In neural computing, we approximate biological neurons to
rather more basic **artificial neurons**. Artificial neurons mimic the
functionality of the biological neuron. In turn, these artificial neurons form
the basis of any neural network model.

Neural computing is
said to have started with the work of McCulloch & Pitts (published 1943),
who set out to explain nervous systems in logical terms. They reduced the
electro-chemical complexity of the biological neuron to the following form
(biological equivalent in parentheses):

Each processing element, PE (cell body), sums the binary
inputs (0/1 or +-1) coming though its input channels (dendrites). If this
summation exceeded a threshold energy (the activation energy),
the PE would respond by firing a binary value through its output channels
(axon, terminal fibres). Otherwise, the PE remains inactive. Each output
channel leads to another input channel via a weighted channel (synapse), which
modifies the binary signals moving to the next neuron.

The M&P model
describes a typical artificial neuron. Note that the M&P neuron works with
binary (or bipolar) values for inputs and outputs, demonstrating the
‘all-or-nothing’ firing mechanism of biological neurons.

Since the time of
M&P neurons, refinements have been made to this model. In order to define
precisely our mathematical model of a neuron, let us first state the basic
elements that must be incorporated:

1. A set of
connecting **links** (the biological synapses) that connect and transfer the
outputs of neurons to the inputs of other neurons. Each link has a
corresponding **synaptic weight**, which alters the value (or strength) of
the signal passing through it. For instance, the output signal from neuron x_{j}
connected to the input at neuron x_{i} will use a link with a
corresponding weight w_{ij} (note that the subscript first defines the
neuron the link is connected *to*, and then the neuron that the link has
come from). The value passing from x_{j} would be multiplied by w_{ij}
before being used as an input to x_{i} (this modified value is hence
referred to as a **weighted input**). Weights may take a positive or
negative value, and links between various neurons are usually identified using
their weights (for instance the link from x_{1} to x_{2} would
be w_{21}).

2. Each neuron
contains a **linear combiner**, which has the effect of adding all the
weighted input values fed into it. This value is commonly known as the **induced
local field, u** of the neuron.

3. The induced
local field is subjected to the neuron’s **activation function, f(x)**,
which could be any mathematical function. This serves to limit the amplitude of
the output from the neuron (usually the amplitude range for the output of a
neuron is restricted to [0,1] or [-1,1], and sometimes a simple binary/bipolar
output).
The final value, f(u), constitutes the neuron’s output.

This particular neuron, x_{k}, has n inputs (x_{1}
... x_{n}), which are being modified by the respective n weights (w_{k1}
... w_{kn}). These weighted values are then summed in the linear
combiner. This value, the induced local field, u_{k}, is then passed
onto the activation function, f(x), before being fed out as the output for the
neuron, y_{k}.

The function of each neuron can therefore be expressed
as:

u_{k}=SUM(w_{kj}x_{j})

y_{k}
= f(u_{k})

*u*_{k} is the induced local field of the neuron

w_{ik} is the weight of the connection from
input x_{i} to the neuron

N is the total number of inputs.

f(x) is the
activation function of the neuron.

y_{k} is the output from the neuron

In some neuronal
models, a **bias**, b_{k}, is added to the combined weighted input
of the neuron (i.e. added at the summing stage). Therefore using bias is
equivalent to the effect of applying an **affine transformation** to the
induced local field, u_{k}, of the neuron.

The model for a
neuron with bias is:

u_{k}=SUM(w_{kj}x_{j} + b_{k})

y_{k}
= f(u_{k})

Using bias allows
the neuron to increase or decrease the net input for its activation function,
depending on whether *b*_{k} > 0 or *b*_{k} < 0
respectively.