(2015)
An attractor network of weakly-coupled excitable neurons for
general purpose of edge detection.
*University of Southampton, Physical Sciences and Engineering, Doctoral Thesis*, 133pp.

## Abstract

The prospect of emulating the impressive computational capacities of biological systems has led to much interest in the design of analog circuits, potentially implementable in VLSI CMOS technology, that are guided by biologically motivated models. However, system design inevitably encounters the contrary constraints of size(or complexity) and computational power (or performance). From a high level design point of view, we believe that theoretical analysis of the model properties will undoubtedly benefit the implementation at a lower level. This thesis focuses on this simple aim to provide an extensive study of task-specific models based on dynamical systems in order to reduce model complexity or enhance the performance of algorithms.

In many examples, it is the self-evolving dynamics of the model that allows the intrinsic parallel computations of algorithms, which are traditionally expressed by differential equations and systems. For instance, simple image processing tasks, such as the detection of edges in binary and grayscale images, have been performed by a reaction diffusion equation using the FitzHugh-Nagumo model as the reaction term in the previous work done by Kurata et al. (2008); Nomura et al. (2003, 2008, 2011b,a). Once the initial condition is correctly assigned according to a processed image, system states of this model will automatically evolve to the final result.

From an application of view, the spatial distribution of system state can be regarded as a grid network with a proper discrete pattern; each network node becomes a FitzHugh-Nagumo type of neuron, while the diffusion term turns out to be the nearest couplings among them, where the coupling strength k is proportional to the original coefficient of diffusion D. So, one neuron (node) in the network deals with one pixel in the processed image. However, in previous study, this one-to-one mapping of image pixels to component neurons makes the size of the network a critical factor in any such implementation. The wrong edges are found due to the intrinsic mechanism of the algorithm when the diffused the processed image are used to pick out edges among the grayscale intensity levels and their most successful method solves this problem by a doubling of the size of the network.

In the thesis, we propose two main improvements of the original algorithm in order for the smaller complexity and the better performance. We treat dynamics of the coupled system for the purpose of edge detection as a k-perturbation of the uncoupled one. Based on stability analysis of system state for both uncoupled and coupled cases, the system used for edge detection is identified as a Multiple Attractor type network and the final edge result corresponds to an attractor in high dimensional space. Hence, we conclude that the edge detection problem maps an image to an initial condition that is correctly located within the attraction domain of an expected attractor. For the first improvement, in order to get rid of the wrong edges, we provide a way of quantify the excitability of uncoupled neurons based on the Lyapunov exponents so that the boundary of attraction domain of the attractors can be well estimated. Moreover, an anisotropic diffused version of processed image is used for the further enhancement on the performance. For the second improvement, in order for diffusion of the processed image being accomplished by the hardware, we introduce a self-stopping mechanism to the original equation. Moreover, we link the basic design rules on system parameter settings to the fundamental theorem of WCNN (weakly coupled neural network) (Hoppensteadt and Izhikevich, 1997), which states that an uncoupled neuron must be near a threshold (bifurcation point) in order for rich dynamics to be presented in the coupled network. We apply our techniques to detect edges in data sets of artificially generated images (both noise-free and noise polluted) and real images, demonstrating performance that is as good if not better that the results of (Nomura et al., 2011b,a) with a smaller size of the network.

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