Pith. sign in

REVIEW 2 major objections 5 minor 108 references

Energy landscape analysis based on the Ising model: Tutorial review

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This tutorial argues that energy landscape analysis with the Ising model turns multivariate time series into trajectories of a state among basins, provided the data behave like a reversible nearest-neighbor random walk on the hypercube.

desk verdict Solid, honest tutorial review of Ising-model energy landscape analysis; the main soft spot is that the 'ball dynamics' are an assumed single-spin-flip kinetics, not a consequence of the fitted equilibrium model. read the letter →

arxiv 2411.16979 v2 pith:T27XZECC submitted 2024-11-25 cond-mat.dis-nn

Naoki Masuda , Saiful Islam , Si Thu Aung , Takamitsu Watanabe This is my paper · ORCID
classification cond-mat.dis-nn
keywords energylandscapeanalysisIsingmodelpairwisemaximumentropymultivariatetimeseriesdisconnectivitygraphbasindetailedbalancestatetransitions
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

This paper is a tutorial review of energy landscape analysis (ELA) built on the Ising model. It argues that ELA is a domain-general method for interpreting multivariate time series as a ball moving among basins of an energy landscape, where each basin is a locally stable pattern of binarized activity. The paper walks through every step: binarizing each variable, fitting the pairwise maximum entropy model (PMEM) by matching single-variable and pairwise joint activities, identifying local minima on the hypercube of activity patterns, building disconnectivity and basin graphs, and extracting numerical features. Its central claim is that once the PMEM is estimated, it implies a particular equilibrium stochastic dynamics, so transitions between basins can be read as the system's behavior. The authors therefore emphasize validation throughout, including accuracy of fit, threshold robustness, randomization-based pruning of insignificant local minima, random-walk simulations, and test-retest reliability.

What carries the argument

The central object is the pairwise maximum entropy model, also called the Ising model or Boltzmann machine, with energy $E(\sigma) = -\sum_i h_i\sigma_i - \frac{1}{2}\sum_{i,j}J_{ij}\sigma_i\sigma_j$ and Boltzmann probability $P(\sigma)\propto e^{-E(\sigma)}$. The parameters $h_i$ and $J_{ij}$ are fitted by matching the empirical means $\langle\sigma_i\rangle$ and pairwise joint activities $\langle\sigma_i\sigma_j\rangle$. The dynamics are carried by the detailed balance condition $T_{\sigma\to\sigma'}/T_{\sigma'\to\sigma} = e^{E(\sigma)-E(\sigma')}$, which turns the energy landscape into a nearest-neighbor random walk on the hypercube of activity patterns. From this walk, the disconnectivity graph encodes the hierarchy of local minima and energy barriers, while the basin graph assigns every activity pattern to the local minimum reached by steepest descent.

What would settle it

Take a candidate multivariate time series, binarize it, and count how often consecutive binary states differ in more than one variable; if such multi-bit jumps persist when the sampling interval is shortened toward the timescale of the underlying continuous signals, then the data are not a nearest-neighbor walk on the hypercube, and the ELA's energy-landscape interpretation fails.

Watch

Extended reading notes

Core claim

The paper establishes the logic and validity conditions of Ising-model-based energy landscape analysis. The core claim is that a multivariate time series, after binarization, can be summarized by a pairwise maximum entropy model whose Boltzmann distribution $P(\sigma) = e^{-E(\sigma)}/\sum_k e^{-E(\sigma^{(k)})}$ defines an energy $E(\sigma) = -\sum_i h_i\sigma_i - \frac{1}{2}\sum_{i,j} J_{ij}\sigma_i\sigma_j$ over the $2^N$ activity patterns. From this energy, the local minima, their basins, and the minimal energy barriers between them are computed, yielding disconnectivity graphs and basin graphs that describe the system's states. The paper argues that the PMEM dictates a nearest-neighbor random walk on the hypercube with detailed balance, $T_{\sigma\to\sigma'}/T_{\sigma'\to\sigma} = e^{E(\sigma)-E(\sigma')}$, so the time series can be interpreted as a trajectory through basins. This interpretation is valid only when the binarized data are long enough and behave like such a walk; otherwise the energy landscape and its dynamics are not justified.

Load-bearing premise

The dynamic interpretation of ELA assumes that the observed binarized states evolve as a nearest-neighbor random walk on the hypercube with detailed balance, meaning only one variable flips at a time and transitions are reversible; if synchronous threshold crossings occur or the time resolution is too low, the inferred basins and transitions do not represent the real dynamics.

Editorial extensions

If this is right

  • Any multivariate time series with roughly $N \ge 7$ variables and sufficiently long recording time can be analyzed with ELA, giving comparable energy-landscape features across scientific domains.
  • The randomization-based pruning procedure provides a statistical criterion for deciding which local minima are significant, reducing the arbitrariness of earlier visual inspection.
  • The detailed-balance random walk yields quantitative predictions for basin transition probabilities and dwell times, which can be checked directly against the observed data.
  • The phase-diagram method allows positioning a data set relative to the paramagnetic, spin-glass, and ferromagnetic phases, connecting empirical dynamics to criticality.
  • The variational Bayes extension enables per-individual energy landscapes estimated from pooled group data, easing the exponential data-length requirement of the basic method.

