# On the Nature of Functional Differentiation: The Role of Self-Organization with Constraints

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## Abstract

**:**

## 1. Introduction

## 2. The Difference between Self-Organization and Self-Organization with Constraints

## 3. Dynamic Heterarchy

## 4. Heterarchy in the Model of Functional Differentiation

#### 4.1. Heterarchy in EDS

**Hypothesis**

**1.**

#### 4.2. Heterarchy in ERC

**Hypothesis**

**2.**

**Hypothesis**

**3.**

**Hypothesis**

**4.**

## 5. Superposition Theorem, Epigenetic Landscape, and Functional Differentiation

#### 5.1. The Kolmogorov–Arnold–Sprecher Superposition Theorem

**Theorem**

**1.**

#### 5.2. Epigenetic Landscape with Indices of Phenotype Yielding Functional Differentiation

**Working**

**hypothesis:**

**Figure 5.**Schematic drawing of the mechanism of functional differentiation: three-dimensional landscape representation (see also [47]). The three colored balls represent three types of transcription factors that may behave in, for example, oscillatory states of concentration. The change in the landscape toward the left-lower part of the figure represents the differentiation of three types of neuronal cells that are realized via the mutual inhibition of transcription factors. Differentiation may be triggered by the inhibition triggered by one of the other factors. Here, neither the reprogramming (i.e., rejuvenation) nor the progenitor cell states shown in Waddington’s epigenetic landscape [48] are drawn.

**Hypothesis**

**5.**

## 6. Summary and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Schematic drawing of three typical differentiated states: (

**a**) a passive state expressing a glial passive susceptible state, (

**b**) an excitable state expressing a spiking neuron, and (

**c**) an oscillatory state expressing an oscillatory neuron or glial oscillation (see [3] for numerical results of activity).

**Figure 3.**Numerical evidence of the emergence of a heterarchical structure in the evolved network of dynamical systems. (

**a**) Randomly chosen network topology. Layers are defined by the closeness of units to the receiver unit, 0. In (

**b**–

**d**), the red (blue) directional lines indicate excitatory (inhibitory) connections, with the shades of color indicating the connection strength: the scale shown at a lower place of each figure. (

**b**) Evolved network. Evolution was applied to the connection strength, preserving network topology. (

**c**) Drastic reduction of information quantity in unit 3 after erasing all feedback connections (green curve in the panel on the right: the abscissa denotes time and the ordinate denotes mutual information between unit 3 and the input), in relation to the time change of information quantity in unit 3 of the original evolved network (a purple curve). (

**d**) Change of information quantity in unit 3 after erasing only the feedback connection from unit 7 to unit 3 (green curve in the panel on the right). Information quantity was dropped, but qualitative behaviors did not change, which implies the preservation of quality (values) of information processing. Here, mutual information was calculated as time-dependent mutual information between two arbitrary units, which measures the dynamic change of shared information (see [24] for a detailed technique).

**Figure 4.**Numerical evidence of the heterarchical structure of an ERC. (

**a**) Change in the internal network structure, consisting of an input and output layer from an initial random network to an evolved network. The network change proceeded with the change of wiring topology and connection weights, according to the optimization algorithm, such as the minimization of errors (present case, see [2] for ERC and [34,35] for predictive coding formulations), minimization of energy cost, or maximization of information (see, e.g., [3,18,24,25,36,37]). The colors of the nodes indicate the degree of information quantity shared with the spatial or temporal output neurons: the higher the shared information with spatial (temporal) output neurons, the deeper the reddish (bluish) color of the node. The colors of the edges are as follows: red for feedforward connections; blue for feedback connections; green for connections within the input layer; and purple for connections within the output layer. The thickness of the lines indicates the magnitude of the connection weights. (

**b**) Change in the accuracy of the realization of functional differentiation with the change in the scale factor of feedback connections. The red (blue) curve denotes the accuracy of the spatial (temporal) neuron.

**Figure 6.**Numerical construction of the inner functions of Equation (4), ψ(c

_{i}), (i = 1,2,⋯,n). k = 3. An approximation of a single-variable function with a finite precision is shown, which can be an elementary function constituting a given n-variable function. In the present theory, this type of function is viewed as the states of transcription factors of stem cells. (

**a**) n = 2. Blue, orange, and green indicate γ = 6, γ = 10, and γ = 20, respectively; (

**b**) n = 2, γ = 10; (

**c**) n = 3, γ = 10.

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Tsuda, I.; Watanabe, H.; Tsukada, H.; Yamaguti, Y.
On the Nature of Functional Differentiation: The Role of Self-Organization with Constraints. *Entropy* **2022**, *24*, 240.
https://doi.org/10.3390/e24020240

**AMA Style**

Tsuda I, Watanabe H, Tsukada H, Yamaguti Y.
On the Nature of Functional Differentiation: The Role of Self-Organization with Constraints. *Entropy*. 2022; 24(2):240.
https://doi.org/10.3390/e24020240

**Chicago/Turabian Style**

Tsuda, Ichiro, Hiroshi Watanabe, Hiromichi Tsukada, and Yutaka Yamaguti.
2022. "On the Nature of Functional Differentiation: The Role of Self-Organization with Constraints" *Entropy* 24, no. 2: 240.
https://doi.org/10.3390/e24020240