Non linear neural networks?
Sigmoid does add nonlinearity -- and the nonlinearity "builds" as you add more layers.  Neural networks can recognize arbitrary decision boundaries (given enough neurons and enough layers).

I think your visualization of what the sigmoid activation does is a bit off.  Take a look at http://playground.tensorflow.org/ and play with some of the tuning knobs to see the kinds of decision boundaries that can be achieved -- that site is a helpful way to get some intuition.

So, the answer to your question is: we don't need a separate version of neural networks.  Standard neural networks already can support arbitrary decision boundary shapes.



[ About the kernel trick: The kernel trick can be used here, but is possibly a bit more expensive than in the case of SVM's; for SVM's, there's a particular mathematical trick that avoids the need to generate all of the "extra features", but that trick is specific to SVM's.  You can still construct derived features and add them to the feature vector before applying a neural network, though, if you want -- nothing prevents you from doing it.  But the kernel trick isn't needed, because neural networks can learn non-linear functions.  In contrast, without the kernel trick, SVMs can only learn linear decision boundaries, so it's needed more there. ]