Abstract
Most neurons in the central nervous system exhibit all-or-none firing behavior. This makes Boolean Functions (BFs) tractable candidates for representing computations performed by neurons, especially at finer time scales, even though BFs may fail to capture some of the richness of neuronal computations such as temporal dynamics. One biologically plausible way to realize BFs is to compute a weighted sum of products of inputs and pass it through a heaviside step function. This representation is called a Higher Order Neuron (HON). A HON can trivially represent any n-variable BF with \(2^n\) product terms. There have been several algorithms proposed for obtaining representations with fewer product terms. In this work, we propose improvements over previous algorithms for obtaining parsimonious HON representations and present numerical comparisons. In particular, we improve the algorithm proposed by Sezener and Oztop [1] and cut down its time complexity drastically, and develop a novel hybrid algorithm by combining metaheuristic search and the deterministic algorithm of Oztop [2].
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Notes
- 1.
An earlier version of the paper was posted as arXiv:1504.01167.
- 2.
The vector is constructed as \([1, x_1, x_2, x_2x_1,x_3,x_3x_1,x_3x_2x_1,...,x_{n}...x_2x_1]\). Assignments to (\(x_1, x_2, x_3,...,x_{n}\)) are ordered as (0’s represent 1’s and 1’s represent -1’s): 000...0, 100...0, 010...0, 110...0, ..., 111...1.
- 3.
By inspection, we choose \(k_{max}\) as 5 and further constraint the space to odd-integers.
- 4.
The same equivalence classes and numbering used in [1] is adopted.
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Sezener, C.E., Oztop, E. (2017). Algorithms for Obtaining Parsimonious Higher Order Neurons. In: Lintas, A., Rovetta, S., Verschure, P., Villa, A. (eds) Artificial Neural Networks and Machine Learning – ICANN 2017. ICANN 2017. Lecture Notes in Computer Science(), vol 10613. Springer, Cham. https://doi.org/10.1007/978-3-319-68600-4_18
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