Reading between the lines

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

  • Because the PMEM ignores the temporal order of the data, ELA is best understood as a description of the equilibrium distribution with an implied dynamics, rather than a fitted dynamical model; treating it as the latter on strongly non-stationary data would likely misattribute transitions.
  • A natural extension the paper only sketches is sliding-window ELA, where energy landscapes are estimated on overlapping time windows to track barrier shrinkage or basin loss as early-warning signals for regime shifts.
  • A direct stress test would be to generate data from a hidden Markov model with state-dependent transition rates and ask whether ELA's basin-transition frequencies recover those rates; the paper's own comparison with clustering methods suggests such tests are feasible and informative.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. This manuscript is a tutorial review of energy landscape analysis (ELA) based on the Ising model / pairwise maximum entropy model (PMEM) for multivariate time series data. It walks through the workflow: binarization, PMEM estimation via maximum likelihood, assessment of fit via rD, construction of disconnectivity graphs and basin graphs, feature extraction, and downstream group comparisons. It then covers advanced topics: phase-diagram analysis, validation methods (threshold variation, randomization, random-walk simulations, test-retest reliability), variational Bayes estimation for multi-participant data, clustering and hidden-Markov alternatives, available software, and illustrative applications. The paper is explicitly tutorial in intent and includes running examples (e.g., the Midnight Scan Club fMRI data) that make the steps concrete.

Significance. The review fills a practical gap: ELA has been used mainly in fMRI research, and this paper provides a much more detailed, beginner-oriented account of the method than the earlier short review [54]. Its most valuable features are the honest treatment of arbitrary choices (e.g., the rD > 0.85 guideline in Section 2.3 and the pruning threshold in Section 3.2.2), the explicit warning about data hunger and binarization loss (Section 3.3), and the concrete validation protocols. If the kinetic-assumption issue raised below is addressed, the manuscript will be a reliable entry point for researchers outside neuroscience. The paper does not claim new results; its contribution is didactic synthesis and critical discussion, which is appropriate for a tutorial review.

major comments (2)
  1. [§2.2 and Abstract] The statement in Section 2.2 that 'the PMEM implies a particular equilibrium stochastic dynamics' and that 'once a PMEM is estimated, it dictates the probability or rate of the move from any activity pattern σ to another' is too strong. The PMEM estimation in Eqs. (4)-(12) matches only single-variable and pairwise marginals; the resulting Boltzmann distribution (Eq. (6)) is the stationary distribution of many different Markov chains, including the Metropolis single-spin-flip walk of Section 3.2.3, Glauber dynamics, and non-reversible chains, and these alternatives yield different basin-to-basin transition rates and dwell times. The kinetic model in Section 3.2.3 (uniform neighbor proposal plus Metropolis acceptance) is an additional, unestimated modelling assumption rather than a consequence of the PMEM fit. Because the abstract promises that the method 'allows one to capture dynamics of the data as trajectories of a ball,' the manuscript should be revised to state that ELA imposes a reversible nearest-neighbour dynamics whose stationary distribution is the fitted PMEM, and that the validity of this kinetic assumption should be tested for each data set.
  2. [§3.2.3] The random-walk validation described in this section is the correct safeguard, but the text should explicitly state that the comparison of basin-level transition probabilities validates the kinetic assumption (nearest-neighbour reversible Metropolis dynamics) in addition to the PMEM fit. As written, the section says a large discrepancy implies 'the ELA may not be a suitable method,' which conflates two distinct claims: the landscape (stationary distribution) may still describe the data well while the imposed dynamics do not. Clarifying this distinction would help users decide whether to attribute any observed discrepancy to the model fitting or to the kinetic assumption.
minor comments (5)
  1. [§2.1] Section 2.1: 'multivaraite' should be 'multivariate.'
  2. [§3.2.2] Section 3.2.2: 'discoonnectivity' should be 'disconnectivity.'
  3. [§3.5] Section 3.5: 'clulster' should be 'cluster.'
  4. [§3.2.4] Section 3.2.4: the heading 'T est-retest reliability' contains an unintended space.
  5. [§2.6] Section 2.6: 'an local minimum' should be 'a local minimum.'

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the review is self-contained; the dynamic interpretation rests on an explicit kinetic assumption that is validated rather than assumed into existence.

full rationale

This is a tutorial review, not a new derivation, and the workflow it describes is internally consistent. The PMEM is fitted to the empirical first and second moments (Eqs. (4)-(12)), and the disconnectivity graph, basin graph, and landscape features are deterministic functions of the fitted energy E(σ). No fitted parameter is renamed as a prediction, and no result is imported from the authors' prior work as the proof of a new claim; self-citations such as [19], [32], and [54] describe previously published methods and are used as references, not as load-bearing evidence. The nearest candidate for circularity is the dynamic interpretation: Section 2.2 states that 'the PMEM implies a particular equilibrium stochastic dynamics,' and Eq. (18) indeed defines transition ratios so that the stationary distribution is the Boltzmann distribution of Eq. (6) by construction. However, the paper itself treats this as a modeling assumption to be checked: Section 3.2.3 explicitly recommends comparing random-walk transition probabilities with empirical transition probabilities, and Section 2.4 warns that synchronous threshold crossings or low time resolution undermine the nearest-neighbor random-walk view. Thus the 'ball' trajectories are a constructed, falsifiable dynamic model with the fitted landscape as input, not a claim that the landscape alone forces the transition rates. The overstatement in the abstract and Section 2.2 is a validity and identifiability concern rather than a circular reduction.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The tutorial's guidance depends on several user-chosen parameters (thresholds, hyperparameters) and on modeling assumptions inherited from the ELA framework. No new theoretical entities are postulated; the 'energy' is explicitly described as a mathematical construct.

