Abstract
In this paper, we consider a biologically-inspired Boolean function P n D that models a simple task of detecting global spatial patterns on a twodimensional map. We prove that P n D is computable by a threshold circuit of size (i.e., number of gates) O( p n logn), which is an improvement on the previous upper bound O(n), while our circuit has larger depth O( p n) and total wire length O(n log 2 n). Moreover, we demonstrate that the size of our circuit is nearly optimal up to a logarithmic factor: we show that any threshold circuit computing P n D has size ( p n= logn).
Highlights
A threshold circuit is a combinatorial circuit comprising logic gates computing linear threshold function, and comprises one of the most well-studied computational models in circuit complexity theory
Threshold circuits have been studied from another perspective: threshold circuits are considered to provide a theoretical model of neural networks in the brain [6, 7, 10]
It is known that a threshold gate, the basic element of a threshold circuit, captures the basic input-output characteristic of a biological neuron
Summary
A threshold circuit is a combinatorial circuit comprising logic gates computing linear threshold function, and comprises one of the most well-studied computational models in circuit complexity theory. (1) Conventional complexity theory focuses on a different set of computational problems such as the arithmetic operations above mentioned, and (2) standard complexity measures such as circuit size and depth are not tailored to resources that are of primary interest in neuromorphic engineering and the analysis of neural circuits in biological organisms. Motivated by this viewpoint, Legenstein and Maass attempted to resolve the situation by introducing the following setting.
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