Single Neuron for Solving XOR like Nonlinear Problems

Table 1

List of variables used in this article with their meaning.

Variables	Meaning
π-neuron	Multiplicative neuron model
πt-neuron	Symbol of translated multiplicative neuron model
Π	Product operator
π‒t	The mathematical form of the pt-neuron model
f	Activation function
xi	‘i^th’ input to the neuron
y	The final output of the pt-neuron model through the activation function ‘f’
bπ‒t	Scaling factor associated with the pt-neuron model
ti	The coordinates of the center of the decision surfaces (‘i^th’ threshold value)
δ(n)	The local gradient in the backpropagation algorithm at ‘n^th’ iteration
ϕ(x)	Sigmoid activation function
Ɛ(n)	The error energy (the instantaneous sum of the error squares at ‘n^th’ iteration)
∇Ɛ	Error gradient
e(n)	The error value in the backpropagation algorithm at ‘n^th’ iteration
ϕ′	Derivative of the sigmoid activation function
b	Scaling factor associated with the enhanced pt-neuron model (proposed)
	The mathematical form of the proposed model
O	The final output of the proposed model through the activation function ‘ϕ’
ℝ	Set of the real numbers
L1, L2, L∞	Respective loss functions
ℒ	L1 loss function
W	Weight matrix
G	Gaussian probability distribution
p	Number of samples required in an XOR dataset for appropriate training
μ	Mean or average value
σ	Standard deviation value