An artificial neuron is a mathematical function conceived as a model of biological neurons in a neural network. Artificial neurons are the elementary units of artificial neural networks. [1] The artificial neuron is a function that receives one or more inputs, applies weights to these inputs, and sums them to produce an output.
The design of the artificial neuron was inspired by neural circuitry. Its inputs are analogous to excitatory postsynaptic potentials and inhibitory postsynaptic potentials at neural dendrites, or activation, its weights are analogous to synaptic weight, and its output is analogous to a neuron's action potential which is transmitted along its axon.
Usually, each input is separately weighted, and the sum is often added to a term known as a bias (loosely corresponding to the threshold potential), before being passed through a non-linear function known as an activation function or transfer function [ clarification needed ]. The transfer functions usually have a sigmoid shape, but they may also take the form of other non-linear functions, piecewise linear functions, or step functions. They are also often monotonically increasing, continuous, differentiable and bounded. Non-monotonic, unbounded and oscillating activation functions with multiple zeros that outperform sigmoidal and ReLU-like activation functions on many tasks have also been recently explored. 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. [2]
The artificial neuron transfer function should not be confused with a linear system's transfer function.
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. [3] Artificial neurons can also refer to artificial cells in neuromorphic engineering that are similar to natural physical neurons.
For a given artificial neuron k, let there be m + 1 inputs with signals x0 through xm and weights wk0 through wkm. Usually, the x0 input is assigned the value +1, which makes it a bias input with wk0 = bk. This leaves only m actual inputs to the neuron: from x1 to xm.
The output of the kth neuron is:
Where (phi) is the transfer function (commonly a threshold function).
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 unit, Nv neuron, binary neuron, linear threshold function, or McCulloch–Pitts (MCP) neuron. The original MCP neural network are described in detail in Chapter 3 of. [4]
In short, a MCP neuron is a restricted kind of artificial neuron, operating in discrete time-steps. Each has zero or more inputs, and let them be written as . It has one output, let it be written as . Each input can be either excitatory or inhibitory. The output could either be quiet or firing. It also has a threshold
In a MCP neural network, all the neurons operate in synchronous discrete time-steps of . At time , the output of the neuron is if the number of firing excitatory inputs is at least equal to the threshold, and no inhibitory inputs are firing; otherwise.
Each output can be the input to an arbitrary number of neurons, including itself (that is, self-loops are possible). However, an output cannot twice connect with a single neuron. Self-loops do not cause contradictions, since the network operates in synchronous discrete time-steps.
As a simple example, consider a single neuron with threshold 0, and a single inhibitory self-loop. Its output would oscillate between 0 and 1 at every step, acting as a "clock".
Any finite state machine can be simulated by a MCP neural network, as proven in. [4] Furnished with an infinite tape, MCP neural networks can simulate any Turing machine. [5]
Artificial neurons are designed to mimic aspects of their biological counterparts. However a significant performance gap exists between biological and artificial neural networks. In particular single biological neurons in the human brain with oscillating activation function capable of learning the XOR function have been discovered. [6]
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.) at which an axon fires converts directly into the rate at which neighboring cells get signal ions introduced into them. 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). It is this conversion that allows computer scientists and mathematicians to simulate biological neural networks using artificial neurons which can output distinct values (often from −1 to 1).
Research has shown that unary coding is used in the neural circuits responsible for birdsong production. [7] [8] The use of unary in biological networks is presumably due to the inherent simplicity of the coding. Another contributing factor could be that unary coding provides a certain degree of error correction. [9]
There is research and development into physical artificial neurons – organic and inorganic.
For example, some artificial neurons can receive [10] [11] and release dopamine (chemical signals rather than electrical signals) and communicate with natural rat muscle and brain cells, with potential for use in BCIs/prosthetics. [12] [13]
Low-power biocompatible memristors may enable construction of artificial neurons which function at voltages of biological action potentials and could be used to directly process biosensing signals, for neuromorphic computing and/or direct communication with biological neurons. [14] [15] [16]
Organic neuromorphic circuits made out of polymers, coated with an ion-rich gel to enable a material to carry an electric charge like real neurons, have been built into a robot, enabling it to learn sensorimotorically within the real world, rather than via simulations or virtually. [17] [18] Moreover, artificial spiking neurons made of soft matter (polymers) can operate in biologically relevant environments and enable the synergetic communication between the artificial and biological domains. [19] [20]
The first artificial neuron was the Threshold Logic Unit (TLU), or Linear Threshold Unit, [21] first proposed by Warren McCulloch and Walter Pitts in 1943. The model was specifically targeted as a computational model of the "nerve net" in the brain. [22] As a transfer function, it employed a threshold, equivalent to using the Heaviside step function. 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 perceptron, developed by Frank Rosenblatt. This model already considered more flexible weight values in the neurons, and was used in machines with adaptive capabilities. The representation of the threshold values as a bias term was introduced by Bernard Widrow in 1960 – see ADALINE.
