On Macrostates in Complex Multi-Scale Systems

Abstract

A characteristic feature of complex systems is their deep structure, meaning that the definition of their states and observables depends on the level, or the scale, at which the system is considered. This scale dependence is reflected in the distinction of micro-and macro-states, referring to lower and higher levels of description. There are several conceptual and formal frameworks to address the relation between them. Here, we focus on an approach in which macrostates are contextually emergent from (rather than fully reducible to) microstates and can be constructed by contextual partitions of the space of microstates. We discuss criteria for the stability of such partitions, in particular under the microstate dynamics, and outline some examples. Finally, we address the question of how macrostates arising from stable partitions can be identified as relevant or meaningful.

Time Series Complexities and Their Relationship to Forecasting Performance

“…Complexity science is a multidisciplinary field in charge of studying dynamical systems composed of several parts, whose behavior is nonlinear, and that cannot be studied neither by the laws of linear thermodynamics nor by modelling parts in isolation [ 29 , 30 ]. A key aspect of these systems is that individual parts’ interactions will heavily determine the future states of the overall system, and shall induce spatial, functional, or temporal structures all alone (i.e., self-organizing) [ 27 ].…”

Dynamic Computation in Visual Thalamocortical Networks

“…Common to most of these frameworks is their attempt to uncover symbolic structure in continuous, nonstationary dynamics by offering ways to identify separable activity patterns and ensembles. They also share an important underlying assumption: that neural computation is mediated by reproducible [ 16 ] spatiotemporal patterns that are both stable enough to be causally effective [ 76 ] and varied enough to support a rich behavioral and cognitive repertoire (i.e., they are complex [ 77 , 78 ]). The specifics, however, differ between accounts; depending on context, states have been defined, for instance, as fixed points [ 18 ], attractor ruins (defined and discussed in [ 12 , 17 , 52 , 71 ]), or saddle points in heteroclinic channels [ 13 ].…”

Correlations and How to Interpret Them

“…In this way, the relation between a thermal equilibrium state and the KMS state provides the sound foundation for the bridge law expressed above. (For a good overview and technical details see [42], Chapters 5-6, see also [13], Section 2.2, for an in-depth discussion accessible for the more general reader. )…”

“…All these quantitative measures are monotonic functions of randomness and are designed as context-free as possible, not to address questions of context or meaning (Shannon and Weaver [12]. Alternative measures that are convex functions of randomness highlight the fact that complex superpositions of regularity and randomness can be used to highlight the semantic (and pragmatic) content of meaningful behavior (for more details see [13]).…”