INTRODUCING A REAL-TIME SCALE INTO THE BAK-SNEPPEN MODEL

Recently, a simple model of evolution has been proposed by Bak and Sne ppen [Phys. Rev. Lett. 71, 4083 (1993)]. This model self-organizes int o a critical state for nearest- and random-neighbor interactions. The Bak-Sneppen (BS) model has no explicit time scale, because time steps are always identified with an evolutionary step. Therefore, we introdu ce at each time step a local stochastical update rule. Hence it is pos sible to observe time steps in which no species are removed from the s ystem. In the following, the durations of time steps in which no furth er evolution occurs are called interevent intervals. We study a random -neighbor version of the model and derive the steady-state distributio n of the fitnesses. The distributions are the same for synchronous and asynchronous updating rules and resemble the solutions obtained for t he mean field BS model. We give an interpretation of the modified BS m odel as a neural network with random connections. For a concrete choic e of the stochastical updating rule, we derive the distribution of the interevent or interspike intervals. It turns out that for parallel up dating we get a power law decay, whereas in the case of random sequent ial updating the distribution is simply an exponential in the limit N --> infinity. N is the system size. All analytical results are support ed by numerical simulations.