# Quantifying Morphological Computation

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## Abstract

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## 1. Introduction

The last sentence of this quote nicely summaries that the function of the wing is determined by the interaction of the environment with the physical properties of the wing. It is the quantification of this sort of contribution of the morphology to the behaviour of a system that is in the focus of this work. The difference to previous literature by Paul [8] and Lundh [9], who investigated morphological computational with respect to either the actuation (Paul) or the sensors (Lundh) only, is that we will measure morphological computation of embodied agents acting in the sensori-motor loop, including both sensors and actuators.However, active muscular forces cannot entirely control the wing shape in flight. They can only interact dynamically with the aerodynamic and inertial forces that the wings experience and with the wing’s own elasticity; the instantaneous results of these interactions are essentially determined by the architecture of the wing itself: its planform and relief, the distribution and local mechanical properties of the veins, the local thickness and properties of the membrane, the position and form of lines of flexion. The interpretation of these characters is the core of functional wing morphology.([7] [see p. 188])

## 2. Sensori-Motor Loop

**Figure 1.**Schematics and causal graph of the sensori-motor loop. (

**a**) shows the conceptual understanding of the sensori-motor loop. A cognitive system consists of a controller, a sensor and actuator system, and a body that is situated in an environment. The basic understanding is that the controller sends signals to the actuators that affect the environment. Information about the environment and also about internal states is sensed by the sensors, and the loop is closed when this information is passed to the controller. (

**b**) shows the representation of the sensori-motor loop as a causal graph. Here ${w}_{t}$ represents the world state at time t. The world is everything that is physical, i.e., the environment and the morphology. The variables ${s}_{t}$ and ${a}_{t}$ are the signals provided by the sensors or passed to the actuators, respectively. They are not to be mistaken with the sensors and actuators, which are part of the morphology, and hence, part of the world. (

**a**) Schematics of the sensori-motor loop; (

**b**) Representation of the sensori-motor loop as a causal graph.

## 3. Measuring Morphological Computation

- Descriptive definition of morphological computation.
- Framing of two concepts that follow from step 1.
- Formal definition of the two concepts given in step 2.
- Intrinsic adaptations of the definitions given in step 3.

**Figure 2.**Outline of the third section of this work, in which the measures for morphological computation are derived. The first step is to provide a description of morphological computation. From this description two concepts are presented in the second step, which are then formalised in the third step. The last step adapts the formalisations to measures that operate on intrinsically available information only.

#### 3.1. Morphological Computation

#### 3.2. Concepts of Measuring Morphological Computation

**Concept 1 (Negative effect of the action)**

**Concept 2 (Positive effect of the world)**

#### 3.3. Formalising the Concepts

**Definition 1 (Morphological Computation as negative effect of the action)**

**Definition 2 (Morphological Computation as positive effect of the world)**

#### 3.3.1. Concept 1, Associative Measure

**Definition 3 (Associative measure of the negative effect of the action)**

#### 3.3.2. Concept 1, Causal Measure

**Figure 3.**Reactive system. In the context of this work, a reactive system is defined by a direct coupling of the sensors and actuators. There is no form of memory present in the system. (

**a**) Schematics of the sensori-motor loop for a reactive system; (

**b**) Representation of the sensori-motor loop for a reactive system as a causal graph.

**Figure 4.**Visualisation of the causal measure ${\mathrm{C}}_{\mathrm{A}}$. The causal graph used in (

**a**), (

**b**) and (

**c**) is the reduction of sensori-motor loop shown in Figure 3(b) to two consecutive time steps. (

**a**) shows that the causal information flow $CIF(S\to {S}^{\prime})$ measures all causal information from S to ${S}^{\prime}$, including the information that flows over A. (

**b**) shows that the causal information flow $CIF(A\to {S}^{\prime})$ only captures the information flowing from A to ${S}^{\prime}$. Both can be used to approximate the causal information from S to ${S}^{\prime}$ that does not pass through A, denoted by $CIF(S\to {S}^{\prime}\backslash A)$, as shown (

**c**). (

**a**) $CIF(S\to {S}^{\prime})$; (

**b**) $CIF(A\to {S}^{\prime})$; (

**c**) $CIF(S\to {S}^{\prime}\backslash A)$.

**Definition 4 (Causal measure of the negative effect of the action for a reactive system)**

**Definition 5 (Causal measure of the negative effect of the action for a non-reactive system)**

#### 3.3.3. Concept 2, Associative Measure

**Definition 6 (Associative measure of the positive effect of the world)**

#### 3.3.4. Concept 2, Conditional Independence

**Definition 7 (Conditional dependence of the world on itself)**

**Figure 5.**Visualisation of the conditional independence measure ${\mathrm{C}}_{\mathrm{W}}$. The left-hand side shows how the conditional probability distributions $p\left({s}^{\prime}\right|s)$ can be calculated from the world model $p\left({s}^{\prime}\right|s,a)$ and the policy $p\left(a\right|s)$. The right-hand side shows how $p\left({s}^{\prime}\right|s)$ changes, if one assumes that the world does not influence itself (gray arrow between W and ${W}^{\prime}$), and if this is reflected in the internal world model (gray arrow between S and ${S}^{\prime}$). This difference between $p\left({s}^{\prime}\right|s)$ and $\tilde{p}\left({s}^{\prime}\right|s)$ is a measure for morphological computation

