Literally, an epicenter is the point of origin of an earthquake, the part that has the strongest vibrations. Figuratively, especially when applied to culture, an "epicenter" is the center, the focal point, the place with the richest, highest concentration of something, such as culture.
Example: In a recent review of Sapporo University for potential exchange students to Japan, I wrote, "Of course, being removed from the cultural epicenters of Japan means making some compromises. Certain things will probably be more expensive in Hokkaido than they would be in Honshu. Hotels don’t seem to have free Internet, for instance."
To conclude the article, I wrote, "It’s an interesting mixture of nature and people, occupying a different place on the slider than options in Tokyo, Osaka, and Kyoto." Tokyo, Osaka and Kyoto are the cultural epicenters of Japan.