Hierarchical Timescales and Nested Process Composition

Hierarchical structures and processes are central to many theories of cognition, and hierarchical temporal structure is widespread in time series of neural and behavioral activity. We present a mathematical model that shows how hierarchical processes are bound to yield hierarchical temporal structure in the form of 1/f scaling if processes are cyclic and their timescales are drawn from the reciprocal distribution—that is, the probability of a given process is in direct proportion to its cycle frequency. The model also shows how commonly observed deviations from 1/f scaling can come from different kinds of perturbation to processes and their measurements. We derive a mechanism of nested process composition from the reciprocal distribution, whereby cycles are copied and divided or merged recursively to create a hierarchy of processes across a range of timescales. Nested process composition accounts for 1/f scaling as well as neuronal avalanches and Lévy flights in terms of the formation of hierarchical processes. We discuss how self-organized criticality may support nested process composition, and how the mechanism relates to language and cognition.