In the late 1980s, 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. [23] [24]
The transfer function (activation 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; a non-linear function is therefore necessary to gain the advantages of a multi-layer network.[ citation needed ]
Below, u refers in all cases to the weighted sum of all the inputs to the neuron, i.e. for n inputs,
where w is a vector of synaptic weights and x is a vector of inputs.
The output y of this transfer function is binary, depending on whether the input meets a specified threshold, θ. The "signal" is sent, i.e. the output is set to one, if the activation meets the threshold.
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 analysis, and they can all be used in neural networks with this linear neuron. The bias term allows us to make affine transformations to the data.
See: Linear transformation, Harmonic analysis, Linear filter, Wavelet, Principal component analysis, Independent component analysis, Deconvolution.
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.
In the context of artificial neural networks, the rectifier or ReLU (Rectified Linear Unit) is an activation function defined as the positive part of its argument:
where x is the input to a neuron. This is also known as a ramp function and is analogous to half-wave rectification in electrical engineering. This activation function was first introduced to a dynamical network by Hahnloser et al. in a 2000 paper in Nature [25] with strong biological motivations and mathematical justifications. [26] It has been demonstrated for the first time in 2011 to enable better training of deeper networks, [27] compared to the widely used activation functions prior to 2011, i.e., the logistic sigmoid (which is inspired by probability theory; see logistic regression) and its more practical [28] counterpart, the hyperbolic tangent.
A commonly used variant of the ReLU activation function is the Leaky ReLU which allows a small, positive gradient when the unit is not active:
where x is the input to the neuron and a is a small positive constant (in the original paper the value 0.01 was used for a). [29]
The following is a simple pseudocode implementation of a single TLU which takes boolean inputs (true or false), and 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.
class TLU defined as:data member threshold : number data member weights : list of numbers of size X function member fire(inputs : list of booleans of size X) : boolean defined as:variable T : number T ← 0 for each i in 1 to X doif inputs(i) is true then T ← T + weights(i) end ifend for eachif T > threshold thenreturn true else:return false end ifend functionend class
In machine learning, the perceptron is an algorithm for supervised learning of binary classifiers. A binary classifier is a function which can decide whether or not an input, represented by a vector of numbers, belongs to some specific class. It is a type of linear classifier, i.e. a classification algorithm that makes its predictions based on a linear predictor function combining a set of weights with the feature vector.
Hebbian theory is a neuropsychological theory claiming that an increase in synaptic efficacy arises from a presynaptic cell's repeated and persistent stimulation of a postsynaptic cell. It is an attempt to explain synaptic plasticity, the adaptation of brain neurons during the learning process. It was introduced by Donald Hebb in his 1949 book The Organization of Behavior. The theory is also called Hebb's rule, Hebb's postulate, and cell assembly theory. Hebb states it as follows:
Let us assume that the persistence or repetition of a reverberatory activity tends to induce lasting cellular changes that add to its stability. ... When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A’s efficiency, as one of the cells firing B, is increased.
A Hopfield network is a form of recurrent artificial neural network and a type of spin glass system popularised by John Hopfield in 1982 as described by Shun'ichi Amari in 1972 and by Little in 1974 based on Ernst Ising's work with Wilhelm Lenz on the Ising model. Hopfield networks serve as content-addressable ("associative") memory systems with binary threshold nodes, or with continuous variables. Hopfield networks also provide a model for understanding human memory.
In machine learning, backpropagation is a gradient estimation method used to train neural network models. The gradient estimate is used by the optimization algorithm to compute the network parameter updates.
A feedforward neural network (FNN) is one of the two broad types of artificial neural network, characterized by direction of the flow of information between its layers. Its flow is uni-directional, meaning that the information in the model flows in only one direction—forward—from the input nodes, through the hidden nodes and to the output nodes, without any cycles or loops, in contrast to recurrent neural networks, which have a bi-directional flow. Modern feedforward networks are trained using the backpropagation method and are colloquially referred to as the "vanilla" neural networks.
A neural circuit is a population of neurons interconnected by synapses to carry out a specific function when activated. Multiple neural circuits interconnect with one another to form large scale brain networks.
ADALINE is an early single-layer artificial neural network and the name of the physical device that implemented this network. The network uses memistors. It was developed by professor Bernard Widrow and his doctoral student Ted Hoff at Stanford University in 1960. It is based on the perceptron. It consists of a weight, a bias and a summation function.
Oja's learning rule, or simply Oja's rule, named after Finnish computer scientist Erkki Oja, is a model of how neurons in the brain or in artificial neural networks change connection strength, or learn, over time. It is a modification of the standard Hebb's Rule that, through multiplicative normalization, solves all stability problems and generates an algorithm for principal components analysis. This is a computational form of an effect which is believed to happen in biological neurons.
Neural backpropagation is the phenomenon in which, after the action potential of a neuron creates a voltage spike down the axon, another impulse is generated from the soma and propagates towards the apical portions of the dendritic arbor or dendrites. In addition to active backpropagation of the action potential, there is also passive electrotonic spread. While there is ample evidence to prove the existence of backpropagating action potentials, the function of such action potentials and the extent to which they invade the most distal dendrites remain highly controversial.
