Now lets see the case of two-dimensional neuron. In the geometrical interpretation all input vectors are in the OXY plane and the neuron output is the third dimension. So the activation function is a surface in the 3 dimensional space, the example of the sigmoidal function is shown in the figure below.

fig 3. Activate function for dual input neuron.

Normalization of the input vectors causes that all of them are moved to the edge of a unitary radius circle, with one exception of point (0,0), which stay on its place. Now we might think about how the bias works in the dual input neuron. At first lets take a look to the activation function only. As we know from the previous chapter, bias input is responsible for moving of activation function toward the straight line. In the two dimensional case bias moves the activate function towards the direction that is perpendicular to the line given by the equation:

Examples of activation function for neuron with and without a bias are shown in the figure below.

fig 4. Biased and non biased activate function.

Taking into account neuron with bias addition of an extra weight cause move of input vectors from the two to the three dimensional space. All of points are situated on the sphere; however for the positive bias on top half of sphere, and for negative bias on a bottom sphere part. This is result of the input vectors normalization, namely the third coordinate is constant and that causes points separation for negative and positive bias. The (0,0) point changes to (0,0,1) “the highest” point of sphere or to (0,0,-1), “the lowest” point of sphere.
           Use of bias is necessary to reach any result in some cases. Example of the solution of the one problem using biased and non-biased neuron is shown below.

fig 5a. Solution for non-biased neuron is impossible.

fig 5b. Solution for biased neuron exists.

In these figures we can see that the points are chosen in such a way that for non biased neuron we cannot drive straight line through the center of coordinate system that separates different values of neuron response. Neuron responses for each point was marked by circles colored dependently on the neuron response value. Conclusion is that the non-biased neuron cannot make correct classification of the selected points, so the neuron cannot be leart it. However for the biased neuron the points are moved to the edge of a sphere and because of that they might be separated by plane that passes through the center of coordinates system. Conclusion is that the biased neuron can separate these points.