Coarse-graining and macroscopic variables.

Consider a discrete-time multivariate stochastic process X taking values in a state space , which we will refer to as the microlevel or microscopic scale of the system, where is the discrete time index. (We generally set specific state values in lower case, and random variables in upper case. When referring to a stochastic process as a whole, we write just X.) Generally, will have some additional mathematical structure, e.g., topological, metric, differentiable, linear, etc. Later we shall specialise to the case where , i.e., N-dimensional Euclidean space with the usual metric and linear vector space structure. In that case, we use standard vector-matrix notation and set vector quantities in bold type; so the microscopic system becomes a multivariate vector process defined by with superscripts indexing components.

Now consider a surjective, structure-preserving mapping from the microscopic state-space to a macroscopic state space . Surjectivity means that for every , there exists at least one element such that . We refer to such a mapping as a coarse-graining. A coarse-graining effects a partitioning of the microscopic state space as a disjoint union: . However, distinct coarse-grainings across scales do not necessarily partition the dimensions of the microscopic state space into non-overlapping sets, meaning that the dimensions could belong to multiple coarse-grainings. This also results in the possibility that coarse-grainings can be nested.

The macroscopic state space will typically be of smaller cardinality or dimension—which we refer to as the scale of the coarse-graining—so coarse-grainings may be considered as dimensional reductions. The coarse-graining M naturally maps the microscopic process X to the macroscopic process (or macroscopic variable) Y defined by ; this will in general entail a loss of information. In the Euclidean case , with , in order to preserve the vector space structure we consider only linear coarse-grainings, so that M becomes an full-rank. Note that the linear mapping is surjective iff M has full rank n.

The DI framework.

DI is a data-driven information-theoretic principle aimed at the identification of emergent coarse-grained macroscopic variables from discrete-valued or continuous-valued time-series data [30]. Intuitively, a macroscopic variable Y given by is dynamically independent if—notwithstanding its deterministic dependence on the microscopic process—it behaves like a dynamical process in its own right, following its own dynamical laws, distinct from the laws governing the dynamics of the microlevel variable X. Note that in general an arbitrary macroscopic variable will not have the property of dynamical independence from the micro-level base.

DI is defined in a predictive sense: Y is dynamically independent of X if knowledge of the history of X does not enhance prediction of Y beyond the extent to which Y already self-predicts. Discovering the macroscopic variables, Y, may be framed as an optimisation problem; namely to minimise, across all coarse-grainings, the objective function of dynamical dependence (DD), an information-theoretic measure of departure from dynamical independence for macroscopic variables:

Definition 1. Dynamical dependence is defined as the transfer entropy [61, 62]—a nonparametric measure of information flow—from the historical past of the microscopic process X to the present state of the macroscopic process, Y at time t:

(1a)

(1b)

We use the notation − , where is not a single value but represents a set of lagged time points, , referring to the history of the process. In the case of an infinite history , this set would extend indefinitely.

Next, we impose a supervenience relation, which asserts that:

(2)

This indicates that there is no additional information in the macroscopic process beyond what is already captured in the history of the microscopic process. Note that a coarse-grained macroscopic variable trivially satisfies (2).

If the process X (and hence also Y) is stationary—i.e., its statistical structure does not change over time—then the DD is not time-dependent, and we drop the subscript t. We assume stationarity for all processes from here on. In practice, to compute dynamical dependence from neurophysiological data such as that simulated by biophysical neural models, we employ an approximation method based on linear state-space modelling (see S2 Appendix of the Supporting Information).

Thus a macrovariable is perfectly dynamically independent if and only if (iff) its dynamical dependence on the microlevel vanishes identically (is zero); i.e., its capacity to predict its future based only on its own history is not enhanced by knowledge of the history of the microscopic process. We define this with a transfer-entropic identity:

Definition 2. A macrovariable Y given by the coarse-graining is dynamically indep- endent of the microscopic process X iff

(3)

This formalises the intuition of an emergent macrovariable as one which behaves as a dynamical process in its own right, independently of the micro-level dynamics. In the language of Bertschinger et al. [63, 64], the macroscopic variable is informationally (or dynamically) closed with respect to the microscopic dynamics. Dynamical independence encapsulates the specific notion of emergence discussed in this article, and dynamical dependence stands as a quantitative information-theoretic measure of the degree of (non-)emergence. That is, emergence is not an all-or-none categorical distinction, but rather, in our framework emergence is a continuous property of a macroscopic variable which is quantified by the dependence of the macroscopic process on the microlevel multivariate process.

In general, given a microscopic process X, there may well be no perfectly dynamically-independent macroscopic variables (i.e., variables for which Eq 3 holds identically) at some, or indeed at any, spatial scale. Instead via an optimisation strategy, we search for macrovariables with minimal DD—i.e., macroscopic variables with a significant degree of emergence. In practice, at any given scale, we attempt to identify macroscopic variables that minimise the DD over the space of all possible structure-preserving coarse-grainings at that scale. Importantly, in an empirical scenario, minimal DD may be quantified in a principled manner: namely, that at some predefined significance level, and accounting appropriately for multiple hypotheses, we cannot reject the null hypothesis that an optimised DD value is zero.

For non-trivial problems like the biophysical neural model considered in this study, optimisation is generally analytically intractable, so we are bound to deploy numerical methods. DD minimisation—which will generally take the form of multiple optimising runs with random initial configurations—will, furthermore, be limited by time and computational resources, so will not be sure to guarantee that globally minimal-DD macrovariables are identified. Indeed, it turns out that numerical optimisation runs are in practice likely to terminate at local suboptima of the DD objective function. We therefore adopt the following pragmatic criterion for the discovery of emergent macroscopic variables:

Definition 3. Emergent n-macros are macroscopic variables with minimal dynamical dependence over all optimisation runs at scale n.

