# Four-Types of IIT-Induced Group Integrity of Plecoglossus altivelis

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## Abstract

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## 1. Introduction

## 2. Results

#### 2.1. Brief Summary of IIT 3.0

- MIP: The method of partitioning the system into two disconnected halves such that there is a minimal degree of information loss in the causal power of the system (the partition is unidirectional; thus, it only cuts the connections from one half to the other, but not the other way around). The cut is not bidirectional, but unidirectional.
- Integrated information $\Phi $: A degree of information loss in the causal power of the system when an MIP cut is applied. High $\Phi $ values represent a strong integrated system. The reason for this is no matter how the system is divided, information loss always exceeds the value of the MIP being applied.

#### 2.2. Definition of the Local Parameter Settings

#### 2.2.1. $\Phi $ Values for Local Parameter Settings Across Timescales

#### 2.2.2. High Average Group Integrity $\langle \Phi \left(N\right)\rangle $ in the Four- and Five-Fish Schools When They Have Blind Spots

#### 2.2.3. High Average Group Integrity $\langle \Phi \left(N\right)\rangle $ in The Two-Fish Schools When Their Visual Fields Set Narrow

#### 2.3. $\Phi $ Values for Global Parameter Settings Across Timescales

#### 2.4. Correlation between $\langle \Phi \left(N\right)\rangle $ and ${\sigma}^{2}(\Phi \left(N\right))$

#### 2.5. New Classification of Schools as Different Autonomous Systems

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Ethics Statement

#### 4.2. $\Phi $ Computation

#### 4.3. Experimental Settings

#### 4.4. Data Summary

#### 4.5. Timescales

#### 4.6. The Definition of ON and OFF State for Each Parameter

#### 4.6.1. Local Parameters

- Distance function ${D}_{i}^{t}({\mathit{x}}_{1}\left(t\right),{\mathit{x}}_{2}\left(t\right),\cdots ,{\mathit{x}}_{n}\left(t\right))$: ${\mathbb{R}}^{d}\times {\mathbb{R}}^{d}\times \cdots \times {\mathbb{R}}^{d}\stackrel{}{\to}\{0,1\}$For each individual, i, we obtain a set ${S}_{i}^{t}=\left\{j\right|d({\mathit{x}}_{i}\left(t\right),{\mathit{x}}_{j}\left(t\right))<{\xi}_{D},j\ne i\}$ of all other individuals within a specified distance, ${\xi}_{D}$. Here $d(\mathit{x},\mathit{y})$ gives the Euclidean distance between $\mathit{x}$ and $\mathit{y}$. Then, ${D}_{i}^{t}({\mathit{x}}_{1}\left(t\right),{\mathit{x}}_{2}\left(t\right),\dots ,{\mathit{x}}_{n}\left(t\right))=1$ when $|{S}_{i}^{t}|>0$ and is 0 otherwise, where $\left|S\right|$ denotes the number of elements of the set, S.
- Visual field function ${B}_{i}^{t}({\mathit{x}}_{1}\left(t\right),{\mathit{x}}_{2}\left(t\right),\dots ,{\mathit{x}}_{n}\left(t\right),{\mathit{v}}_{1}\left(t\right),{\mathit{v}}_{2}\left(t\right),\cdots ,{\mathit{v}}_{n}\left(t\right)):{\mathbb{R}}^{d}\times {\mathbb{R}}^{d}\times \cdots \times {\mathbb{R}}^{d}\stackrel{}{\to}\{0,1\}$For each individual we form the set ${O}_{i}^{t}=\left\{j\right|$ arg(${\mathit{v}}_{i}\left(t\right)$, ${\mathit{x}}_{i}\left(t\right)-{\mathit{x}}_{j}\left(t\right)$) $<{\xi}_{VF}$, $j\ne i\}$ of all other individuals whose velocity vectors point in a direction within an angle ${\xi}_{VF}$ of the focal individual. The function arg(${\mathit{x}}_{1}\left(t\right)$, ${\mathit{x}}_{2}\left(t\right)$) gives the angle between the two vectors. Then, ${B}_{i}^{t}({\mathit{x}}_{1}\left(t\right),{\mathit{x}}_{2}\left(t\right),\dots ,{\mathit{x}}_{n}\left(t\right),{\mathit{v}}_{1}\left(t\right),{\mathit{v}}_{2}\left(t\right),\cdots ,{\mathit{v}}_{n}\left(t\right))=1$ when $|{O}_{i}^{t}|>0$ and is 0 otherwise.