In neuroscience and computer science, synaptic weight refers to the strength or amplitude of a connection between two nodes, corresponding in biology to the amount of influence the firing of one neuron has on another. The term is typically used in artificial and biological neural network research.
Biological neuron models, also known as spiking neuron models, are mathematical descriptions of neurons. In particular, these models describe how the voltage potential across the cell membrane changes over time. In an experimental setting, stimulating neurons with an electrical current generates an action potential, that propagates down the neuron's axon. This axon can branch out and connect to a large number of downstream neurons at sites called synapses. At these synapses, the spike can cause release of a biochemical substance (neurotransmitter), which in turn can change the voltage potential of downstream neurons, potentially leading to spikes in those downstream neurons, thus propagating the signal. As many as 85% of neurons in the neocortex, the outermost layer of the mammalian brain, consist of excitatory pyramidal neurons, and each pyramidal neuron receives tens of thousands of inputs from other neurons. Thus, spiking neurons are a major information processing unit of the nervous system.
In neurophysiology, a dendritic spike refers to an action potential generated in the dendrite of a neuron. Dendrites are branched extensions of a neuron. They receive electrical signals emitted from projecting neurons and transfer these signals to the cell body, or soma. Dendritic signaling has traditionally been viewed as a passive mode of electrical signaling. Unlike its axon counterpart which can generate signals through action potentials, dendrites were believed to only have the ability to propagate electrical signals by physical means: changes in conductance, length, cross sectional area, etc. However, the existence of dendritic spikes was proposed and demonstrated by W. Alden Spencer, Eric Kandel, Rodolfo Llinás and coworkers in the 1960s and a large body of evidence now makes it clear that dendrites are active neuronal structures. Dendrites contain voltage-gated ion channels giving them the ability to generate action potentials. Dendritic spikes have been recorded in numerous types of neurons in the brain and are thought to have great implications in neuronal communication, memory, and learning. They are one of the major factors in long-term potentiation.
Models of neural computation are attempts to elucidate, in an abstract and mathematical fashion, the core principles that underlie information processing in biological nervous systems, or functional components thereof. This article aims to provide an overview of the most definitive models of neuro-biological computation as well as the tools commonly used to construct and analyze them.
Sparse distributed memory (SDM) is a mathematical model of human long-term memory introduced by Pentti Kanerva in 1988 while he was at NASA Ames Research Center.
Network of human nervous system comprises nodes that are connected by links. The connectivity may be viewed anatomically, functionally, or electrophysiologically. These are presented in several Wikipedia articles that include Connectionism, Biological neural network, Artificial neural network, Computational neuroscience, as well as in several books by Ascoli, G. A. (2002), Sterratt, D., Graham, B., Gillies, A., & Willshaw, D. (2011), Gerstner, W., & Kistler, W. (2002), and Rumelhart, J. L., McClelland, J. L., and PDP Research Group (1986) among others. The focus of this article is a comprehensive view of modeling a neural network. Once an approach based on the perspective and connectivity is chosen, the models are developed at microscopic, mesoscopic, or macroscopic (system) levels. Computational modeling refers to models that are developed using computing tools.
Compartmental modelling of dendrites deals with multi-compartment modelling of the dendrites, to make the understanding of the electrical behavior of complex dendrites easier. Basically, compartmental modelling of dendrites is a very helpful tool to develop new biological neuron models. Dendrites are very important because they occupy the most membrane area in many of the neurons and give the neuron an ability to connect to thousands of other cells. Originally the dendrites were thought to have constant conductance and current but now it has been understood that they may have active Voltage-gated ion channels, which influences the firing properties of the neuron and also the response of neuron to synaptic inputs. Many mathematical models have been developed to understand the electric behavior of the dendrites. Dendrites tend to be very branchy and complex, so the compartmental approach to understand the electrical behavior of the dendrites makes it very useful.
An artificial neural network's learning rule or learning process is a method, mathematical logic or algorithm which improves the network's performance and/or training time. Usually, this rule is applied repeatedly over the network. It is done by updating the weights and bias levels of a network when a network is simulated in a specific data environment. A learning rule may accept existing conditions of the network and will compare the expected result and actual result of the network to give new and improved values for weights and bias. Depending on the complexity of actual model being simulated, the learning rule of the network can be as simple as an XOR gate or mean squared error, or as complex as the result of a system of differential equations.
CoDi is a cellular automaton (CA) model for spiking neural networks (SNNs). CoDi is an acronym for Collect and Distribute, referring to the signals and spikes in a neural network.
An artificial neural network (ANN) combines biological principles with advanced statistics to solve problems in domains such as pattern recognition and game-play. ANNs adopt the basic model of neuron analogues connected to each other in a variety of ways.
A neural network is a group of interconnected units called neurons that send signals to one another. Neurons can be either biological cells or mathematical models. While individual neurons are simple, many of them together in a network can perform complex tasks. There are two main types of neural network.