We consider the emergent n-macros across all candidate spatial scales and we define the localisation perspective of emergent dynamical structure as:

Definition 4. Emergent dynamical structure (Localised Perspective) is defined by the set of all emergent n-macros which together reveal the emergent dynamical structure across spatial scales of the whole system.

We note, for example, that dynamical independence is transitive [30], and that emergent macrovariables may thus potentially (but not necessarily) be nested, leading to the possibility of heterarchical emergence structures.

These definitions of an emergent n-macro and the emergent dynamical structure aligns with the ansatz proposed in the original publication [30]. The dynamical structure is revealed by identifying the degree to which the microlevel node contribution to emergent n-macros is distributed. This definition allows us to derive the localisation of n-macros in the original biophysical neural model.

In the context of our study, emergent macrovariables are defined as linear subspaces of the microscopic state space, where each dimension corresponds to a neural source or node in the network. The term localisation refers to the degree to which these macrovariables are aligned with specific axes of the microscopic space. Specifically, a macrovariable is considered localised when it aligns closely with one or a few of the original coordinate axes—meaning it has smaller subspace angles with these axes. This alignment indicates that the macrovariable primarily captures the dynamics of specific microlevel components, rather than being a distributed combination of many nodes. This concept of localisation is crucial for understanding how emergent dynamical structure forms within the network.

The localisation of emergent n-macros within the microscopic state space does not resemble partitioning as understood in network theory–where systems are divided into discrete, non-overlapping modules. Partitioning, in the network-theoretic sense, typically refers to the identification of modular structures based on static connectivity patterns, such as communities or clusters within a graph, often derived from measures like modularity or edge density.

In contrast, the emergent macroscopic variables identified through DI analysis are not mere static partitions or modules. Rather, they represent dynamical subsystems–low-dimensional processes arising from the complex interplay of microlevel dynamics, capturing dependencies that may span traditional module boundaries. It is important to note that, in general, n-macros will not simply correspond to subsets of network nodes. However, in the present study, they are linked to the linear subspace spanned by particular subsets of nodes under particular dynamical conditions. This by design construction is motivated by didactic considerations: when n-macros are associated with subsets of nodes, dynamical independence is clearly defined by the absence of incoming directed connections, making them straightforward to construct. Though this is not always the case, a point that we see by the minimisation of dynamical dependence in noisy regimes. Yet, the former scenario is not universally applicable, which is why our approach also involves examining subspace angles and other measures to capture the broader, inherently dynamic and distributed nature of functional organisation.

Thus, while traditional network partitions capture static network-theoretic dependencies, emergent n-macros as revealed by DI analysis capture functional organisation that is dynamic, distributed, and potentially non-local. Accordingly, DI analysis applied to neurophysiological time-series will not necessarily reveal clearly modularised emergent dynamical structure at any single spatial scale. Instead, the present study develops a method to assess whether functional integration and segregation modulate emergent dynamical structure–via local contributions to dynamical subsystems–across all spatial scales, and how this modulation occurs.

Dynamical dependence minimisation in linear systems.

We operationalise DI and the identification of emergent n-macros in simulated neurophysiological data using a linear approximation (see S1 Appendix and S2 Appendix of the Supporting Information as to the appropriateness of the linear approximation). The microscopic state space is thus the Euclidean space , and coarse-grainings at spatial scale n are full-rank linear mappings ( matrices) .

For the coarse-grained macrovariable , the dynamical dependence (Eq 1a) is invariant under nonsingular (invertible) linear transformations of the macroscopic state space . Thus we may consider two coarse-grainings to specify the same macrovariable if they are related by for some nonsingular linear transformation . The space of all possible linear coarse-grainings at scale n < N—the space over which DD will be minimised—may consequently be identified with the set of n-dimensional subspaces of . This defines a mathematical object known as a Grassmannian manifold [65, 66], written . A coarse-graining M may be visualised as an n-dimensional subspace in the original microscopic state space , on which the dynamics of the macroscopic variable play out. Intuitively, the coarse-graining map M projects the N-dimensional microscopic dynamics down onto the macrovariable , which resides on the associated n-dimensional subspace [30, 67].

Grassmannian manifolds are a type of homogeneous Riemannian manifold; they are non-Euclidean spaces with a distinctive metric geometry and symmetries, on which we can do calculus. Given a microscopic process , then, we can minimise the DD —considered now as the objective function on parametrised by the matrix M—using standard methods like gradient descent [65, 68] on the Grassmannian. The subspace associated with a coarse-graining is spanned by the n basis vectors defined by the columns of the transposed coarse-graining matrix . Exploiting the invariance of DD under transformations of the macroscopic space , it turns out to be convenient to parametrise the Grassmannian by orthogonal matrices M; i.e., matrices for which . These matrices constitute the Stiefel manifold of orthonormal bases, written —also a homogeneous Riemannian manifold—and we consequently refer to the parametrisation of the Grassmannian by orthogonal matrices as the Stiefel parametrisation. We note that for n > 1 this parametrisation is not one-to-one; has dimension − , which for n > 1 is larger than the dimension n(N–n) of . In practice, optimising over the Grassmannian and Stiefel manifolds produce similar results [67], and the latter is simpler to implement.