free parameters (5)
  • Binarization threshold θ_i = typically median of each time series
    Chosen to convert continuous signals to binary; the paper notes it is arbitrary (Section 2.2).
  • Learning rate ε in gradient ascent = not specified
    User-chosen step size for PMEM estimation (Section 2.2).
  • Significance threshold for pruning local minima = ⟨L⟩ + 2×std(L)
    Arbitrary cutoff in randomization test; the paper calls it 'arbitrary and probably too generous' (Section 3.2.2).
  • Number of random surrogates c = 10^3 in example
    Choice for computing null distribution in significance testing (Section 3.2.2).
  • Prior precision initialization in variational Bayes = 6 for h_i, 30 for J_ij
    Hyperparameter choices from refs [20,21] used in the tutorial (Section 3.4).
assumptions (4)
  • domain assumption The distribution of activity patterns is well-approximated by a pairwise maximum entropy model (PMEM).
    This is the foundational modeling assumption of ELA, introduced in Section 2.2.
  • domain assumption The binarized time series is generated by a reversible Markov chain on the hypercube satisfying detailed balance, with at most one spin flip per step.
    Assumed for interpreting ELA dynamics; stated in Sections 2.4 and 3.2.3.
  • domain assumption The data are stationary and time order can be ignored when estimating the PMEM.
    The paper states ELA is not a time-series method because it only fits the marginal distribution (Section 2.2).
  • standard math The empirical frequencies of patterns reflect the model distribution closely enough for parameter estimation.
    Standard likelihood-based statistical assumption underlying PMEM fitting (Section 2.2).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Energy landscape analysis based on the Ising model: Tutorial review." pith.science (2026). https://pith.science/paper/T27XZECC

@misc{pith2026241116979,
  author       = {Pith},
  title        = {Pith review of: Energy landscape analysis based on the Ising model: Tutorial review},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T27XZECC}},
  note         = {Machine review of arXiv:2411.16979}
}
read the original abstract

We review a class of energy landscape analysis method that uses the Ising model and takes multivariate time series data as input. The method allows one to capture dynamics of the data as trajectories of a ball from one basin to a different basin to yet another, constrained on the energy landscape specified by the estimated Ising model. While this energy landscape analysis has mostly been applied to functional magnetic resonance imaging (fMRI) data from the brain for historical reasons, there are emerging applications outside fMRI data and neuroscience. To inform such applications in various research fields, this review paper provides a detailed tutorial on each step of the analysis, terminologies, concepts underlying the method, and validation, as well as recent developments of extended and related methods.

Figures

Figures reproduced from arXiv: 2411.16979 by the authors.

Figure 1
Figure 1. One-dimensional schematic of energy landscapes. (a) An energy landscape with two local minima of energy, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Two-dimensional schematic of an energy landscape. Four local minima are present. The dotted lines show the [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Steps of the ELA. (a) The input data is multivariate time series with [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: A schematic of the local minimum of energy. A system of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Networks of activity patterns for N ∈ {1, . . . , 5}. They are hypercubes having 2N nodes. Each node represents an activity pattern and has N neighbors. 2, ten nodes in distance 3, five nodes in distance 4, and one node in distance 5. The single node in distance 5 is t…
Figure 6
Figure 6. Figure 6: Time series of activity pattern as a discrete-time observation of a continuous-time walk. (a) Absence and (b) [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: Disconnectivity graphs. (a) Disconnectivity graph from fMRI data obtained from the brain of healthy human [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: Local minima and paths between them. (a) The network of activity patterns in the case of [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: Basin graph. (a) The steepest descent path from [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: Schematic of the phase diagram of the SK model. A more common parameter in physics literature is the [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

108 extracted references · 79 canonical work pages

  1. [54]

    Energy landscape analysis of neuroimaging data

    Ezaki T, Watanabe T, Ohzeki M, Masuda N. Energy landscape analysis of neuroimaging data. Philos Trans R Soc A. 2017; 375(2096):20160287

    work page 2017

  2. [1]

    Theory of protein folding: The energy landscape perspective

    Onuchic JN, Luthey-Schulten Z, Wolynes PG. Theory of protein folding: The energy landscape perspective. Annu Rev Phys Chem. 1997; 48:545–600

    1997

  3. [2]

    Energy Landscapes

    Wales DJ. Energy Landscapes. Cambridge, UK: Cambridge University Press; 2003

    2003

  4. [3]

    The energy landscape as a unifying theme in molecular science

    Wales DJ. The energy landscape as a unifying theme in molecular science. Philos Trans R Soc A. 2005; 363(1827):357–377

    2005

  5. [4]

    The energy landscape perspective: encoding structure and function for biomolecules

    R¨ oder K, Wales DJ. The energy landscape perspective: encoding structure and function for biomolecules. Front Mol Biosci. 2022; 9:820792

    2022

  6. [5]

    Energy landscapes of resting-state brain networks

    Watanabe T, Hirose S, Wada H, Imai Y, Machida T, Shirouzu I, et al. Energy landscapes of resting-state brain networks. Front Neuroinform. 2014; 8:12

    2014

  7. [6]

    Energy landscape and dynamics of brain activity during human bistable perception

    Watanabe T, Masuda N, Megumi F, Kanai R, Rees G. Energy landscape and dynamics of brain activity during human bistable perception. Nat Commun. 2014; 5:4765

    2014

  8. [7]

    Weak pairwise correlations imply strongly correlated network states in a neural population

    Schneidman E, Berry MJ, Segev R, Bialek W. Weak pairwise correlations imply strongly correlated network states in a neural population. Nature. 2006; 440(7087):1007–1012

    2006

Show all 108 references
  1. [8]

    The structure of multi-neuron firing patterns in primate retina

    Shlens J, Field GD, Gauthier JL, Grivich MI, Petrusca D, Sher A, et al. The structure of multi-neuron firing patterns in primate retina. J Neurosci. 2006; 26(32):8254–8266

    2006

  2. [9]

    A maximum entropy model applied to spatial and temporal correlations from cortical networks in vitro

