Introduction

Critical phenomena are widely observed in living organisms1,2,3,4. The basic interpretation of a critical phenomenon is that the system is situated between ordered and disordered states1,3,5,6,7. Although critical theoretical phenomena originate from statistical physics, they have various applications in living systems8,9,10,11. Many researchers suggest that living systems efficiently utilise these critical states for optimal information transfer12,13, high computational power9,14,15, and adaptive behaviour16,17,18. Moreover, some researchers have proposed designing artificial systems that harness this computational power, such as robots and computers11,19.

The criticality observed in nature, referred to as ‘empirical criticality’ in this paper, has traditionally been interpreted through the lens of theoretical criticality. However, directly applying theoretical criticality to empirical systems poses significant challenges. First, traditional approaches often struggle to define a suitable unit of analysis for criticality, an issue across all methods of measuring a system’s criticality20,21. For example, determining the threshold for group size in collective behaviour20 or defining cell boundaries can be problematic21. Second, critical phenomena in nature frequently exhibit a nested structure, where small- and large-scale criticalities coexist22,23,24. Traditional theoretical frameworks are limited in addressing this ‘nested criticality’.

These limitations arise from two types of fluctuation: internal and external. Internal fluctuation is generated by the motivations and interactions of agents within the system25,26,27,28,29,30,31,32,33,34. It is heterogeneous across time and space and has context-dependent characteristics. In contrast, external fluctuation is probabilistic or random, independent of the system, and typically treated as uniform throughout the system1,8,9,35,36,37.

Many researchers describe traditional theoretical criticality as a balance between deterministic interactions and context-independent external fluctuation8,9,14,34,38. In physical systems, external fluctuation acts as a driver that facilitates the transition from order to disorder through system interactions8,9,14,34. For example, in spin systems, external fluctuation (e.g. temperature) plays a crucial role in promoting dynamic changes and triggering phase transitions at a critical point6,39. However, in biological systems, external fluctuation is not purely independent but becomes intricately intertwined with the internal interactions of the system25,26,27,28. Biological systems are highly context-sensitive. Failing to account for internal fluctuation makes it difficult to explain why biological systems often exhibit nested criticality or to determine system boundaries, as these phenomena are strongly influenced by context.

Integrated information theory (IIT) was proposed to mathematically define human consciousness40,41,42,43,44. The basic concept of IIT suggests that the degree of consciousness can be measured as the difference between the entire system and the sum of its parts (i.e. Φ). Although IIT originated from neurological studies, its applications extend to general systems45,46,47,48,49,50. For example, as a purely theoretical application, the degree of integration (Φ) effectively measures critical state51, such as the edge of chaos47. Niizato et al. applied IIT to empirical systems (e.g. fish school20,49 and body information50), discovering characteristic structures that were not observed in other information-theoretic analyses.

The advantages of applying IIT to a living system are twofold. First, as discussed, system integration (i.e. Φ) measures the degree of criticality47,51. The high accuracy in identifying the phase transition point is helpful for examining the information structure of system criticality using IIT.

Second, IIT does not discriminate between internal and external fluctuation in advance42,44,52. This property has several implications for analysing living systems, as follows. Unlike traditional methods, we do not need to assume a priori unit for the analysis in the IIT paradigm. IIT naturally handles internal and external fluctuation, addressing the challenge of distinguishing between these types of fluctuation, which has been a major limitation of previous methods. By applying the minimum information partition (MIP)42,44, we can divide the system into two halves with the most irrelevant correlation (i.e. two divided subsystems act as external fluctuation to each other). The measure of connectedness defined by MIP is Φ. Therefore, if the system contains irrelevant subsystems (i.e. Φ = 0), we can identify and omit them from our analysis using MIP. Furthermore, we can define the subsystem generated by context-dependent fluctuation as the main complex. Since these main complexes change over time, we can track multi-scale interactions within the system.

This study aimed to reveal the essential factors for empirical criticality, which have not been confirmed in theoretical criticality, by considering collective animal behaviour. We applied IIT to actual fish data (Plecoglossus altivelis). We used two variables for IIT computation: orientation and speed, which are critical properties12. Throughout this study, we assumed that group integrity (in this paper, we use the term ‘integrity’ to distinguish it from ‘integration’, which is used in the context of consciousness research42,44,52) represented the degree of the critical state. In this study, we discuss the essential differences between theoretical and empirical critical phenomena by making detailed comparisons with the self-propelled particle (SPP) model5,39 and the BOID model53, both widely used in analysing collective behaviour. Furthermore, by applying IIT to actual collective animal behaviour, we can better understand the information process of criticality (i.e. the relationship between collective states and their information structure). The IIT analysis confirmed the traditional argument on criticality while revealing a new aspect of criticality, defined as local criticality, which can coexist with traditional criticality.