For the class of linear state-space models for the microscopic dynamics, moreover, both the DD and its gradient may be calculated explicitly; see S3 Appendix of the Supporting Information. In this study, we used gradient descent with an adaptive step-size [69] to minimise DD. Typically, minimisation is initiated at a uniform random M for a specific scale n, and allowed to run until it converges to a specified tolerance. However, as the optimisation landscape described by the DD is quite deceptive, with multiple local suboptima (minima), we repeat randomised optimisation runs many times to obtain acceptable minima for the specified scale.

Statistical significance of dynamical dependence.

In the present work, we focus on the exploration of dynamical dependence values (i.e., transfer entropy or Granger causality in linear settings) quantifying the degree of emergence without imposing a specific statistical cutoff. Having said this, it is worth noting that statistically principled thresholds can be established if needed (see Reference [30], Sec. III E). Given the linear GC operationalisation of dynamical independence [30], one can employ standard statistical tests (e.g., , F, Wald tests) to evaluate the null hypothesis that dynamical dependence is zero. One can interpret failure to reject this null hypothesis (at a given significance level) as a definition of ‘small enough’ dynamical dependence, corresponding to a principled threshold for asserting near dynamical independence. We note that in frequentist statistics, failure to reject the null hypothesis is not evidence for dynamical dependence being zero, hence we interpret this situation as an operational rather than inferential criterion.

In short, one can draw a direct analogy to correlation: just as a correlation statistic quantifies the extent to which two variables deviate from perfect statistical independence, the dynamical dependence measure reflects how far a macrovariable departs from perfect dynamical independence. In both cases, these measures serve as effect sizes for the departure from independence. However, the use of statistical significance tests (e.g., for correlation) is what enables rigorous inference on whether such deviations are meaningful, and similar inference-based methods can be applied to dynamical dependence in empirical contexts.

Characterisation the composition of linear macroscopic variables.

A putatively emergent macroscopic variable is perhaps best thought of as a dimensionally-reduced subsystem of the global microscopic dynamics. Computational neuroscience has a tendency to present functional structure in terms of networks. We stress that in DI, macroscopic variables are, in general, not networks in any meaningful sense. As explained above, in the linear case a macrovariable may be considered a projection of the microscopic process onto a subspace in the original microscopic Euclidean state space. In the context of modelling multi-region neurophysiological time-series data, we may get a sense of the extent to which different nodes (corresponding to neural signals in specific anatomical brain regions) in the brain network participate in, or contribute to, a macroscopic subsystem that we term an n-macro. We do this by invoking geometric intuition: consider, for example, a 2-dimensional plane in a 3-dimensional Euclidean space. The plane is uniquely identified if we know the angles between it and each of the three x,y,z coordinate axes in the 3-dimensional space (Fig 1d).

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Fig 1. Geometric interpretation and data visualisation equivalent describing the DI algorithmic pipeline.

(a) a time-series capturing the dynamics of 3 variables, brain regions, or electrodes. (b) A state-space model is estimated from the time-series dynamics, from which we calculate the pairwise-conditional Granger-causal statistics representing directed information flow. The numeric labels (e.g., 1.75, 4.30, 0.54) indicate illustrative values of the three variables (along the dashed line in panel (a)) and are included here simply to schematically demonstrate how these values appear in the state-space representation. (c) A 2D plane embedded in the 3-dimensional system. This plane represents the lower-dimensional macroscopic variable—a 2-macro—capturing the projection of the 3D dynamics onto a 2D space (dimensional reduction). In optimisation, the plane shifts incrementally toward decreasing levels of dynamical dependence, moving closer to a local or global minimum. (d) The minimally dynamical dependent plane is selected as the emergent 2-macro. To determine the single-node contribution to the emergent 2-macro distances between the axes of the original 3D system, and the axes of our emergent 2-macro are computed. A projection of the emergent 2-macro back onto the G-causal graph results in a visualisation of the degree of contribution from each node.

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In the multi-channel recording scenario, microscopic coordinate axes correspond to recording channels (Fig 1a and 1b). Consider again the 3D example: if, say, the angle between axis y (corresponding to channel y) and the 2D plane associated with a given macroscopic variable at scale is close to the maximum , this tells us that neural activity in region y is projected away by the corresponding coarse-graining; region y does not participate strongly in the macroscopic process. As the angle approaches zero, participation is maximised.

With some caveats, this generalises to arbitrary dimensions: to quantify participation of brain regions in a macroscopic dynamical subsystem we calculate the angles between each of the region axes and the macroscopic subspace; then small angles correspond to high contribution, and vice versa (see S5 Appendix of the Supporting Information for details).

Subspace angles have another important use, as a metric to measure the similarity, or co-planarity between macrovariables. For example, when optimising dynamical dependence for (nearly-)DI n-macros, two gradient-descent runs may yield coarse-grainings and , respectively, which we suspect may be identical, or nearly so. Measuring the angle between and can help us decide how similar they are: angles close to zero indicate high similarity. In the case where we have subspaces and of different dimensions , a subspace angle near zero indicates that is nested in ; that is, the dynamics may be viewed as a self-contained subsystem of the dynamics. Within the current study we only consider subspace angles from microlevel constituents to n-macros and leave the comparison across higher-order scales for future research.

For a summary of key terms used within this paper, please see S6 Appendix of the Supporting Information.

Local dynamics.

The SJ3D neural mass model consists of an excitatory and an inhibitory neural mass, which are interconnected through the fast excitatory variable and the fast inhibitory variable (detailed in Table A in S4 Appendix). These variables, along with the connectivity between the excitatory masses, define the dynamics at each node. The SJ3D model can generate various ensemble dynamics, including excitable regimes, oscillations, and transient spike-bursts [54]. Fig 5 illustrates the construction of the NMM.

