 This article will explain the actual concepts of Backward Propagation Neural Networks  in such a way that even a person with zero knowledge in neural networks can understand the required theory and concepts very easily. The related project demonstrates the designing and implementation of a fully working 'BackProp' Neural Network library, i.e, the Brain Net library as I call it. You can find the theory, illustration and concepts here  along with the explanation of the neural network library project  in this article. Also, find the full source code of the library and related demo projects (a simple pattern detector, a hand writing detection pad, an xml based neural network processing language etc) in the associated zip file. 
Contents
 Solution Architect: "Well, you learned something about neural networks?"
 (Dumb?) Developer: "No, I'm smart enough. I love using other's code."
 Solution Architect: "But, if you don't understand the concepts, how you can optimize and reuse other's code?"
 (Dumb?) Developer: "Err.. I feel that most others can code better than me, so why should I optimize?"
In my previous article, the focus was on what a neural network can do. In this article, we will see what a neural network is, and how to create one yourself. I will go a little deeper. After reading this article, you will be able to
 Understand the basic theory behind neural networks (backward propagation neural networks in particular)
 Understand how neural networks actually 'work'
 Understand in more detail, the design and source code of BrainNet library.
 Understand in more detail, how to use BrainNet Library in your projects.
 Think about new possibilities of neural network programming
 Put forward some concepts to optimize and generalize BrainNet library.
Now, let me answer some questions I got in past.
 Q) Why you selected an object oriented programming model for this Neural Network Library?
 Answer  The focus is on the understandability of basic concepts, not on performance.
 Q) Is this neural network library fully optimized?
 Answer  Not yet, we are still in the beta stage. The focus is on readability, so the code is flattened so that even a beginner can understand it. Suggestions and modifications are always welcome. Send your modifications, hacks and suggestions to amazedsaint@gmail.com
 Q) Whether this library can be used in projects?
 You can use it  as long as your usage confronts to the specifications in the associated license notice (see the source code). Anyway, I request you to send me a notification (and the modified code), if you hack it or use it in any of your projects.
This article is complete by itself. It explains what is a neural network, and how to create one your own. How ever, to get an idea regarding what a neural network can do, and to get a user level experience  please read the first part of this article.
The first article in this article series is titled "BrainNet Neural Network Library  Part I  Learn Neural Network Programming step by step And Develop a Simple Handwriting Detection System". You can read it and download the source code, [ Click Here ].
If you are really a beginner, it will help you a lot, and may provide you a step by step approach towards understanding neural networks.
This is my second article about Neural Networks in general and the BrainNet Neural Network Library in particular. This article explains Neural Networks and their working in more detail, and in a very simple way. Then I will explain the design concepts of BrainNet library.
 Tip  In this article, the theory about neural networks is explained in the most simplest (human readable) manner, so that even a person with zero background in neural networks can understand it. So, if you already know some theory about neural networks, you may consider skipping the theory part, and go ahead to the designing part which describes the design concepts of BrainNet library. 

One fascinating thing about artificial neural networks is that, they are mainly inspired by the human brain. This doesn't mean that Artificial Neural Networks are exact simulations of the biological neural networks inside our brain  because the actual working of human brain is still a mystery. The concept of artificial neural networks emerged in its present form our very limited understanding about our own brain ("I know that I know nothing").
Brain Net Neural Network library is designed and implemented using Object Oriented Concepts.
 Alert  You can visit the articles section of my website here  http://amazedsaint.blogspot.com/  for reading other articles in this series. You can also see there a full tutorial about OOP concepts in .NET and VB.NET programming. This may help you to understand this article better  especially if you don't have a good understanding regarding the Object Oriented Programming concepts. Also, read about design patterns there. 

