Feature extraction from a scatter plot
As @David Masip mentioned, Principal Component Analysis would be a good method to use here. Essentially PCA is a method by which a mapping is found between a high dimensional space to a lower dimenional space while keeping as much variation in the data as possible - perfect for dimensionality reduction of high dimensional data.

However, you mention that you want to use this reduced data to train a neural network model. It may be best to first train the neural net model and see how well it performs, as neural nets are usually very good at identifying interactions between features as well as other hidden structures in the data. If it doesn't perform well, then one approach to improve performance may be to use PCA - although this is highly dependent on your use case, content / type / amount of data, neural network architecture etc.

p.s PCA is also good to visualize high dimensional data (reduce the dimensionality to 2 or 3 dimensions, then plot it. This is better than plotting only 2 features at a time as you have done above).