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Dear Dr. Marquand,

We thank you and the reviewers for your time and the insightful and constructive assessment of our manuscript, ”Connectome-Based Attractor Dynamics Underlie Brain Activity in Rest, Task, and Disease,” and for the opportunity to revise it.

We apologize for the lengthy revision process. The reviewers’ comments about the alignment of the first two attractors with the principal components of the time-series data highlighted important connections to our ongoing theoretical work. We therefore first completed a related theoretical study, allowing us to integrate its key insights into this revision. In that recent theoretical work (Spisak & Friston, 2025), we showed that the specific type of attractor networks emerging from first principles of self-organization (i.e. the free energy principle) displays the same functional form as the model proposed in the present manuscript. The theoretical framework also predicts that free-energy-minimizing attractor exhibit a special characteristic: they naturally develop approximately orthogonal attractor representations. This may yield an appealing answer to the reviewers’ question: the close alignment between attractors and principal components may be a necessary consequence of this “self-orthogonalization” property, suggesting that the brain, as an attractor network, approximates the computationally efficient structure of the so-called Kanter-Sompolinsky projector neural networks (Kanter & Sompolinsky, 1987), where eigenvectors of the coupling matrix and attractors become equivalent.

We therefore extended our study with a high-level summary of the new framework and a new analysis that directly tests this theoretical prediction by quantifying the similarity between empirically reconstructed brain attractors and the eigenvectors of the coupling matrix, as well as the similarity of the whole network to a Kanter-Sompolinsky projector network with the same attractors. Our results present first empirical evidence of “self-orthogonalization” in large-scale brain attractor dynamics and explains it as an emergent property of free-energy minimization.

Integrating this new theoretical and empirical results into the present manuscript highlights that our approach is more than an ad-hoc phenomenological model; it is now directly built upon a general formal framework rooted in first principles of self-organization in sparsely coupled dynamical systems. This establishes formal links between the computations performed by the attractor states and Bayesian inference and predictive coding, which we now briefly discuss in the revised manuscript.

In addition to incorporating these new results, we have thoroughly revised the manuscript to address all other comments from the reviewers. We present our point-by-point responses below. Note that - other than adding the new results about orthogonality - we didn’t have to change any of the analyses in the manuscript, as those were already well aligned with the new theoretical framework.

We are grateful for the insightful comments, which, as outlined above, have been transformative for this work. We apologize again for the extended revision period and hope the revisions and our accompanying responses will make our manuscript suitable for publication in eLife. We would be glad to respond to any further questions you or the referees may have.

Yours sincerely,

Tamas Spisak, on behalf of all co-authors

References

Kanter, I., & Sompolinsky, H. (1987). Associative recall of memory without errors. Physical Review A, 35(1), 380–392. 10.1103/physreva.35.380 Spisak T & Friston K (2025). Self-orthogonalizing attractor neural networks emerging from the free energy principle. arXiv preprint arXiv:2505.22749.

Public Reviews:

Author response preface:

We thank the reviewers for their insightful comments. The feedback and revision process was transformative for this work, prompting us to introduce formal links between our analyses and a new theoretical framework of self-orthogonalizing attractor networks (Spisak & Friston, 2025), opening several new avenues in terms of interpretation and shedding light on the computational principles underlying attractor dynamics in the brain. In our point-by-point responses to the reviewers’ comments below - next to explaining how we addressed each of the specific comments - we also take special care to detail how the work benefits from explicating the links to the more general theoretical framework.

Reviewer #1 (Public Review):

Summary:

Englert et al. proposed a functional connectome-based Hopfield artificial neural network (fcHNN) architecture to reveal attractor states and activity flows across various conditions, including resting state, task-evoked, and pathological conditions. The fcHNN can reconstruct characteristics of resting-state and task-evoked brain activities. Additionally, the fcHNN demonstrates differences in attractor states between individuals with autism and typically developing individuals.

Strengths:

(1) The study used seven datasets, which somewhat ensures robust replication and validation of generalization across various conditions.

