Complex Processes from Dynamical Architectures with Time-Scale Hierarchy

Abstract

The idea that complex motor, perceptual, and cognitive behaviors are composed of smaller units, which are somehow brought into a meaningful relation, permeates the biological and life sciences. However, no principled framework defining the constituent elementary processes has been developed to this date. Consequently, functional configurations (or architectures) relating elementary processes and external influences are mostly piecemeal formulations suitable to particular instances only. Here, we develop a general dynamical framework for distinct functional architectures characterized by the time-scale separation of their constituents and evaluate their efficiency. Thereto, we build on the (phase) flow of a system, which prescribes the temporal evolution of its state variables. The phase flow topology allows for the unambiguous classification of qualitatively distinct processes, which we consider to represent the functional units or modes within the dynamical architecture. Using the example of a composite movement we illustrate how different architectures can be characterized by their degree of time scale separation between the internal elements of the architecture (i.e. the functional modes) and external interventions. We reveal a tradeoff of the interactions between internal and external influences, which offers a theoretical justification for the efficient composition of complex processes out of non-trivial elementary processes or functional modes.

Does changing Fitts’ index of difficulty evoke transitions in movement dynamics?

“…For instance, Schöner [25] has proposed that a fixed point (at location A) vanishes so as to temporally give way to a limit cycle-causing the movement, after which the limit cycle vanishes and the fixed point (at location B) re-occurs. Alternatively, a fixed point may be driven through phase space, more or less continuously changing the phase flow so that the system is 'dragged behind it" [29]. This scenario constitutes an interpretation of equilibrium point models [30] in the framework of dynamical systems [31].…”

“…In this case, the trajectories in the phase space can be expected to be 'wiggly' and reveal little local convergence (i.e., overlaid trajectories can be expected to have a similar thickness throughout). Yet another possible realization involves a competition in which an active fixed point at location A vanishes while simultaneously another at location B comes into existence [29]. Indeed, while the discrete Fitts' task must involve fixed points, what remains unknown is: i) whether they are similar under D and W induced ID scaling, ii) if ID changes evoke a transition between mechanisms, and iii) if the fixed points assumed in the discrete task are the same as those found for the (W induced ID scaling) cyclic task.…”

Functional architectures and structured flows on manifolds: A dynamical framework for motor behavior.

“…The resulting dynamics is determined by the SFM emerging from a competition among the SFM in the dynamic repertoire. Perdikis et al (2011aPerdikis et al ( , 2011b, building on documented examples of distinct movement classes (see above), investigated whether systematic transitions from one elementary process to another can explain the emergence of non-stationary multitime scale behaviors. The functional architectures they proposed comprise functional modes and additional dynamics operating on distinct time scales (referred to as operational signals).…”

“…in phase space potentially subjected to additional operations (see Perdikis et al 2011a, for details):…”

“…This continuous process is illustrated in the video in the supplemental materials. There, it can be seen that soon 8 Under the assumption that the control demands associated with using functional modes, once established in one's dynamical repertoire, are negligible relative to the task-varying utilization of operational signals, it can be shown that different degrees of control are required for distinct functional architectures (Perdikis et al, 2011a). This document is copyrighted by the American Psychological Association or one of its allied publishers.…”