Before understanding how neurons and neural networks actually work, let us revisit the structure of a neural network. As I mentioned earlier, a neural network consists of several layers, and each layer has a number of neurons in it. Neurons is one layer is connected to multiple or all neurons in the next layer. Input is fed to the neurons in input layer, and output is obtained from the neurons in the last layer.
Fig: A Fully Connected 442 neural network with 4 neurons in input layer, 4 neurons in hidden layer and 2 neurons in output layer.
An artificial neural network can learn from a set of samples.
For training a neural network, first you provide a set of inputs and outputs. For example, if you need a neural network to detect fractures from an XRay of a born, first you train the network with a number of samples. You provide an XRay, along with the information that whether that particular XRay has a fracture or not. After training the network a number of times with a number of samples like this (probably thousands of samples), it is assumed that the neural network can 'detect' whether a given XRay indicates a fracture in the born (This is just an example). The concept of training a network is detailed in my first article. Later, in this article, we will discuss the theory behind network learning.
As we already discussed, the basic component in a neural network is a neuron. First of all, let us have a very brief look towards biological neurons, and their corresponding artificial models.
First of all, let us have a look at a biological neuron. Frankly, I don't have much knowledge regarding the actual structure of a biological neuron  how ever, the following information is more than enough at this stage for us to get in to the groove. A biological neuron will look some what similar to this.
The four basic components of a biological neuron are

Dendrites  Dendrites are hair like extensions of a neuron, and each dendrite can bring some input to the neuron (from neurons in the previous layer). These inputs are given to the soma.

Soma  Soma is responsible for processing these inputs, and the output is provided to other neurons through the axon and synapses.

Axon  The axon is responsible for carrying the output of soma to other neurons, through the synapses

Synapses  Synapses of one neuron is connected to the dendrites of neurons in the next layer. The connections between neurons is possible because of synapses and dendrites.
A single neuron is connected to multiple neurons (mostly, all neurons) in the next layer. Also, a neuron in one layer can accept inputs from more than one neuron (mostly, all neurons) in the previous layer.
Now, let us have a look at the model of an artificial neuron.
An artificial neuron consists of various inputs, much like the biological neuron. Instead of Soma and Axon, we have a summation unit and a transfer function unit. The output of one neuron can be given as input to multiple neurons.
Please note that for an artificial neuron, we have a weight value associated with each input. Now, let us have a look at the working of a neuron.
Summation Unit

When inputs are fed to the neuron, the summation unit will initially find the netvalue. For finding the Net Value, the product of each input value and corresponding connection weight is calculated.

i.e, input value x(i) of each input to the neuron is multiplied with the associated connection weight w(i). In simplest case, these products are summed and fed to the transfer function. See the pseudo code below, it is simpler to understand.
Also, a neuron has a bias value, which affects the net value. A bias of a neuron is set to a random value, when the network is initialized. We will change the connection weights and bias of all neurons in the network (other than neurons in the input layer), during training phase.
I.e, if x is the input, and w is the associated weight, then pseudo code for net value calculation is as follows.
netValue=0
for i=0 to neuron.inputs.count1
netValue=netValue + x(i) * w(i)
next
netValue=netValue + Bias
Transfer Function
Transfer function is a simple function, that uses the net value to generate an output. This output is then propagated to the neurons in the next layer. We can use various types of transfer functions as shown below.
Hard Limit Transfer Function: For example, a simple hard limit function will output 1 if net value is greater than 0.5, and will output 0 if the net value is lesser than 0.5  as shown.
if (netValue<0.5)
output = 0
else
output = 1
Sigmoid Transfer Function: Another type of transfer function is a sigmoid transfer function. A sigmoid transfer function will take a net value as input and produce an output between 0 and 1 as shown.
output = 1 / (1 + Exp(netValue))
The implementation of summation unit and transfer function unit may vary in different networks.
This, a neural network is constructed from such basic models, called neurons, arranged together in layers, and connected to each other as explained earlier. Now let us see how all these neurons work together, inside a neural network.
Working with a neural network includes
 Training the network  by providing inputs and corresponding outputs.
 In this phase, we train a neural network with samples to perform a particular task.
 Running the network  by providing the input to obtain the output.
 In this phase, we will provide an input to the network, and obtain the output. The output may not be accurate always. Generally speaking, the accuracy of the output during running phase depends a lot on the samples we provided during the training phase, and the number of times we trained the network.
This section explains how the training takes place, in a back ward propagation neural network. In a backward propagation neural network, there are several layers, and each neuron in each layer is connected to all neurons in the next layer. For each connection, a random weight is assigned when the network is initialized. Also, a random bias value is assigned to each neuron during initialization.
Training is the process of adjusting the connection weights and bias of all neurons in the network (other than neurons in the input layer), to enable the network to produce expected output for all input sets.
Now, let us see how the training actually happens. Consider a small 221 network. Now, we are going to train this network with AND truth table. As you know, AND truth table is

