A statistical theory of an excitable membrane is proposed under the assumptions that the membrane is composed of an ensemble of a number of active patches, and that the flip-flop transition of an active patch is governed by the dissipative interaction caused by an electric eddy current. By the method of expanding the master equation in the system size together with the Gaussian approximation, the time course of an excitation and the steady N-shaped relation between current and voltage are derived from the ensemble theory of open systems. The fluctuation of electric current under a fixed voltage becomes anomalously large in the marginal state at the threshold potential. The fluctuation associated with an action potential increases sharply at the jump-up transition, but it is not so large at the flip-back catastrophe. The fluctuation-dissipation relation at the steady state of the membrane system is discussed on the basis of a variational principle.