# Model Unity and the Unity of Consciousness: Developments in Expected Float Entropy Minimisation

## Abstract

**:**

## 1. Introduction

#### 1.1. The Fundamental Postulate of EFE Minimisation

**Postulate**

**1**

#### 1.2. Expected Float Entropy Minimisation

#### 1.3. Model Unity

**Definition**

**1**

**Lemma**

**1**

**Proof.**

## 2. Materials and Methods

## 3. Results

**Example**

**1.**

**Suggested scenario:**For a suitable system of nodes of Bob’s visual cortex, Example 1 may suggest that the definitions of this article would yield a result of model unity for such a system. Note that the nodes involved need not be individual neurons and can be tuples of neurons or larger structures. Accordingly, the system as a whole would also yield an associated primary relational model that would provide a minimum expected entropy interpretation of system states.

**Example**

**2.**

**Suggested scenario:**For a system with two similarly distributed and yet independent subsystems, where one subsystem is a suitable system of nodes of Alice’s visual cortex and the other subsystem is a suitable system of nodes of Bob’s visual cortex, Example 2 suggest that the definitions of this article would yield a result of model disunity at the level of primary models, that is the system as a whole would not have model unity. Note that the nodes involved need not be individual neurons and can be tuples of neurons or larger structures. Examples 1 and 2 also suggest that the two independent subsystems would each individually as separate systems have model unity. Accordingly, each of the individual subsystems would yield its own associated primary relational model that would provide a minimum expected entropy interpretation of subsystem states for that subsystem. The pair of primary models together would provide a minimum expected entropy interpretation of system states for the system when taken as a whole. Therefore, the two subsystems would have separate individual interpretations according to Postulate 1.

**Example**

**3.**

**Suggested scenario:**For a suitable system of nodes of Bob’s brain with two similarly distributed subsystems where the activity in one subsystem lacks regularity with the activity in the other subsystem but the subsystems need not be independent, Example 3 suggests that the definitions of this article would yield a result of model disunity at the level of primary models, that is the system as a whole would not have model unity at the primary model level. Note that the nodes involved need not be individual neurons and can be tuples of neurons or larger structures. Examples 1 and 3 also suggest that the two subsystems would each individually as separate subsystems have model unity. Accordingly, each of the individual subsystems would yield its own associated primary relational model that would provide a minimum expected entropy interpretation of subsystem states for that subsystem. Secondary models and higher may involve relationships between features that are present when the system states are interpreted in the context of the primary relational models and therefore there may be model unity at the level of higher models. Alternatively, if the mode of the system changed (note, it may be that the distribution P changes with mode) such that the activity in one subsystem becomes more regular with the activity in the other subsystem, then the system as a whole may gain model unity at the primary model level. This could have relevance to phenomena such as attention.

**Example**

**4.**

**Suggested scenario:**For a system with two subsystems, where one subsystem is a suitable system of nodes of Bob’s visual cortex and the other subsystem is a suitable system of nodes of Bob’s auditory cortex, Example 4 suggest that the definitions of this article would yield a result of model disunity at the level of primary models, that is the system as a whole would not have model unity at the primary model level. Note that the nodes involved need not be individual neurons and can be tuples of neurons or larger structures. Examples 1 and 4 also suggest that the two subsystems would each individually have model unity. Accordingly, each of the individual subsystems would yield its own associated primary relational model that would provide a minimum expected entropy interpretation of subsystem states for that subsystem. Secondary models and higher would involve relationships between features that are present when the system states are interpreted in the context of the primary relational models and therefore there may be model unity at the level of higher models. Because the primary model for Bob’s visual system is different to the primary model for Bob’s auditory system, the states of the two systems have different interpretations and so visual experience is different to auditory experience.

