| Biological systems possess the ability to adapt quickly and adequately to both environmental and internal changes by emergence and disappearance of attractor states. This vital ability cannot be explained in terms of conventional stochastic processes because such processes are characterized by a trade-off between flexibility and accuracy, that is, they either show short transition times (large Kramers escape rates) to broad steady-state distributions or long transition times to sharply peaked distributions. To develop a stochastic theory for systems exhibiting both flexibility and accuracy we discuss effects of noise multiplied with accordant statistical measures or even the entire probability density. The coupling between the probability density of an ensemble and every individual realization may be seen as statistical feedback that can be assessed using generalized entropies and nonlinear Fokker-Planck equations. For ergodic systems a feedback via the probability density might be estimated using a finite-size buffer memorizing a realization's (and its neighbours) short-term history. |
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