AND TRUTH TABLE
 A  B  Output 
0
 0  0  0  1  0  1  0  0  1  1  1 

Fig: A 221 Neural Network and Truth Table Of AND
In the above network, N1 and N2 are neurons in input layer, N3 and N4 are neurons in hidden layer, and N5 is the neuron in output layer. The inputs are fed to N1 and N2. Each neuron in each layer is connected to all neurons in next layer. We call the above network a 221 network, based on the number of neurons in each layer.
 Tip  The concepts we are going to discuss here is largely biased towards a commonly used neural network model called Backward Propagation Neural Networks. How ever, you should understand that various other models also exist  like Counter Propagation Neural Networks, Kohanen's Self Organizing Maps etc. 

The above diagram will be used to illustrate the process of training.
First, let us see how we train our 221 network, the first condition in the truth table, i.e, when A=0, B=0 then output=0.
Step 1  Feeding The Inputs
Initially, we will feed the inputs to the neural network. This is done by simply setting the output of neurons in Layer 1, as the input values we need to feed. I.e, as per the above example, our inputs are 0,0 and output is 0. we will set the output of Neuron N1 as 0, and the output of N2 is set to 0.
Have a look at this pseudo code, and it will make things clear. Inputs is the input array. The number of elements in Input array should match the number of neurons in input layer.
i = 0
For Each neuron In InputLayer
someNeuron.OutputValue = Inputs(i)
i = i + 1
Next
Step 2  Finding the output of the network
We have already seen how we calculate the output of a single neuron. As per our above example, the output of neurons N1 and N2 will act as the inputs of N3 and N4.
Finding the output of neural network involves, calculating the outputs of all hidden layers and output layer. As we discussed earlier, a neural network can have a number of hidden layers.
For each layer in HiddenLayers
For Each neuron In layer.Neurons
neuron.UpdateOutput()
Next
Next
For Each neuron In OutputLayer.Neurons
neuron.UpdateOutput()
Next
UpdateOutput() function of a single neuron works exactly as we discussed earlier. First, net value is calculated by the summation unit, and then it is provided to a transfer function to obtain the output of the neuron. Pseudo code is again shown below.
Summation Unit works like this:
Dim netValue As Single = bias
For Each InputNeuron connected to ThisNeuron
netValue = netValue + (Weight Associated With InputNeuron * _
Output of InputNeuron)
Next
I.e, as per our above example, let us calculate the net value of neuron N3. We know that N1 and N2 are connected to N3.
 Net Value Of N3 = N3.Bias + (N1.Output * Weight Of Connection From N1 to N3) + (N2.Output * Weight Of Connection From N2 to N3)
Similarly, to calculate the net value of N4,
 Net Value Of N4 = N4.Bias + (N1.Output * Weight Of Connection From N1 to N4) + (N2.Output * Weight Of Connection From N2 to N4)
Activation Unit Or Transfer Unit:
Now, let us see how we are generating the output, using Transfer unit. Here, we are using the sigmoid transfer function. This is exactly as we discussed earlier.
Output of Neuron = 1 / (1 + Exp(  NetValue )
Now, the output of N3 and N4 will be passed to each neuron in the next layer as inputs. This process of propagating the output of one layer as the input to the next layer is called forward propagation part in the training phase.
Thus, after step 2, we just found the output of each neuron in each layer  starting from the first hidden layer to the output layer. The output of the network is simply the output of all neurons in the output layer.
Step 3  Calculating The Error or Delta
In this step, we will calculate the error of the network. Error or Delta can be stated as the difference between the expected output and the obtained output. For example, when we find the output value of the network for the first time, most probably the output will be wrong. We need to get 0 as the output for inputs A=0 and B=0. But the output may be, some other value like 0.55, based on the random values assigned to the bias and connection weights of each neuron.
Now let us see, how we can calculate the error. Let us see how to calculate the error or delta of each neuron in all the layers.
 First we will calculate the error or delta of each neuron in the output layer.
 The delta value thus calculated will be used to calculate the error or delta of neurons in the previous layer (i.e, the last hidden layer)
 The delta value of all neurons in the last hidden layer is used to calculate the error or delta of all neurons in the previous layer (i.e, second last hidden layer)
 This process is continued, till we reach the first hidden layer (delta of input layer is not calculated).
Please note one interesting point. In Step 2, we are propagating values forward  starting from the first hidden layer to the output layer, for finding the output. In Step 3, we are starting from the output layer, and propagating the error values backward  and hence, this neural network is called as a Backward Propagation neural network.
Time to see how things actually work. The general equation for finding the delta of a neuron is
Neuron.Delta = Neuron.Output * (1  Neuron.Output) * ErrorFactor
Now, let us see how the error factor is calculated for each neuron. The Error Factor of neurons in output layer can be calculated directly (since we know the expected output of each neuron in output layer).
For a neuron in output layer,
ErrorFactor Of An Output Layer Neuron = _
ExpectedOutput  Neuron
i.e, with respect to our above example, if the output of N5 is 0.5 and the expected output is 0, then error factor = 0  0.5 =  0.5
For a neuron in hidden layer, error factor calculation is some what different. To calculate the error factor of a neuron in hidden layer,
 First the delta of each neuron to which this neuron is connected is multiplied with the weight of this connection
 These products are summed up together to obtain the error factor of a hidden layer neuron
Simply speaking, a neuron in a hidden layer is using the delta of all connected neurons in next layer, along with the corresponding connection weights, to find the error factor. This is because, we don't have any direct parameters for calculating the error of neurons in the hidden layer (as we did in the output layer neurons).
 Remember  To calculate the output of a neuron, we used the outputs of connected neurons in previous layer, along with the corresponding connection weights. 

