Reconstructing Dynamical Systems from Amplitude Measures of Spiky Time-Series

AND MOTIVATION Many examples exist in nature where the output of a dynamic system is a spiky time series. Well known examples of such systems include the Rössler system [1], which was developed to model the Earth’s magnetic field, the Hindmarsh-Rose [2, 3] system which has been proposed to model neuron membrane electrical activity, Chay-Keizer equations which model pancreatic cells, and many other. These hypothetical mathematical models of the underlying pulse generating systems were developed empirically by developing sets of equations that could account for all the different operating regimes of the system. Of interest is to perform the inverse of such empirical modeling that is extract a model of the underlying dynamics from the output data. The data are typically single time series, for example a recording of neuronal impulses. The modeling from data would allow us: • to verify the accuracy of the empirically derived models • to identify alternate, simpler models, that are topologically equivalent to the existing empirical models of physical processes. Simplier models are desirable because they are easier to use in numerical simulations. Alternate models also have theoretical importance because they can be used to draw correspondence between the different physical systems and may also reveal physcial structures that are not obvious in more complex system representations. Attempts to recover multidimensional models from spiky data have not been successful. Moreover, theoretical explanations have been offered for why such recovery is not possible. In this paper, for the first time, a global 3-D model is derived directly from a spiky time series. We use the z-component of the Rössler system as an example to perform model recovery. The reasons for our choice of the system and its relevance to neuronal models are discussed.