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Fig 5. Network architecture of a Stefanescu-Jirsa model.

is the inhibitory-to-excitatory mass connectivity, is the excitatory-to-inhibitory connection, and is the excitatory-to-excitatory connectivity. The left-most panel indicates that each neural mass model, consisting of both, excitatory and inhibitory neural masses represents a single node in a region-based brain network simulation. Going from the central panel to the right-most indicates the reduced model of the mean-field approximations representing the ensemble activity on the excitatory (pink) and inhibitory (green) neural masses, respectively. Created in BioRender. Milinkovic, B (2025) https://BioRender.com/f40j657

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A key advantage of the SJ3D worth noting is its multi-modal construction. Modes represent different dynamical regimes in which the local dynamics can exhibit and represent distinct population dynamics which give rise to rich heterogeneous activity exhibited locally [9, 70, 71].

Global dynamics.

Our simulations are based on a well-established evolution equation governing the dynamics of brain network models, adapted and modified from The Virtual Brain (TVB) for this study (details in S4 Appendix of the Supporting Information). The structural connectivity, which defines the anatomical backbone of the network model, is represented by a weight matrix and a tract-length matrix. In the context of a whole-brain model (WBM), the weights matrix quantifies the strength of pairwise anatomical connections between brain regions, while the tract-length matrix measures the axonal fibre lengths between these regions. Tract-length matrices, which encode conduction delays and attenuations across white-matter pathways, critically shape the global dynamics of large-scale brain network models by modulating the timing among interconnected regions [7, 8]. Longer tracts generally introduce greater delays, potentially shifting synchrony or altering phase relationships, and thus influencing the emergent patterns of whole-brain activity [9].

The 5-node biophysical neural model used here is derived from an empirically-informed TVB structural connectome, which incorporates a biologically realistic tract-length matrix. This matrix is created through homotopical morphing, a computational technique that optimizes and aligns primate tracer imaging data with the human anatomical connectome, based on a Desikan-Killiany parcellation atlas [72]. To evaluate the plausibility of the emergent dynamical structure, constrained by network connectivity, we also perform the same analysis on an uncoupled structural connectome as a control condition. Throughout the experiments, the network connectivity was kept constant across both the coupled and uncoupled regimes, serving as a ground-truth model that constrains the dynamics revealed by varying degrees of functional integration and segregation (see Fig 6).

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Fig 6. Structural connectivities defining the 5-node biophysical neural models implemented in this study.

Weights matrix (left) for the coupled (a) and uncoupled (b) regimes, where each entry indicates the coupling strength between two nodes. Higher values denote stronger interactions. Tracts matrices (right) showing the corresponding inter-node tract lengths (in mm), indicating the physical distance over which signals travel. The numerical values in both matrices originate from diffusion-based tractography analyses, as further described in the main text.

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To reiterate, two global parameters known to influence macroscopic brain dynamics [2, 58, 73] were manipulated: global coupling and dynamical noise. Functional integration is defined by the global coupling factor (G in Eq (1) of S4 Appendix), which scales the influence of incoming activity from other nodes in the network. Conversely, dynamical noise, represented by the term in Eq (1) of S4 Appendix, reflects functional segregation by reducing the signal-to-noise ratio, thereby dampening the impact of external nodes on local dynamics.

Dynamical dependence peaks in a parameter regime balancing functional integration and segregation.

Fig 7 illustrates that in the coupled network, there exists a parameter regime where the DD of emergent 2-macros and 3-macros is higher, indicating lower emergence of macroscopic dynamics—a pattern absent in the uncoupled network. Notably, this regime exhibits a consistent relationship where increases in both functional integration (global coupling) and functional segregation (dynamical noise) coincide. This balance results in the emergent dynamical structure being maximised through the localisation of contributions from specific microlevel nodes to the emergent n-macros, as dynamical noise increases proportionally to global coupling (see Figs 8 and 9).

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Fig 8. The single-node contribution to emergent 2-macros across simulations.

Global coupling varies along the y-axis, and dynamical noise varies along the x-axis. Here, and on the subsequent figures identifying node contribution values on both axes are plotted on a logarithmic scale. Higher contributions are indicated by darker colours. The degree of localisation of single-node contributions within the emergent 2-macro is most distinct at a balance point between functional integration and segregation.

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Fig 9. The single-node contribution to emergent 3-macros across simulations.

Global coupling varies along the y-axis, and dynamical noise varies along the x-axis. Higher contributions are indicated by darker colours. The localisation of single-node contributions within the emergent 3-macro is evident at balanced points between functional integration and segregation, with a loss of localisation in extreme regimes.

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This pattern suggests that the observed regularity across simulations is not merely a result of individual parameter values but rather emerges from the interaction between global coupling and dynamical noise. Their combined influence shapes the system’s dynamical structure, leading to higher DD (lower emergence) and maximised localisation of contributions at this balanced parameter regime. This finding underscores the importance of considering the interplay between functional integration and segregation when assessing both the emergence and the organisational structure of complex macroscopic dynamics.

Identifying structure: Spatial localisation of single-node contributions in emergence n-macros across functional integration and segregation.

We next performed a single-node analysis to examine the contributions of individual nodes to the emergent 2-macros across simulations. As shown in Fig 8 nodes 4 and 5 exhibit high contributions to the emergent 2-macro, while nodes 1, 2, and 3 show negligible contributions, particularly at the parameter regime where functional integration and segregation are balanced. This distinct organisation of node contributions towards an emergent 2-macro which is localised across 2 microlevel nodes is closely associated with the parameter regime characterised by higher DD (lower emergence) values (Fig 7). This indicates that at the balance point, the emergent dynamical structure is maximised through the localisation of contributions from specific nodes.