    Tang A, Jackson D, Hobbs J, Chen W, Smith JL, Patel H, et al. A maximum entropy model applied to spatial and temporal correlations from cortical networks in vitro. J Neurosci. 2008; 28(2):505–518

    2008

  3. [10]

    A small world of neuronal synchrony

    Yu S, Huang D, Singer W, Nikoli´ c D. A small world of neuronal synchrony. Cereb Cortex. 2008; 18(12):2891– 2901. 23

    2008

  4. [11]

    Maximum entropy approaches to living neural networks

    Yeh FC, Tang A, Hobbs JP, Hottowy P, Dabrowski W, Sher A, et al. Maximum entropy approaches to living neural networks. Entropy. 2010; 12(1):89–106

    2010

  5. [12]

    Are biological systems poised at criticality? J Stat Phys

    Mora T, Bialek W. Are biological systems poised at criticality? J Stat Phys. 2011; 144(2):268–302

    2011

  6. [13]

    A pairwise maximum entropy model accurately describes resting-state human brain networks

    Watanabe T, Hirose S, Wada H, Imai Y, Machida T, Shirouzu I, et al. A pairwise maximum entropy model accurately describes resting-state human brain networks. Nat Commun. 2013; 4:1370

    2013

  7. [14]

    Brain network dynamics in high-functioning individuals with autism

    Watanabe T, Rees G. Brain network dynamics in high-functioning individuals with autism. Nat Commun. 2017; 8:16048

    2017

  8. [15]

    The energy landscape underpinning module dynamics in the human brain connectome

    Ashourvan A, Gu S, Mattar MG, Vettel JM, Bassett DS. The energy landscape underpinning module dynamics in the human brain connectome. NeuroImage. 2017; 157:364–380

    2017

  9. [16]

    Energy landscape analysis of the subcortical brain network unravels system properties beneath resting state dynamics

    Kang J, Pae C, Park HJ. Energy landscape analysis of the subcortical brain network unravels system properties beneath resting state dynamics. NeuroImage. 2017; 149:153–164

    2017

  10. [17]

    Age-related changes in the ease of dynamical transitions in human brain activity

    Ezaki T, Sakaki M, Watanabe T, Masuda N. Age-related changes in the ease of dynamical transitions in human brain activity. Hum Brain Mapp. 2018; 39(6):2673–2688

    2018

  11. [18]

    Graph-theoretical analysis for energy landscape reveals the organization of state transitions in the resting-state human cerebral cortex

    Kang J, Pae C, Park HJ. Graph-theoretical analysis for energy landscape reveals the organization of state transitions in the resting-state human cerebral cortex. PLoS ONE. 2019; 14(10):e0222161

    2019

  12. [19]

    Closer to critical resting-state neural dynamics in individuals with higher fluid intelligence

    Ezaki T, Fonseca dos Reis E, Watanabe T, Sakaki M, Masuda N. Closer to critical resting-state neural dynamics in individuals with higher fluid intelligence. Commun Biol. 2020; 3:52

    2020

  13. [20]

    Empirical Bayes estimation of pairwise maximum entropy model for nonlinear brain state dynamics

    Jeong SO, Kang J, Pae C, Eo J, Park SM, Son J, et al. Empirical Bayes estimation of pairwise maximum entropy model for nonlinear brain state dynamics. NeuroImage. 2021; 244:118618

    2021

  14. [21]

    Bayesian estimation of maximum entropy model for individualized energy landscape analysis of brain state dynamics

    Kang J, Jeong SO, Pae C, Park HJ. Bayesian estimation of maximum entropy model for individualized energy landscape analysis of brain state dynamics. Hum Brain Mapp. 2021; 42(10):3411–3428

    2021

  15. [22]

    Modeling heterogeneous brain dynamics of depression and melancholia using energy landscape analysis

    Regonia PR, Takamura M, Nakano T, Ichikawa N, Fermin A, Okada G, et al. Modeling heterogeneous brain dynamics of depression and melancholia using energy landscape analysis. Front Psychiatry. 2021; 12:780997

    2021

  16. [23]

    Brain state-based detection of attentional fluctuations and their modulation

    Yamashita A, Rothlein D, Kucyi A, Valera EM, Esterman M. Brain state-based detection of attentional fluctuations and their modulation. NeuroImage. 2021; 236:118072

    2021

  17. [24]

    Variable rather than extreme slow reaction times distinguish brain states during sustained attention

    Yamashita A, Rothlein D, Kucyi A, Valera EM, Germine L, Wilmer J, et al. Variable rather than extreme slow reaction times distinguish brain states during sustained attention. Sci Rep. 2021; 11:14883

    2021

  18. [25]

    Dynamic transitions between brain states predict auditory attentional fluctuations

    Kondo HM, Terashima H, Ezaki T, Kochiyama T, Kihara K, Kawahara JI. Dynamic transitions between brain states predict auditory attentional fluctuations. Front Neurosci. 2022; 16:816735

    2022

  19. [26]

    The longitudinal neural dynamics changes of whole brain connectome during natural recovery from poststroke aphasia

    Fan L, Li C, Huang ZG, Zhao J, Wu X, Liu T, et al. The longitudinal neural dynamics changes of whole brain connectome during natural recovery from poststroke aphasia. NeuroImage Clin. 2022; 36:103190

    2022

  20. [27]

    Triple-network analysis of Alzheimer’s disease based on the energy landscape

    Li Y, An S, Zhou T, Su C, Zhang S, Li C, et al. Triple-network analysis of Alzheimer’s disease based on the energy landscape. Front Neurosci. 2023; 17:1171549

    2023

  21. [28]

