Partial Information Rate Decomposition

Daniele Marinazzo; Gorana Mijatovic; Laura Sparacino; Leonardo Ricci; Luca Faes; Sebastiano Stramaglia; Yuri Antonacci

REVIEW 3 major objections 5 minor 30 references

Partial Information Rate Decomposition

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read PIRD is a dynamic extension of partial information decomposition that splits the mutual information rate of Gaussian processes into unique, redundant, and synergistic parts via a frequency-wise minimum redundancy rule.

desk verdict A genuine spectral PID for dynamic Gaussian systems, but the N>2 generalization rests on a deferred proof; worth refereeing. read the letter →

arxiv 2502.04550 v3 pith:XLWCTIMZ submitted 2025-02-06 stat.ME

Luca Faes , Laura Sparacino , Gorana Mijatovic , Yuri Antonacci , Leonardo Ricci , Daniele Marinazzo , Sebastiano Stramaglia This is my paper · ORCID

classification stat.ME MSC 62M1062B1094A17

keywords partialinformationdecompositionmutualratespectralredundancysynergyGaussianprocessesvectorautoregressivemodelsclimatetimeseries

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces Partial Information Rate Decomposition (PIRD), a framework that decomposes the mutual information rate between a target random process and a set of source processes into unique, redundant, and synergistic information rates. It argues that the standard static partial information decomposition (PID) misrepresents high-order dynamic interactions whenever the analyzed time series carry temporal correlations, because it treats each time point as an independent observation. PIRD instead applies the PID lattice to information rates and defines redundancy as the frequency-wise minimum of spectral mutual information rates, so that Möbius inversion recovers the individual atoms. For stationary Gaussian processes the decomposition is computable from the joint power spectral density, and it reduces to static PID in the i.i.d. case and to a PID of joint transfer entropy in strictly causal cases. A sympathetic reader should care because PIRD offers a principled dynamic replacement for a method that is widely applied, often incorrectly, to time series from networks.

What carries the argument

The load-bearing object is the spectral redundancy rate $i^\cap_{X_\alpha;Y}(\omega)=\min_{j=1,\dots,J} i_{X_{\alpha_j};Y}(\omega)$, the pointwise minimum at each frequency of the spectral mutual information rates between the target and each source in the atom. Integrating this quantity over the normalized angular frequency yields the redundancy rate $I^\cap_{X_\alpha;Y}$, and Möbius inversion of Eq. (4) on the PID lattice turns it into the unique, redundant, and synergistic information-rate atoms. The spectral mutual information rate itself is computed from the power spectral density of the joint Gaussian process, which is what makes the framework operational for vector autoregressive models and linear Gaussian systems.

What would settle it

Fit a three-source stationary Gaussian vector autoregressive model in which two sources carry the same message to the target in two disjoint frequency bands and the third source is a frequency-dependent linear mixture of the other two; compute the redundancy rate of Eq. (6) for every lattice atom, apply the Möbius inversion of Eq. (4), and check whether any atom is negative. A single negative atom, or any violation of monotonicity of $I^\cap_{X_\alpha;Y}$ under adding sources, would show that the spectral minimum redundancy function is not valid for $N>2$.

Watch

Extended reading notes

Core claim

On its own terms, the paper claims that for a jointly stationary Gaussian process $S=\{X_1,\dots,X_N,Y\}$, the mutual information rate $I_{X;Y}$ can be written as a sum over the PID lattice of information-rate atoms, with each atom obtained by Möbius inversion of a redundancy rate function $I^\cap_{X_\alpha;Y}$. The redundancy rate is defined spectrally as the integral over frequency of the pointwise minimum, across the sources in an atom, of the spectral mutual information rates $i_{X_{\alpha_j};Y}(\omega)=\frac{1}{2}\log\frac{|P_{X_{\alpha_j}}(\omega)|P_Y(\omega)}{|P_{[X_{\alpha_j}Y]}(\omega)|}$. In simulations of a two-source network with only time-lagged effects, the paper shows that static PID reports the lag-zero information as predominantly redundant, while PIRD reveals increasing total information rate and expected net synergy as common-target coupling grows. In the climate application with SOI as target, PIRD brings out redundant and synergistic contributions of the North Tropical Atlantic index that static PID makes negligible, and surrogate data show that temporal correlations cannot be ignored.