Result

Brief review of IIT

First, we explain certain IIT concepts for readers unfamiliar with the theory (we used IIT version 2.0, as four versions have been proposed). Although many concepts exist within IIT, we focused on three key aspects: the MIP, main complexes, and integrated information (Φ). As discussed in the Introduction, a major issue of IIT is intrinsic information; it depends only on internal variables, with no external system involved. These internal variables also change in response to both external and internal stimuli. Typical information theories in biology assume a relationship between external inputs and outputs. In this setting, the observer’s interest lies in the input–output relationship via the system of interest. By contrast, IIT focuses on system dynamics, which concerns the mutual information between past and present states. Let \({X}^{t}\) be an internal variable vector (\(X=\left\{{X}_{1},{X}_{2},\dots ,{X}_{n}\right\}\)) at time t. This mutual information is expressed as follows:

$$\begin{array}{*{20}c} {I\left( {X^{t - \tau } ;X^{t} } \right) = H\left( {X^{t - \tau } } \right) - H\left( {X^{t - \tau } |X^{t} } \right)} \\ \end{array}$$
(1)

where \(H\left({X}^{t-\tau }\right)\) represents the entropy of the past states and \(H\left({X}^{t-\tau }|{X}^{t}\right)\) is the conditional entropy of past states given the present states, with \({X}^{t-\tau }\) representing the state of \({X}^{t}\), τ steps before.

However, mutual information includes data irrelevant to integrity, such as event correlations and historical effects. IIT applies the MIP to the system to eliminate these effects. We must search for the weakest link among all possible cuts (i.e. dividing the system into two subsets), which grows exponentially as the system size increases. This computational complexity forces us to limit the application of IIT to smaller systems. For example, if a system has two elements (i.e. {a, b}), there is only one possible cut: {a} and {b}. However, for a system of three elements (i.e. {a, b, c}), the possible cuts increase to three: {a, b} and {c}, {a, c} and {b}, and {b, c} and {a}. We explore all possible cuts for the given system and identify the weakest information link between two subsets.

In this study, we used the mismatch decoding method proposed by Oizumi et al.44, which offers a relatively high computational speed. System integrity Φ∗ is expressed as

$$\begin{array}{*{20}c} {{\Phi }^{*} = I\left( {X^{t - \tau } ;X^{t} } \right) - I^{*} \left[ {X;\tau ,\pi \in {\mathcal{P}}_{S} } \right]} \\ \end{array}$$
(2)

where \(I\left({X}^{t-\tau };{X}^{t}\right)\) is the mutual information between the current \({X}^{t}\) and past states \({X}^{t-\tau }\), S is the set of all nodes of a given system, \({\mathcal{P}}_{S}\) is the set of all bi-partitions (total 2|S|-1 − 1 partitions), π is an element of the set \({\mathcal{P}}_{S}\), and \({I}^{*}\left[X;\tau ,\pi \in {\mathcal{P}}_{S}\right]\) is a ‘hypothetical’ mutual information, indicating the mismatched decoding in the partitioned probability distribution by π. More precisely, \({I}^{*}\left[X;\tau ,\pi \in {\mathcal{P}}_{S}\right]\) is expressed as the partition \({{\text{max}}_{\beta }I}^{*}\left[X;\beta ,\tau ,{\mathcal{P}}_{S}\right]\) minimises Φ∗ (see listed studies39,41 for further details about this expression). Intuitively, \({I}^{*}\left[X;\tau ,\pi \in {\mathcal{P}}_{S}\right]\) represents the hypothetical mutual information when the system S is divided into two subsets, S1 and S2 (note \(S={S}_{1}\cup {S}_{2}\)), with the assumption that no information is exchanged between the two subsystems (i.e. two subsystems are probabilistically independent). Since S1 and S2 are parts of the system S, they are referred to as subsystems hereafter. If no information is exchanged between the two subsystems under this assumption, the resulting mutual information \({I}^{*}\left[X;\tau ,\pi \in {\mathcal{P}}_{S}\right]\) would equal the original mutual information \(I\left({X}^{t-\tau };{X}^{t}\right)\), and Φ∗ would become zero. In other words, a hypothetical distribution \({I}^{*}\left[X;\tau ,\pi \in {\mathcal{P}}_{S}\right]\) must be chosen so that Φ∗ is minimised under the partition π. The parameter β is introduced to achieve this minimisation44,46.

Because Φ ∗ depends on a partition π (\(\in {\mathcal{P}}_{S}\)), the MIP is the partition that minimises the integrated information.

$$\begin{array}{*{20}c} {\pi_{{{\text{MIP}}}} \equiv \arg \mathop {\min }\limits_{{\pi \in {\mathcal{P}}_{S} }} {\Phi }^{*} \left( \pi \right)} \\ \end{array}$$
(3)

The integrated information for πMIP is expressed as Φ*(πMIP). We denote Φ*(πMIP) as ΦMIP. Notably, if ΦMIP is equal to zero, the parts of the system are mutually independent; that is, the system contains no interaction between two subsystems. In this sense, ΦMIP characterises the irreducibility of a system into its subsystems.

