I should clarify a couple things since your question was directed to the "inverse problem," which, as I said, is a descriptive term, but it has to do with difficulty interpreting where in the brain the signal is coming from, due to the multitude of brain currents picked up by the EEG. But my first post responded to what I thought was a question about any inverse relationship between an EKG and EEG that differed in some manner. Also, a heart-type signal CAN come thru the EEG, again this is an interpretation situation, which a monitor for the heart will help distinguish that signal origin from a brain wave. Used to be, all docs had was monitors. But since we now have CAT scans and MRIs, we can SEE internal structures in the brain and where, for example, an abnormal signal from an EEG is being picked up, like an abnormal growth might cause an electrical discharge, but not know where it comes from necessarily. Also, when an EEG is done, as long as a monitor is also on the heart, any signals coming from that organ can be compared to the EEG tape and be interpreted as a non-brain signal. I'm just telling you this in case you were talking about interpreting "problematic" EEG signals, as to their origin, rather than an inverse relationship between the results of an EEG and EKG, which is what I thought you were asking about originally, as tho you had a good result with one and a bad result with the other.
EEGs normally filter out ECG signals. Both EEG and ECG are based on measuring voltage, not current. They also measure the voltage using a "differential amplifier." Unlike the voltage in a US home with is 110 v RMS (root mean square) with respect to ground, human bodies have no ground, the the baseline "floats" all over the place.
The heart signal is 150 milli volts. The monitoring frequency response of 0.5-40 Hz. The diagnostic frequency response range of 0.05-100 Hz.
EEG voltages at the skins surface are typically 10-100 micro volts (more than and the frequencies are listed below:
EEG in to the following frequency bands:
Delta 0.5-4Hz
Theta 4-8Hz
Alpha 8-13Hz
Beta 13-30Hz
Gamma 36-44Hz
Both systems are designed to reject the other signal by measuring electrode-to-electrode and using high pass and low pass filters to reject unwanted signals. You will notice that the EEG signals are all below the 50-60 Hz poweline frequencies and are easy to "clean up" with a low pass filter. ECG is a bit harder since the 50-60 Hz frequency range falls right in the middle of the diagnostic frequency response. In this case, notch filters are used to attenuate the 48-62 Hz frequency range.
Hope that helps,
Bob