Load-bearing premise

The paper assumes the spectral pointwise minimum in Eq. (6) is a valid redundancy measure across the whole lattice: nonnegative, symmetric, equal to the source's own information rate for a single source, and decreasing when sources are added; the proof is deferred to the companion paper, and if the assumption fails the decomposed atoms can become negative or meaningless.

Editorial extensions

If this is right

Static PID applied to zero-lag variables can mischaracterize temporally correlated dynamics as redundant even when the true dynamic interaction is synergistic; PIRD includes the full temporal structure and corrects this.
In strictly causal systems with no target-to-source feedback, PIRD reduces to the PID of the joint transfer entropy from all sources to the target, linking dynamic information decomposition to established causality measures.
Because redundancy is defined frequency by frequency, PIRD can be restricted to predefined oscillation bands, enabling band-specific analysis of oscillatory networks.
For stationary Gaussian processes the decomposition is estimated from VAR parameters, so it is computationally light and directly applicable to real multivariate time series.
Surrogate tests in the paper show PIRD values can be either larger or smaller than static PID, so time-lagged and instantaneous effects cannot be separated by a simple adjustment.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the frequency-wise minimum suggests a band-limited redundancy measure, so one could ask whether redundant information is carried by slow or fast oscillations in a given network.
Beyond the paper, a natural stress test is to apply PIRD to nonlinear stationary processes such as coupled chaotic maps, where the Gaussian spectral formula is an approximation, and check whether all atoms remain nonnegative.
Beyond the paper, the conservative spectral minimum implies that when two sources carry identical information at disjoint frequencies, PIRD will label that information unique rather than redundant, a behavior worth explicit study in frequency-multiplexed systems.
Beyond the paper, one could define a significance test for individual PIRD atoms using block-permutation surrogates that preserve the spectral density, which would make the decomposition hypothesis-testable in practice.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

PIRD is the first dynamic PID for Gaussian processes that I'd actually consider using. The idea is simple and correct in outline: translate the Williams-Beer redundancy lattice from random variables to stationary processes by defining redundancy at the level of spectral MIR, taking the pointwise minimum across frequencies. This gives a decomposition that reduces to static PID in the i.i.d. case and to joint transfer entropy PID in the strictly causal case. The authors show with a VAR model that zero-lag PID can be badly misled by temporal correlations, and the climate example is a reasonable demonstration.

What's genuinely new is the spectral minimum redundancy rate; the lattice translation itself is straightforward once you have that. The paper is honest about deferring the axiom proof to a companion preprint, and it does ship code and a reproducible simulation setup. That counts.

The soft spots are real but not fatal. The biggest is that the axioms of the redundancy rate -- symmetry, self-redundancy, monotonicity, and the implied nonnegativity of Möbius atoms -- are asserted, not proven here. For N=2 you can verify nonnegativity by hand, but the paper doesn't. For N>2, a monotone function on the lattice is not enough; Möbius inversion of a monotone but not modular function can go negative. So the claimed generalization to N>2 is unsupported in this manuscript. The simulations and the climate application only use two sources, so they never exercise the nontrivial lattice. This is the load-bearing gap, and the stress test is right about it.

The circularity concern is overblown. The decomposition follows by definition from the redundancy function; that's how all PID works. The validation is qualitative -- expected trends in a designed VAR -- but that's typical for methods papers and not a flaw.

The surrogate analysis in the climate section is appropriate and partially addresses error-bar concerns; I wouldn't push on that. The missing N>2 validation and the proof discrepancy are the honest reasons for a conditional rather than clean verdict.

Who's it for: anyone doing PID on time series in neuroscience, physiology, or climate. It deserves a serious referee. I'd accept it for review and ask the authors to either include the axiom proof or make the companion paper part of the submission, and to add one N>2 example. If the companion proof is watertight, this is a solid methods contribution.

