PSYCH 4320 / COGST 4310 / BioNB 4330

Consciousness and Free Will

Theme V

  Week 11: levels and boundary problems, II

Week 11: levels and boundary problems, II

multiscale homology (Fekete et al., 2016, suppl. fig.2)

(A) Increasing the maximal allowed point to point distance, the first edges are added. As the points are equally spaced on the smaller circles, this causes the emergence of eight open circles, which are associated with 8 1-D holes. At the same time the number of connected components decreases to 8.

(B) As the scale is increased, more edges add to the complex, and with them the corresponding surfaces (faces). The complex now still has 8 connected components, but no longer holes of higher dimensions.

(C) As the scale is increased further, the eight clusters connect. Now there is a single connected component in the complex, and a single 1-D hole resulting from the larger circle.

(D) Finally, the complex completes, the ring structure no longer exists, only a simply connected convex shape.

(E) 0-D holes (counting connected components) as a function of scale.

(F) 1-D holes as a function of scale. There are two non-zero scale intervals in which 2D structure manifests. Note that the larger scale feature persists for a longer interval. This is typical of multiscale structure, which at times can also exhibit scale-free behavior that manifests itself in a power law of persistence intervals.

the key definitions

A proper part of a system \(S\) under property \(\phi\) is \(s\subset S\) such that \(s +_{\phi} \bar{s} \neq_{\phi} S\).
In other words, \(s\) exhibits different dynamics when operating "on its own" as compared to operating within the putative system \(S\); and \(\phi\) not indifferent to the greater context of \(S\).

A normal cut of a system \(S\) is a bipartition into \(s\) and its complement \(\bar{s}\) such that \(s \cup \bar{s} = S\) and neither of them is empty.

A property \(\phi\) is conjoint if every part of \(S\) is a proper part.

Conjoint properties may be:

  1. subadditive if for every normal cut, \(s +_{\phi} \bar{s} >_{\phi} S\)
  2. superdditive if for every normal cut, \(s +_{\phi} \bar{s} <_{\phi} S\)
  3. alloadditive: a conjoint property that is neither subadditive nor superadditive
A conjoint property is systemic for \(S\) if \(S\) is not a proper part of some greater system under this property. With this definition in mind, we can say that the member of any subset of units that contribute to realizing a systemic property "cares" about each and every other unit or collection of units within the system.

a systemic, superadditive property (app.2, example 2)

(A) Consider the system 𝑋1𝑋2𝑋3𝑋4, arranged on a ring with a coupling parameter of 0.3.

(B) This gives rise to a circular trajectory space.

(C) A normal cut would result if some of the connections are set to 0. An example: 𝑋1𝑋2𝑋3|𝑋4. This compromises this ring connectivity pattern, resulting in an unstructured trajectory space

Any part of 𝑋1𝑋2𝑋3𝑋4 is a proper part. Thus for this architecture, the homology of the trajectory space is a super-additive conjoint property. As in this case it is the complete system, it is not a proper part, and hence systemic.