Differentiability implies continuity in neuronal dynamics

Physica D: Nonlinear Phenomena

Volumn 148, Issue 1-2, 2001, Pages 175-181

  Francis, Joseph T ^a,b   So, Paul ^a   Gluckman, Bruce J ^a   Schiff, Steven J ^a,b  

^a GEORGE MASON UNIVERSITY   (United States)

^b GEORGE WASHINGTON UNIVERSITY SCHOOL OF MEDICINE AND HEALTH SCIENCES   (United States)

Author keywords

5.45.Tp; 87.17.Nn; 87.18.Sn; Continuity; Determinism; Differentiability; Jacobian; Neuronal network; Periodic orbit

Indexed keywords

EID: 0040333700     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(00)00189-5     Document Type: Article

References (31)

1. Artuso R., Aurell E., Cvitanovic P. Recycling of strange sets. I. Cycle expansions. Nonlinearity. 3:1990;325-360.
2. Artuso R., Aurell E., Cvitanovic P. Recycling of strange sets. II. Applications. Nonlinearity. 3:1990;361-386.
3. G.E.P. Box, G.M. Jenkins, Time Series Analysis, Revised Edition, Holden-Day, Oakland, CA, 1976.
4. Braun H.A., Dewald M., Schafer K., Voigt K. Low-dimensional dynamics in sensory biology. 2: Facial cold receptors of the rat. J. Comp. Neurosci. 7:1999;17-32.
5. Braun H.A., Schafer K., Voigt K., Peters R., Bretschneider F., Pei X., Wilkens L., Moss F. Low-dimensional dynamics in sensory biology. 1: Thermally sensitive electroreceptors of the catfish. J. Comp. Neurosci. 4:1997;335-347.
6. Cvitanovic P. Invariant measurement of strange sets in terms of cycles. Phys. Rev. Lett. 61:1988;2729-2732.
7. Ditto W.L., Rauseo S.N., Spano M.L. Experimental control of chaos. Phys. Rev. Lett. 65:1990;3211-3214.
8. E.J. Gumbel, Statistics of Extremes, Columbia University Press, New York, 1958.
9. Gluckman B.J., So P., Schiff S.J. Significance beyond surrogate data in unstable periodic orbit extraction. Am. Phys. Soc. Abstr. 44:1999;110.
10. M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, New York, 1990.
11. Hopfield J.J. Pattern recognition computation using action potential timing for stimulus representation. Nature. 376:1995;33-36.
12. Hill G.W. Researches in the lunar theory. Am. J. Math. 1:1878;245-260.
13. H. Kantz, T. Schreiber, Nonlinear Time Series Analysis, Cambridge University Press, Cambridge, 1997.
14. Kaplan D.T. Exceptional events as evidence for determinism. Physica D. 73:1994;38-48.
15. Mainen Z.F., Sejnowski T.J. Reliability of spike timing in neocortical neurons. Science. 268:1995;1503-1506.
16. Ott E., Grebogi C., Yorke J.A. Controlling chaos. Phys. Rev. Lett. 64:1990;1196-1199.
17. Pawelzik K., Schuster H.G. Unstable periodic orbits and prediction. Phys. Rev. A. 43:1991;1808-1812.
18. Pei X., Moss F. Characterization of low-dimensional dynamics in the crayfish caudal photoreceptor. Nature. 379:1996;619-621.
19. Pierson D., Moss F. Detecting periodic unstable points in noisy chaotic and limit cycle attractors with applications to biology. Phys. Rev. Lett. 75:1995;2124-2127.
20. Le Van Quyen M., Martinerie J., Adam C., Varela F.J. Unstable periodic orbits in human epileptic activity. Phys. Rev. E. 56:1997;3401-3411.
21. Rutecki P.A., Lebeda F.J., Johnson D. Epileptiform activity induced by changes in extracellular potassium in hippocampus. J. Neurophysiol. 54:1985;1363-1374.
22. Schiff S.J., Jerger K., Duong D.H., Chang T., Spano M.L., Ditto W.L. Controlling chaos in the brain. Nature. 370:1994;615-620.
23. Schmelcher P., Diakonos F.K. Detecting unstable periodic orbits in chaotic dynamical systems. Phys. Rev. Lett. 78:1997;4733-4736.
24. So P., Francis J.T., Netoff T.I., Gluckman B.J., Schiff S.J. Periodic orbits: a novel language for neuronal dynamics. Biophys. J. 74:1998;2776-2785.
25. So P., Ott E., Sauer T., Gluckman B.J., Grebogi C., Schiff S.J. Extracting unstable periodic orbits from chaotic time series data. Phys. Rev. E. 55:1997;5398-5417.
26. So P., Ott E., Schiff S.J., Kaplan D.T., Sauer T., Grebogi C. Detecting unstable periodic orbits in chaotic experimental data. Phys. Rev. Lett. 76:1996;4705-4708.
27. D.A. Sprecher, Elements of Real Analysis, Dover, New York, 1970.
28. de Ruyter van Steveninck R.R., Lewen G.D., Strong S.P., Koberle R., Bialek W. Reproducibility and variability in neural spike trains. Science. 275:1997;1805-1808.
29. Takens F. Detecting strange attractors in turbulence. Lecture Notes Math. 898:1981;366-381.
30. Theiler J., Eubank S., Longtin A., Galdrikian B., Farmer J.D. Physica D. 58:1992;77-94.
31. H.C. Tuckwell, Introduction to Theoretical Neurobiology, Vol. 2, Cambridge University Press, Cambridge, 1988.