Referee Report

3 major / 5 minor

Summary. The paper proposes Partial Information Rate Decomposition (PIRD), a dynamic extension of Partial Information Decomposition for stationary random processes. It defines the information shared between a scalar target process Y and N source processes through a redundancy rate constructed as the frequency-wise minimum of spectral mutual information rates (Eqs. 5-6), and then obtains unique, redundant, and synergistic information-rate atoms by Möbius inversion over the PID redundancy lattice (Eqs. 2-4). For Gaussian processes, spectral mutual information rates are computed from power spectral densities via Eq. (7). The framework is illustrated on a two-source Gaussian network and on six triplets of climate indices, and it is claimed to reduce to static PID for i.i.d. processes and to transfer-entropy-based PID in strictly causal cases.

Significance. If the construction is valid, PIRD is a useful and computationally tractable dynamic counterpart to PID for stationary Gaussian processes, and the frequency-domain formulation offers the attractive possibility of band-limited analyses. The paper has concrete strengths: a closed-form spectral implementation, explicit reduction to known static and causal special cases, an openly available codebase, and a real-data demonstration on climate indices. The principal weakness is that the axiomatic validity of the spectral-minimum redundancy rate—especially nonnegativity of all Möbius atoms for N>2—is asserted and deferred to a companion paper rather than proved or demonstrated in the present manuscript.

major comments (3)

[Definition of redundancy rate, Eqs. (5)-(6) and following paragraph] The claim that the frequency-wise minimum in Eq. (6) defines a redundancy rate that is nonnegative and satisfies symmetry, self-redundancy, and monotonicity is the load-bearing premise of the entire construction, because Eq. (4) then produces all atoms by Möbius inversion. This property is not proved in the manuscript; the text says only that the rate is shown in [14] to be bounded by the minimum time-domain MIR and that it preserves nonnegativity and satisfies the main axioms. For N=2 the needed nonnegativity follows from pointwise inequalities, but for N>2 the redundancy lattice is not a chain, and monotonicity of a lattice function does not by itself guarantee nonnegative Möbius atoms—on the Boolean lattice, for example, the monotone function f(S)=1_{|S|>=2} has a negative atom at the full set. Thus the central N>2 generalization is currently unsupported by the present text. Please include a self-contained proof, or state the exact theorem with full hypotheses and a proof outline, so that the Möbius inversion in Eq. (4) is justified for all lattice nodes.
[Simulation and application sections] All numerical evidence in the paper uses N=2 sources: the simulated network has two source processes, and the climate analysis is restricted to triplets (a target plus two sources). The nontrivial region of the redundancy lattice, in which antichains with more than two entries appear and the Möbius inversion is genuinely alternating, is therefore never exercised. I request at least one N=3 or N=4 Gaussian example with known model parameters, reporting all atoms and checking both nonnegativity and the marginal constraints in Eqs. (2)-(3). This is the minimum validation needed to make the claimed generalization beyond N=2 credible.
[Summary/limitations paragraph] The statement that the Gaussian restriction does not invalidate the analysis of nonlinear systems because stationary random processes have a linear (albeit infinite-order) representation is too strong. The Wold representation is linear in the innovations, but the innovations need not be Gaussian, and Eq. (7) computes a spectral mutual information rate that is exactly the MIR only under joint Gaussianity. For non-Gaussian stationary processes, the power spectral density does not determine the mutual information rate. Please restrict the claim to Gaussian processes or provide a separate argument explaining why the spectral MIR formula remains valid beyond Gaussianity.

minor comments (5)

[Notation in Eqs. (2)-(4)] The verbal description of A as the collection of subsets of sources such that no subset is a superset of any other is imprecise; what is meant is the lattice of antichains of nonempty subsets of {1,...,N}. Please correct the definition and use notation that distinguishes a source subset such as {i,j} from an antichain such as {{1},{2}}.
[Companion paper [14]] Properties that are central to the method, including the inequality I^∩_{Xα;Y} <= min_j I_{Xαj;Y}, the frequency-band decomposition, and the proof of the redundancy axioms, are repeatedly attributed to [14]. Please state explicitly which claims are proven in the present manuscript and which are proven only in the companion paper.
[Fig. 1 caption] The notation R(Y(tn); X(tn)) and S(Y(tn); X(tn)) is easy to confuse with the source processes X1 and X2; consider denoting the static redundant and synergistic components with subscripts, for example R_static and S_static, to avoid ambiguity.
[Climate application] The statement that the importance of NTA is elicited only using PIRD rests on point estimates; no confidence intervals or statistical tests are reported for the PID values, so this claim should be softened or accompanied by interval estimates for both PID and PIRD.
[Typo and assumptions] There is a typo in the ENSO application section: 'athmospheric' should be 'atmospheric'. In addition, Eq. (1) should state the stationarity and regularity assumptions under which the limit defining the mutual information rate exists.