Main complex

In general, we can compute ΦMIP for any subsystem within the system and not only for set S. We denote each \({\Phi }_{\text{MIP}}^{T}\) for subsystem T as ΦMIP, where \(T\subset S\). A ‘complex’ is a subsystem \(C(\subset S)\), where \({\Phi }_{\text{MIP}}^{C}>{\Phi }_{\text{MIP}}^{T}\) for all supersets T of C. Note that the entire set S always satisfies complex conditions. Based on this definition, we define the main complexes as those with a local maximum ΦMIP.

Definition

(Main complex): A main complex is a complex M satisfying \({\Phi }_{\text{MIP}}^{M}>{\Phi }_{\text{MIP}}^{R}\) for the subset \(R\subset M\).

This definition implies that if two main complexes exist (say, A and B), then there is no inclusive relation. This is because the main complex A, for instance, has maximal among all the supersets and subsets of the set A. IIT researchers consider these complexes to be the information core of the system, which may be related to our conscious experience. Although such information cores play a vital role in living systems, the validity of this assumption remains to be demonstrated45,49.

The main complex is among the most critical concepts of IIT. The main complex is a subset with a local maximum ΦMIP. Based on its definition, several main complexes exist in a system. Tononi et al. suggested that the main complex, having a global maximum ΦMIP, can serve as a consciousness based on their exclusive principle43. In the present study, we do not assert that the ΦMIP correspond to the consciousness of the fish group; instead, we apply it as a measure of the system integrity, particularly the degree of criticality.

Application of the IIT to a fish school

Data

We tracked the trajectories of Plecoglossus altivelis fish schools with N = 10, using seven samples (8–12 min recording length). Because the frame rate was 1/20 s, there were approximately 12,000 frames for the position data (i.e. \({{\varvec{x}}}_{i}\left(t\right)=\left({x}_{i}\left(t\right), {y}_{i}\left(t\right)\right)\)).

Variable settings for IIT 2.0

We applied IIT 2.0 to the direction of the fish (i.e. turning rate) and speed (i.e. acceleration). We computed each velocity vector \({{\varvec{v}}}_{i}\left(t\right)\) from the obtained positional information, \({{\varvec{x}}}_{i}\left(t\right)\), that is, \({{\varvec{v}}}_{i}\left(t\right)={{\varvec{x}}}_{i}\left(t\right)-{{\varvec{x}}}_{i}\left(t-\Delta t\right)\), where ∆t was 1/20 s. The turning rate, \(d{\theta }_{i}\left(t\right)\) is defined as the rotation angle from \({\widehat{{\varvec{v}}}}_{i}\left(t-\Delta t\right)\) to \({\widehat{{\varvec{v}}}}_{i}\left(t\right)\), where hut is the unit vector of v. The acceleration \(d{s}_{i}\left(t\right)\) is \(\Vert {{\varvec{v}}}_{i}\left(t\right)-{{\varvec{v}}}_{i}\left(t-\Delta t\right)\Vert\). Therefore, the group vector information is expressed as \({\varvec{d}}{\varvec{\theta}}(t)=\left[d{\theta }_{1}\left(t\right),\boldsymbol{ }d{\theta }_{2}\left(t\right),\boldsymbol{ }\dots ,d{\theta }_{10}\left(t\right)\right]\), which generate \({\Phi }_{\text{MIP}}^{\text{dir}}\), and \({\varvec{d}}{\varvec{s}}(t)=\left[d{s}_{1}\left(t\right),\boldsymbol{ }d{s}_{2}\left(t\right),\boldsymbol{ }\dots ,d{s}_{10}\left(t\right)\right]\), which generate \({\Phi }_{\text{MIP}}^{\text{sp}}\). Each ΦMIP was computed from the time series of these vectors (Fig. 1A). We set the maximum interval (i.e. window size, Tmax) for the analysis. The vector \(\left[{\varvec{d}}{\varvec{\theta}}\left(t\right),\boldsymbol{ }{\varvec{d}}{\varvec{\theta}}\left(t-\Delta t\right),\boldsymbol{ }\dots ,\boldsymbol{ }{\varvec{d}}{\varvec{\theta}}\left(t-{T}_{\text{max}}\Delta t\right)\right]\) was set for the analysis. We calculated ΦMIP for three conditions: \({T}_{\text{max}}=\{200, \text{400,600}\}\), that is, 10, 20, and 30 s.

Fig. 1
figure 1

IIT computation methods and concepts (A) Constructing dθ for IIT computation. We used the same method for ds. (B) IIT concepts. (1) MIP: MIP-cut divides the fish school into two halves (coloured red and blue), which show the weakest information link. The loss of information induced by this cut implies ΦMIP. In this study, we set ∆t = 0.15 and Tmax = 400. (2) We have Maximum main complex: the subset T has the maximum \({\Phi }_{\text{MIP}}^{T}\) among any complexes (superset). In this case, T (green) and M (red) are the main complexes (\({\Phi }_{\text{MIP}}^{T}\) and \({\Phi }_{\text{MIP}}^{M}\) are local maxima). In particular, the maximum main complex (MMC, in short) is given as T (green).