    Energy of functional brain states correlates with cognition in adolescent-onset schizophrenia and healthy persons

    Theis N, Bahuguna J, Rubin JE, Banerjee SS, Muldoon B, Prasad KM. Energy of functional brain states correlates with cognition in adolescent-onset schizophrenia and healthy persons. Human Brain Mapp. 2025; 46:e70129

    2025

  22. [29]

    Distinct frontoparietal brain dynamics underlying the co-occurrence of autism and ADHD

    Watanabe D, Watanabe T. Distinct frontoparietal brain dynamics underlying the co-occurrence of autism and ADHD. eNeuro. 2023; 10(5):ENEURO.0146-23.2023

    2023

  23. [30]

    Linear models replicate the energy landscape and dynamics of resting-state brain activity

    Hosaka Y, Hieda T, Hayashi K, Jimura K, Matsui T. Linear models replicate the energy landscape and dynamics of resting-state brain activity. bioRxiv. preprint. 2024 [posted Sep 3, 2024; cited Nov 12, 2024] Available from: https://doi.org/10.1101/2024.05.21.595246 doi: 10.1101/...

    2024 doi

  24. [31]

    Aberrant brain dynamics of large-scale functional networks across schizophrenia and mood disorder

    Ishida T, Yamada S, Yasuda K, Uenishi S, Tamaki A, Tabata M, et al. Aberrant brain dynamics of large-scale functional networks across schizophrenia and mood disorder. NeuroImage Clin. 2024; 41:103574. 24

    2024

  25. [32]

    Reliability of energy landscape analysis of resting- state functional MRI data

    Khanra P, Nakuci J, Muldoon S, Watanabe T, Masuda N. Reliability of energy landscape analysis of resting- state functional MRI data. Eur J Neurosci. 2024; 60(12):4265–4290

    2024

  26. [33]

    Aberrant brain dynamics in individuals with clinical high risk of psychosis

    Kindler J, Ishida T, Michel C, Klaassen AL, St¨ uble M, Zimmermann N, et al. Aberrant brain dynamics in individuals with clinical high risk of psychosis. Schizophr Bull Open. 2024; 5(1):sgae002

    2024

  27. [34]

    Associations of conservatism and jumping to conclusions biases with aberrant salience and default mode network

    Miyata J, Sasamoto A, Ezaki T, Isobe M, Kochiyama T, Masuda N, et al. Associations of conservatism and jumping to conclusions biases with aberrant salience and default mode network. Psychiatry Clin Neurosci. 2024; 78(4):322–331

    2024

  28. [35]

    Energy landscape analysis of brain network dynamics in Alzheimer’s disease

    Xing L, Guo Z, Long Z. Energy landscape analysis of brain network dynamics in Alzheimer’s disease. Front Aging Neurosci. 2024; 16:1375091

    2024

  29. [36]

    Characterising Alzheimer’s disease with EEG-based energy landscape analysis

    Klepl D, He F, Wu M, De Marco M, Blackburn DJ, Sarrigiannis PG. Characterising Alzheimer’s disease with EEG-based energy landscape analysis. IEEE J Biomed Health Inform. 2022; 26(3):992–1000

    2022

  30. [37]

    Causal roles of prefrontal cortex during spontaneous perceptual switching are determined by brain state dynamics

    Watanabe T. Causal roles of prefrontal cortex during spontaneous perceptual switching are determined by brain state dynamics. eLife. 2021; 10:e69079

    2021

  31. [38]

    Brain connectivity signature extractions from TMS invoked EEGs

    Gupta D, Du X, Summerfelt A, Hong LE, Choa FS. Brain connectivity signature extractions from TMS invoked EEGs. Sensors. 2023; 23(10):4078

    2023

  32. [39]

    Energy landscape of resting magnetoen- cephalography reveals fronto-parietal network impairments in epilepsy

    Krzemi´ nski D, Masuda N, Hamandi K, Singh KD, Routley B, Zhang J. Energy landscape of resting magnetoen- cephalography reveals fronto-parietal network impairments in epilepsy. Netw Neurosci. 2020; 4(2):374–396

    2020

  33. [40]

    The energy landscape of neurophys- iological activity implicit in brain network structure

    Gu S, Cieslak M, Baird B, Muldoon SF, Grafton ST, Pasqualetti F, et al. The energy landscape of neurophys- iological activity implicit in brain network structure. Sci Rep. 2018; 8:2507

    2018

  34. [41]

    Energy landscape analysis of cardiac fibrillation wave dynamics using pairwise maximum entropy model

    Song E. Energy landscape analysis of cardiac fibrillation wave dynamics using pairwise maximum entropy model. arXiv. preprint. arXiv:1809.09798. 2018 [cited Nov 12, 2024]. Available from: https://arxiv.org/abs/1809.09798

    2018 arXiv

  35. [42]

    Energy landscape analysis and time-series clustering analysis of patient state multistability related to rheumatoid arthritis drug treatment: the KURAMA cohort study

    Yamamoto K, Sakaguchi M, Onishi A, Yokoyama S, Matsui Y, Yamamoto W, et al. Energy landscape analysis and time-series clustering analysis of patient state multistability related to rheumatoid arthritis drug treatment: the KURAMA cohort study. PLoS ONE. 2024; 19(4):e0302308

    2024

  36. [43]

    Yonezawa S, Haruki T, Koizumi K, Taketani A, Oshima Y, Oku M, et al. Establishing monoclonal gammopathy of undetermined significance as an independent pre-disease state of multiple myeloma using Raman spectroscopy, dynamical network biomarker theory, and energy landscape analy...

