Life

As we have seen above, while the approach to the critical state can take a long time, once the system achieves crticality small perturbations can trigger avalanches of all sizes. That is, long periods of "quiet" can be interrupted by large events. Interestingly, prior to the idea of SOC, the paleontologists Gould and Eldredge had proposed the idea of "punctuated equilibrium" to describe how evolution seemed to occur: Long periods of stasis punctuated by extinction and evolution events of all sizes. Clearly the independently proposed idea of SOC is similar and supposedly of applicability to other complex systems. Different species compete for resources, forming a large and complex evolving system. Thus it seems possible that the SOC scenario is realized for evolution. Indeed the extinction rate for the last 500 million years, from data collected by Sepkoski, does seem to follow a power law. What is more, such power law behaviour is also seen in various computer simulations of evolution, such as Ray's "Tierra" model and the simple Bak-Sneppen model. The Bak-Sneppen model consists of a one dimensional cellular automaton with N species placed on a circle with nearest neighbour interactions. In this model, evolution is approximated to act at the level of a species rather than at the level of the individual; thus each species is represented by a single fitness level $0<f<1$. At each discrete time step, the species with the lowest fitness level is made extinct and is replaced by a new species with a randomly assigned fitness. In addition, the two neighbouring species are also replaced by new species with randomly assigned fitness values. The frequency power spectrum of the changes experienced by any given species in time is found to follow a power law. An interesting feature of the Bak-Sneppen model is evolution by the mechanism of "elimination of the least fit" rather than the popular folklore of "survival of the fittest". This extremal dynamics whereby the weakest link in a complex network is removed, or breaks down, is probably a realistic modelling of most natural phenomena. Indeed, Chialvo and Bak [3] have used extremal dynamics in their model of a brain that learns from mistakes. In their neural network model, whenever mistakes are made, all the synapses that contributed to that decision are punished. This is in contrast to usual Hebbian rules where neurons that perform well are strengthened. Chiavlo and Bak show that their model is able to achieve quick learing of new patterns because their network is closer to a "critical state" rather than the sub-critical (highly stable) states of tradiational models.