(2) The proposed fcHNN improves upon existing activity flow models by mimicking artificial neural networks, thereby enhancing the representational ability of the model. This advancement enables the model to more accurately reconstruct the dynamic characteristics of brain activity.

(3) The fcHNN projection offers an interesting visualization, allowing researchers to observe attractor states and activity flow patterns directly.

We are grateful to the reviewer for highlighting the robustness of our findings across multiple datasets and for appreciating the novelty and representational advantages of our fcHNN model (which has been renamed to fcANN in the revised manuscript).

Weaknesses:

(1) The fcHNN projection can offer low-dimensional dynamic visualizations, but its interpretability is limited, making it difficult to make strong claims based on these projections. The interpretability should be enhanced in the results and discussion.

We thank the reviewer for these important points. We agree that the interpretability of the low-dimensional projection is limited. In the revised manuscript, we have reframed the fcANN projection primarily as a visualization tool and moved the corresponding part of Figure 2 to the Supplementary Material (Supplementary Figure 2). We have also implemented a substantial revision of the manuscript, which now directly links our analysis to a novel theoretical framework of self-orthogonalizing attractor networks (Spisak & Friston, 2025), opening several new avenues in terms of interpretation and shedding light on the computational principles underlying attractor dynamics in the brain.

(2) The presentation of results is not clear enough, including figures, wording, and statistical analysis, which contributes to the overall difficulty in understanding the manuscript. This lack of clarity in presenting key findings can obscure the insights that the study aims to convey, making it challenging for readers to fully grasp the implications and significance of the research.

We have thoroughly revised the manuscript for clarity in wording, figures, and statistical reporting, addressing the specific points raised in the recommendations. Specifically, we now clearly define our research questions and the corresponding analyses and null models in the revised manuscript, both in the main text and in two new tables (Tables 1 and 2).

todo: improved the methods description...

Reviewer #2 (Public Review):

Summary:

Englert et al. use a novel modelling approach called functional connectome-based Hopfield Neural Networks (fcHNN) to describe spontaneous and task-evoked brain activity and the alterations in brain disorders. Given its novelty, the authors first validate the model parameters (the temperature and noise) with empirical resting-state function data and against null models. Through the optimisation of the temperature parameter, they first show that the optimal number of attractor states is four before fixing the optimal noise that best reflects the empirical data, through stochastic relaxation. Then, they demonstrate how these fcHNN-generated dynamics predict task-based functional activity relating to pain and self-regulation. To do so, they characterise the different brain states (here as different conditions of the experimental pain paradigm) in terms of the distribution of the data on the fcHNN projections and flow analysis. Lastly, a similar analysis was performed on a population with autism condition. Through Hopfield modeling, this work proposes a comprehensive framework that links various types of functional activity under a unified interpretation with high predictive validity.

Strengths:

The phenomenological nature of the Hopfield model and its validation across multiple datasets presents a comprehensive and intuitive framework for the analysis of functional activity. The results presented in this work further motivate the study of phenomenological models as an adequate mechanistic characterisation of large-scale brain activity.

Following up on Cole et al. 2016, the authors put forward a hypothesis that many of the changes to the brain activity, here, in terms of task-evoked and clinical data, can be inferred from the resting-state brain data alone. This brings together neatly the idea of different facets of brain activity emerging from a common space of functional (ghost) attractors.

The use of the null models motivates the benefit of non-linear dynamics in the context of phenomenological models when assessing the similarity to the real empirical data.

We thank the reviewer for recognizing the comprehensive and intuitive nature of our framework and for acknowledging the strength of our hypothesis that diverse brain activity facets emerge from a common resting state attractor landscape.

Weaknesses:

While the use of the Hopfield model is neat and very well presented, it still begs the question of why to use the functional connectome (as derived by activity flow analysis from Cole et al. 2016). Deriving the functional connectome on the resting-state data that are then used for the analysis reads as circular.