- Turning rate function ${T}_{i}^{t}({\mathit{v}}_{i}\left(t\right),{\mathit{v}}_{i}(t-\Delta t)):{\mathbb{R}}^{d}\times {\mathbb{R}}^{d}\stackrel{}{\to}\{0,1\}$The turning rate function returns 1 when an individual’s turning rate exceeds a specified threshold, $\delta $. That is, ${T}_{i}^{t}({\mathit{v}}_{i}\left(t\right),{\mathit{v}}_{i}(t-\Delta t))=1$ when arg(${\mathit{v}}_{i}\left(t\right)$, ${\mathit{v}}_{i}(t-\Delta t))\ge {\xi}_{TR}$ and is 0 otherwise. The time step used in this study is from $\Delta t=0.05$ to $\Delta t=1.0$ s.To obtain the states of the school, we take a conjunction of this result, that is, ${D}_{i}^{t}({\mathit{x}}_{1}\left(t\right),{\mathit{x}}_{2}\left(t\right),\cdots ,{\mathit{x}}_{n}\left(t\right))\wedge {B}_{i}^{t}({\mathit{v}}_{1}\left(t\right),{\mathit{v}}_{2}\left(t\right),\cdots ,{\mathit{v}}_{n}\left(t\right))\wedge {T}_{i}^{t}({\mathit{v}}_{i}\left(t\right),{\mathit{v}}_{i}(t-\Delta t))$ for each individual, i. The conjunction is given as $\wedge :{\{0,1\}}^{2}\stackrel{}{\to}\{0,1\}$ where $1\wedge 1=1$ and is 0 otherwise. Thus the state of each individual i at time t is ${s}_{i}(t;{\xi}_{D},{\xi}_{VF},{\xi}_{TR})\in \{0,1\}$ which depends on the three parameter values $({\xi}_{D},{\xi}_{VF},{\xi}_{TR})$. The state of the school at time t is a vector $\mathbf{s}\left(t\right)=({s}_{1}\left(t\right),{s}_{2}\left(t\right),\dots ,{s}_{n}\left(t\right))\in {\{0,1\}}^{n}$, where the parameter dependence has been omitted for simplicity.

#### 4.6.2. Global Parameters

- Average direction function $Av{d}_{i}^{t}(\mathit{V}\left(t\right),{\mathit{v}}_{i}\left(t\right)):{\mathbb{R}}^{d}\times {\mathbb{R}}^{d}\stackrel{}{\to}\{0,1\}$$\mathit{V}\left(t\right)$ is the average of $\{{\mathit{v}}_{1}\left(t\right),{\mathit{v}}_{2}\left(t\right),\dots ,{\mathit{v}}_{n}\left(t\right)\}$. If an individual’s direction of motion deviates from the average by more than a threshold amount, ${\Xi}_{AD}$, then the individual is in the OFF state: that is, $Av{d}_{i}^{t}(\mathit{V}\left(t\right),{\mathit{v}}_{i}\left(t\right))=1$ when arg($\mathit{V}\left(t\right)$, ${\mathit{v}}_{i}\left(t\right))\le {\Xi}_{AD}$, and is 0 otherwise.
- Centre of mass function $Co{m}_{i}^{t}(\mathit{X}\left(t\right),{\mathit{x}}_{i}\left(t\right)):{\mathbb{R}}^{d}\times {\mathbb{R}}^{d}\stackrel{}{\to}\{0,1\}$$\mathit{X}\left(t\right)$ is the average of $\{{\mathit{x}}_{1}\left(t\right),{\mathit{x}}_{2}\left(t\right),\cdots ,{\mathit{x}}_{n}\left(t\right)\}$. If an individual is further from $\mathit{X}\left(t\right)$ compared with a specified threshold ${\Xi}_{CM}$ then the individual is in the OFF state: that is, $Co{m}_{i}^{t}(\mathit{X}\left(t\right),{\mathit{x}}_{i}\left(t\right))=1$ when $d\left(\mathit{X}\right(t)$, ${\mathit{x}}_{i}\left(t\right))\le {\Xi}_{CM}$ and is 0 otherwise.To obtain the state of the school, we take a conjunction of these results to obtain a state for each individual, which depends on the pair $({\Xi}_{AD},{\Xi}_{CM})$:, ${s}_{i}(t;{\Xi}_{AD},{\Xi}_{CM})=Av{d}_{i}^{t}(\mathit{V}\left(t\right),{\mathit{v}}_{i}\left(t\right))\wedge Co{m}_{i}^{t}(\mathit{X}\left(t\right),{\mathit{x}}_{i}\left(t\right))\in \{0,1\}$. The state of the school at time t is then a vector $\mathit{s}\left(t\right)=({s}_{1}\left(t\right),{s}_{2}\left(t\right),\dots ,{s}_{n}\left(t\right))\in {\{0,1\}}^{n}$, where the parameter dependence has been omitted for simplicity.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

