I'll preface this by saying I haven't been to London and I'm not very familiar with Brick Lane, but after looking at some info. online, here goes:
From what I could find, it seems Brick Lane is neighborhood with a vibrant ethnic aspect to it, especially Bangladeshi. There are a couple of areas in Paris that are infused with different cultures, that you might say are equivalent. If, though, you're looking for bohemian setting, you'll want to go to Montmartre, which I'll get into in a second. So, as far as foreign cultures in Paris I'd say there's a section in the northeast (19th and 20th arrondissements) called Belleville (metro Belleville). Now, as for things to do there I know of mainly the food markets they have there (possibly flea markets too) with specialty in African and Asian food as those are the main ethnicities of the neighborhood. Otherwise, there's the Marais, which was traditionally a neighborhood with a lot of Jewish culture, as well as being a generally hip, cool, and historic area to walk around in and check out a variety of small shops. This area is in the east-central region of Paris.
There's also rue Oberkampf in the 11th, just northeast of the Marais, with a good number of bars and small live music clubs.
The traditional grandaddy, though, in terms of bohemian atmosphere is Montmartre, in the north of Paris-- the 18th arrond., that is located on a hill that overlooks Paris. You can get off at Metro Anvers and walk to the funicular that takes you up the hill and to the famous Sacre Coeur basilica. It is a bit touristy around this part, but worth checking out, especially if you walk around the streets surrounding the Place du Tertre (again a touristy spot, but cool to check out anyway). If you've seen the movie Moulin Rouge, the bohemian folk hang out in Montmartre, and just next door in the seedier area of Pigalle, where the Moulin Rouge actually is located (rue Pigalle).