Which sign or symptom is most likely caused by a problem with the autonomic component of the nervous system?

The slowing of the heart rate is indicative of an issue with the autonomic component of the nervous system, specifically the parasympathetic branch. The autonomic nervous system regulates involuntary bodily functions, including heart rate, blood pressure, and digestion. When the parasympathetic system is activated, it releases neurotransmitters such as acetylcholine that can slow down the heart rate.

This is in contrast to the other options. The inability to swallow suggests a problem with the motor function of the cranial nerves or central nervous system, not primarily the autonomic system. The inability to move facial muscles also points to issues with the motor system or potential damage to relevant cranial nerves rather than the autonomic nervous system. Lastly, trouble remembering his name is indicative of cognitive or neurological issues, which also do not primarily involve the autonomic nervous system. Therefore, the correct answer reflects the specific role the autonomic system plays in regulating heart function.