    2024

  37. [44]

    Energy landscape analysis elucidates the multistability of ecological communities across environmental gradients

    Suzuki K, Nakaoka S, Fukuda S, Masuya H. Energy landscape analysis elucidates the multistability of ecological communities across environmental gradients. Ecol Monogr. 2021; 91(2):e01469

    2021

  38. [45]

    Alternative stable states of microbiome structure and soil ecosystem functions

    Fujita H, Yoshida S, Suzuki K, Toju H. Alternative stable states of microbiome structure and soil ecosystem functions. Environ Microbiome. 2025; 20:28

    2025

  39. [46]

    Alternative stable states, nonlinear behavior, and predictability of microbiome dynamics

    Fujita H, Ushio M, Suzuki K, Abe MS, Yamamichi M, Iwayama K, et al. Alternative stable states, nonlinear behavior, and predictability of microbiome dynamics. Microbiome. 2023; 11:63

    2023

  40. [47]

    Computational estimation of sediment symbiotic bacterial structures of seagrasses overgrowing downstream of onshore aquaculture

    Miyamoto H, Kawachi N, Kurotani A, Moriya S, Suda W, Suzuki K, et al. Computational estimation of sediment symbiotic bacterial structures of seagrasses overgrowing downstream of onshore aquaculture. Environ Res. 2023; 219:115130

    2023

  41. [48]

    Estimation of silent phenotypes of calf antibiotic dysbiosis

    Okada S, Inabu Y, Miyamoto H, Suzuki K, Kato T, Kurotani A, et al. Estimation of silent phenotypes of calf antibiotic dysbiosis. Sci Rep. 2023; 13:6359

    2023

  42. [49]

    Potential energy landscape and robustness of a gene regulatory network: Toggle switch

    Kim KY, Wang J. Potential energy landscape and robustness of a gene regulatory network: Toggle switch. PLoS Comput Biol. 2007; 3(3):e60

    2007

  43. [50]

    Energy landscape of social balance

    Marvel SA, Strogatz SH, Kleinberg JM. Energy landscape of social balance. Phys Rev Lett. 2009; 103(19):198701

    2009

  44. [51]

    Energy landscape reveals that the budding yeast cell cycle is a robust and adaptive multi-stage process

    Lv C, Li X, Li F, Li T. Energy landscape reveals that the budding yeast cell cycle is a robust and adaptive multi-stage process. PLoS Comput Biol. 2015; 11(4):e1004156. 25

    2015

  45. [52]

    Exploring the free energy landscape: From dynamics to networks and back

    Prada-Gracia D, G´ omez-Garde˜ nes J, Echenique P, Falo F. Exploring the free energy landscape: From dynamics to networks and back. PLoS Comput Biol. 2009; 5(6):e1000415

    2009

  46. [53]

    CELLoGeNe—An energy landscape framework for logical networks controlling cell decisions

    Andersson E, Sj¨ o M, Kaji K, Olariu V. CELLoGeNe—An energy landscape framework for logical networks controlling cell decisions. iScience. 2022; 25(1):104743

    2022

  47. [55]

    Inferring pairwise interactions from biological data using maximum-entropy probability models

    Stein RR, Marks DS, Sander C. Inferring pairwise interactions from biological data using maximum-entropy probability models. PLoS Comput Biol. 2015; 11(4):e1004182

    2015

  48. [56]

    Inverse statistical problems: From the inverse Ising problem to data science

    Nguyen HC, Zecchina R, Berg J. Inverse statistical problems: From the inverse Ising problem to data science. Adv Phys. 2017; 66(3):197–261

    2017

  49. [57]

    Machine learning and the physical sciences

    Carleo G, Cirac I, Cranmer K, Daudet L, Schuld M, Tishby N, et al. Machine learning and the physical sciences. Rev Mod Phys. 2019; 91(4):045002

    2019

  50. [58]

    A high-bias, low-variance introduction to Machine Learning for physicists

    Mehta P, Bukov M, Wang CH, Day AGR, Richardson C, Fisher CK, et al. A high-bias, low-variance introduction to Machine Learning for physicists. Phys Rep. 2019; 810:1–124

    2019

  51. [59]

    The topology of multidimensional potential energy surfaces: Theory and application to peptide structure and kinetics

    Becker OM, Karplus M. The topology of multidimensional potential energy surfaces: Theory and application to peptide structure and kinetics. J Chem Phys. 1997; 106(4):1495–1517

    1997

  52. [60]

    Generalized iterative scaling for log-linear models

    Darroch JN, Ratcliff D. Generalized iterative scaling for log-linear models. Ann Math Stat. 1972; 43(5):1470– 1480

    1972

  53. [61]

    Statistical analysis of non-lattice data

    Besag J. Statistical analysis of non-lattice data. J R Stat Soc Ser D. 1975; 24(3):179–195

    1975

  54. [62]

    Statistical physics of pairwise probability models

    Roudi Y, Aurell E, Hertz JA. Statistical physics of pairwise probability models. Front Comput Neurosci. 2009; 3:22

    2009

  55. [63]

    Correlation networks: Interdisciplinary approaches beyond thresholding

    Masuda N, Boyd ZM, Garlaschelli D, Mucha PJ. Correlation networks: Interdisciplinary approaches beyond thresholding. arXiv. preprint. arXiv:2311.09536. 2023 [cited Nov 12, 2024]. Available from: https://arxiv.org/abs/2311.09536

    2023 arXiv

  56. [64]

    Precision functional mapping of individual human brains

    Gordon EM, Laumann TO, Gilmore A W, Newbold DJ, Greene DJ, Berg JJ, et al. Precision functional mapping of individual human brains. Neuron. 2017; 95(4):791–807

    2017

  57. [65]

