Why feature maps are indexed by two indices? I'm reading about convolutional neural networks. As I understood a feature map is a set of neurons (i.e like a single hidden layer in traditional ANN). So why feature maps are indexed by (i,j)? Shouldn't a feature map be indexed only by one index because it's just a set of neurons?
In the following picture layer C1 has 6 feature maps. The picture says that a feature map is 28 by 28. But why?! Why 2D?!

 A: There is actually no reason for that layer to be 1-dimensional. Applying a convolution on 2D input would typically result in a 2D result. For intuition, observe this diagram. Furthermore since LeNet-5 was (at least initially) meant for image recognition, this can make intuitive sense from yet another angle - the role of the convolution in this case, is intuitively to "smooth out" the image in a sense. Clearly "smoothing out" should result in 2D; resulting in a 1D representation would be too much of a smooth...... 
You can get good intuition into that last part, on this LeNet demo page - refering to the animated diagram there, you can see the resulting convolutions (on the left) per input to the right. Addressing intuition yet again, the convolution somehow reduces the reliance of the learning process upon the exact minute details of the input image, something that needs to happen in many ML algorithms.
Now why isn't the result the same amount of cells, in this case, but 4 less? as far as I recall there's this thing about how to treat or relate to edge cells of the output - as the convolution uses, for each input pixel, surrounding cells from all sides, edge pixels can be seen as somewhat of a special case. 
I hope at least part of this answer can be validated by experts on the related topics, but I think this should already help.
