First off, I'm assuming you are using pre-2.1 StreamInsight based on your use of the term "output adapter".
From what you've posted, I would strongly recommend that your adapters do either input or output, but not both. This cuts down on the complexity, makes the implementation easier, and depending on how you wrote the adapter, you now have a reusable piece of code in your solution.
If you are wanting to send data from StreamInsight to your neural network prediction engine, you will need to write an output adapter to do that. Then I would create an input adapter that will get the results from the neural network prediction engine and enqueue the data into StreamInsight. After creating your stream from the neural network prediction engine input adapter, you can use dynamic query composition to share the stream to a Windows Azure storage output adapter and your next query.
If your neural network prediction engine can "push" data to your input adapter, that would be the way to do. If not, you'll have to poll for results.
There is a lot more to this, but it's difficult to drill in to more specifics without more details.
Hope this helps.