Interspike interval attractors from chaotically driven neuron models

Physica D: Nonlinear Phenomena

Volumn 104, Issue 2, 1997, Pages 184-204

  Racicot, Daniel M ^a   Longtin, André ^a  

^a UNIVERSITY OF OTTAWA   (Canada)

Author keywords

Attractors; Chaos; Embedding; FitzHugh Nagumo; Integrate and fire; Interspike interval; Neuron model; Nonlinear prediction; Point processes; Threshold

Indexed keywords

EID: 0001255734     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(97)00296-0     Document Type: Article

References (35)

1. A. Lasota, M.C. Mackey and J. Tyrcha, The statistical dynamics of recurrent biological events, J. Math. Biol. 30 (1992) 775-800.
2. P.E. Rapp, A.M. Albano, I.D. Zimmerman and M.A. Jiménez-Montano, Phase-randomized surrogates can produce spurious identifications of non-random structure, Phys. Lett. A 192 (1994) 27-33.
3. R. Shaw, The Dripping Faucet as a Model Chaotic System (Aerial Press, Santa Cruz, CA, 1984).
4. H. Tuckwell, Stochastic Processes in the Neurosciences (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1989).
5. B. McA. Sayers, Inferring significance from biological signals, in: Biomedical Engineering Systems, eds. M. Clynes and J. Milsum (McGraw-Hill, New York, 1970) Ch. 4, pp. 84-164.
6. R. Eckhorn, O.-J. Grusser, J. Kröller, K. Pellnitz and B. Pöpel, Efficiency of different neuronal codes: Information transfer calculations for three different neuronal systems, Biol. Cybern. 22 (1976) 49-60.
7. W. Bialek, F. Rieke, R.R. de Ruyter van Steveninck and D. Warland, Reading a neural code, Science 252 (1991) 1854-1857.
8. L.F. Abbott, Decoding neuronal firing and modelling neural networks, Quart. Rev. Biophys. 27 (1994) 291-331.
9. F. Gabbiani and C. Koch, Coding of time-varying signals in spike trains of integrate-and-fire neurons with random threshold, Neural Comput. 8 (1996) 44-66.
10. A.V. Holden, Models of the Stochastic Activity of Neurones, Lecture Notes in Biomathematics, Vol. 12 (Springer, Berlin, 1976).
11. F. Takens, Detecting strange attractors in turbulence, in: Dynamical Systems and Turbulence (Warwick, 1980), eds. D.A. Rand and L.-S. Young; Lecture Notes in Mathematics, Vol. 898 (Springer, Berlin, 1981) pp. 366-381.
12. T. Sauer, J.A. Yorke and M. Casdagli, Embedology, J. Statist. Phys. 65 (1991) 579-616.
13. T. Sauer, Reconstruction of dynamical systems from interspike intervals, Phys. Rev. Lett. 72 (1994) 3811-3814.
14. L. Glass and M.C. Mackey, A simple model for phase locking of biological oscillators, J. Math. Biol. 7 (1979) 339-352.
15. J.P. Keener, F.C. Hoppensteadt and J. Rinzel, Integrate and fire models of nerve membrane response to oscillatory input, SIAM J. Appl. Math. 41 (1981) 503-517.
16. R. FitzHugh, Mathematical models of excitation and propagation in nerve, in: Biological Engineering, ed. H.P. Schwann (McGraw-Hill, New York, 1962) Ch. 1, pp. 1-85.
17. J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE. 50 (1962) 2061-2070.
18. P.E. Rapp, I.D. Zimmerman, A.M. Albano, G.C. deGuzman and N.N. Greenbaun, Dynamics of spontaneous neural activity in the simian motor cortex: The dimension of chaotic neurons, Phys. Lett. A 110 (1985) 335-338.
19. N. Takahashi, Y. Hanyu, T. Musha, R. Kubo and G. Matsumoto, Global bifurcation structure in periodically stimulated giant axons of squid, Physica D 43 (1990) 318-334.
20. H. Preissl, A. Aertsen and G. Palm, Are fractal dimensions a good measure for neuronal activity, in: Parallel Processing in Neural Systems and Computers, eds. R. Eckmiller, R. Hartmann and G. Hausske (Elsevier, Amsterdam, 1990) pp. 83-86.
21. A. Longtin, Nonlinear forecasting of spike trains from sensory neurons, Int. J. Bifurc. and Chaos 3 (3) (1993) 651-660.
22. S.J. Schiff, K. Jerger, T. Chang, T. Sauer and P.G. Aitken, Stochastic versus deterministic variability in simple neuronal circuits: II. hippocampal slice, Biophys. J. 67 (1994) 684-691.
23. J.P. Segundo, G. Sugihara, P. Dixon, M. Stiber and L. Bersier, The spike trains of inhibited pacemaker neurons seen through the magnifying glass of nonlinear analyses, preprint.
24. U. an der Heiden and M.C. Mackey, The dynamics of production and destruction: Analytical insight into complex behavior, J. Math. Biol. 16 (1982) 75-101.
25. P. Alström and M.T. Levinsen, Phase-locking structure of integrate-and-fire models with threshold modulation, Phys. Lett. A 128 (1988) 187-192.
26. D. Pierson and F. Moss, Detecting periodic unstable points in noisy chaotic and limit cycle attractors with applications to biology, Phys. Rev. Lett. 75 (1995) 2124-2127.
27. K. Aihara, T. Numajiri, G. Matsumoto and M. Kotani, Structures of attractors in periodically forced neural oscillators, Phys. Lett. A 116 (1986) 313-317.
28. D.R. Chialvo and J. Jalife, Non-linear dynamics of cardiac excitation and impulse propagation, Nature 330 (1987) 749-752.
29. J.D. Farmer and J.J. Sidorowich, Predicting chaotic time series, Phys. Rev. Lett. 59 (1987) 845-848.
30. A. Longtin and D.M. Racicot, Assesment of linear and nonlinear correlations between neural firing events, in: Nonlinear Dynamics and Time Series: Building a Bridge between the Natural and Statistical Sciences, eds. C.D. Cutler and D.T. Kaplan, Fields Institute Communications, Vol. II (American Mathematical Society, Providence, RI, 1997).
31. A.A. Tsonis and J.B. Elsner, Nonlinear prediction as a way of distinguishing chaos from random fractal sequences, Nature 359 (1992) 217-220.
32. J. Theiler, S. Eubank, A. Longtin, B. Galdrikian and J.D. Farmer, Testing for nonlinearity in time series: The method of surrogate data, Physica D 58 (1992) 77-94.
33. D.M. Racicot and A. Longtin, Reconstructing dynamics from neural spike trains, Proc. IEEE 17th Ann. Conf. Engr. Med. Biol. Soc. (1995).
34. D.J. Wales, Calculating the rate of loss of information from chaotic time series by forecasting, Nature 350 (1991) 485-488.
35. J. Rinzel, Electrical excitability of cells, theory and experiment: Review of the Hodgkin-Huxley foundation and an update, Bull. Math. Biol. 52 (1990) 5-23.