#### 3.4. Relation to the Information Bottleneck Method

**Figure 6.**Visualisation of the relation of the concepts to the Information Bottleneck Method. (

**a**) shows the general concept of morphological computation. The world states are highly correlated (red arrow), whereas the world and action are only weakly correlated (blue arrow). (

**b**) shows the concept of the Information Bottleneck Method. The difference is that a strong correlation between the action A and ${W}^{\prime}$ and a weak correlation between W and A are required. For a discussion, please read the text below. (

**a**) Concept of morphological computation; (

**b**) Concept of the Information Bottleneck Method.

## 4. Experiments

#### 4.1. Binary Model Experiment

**Figure 7.**Visualisation of the binary model. This figure shows the causal graph that is used in the binary model experiment (see text below). It is the graph representing the single step sensori-motor loop of a reactive system, where the indices of the transition maps refer to their parametrisation.

**Figure 8.**Visualisation of the four discussed cases of the world model. High values are indicated by an arrow pointing upwards (↑), and low values with an arrow pointing downwards (↓), i.e., $\varphi \uparrow ,\varphi \downarrow $ refers to the case where $\varphi \gg 0$ and $\varphi \approx 0$, resulting in a high influence of $W\to {W}^{\prime}$ and low influence of $A\to {W}^{\prime}$. The figures are discussed in the text below.

**Figure 9.**Numerical results for the measures ${\mathrm{MC}}_{\mathrm{A}}$ and ${\mathrm{MC}}_{\mathrm{W}}$ and the binary model. The three figures in the first row show the results for ${\mathrm{MC}}_{\mathrm{A}}$ with increasingly deterministic policy π (from left to right). The second row shows the results for ${\mathrm{MC}}_{\mathrm{W}}$, also with increasingly deterministic policy. For all plots, the base axes are given by world kernel parameters ψ and φ. The plots confirm the considerations discussed in the text below.

**Figure 10.**Comparison of the two measures ${\mathrm{ASOC}}_{\mathrm{A}}$ and ${\mathrm{ASOC}}_{\mathrm{W}}$. The results confirm that the measures are intrinsic adaptations of the measures ${\mathrm{MC}}_{\mathrm{A}}$ and ${\mathrm{MC}}_{\mathrm{W}}$ (see Figure 9).

**1st Case: $\varphi \gg 0$, $\psi \approx 0$.**

**Figure 11.**Comparison of the two measure ${\mathrm{C}}_{\mathrm{A}}$ and ${\mathrm{C}}_{\mathrm{W}}$. The results confirm the measures are intrinsic adoptions of the measures ${\mathrm{MC}}_{\mathrm{A}}$ and ${\mathrm{MC}}_{\mathrm{W}}$ (see Figure 9).

**2nd Case: $\varphi \approx 0$, $\psi \gg 0$.**

**3rd Case: $\varphi \approx 0$, $\psi \approx 0$.**

**4th Case: $\varphi \gg 0$, $\psi \gg 0$.**

## 4.2. Rotator

**Figure 12.**Results of the intrinsic measure in the rotating pendulum experiment. From left to right ${\mathrm{ASOC}}_{\mathrm{A}}$, ${\mathrm{C}}_{\mathrm{A}}$, ${\mathrm{ASOC}}_{\mathrm{W}}$, and ${\mathrm{C}}_{\mathrm{W}}$. The plane in each plot is defined by the noise η and the threshold parameter β. The values are averaged over 10 runs, for $\eta \in \{0,0.025.0.05,\dots ,0.5\}$ and $\beta \in \{0,0.01,0.02,\dots ,2.0\}$. The results are discussed in the text below.

**Figure 13.**Transients. Please see the legend to each plot and read the text below for a discussion. The values were averaged over 100 runs.

## 5. Discussion

**Figure 14.**Braitenberg Vehicle 3. The figure shows two Braitenberg vehicles, each equipped with two light sensors and two actuators. The polarity of the sensor-motor couplings determines the behaviour. The Braitenberg vehicle on the left-hand side is repelled by light, whereas the Braitenberg vehicle on the right-hand side is attracted by light.

## 6. Conclusions

## Acknowledgements

## Conflict of Interest

## A. Appendix

#### A.1. Derivation of the Causal Measure 1

#### A.2. Sampling World Model, Policy and Input Distribution in Pendulum Experiments

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Zahedi, K.; Ay, N.
Quantifying Morphological Computation. *Entropy* **2013**, *15*, 1887-1915.
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**AMA Style**

Zahedi K, Ay N.
Quantifying Morphological Computation. *Entropy*. 2013; 15(5):1887-1915.
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**Chicago/Turabian Style**

Zahedi, Keyan, and Nihat Ay.
2013. "Quantifying Morphological Computation" *Entropy* 15, no. 5: 1887-1915.
https://doi.org/10.3390/e15051887