IIT | Integrated Information Theory |

SOC | Self-organising Criticality |

MIP | Minimum information partition |

TPM | Transition probability matrix |

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**Figure 1.**Timescale rules. The video recordings present each trajectory. The frame rate of our device is 120 fps. The velocity vectors of the time scale, $\Delta t$, define every $\Delta t$ step-length when the time scale is $\Delta t=\delta t/120$. The timescales are $\Delta t=5/120$, $\Delta t=10/120$, $\Delta t=20/120$, $\Delta t=40/120$, $\Delta t=80/120$, and $\Delta t=120/120$ s. Long-time scales eliminate subtle noise-like movements in a school (or support the predictability of other fish’s movements).

**Figure 2.**The definition of ON and OFF states for local parameter settings. Three parameters are used to determine a school’s state (Yellow: distance, Green: visual field, and Purple: turning rate). The coloured individuals are in the ON state. We take a conjunction of the three school states to obtain the final school state at time t. Subsequently, we compute $\Phi $ from a time series of these states using PyPhi. The meanings of the above figures is as follows: Distance means fish 1 and 2 are in the ON state because their interaction radius includes each other. Visual Field means fish 2 (fish 4) is in the ON state because fish 1 (fish 2) is included in its visual field. Turning rate means the sector represents the threshold of this parameter. The bold vector represents the velocity vector at time $t-1$ and the dotted vector represents the velocity vector at time t. Fish 2 and 3 are in the ON state because of the dotted direction out of their own threshold sector.

**Figure 3.**Three-dimensional distribution of the mean $\langle \Phi \left(N\right)\rangle $ values with respect to the three parameters according to the group size for all experimental data ($\Delta t=20/120$ s). The ball size and shaded colours represent the $\langle \Phi \left(N\right)\rangle $ strength. Owing to visibility, we only show the points over 0.4*${\langle \Phi \left(N\right)\rangle}_{MAX}$, where ${\langle \Phi \left(N\right)\rangle}_{MAX}$ implies the $\langle \Phi \left(N\right)\rangle $ values of the maximum cell for each group size. We measured the $\langle \Phi \left(N\right)\rangle $ values over the main complexes and full subsystems throughout our analysis. All the graphs only refer to the main complexes. For similar distributions, please refer to [77]. Figures S1 and S2 include all the mean $\langle \Phi \left(N\right)\rangle $ and mean ${\sigma}^{2}(\Phi \left(N\right))$ values for all the timescales.