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Setting a parameter for IIT 2.0 application

Oizumi et al. proposed approximation methods for the computational challenges of the IIT44,54,55. In this study, we applied their ‘Practical Φ Toolbox for MATLAB’ to the fish data. An exhaustive method (computing all possible MIPs) could be applied within a realistic computation time for such a small system.

Because the time delay τ (IIT has only two constraints: partition π and time delay τ) was chosen as a suitable parameter, we computed ΦMIP for τ ranging from 1 (i.e. 1/20 s) to 100 (i.e. 5 s) frames. The time delay τ is the point at which the mean values of both \({\Phi }_{\text{MIP}}^{\text{dir}}\) and \({\Phi }_{\text{MIP}}^{\text{sp}}\) were maximised across all the datasets with both peaks were located simultaneously. We selected the value of τ that maximises ΦMIP as we expected strong interactions within fish schools. τ comprised approximately three frames (i.e. 0.15 s). This value was suitable because it was the same as the Plecoglossus altivelis’s response time (Figure S5).

Basic interpretation of IIT 2.0 in the group dynamics

Let us refocus on fish schools from the perspective of IIT. In group behaviour, the IIT describes an agent’s connectedness as a group. If all fish move randomly, ΦMIP is expected to be zero because removing one fish does not affect the group’s behaviour. Similarly, if all fish move in the same direction and at the same speed, ΦMIP becomes zero because there is no information exchange under the static state. ΦMIP = 0 indicates that the group can be divided into two independent subgroups by MIP (Fig. 1B). By contrast, the system must incorporate heterogeneous interactions to achieve a high ΦMIP. From an intrinsic perspective, the ‘differences that make a difference’ information matters. Therefore, highly integrated collective behaviour is neither ordered nor disordered. This is referred to as the ΦMIP of the ‘group integrity’ to distinguish it from the context of consciousness. ΦMIP evaluates these complex but incomparable group dynamics.

The main complex in the group corresponds to the core of information processing within the group (Fig. 1B). In general, the main complex M is an appropriate subset of the entire set S. Since the highest ΦMIP is related to the critical state (Figure S1 and S2; we show that a normal critical state, such as in the SPP model, has no parts for group criticality —we will discuss this issue later), this small group is expected to be highly susceptible to external perturbations. The interactions between these mutually susceptible subgroups determine how the main complexes are allocated to the entire group. Such a separation is based on the unique methodology of the IIT, wherein the theory does not discriminate between internal and external fluctuation in advance. By applying MIP to the system, we can identify the independent subsystems (i.e. the external fluctuation to each other), while simultaneously measuring the group integrity (i.e. the internal fluctuation induced by interaction) within each subsystem.

Furthermore, we applied two Φs considering different information perspectives: orientation (i.e. \({\Phi }_{\text{MIP}}^{\text{dir}}\)) and speed (i.e. \({\Phi }_{\text{MIP}}^{\text{sp}}\)), for two reasons: First, the criticality of orientation and speed is observed in animal groups12; however, the relation between these two critical dynamics is still uncertain. Examining group integrity from both perspectives facilitated the exploration of the relationship between these two criticalities from an IIT standpoint. Second, the group requires a speed parameter to achieve group formation (e.g. schooling, milling, and swarming). We examined how these two ΦMIP relate to group formation (Figures S3 and S4).

Two types of group integrity \(\Phi_{MIP}^{dir}\) and \(\Phi_{MIP}^{sp}\)

First, IIT was applied to the experimentally obtained data from schools of Plecoglossus altivelis to investigate how ΦMIP changes. Since \({\Phi }_{\text{MIP}}^{C}\) is obtained for each main complex C (Fig. 1B right), we define two quantities to estimate the group integrity: (1) the sum of all ΦMIP corresponding to the main complexes at a time t. Mathematically, this value can be defined as \(\sum_{C\in {\mathcal{M}}_{t}}{\Phi }_{\text{MIP}}^{C}\), where \({\mathcal{M}}_{t}\) is a set of main complexes at a time t. We denote this value as \(\sum {\Phi }_{\text{MIP}}\), in short. (2) the ΦMIP of only the main complex, which holds the highest value at a time t. Mathematically, this value can be defined as \(\text{max}\{{\Phi }_{\text{MIP}}^{C}|C\in {\mathcal{M}}_{t}\}\). We denote this value as max{ΦMIP}, in short.

Figure 2A shows the time series of \(\sum {\Phi }_{\text{MIP}}^{\text{dir}}\) and \(\sum {\Phi }_{\text{MIP}}^{\text{sp}}\). Both values exhibit dynamic changes over time. Figure 2B shows the correlation between \(\sum {\Phi }_{\text{MIP}}^{\text{dir}}\) and \(\sum {\Phi }_{\text{MIP}}^{\text{sp}}\) for all data. Although a weak correlation between the two integrities is observed, a consistent information flow between them cannot be confirmed (Table S1. However, upon increasing Tmax, we observe a statistically significant information flow). Therefore, the relationship between these two information processes can be considered independent.

