I'm no expert in Illustrator, but in my initial self-teaching with it, I was doing this exact project, but with sports logos (to create that deep neon sign effect).
I was able to easily do this with Gaussian Blur and Outer Glow effect in Illustrator CC 2015 by performing an image trace>expand, applying a clear fill and thin stroke to all remaining traced lines, then saving the image as a vector.
Once done, I created a new document and added a rectangle, swapping default stroke and fill colors so it would be black. I added the svg and selected each path separately, making the fill clear and the stroke about 10 pt. Changed the color to something lighter than black, then duplicated the stroke about 8 times. From the bottom up, I reduced each stroke by one and in the color panel, brought up the HSB options...reducing the saturation by 10% each going up the list of strokes.
Eventually, I had that almost white middle and bright colored exterior. I added the outer glow effect at 80% opacity and 3-5 pt (remember, the stroke here will flair out that many pts in both directions of the glow). Once done, I saved that as a graphic style.
After that, I applied that style to every other path in the logo...changing only the colors of the strokes and the glow effect along the way.
Play around with the glow opacity and stroke saturation to get the effect you want. One thing you might also try is to brighten up the inner strokes to achieve the brightness level you want. I played around with that after reading this and I could almost hang my neon logo in Casters office and it would blend in perfectly!