Measure currents (~50 mA) at every 10 microseconds interval and store the data
Maybe consider a cheap-end PicoScope like the 2204A-D2: -



It costs £79 in Farnell but it might be a tad cheaper elsewhere. It comes with drivers that are described as: -


  The drivers support data streaming, a mode that captures gap-free
  continuous data over the USB port directly to the PC’s RAM or hard
  disk at a rate of 1 to 9.6 MS/s, so you are not limited by the size of
  the scope’s buffer memory. Sampling rates in streaming mode are
  subject to PC specifications and application loading.


Sounds like it would do the job. If you want cheaper maybe go to this page. It looks like the cheapest one only costs $26: -



But I can't vouch for its drivers providing streaming functionality.