"How can the brain think that the brain itself hurts (e.g., during a headache)? Are headaches and migraines (excluding secondary headaches) essentially psychosomatic in nature?"

Before I answer your question, I need to say one important thing: ouch!

That’s right – as I sat down at my keyboard to answer your question, I stubbed my toe quite painfully on my desk. Fortunately, this is a great time to explain how the brain processes pain, which will help answer your question.