## 4. Discussion

#### 4.1. Model Unity in the Context of Other Theories

#### 4.2. Maximal Subsystems with Model Unity

**Lemma**

**2**

**Proof.**

**Corollary**

**1.**

**Proof.**

#### 4.3. Node Base at the Level of Primary Models

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EFE | Expected Float Entropy |

GNW | Global Neuronal Workspace |

IIT | Integrated Information Theory |

MCS | Mathematical Consciousness Science |

QII | Quantum Integrated Information |

## References

- Bayne, T.; Chalmers, D.J. What is the unity of consciousness? In The Unity of Consciousness: Binding, Integration, and Dissociation; Cleeremans, A., Frith, C., Eds.; Oxford University Press: Oxford, UK, 2003; pp. 23–58. [Google Scholar]
- Oizumi, M.; Albantakis, L.; Tononi, G. From the Phenomenology to the Mechanisms of Consciousness: Integrated Information Theory 3.0. PLoS Comput. Biol.
**2014**, 10, e1003588. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wiese, W. How to solve the problem of phenomenal unity: Finding alternatives to the single state conception. Phenom. Cogn. Sci.
**2016**, 16, 811–836. [Google Scholar] [CrossRef][Green Version] - Boccella, K. 20 Years after Surgery, a Full Life with Half a Brain; The Philadelphia Inquirer: Philadelphia, PA, USA, 2016. [Google Scholar]
- Mason, J.W.D. Quasi-conscious multivariate systems. Complexity
**2016**, 21, 125–147. [Google Scholar] [CrossRef][Green Version] - Mason, J.W.D. From Learning to Consciousness: An Example Using Expected Float Entropy Minimisation. Entropy
**2019**, 21, 60. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mason, J.W.D. Consciousness and the structuring property of typical data. Complexity
**2013**, 18, 28–37. [Google Scholar] [CrossRef][Green Version] - Kleiner, J. Mathematical Models of Consciousness. Entropy
**2020**, 22, 609. [Google Scholar] [CrossRef] [PubMed] - Nelder, J.A.; Mead, R. A Simplex Method for Function Minimization. Comput. J.
**1965**, 7, 308–313. [Google Scholar] [CrossRef] - Dehaene, S.; Kerszberg, M.; Changeux, J.P. A neuronal model of a global workspace in effortful cognitive tasks. Proc. Natl. Acad. Sci. USA
**1998**, 95, 14529–14534. [Google Scholar] [CrossRef] [PubMed][Green Version] - Grindrod, P.; Lester, C. Cortex-Like Complex Systems: What Occurs Within? Front. Appl. Math. Stat.
**2021**, 7, 51. [Google Scholar] [CrossRef] - Friston, K. A Free Energy Principle for Biological Systems. Entropy
**2012**, 14, 2100–2121. [Google Scholar] [CrossRef] - Kleiner, J.; Kremnizer, K. Collapse and the Closure of the Physical. Forthcoming.
- Kremnizer, K.; Ranchin, A. Integrated Information-Induced Quantum Collapse. Found. Phys.
**2015**, 45, 889–899. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**An illustration of the fundamental postulate of EFE minimisation. Three system states are shown, ${S}_{1},{S}_{2},{S}_{3}$ and five examples of interpretations ${I}_{1}$ to ${I}_{5}$. ${I}_{1}$ has unnecessary abrupt discontinuities due to its geometric properties. ${I}_{2}$ has unnecessary abrupt discontinuities due to the choice of relationships between the colors. ${I}_{3}$ combines the abrupt discontinuities of both ${I}_{1}$ and ${I}_{2}$. ${I}_{4}$ is nice and continuous but has lost all the details. In this case all of the colors are equally related and there is no relational distinction between them. ${I}_{5}$ looks to be a good candidate for a minimal expected entropy interpretation of system states.

**Figure 2.**The sampling of a digital photograph using six nodes, $\left|S\right|=6$, and a four shade gray scale, $\left|V\right|=4$. This gives $|{\mathsf{\Omega}}_{S,V}|={4}^{6}=4096$. Top-left to bottom-right: the first image is the original, the image is then desaturated and then a contrast enhancement is applied (the contrast enhancement is not needed but may allow the use of a smaller number of samples which is computationally desirable), next the image is posterised (the number of shades is reduced to four giving a four-state node repertoire) and finally the sample is taken. The same sampling method is used for hundreds of images giving hundreds of system states in our sample.

**Figure 3.**A plot of all $\mu (\mathcal{X},T)$ values for Example 1 with the elements of $\widehat{S}$ distributed along the x-axis and ordered by increasing value of $\mu $.