**Figure 4.**(

**a**) The mean distribution over $\frac{1}{2}{\langle \Phi \left(N\right)\rangle}_{MAX}$, where ${\xi}_{VF}>\pi $ and ${\xi}_{TR}<0.05$. Each colour corresponds to the group size, and each shape corresponds to the time scale: $\Delta t=5/120$ s (circle), 10/120 s (upward triangle), 20/120 s (downward triangle), 40/120 s (rectangle), 80/120 s (pentagon), and 120/120 s (hexagon). ${\langle \Phi \left(N\right)\rangle}_{MAX}$ represents the maximum $\langle \Phi \left(N\right)\rangle $ value of all the cells in the $\langle \Phi \left(N\right)\rangle $ distribution. The distributions of $N=2$ and $N=3$ for all the scales distributed on the complete visual fields (${\xi}_{VF}=2\pi $). In contrast, the distributions of $N=4$ and $N=5$ for all the scales distributed on the lower right field (${\xi}_{VF}<2\pi $). (

**b**) The box plot for the mean normalised $\langle \Phi \left(N\right)\rangle $ values, where $\pi <{\xi}_{VF}<2\pi $ for all datasets. Each datum is divided by the ${\langle \Phi \left(N\right)\rangle}_{MAX}$ in the region of $\pi <{\xi}_{VF}<2\pi $ and ${\xi}_{VF}<\pi $ (this graph uses all the $\langle \Phi \left(N\right)\rangle $ values: no restriction such as $\frac{1}{2}{\langle \Phi \left(N\right)\rangle}_{MAX}$). The $\langle \Phi \left(N\right)\rangle $ values of four- and five-fish schools are significantly higher than those of two- and three-fish schools. For comparison, Figure S3 presents the same box plot for high turning rates of ${\xi}_{TR}\ge 0.05$.

**Figure 5.**(

**a**) The mean distribution over $\frac{1}{2}{\langle \Phi \left(N\right)\rangle}_{MAX}$, where ${\xi}_{VF}<\pi $ and ${\xi}_{TR}<0.05$. Each colour corresponds to the group size, and each shape corresponds to the time scale: $\Delta t=5/120$ s (circle), 10/120 s (upward triangle), 20/120 s (downward triangle), 40/120 s (rectangle), 80/120 s (pentagon), and 120/120 s (hexagon). No value exceeds $\frac{1}{2}{\langle \Phi \left(N\right)\rangle}_{MAX}$ for the four- and five-fish schools. Some of the three-fish school exceed this value; however, only in a few samples. (

**b**) The box plot for the mean normalised $\Phi $ values, where $0<{\xi}_{TR}<\pi $ for all datasets. The data were divided with the maximum ${\langle \Phi \left(N\right)\rangle}_{MAX}$ in $0<{\xi}_{VF}<\pi $ and ${\xi}_{TR}<0.05$ (this graph uses all the $\langle \Phi \left(N\right)\rangle $ values: no restriction such as $\frac{1}{2}{\langle \Phi \left(N\right)\rangle}_{MAX}$). The $\langle \Phi \left(N\right)\rangle $ values in the two-fish school were determined as ‘chasing’, which is the opposite of ‘leadership’. For comparison, Figure S4 depicts the box plot for the high turning rate, ${\xi}_{TR}\ge 0.05$, under the same condition. The statistical test is included in Table S1.

**Figure 6.**Definition of ON and OFF states for global parameter settings. Two parameters determine a school’s state (Yellow: Centre of Mass and Blue: Average Direction). Coloured individuals are in the ON state. We calculate the conjunction of the two school states and obtain the final school state at time t. For the left figure, the fish 2 and 3 are ON because they are in the radius of centre of mass. For the left figure, the fish 1, 2, and 3 are ON because their direction (bold black arrow) are diverted from the average direction (bold red arrow). Subsequently, we compute $\Phi $ from a time series of these states using PyPhi. We assume that the network structure is similar to that of the local parameter settings, i.e., the fully connected network without self-loop.

**Figure 7.**Heat maps on the global parameter settings for each group size ($\Delta t=20/120$ s). We took the average of $\langle \Phi \left(N\right)\rangle $ for all datasets (colour bar). The horizontal axis shows the distance from the centre of mass (${\Xi}_{CM}$), and the vertical axis shows the difference from average direction (${\Xi}_{AD}$). The cells from ${\Xi}_{CM}=600$ to ${\Xi}_{CM}=3000$ were omitted owing to space limitations. All timescale figures are listed in Figure S5.