The degree of single-node contribution is represented as follows: a value of 0 (lighter colour) indicates that the node is not implicated in the emergent 2-macro, and a value of 1 (darker colour) indicates that the node is fully implicated. For a detailed explanation see the worked example in above.

Further analysis, as illustrated in Fig 8, reveals that in simulations where the parameter regime is dominated by either excessive functional integration or segregation, all nodes show varied degrees of contribution to the emergent 2-macro. This variability indicates a breakdown in the localisation of single-node contributions, which underpins the dynamical structure of the emergent n-macros. Notably the dynamical structure of the n-macros appears randomly distributed across the entire network.

By examining single-node contributions across all parameter regimes, we assess the degree of localisation within each n-macro, providing insight into the integrity of the emergent dynamical structure. Importantly, distinct localisation is most apparent in parameter regimes associated with higher dynamical dependence (lower emergence)—that is, at a balanced point of the co-existence of functional integration and segregation. In contrast, in regimes with extreme dynamical noise or global coupling, this localisation diminishes, leading to a more distributed contribution of nodes across the entire network and a subsequent weakening of the dynamical structure.

Similarly, Fig 9 shows that single-node contributions to an emergent 3-macro exhibit a distinct localised pattern at balanced points between functional integration and segregation. In this regime, nodes 1, 2, and 3 contribute significantly to the emergent 3-macro, while nodes 4 and 5 show minimal to no contribution, as reflected by the lighter colours in their respective matrices. However, when the parameter regime shifts towards extreme functional integration or segregation, this localisation of contributions weakens. The result is a more distributed contribution across all nodes, leading to a diminished and less coherent dynamical structure.

Finally, given that the uncoupled network serves as a control, we should expect the minimal DD-valued n-macros in such a system to reflect subspaces that span some combination of the original coordinate axes. That is, they are localised or distributed across some set of nodes. In fact, without the need for simulation, we can theoretically predict that for a macro dimension n, any subspace formed by linear combinations of n of the N original coordinate axes (microscopic state space) should exhibit close-to-zero DD values. Indeed, this expectation is supported by the results obtained in the lower graphs in Fig 7, where the 2- and 3-dimensional emergent macros identified by the optimisation process in the uncoupled system correspond to these close-to-zero DD subspaces.

Further, Fig 10 illustrates that in the uncoupled network, the contributions of microscopic nodes to the emergent 2- and 3-macros vary randomly across the entire parameter space, resulting in the absence of distinct localisation of the dynamical structure of emergent macros, even when functional integration and segregation are balanced. These findings suggest that the anatomical connectivity between nodes in the brain network is crucial in determining the localisation of dynamics and the contribution of nodes to emergent dynamical structure at the macroscopic scale. In the absence of coupling, the emergent dynamical structure is not maximised, and the DD values are close to zero, indicating higher emergence (lower DD) but without the distinct localisation that characterises the maximised emergent dynamical structure in the coupled network.

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Fig 10. The single-node contribution to emergent 3-macros across simulations.

Global coupling varies along the y-axis, and dynamical noise varies along the x-axis. Higher contributions are indicated by darker colours. The contributions of nodes to the emergent 3-macros vary randomly across the parameter space, indicating the absence of distinct localisation in the uncoupled network.

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Statistical significance of single-node contributions to emergent n-macros.

Thus far, we have explored how and when macroscopic dynamical structure emerges through the localisation of single-node contributions to emergent n-macros. To assess whether these single-node contributions to emergent 2-macros and 3-macros across the parameter space are statistically significant compared to the uncoupled brain network (control condition), we performed a Wilcoxon rank-sum test. Specifically, this analysis compares the distribution of node weights across all simulations for the coupled and uncoupled regimes (Figs 8, 9, and 10). The statistical comparison is conducted across the entire parameter space explored, not just within the parameter regime where we qualitatively observe distinct localisation of the emergent macroscopic dynamical structure. By performing the statistical comparison across the full parameter space, we ensure a robust and comprehensive assessment of the significance of single-node contributions to emergent macroscopic dynamical structures, thereby avoiding potential biases that could arise from selectively focusing on regions where qualitative observations suggest distinct localisation.

Consulting Fig 11, the Wilcoxon rank-sum test reveals a significant difference in the contribution of node 4 to an emergent 2-macro in the coupled brain network compared to the uncoupled brain network used as a control condition (). No significant contribution was observed from any other nodes. Interestingly, despite the qualitative illustration in Fig 8 showing distinctly higher contributions from node 5 to the emergent 2-macro, this node did not exhibit statistical significance when compared to the control Fig 10. This discrepancy might be due to the statistical analysis being conducted across the entire parameter space, rather than being confined to the regime where distinct localisation is observed. However, it is curious that node 4 still shows significance under the same conditions, suggesting that the lack of significance for node 5 may not be fully explained by this alone.

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Fig 11. Wilcoxon rank sum test for single-node contribution.

(Left) Emergent 2-macro and (Right) Emergent 3-macro in the coupled network versus the uncoupled network, evaluated across a parameter sweep of global coupling and dynamical noise. Statistical significance is indicated by star symbols in the plots: one star () denotes a pâ€#144;value less than 0.05, two stars () represent a p-value less than 0.01, and three stars () indicate a p-value less than 0.001. Note that the z-score is near zero for node 5 in the left plot

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Similarly, a Wilcoxon rank-sum test reveals a significant difference in the contributions of nodes 1 () , 2 (), and 3 () to an emergent 3-macro in the coupled network compared to the uncoupled network. In contrast, nodes 4 and 5 do not show significant contributions.

