Abstract. We provide an asymptotically justified derivation of activity measure evolution equations (AMEE) for a finite size neural network. The approach takes into account the dynamics for each isolated neuron in the network being modeled by a biophysical model, i.e. Hodgkin-Huxley equations or their reductions. By representing the interacting network as self and pairwise interactions, we propose a general definition of spatial projections of the network, called activity measures, that quantify the activity of a network. We show that the evolution equations that govern the dynamics of the activity measure shadow the activity measure of the network (i.e. the two quantities stay close to each other for all times) for general interactions and various asymptotic dynamics. The AMEE effectively serve as a dimensionality reduction technique for the complex network when spatial synchrony and coherence are present and allow to a priori predict network dynamics that would not be guessed from individual neuron behavior. To demonstrate an explicit derivation of such a reduction, we consider the mean measure for a network of interacting FitzHugh-Nagumo neurons.Computational results comparing the full network dynamics with the mean AMEE model of identical and nonidentical FitzHugh-Nagumo and FitzHugh-Rinzel neurons validate the shadowing theorems and expose the various resulting AMEE models that allow to describe the mean of the network.

Accurate and real-time video surveillance techniques for removing background variations in a video stream, which are highly correlated between frames, are at the forefront of modern data-analysis research. The objective in such algorithms is to highlight foreground objects of potential interest. Background/foreground separation is typically an integral step in detecting, identifying, tracking, and recognizing objects in video sequences. Most modern computer vision applications demand algorithms that can be implemented in real-time, and that are robust enough to handle diverse, complicated, and cluttered backgrounds. Competitive methods often need to be flexible enough to accommodate changes in a scene due to, for instance, illumination changes that can occur throughout the day, or location changes where the application is being implemented. Given the importance of this task, a variety of iterative techniques and methods have already been developed in order to perform background/foreground separation [4,8,11,15,23,24] (See also, for instance, the recent reviews by Bouwmans [2] and Benezeth et al. [1], which compare error and timing of various methods).One potential viewpoint of this computational task is as a matrix separation problem into low-rank (background) and sparse (foreground) components. Recently, this viewpoint has been advocated by Candès et al. in the framework of robust principal component analysis 19-1 (RPCA) [4]. By weighting a combination of the nuclear and the L 1 norms, a convenient convex optimization problem (principal component pursuit) was demonstrated, under suitable assumptions, to exactly recover the low-rank and sparse components of a given data-matrix (or video for our purposes). It was also compared to the state-of-the-art computer vision procedure developed by De La Torre and Black [10]. We advocate a similar matrix separation approach, but by using the method of dynamic mode decomposition (DMD) [5,17,[20][21][22]26] (see also Kutz [9] for a tutorial review). This method, which essentially implements a Fourier decomposition of correlated spatial activity of the video frames in time, distinguishes the stationary background from the dynamic foreground by differentiating between the nearzero Fourier modes and the remaining modes bounded away from the origin, respectively [7]. Originally introduced in the fluid mechanics community, DMD has emerged as a powerful tool for analyzing the dynamics of nonlinear systems [5,17,[20][21][22]26].In the application of video surveillance, the video frames can be thought of as snapshots of some underlying complex/nonlinear dynamics. The DMD decomposition yields oscillatory time components of the video frames that have contextual implications. Namely, those modes that are near the origin represent dynamics that are unchanging, or changing slowly, and can be interpreted as stationary background pixels, or low-rank components of the data matrix. In contrast, those modes bounded away from the origin are changing on O(1) timescales or faster, and represent ...

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