A quantitative study has been made of the conditions needed to produce paroxysmal or epileptiform afterdischarge in cat''s neurologic ally isolated cerebral cortex; the relationships between various parameters of stimulation and the production of afterdischarges have been determined. The establishment in the cortex of a focus for the paroxysmal afterdischarge is dependent in an all-or-nothing manner upon the excitation of a critical minimum number of neurones. This critical number of neurones will act as a focus for afterdischarge only if a minimum density of neurones per unit area of cortex is excited by the stimulus. The neurones at the focus must be excited to give a critical minimum number of all-or-nothing driven responses before the focus is capable of initiating the afterdischarge. Experiments done with varying patterns of stimulation have shown that afterdischarge results from the recovery of neurones from an exhausted state; it breaks out only during recovery and not during a period of driven activity. The conditioning stimulation must drive the neurones to or below a critical state of exhaustion before an afterdischarge can occur. A number of simple assumptions have been used to develop a theoretical formula providing satisfactory prediction of the duration of afterdischarge following a known number and frequency of conditioning stimuli. This formula has also enabled the estimation of certain properties of the neurones which constitute the focus for afterdischarge. The average refractory period of these neurones at the stimulus strengths used is fifteen milliseconds, and they recover exponentially from their state of exhaustion with a time-constant of 5.6 seconds. The theoretical minimum number of driven discharges required to produce an afterdischarge is predicted by the formula to be twenty-one, a value that agrees well with experimental results. Our results support the hypothesis that the paroxysmal afterdischarge is caused by the differential repolarization of neurones at the excited focus.