**Figure 8.**Difference between the average top $\langle \Phi \left(N\right)\rangle $ value of the local and global group integrities. The horizontal axis shows the timescales ($\Delta t=\delta t/120$, where $\delta t=5,10,20,40,80,120$). The negative and positive values on the vertical axis represent that the global integrity over- or under-estimates the local integrity, respectively. The peak value of $N=5$ at $\Delta t=20/120$ is significantly greater than at the other peaks (Table 1).

**Figure 9.**Correlation relation between $\langle \Phi \left(N\right)\rangle $ and its Fano scales, ${\sigma}^{2}(\Phi \left(N\right))/\langle \Phi \left(N\right)\rangle $ (normalized variance) for the local parameter setting. Each colour corresponds to timescales. The correlation coefficients are 0.66 ($\Delta t=5/120$ s), 0.71 ($\Delta t=10/120$ s), 0.61 ($\Delta t=20/120$ s), 0.72 ($\Delta t=40/120$ s), 0.75 ($\Delta t=80/120$ s), and 0.61 ($\Delta t=120/120$ s). For all the Pearson correlation tests, $n=3200$ (all data points for each scale: Figure S1) and $p<{10}^{-30}$. For the global parameter settings, see Figure S6.

**Figure 10.**The classification from $\Phi $ values related to the school’s behaviour. (

**a**) analysis of 2-fish schools. (

**b**) analysis of 3-fish schools. (

**c**) analysis of 4-fish schools. (

**d**) analysis of 5-fish schools. We only compared the representative $\Phi $ values for each collective state from Table 2 parameter ranges. We fixed the turning rate parameter ${\xi}_{TR}=0.001$ (for other parameter settings, see Figure S9). We averaged $\Phi $ values for the same number of ON states (e.g., 01 and 10, etc.). For instance, over $99\%$ collective states of 3, 4, and 5-fish’s school are all in the ON state (e.g., 111 for 3-fish schools) or the single OFF states (e.g., 011, 101, and 110 for 3-fish schools). We list all collective states for only the two-fish school because the rate of the collective states (11, 01, and 10) is approximately $20\%\sim 50\%$. The horizontal circles represent the corresponding collective state in the same number of ON states (ON: white, OFF: black). The $\Phi $ values averaged the above-mentioned data over the distance parameters (${\xi}_{D}=100$ for two-fish, $400\le {\xi}_{D}\le 1000$ for three-, four-, five-fish school) for each visual field parameter. The right box figure represents the information flow indicated from the left figure. The blue circle is the sub-group divided by the MIP cut (red dotted line). The red arrow is the cut flow and the black arrow is the opposite flow. The thickness of the arrow represents the intensity of the flow. From the MIP definition, the thickness of the black arrow is always greater than that of the red arrow. The statistical test is included in Table S1.

**Table 1.**Tukey–Kramer method for Figure 8 at $\Delta t=20/120$ (s). The notation, ${D}_{G-L}(\Phi \left(N\right))$, represents the difference between ${\Phi}_{local}\left(N\right)-{\Phi}_{global}\left(N\right)$ for group size N. We compute all pairs of the difference between different sizes at $\Delta t=20/120$ s for this statistical test.

${\mathit{D}}_{\mathit{G}-\mathit{L}}\left(\mathbf{\Phi}\left(\mathit{N}\right)\right)-{\mathit{D}}_{\mathit{G}-\mathit{L}}\left(\mathbf{\Phi}\left(\mathit{M}\right)\right)$ | p-Value | |
---|---|---|

$N=2-M=3$ | −0.049 | 0.999 |

$N=2-M=4$ | 0.193 | 0.69 |

$\mathit{N}=\mathbf{2}-\mathit{M}=\mathbf{5}$ | 2.31 | <${\mathbf{10}}^{-\mathbf{7}}$ |

$N=3-M=4$ | 0.242 | 0.48 |

$\mathit{N}=\mathbf{3}-\mathit{M}=\mathbf{5}$ | 2.36 | <${\mathbf{10}}^{-\mathbf{8}}$ |

$\mathit{N}=\mathbf{4}-\mathit{M}=\mathbf{5}$ | 2.12 | <${\mathbf{10}}^{-\mathbf{7}}$ |

**Table 2.**The matching rate of the single OFF state individual and MIP cut or the positional leadership (PL). ${\xi}_{D}$ is set as follows. for $N=2$, ${\xi}_{D}=100$, for $N=3,4,5$, $400\le {\xi}_{D}\le 1000$. ${\xi}_{TR}$ is fixed at 0.001 (rad/step) because values are greater than other parameters (for ${\xi}_{TR}=0$ or $0.005$, see Table S1).

