Abstract
Takens showed in 1981 that dynamical state information can be reconstructed from an experimental time series. Recently it has been shown that the same is true from a series of interspike interval (ISI) measurements. This method of system analysis allows prediction of the spike train from its history. The underlying assumption is that the spike train is generated by an integrate- and-fire model. We show that it is possible to use system reconstruction based on spiking behavior alone to control unstable periodic trajectories of the system in two separate ways: by making small (subthreshold) changes in a system parameter, and by using an on-demand pacing protocol with superthreshold pulses.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Ding, C. Grebogi, E. Ott, T. Sauer, J.A. Yorke, Estimating correlation dimension from a chaotic time series: when does plateau onset occur? Physica D 69, 404 (1993).
E. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 20, 130–141 (1963).
M.C. Mackey, L. Glass, Oscillations and chaos in physiological control systems, Science 197, 287 (1977).
M. Miller, D. Snyder, Random Point Processes in Time and Space. Springer-Verlag, New York (1991). P.A.W. Lewis, ed. Stochastic Point Processes: Statistical Analysis, Theory, and Applications. Wiley-Interscience, New York (1972).
E. Ott, C. Grebogi, E. Ott, Controlling chaos. Phys. Rev. Lett. 64, 1196 (1990).
E. Ott, T. Sauer, J.A. Yorke, Coping with Chaos: Analysis of Chaotic Data and the Exploitation of Chaotic Systems. Wiley Interscience, New York, 418 pp. (1994).
N. Packard, J. Crutchfield, J.D. Farmer, R. Shaw, Geometry from a time series, Phys. Rev. Lett. 45, 712 (1980).
D. Prichard, J. Theiler, Generating surrogate data for time series with several simultaneously measured variables, Phys. Rev. Lett. 73, 951 (1994).
T. Sauer, Reconstruction of dynamical systems from interspike intervals, Phys. Rev. Lett. 72, 3811 (1994).
T. Sauer, J.A. Yorke, and M. Casdagli, Embedology, J. Stat. Phys. 65, 579 (1991).
S.J. Schiff, K. Jerger, T. Chang, T. Sauer, P.G. Aitken, Stochastic versus deterministic variability in simple neuronal circuits. II. Hippocampal slice, Biophys. J. 67, 684 (1994).
F. Takens , Detecting strange attractors in turbulence, Lecture Notes in Math. 898, Springer-Verlag (1981).
J. Theiler, S. Eubank, A. Longtin, B. Galdrakian, J.D. Farmer, Testing for nonlinearity in time series: the method of surrogate data, Physica D 58, 77 (1992).
H. Tuckwell, Introduction to Theoretical Neurobiology, vols 1,2. Cambridge University Press (1988).
Babloyantz, A. 1989. Some remarks on nonlinear data analysis. Pages 51–62 in N.B. Abraham, A.M. Albano, A. Passamante and P.E. Rapp (editors). Measures of Complexity and Chaos. New York, Plenum Press.
MacDonald, G.M., T.W.D. Edwards, K.A. Moser, R. Pienitz, and J.P. Smol. 1993. Rapid response of treeline vegetation and lakes to past climate warming. Nature 361: 243–245.
Schaffer, W. 1984. Stetching and folding in lynx fur returrns: evidence for a strange attractor in nature? The American Naturalist 124: 798–820.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Birkhauser Boston
About this paper
Cite this paper
Sauer, T. (1997). Attractor Reconstruction and Control Using Interspike Intervals. In: Judd, K., Mees, A., Teo, K.L., Vincent, T.L. (eds) Control and Chaos. Mathematical Modelling, vol 8. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2446-4_2
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2446-4_2
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-7540-4
Online ISBN: 978-1-4612-2446-4
eBook Packages: Springer Book Archive