For Each Neuron N to which ThisNeuron Is Connected
errorFactor = errorFactor + (N.DeltaValue * _
Weight Of Connection From ThisNeuron To N)
Next
To illustrate this, consider a neuron x1 (ThisNeuron), which is a hidden layer neuron. X1 is connected to neurons y1, y2, y3 and y4  and these are neurons in next layer.
i.e, to make things simple,
 Error Factor of X1 = (Y1.Delta * Weight Of Connection From X1 To Y1) + (Y2.Delta * Weight Of Connection From X1 To Y2) + (Y3.Delta * Weight Of Connection From X1 To Y3) + (Y4.Delta * Weight Of Connection From X4 To Y4)
Now, as we discussed earlier, the Delta of a X1 can be calculated as,
 X1.Delta = X1.Output * (1  X1.Output) * ErrorFactor Of X1
Thus, after finishing step 3, we have the Delta of all neurons.
Step 4  Adjusting The Weights and Bias
After calculating the delta of all neurons in all layers, we should correct the weights and bias with respect to the error or delta, to produce a more accurate output next time. Connection Weights and Bias, together are called free parameters. Remember that a neuron should update more than one number of weights  because, as we already discussed, there is a weight associated with each connection to a neuron.
See the pseudo code for updating the free parameters of all neurons in all layers
For each layer in HiddenLayers
For Each neuron In layer.Neurons
neuron.UpdateFreeParams()
Next
Next
For Each neuron In OutputLayer.Neurons
neuron.UpdateFreeParams()
Next
UpdateFreeParams() function simply does two things.
 Find the new bias of a neuron, based on the delta we calculated above
 Update the connection weights based on the delta we calculated above
Finding the new bias value of a neuron is pretty simple. See the pseudo code. If Learning Rate is a constant (for e.g, Learning Rate=0.5)
New Bias Value = Old Bias Value + _
LEARNING_RATE * 1 * Delta
Now let us see how to update the connection weights. The new weight associated with an input neuron can be calculated as shown below.
New Weight = Old Weight + LEARNING_RATE * 1 * Output Of InputNeuron * Delta
As a neuron can have more than one input, the above step should be performed for all input neurons connected to this neuron.
I.e,
For Each InputNeuron N connected to ThisNeuron
New Weight of N = Old Weight of N + _
LEARNING_RATE * 1 * N.Output * ThisNeuron.Delta
Next
Now, after step 4, we have a better network. This process is repeated for all other entries in the AND truth table  for probably more than thousand number of times, to train the network 'well'.
Running the network involves,
 Providing the inputs to the network exactly as described earlier in Step 1 above
 Calculating the outputs as explained in Step 2 above
How ever, it is important to note that the network should be trained with sufficient samples (and sufficient number of times), to obtain desired results. Anyway, it is almost impossible to say that the output of a neural network will be 100% accurate for any input.
Now, let us see how these concepts are implemented in BrainNet Neural Network Library.
The fundamental challenge for any solution developer is to create, build or assemble a working program from his abstract concepts about a system. The quality of this transformation depends a lot on how well he understand the system. At this point, I would like to mention that Brain Net Library is actually not designed after a complete and thorough understanding of various existing neural network models and emerging possibilities in the area of neural networks. Hence, I suspect that the present design of this framework is mostly biased towards Backward Propagation systems I explained earlier  though it can be modified to create other neural network models also.
We are simply mapping the above concepts to the library. Hence, the following code and explanation is very easy to understand, if you read the above concepts regarding Neural Networks.
Now, let us have a look at some of the interfaces and classes in BrainNet library.
 Remember  If you need to brush up some Object Oriented Designing and UML concepts, have a look at my article regarding design patterns [ Click Here ]. 