Localisation of emergent n-macros: Defining the emergent dynamical structure.

Our results demonstrate that when the co-existence of functional integration and segregation are finely balanced, the emergent macroscopic dynamical structure is maximised through the localisation of single-node contributions to the emergent n-macros. Specifically, at these balanced points, the whole-system dynamics exhibit distinct patterns of single-node contributions to the emergent n-macros, rather than random or evenly distributed patterns. This maximised localisation defines the emergent dynamical structure at the balanced points of integration and segregation.

This distinct structure is evident from Figs 8 and 9, where nodes 1, 2, and 3 show negligible contributions to the emergent 2-macro, while nodes 4 and 5 contribute significantly to the emergent 2-macro at the balanced points. Conversely, for the emergent 3-macro, nodes 4 and 5 show negligible contributions, while nodes 1, 2, and 3 contribute significantly. Statistically, we do not have a definitive explanation for why node 5 does not show significant contributions in certain cases while node 4 does (see Fig 11). We suspect that there might be a hidden bias in the optimisation procedure used for the DI analysis, particularly in the case of the uncoupled network. In theory, the optimisation should identify 2-macros that are combinations of pairs of coordinate axes uniformly across all pairs. However, it is possible that the procedure is biased towards certain pairs more than others. This issue requires further investigation into the optimisation procedure across the coupled and uncoupled networks to fully understand the underlying reasons. Moreover, when moving away from the balanced points—either by increasing functional integration or segregation—the lack of distinct, localised single-node contributions to emergent n-macros leads to varied, distributed contributions from all micro-level nodes. This distributed nature induces a loss of distinct dynamical structure over the micro-level nodes that is otherwise observed at the balanced points. Interestingly, this loss of distinct structure is accompanied by relatively lower values of dynamical dependence (DD) for emergent n-macros, suggesting higher emergence of macroscopic dynamics.

While it might initially seem counterintuitive that a distinct, localised emergent dynamical structure occurs alongside lower emergence of macroscopic dynamics (as indicated by higher DD values), this observation remains consistent with the DI framework. At the balanced points, the DD is higher, indicating that the macroscopic dynamics are less emergent and more dependent on the underlying micro-level dynamics. The distinct structure does not necessarily imply that the emergent n-macros should possess a greater degree of dynamical (informational) closure. Crucially, the dynamical closure of the emergent n-macros is determined in relation to the microscopic processes alone, and not in comparison to other n-macros discovered across spatial scales.

Furthermore, lower dynamical dependence (higher emergence) does not necessarily mean that the emergent n-macro predicts itself well, i.e., that it is self-determining or autonomous (see [30]). Rather, it indicates that the macroscopic dynamics are more independent from the microscopic base. Through the same optimisation procedure, one could, in principle, reveal an entirely Gaussian, white-noise process as an n-macro that is independent of the micro-level constituents. Consequently, these ’noisy’ macroscopic variables might not be dynamically relevant for the whole-system dynamical structure.

However, while such a macroscopic variable might initially seem of limited interest, identifying these white-noise macros could be useful depending on the empirical question. They can potentially be factored out, allowing researchers to focus on the core, non-trivial dynamical structures that are more informative. From the results obtained, we suggest that the lower dynamical dependence (higher emergence) values observed when increasing dynamical noise (mediating functional segregation) could be driven by the emergence of such noise-dominated macros.

Consequently, we speculate that emergent macroscopic variables accompanied by relatively higher dynamical dependence (lower emergence) could be expected at the balanced points. This suggests that for the emergent n-macros to hold any dynamical relevance within the whole-system dynamics, some degree of dependence between the macroscopic process and the microscopic process may be necessary. However, it is crucial to carefully assess the functionality and significance of the n-macros, recognising that not all identified macros may contribute meaningfully to the overall system dynamics.

Lastly, Fig 10 indicates that the parameter regime associated with the emergence of macroscopic dynamical structure is absent in an uncoupled network. This finding underscores the importance of interactions between micro-level constituents in driving the emergent macroscopic patterns of activity in neural models. In this study, we focused on the contribution levels of each node to the dynamics of the emergent n-macros, comparing these contributions with those in the uncoupled network using a Wilcoxon rank-sum test.

Beyond indices: Uncovering the dynamical structure of complexity and emergence.

Our approach informs existing intuitions about organisational complexity in both computational [77, 78] and biological [79, 80] systems. Furthermore, it explores the association between organisational structure and the optimal balance between integration and segregation [60, 81, 82]. Our methodology uniquely provides a robust operational approach to identify the dominant dynamical structures underlying global brain states, which are often indexed by measures of criticality [6, 21, 83] and neural complexity [60, 84–87].

An attempt to clarify the relationship between the integration-segregation balance and criticality has been attempted [25]. In general, criticality is commonly associated with systems at a phase transition, often characterised by power-law dynamics [88]. Theoretically, criticality refers to the point at which a system transitions between different phases, typically marked by scale-free properties [6]. However, the term “criticality” can be somewhat ambiguous, as it is empirically measured in various ways [6, 83], often alongside measures of complexity or signal diversity [89].

Both measures of criticality and complexity serve as indices that refer to underlying organisational structure within the system. For instance, the power-law structure, such as the 1/f curve, indicates that the system exhibits scale-free organisation. Similarly, measures of neural complexity aim to index the degree of statistical structure within in neurophysiological data.