Circularity Check

1 steps flagged · score 4.0 of 10

The algebraic decomposition is not circular, but the N>2 validity of PIRD rests on a load-bearing self-citation to the companion paper [14] for the redundancy axioms.

self citation load bearing [Paragraph after Eq. (7), redundancy-rate definition section]
"Nevertheless, the proposed redundancy rate preserves non-negativity, satisfies the main axioms of redundancy measures (i.e. symmetry, self-redundancy, and monotonicity) [1], and offers the interesting possibility to perform PIRD focusing on predefined frequency bands with practical meaning [14]."

The central claim that PIRD yields valid nonnegative atoms for any number of sources depends on the redundancy rate satisfying the PID axioms. The paper does not prove this property here; it asserts it and defers to [14], a companion preprint by the same authors. The Mobius inversion in Eq. (4) produces meaningful atoms only if that axiom claim is true, and monotonicity alone does not guarantee nonnegative Mobius coefficients on the full redundancy lattice. Thus the load-bearing premise of the derivation is carried by a self-citation rather than by a proof or an independently verified result inside this paper.

full rationale

No equation-level circularity is present in the main construction: the PIRD atoms are obtained from the spectral-minimum redundancy rate by Mobius inversion, which is an algebraic identity given the definitions in Eqs. (4)-(6), not a fitted parameter renamed as a prediction. The simulations use true parameters imposed in Eq. (8) and compare against qualitative expectations from network structure, so they are not circular validation. The climate application is demonstrative rather than a test of the framework. The only circularity-adjacent issue is the load-bearing reliance on the authors' companion paper [14] for the redundancy axioms and for the N>2 nonnegativity guarantee, with no proof reproduced in the present manuscript. Because the framework still has independent content and the central decomposition is definitional rather than fitted, the appropriate score is 4.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The central claim rests on one ad-hoc definitional choice, the spectral min-MI redundancy rate; one explicit domain assumption, stationary Gaussian processes; and two standard background assumptions, lattice PID theory and Wold decomposition. No data-derived constants are used to construct the framework. The real-data PIRD values do depend on estimated VAR parameters, but that is an estimation step, not part of the formalism.

assumptions (4)

ad hoc to paper The pointwise minimum of spectral MIR functions in Eq. (6) is a valid redundancy rate for the PID lattice.
This is the central definitional choice. The text states it preserves non-negativity and satisfies the main axioms, but the proof is deferred to companion paper [14].
domain assumption The analyzed processes are jointly stationary and Gaussian for the spectral MIR formula in Eq. (7) to apply.
Explicitly stated in the implementation section; the real-data analysis also assumes Gaussianity, as in prior ENSO studies [25,26].
standard math The Williams-Beer lattice formalism for PID extends unchanged to entropy rates via Eqs. (2)-(4).
The lattice partial order and Möbius inversion are taken from [1]; the paper relies on this extension without proving the lattice axioms for rates.
standard math Stationary processes admit a linear representation, so spectral methods remain relevant even when the generating mechanism is nonlinear.
Invoked in the summary via Wold decomposition and Hannan [29]; this is standard time-series theory but does not make non-Gaussian processes Gaussian.

invented entities (1)

Spectral redundancy rate function for atom alpha
purpose: Defines redundant information rate as the frequency-wise minimum of spectral mutual information rates, enabling Möbius inversion into PIRD atoms.
This is a definitional construct specific to PIRD; its validity depends on satisfying PID axioms, not on external measurement. No independent empirical handle is provided beyond the simulations and climate example in the paper.

