Is it possible for tooth decay to cause neurological side effects?

Not likely . Not unless the tooth decay has reached the pulp or living section of the tooth. You would have what is better known as a tooth abscess at that point ...Depending whether it is acute or chronic may determine the level of pain which in turn could possibly lead to neurological side effects! would certainly expedite the time one would seek professional help ....
Yes, it's possible. In the early stages of tooth decay, it's unlikely to cause neurological side effects. But if tooth decay is neglected long enough, the tooth can abscess. A tooth abscess is a nasty infection that can lead to the blood stream and spread to all parts of the body including the nervous system and brain. So yes, it is possible.
Talk to an MD. I am not aware of any such connections. You should probably consult a neurologist.