While remaining agnostic to specific measures of criticality or complexity, our work seeks to vary the two parameters that are believed to influence both, with the aim of providing a complementary method that goes beyond empirical indices to identify the underlying dynamical structure and its level of emergence. Further, we would like to emphasise that the association between emergence, criticality, and complexity is an historical association rather than a principled one. This is primarily due to both taking emergent phenomena as their object of inquiry. Although criticality identifies critical regimes as their emergent phenomena of inquiry, within our framework dynamical dependence has a quite different definition, primarily based on dynamically independent subsystems (macroscopic variables). An exciting avenue for future research will be to compare our method with other criticality or neural complexity measures—which actually peak during a balanced point of integration and segregation—to explore how these measures deviate from each other in a principled and practical way. Characterising the emergent dynamical structure of global brain states in relation to these indices, using both observational and perturbed datasets, could offer significant advancements in our understanding of complexity in neuroscience.

Building on existing studies that explore the macrostates of brain activity [31, 37, 39, 90, 91], as well as those quantifying synergy between brain region pairs [49, 51, 92], our approach offers a complementary perspective by uncovering the macroscopic dynamical structure derived from microscopic interactions and quantifying its dependence on these interactions. Unlike other views of emergence, such as synergy, which captures a specific type of higher-order interactions, DI provides a framework for understanding dynamical closure at the macroscopic level. Thereby identifying dynamical subsystems that might serve as independent communication subspaces for regions. This approach not only reveals emergent macroscopic variables across spatial scales but also highlights the system-wide organisation that arises from the underlying dynamics. Integrating these tools to form a comprehensive understanding of emergence in the brain represents an intriguing direction for future research. Further research will need to expand the application of our framework to larger artificial networks or neurophysiological data.

Bridging emergence, coarse-graining, dynamical closure, and dimensionality reduction.

This work contributes to the broader effort to (i) quantify and detect coarse-grained macroscopic dynamics, and (ii) provide a precise methodology for further application to large-scale neurophysiological data. In particular, we consider the relation of our approach to effective information-based measures of causal emergence [46, 75], informational closure [63, 64, 93], and common dimensionality reduction techniques [67, 76].

First, we examine the distinction between coarse-graining in the DI framework and coarse-graining within the context of effective information-based causal emergence [46, 75, 94, 95]. In causal emergence, coarse-graining involves recasting a complex network into non-overlapping partitions using a hard-partitioning function and evaluating the effective information of the resulting higher-order network. Effective information is measured by balancing the average degeneracy and determinism within the network. A partitioned graph that exhibits higher effective information than the original is considered causally emergent. This approach can be applied even without interventionist methods of causality [96, 97], extending across various bidirectional networks [75, 98].

In contrast, coarse-graining within the DI framework focuses on the degree of informational or causal closure of a macroscopic variable, aligning more closely with the principles of statistical mechanics. Instead of partitioning the network into sub-networks, this approach involves partitioning the microscopic state-space (which partitions the dynamics, not the nodes) into subsets , where Y represents the macroscopic states. This method captures how individual microlevel nodes contribute to these macroscopic variables, which is much closer aligned to statistical mechanics descriptions macroscopic (ensemble) properties emerging from microscopic interactions [8, 99]. DI serves as a dimensionality-reduction technique that reveals low-dimensional dynamics as self-contained systems, without transforming the network into hierarchical structures. Given its ability to capture the distributed nature of macroscopic variables and their degree of dependence on the microlevel, DI aligns more closely with a heterarchical perspective of dynamical structure [100].

In support of a dynamical closure perspective on the emergence of macroscopic dynamics in the brain, it is important to recognise that a core principle of brain organisation is its function as a highly distributed information system [102–104], where local (microlevel) functional units integrate to generate macrolevel dynamics. These dynamics are not fixed but fluctuate over time, involving the transient recruitment of various microlevel regions [104]. This suggests that the brain’s global dynamics are driven by overlapping compositions of regions, without clear distinction between the microlevel parts and their relation to the whole. Regions within the brain are dynamically organised in response to functional and computational demands, reflecting the adaptive and non-static nature of brain dynamics across spatial scales. Although we worked on a 5-node simulated network in which these challenges are not present, our approach can adapt to the challenges of larger or real brain networks because it emphasises dynamical closure, focuses on the emergent dynamical structure, and examines to what degree microlevel nodes contribute to macroscopic dynamics—allowing for us to assess the degree of localised or distributed activity.

DI distinguishes itself from other formal approaches to informational closure, such as those by [105] and [64], by extending the concept beyond absolute informational closure (perfect dynamical independence) and accommodating non-Markovian dynamics. This allows DI to capture the nuanced interdependence between macroscopic and microscopic processes.

While [93] explore informational closure as a framework for conscious processing, our approach remains neutral on such interpretations. Additionally, unlike Chang and colleagues’ consideration of direct information flow between macroscopic and environmental variables, our framework imposes a supervenience condition (See Eq 2), ensuring that no new information emerges at the macroscopic level beyond what is determined by the microscopic processes. This means that the microscopic fully dictates the information shared with the macroscopic dynamics.

Finally, although DI results in dimensionality reduction, it offers distinct advantages over commonly used techniques such as Principal Components Analysis (PCA) and t-distributed Stochastic Neighbour Embedding (t-SNE). PCA identifies components of maximum variance, and t-SNE preserves local relationships between data points [67, 76, 106]. However, neither method explicitly accounts for the time-dependence of neural activity to capture low-dimensional dynamics. In contrast, DI operates as an information-theoretic dimensionality reduction technique, directly mediated by microlevel interactions by the temporal structure of the time series.