Fig. 2
figure 2

\({\Phi }_{\text{MIP}}^{\text{dir}}\) and \({\Phi }_{\text{MIP}}^{\text{sp}}\) from actual fish data. (A) Time series of \(\sum {\Phi }_{\text{MIP}}^{\text{dir}}\) and \(\sum {\Phi }_{\text{MIP}}^{\text{sp}}\). (B) Correlation between \(\sum {\Phi }_{\text{MIP}}^{\text{dir}}\) and \(\text{log}(\sum {\Phi }_{\text{MIP}}^{\text{sp}})\). Each colour corresponds to the data set number. (C) Violin plot of Hurst exponent (\(\langle H\rangle\)) of \(\sum {\Phi }_{\text{MIP}}^{\text{dir}}\) and \(\sum {\Phi }_{\text{MIP}}^{\text{sp}}\). There is no significant difference between them.

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However, the two information processes also shared a common characteristic. The most notable aspect of these two-time series was long-range correlation (i.e. non-Brownian); the time series was affected by past behaviour. Figure 2C shows the generalised Hurst exponent (H). If the time series is Brownian, the exponent H is approximately 0.5. The exponent H is approximately zero if the time series is white noise. The exponent of the real-time series of both datasets is approximately 0.25 (Fig. 2C). This value indicates that short-term fluctuations in \(\sum {\Phi }_{\text{MIP}}\) appear random while exhibiting long-term regularity similar to Brownian noise. Such noise is commonly referred to as pink noise, characterised by scale-invariance and long autocorrelation (See Table S2. This scale invariance, \({f}^{-\alpha }\), is also observed in the power spectrum analysis). In contrast, the time series of max{ΦMIP} is Brownian (Table 1). This Brownian noise in max{ΦMIP} arises from the fact that max{ΦMIP} only retains the historical effect of the peak ΦMIP, not of the MMC, M. This difference implies that the actual fish school has a historical effect as a whole (i.e. a set of all main complexes) but not for a specific subgroup, such as the MMC.

Table 1 Statistical tests for \(\langle H\rangle\) for each hypothesis. Note that H = 0 represents the white noise and H = 0.5 indicates Brownian noise (see Table S2 for the power spectrum analysis.)
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Difference between the theoretical models and Plecoglossus altivelis in terms of the integrity

Comparison with self-propelled particle model

The self-propelled particle (SPP) model is widely used to study collective behaviour5,12,25. In this model, each agent increases overall alignment by averaging the directions of its neighbours. Additionally, the SPP model applies noise to the averaging process of each agent (detail in the Supporting Information). This noise functions as a parameter to replicate various group formations5. Notably, the system undergoes a phase transition when this noise parameter reaches a certain critical point38. Many researchers have considered the criticality observed in the SPP model as a standard for understanding the criticality of collective behaviour5,12,25. However, most studies have not identified the differences between the criticality of the SPP model and the criticality observed in natural collective behaviour. The aim of this section is to clarify this difference from the perspective of IIT.

Figure 3A shows the frequency distribution of the MMC size (i.e. |M|, where \(M=\text{argmax}\left\{C|{\Phi }_{\text{MIP}}^{\text{C}}\right\}\)) in the direction, speed, and SPP (10 agents). As listed in Figure S1, the maximum \(\text{max}\left\{{\Phi }_{\text{MIP}}^{\text{dir}}\right\}\) was located near the critical noise parameter. This result suggested the validity of our selection of the turning rate dθ as IIT variable and that \(\text{max}\left\{{\Phi }_{\text{MIP}}^{\text{dir}}\right\}\) could be a measure of the degree of criticality in SPP. Furthermore, a stark contrast was observed between the Plecoglossus altivelis and SPP model. The MMC size of the SPP was not fragmented (10/10 in most cases). The critical state of the SPP indicated that the system is indecomposable (non-divisible) from the IIT perspective (i.e. the size of the MMC is the same as that of the whole group). This result matched the critical-state property.

Fig. 3
figure 3

Data comparison with SPP. (A) Frequency distribution of the MMC for direction (orange), speed (green), and SPP (blue). The group size is 10. The inset figures represent an example of MMC distribution (coloured orange). (B) Mean \(\text{max}\left\{{\Phi }_{\text{MIP}}\right\}\) for each MMC size (the direction: orange, speed: green, and SPP: blue).

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In contrast, the MMC size of the actual fish data was widely fragmented. The MMC sizes ranged from 2 (minimum) to 10 (maximum). Figure 3B shows the \(\text{max}\left\{{\Phi }_{\text{MIP}}^{\text{dir}}\right\}\) for each MMC. All \({\Phi }_{\text{MIP}}^{\text{dir}, C}\) of each MMC were significantly larger than those of SPPs. This result suggests that each fragmented MMC of actual fish exhibited a critical state. Although our analysis did not prove that the high \({\Phi }_{\text{MIP}}^{\text{sp}}\) s were also critical states (in the original SPP model, the velocity was set to be constant), our results suggest the coexistence of several degrees of critical states at various levels (i.e. subgroups and behaviours). The critical state observed in actual fish may not be a theoretically homogeneous phenomenon (observed in the SPP model) but heterogeneous.

