Two or more of the neurons shown earlier can be combined in a layer, and a particular network could contain one or more such layers. First consider a single layer of neurons.

A one-layer network with *R* input elements
and *S* neurons follows.

In this network, each element of the input vector **p** is connected to each neuron input through
the weight matrix **W**. The *i*th
neuron has a summer that gathers its weighted inputs and bias to form
its own scalar output *n*(*i*).
The various *n*(*i*) taken together
form an *S*-element net input vector **n**. Finally, the neuron layer outputs form a
column vector **a**. The expression for **a** is shown at the bottom of the figure.

Note that it is common for the number of inputs to a layer to
be different from the number of neurons (i.e., *R* is
not necessarily equal to *S*). A layer is not constrained
to have the number of its inputs equal to the number of its neurons.

You can create a single (composite) layer of neurons having different transfer functions simply by putting two of the networks shown earlier in parallel. Both networks would have the same inputs, and each network would create some of the outputs.

The input vector elements enter the network through the weight
matrix **W**.

$$W=\left[\begin{array}{cccc}{w}_{1,1}& {w}_{1,2}& \dots & {w}_{1,R}\\ {w}_{2,1}& {w}_{2,2}& \dots & {w}_{2,R}\\ & & & \\ {w}_{S,1}& {w}_{S,2}& \dots & {w}_{S,R}\end{array}\right]$$

Note that the row indices on the elements of matrix **W** indicate the destination neuron of the weight,
and the column indices indicate which source is the input for that
weight. Thus, the indices in *w*_{1,2} say
that the strength of the signal *from* the second
input element *to* the first (and only) neuron
is *w*_{1,2}.

The *S* neuron *R*-input one-layer
network also can be drawn in abbreviated notation.

Here **p** is an *R*-length
input vector, **W** is an *S* × *R* matrix, **a** and **b** are *S*-length
vectors. As defined previously, the neuron layer includes the weight
matrix, the multiplication operations, the bias vector **b**, the summer, and the transfer function blocks.

To describe networks having multiple layers, the notation must be extended. Specifically, it needs to make a distinction between weight matrices that are connected to inputs and weight matrices that are connected between layers. It also needs to identify the source and destination for the weight matrices.

We will call weight matrices connected to inputs *input weights; *we will call weight matrices
connected to layer outputs *layer weights.* Further,
superscripts are used to identify the source (second index) and the
destination (first index) for the various weights and other elements
of the network. To illustrate, the one-layer multiple input network
shown earlier is redrawn in abbreviated form here.

As you can see, the weight matrix connected to the input vector **p** is labeled as an input weight matrix (**IW**^{1,1}) having a
source 1 (second index) and a destination 1 (first index). Elements
of layer 1, such as its bias, net input, and output have a superscript
1 to say that they are associated with the first layer.

Multiple Layers of Neurons uses layer weight (**LW**) matrices as well as input weight (**IW**) matrices.

A network can have several layers. Each layer has a weight matrix **W**, a bias vector **b**,
and an output vector **a**. To distinguish
between the weight matrices, output vectors, etc., for each of these
layers in the figures, the number of the layer is appended as a superscript
to the variable of interest. You can see the use of this layer notation
in the three-layer network shown next, and in the equations at the
bottom of the figure.

The network shown above has *R*^{1} inputs, *S*^{1} neurons
in the first layer, *S*^{2} neurons
in the second layer, etc. It is common for different layers to have
different numbers of neurons. A constant input 1 is fed to the bias
for each neuron.

Note that the outputs of each intermediate layer are the inputs
to the following layer. Thus layer 2 can be analyzed as a one-layer
network with *S*^{1} inputs, *S*^{2} neurons,
and an *S*^{2} × *S*^{1} weight
matrix **W**^{2}.
The input to layer 2 is **a**^{1};
the output is **a**^{2}.
Now that all the vectors and matrices of layer 2 have been identified,
it can be treated as a single-layer network on its own. This approach
can be taken with any layer of the network.

The layers of a multilayer network play different roles. A layer
that produces the network output is called an *output layer*. All other layers are called *hidden layers*. The three-layer network shown
earlier has one output layer (layer 3) and two hidden layers (layer
1 and layer 2). Some authors refer to the inputs as a fourth layer.
This toolbox does not use that designation.

The architecture of a multilayer network with a single input
vector can be specified with the notation *R* − *S*^{1} − *S*^{2} −...− *S ^{M}*,
where the number of elements of the input vector and the number of
neurons in each layer are specified.

The same three-layer network can also be drawn using abbreviated notation.

Multiple-layer networks are quite powerful. For instance, a network of two layers, where the first layer is sigmoid and the second layer is linear, can be trained to approximate any function (with a finite number of discontinuities) arbitrarily well. This kind of two-layer network is used extensively in Multilayer Shallow Neural Networks and Backpropagation Training.

Here it is assumed that the output of the third layer, **a**^{3}, is the network
output of interest, and this output is labeled as **y**.
This notation is used to specify the output of multilayer networks.

Network inputs might have associated processing functions. Processing functions transform user input data to a form that is easier or more efficient for a network.

For instance, `mapminmax`

transforms
input data so that all values fall into the interval [−1, 1].
This can speed up learning for many networks. `removeconstantrows`

removes
the rows of the input vector that correspond to input elements that
always have the same value, because these input elements are not providing
any useful information to the network. The third common processing
function is `fixunknowns`

, which
recodes unknown data (represented in the user's data with `NaN`

values)
into a numerical form for the network. `fixunknowns`

preserves
information about which values are known and which are unknown.

Similarly, network outputs can also have associated processing functions. Output processing functions are used to transform user-provided target vectors for network use. Then, network outputs are reverse-processed using the same functions to produce output data with the same characteristics as the original user-provided targets.

Both `mapminmax`

and `removeconstantrows`

are often associated
with network outputs. However, `fixunknowns`

is
not. Unknown values in targets (represented by `NaN`

values)
do not need to be altered for network use.

Processing functions are described in more detail in Choose Neural Network Input-Output Processing Functions.