Here is My Question:
Can a persistent cough, especially at night, be a symptom of MS, and if so is there anything that can be done to treat it?
Lesions in/near the area postrema (the area where the brainstem and spinal cord meet) can cause nausea, hiccups, and cough. Such a symptom is unusual in MS and more common in rarer neurologic disease that can mimick MS. When this part of the nervous system is affected, it can cause intractable symptoms throughout the day as well.
More common explanations for persistent coughing include certain medications (called ACE inhibitors), seasonal allergies, post-nasal drip, acid reflux disease, asthma, etc. The cause of your symptom can't be diagnosed over a blog (requires a clinic evaluation and examination). I would suggest speaking with your PCP first. Treatment varies depending on the identified cause of your cough.
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