**Figure 4.**A graph depiction of the primary model in Table 1 showing strongest relationships (solid lines) and intermediate relationships (dash lines).

**Figure 5.**A plot of all $\mu (\mathcal{X},T)$ values for Example 2 with the elements of $\widehat{S}$ distributed along the x-axis and ordered by increasing value of $\mu $.

**Figure 6.**Left: A plot of all $\mu (\mathcal{X},{T}_{{X}_{\mathrm{Alice}}})$ values over the elements of ${\widehat{X}}_{\mathrm{Alice}}$. Right: A plot of all $\mu (\mathcal{X},{T}_{{X}_{\mathrm{Bob}}})$ values over the elements of ${\widehat{X}}_{\mathrm{Bob}}$.

$\mathfrak{U}$ | 0 | 147.224 | 294.449 | 441.673 | ||

0 | 1 | 0.41970 | 0.09353 | 0.00003 | ||

147.224 | 0.41970 | 1 | 0.48171 | 0.24282 | ||

294.449 | 0.09353 | 0.48171 | 1 | 0.46463 | ||

441.673 | 0.00003 | 0.24282 | 0.46463 | 1 | ||

$\mathfrak{R}$ | node 1 | node 2 | node 3 | node 4 | node 5 | node 6 |

node 1 | 1 | 0.98568 | 0.72952 | 0.98239 | 0.74209 | 0.68606 |

node 2 | 0.98568 | 1 | 0.87753 | 0.75186 | 0.93862 | 0.78366 |

node 3 | 0.72952 | 0.87753 | 1 | 0.73635 | 0.84329 | 0.99606 |

node 4 | 0.98239 | 0.75186 | 0.73635 | 1 | 0.99130 | 0.83291 |

node 5 | 0.74209 | 0.93862 | 0.84329 | 0.99130 | 1 | 0.98446 |

node 6 | 0.68606 | 0.78366 | 0.99606 | 0.83291 | 0.98446 | 1 |

${\mathfrak{U}}_{{\mathit{X}}_{\mathbf{Alice}}}$ | 0 | 147.224 | 294.449 | 441.673 |

0 | 1 | 0.23437 | 0.04687 | 0.01562 |

147.224 | 0.23437 | 1 | 0.39062 | 0.07812 |

294.449 | 0.04687 | 0.39062 | 1 | 0.45312 |

441.673 | 0.01562 | 0.07812 | 0.45312 | 1 |

${\mathfrak{R}}_{{\mathit{X}}_{\mathrm{Alice}}}$ | node 1 | node 2 | node 3 | |

node 1 | 1 | 0.98437 | 0.73437 | |

node 2 | 0.98437 | 1 | 0.85937 | |

node 3 | 0.73437 | 0.85937 | 1 |

${\mathfrak{U}}_{{\mathit{X}}_{\mathbf{Bob}}}$ | 0 | 147.224 | 294.449 | 441.673 |

0 | 1 | 0.21875 | 0.03125 | 0.03125 |

147.224 | 0.21875 | 1 | 0.40625 | 0.09375 |

294.449 | 0.03125 | 0.40625 | 1 | 0.34375 |

441.673 | 0.03125 | 0.09375 | 0.34375 | 1 |

${\mathfrak{R}}_{{\mathit{X}}_{\mathrm{Bob}}}$ | node 1 | node 2 | node 3 | |

node 1 | 1 | 0.84375 | 0.71875 | |

node 2 | 0.84375 | 1 | 0.96875 | |

node 3 | 0.71875 | 0.96875 | 1 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mason, J.W.D.
Model Unity and the Unity of Consciousness: Developments in Expected Float Entropy Minimisation. *Entropy* **2021**, *23*, 1444.
https://doi.org/10.3390/e23111444

**AMA Style**

Mason JWD.
Model Unity and the Unity of Consciousness: Developments in Expected Float Entropy Minimisation. *Entropy*. 2021; 23(11):1444.
https://doi.org/10.3390/e23111444

**Chicago/Turabian Style**

Mason, Jonathan W. D.
2021. "Model Unity and the Unity of Consciousness: Developments in Expected Float Entropy Minimisation" *Entropy* 23, no. 11: 1444.
https://doi.org/10.3390/e23111444