Have a look at this model below. Please not that this model holds only the major interfaces and classes with in the model.
Fig: An Partial Model of BrainNet Framework
As we discussed earlier, a Neural Network consists of various Neuron Layers, and each Neuron Layer has various Neurons. A Neuron has a strategy  which decides how it should perform tasks like summation, activation, error calculation, bias adjustment, weight adjustment etc.
To brief the UML diagram above,
 INeuron, INeuronStrategy, INeuralNetwork and INetworkFactory are interfaces
 A Neuron should implement the INeuron interface
 A Neural Network should implement the INeuralNetwork interface
 A Neuron has a strategy, and a strategy should implement the INeuronStrategy interface. We have a concrete implementation of INeuronStrategy, called BackPropNeuronStrategy (for a backward propagation neural network).
 A Neural Network is initialized and connections betweens layers are made by a neural network factory. A Factory should implement the INetworkFactory interface. We have a concrete implementation of INetworkFactory, called BackPropNetworkFactory, for creating Backward Propagation neural networks.
The major interfaces in the model are briefed below.
INetworkFactory

An interface to define a neural network factory

INeuron

The interface for defining a neuron

INeuronStrategy

The interface for defining the strategy of the neuron

INeuralNetwork

The interface for defining a neural network

The major classes in the model are briefed below.
BackPropNeuronStrategy

A backward propagation neuron strategy. This is a concrete implementation of INeuronStrategy

NetworkHelper

The class is to help the user to initialize and train the network. It maintains a list of training data elements.

NeuralNetwork

A generic neural network. This is a concrete implementation of INeuralNetwork

NeuralNetworkCollection

A collection of neural networks

Neuron

A concrete implementation of INeuron

NeuronCollection

A collection of INeurons

NeuronConnections

This is a hash table to keep track of all neurons connected to/from a neuron, along with the related weights


