Will not having a publication affect my PhD chances even though I have research experience?
This is a difficult question to answer and I know little about the intersection of CS and neuroscience. I do know a little bit about the intersection of CS and social sciences (i.e. HCI/information sciences) so I will write about that. In addition, I am the student representative on the admissions committee of our department so I have some limited experience with that process as well.

You do not need a publication to apply for PhD in the US. For other countries, it might be different and I do not have that information. Potentially, you will be more competitive if you have a publication. But remember, that all publications are not created equal. If you have a first author paper (or otherwise depending upon the publication standards in your field) in a top ranked journal/conference in your field, then you are competitive (assuming that other parts of your application pass muster).

If that publication is in a lesser known journal/conference, you still have displayed some knowledge about how to conduct and publish research - which in the end is better than nothing.

However, in my experience, I have seen that detailed letters of recommendation from authors known to members of the admissions committee are usually given way more weight than any published research.

Also, take my words with a pinch of salt. Everything varies from field to field and from year to year.