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Cite this review

Pith. "Pith review of Partial Information Rate Decomposition." pith.science (2026). https://pith.science/paper/XLWCTIMZ

@misc{pith2026250204550,
  author       = {Pith},
  title        = {Pith review of: Partial Information Rate Decomposition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XLWCTIMZ}},
  note         = {Machine review of arXiv:2502.04550}
}

read the original abstract

Partial Information Decomposition (PID) is a principled and flexible method to unveil complex high-order interactions in multi-unit network systems. Though being defined exclusively for random variables, PID is ubiquitously applied to multivariate time series taken as realizations of random processes with temporal statistical structure. Here, to overcome the incorrect depiction of high-order effects by PID schemes applied to dynamic networks, we introduce the framework of Partial Information Rate Decomposition (PIRD). PIRD is first formalized applying lattice theory to decompose the information shared dynamically between a target random process and a set of source processes, and then implemented for Gaussian processes through a spectral expansion of information rates. The new framework is validated in simulated network systems and demonstrated in the practical analysis of time series from large-scale climate oscillations.

Figures

Figures reproduced from arXiv: 2502.04550 by the authors.

**Figure 2.** Decomposition of the multivariate infor [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Reference graph

Works this paper leans on

30 extracted references · 12 canonical work pages

[14]

T. E. Duncan, SIAM Journal on Applied Mathematics 19, 215 (1970)

1970
[1]

HONEST - High-Order Dynamical Networks in Computational NeuroscienceandPhysiology: anInformation-Theoretic Framework

2) 3) 4) 5) 6)1) 2) 3) 4) 5) 6) IY;X ,X1 2 PIRD: original PIRD: surrogates PID a) b) c) Figure 2. Decomposition of the multivariate infor- mation shared instantaneously and dynamically in six triplets of climate time series (target Y: SOI; sources {X1, X2}: 1) {N IN O34, T SA}, 2) {N IN O34, P DO}, 3) {N IN O34, N T A}, 4) {T SA, P DO}, 5) {T SA, N T A}, ...

work page 2022
[2]

P. L. Williams and R. D. Beer, arXiv preprint arXiv:1004.2515 (2010)

arXiv 2010
[3]

Wibral, V

M. Wibral, V. Priesemann, J. W. Kay, J. T. Lizier, and W. A. Phillips, Brain and cognition112, 25 (2017). 5

work page 2017
[4]

Cang and Q

Z. Cang and Q. Nie, Nature communications11, 2084 (2020)

2020
[5]

F. E. Rosas, P. A. Mediano, H. J. Jensen, A. K. Seth, A. B. Barrett, R. L. Carhart-Harris, and D. Bor, PLoS computational biology 16, e1008289 (2020)

work page 2020
[6]

A. I. Luppi, P. A. Mediano, F. E. Rosas, N. Holland, T. D. Fryer, J. T. O’Brien, J. B. Rowe, D. K. Menon, D. Bor, and E. A. Stamatakis, Nature Neuroscience25, 771 (2022)

2022
[7]

Wollstadt, S

P. Wollstadt, S. Schmitt, and M. Wibral, J. Mach. Learn. Res. 24, 1 (2023)

work page 2023

Show all 30 references

[8]

Dissanayake, F

P. Dissanayake, F. Hamman, B. Halder, I. Su- cholutsky, Q. Zhang, and S. Dutta, arXiv preprint arXiv:2411.07483 (2024)

2024 arXiv
[9]

J. W. Kay, J. M. Schulz, and W. A. Phillips, Entropy 24, 1021 (2022)

2022
[10]

T. F. Varley, O. Sporns, S. Schaffelhofer, H. Scher- berger, and B. Dann, Proceedings of the National Academy of Sciences120, e2207677120 (2023)

2023
[11]

T. F. Varley, M. Pope, J. Faskowitz, and O. Sporns, Communications biology 6, 451 (2023)

2023
[12]

Koçillari, M

L. Koçillari, M. Celotto, N. A. Francis, S. Mukherjee, B. Babadi, P. O. Kanold, and S. Panzeri, Brain Infor- matics 10, 34 (2023)

2023
[13]

A. I. Luppi, P. A. Mediano, F. E. Rosas, J. Allanson, J. Pickard, R. L. Carhart-Harris, G. B. Williams, M. M. Craig, P. Finoia, A. M. Owen,et al., Elife12, RP88173 (2024)

2024
[15]

Sparacino, G

L. Sparacino, G. Mijatovic, Y. Antonacci, L. Ricci, D. Marinazzo, S. Stramaglia, and L. Faes, arXiv preprint arXiv:2502.04555 (2025)