Comparison with synergy-based causal emergence.

A separate but related flavour of causal emergence—utilising a Partial Information Decomposition (PID) framework [85]—captures emergence based on synergistic capacity [47]. Specifically, Rosas et al. propose a measure of emergence defined through the synergistic component of interactions within multivariate systems, explicitly quantifying causal emergence in terms of synergy within the PID framework. This approach has gained increasing attention among practitioners, demonstrated by recent applications to neural data [49, 51, 101].

Like Dynamical Independence (DI), causal emergence conceptualises emergence as a fundamentally multivariate phenomenon, and measures it on a continuous scale. Note that within our DI framework, zero dynamical dependence () corresponds to perfect dynamical independence or perfect emergence, and increasing dynamical dependence values indicate reduced emergence.

The principal difference between DI and causal emergence stems from their distinct theoretical foundations rather than from specific methodological choices. While causal emergence is fundamentally mereological, emphasising part-to-whole relationships, DI is based on informational or dynamical closure, highlighting the degree of independence of emergent dynamical subsystems from the micro-level dynamics. This foundational distinction leads to contrasting operationalisations: causal emergence evaluates how macroscopic variables improve predictions of micro-level processes (macro-to-micro), whereas DI identifies macroscopic variables that minimise predictive dependence on micro-level processes (micro-to-macro).

To summarise, causal emergence measures emergence by the gain in synergy, with higher values indicating more emergence, whereas DI is operationalised through dynamical de-pendence (closure), with lower values corresponding to greater emergence. This inverse relationship underlines the deliberate analogy between dynamical independence and statistical independence emphasised in Methods Section Statistical significance of dynamical dependence.

Limitations and open questions.

DI, in it’s current framework, is based on Granger causality. Although Granger causality is well-defined for non-stationary processes, it is notoriously difficult to estimate in these cases, and as such is not considered here. Thus we assume the wide-sense stationarity of neurophysiological data, and might not be able to capture physiologically-relevant non-stationarities in the time-series strongly correlated with specific brain states. An example can be illustrated by brief neural oscillations (e.g., sleep spindles), or waves (e.g., K-complexes, or epileptic activity) as well as transient responses to external or internal stimuli. However, linear approximations have been argued to optimally capture macroscopic neural dynamics [107], particularly in resting-state activity.

Furthermore, fitting a VAR or SS model for G-causality estimation assumes a linear model rather than a linear process, which is a subtle but important distinction. While the model assumes linearity in its structure, this does not necessarily mean that the underlying time series must be linear. The presence of non-linearities in the data does not inherently imply that the fitted model will be unstable (see [44] for an in-depth discussion); rather, it suggests that the model may not fully capture the effects of those non-linearities. In fact, if the SS or VAR model is stable, then Granger causality inference and DI analysis remain well-defined.

A more precise understanding is that the model, if stable, induces a linear representation of the underlying data. The key issue then becomes whether this induced linear model can sufficiently account for the non-linearities present in the time series. Wold’s decomposition theorem [108] guarantees that any stationary process can be decomposed into a linear model, although this model may require an infinite order, making it potentially non-parsimonious for time series generated by a nonlinear process [109]. Thus, the impact of nonlinearity on G-causality estimation is nuanced and complex. The critical question remains whether the linear model, when applied to nonlinear data, provides a sufficiently accurate representation of the underlying dynamics to make valid inferences. While some research suggests advantages of linear models [107], this remains an important area for future research.

One limitation of our study is that the neural networks we employed are not functionally specialised–they are not designed to perform specific tasks or processes. In biological brain networks, functional specialisation is a fundamental characteristic that influences how integration and segregation manifest in neural dynamics. Therefore, it is possible that functionally specialised systems might exhibit different patterns of dynamical dependence and emergent dynamical structures. To address this, future research could explore simulations of networks with functional specialisation, perhaps by incorporating task-specific modules or connectivity patterns. Testing our methods in such simulated environments would help determine whether the observed results generalise to systems that more closely resemble the functional organisation of the brain.

Our study employs a relatively small 5-node neural model, which allows for detailed exploration of emergent n-macros. This work in part serves to motivate future applications of these methods to larger, functionally specialised systems. In ongoing work, we have begun to extend our techniques to whole-brain EEG data, performing optimisation on 46-region source-reconstructed EEG data from 14 subjects under anaesthetic conditions, requiring approximately two hours of computation time per subject, Furthermore, by employing the CPSD approach instead of the state-space modeling approach (see S1 Appendix of the Supporting Information) on 68-region source-reconstructed EEG data spanning a full sleep cycle (approximately 6–8 hours), we achieved 100 optimisations for each macroscopic scale (ranging from 2 to 10) per participant in about 90–120 minutes. These computations were conducted on the SPARTAN HPC cluster, with participants processed in serial order and macroscopic variables handled in parallel. This progression not only addresses the computational challenges associated with larger models but also brings us closer to understanding emergent dynamical structures in more realistic neural systems.

Ultimately, the dynamical relevance and implications of emergent n-macros for the systems under study remain an open empirical question. It is plausible to consider that emergent n-macros with many equally implicated contributions from microlevel nodes could function as low-dimensional communication subspaces through which higher-order interactions are mediated [110]. These n-macros, by capturing distributed contributions across the network, may represent the channels through which complex, coordinated dynamics occur at a macroscopic level. With DI we can establish the degree of localisability or distribution of these communication subspaces (n-macros). This concept of communication subspaces presents a compelling direction for future research.