Learning the activation function in a neural network?
Monotonicity is necessary! A neuron (perceptron) is either activated or not activated given an input.  If you're trying to learn "red" and red should activate the neuron; the more red it's the most likely it should be activated (This monotonic gradient is what this activation function represents).  If at certain redness it activates but at more redness it deactivates it makes no sense e.g. it can't learn red (The case for non-monotonic functions).

That they're differentiable functions makes it practical for the back propagation algorithm.

What they explain in your linked article is that they're trying to better approximate other functions e.g. sinusoidal.  They're not trying sinuosoidal as an activation function. 
They're combining several monotonic functions to create yet another monotonic function, that is also differentiable.