2025 arXiv
[16]

A. B. Barrett, Physical Review E91, 052802 (2015)

2015
[17]

R. A. Ince, Entropy19, 318 (2017)

2017
[18]

A. J. Gutknecht, M. Wibral, and A. Makkeh, Proceed- ings of the Royal Society A477, 20210110 (2021)

2021
[19]

D. A. Ehrlich, K. Schick-Poland, A. Makkeh, F. Lanfer- mann, P. Wollstadt, and M. Wibral, Physical Review E 110, 014115 (2024)

2024
[20]

Finn and J

C. Finn and J. T. Lizier, Entropy20, 297 (2018)

2018
[21]

Sparacino, Y

L. Sparacino, Y. Antonacci, G. Mijatovic, and L. Faes, Neurocomputing 630, 129675 (2025)

2025
[22]

Chicharro, Biological cybernetics105, 331 (2011)

D. Chicharro, Biological cybernetics105, 331 (2011)

2011
[23]

Mijatovic, Y

G. Mijatovic, Y. Antonacci, M. Javorka, D. Marinazzo, S. Stramaglia, and L. Faes, IEEE Transactions on Net- work Science and Engineering (2025)

2025
[24]

M. J. McPhaden, S. E. Zebiak, and M. H. Glantz, sci- ence 314, 1740 (2006)

2006
[25]

Chang, R

P. Chang, R. Saravanan, and L. Ji, Geophysical Re- search Letters 30 (2003)

2003
[26]

Silini, G

R. Silini, G. Tirabassi, M. Barreiro, L. Ferranti, and C. Masoller, Climate Dynamics61, 79 (2023)

2023
[27]

Stramaglia, L

S. Stramaglia, L. Faes, J. M. Cortes, and D. Marinazzo, Physical Review Research6, L032007 (2024)

2024
[28]

Lütkepohl, New introduction to multiple time series analysis (Springer Science & Business Media, 2005)

H. Lütkepohl, New introduction to multiple time series analysis (Springer Science & Business Media, 2005)

2005
[29]

L. Faes, S. Erla, and G. Nollo, Computational and mathematical methods in medicine 2012, 140513 (2012)

2012
[30]

E. J. Hannan, inDevelopments in Statistics , Vol. 2 (El- sevier, 1979) pp. 83–121

1979

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [14]

T. E. Duncan, SIAM Journal on Applied Mathematics 19, 215 (1970)

1970

[2] [1]

HONEST - High-Order Dynamical Networks in Computational NeuroscienceandPhysiology: anInformation-Theoretic Framework

2) 3) 4) 5) 6)1) 2) 3) 4) 5) 6) IY;X ,X1 2 PIRD: original PIRD: surrogates PID a) b) c) Figure 2. Decomposition of the multivariate infor- mation shared instantaneously and dynamically in six triplets of climate time series (target Y: SOI; sources {X1, X2}: 1) {N IN O34, T SA}, 2) {N IN O34, P DO}, 3) {N IN O34, N T A}, 4) {T SA, P DO}, 5) {T SA, N T A}, ...

work page 2022

[3] [2]

P. L. Williams and R. D. Beer, arXiv preprint arXiv:1004.2515 (2010)

arXiv 2010

[4] [3]

Wibral, V

M. Wibral, V. Priesemann, J. W. Kay, J. T. Lizier, and W. A. Phillips, Brain and cognition112, 25 (2017). 5

work page 2017

[5] [4]

Cang and Q

Z. Cang and Q. Nie, Nature communications11, 2084 (2020)

2020

[6] [5]

F. E. Rosas, P. A. Mediano, H. J. Jensen, A. K. Seth, A. B. Barrett, R. L. Carhart-Harris, and D. Bor, PLoS computational biology 16, e1008289 (2020)

work page 2020

[7] [6]

A. I. Luppi, P. A. Mediano, F. E. Rosas, N. Holland, T. D. Fryer, J. T. O’Brien, J. B. Rowe, D. K. Menon, D. Bor, and E. A. Stamatakis, Nature Neuroscience25, 771 (2022)

2022

[8] [7]

Wollstadt, S

P. Wollstadt, S. Schmitt, and M. Wibral, J. Mach. Learn. Res. 24, 1 (2023)

work page 2023