Much of the above describes how the professor can participate in discussions herself. If you've taught in classrooms, this makes perfect sense: "When you teach in person," Flower Darby points out, "you wouldn't launch a class discussion and then walk out of the room, leaving the students to conduct their own discussion."
Other benefits aside, this creates a powerful teaching presence in a course where asynchronous discussions is a major site for social presence. If students read your input together with their fellow classmates, they get a sense that you are in the course as a participant, committed to their learning, instead of just monitoring it or occasionally visiting it.
As indicated above this requires careful balance so that a professor avoids over-burdening herself. On the other hand, instructor participation in discussions can be so powerful that it's worth factoring it into your syllabus, perhaps at the expense of other things you think you "need" to assign or assess.
You needn't simply let discussions fade out in your class. Here's some ideas to get further value from previous discussions:
If you craft weekly messages that introduce new concepts, where possible make reference to points raised by specific students in the discussion. Those students feel like they made a special contribution, and everyone gets the sense that discussions matter in the larger scheme of the course.
Maybe write, or assign students to write, a short summary that captures the main points or considerations raised in your asynchronous discussions. This can be just a few sentences long. You can assign one or more students to do this each week. This can bring a sense of closure and enduring value to discussions.