# Morphological Computation: Synergy of Body and Brain

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## Abstract

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## 1. Introduction

## 2. A Synergistic Perspective on Morphological Computation

#### 2.1. Causal Model of the Sensorimotor Loop

#### 2.2. Quantifying Morphological Computation as Synergistic Information

#### 2.3. Synergistic Information Based on the Decomposition of the Multivariate Mutual Information

#### 2.4. Synergistic Information as the Difference between the Whole and the Sum of Its Parts

#### 2.5. Maximum Entropy Estimation with the Iterative Scaling Algorithm

## 3. Parametrised Model of the Sensorimotor Loop

#### 3.1. Binary Model of the Sensorimotor Loop

#### 3.1.1. Non-Binary Model of the Sensorimotor Loop

## 4. Numerical Simulations

#### 4.1. Results for the Binary Sensorimotor Loop

#### 4.2. New Measure for Unique Information

## 5. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Sensorimotor Loop. Left-hand side: schematics of the sensorimotor loop (redrawn from [22]), Right-hand side: causal diagram of a reactive system.

**Figure 2.**Visualisation of the two concepts ${\mathrm{MC}}_{\mathrm{A}}$ and ${\mathrm{MC}}_{\mathrm{W}}$. Left-hand side: causal diagram for a reactive system. Centre: causal diagram assuming no effect of the action A on the next world state ${W}^{\prime}$. Right-hand side: causal diagram assuming no effect of the previous world state W on the next world state ${W}^{\prime}$.

**Figure 3.**Quantifying complexity. Left-hand side: full model of two input and two output variables. Right-hand side: split model, as proposed by [35].

**Figure 4.**Quantifying synergy. Left-hand side: full model of two input and two output variables. Right-hand side: split model, as proposed by [36].

**Figure 5.**$CI({W}^{\prime}:W;A)$ (left-hand side) and ${\mathrm{MC}}_{\mathrm{SY}}$ (right-hand side) without synergistic information present in the model, i.e., $\chi =0$. The comparison of the plots reveal that $CI({W}^{\prime}:W;A)$ has regions with non-zero values, although no synergistic information is present. By that we mean that the higher order interaction term $\chi {w}^{\prime}wa$ is set to zero (see Equation (21)). No false positives are found for ${\mathrm{MC}}_{\mathrm{SY}}$ in this case.

**Figure 6.**$CI({W}^{\prime}:W;A)$ (

**top**) and ${\mathrm{MC}}_{\mathrm{SY}}$ (

**bottom**) for with ${w}^{\prime},w,s,a\in \mathsf{\Omega}=\{-1,1\}$.

**Figure 7.**${\mathrm{MC}}_{\mathrm{SY}}$ with ${w}^{\prime},w,s,a\in \mathsf{\Omega}=\{1,2,\dots ,4\}$ (

**top**) and ${w}^{\prime},w,s,a\in \mathsf{\Omega}=\{1,2,\dots ,8\}$ (

**bottom**).

**Figure 8.**From left to right $\chi \in \{0,1.25,2.5,3.75,5.0\}$. From top to bottom: ${\mathrm{MC}}_{\mathrm{W}}$, ${\mathrm{MC}}_{\mathrm{SY}}$, ${\mathrm{MC}}_{\mathrm{P}}$, for binary system, i.e., ${w}^{\prime},w,s,a\in \mathsf{\Omega}=\{-1,1\}$. Note that each plot in the upper row (${\mathrm{MC}}_{\mathrm{W}}$) is the sum the corresponding plot in the second row (${\mathrm{MC}}_{\mathrm{SY}}$) and the final row (${\mathrm{MC}}_{\mathrm{p}}$).

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**MDPI and ACS Style**

Ghazi-Zahedi, K.; Langer, C.; Ay, N.
Morphological Computation: Synergy of Body and Brain. *Entropy* **2017**, *19*, 456.
https://doi.org/10.3390/e19090456

**AMA Style**

Ghazi-Zahedi K, Langer C, Ay N.
Morphological Computation: Synergy of Body and Brain. *Entropy*. 2017; 19(9):456.
https://doi.org/10.3390/e19090456

**Chicago/Turabian Style**

Ghazi-Zahedi, Keyan, Carlotta Langer, and Nihat Ay.
2017. "Morphological Computation: Synergy of Body and Brain" *Entropy* 19, no. 9: 456.
https://doi.org/10.3390/e19090456