# Geometry of binary threshold neurons and their networks

An artificial neuron is a mathematical function conceived as a model of biological neuronsa neural network. Artificial neurons are elementary units in an artificial neural network.

The artificial neuron receives one or more inputs representing excitatory postsynaptic potentials and inhibitory postsynaptic potentials at neural dendrites and sums them to produce an output or activationrepresenting a neuron's action potential which is transmitted along its axon.

Usually each input is separately weightedand the sum is passed through a non-linear function known as an activation function or transfer function [ clarification needed ]. The transfer functions usually have a sigmoid shapebut they may also take the form of other non-linear functions, piecewise linear functions, or step functions. They are also often monotonically increasingcontinuousdifferentiable and bounded.

The thresholding function has inspired building logic gates referred to as threshold logic; applicable to building logic circuits resembling brain processing. For example, new devices such as memristors have been extensively used to develop such logic in recent times.

The artificial neuron transfer function should not be confused with a linear system's transfer function. This leaves only m actual inputs to the neuron: The output is analogous to the axon of a biological neuron, and its value propagates to the input of the next layer, through a synapse.

It may also exit the system, possibly as part of an output vector. It has no learning process as such. Its transfer function weights are calculated and threshold value are predetermined. Depending on the specific model used they may be called a semi-linear unitNv neuronbinary neuronlinear threshold functionor McCulloch—Pitts MCP neuron. Simple artificial neurons, such as the McCulloch—Pitts model, are sometimes described as "caricature models", since they are intended to reflect one or more neurophysiological observations, but without regard to realism.

Unlike most artificial neurons, however, biological neurons fire in discrete pulses. Each time the electrical potential inside the soma reaches a certain threshold, a pulse is transmitted down the axon. This pulsing can be translated into continuous values. The rate activations per second, etc. The faster a biological neuron fires, the faster nearby neurons accumulate electrical potential or lose electrical potential, depending on the "weighting" of the dendrite that connects to the neuron that fired.

Research has shown that unary coding is used in the neural circuits responsible for birdsong production. Another contributing factor could be that unary coding provides a certain degree of error correction.

The model was specifically targeted as a computational model of the "nerve net" in the brain. Initially, only a simple model was considered, with binary inputs and outputs, some restrictions on the possible weights, and a more flexible threshold value.

Since the beginning it was already noticed that any boolean function could be implemented by networks of such devices, what is easily seen from the fact that one can implement the AND and OR functions, and use them in the disjunctive or the conjunctive normal form.

Researchers also soon realized that cyclic networks, with feedbacks through neurons, could define dynamical systems with memory, but most of the research concentrated and still does on strictly feed-forward networks because of the smaller difficulty they present.

One important and pioneering artificial neural network that used the linear threshold function was the perceptrondeveloped by Frank Rosenblatt. This model already considered more flexible weight values in the neurons, and was used in machines with adaptive capabilities. In the late s, when research on neural networks regained strength, neurons with more continuous shapes started to be considered. The possibility of differentiating the activation function allows the direct use of the gradient descent and other optimization algorithms for the adjustment of the weights.

Neural networks also started to be used as a general function approximation model. The best known training algorithm called backpropagation has been rediscovered several times but its first development goes back to the work of Paul Werbos. The transfer function of a neuron is chosen to have a number of properties which either enhance or simplify the network containing the neuron.

Crucially, for instance, any multilayer perceptron using a linear transfer function has an equivalent single-layer network [ citation needed ] ; a non-linear function is therefore necessary to gain the advantages of a multi-layer network.

Below, u refers in all cases to the weighted sum of all the inputs to the neuron, i. The "signal" is sent, i. This function is used in perceptrons and often shows up in many other models.

It performs a division of the space of inputs by a hyperplane. It is specially useful in the last layer of a network intended to perform binary classification of the inputs.

It can be approximated from other sigmoidal functions by assigning large values to the weights. In this case, the output unit is simply the weighted sum of its inputs plus a bias term. A number of such linear neurons perform a linear transformation of the input vector.

This is usually more useful in the first layers of a network. A number of analysis tools exist based on linear models, such as harmonic analysisand they can all be used in neural networks with this linear neuron.

The bias term allows us to make affine transformations to the data. Linear transformationHarmonic analysisLinear filterWaveletPrincipal component analysisIndependent component analysisDeconvolution. A fairly simple non-linear function, the sigmoid function such as the logistic function also has an easily calculated derivative, which can be important when calculating the weight updates in the network.

It thus makes the network more easily manipulable mathematically, and was attractive to early computer scientists who needed to minimize the computational load of their simulations. It was previously commonly seen in multilayer perceptrons. However, recent work has shown sigmoid neurons to be less effective than rectified linear neurons. The reason is that the gradients computed by the backpropagation algorithm tend to diminish towards zero as activations propagate through layers of sigmoidal neurons, making it difficult to optimize neural networks using multiple layers of sigmoidal neurons.

The following is a simple pseudocode implementation of a single TLU which takes boolean inputs true or falseand returns a single boolean output when activated. An object-oriented model is used. No method of training is defined, since several exist. If a purely functional model were used, the class TLU below would be replaced with a function TLU with input parameters threshold, weights, and inputs that returned a boolean value.

From Wikipedia, the free encyclopedia. This section needs expansion. You can help by adding to it. It has been suggested that this section be split out into another article titled transfer function. It has been suggested that this section be split out into another article titled Threshold Logic Unit.

Weakly connected neural networks. Neural network models of birdsong production, learning, and coding PDF. New Encyclopedia of Neuroscience: Archived from the original PDF on Retrieved 12 April Discrete Mathematics of Neural Networks: Aggarwal 25 July A logical calculus of the ideas immanent in nervous activity.

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