Your asp.net application is running with the credentials being used by the application pool that owns the process. If the application pool is running using the "Built-In Service Account", then the user will be local to that computer (when you add the account, select Location, select your computer, then enter IIS APPPOOL\<Application pool name>
(where "" is the name of your application pool).
If you open Sql Server Management Studio [Express] you can find your database, drill into "Security", right click on "Users", select a User type of "Windows User" then select the user indicated above.
Note: The Network Service
account exists in Windows 7 and 8. It is a local machine account.