A new physical and mathematical model for the epilepsies is proposed, based on the theory of bond percolation on finite lattices. Within this model, the onset of seizures in the brain is identified with the appearance of spanning clusters of neurons engaged in the spurious and uncontrollable electrical activity characteristic of seizures. It is proposed that the fraction of excitatory to inhibitory synapses can be identified with a bond probability, and that the bond probability is a randomly varying quantity displaying Gaussian statistics. The consequences of the proposed model to the treatment of the epilepsies is explored. The nature of the data on the epilepsies which can be acquired in a clinical setting is described. It is shown that such data can be analyzed to provide preliminary support for the bond percolation hypothesis, and to quantify the efficacy of anti-epileptic drugs in a treatment program. The results of a battery of statistical tests on seizure distributions are discussed. The physical theory of the electroencephalogram (EEG) is described, and extant models of the electrical activity measured by the EEG are discussed, with an emphasis on their physical behavior. A proposal is made to explain the difference between the power spectra of electrical activity measured with cranial probes and with the EEG. Statistical tests on the characteristic EEG manifestations of epileptic activity are conducted, and their results described. Computer simulations of a correlated bond percolating system are constructed. It is shown that the statistical properties of the results of such a simulation are strongly suggestive of the statistical properties of clinical data. The study finds no contradictions between the predictions of the bond percolation model and the observed properties of the available data. Suggestions are made for further research and for techniques based on the proposed model which may be used for tuning the effects of anti-epileptic drugs. / Ph. D.