The INeuron interface provides an abstract interface that should be implemented to create a concrete neuron. I request you to refresh the concepts of an artificial neuron we discussed earlier.
The elements in INeuron interface is detailed below.
Public Interface INeuron
Property BiasValue() As Single
Property OutputValue() As Single
Property DeltaValue() As Single
ReadOnly Property ForwardConnections() As NeuronCollection
ReadOnly Property Inputs() As NeuronConnections
Property Strategy() As INeuronStrategy
Sub UpdateOutput()
Sub UpdateDelta(ByVal errorFactor As Single)
Sub UpdateFreeParams()
End Interface
A concrete neuron will implement the INeuron interface. Neuron class is a concrete implementation of INeuron. The Strategy property of a Neuron holds its current strategy. Inputs property holds the references of Neurons (in previous layer) connected to this neuron. ForwardConnections holds references to the neurons (in next layer) to which this neuron is connected.
Now, have a look at the Neuron class by extracting the source code zip of BrainNet library. Let us inspect three major functions implemented in the Neuron class  UpdateOutput, UpdateDelta and UpdateFreeParams. These functions are called by the NeuralNetwork class, by training and running the network. We will see later how the functions in NeuralNetwork class call these functions.
These functions uses the current strategy object of the neuron to perform operations.
 UpdateDelta  Find the new delta of this neuron using the current strategy. Error factor (remember that this will vary based on the layer of a neuron) will be passed to the UpdateDelta function, from the functions in Neural Network class.
 UpdateOutput  Find the new output of the neuron, by finding the net value, and then by invoking the activation function  as defined in the current strategy.
 UpdateFreeParams  Updating free parameters includes calling the functions according to the current strategy of this neuron to find new bias and to update weights.
Public Sub UpdateDelta(ByVal errorFactor As Single) Implements _
NeuralFramework.INeuron.UpdateDelta
If _strategy Is Nothing Then
Throw New StrategyNotInitializedException("", Nothing)
DeltaValue = Strategy.FindDelta(OutputValue, errorFactor)
End Sub
Public Sub UpdateOutput() _
Implements NeuralFramework.INeuron.UpdateOutput
If _strategy Is Nothing Then
Throw New StrategyNotInitializedException("..", Nothing)
Dim netValue As Single = Strategy.FindNetValue(Inputs, BiasValue)
OutputValue = Strategy.Activation(netValue)
End Sub
Public Sub UpdateFreeParams() _
Implements NeuralFramework.INeuron.UpdateFreeParams
If _strategy Is Nothing Then
Throw New StrategyNotInitializedException("..", Nothing)
BiasValue = Strategy.FindNewBias(BiasValue, DeltaValue)
Strategy.UpdateWeights(Inputs, DeltaValue)
End Sub
How a Neuron actually functions is decided by the strategy of a neuron. A concrete strategy should implement the INeuronStrategy interface. This interface is shown below. BackPropNeuronStrategy is a concrete implementation of INeuronStrategy interface.
The elements in INeuronStrategy interface, along with description is given below.
Public Interface INeuronStrategy
Function FindDelta(ByVal output As Single, _
ByVal errorFactor As Single) As Single
Function Activation(ByVal value As Single) As Single
Function FindNetValue(ByVal inputs As NeuronConnections, _
ByVal bias As Single) As Single
Function FindNewBias(ByVal bias As Single, _
ByVal delta As Single) As Single
Sub UpdateWeights(ByRef connections As NeuronConnections, _
ByVal delta As Single)
End Interface
Have a look at the BackPropNeuronStrategy class, in the code, and see how these functions are implemented as we described earlier. It is pretty easy to understand.
Now, let us see how the Neural Network is implemented. Any concrete neural network should implement the INeuralNetwork interface. INeuralNetwork interface is shown below.
Public Interface INeuralNetwork
Sub TrainNetwork(ByVal t As TrainingData)
Sub ConnectNeurons(ByVal source As INeuron, _
ByVal destination As INeuron, ByVal weight As Single)
Sub ConnectNeurons(ByVal source As INeuron, _
ByVal destination As INeuron)
Sub ConnectLayers(ByVal layer1 As NeuronLayer, _
ByVal layer2 As NeuronLayer)
Sub ConnectLayers()
Function RunNetwork(ByVal inputs As ArrayList) As ArrayList
Function GetOutput() As ArrayList
ReadOnly Property Layers() As NeuronLayerCollection
ReadOnly Property InputLayer() As NeuronLayer
ReadOnly Property OutputLayer() As NeuronLayer
End Interface