    Archetypal energy landscapes

    Wales DJ, Miller MA, Walsh TR. Archetypal energy landscapes. Nature. 1998; 394(6695):758–760

    1998

  58. [66]

    Random walk over basins of attraction to construct Ising energy landscapes

    Zhou Q. Random walk over basins of attraction to construct Ising energy landscapes. Phys Rev Lett. 2011; 106(18):180602

    2011

  59. [67]

    Spin Glasses

    Fischer KH, Hertz JA. Spin Glasses. Cambridge, UK: Cambridge University Press; 1991

    1991

  60. [68]

    Chaotic nature of the spin-glass phase

    Bray AJ, Moore MA. Chaotic nature of the spin-glass phase. Phys Rev Lett. 1987; 58(1):57–60

    1987

  61. [69]

    Chaos in temperature in the Sherrington-Kirkpatrick model

    Rizzo T, Crisanti A. Chaos in temperature in the Sherrington-Kirkpatrick model. Phys Rev Lett. 2003; 90(13):137201

    2003

  62. [70]

    Free-energy fluctuations and chaos in the Sherrington-Kirkpatrick model

    Aspelmeier T. Free-energy fluctuations and chaos in the Sherrington-Kirkpatrick model. Phys Rev Lett. 2008; 100(11):117205

    2008

  63. [71]

    Thermodynamics and signatures of criticality in a network of neurons

    Tkaˇ cik G, Mora T, Marre O, Amodei D, Palmer SE, Berry MJ II, et al. Thermodynamics and signatures of criticality in a network of neurons. Proc Natl Acad Sci USA. 2015; 112(37):11508–11513

    2015

  64. [72]

    Network-dependent modulation of brain activity during sleep

    Watanabe T, Kan S, Koike T, Misaki M, Konishi S, Miyauchi S, et al. Network-dependent modulation of brain activity during sleep. NeuroImage. 2014; 98:1–10

    2014

  65. [73]

    Testing for nonlinearity in time series: the method of surrogate data

    Theiler J, Eubank S, Longtin A, Galdrikian B, Farmer JD. Testing for nonlinearity in time series: the method of surrogate data. Physica D. 1992; 58(1-4):77–94

    1992

  66. [74]

    Surrogate time series

    Schreiber T, Schmitz A. Surrogate time series. Physica D. 2000; 142(3-4):346–382. 26

    2000

  67. [75]

    An introduction to Monte Carlo methods

    Walter JC, Barkema GT. An introduction to Monte Carlo methods. Physica A. 2015; 418:78–87

    2015

  68. [76]

    Random walks and diffusion on networks

    Masuda N, Porter MA, Lambiotte R. Random walks and diffusion on networks. Phys Rep. 2017; 716-717:1–58

    2017

  69. [77]

    Pattern Recognition and Machine Learning

    Bishop CM. Pattern Recognition and Machine Learning. New York, NY: Springer; 2006

    2006

  70. [78]

    Decomposition of spontaneous brain activity into distinct fMRI co-activation patterns

    Liu X, Chang C, Duyn JH. Decomposition of spontaneous brain activity into distinct fMRI co-activation patterns. Front Syst Neurosci. 2013; 7:101

    2013

  71. [79]

    Co-activation patterns in resting-state fMRI signals

    Liu X, Zhang N, Chang C, Duyn JH. Co-activation patterns in resting-state fMRI signals. NeuroImage. 2018; 180:485–494

    2018

  72. [80]

    Modelling state-transition dynamics in resting-state brain signals by the hidden Markov and Gaussian mixture models

    Ezaki T, Himeno Y, Watanabe T, Masuda N. Modelling state-transition dynamics in resting-state brain signals by the hidden Markov and Gaussian mixture models. Eur J Neurosci. 2021; 54(8):5404–5416

    2021

  73. [81]

    Co-activation pattern alterations in autism spectrum disorder—A volume-wise hierarchical clustering fMRI study

    Paakki JJ, Rahko JS, Kotila A, Mattila ML, Miettunen H, Hurtig TM, et al. Co-activation pattern alterations in autism spectrum disorder—A volume-wise hierarchical clustering fMRI study. Brain Behav. 2021; 11(8):e02174

    2021

  74. [82]

    State-transition dynamics of resting-state functional magnetic resonance imaging data: model comparison and test-to-retest analysis

    Islam S, Khanra P, Nakuci J, Muldoon SF, Watanabe T, Masuda N. State-transition dynamics of resting-state functional magnetic resonance imaging data: model comparison and test-to-retest analysis. BMC Neurosci. 2024; 25:14

    2024

  75. [83]

    Microstates in resting-state EEG: Current status and future directions

    Khanna A, Pascual-Leone A, Michel CM, Farzan F. Microstates in resting-state EEG: Current status and future directions. Neurosci Biobehav Rev. 2015; 49:105–113

    2015

  76. [84]

    EEG microstates as a tool for studying the temporal dynamics of whole-brain neuronal networks: a review

    Michel CM, Koenig T. EEG microstates as a tool for studying the temporal dynamics of whole-brain neuronal networks: a review. NeuroImage. 2018; 180:577–593

    2018

  77. [85]

    Fast transient networks in spontaneous human brain activity

    Baker AP, Brookes MJ, Rezek IA, Smith SM, Behrens T, Probert Smith PJ, et al. Fast transient networks in spontaneous human brain activity. eLife. 2014; 3:e01867

    2014

  78. [86]

    +microstate: a MATLAB toolbox for brain microstate analysis in sensor and cortical EEG/MEG

    Tait L, Zhang J. +microstate: a MATLAB toolbox for brain microstate analysis in sensor and cortical EEG/MEG. NeuroImage. 2022; 258:119346