Interestingly, we also found that \({\Phi }_{\text{MIP}}^{\text{dir}, S}\) of an entire set S for all samples (note that set S is not necessarily the main complex) still exhibited larger values than those of the SPP critical states (\(\langle {\Phi }_{\text{MIP}}^{\text{dir}, S}\rangle =0.038\). Welch t-test: t(148.6) =  − 19.2, p < 10−30). The integrity (\({\Phi }_{\text{MIP}}^{\text{dir}, S}\)) for an entire set S also exhibited criticality (the time series of \({\Phi }_{\text{MIP}}^{\text{dir}, S}\) was not Brownian, as in Table 1). The high integrity \({\Phi }_{\text{MIP}}^{\text{dir}, S}\) of the entire group size was consistent with the classical result, which states that the entire group fluctuation was in a critical state. Thus, heterogeneous and homogeneous criticalities can coexist.

Comparison with Boid model

Next, we examine the relationship between max{ΦMIP} and group formation by comparing the results with the Boid model. The Boid model is another widely used framework for studying collective behaviour. In addition to the alignment (averaging directions) used in the SPP model, the Boid model also includes attraction (moving toward distant agents) and repulsion (moving agents apart when too close). By adjusting the parameters for alignment and attraction, it is possible to reproduce various formations observed in collective behaviour, such as swarming, milling, and schooling. This section aims to examine the relationship between group formations and group integrities by comparing Couzin’s model with experimental data53. We will consider the differences between the model and observed data (for more details on parameter settings, see the Supporting information).

The following parameters define group formation:

$$\begin{array}{*{20}c} {P\left( t \right) = \frac{1}{N}\left\| {\mathop \sum \limits_{i = 1}^{N} \widehat{{{\varvec{v}}_{i} }}\left( t \right)} \right\|} \\ \end{array}$$
(4)
$$\begin{array}{*{20}c} {M\left( t \right) = \frac{1}{N}\left\| {\mathop \sum \limits_{i = 1}^{N} \hat{\user2{r}}_{ic} \left( t \right) \times \hat{\user2{v}}_{i} \left( t \right)} \right\|} \\ \end{array}$$
(5)

where \(\widehat{{\varvec{v}}}\) is the unit velocity vector, \({\widehat{{\varvec{r}}}}_{ic}\left(t\right)\) is the unit relative position vector from the centre of mass. P (i.e. polarity) measures the degree of group alignment (i.e. school formation). M (i.e. milling) measures the degree of group milling. P and M do not simultaneously have high values. Swarming formation corresponds to low P and M.

We found that the peaks \(\text{max}\left\{{\Phi }_{\text{MIP}}^{\text{dir}}\right\}\) and \(\text{max}\left\{{\Phi }_{\text{MIP}}^{\text{sp}}\right\}\) never overlapped in the Boid model (Figures S3 and S4). The peak of \(\text{max}\left\{{\Phi }_{\text{MIP}}^{\text{sp}}\right\}\) occurred during milling, and that of \(\text{max}\left\{{\Phi }_{\text{MIP}}^{\text{dir}}\right\}\) occurred within the transition area between schooling and swarming. From the perspective of the Boid-type model, group formations (P and M) and group integrities (\({\Phi }_{\text{MIP}}^{\text{dir}}\) and \({\Phi }_{\text{MIP}}^{\text{sp}}\)) appeared to exhibit an intimate relationship.

However, Fig. 4 from the actual fish school revealed a different story. The max{ΦMIP} values for both cases correlated (see the Supporting Information for different values: Figure S6 and Figure S7). Furthermore, the \(\text{max}\left\{{\Phi }_{\text{MIP}}^{\text{sp}}\right\}\) distribution was significantly different from the Boid distribution. In contrast to the Boid distribution, the speed integrity exhibited a high \(\text{max}\left\{{\Phi }_{\text{MIP}}^{\text{sp}}\right\}\) around the high-alignment area (i.e. ⟨P⟩ ≈ 1). These results suggest that the information structure of an actual fish school cannot be reproduced using a simple Boid-type model.

Fig. 4
figure 4

\(\text{max}\left\{{\Phi }_{\text{MIP}}^{\text{dir}}\right\}\) and \(\text{max}\left\{{\Phi }_{\text{MIP}}^{\text{sp}}\right\}\) with the group formations (P and M). The colour bar represents the log-scale \(\text{max}\left\{{\Phi }_{\text{MIP}}\right\}\). P is a polarity and M is a milling for each data. We listed other heatmaps for log-scale \(\sum {\Phi }_{\text{MIP}}^{\text{dir}}\) (Figure S5) and MMC size (Figure S7).

