There are already excellent techniques for imaging the anatomy of the human brain, and changes in its functional state over seconds. Furthermore, the function of the individual neuron has been studied at time resolutions of less than a millisecond. There is, at present, no technique which combines this fine time resolution with tomographic imaging, although this may be essential if we are ever to understand the processing of information by the brain. The work described in this thesis used mathematical models and in-vivo measurement to investigate whether applied potential tomography (APT), a recently-developed imaging technique, may be used for imaging neuronal depolarisation in the brain. Two factors suggest that APT may be suitable: first, it can acquire data sets at the required rate; second, there is known to be a decrease in the resistance of neuronal membranes, perhaps by up to a factor of 40, when they depolarise. The practicality of APT depends on whether there is corresponding measurable change in the macroscopic impedance of brain tissue. A mathematical model was used to estimate the magnitude and frequency-dependence of brain impedance changes in two tissues: crustacean peripheral nerve and mammalian cortex. The model predicted that at 30 kHz, a typical working frequency for APT, the resistivity change would be 600 times smaller than at DC, for which the predictions were 3.70/0 for nerve and 0.01% for cortex. Measurements of DC resistivity change during depolarisation of crab nerve (0.2% ~ 2.5%), and afferent stimulation of rabbit cortex (0.01%), were in good agreement with the predicted resistivity changes. Finite element modeling of the head suggested that, with a cosine excitation pattern, the scalp voltage changes would be about 0.001%. Suggestions are given for improving the data collection process to make measurements of these small DC voltage changes more tractable.