    2022

  79. [87]

    Energy Landscape Analysis Toolkit (ELAT) for MATLAB

    Ezaki T. Energy Landscape Analysis Toolkit (ELAT) for MATLAB. GitHub. 2017 [cited Nov 12, 2024]. https://github.com/tkEzaki/energy-landscape-analysis

    2017

  80. [88]

    Energy landscape scripts

    Krzemi´ nski D. Energy landscape scripts. GitHub. 2020 [cited Nov 11, 2024]. https://github.com/dokato/ energy_landscape

    2020

  81. [89]

    About elapy

    Oku M. About elapy. GitHub. 2024 [cited Nov 11, 2024]. https://github.com/okumakito/elapy

    2024

  82. [90]

    rELA—an R package for energy landscape analysis

    Suzuki K. rELA—an R package for energy landscape analysis. GitHub. 2024 [cited May 30, 2024]. https: //github.com/kecosz/rELA

    2024

  83. [91]

    Suzuki K. ela. GitHub. 2024 [cited May 30, 2024]. https://github.com/kecosz/ela

    2024

  84. [92]

    Alzheimer-s-disease-Energy-landscape

    Klepl D. Alzheimer-s-disease-Energy-landscape. GitHub. 2022 [cited Aug 30, 2024]. https://github.com/ dominikklepl/AD-energy-landscape/tree/master/scripts

    2022

  85. [93]

    monetyonsei. VEM MEM. GitHub. 2021 [cited Aug 30, 2024]. https://github.com/monetyonsei/paper/ tree/main/VEM_MEM

    2021

  86. [94]

    Energy landscape analysis with the conventional likelihood maximization and variational Bayes approximation methods

    Khanra P. Energy landscape analysis with the conventional likelihood maximization and variational Bayes approximation methods. GitHub. 2024 [cited May 10, 2024]. https://github.com/pitambarkhanra/energy_ landscape_analysis

    2024

  87. [95]

    MATLAB code for phase diagram analysis

    Ezaki T. MATLAB code for phase diagram analysis. GitHub. 2020 [cited Nov 12, 2024]. https://github. com/tkEzaki/phase_diagram_analysis

    2020

  88. [96]

    State-transition dynamics of resting state fMRI data

    Islam S. State-transition dynamics of resting state fMRI data. GitHub. 2024 [cited Nov 11, 2024]. https: //github.com/sislam99/fmri_state_transition_dynamics

    2024

  89. [97]

    How transferable are features in deep neural networks? Adv Neural Inf Process Syst

    Yosinski J, Clune J, Bengio Y, Lipson H. How transferable are features in deep neural networks? Adv Neural Inf Process Syst. 2014; 27:3320–3328. 27

    2014

  90. [98]

    Transfer learning for scalability of neural- network quantum states

    Zen R, My L, Tan R, H´ ebert F, Gattobigio M, Miniatura C, et al. Transfer learning for scalability of neural- network quantum states. Phys Rev E. 2020; 101(5):053301

    2020

  91. [99]

    Expression profiles of switch-like genes accurately classify tissue and infectious disease phenotypes in model-based classification

    Gormley M, Tozeren A. Expression profiles of switch-like genes accurately classify tissue and infectious disease phenotypes in model-based classification. BMC Bioinform. 2008; 9:486

    2008

  92. [100]

    Functional roles for noise in genetic circuits

    Eldar A, Elowitz MB. Functional roles for noise in genetic circuits. Nature. 2010; 467(7312):167–173

    2010

  93. [101]

    Bimodal distribution of RNA expression levels in human skeletal muscle tissue

    Mason CC, Hanson RL, Ossowski V, Bian L, Baier LJ, Krakoff J, et al. Bimodal distribution of RNA expression levels in human skeletal muscle tissue. BMC Genomics. 2011; 12:98

    2011

  94. [102]

    Switch-like gene expression modulates disease susceptibility

    Aqil A, Li Y, Wang Z, Islam S, Russel M, Kallak TK, et al. Switch-like gene expression modulates disease susceptibility. Preprint bioRxiv. 2024. doi:10.1101/2024.08.24.609537

    2024 doi

  95. [103]

    Early-warning signals for critical transitions

    Scheffer M, Bascompte J, Brock W A, Brovkin V, Carpenter SR, Dakos V, et al. Early-warning signals for critical transitions. Nature. 2009; 461(7260):53-59

    2009

  96. [104]

    Anticipating critical transitions

    Scheffer M, Carpenter SR, Lenton TM, Bascompte J, Brock W, Dakos V, et al. Anticipating critical transitions. Science. 2012; 338(6105):344–348

    2012

  97. [105]

    Early warning signals: the charted and uncharted territories

    Boettiger C, Ross N, Hastings A. Early warning signals: the charted and uncharted territories. Theor Ecol. 2013; 6(3):255-264

    2013

  98. [106]

    Resilience indicators: prospects and limitations for early warnings of regime shifts

    Dakos V, Carpenter SR, van Nes EH, Scheffer M. Resilience indicators: prospects and limitations for early warnings of regime shifts. Philos Trans R Soc B. 2015; 370(1659):20130263

    2015

  99. [107]

    Early warning signals of infectious disease transitions: a review

    Southall E, Brett TS, Tildesley MJ, Dyson L. Early warning signals of infectious disease transitions: a review. J R Soc Interface. 2021; 18(181):20210555

    2021

  100. [108]

    A computational framework for controlling the self-restorative brain based on the free energy and degeneracy principles

    Park HJ, Kang J. A computational framework for controlling the self-restorative brain based on the free energy and degeneracy principles. Front Comput Neurosci. 2021; 15:590019. 28

    2021

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.