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Role of MMC individuals

In this section, we again consider only experimental data. Since we found that the time series of max{ΦMIP} was Brownian, we examined whether the survival time of MMC was also a random process. In other words, how long can an individual a (\(\in S\)), once it belongs to the maximum main complex (MMC) at a given time t, say \({M}_{t}\), continue to remain the element of \({M}_{t+\Delta t}\)? The survival distribution can be approximated using a cumulative Weibull distribution as follows:

$$\begin{array}{*{20}c} {S\left( q \right) = exp\left\{ { - \beta \left( {\frac{q}{\gamma }} \right)^{\alpha } } \right\}} \\ \end{array}$$
(6)

where γ is the average lifespan and β and α are the parameters to be estimated. α determines the shape of the exponential distribution. If α ≈ 1, then the decay is exponential (i.e. there is no historical effect on MMC survival). In contrast, if α < 1, the graph is called ‘stretched exponential’ (i.e. the long-tail distribution but not the power law distribution).

Figure 5 shows the lifespan distribution for the actual and shuffled fish data (an average of 100 samples). The shuffled data represents a random shuffling of the MMC time series from each experimental dataset, eliminating historical dependencies. Indeed, the shuffled distribution had no memory effect (i.e. \({\alpha }_{\text{dir}}^{\text{shuffle}}=0.96\) and \({\alpha }_{\text{sp}}^{\text{shuffle}}=1.30\)). By contrast, the actual fish distributions were highly stretched in both cases (\(\langle {\alpha }_{\text{dir}}\rangle =0.61\pm 0.03\) and \(\langle {\alpha }_{\text{sp}}\rangle =0.61\pm 0.02\)). Therefore, although max{ΦMIP} was a Brownian process, the MMC series was not random and exhibited a highly heterogeneous distribution (see other results in Table S3).

Fig. 5
figure 5

Survival probability P(t) of the MMC lifespan. The shuffle dataset decays exponentially compared with the experimental dataset.

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The heterogeneity of MMC lifespan suggests that being an MMC member for each fish may not be uniform. Certain members were highly allocated to the MMC, whereas others were not. The MMC rate is defined as follows:

$$\begin{array}{c}{\mathfrak{c}}_{i}=\frac{\left|\left\{t\in [1, T]|i\in {M}_{t}\right\}\right|}{T}\end{array}$$
(7)

where \({M}_{t}\) is the MMC at time t and T is the entire time of the data series. We refer to \({\mathfrak{c}}_{i}\) as the ‘core rate’ for fish i. For instance, let be S = {a, b, c} (group size is three) and T = 4 and the time series of MMC is given as \({M}_{1}=\left\{a,b\right\}, {M}_{2}=\left\{a,b,c\right\}, {M}_{3}=\left\{a,b\right\}, {M}_{4}=\left\{a,c\right\}\), then \({\mathfrak{c}}_{a}=1,{\mathfrak{c}}_{b}=0.75, {\mathfrak{c}}_{v}=0.5\). The element a is the high core rate but element c is not so high.

Figure 6A shows the core rate for direction (\({\mathfrak{c}}_{i}^{\text{dir}}\)) and the logarithm of the standard deviation of dθi (dsi) through the entire data series (no correlation was observed with mean dθi: n = 70, r = 0.104, p = 0.392. In contrast, the mean dsi (Fig. 6B) was highly correlated with \({\mathfrak{c}}_{i}^{\text{sp}}\): n = 70, r =  − 0.565, p < 10−6). This negative correlation suggests that the fish belonging to the MMC group moved more stably (i.e. less affected by others’ movement) compared with the low-frequency fish. Thus, most individual activities in fish schools occur in a supercritical state (i.e. referring to the disordered state in terms of the noise parameter). To achieve a high group performance, the group requires unaffected individuals (i.e. low SD(dθi) and SD(dsi)) to connect various demands from other fish. This characteristic of the core individuals is distinct from leadership, as their position never places them at the group head in the moving direction (see the cite on Data availability). The position of core individuals changes dynamically throughout the collective behaviour.

Fig. 6
figure 6

Correlation between the core rate (\({\mathfrak{c}}_{i}\)) and fish movements. (A) The direction deviation and core rate for direction (Pearson’s correlation test: n = 70, r =  − 0.536, p < 10−6). (B) Mean speed and the core rate for speed (Pearson’s correlation test: n = 70, r =  − 0.516, p < 10−6). Table S4 shows the results for other parameter settings.

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Discussion

In this study, we applied IIT to an actual fish school (i.e. Plecoglossus altivelis) and two representative models (SPP and Boid). We selected two variables (i.e. dθ and ds) for the IIT analysis. Throughout the SPP analysis, the peak of the group integrity, ΦMIP, corresponded to the phase transition point defined as the noise parameter. This result also aligned with the previous IIT analysis of other criticality systems, which stated that ΦMIP can measure the degree of criticality of the system.

Having identified the meaning of group integrity compared with the SPP, we found that the MMC of an actual fish school comprised proper subsets. This result indicates that many degrees of criticality (e.g. intensity ΦMIP and size of group integrity |M|) exist within the group. This criticism provides a new perspective on traditional criticality arguments that state that a critical system is an inseparable system. Note that our argument does not contradict classical criticality. Instead, our findings indicate that global critical states (i.e. the entire group S) and local critical states (i.e. the subgroup M) can coexist.