N | Visual Field ${\mathit{\xi}}_{\mathit{V}\mathit{F}}$, $\mathit{\pi}$(rad) | MIP Cut Match Rate (%) | PL Match Rate (%) |
---|---|---|---|

2 | 0.18 | 100 | 100 |

0.36 | 100 | 100 | |

0.56 | 100 | 100 | |

3 | 1.9 | 19 | 81 |

2.0 | 26 | 40 | |

4 | 1.6 | 18 | 92 |

1.8 | 58 | 76 | |

1.9 | 67 | 42 | |

5 | 1.6 | 70 | 91 |

1.8 | 73 | 73 | |

1.9 | 100 | 42 |

**Table 3.**Data summary. These are all the data sets that we used in this paper. Three data sets for $N=2,4,5$ and four data sets for $N=3$. The total time length are approximately 10–15 min.

N | Average Distance (mm) | Average Velocity (mm/s) | Error (S.D.) | Minimum Distance (mm) | Cohesion Rate (%) | Total Time Steps |
---|---|---|---|---|---|---|

2 | 166.3 | 268.8 | 0.18 | 1.90 | 99.7 (99.9) | 106,961 |

90.67 | 271.68 | 0.23 | 0.10 | 99.8 (100) | 99,431 | |

122.0 | 256.08 | 0.18 | 1.60 | 99.7 (100) | 107,206 | |

3 | 170.8 | 301.2 | 0.23 | 1.80 | 95.5 (97.9) | 90,051 |

159.1 | 343.2 | 0.14 | 1.83 | 98.8 (99.8) | 83,654 | |

173.1 | 300.0 | 0.13 | 2.82 | 97.4 (99.0) | 97,446 | |

132.0 | 240.0 | 0.19 | 1.67 | 99.2 (99.9) | 93,931 | |

4 | 164.3 | 270.72 | 0.14 | 1.18 | 98.8 (100) | 106,327 |

141.5 | 190.8 | 0.12 | 1.38 | 99.2 (100) | 103,226 | |

114.9 | 148.56 | 0.38 | 1.83 | 99.8 (100) | 98,126 | |

5 | 143.8 | 259.92 | 0.28 | 0.79 | 98.7 (99.6) | 102,895 |

146.0 | 213.12 | 0.12 | 1.16 | 100 (100) | 97,346 | |

143.7 | 259.2 | 0.28 | 1.44 | 97.2 (100) | 92,116 |

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**MDPI and ACS Style**

Niizato, T.; Sakamoto, K.; Mototake, Y.-i.; Murakami, H.; Tomaru, T.; Hoshika, T.; Fukushima, T.
Four-Types of IIT-Induced Group Integrity of *Plecoglossus altivelis*. *Entropy* **2020**, *22*, 726.
https://doi.org/10.3390/e22070726

**AMA Style**

Niizato T, Sakamoto K, Mototake Y-i, Murakami H, Tomaru T, Hoshika T, Fukushima T.
Four-Types of IIT-Induced Group Integrity of *Plecoglossus altivelis*. *Entropy*. 2020; 22(7):726.
https://doi.org/10.3390/e22070726

**Chicago/Turabian Style**

Niizato, Takayuki, Kotaro Sakamoto, Yoh-ichi Mototake, Hisashi Murakami, Takenori Tomaru, Tomotaro Hoshika, and Toshiki Fukushima.
2020. "Four-Types of IIT-Induced Group Integrity of *Plecoglossus altivelis*" *Entropy* 22, no. 7: 726.
https://doi.org/10.3390/e22070726