Piecewise linear differential equations and integrate-and-fire neurons: Insights from two-dimensional membrane models

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 67, Issue 2 1, 2003, Pages 219081-2190816

  Tonnelier, Arnaud ^a   Gerstner, Wulfram ^a  

^a EPFL   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

BIOLOGICAL MEMBRANES; DIFFERENTIAL EQUATIONS; OSCILLATIONS; PIECEWISE LINEAR TECHNIQUES;

BISTABILITY;

BRAIN MODELS;

EID: 0038394647     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (36)

1. W.M. Yamada, C. Koch, and P.R. Adams, in Methods in Neuronal Modeling: From Ions to Networks, 2nd ed., edited by C. Koch and I. Segev (MIT Press, Cambridge, MA, 1998), pp. 137-170.
2. M.A. Wilson, U.S. Bhalla, J.D. Uhley, and J.M. Brower, in Advances in Neuronal Information Processing Systems, edited by D. Touretzky (Morgan Kaufmann, San Mateo, CA, 1989), pp. 485-492.
3. L. Lapicque, J. Physiol. Pathol. Gen. 9, 620 (1907).
4. L.F. Abbott and C. van Vreeswijk, Phys. Rev. E 48, 1483 (1993).
5. W. Maass and C.M. Bishop, Pulsed Neural Networks (MIT Press, Cambridge, 1999).
6. A.F. FitzHugh, Biophys. J. 1, 445 (1961).
7. J. Nagumo, S. Arimoto, and S. Oshizawa, Proc. IRE 50, 2061 (1962).
8. L. Morris and H. Lecar, Biophys. J. 35, 193 (1981).
9. J.L. Hindmarsh and R.M. Rose, Nature (London) 296, 162 (1982).
10. J. Rinzel, Fed. Proc. 44, 2944 (1985).
11. L.F. Abbott and T.B. Kepler, in Statistical Mechanics of Neural Networks, edited by L. Garrido (Springer-Verlag, Berlin, 1990).
12. J. Rinzel and G.B. Ermentrout, in Methods in Neuronal Modeling: From Ions to Networks, 2nd ed., edited by C. Koch and I. Segev (MIT Press, Cambridge, MA, 1998), pp. 251-291.
13. F.C. Hoppensteadt and E.M. Izhikevich, Weakly Connected Neural Networks (Springer-Verlag, New York, 1997).
14. E.M. Izhikevich, Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 1171 (2000).
15. A.L. Hodgkin, J. Physiol. (London) 107, 165 (1948).
16. G.B. Ermentrout, Neural Comput. 8, 979 (1996).
17. W. Gerstner, in The Handbook of Biological Physics, edited by F. Moss and S. Gielen (Elsevier Science, Amsterdam, 2001), Vol 4, pp. 469-516.
18. W. Gerstner and J.L. van Hemmen, Neural Networks 3, 139 (1992).
19. H.P. McKean, Adv. Math. 4, 209 (1970).
20. A.F. Filippov, Differential Equations with Discontinuous Right-Hand Sides (Kluwer Academic, Dordrecht, 1988).
21. J.P. Aubin and A. Cellina, Differential Inclusions (Springer-Verlag, Berlin, New York, 1984).
22. F. Giannakopoulos and K. Pliete, Nonlinearity 14, 1611 (2001).
23. A. Tonnelier, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 63, 459 (2002).
24. R.R. Llinas, Science 242, 1654 (1988).
25. A.L. Hodgkin and A.F. Huxley, J. Physiol. (London) 117, 500 (1952).
26. E.M. Izhikevich, Neural Networks 14, 883 (2001).
27. One can easily generalize the definitions of the two sets for noncontinuous dynamics of the membrane potential.
28. Note that the leaky integrator neuron presents a resonant behavior when b>0.
29. The case γ=0 is not interesting since a rescaling of w allows to set I = 0.
30. An additional bifurcation occurs for a particular range of γ values (not shown in Fig. 7). This bifurcation is a double homoclinic bifurcation which appears when the two unstable periodic solutions with sliding motion bifurcates from an unstable periodic solution. These two unstable limit cycles surround the two fixed points, respectively, and become simultaneously homoclinic to a degenerate saddle point along the line of discontinuity (as the bifurcation point is reached). We do not analyze this situation since unstable cycles are not directly observable.
31. A.R. Bulsara, R.D. Boss, and E.W. Jacobs, Biol. Cybern. 61, 211 (1989).
32. X. Chen and S.P. Hastings, J. Math. Biol. 38, 1 (1999).
33. S.H. Strogatz, Nonlinear Dynamics and Chaos. With Application to Physics, Biology, Chemistry and Engineering (Addison-Wesley, Reading, MA, 1994).
34. H.G. Othmer and M. Xie, J. Math. Biol. 39, 139 (1999).
35. S. Coombes, Physica D 2820, 1 (2001).
36. W.M. Kistler, W. Gerstner, and J.L. van Hemmen, Neural Comput. 9, 1015 (1997).