There are two interesting functions, TrainNetwork and RunNetwork, for training and running the network. The input to the TrainNetwork function is an object of TrainingData class. The TrainingData class has two properties of type ArrayList  Inputs and Outputs. To train the network, we put the input values to the Inputs array list, and corresponding output values are filled to the Outputs array list.
First of all, feed the inputs to all the neurons in the input layer. Then, the algorithm is like
 Step1: Find the output of hidden layer neurons and output layer neurons
 Step2: Finding Delta
 2.1) find the delta (error rate) of output layer
 2.2) Calculate delta of all the hidden layers, backwards
 Step3: Update the free parameters of hidden and output layers
Have a look at how this goes, inside TrainNetwork function in the NeuralNetwork class, it is commented heavily. Some part of TrainNetwork function is shown below.
Dim i As Long
Dim someNeuron As INeuron
i = 0
For Each someNeuron In InputLayer
someNeuron.OutputValue = t.Inputs(i)
i = i + 1
Next
Dim nl As NeuronLayer
Dim count As Long = 1
For count = 1 To _layers.Count  1
nl = _layers(count)
For Each someNeuron In nl
someNeuron.UpdateOutput()
Next
Next
i = 0
For Each someNeuron In OutputLayer
someNeuron.UpdateDelta(t.Outputs(i)  _
someNeuron.OutputValue)
i = i + 1
Next
Dim layer As Long
Dim currentLayer As NeuronLayer
For i = _layers.Count  2 To 1 Step 1
currentLayer = _layers(i)
For Each someNeuron In currentLayer
Dim errorFactor As Single = 0
Dim connectedNeuron As INeuron
For Each connectedNeuron In _
someNeuron.ForwardConnections
errorFactor = _
errorFactor + (connectedNeuron.DeltaValue * _
connectedNeuron.Inputs.Weight(someNeuron))
Next
someNeuron.UpdateDelta(errorFactor)
Next
Next
For i = 1 To _layers.Count  1
For Each someNeuron In _layers(i)
someNeuron.UpdateFreeParams()
Next
Next
Running the network is pretty simple. For running the network, we just feed the inputs to the first layer, and calculate the outputs, just as explained earlier during the training phase. Here is some part of the RunNetwork function.
Dim someNeuron As INeuron
Dim i As Long = 0
For Each someNeuron In InputLayer
someNeuron.OutputValue = CType(inputs(i), System.Single)
i += 1
Next
Dim nl As NeuronLayer
For i = 1 To _layers.Count  1
nl = _layers(i)
For Each someNeuron In nl
someNeuron.UpdateOutput()
Next
Next
Now, let us see how you can create a network easily. Here is a simple code that shows how to create a network. Let us assume that the input to the method is an array list which holds a list of long values that represent the number of neurons in each layer.
Public Sub CreateNetwork(ByVal neuronsInLayers As ArrayList)
Dim bnn As New NeuralNetwork()
Dim neurons As Long
Dim strategy As New BackPropNeuronStrategy()
For Each neurons In neuronsInLayers
Dim layer As NeuronLayer
Dim i As Long
layer = New NeuronLayer()
For i = 0 To neurons  1
layer.Add(New Neuron(strategy))
Next
bnn.Layers.Add(layer)
Next
bnn.ConnectLayers()
End Function
Or better, you can use the BackPropNetworkFactory class to create a network easily. Have a look at the BackPropNetworkFactory class. It has two overloaded CreateNetwork functions, for creating a neural network.
Some notes.
 This article is much like a 'Developers Guide' of BrainNet neural network library.
 Have a look at my previous article if you haven't done that yet. It is more or less a 'user's guide' for this library  for more information regarding how to use this BrainNet Library in your own projects, and to see the demo projects in action.
Cheers!! Thus, we finished the second article about Neural Networks. Just turn back and make sure that you understood all the points clearly.
Experiment yourself with the library, and try to optimize it a little bit, or even better, create a neural network yourself using this as an example. In my next article,
 I will explain how to create an XML based language yourself, for creating, training and processing neural networks.
 Explain the concept of some classes in the framework that I haven't mentioned in this article (like NXML interpreter, NetworkSerializer etc).
There are some 'Easter Eggs' along with the BrainNet library source code, that I haven't mentioned right now. For example, If you are smart enough, start playing with the nxml tool, already included in the associated zip. The zip file holds the whole code. nxml is a command line tool which may help you to create, train and run a neural network using xml. I'll explain it in detail, in my next article. Anyway, after compiling the project, typing nxml in the command prompt will reveal its usage :)  just if you can't wait till my next article. Another demo project is a simple Handwriting detection pad, which is also available in the source code zip.
Also,
When you play with the library, if you come across any bugs, please report it.