I think that pulsars and variable stars (e.g., the Cepheids) are two different things. For one thing, the cycle is very different. For pulsars, it's anywhere from 1/1000 of a second, to several hours (I think). For the Cepheids, it's days to weaks. Also, the spectrum is very different. Pulsars emit a variety of wavelengths, but with some distinctive bands, and lots of UV radiation, whereas Cepheids emit blackbody radiation.

 

I believe that pulsars are "natural tokamaks", with magnetically confined plasma undergoing nuclear fusion, thus producing the UV radiation, which typically would have been blocked by the surrounding matter, but which isn't, because it's magnetic confinement, not gravitational pressure that's causing the fusion. Then, the rapid pulses are the result of sputtering in the fusion reactor, where fusion disperses the matter, but then it collapses again, and more fusion occurs.

 

Cepheids are a different breed. I believe that they are like the Sun, but the "solar cycle" is just a lot shorter. I believe that this is caused by s-waves below the surface, with crests that elevate matter into oppositely charged matter, triggering electrostatic discharges. The waves resonate, and are subjected to both constructive and destructive interference. The competition between those forces produces the oscillations.