Browser errors and system tray icons will be enough to indicate the status of the wireless network for the user.
The way you describe your application indicates that you are making some kind of continuous AJAX style request, which is often interrupted. Instead of concentrating on basic network connections, I recommend using a more business approach: if your application cannot support the connection (or does not receive a response within a certain period of time), simply inform the user that the connection cannot be established (to increase reliability keep trying again until it succeeds - see, for example, the GMail web interface). The user will have other diagnostic methods, why - your application should not and cannot know the reason for the failure, your application knows that it needs to connect to the server and cannot. (As a bonus, error handling in this way can give you the basis for processing other errors on the server side, for example, displaying a message if data services are not available for maintenance, etc.)
If you download all the data when the page loads, rather than requesting it through scripts after loading, then an error message is displayed in the user's browser indicating that the page cannot be loaded, and you do not need to provide it.
source share