ON THE EQUIVALENCE OF TWO-LAYERED PERCEPTRONS WITH BINARY NEURONS

1995 ◽  
Vol 06 (03) ◽  
pp. 225-231 ◽  
Author(s):  
MARCELO BLATT ◽  
EYTAN DOMANY ◽  
IDO KANTER
Keyword(s):  
Boolean Function ◽  
Global Symmetries ◽  
Input Output ◽  
Output Unit ◽  
Binary Output ◽  
Linearly Independent ◽  
Input Layer ◽  
Binary Input

We consider two-layered perceptrons consisting of N binary input units, K binary hidden units and one binary output unit, in the limit N≫K≥1. We prove that the weights of a regular irreducible network are uniquely determined by its input-output map up to some obvious global symmetries. A network is regular if its K weight vectors from the input layer to the K hidden units are linearly independent. A (single layered) perceptron is said to be irreducible if its output depends on every one of its input units; and a two-layered perceptron is irreducible if the K+1 perceptrons that constitute such network are irreducible. By global symmetries we mean, for instance, permuting the labels of the hidden units. Hence, two irreducible regular two-layered perceptrons that implement the same Boolean function must have the same number of hidden units, and must be composed of equivalent perceptrons.

scholarly journals Purkinje cell outputs selectively inhibit a subset of unipolar brush cells in the input layer of the cerebellar cortex

2021 ◽  
Author(s):  
Chong Guo ◽  
Stephanie Rudolph ◽  
Morgan E. Neuwirth ◽  
Wade G. Regehr
Keyword(s):  
Purkinje Cell ◽  
Cerebellar Cortex ◽  
Mossy Fiber ◽  
Input Output ◽  
Brush Cells ◽  
Unipolar Brush Cells ◽  
Input Layer ◽  
Inhibitory Feedback ◽  
Almost All ◽  
Control Balance

AbstractCircuitry of the cerebellar cortex is regionally and functionally specialized. Unipolar brush cells (UBCs), and Purkinje cell (PC) synapses made by axon collaterals in the granular layer, are both enriched in areas that control balance and eye-movement. Here we find a link between these specializations: PCs preferentially inhibit mGluR1-expressing UBCs that respond to mossy fiber inputs with long lasting increases in firing, but PCs do not inhibit mGluR1-lacking UBCs. PCs inhibit about 29% of mGluR1-expressing UBCs by activating GABAA receptors (GABAARs) and inhibit almost all mGluR1-expressing UBCs by activating GABABRs. PC to UBC synapses allow PC output to regulate the input layer of the cerebellar cortex in diverse ways. GABAAR-mediated feedback is fast, unreliable, noisy, and suited to linearizing input-output curves and decreasing gain. Slow GABABR-mediated inhibition allows elevated PC activity to sharpen the input-output transformation of UBCs, and allows dynamic inhibitory feedback of mGluR1-expressing UBCs.

scholarly journals REALIZATION AND BIFURCATION OF BOOLEAN FUNCTIONS VIA CELLULAR NEURAL NETWORKS

2005 ◽  
Vol 15 (07) ◽  
pp. 2109-2129 ◽  
Author(s):  
FANGYUE CHEN ◽  
GUANRONG CHEN
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Boolean Function ◽  
Boolean Functions ◽  
Cellular Neural Network ◽  
Cellular Neural Networks ◽  
Gene Bank ◽  
Engineering Applications ◽  
Binary Input

In this work, we study the realization and bifurcation of Boolean functions of four variables via a Cellular Neural Network (CNN). We characterize the basic relations between the genes and the offsets of an uncoupled CNN as well as the basis of the binary input vectors set. Based on the analysis, we have rigorously proved that there are exactly 1882 linearly separable Boolean functions of four variables, and found an effective method for realizing all linearly separable Boolean functions via an uncoupled CNN. Consequently, any kind of linearly separable Boolean function can be implemented by an uncoupled CNN, and all CNN genes that are associated with these Boolean functions, called the CNN gene bank of four variables, can be easily determined. Through this work, we will show that the standard CNN invented by Chua and Yang in 1988 indeed is very essential not only in terms of engineering applications but also in the sense of fundamental mathematics.

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