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\bc
{\bf SYSTEM RECONSTRUCTION AND ANALYSIS \\USING INTERSPIKE INTERVALS}
\ec

\bc ROLANDO CASTRO and TIM SAUER\\
\it Institute of Computational Sciences and Informatics \\\it and
Department of Mathematical Sciences \\ \it George Mason
University \\ \it Fairfax, VA 22030, USA \\ %\it tsauer\verb+@+gmu.edu 
\ec

\bc ABSTRACT \ec

\begin{quotation} \footnotesize
\noindent Statistical measures of determinism for spikes train are
described and analyzed. The analysis is focused on two dynamical
invariants: nonlinear forecastability and correlation dimension.  Two
simple, generic, biologically motivated methods for generating spikes
from the system are surveyed: an integrate-and-fire model (IF) and a
threshold-crossing method (TC). The effect on the statistical measures
of threshold level, noise and limited amount of available data are
investigated.
\end{quotation}

\bigskip

\noindent{\bf 1. Introduction}

\bigskip

Substantial progress has been made on chaotic time series analysis
during the last twenty years. Identification and reconstruction of the
chaotic system are specific goals of the researchers, which underlie
the applications of system prediction and control. Using delay
coordinates (Packard et al., 1980; Takens, 1981) to reconstruct the
state space of 
the dynamical system has been a successful idea. A
delay coordinate reconstruction of a compact
attractor from a time series is topologically equivalent to the
attractor for a probability-one choice of measurement functions, as
long as the embedding dimension is greater than twice the box-counting
dimension of the attractor. This topological equivalence is the basis
for a great deal of the statistical methodology for treating chaotic
time series.

There is an analogous problem of relating
interspikes intervals (ISI) to the dynamical system from which they
are measured. This 
problem arises in many physiological systems when direct measurements
of the state variables, to obtain an evenly-sampled time series,  are
not available or possible. Some researchers have attempted to adapt
methods of chaotic time series analysis
to single neuron or neuron population spike trains, others to
the description of more global electrophysiological measurements such
as the EEG.

  Many mathematical models have been proposed for the dynamical
  structure of spiking phenomena in physiological systems
  (Tuckwell, 1988). In this report we survey two of these models and
  discuss the possibility of recovering the dynamical invariants of
  the system from the spike train. In particular we focus on the
  correlation dimension and nonlinear predictability of the
  ISI's, and show how they can be used for control purposes. The goal
  of this article is to study how evidence of 
  deterministic nonlinear dynamics can be carried by spike trains.


\begin{figure}[thbp]
  \begin{center}
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     \epsfxsize= 3in
     \epsfbox{fig01.eps}
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     \caption{The upper trace $x(t)$ is the $x$ coordinate of the Lorenz
attractor graphed as a function of time. The lower trace shows the
times at which spikes are generated  using $S(t) =  x(t) + 25 $ and
$\Theta = 10.$ } 
\label{fig1}
  \end{figure}


Under certain genericity conditions on the underlying dynamics, signal and
threshold, the series $ \{ t_{j} \} $ of ISI's can be used to
reconstruct the attractor. That is, there is a one-to-one
correspondence between the {\it m}-tuples of ISI's and attractor
states, which associates each vector $(t_{i}, t_{i-1} , \cdots ,
t_{i-m+1})$ of ISI's with the corresponding point $x(t_{i})$ on the
attractor. In analogy with Takens theorem (Takens, 1981) and its
generalization (Sauer et al., 1991), the condition $m>2D_{0}$ is sufficient,
where $D_{0}$ is the box counting dimension of the attractor. As with
Takens' theorem, smaller $m$ may be sufficient in particular
cases. 

\bigskip


\noindent{\bf References}

\bigskip

\begin{list}{}{\setlength{\leftmargin}{.3in}\setlength{\itemsep}{0in}
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\item
M. Ding, C. Grebogi, E. Ott, T. Sauer, J.A. Yorke, Estimating correlation
dimension from a chaotic time series: when does plateau occur?, {\it
Physica D} {\bf 69}, 404 (1993).

\item
J.D. Farmer, J. Sidorowich, Exploiting chaos to
predict the future and reduce noise, {\it Evolution, Learning and
Cognition}, ed. Y.C. Lee , World Scientific, Singapore, (1988).

\item
P. Grassberger, An optimized box-assisted algorithm for
fractal dimension, {\it Phys. Rev. A} {\bf 148}, 63 (1990).


\end{list}


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