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as the window was stepped along the time axis (Fig. 3c). We calculated the distance in conguration space between the trajectory for each trial (the presentation of either a rising or a falling frequency-modulated tone) and a trajectory formed by averaging over all trajectories for the other stimulus in that session; that is, a single trial with a rising frequency-modulated tone was compared with the mean over all trials with falling frequency-modulated tones and vice versa. The distance is a dissimilarity function, a measure of the dissimilarity between the compared spatial activity patterns expressed in standard deviations (s.d.) of the mean dissimilarity obtained over the entire trajectory (Fig. 3d). Marked states are dened as peaks in the dissimilarity function with amplitudes at least three s.d. over the baseline of the amplitude distribution. We used a nonlinear projection algorithm30 to display the combinatorial multitude of similarity relations between marked states, resulting in a map of the corresponding 18-dimensional state vectors in the two-dimensional plane that preserves all mutual distances between projected states. This is achieved by minimizing the relative mapping error
Ek 1 D
^
N d*ij 2 dijk2 d*ij
where
D
N
d*ij
^
and d*ij is the euclidean distance between marked state vectors i and j in the original 18-dimensional state space, dijk is the distance between the corresponding vectors projected into the two-dimensional plane after the kth step, and N is the number of vectors projected. The mapping error was minimized with a step-wise steepest-gradient procedure starting with a two-dimensional uniformly random vector distribution in the unit square. The mapping was accepted when the criterion
Ek ! ` # 0:5%
was reached; when this was not achieved, another random starting condition for the steepest-gradient procedure was used. Clustering of the spatial patterns during the marked states after categorization (Fig. 2, right column) was tested using a resampling approach (see Supplementary Information).
Received 12 April; accepted 24 May 2001.
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