Here is an example of this kind of research:

Int J Yoga. 2008 Jul-Dec; 1(2): 66–71.

doi: 10.4103/0973-6131.43543PMCID: PMC3144613

Investigating paranormal phenomena: Functional brain imaging of telepathy

Ganesan Venkatasubramanian, Peruvumba N Jayakumar, Hongasandra R Nagendra,Dindagur Nagaraja, R Deeptha, and Bangalore N Gangadhar

The aim of this study was to elucidate the neural basis of telepathy by examining an individual with this special ability.

Materials and Methods:

Using functional MRI, we examined a famous “mentalist” while he was performing a telepathic task in a 1.5 T scanner. A matched control subject without this special ability was also examined under similar conditions.

Results:

The mentalist demonstrated significant activation of the right parahippocampal gyrus after successful performance of a telepathic task. The comparison subject, who did not show any telepathic ability, demonstrated significant activation of the left inferior frontal gyrus.

Conclusions:

The findings of this study are suggestive of a limbic basis for telepathy and warrant further systematic research.

"Can fMRI be used to demonstrate telepathy?" - The answer is no! Even if we assume participants have telepathic powers, the answer is still no.

The short answer for this is, that fMRI does not "demonstrate" cognitive, psychological or supernatural phenomena. Typically a fMRI experiment consists of an active task and control task, either within or between participants, performed while participants are laying in the scanner. What is being investigated is the difference in neuronal activity between the two conditions, all other factors being kept constant. What can be inferred from the (ideal) experiment, is the difference in neural activity associated or caused by the difference between the two tasks. Say, investigating the contrast between two different visual search tasks in fMRI, would yield the difference in neural correlates between the two tasks - leading to the conclusion, that the two tasks have a different neural realisation. We cannot conclude that the neural activity demonstrate two different tasks. This is reverse inference. This is similar to saying that data cause the experimental paradigm.

Let's say we do a fMRI experiment with telepathic participants doing a 'telepathy task' and a 'non-telepathy task', and find a difference in activity in left inferior frontal gyrus. This does not demonstrate telepathy - only that there is a difference in activity in this particular area between the two tasks. What the interpretation of such result would be I do not know.

When making inference in cognitive neuroscience it is dependent on the assumptions in the experimental paradigm, such as the task being consistent with the cognitive ability that is investigated and not measuring anything else. E.g. if I have two different visual search tasks as described above, the assumption (or hypothesis) I am testing is that they are different, and require different neural activation. Furthermore, I am assuming that my participants are engaging in the task, as instructed, and any measured difference between the two tasks are not caused by any other confounding factor, such as eye blinks and movements, heart beats, muscular movements etc.

The same apply to the example with telepathic participants. If we find a difference in fMRI data between the 'telepathy task' and a 'non-telepathy task' and conclude that this is indeed the neural correlate of telepathy, it rests upon the assumption that 1) the difference is caused by using their telepathic powers, 2) no other factors can explain this difference. Given these assumptions and the data the conclusion is consistent. If we however were to conclude that we have proven telepathy to exist, we would simply have concluded, what we already assumed to be true. This is begging the question.

Dr. Jan Dalkvist at Stockholm University, Biological psychology have studied telepathy for many many years.

Dalkvist, J., & Westerlund, J. (2008). Performance in group telepathy experiments as related to target picture characteristics. www.psychology.su.se/staff/jd/ [PDF]

Other research:

Achterberg J, Cooke K, Richards T, Standish LJ, Kozak L, Lake J. Evidence for correlations between distant intentionality and brain function in recipients: A functional magnetic resonance imaging analysis. J Altern Complement Med. 2005;11:965–71.

Roll WG, Persinger MA, Webster DL, Tiller SG, Cook CM. Neurobehavioral and neurometabolic (SPECT) correlates of paranormal information: Involvement of the right hemisphere and its sensitivity to weak complex magnetic fields. Int J Neurosci. 2002;112:197–224

Richards TL, Kozak L, Johnson LC, Standish LJ. Replicable functional magnetic resonance imaging evidence of correlated brain signals between physically and sensory isolated subjects. J Altern Complement Med. 2005;11:955–63

Thanks for these helpful responses. I am working my way through the references and will respond more fully soon.

I would like to suggest an experimental design, based on the following assumptions. I assume that telepathic communication, if it exists, probably evolved prior to language and served two important survival functions: one as part of an alarm system, and the other to facilitate prosocial behavior, in other words, to transmit information about meaningful and emotionally potent stimuli. Experiments which involve attempts to transmit meaningless or random information seem unlikely to activate such a communication modality.

A second assumption is that there would need to be an organ of perception for this information, and a medium of transmission. If telepathic communication were to be demonstrated, these would be especially thrilling areas of research.

A third assumption is that information coming into this organ of perception would most likely be subject to the same filtering mechanism that applies to other perceptual modalities. Just as we are not consciously aware of everything that strikes our retina or tympanic membrane or skin, so we might be receiving non-auditory, non-visual information from a distance, and be unaware of it. For this reason, experiments which require the receiver to reproduce the information transmitted (which, of course, was the only way to know transmission had occurred prior to brain imaging technologies) may be imposing an insurmountable barrier.

With these assumptions in mind, I propose the following experiment as proof of concept. The receiver is in the fMRI scanner (or other scanner of comparable or greater sensitivity), visually and auditorially isolated from the sender. I don’t know if distance is a critical factor, but for the initial experiment, assuming that our capacities may have dulled over the last ten thousand years, I would prefer the sender and receiver to be fairly close.

After the required preliminaries and baselines, the sender’s arm would be plunged into a bucket of ice water. This is a standard, non-harmful stimulus commonly used in pain physiology experiments, which will set off a strong somatosensory and limbic response in the sender. Any change in the receiver’s scan would be considered a positive, whether or not the receiver had any conscious awareness of it.

You could use EMOTIV EPOG EEG headset.

I have one.

You can buy or borrow one easily.

I have also application for recording emotions.