
9. Brette, R., Guigon, E.: Reliability of spike timing is a general property of spiking model neurons. Neural Computation 15, 279-308 (2003)
10. Carathéodory, C.: Uber die Begrenzung einfach zusammenhangender Gebiete. Math. Ann. 73, 323-370 (1913)
11. Carrillo, H., Ongay, F.A.: On the firing maps of a general class of forced integrate-and-fire neurons. Mathematical Biosciences 172, 33-53 (2001)
12. Cartwright, M. L., Littlewood, J. E.: Some fixed point theorems. Annals of Mathematics 54 (1), 1-37 (1951)
13. Chacron, M., Pakdaman, K., Longtin, A.: Interspike Interval Correlations, Memory, Adaptation, and Refractoriness in a Leaky Integrate-and-Fire Model with Threshold Fatigue. Neural Comp. 15, 253-278 (2003)
14. Chacron, M., Pakdaman, K., Longtin, A.: Chaotic firing in the sinusoidally forced leaky integrate and fire model with threshold fatigue. Preprint (submitted)
15. Coombes, S.: Liapunov exponents and mode-locked solutions for integrate-and-fire dynamical systems. Phys. Lett. A, 255, 49-57 (1999)
16. Coombes, S., Bressloff, P.: Mode locking and Arnold tongues in integrate-and-fire oscillators. Phys. Rev. E 60, 2086-2096 (1999)
17. Coombes, S., Owen, M. R., Smith, G.D.: Mode locking in a periodically forced integrate-and-fire-or-burst neuron models. Phys. Rev. E 64, 041914 (2001)
18. Flaherty, J., Hoppensteadt, F.: Frequency entrainment of a forced van der Pol oscillator. Stud. Appl. Math., 58, 5-15 (1978)
19. Geisler, C., Goldburg, J.: A stochastic model of the repetitive model of neurons. Biophys. J., 7, 53-69 (1966)
20. Glass, L., Mackey, M.: A simple model for phase locking in biological oscillators. J. of Math. Biol., 7, 339-352 (1979)
21. Glass, L., Perez, R.: Fine structure of phase locking. Phys. Rev. Lett. 48 (26), 1772-1775 (1982)
22. Hayashi, H., Ishizuka, S., Ohta, M., Hirakawa, K.: Chaotic behavior in the Ochidium giant neuron under sinusoidal forcing. Phys. Lett. A 88, 435-438 (1982)
23. Herman, M.R.: Measure de Lebesgue et nomber de rotation, Geometry and Topology: Lecture Notes in Mathematics. J. Palis and M. do Carmo eds., Springer Verlag, New York, 1977, pp. 271-293
24. Hockett, K., Hohnes, P.: Josephson's junction, annulus maps, Birkhoff attractors, horseshoes and rotations sets. Ergod. Th. & Dynam. Sys. 6, 205-239 (1986)
25. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press 1995
26. Keener, J.P., Hoppensteadt, F.C., Rinzel, J.: Integrate and Fire Models of Nerve Membrane Response to Oscillatory Input. SIAM J. of App. Math. 41 (3), 503-517 (1981)
27. Keener, J.P.: Chaotic Behavior in piecewise continuous difference equations. Transactions of the AMS, 26, 589-604 (1980)
28. Koch, Ch.: Biophysics of Computation. Oxford University Press, New York, Oxford, 1999
29. Lapicque, L.: Recherches quantitatives sur l'excitation électrique des nerfs traitée comme une polarisation. J. Physiol. Pathol. Gen. 9, 620-635 (1907)
30. LeCalvez, P.: Propriétés des attracteurs de Birkhoff. Ergod. Th. & Dynam. Sys, 8, 241-310 (1988)
31. Mather, J.: Topological proofs of some purely topological consequences of Caratheodory's theory of prime ends, (Th. M. Rassias, G.M. Rassias, eds.) Selected studies, Nort-Holland, Amsterdam, 1982, pp. 225-255
32. Moar, S. G.: Trapping regions for unstable manifolds via Prime Ends, Ph. D. thesis, Department of Mathematics, Northwestern University, 2001
33. Pakdaman, K.: The periodically forced integrate-and-fire model. Phys. Rev. E 63, 1907-1912 (2001)
34. Pakdaman, K., Vibert, J.F.: Modeling excitatory networks. Journal of Sice 34, 788-793 (1995)
35. Pakdaman, K., Vibert, J., Boussard, E., Azmy, N.: Single neuron with recurrent excitation: Effect of the transmission delay. Neural Networks 9, 797-818 (1996)
36. Perez, R., Glass, E.: Bistability, period doubling bifurcations and chaos in a periodically forced oscillator. Phys. Lett., vol 90A, 9, 441-443 (1982)
37. Rabinovich, M., Abarbanel, H.: The role of chaos in neural systems. Neuroscience 87, 5-14 (1998)
38. Rescigno, A., Stein, R., Purple, R., Poppele, R.: A neuronal model for the discharge patterns produced by cyclic inputs. Bull. Math. Biophys. 32, 337-353 (1970)
39. Segundo, J., Perkel, D., Wyman, H., Hegstad, H., Moore, G.: Input output relations in computer simulated nerve cells. Kybernetik 4, 157-175 (1968)