We found no evidence that the two groups’ integrities (\({\Phi }_{\text{MIP}}^{\text{dir}}\) and \({\Phi }_{\text{MIP}}^{\text{sp}}\)) were related. Although we also observed the same type of self-similarity and a positive correlation between the two ΦMIP values, there was no significant information transfer (i.e. transfer entropy) between them. This independence may originate from our definition of the group integrity of speed. Peak \({\Phi }_{\text{MIP}}^{\text{sp}}\) corresponded to a particular state (i.e. milling), whereas peak \({\Phi }_{\text{MIP}}^{\text{dir}}\) corresponded to the transition state. However, there may still be more appropriate variables for speed integration. We leave this issue for future research.

Furthermore, by examining the temporally heterogeneous MMC distribution, we found that high-frequency MMC members had small variances in direction and velocity. Our analysis suggested that lower-frequency MMC members behaved more randomly than higher-frequency members in terms of both direction and speed. Regarding critical phenomena, our results indicate that global criticality can be divided into two subgroups: more affected individuals (i.e. those who quickly respond to internal and external perturbations) and less affected individuals (i.e. those who weakly respond to such perturbations). Global criticality may thus be a mixture of different roles within the group.

To provide a more explicit interpretation of our findings, we can draw valuable insights from recent studies on gene expression networks. It is well established that gene networks exhibit critical behaviours, as demonstrated in Boolean network models replicating the ON/OFF dynamics of gene expression56. However, recent research has shifted focus towards the complex interactions underlying gene expression avalanches57,58. For instance, Tsuchiya et al. challenge the classical understanding of critical phenomena, which posits a global phase transition between a stable subcritical phase and an unstable supercritical phase in response to stimuli58. Instead, they propose that gene expression may involve the coexistence of multiple local critical states. Their work highlights how the synchronisation of different subcritical and supercritical genome attractors triggers sandpile-like avalanches6 in gene expression. Such coexistence of critical points has also been observed in tissue differentiation59.

Our results suggest a parallel between these observations and the dynamics observed in collective behaviour. Specifically, we propose that interactions between supercritical individuals (those more affected) and subcritical groups (those less affected) may drive the sandpile-type information avalanches observed in collective motion60,61. This hypothesis points to a potentially significant mechanism underlying information propagation in collective behaviour, where critical transitions emerge from the interplay among the different degrees of affected agents rather than from the specific threshold of a noise parameter in the classical sense.

However, certain issues remain unresolved, such as the validity of the correspondence between a high ΦMIP and a critical state. In this study, we applied this correspondence to SPP results. However, the information process of the SPP is homogeneous (ΦMIP does not depend on Tmax). Actual fish schools are heterogeneous, and ΦMIP varies depending on Tmax. The relatively heterogeneous process of the Boid model also exhibits a high ΦMIP in the transition phase; however, the high ΦMIP originates from the frequent fission–fusion process in the periodic boundary condition. The fish schools in our data were rarely split. Therefore, we cannot directly identify the high ΦMIP of actual fish using the modelling results. This issue arises not from our method but from the fundamental difference between theoretical and empirical criticalities. Our study offers a possible approach to uncovering the detailed dynamics of critical phenomena in living systems. Based on our findings, we should further explore these relationships to bridge the gap between theoretical and empirical critical phenomena.

Methods

Ethics statement

This study was conducted in strict accordance with the recommendations of the Guide for the Care and Use of Laboratory Animals from the National Institutes of Health. The study protocol was approved by the Committee on the Ethics of Animal Experiments at the University of Tsukuba (Permit Number: 14–386). All efforts were made to minimise animal suffering.

Experimental settings

We studied ayus (Plecoglossus altivelis), also known as sweetfish, which are found throughout Japan and are widely farmed. Juvenile ayus (approximately 7–14 cm in body length) display typical schooling behaviour, although adult ayus tend to exhibit territorial behaviour in environments with low fish density. Juveniles purchased from Tarumiyoushoku (Kasumigaura, Ibaraki, Japan) were housed in a controlled laboratory. Approximately 150 fish were kept in a 0.8 m3 tank of continuously filtered and recycled fresh water with a temperature maintained at 16.4 °C, and they were fed commercial food pellets. Immediately before each experiment, randomly chosen fish were separated to form a school of each size and moved to the experimental arena without pre-training. The experimental arena comprised a 3 × 3 m2 shallow white tank. The water depth was approximately 15 cm; therefore, the schools were approximately two-dimensional. The fish were recorded with an overhead grayscale video camera (Library GE 60; Library Co. Ltd., Tokyo, Japan) at a spatial resolution of 640 × 480 pixels and a temporal resolution of 100 frames per second.

Tracking system

We tracked the trajectories of ayu fish schools with N = 10 using seven samples, each with a recording length of 8–12 min. A semi-automatic tracking system (Move2D; Library GE 60; Library Co. Ltd., Tokyo, Japan) was used to obtain positional information. All occlusion events were processed manually. Each frame was set to 1/20 s.

Data treatment for the IIT application

The fish trajectory data were smoothed before obtaining the orientation and speed from the raw positional data. The numpy.convolve function was applied to the trajectory data for one frame (1/20 s) to reduce noise.