We agree that starting from functional couplings to study dynamics is in stark contrast with the common practice of estimating the interregional couplings based on structural connectome data. We now explicitly discuss how this affects the scope of the questions we can address with the approach, with explicit notes on the inability of this approach to study the structure-function coupling and its limitations in deriving mechanistic insights at the level of biophysical implementation.

The proposed approach is not without limitations. First, as the proposed approach does not incorporate information about anatomical connectivity and does not explitly model biophysical details. Thus, in its present form, the model is not suitable to study the structure-function coupling and cannot yiled mechanistic explanations underlying (altered) polysynaptic connections, at the level of biophysical details.

We are confident, however, that our approach is not circular. At the high level, our approach can be considered as a function-to-function generative model, with twofold aims.

First, we link large-scale brain dynamics to theoretical artificial neural network models and show that the functional connectome display characteristics that render it as an exceptionally “well-behaving” attractor network (e.g. superior convergence properties, as contrasted against appropriate respective null models). In the revised manuscript, we have significantly improved upon this aspect by explicitly linking the fcANN model to the theoretical framework of self-orthogonalizing attractor networks (Spisak & Friston, 2025) (see e.g. lines xx). As part of these efforts, we now provide evidence for the brain’s functional organization approximating a special, computationally efficient class of attractor networks, the so-called Kanter-Sompolinsky projector network. This is exactly, what the theoretical framework of free-energy-minimizing attractor networks predicts. This result is not circular, as the empirical model does not use the key mechanism (the Hebbian/anti-Hebbian learning rule) that induces self-orthogonalization in the theoretical framework. We clarify this in the revised manuscript, e.g. in line xx.

Second, we benchmark ability of the proposed function-to-function generative model to predict unseen data, including data about perturbations on the system. These benchmarks are constructed against well defined null models, which provide reasonable references (e.g. covariance-matched multivariate Gaussian distribution). We have now significantly improved the discussion of these null models in the revised manuscript (Tables 1 and 2, lines xx). We not only show, that our model - when reconstructing resting state dynamics - can generalize to unseen data over and beyond what is possible with the baseline descriptive measure (here: PCA), but also demonstrate the ability of the framework to reconstruct the effects of perturbations on this dynamics (such as task-evoked changes), based solely on the resting state data form another sample.

If the fcHNN derives the basins of four attractors that reflect the first two principal components of functional connectivity, it perhaps suffices to use the empirically derived components alone and project the task and clinical data on it without the need for the fcHNN framework.

We are thankful for the reviewer for highlighting this important point, which encouraged us to develop a detailed understanding of the origins of the close alignment between attractors and principal components (eigenvectors of the coupling matrix) and the corresponding (approximate) orthogonality. Here, we would like to emphasize that the attractor-eigenvector correspondence is by no means a general feature of any arbitrary attractor network. In fact, such networks are a very special class of attractor neural networks (the so-called Kanter-Sompolinsky projector neural network (Kanter & Sompolinsky, 1987)), with a high degree of computational efficiency, maximal memory capacity and perfect memory recall. It has been rigorously shown that in such networks, the eigenvectors of the coupling matrix (i.e. PCA on the timeseries data) and the attractors become equivalent (Kanter & Sompolinsky, 1987). This in turn made us ask the question, what are the learning and plasticity rules that drive attractor networks towards developing approximately orthogonal attractors? We found that this is a general tendency of networks obeying the free energy principle. The formal derivation of this framework in now presented in an accompanying theoretical piece (Spisak & Friston, 2025). In the revised manuscript, we provide a short, high-level overview of these results (lines xx). According to this new theoretical model, attractor states can be understood as a set of priors (in the Bayesian sense) that together constitute an optimal orthogonal basis, equipping the update process (which is akin to a Markov-chain Monte Carlo sampling) to find posteriors that generalize effectively within the spanned subspace. Thus, in sum, understanding brain function in terms of attractor dynamics - instead of PCA-like descriptive projections - provides important links towards a Bayesian interpretation of brain activity. At the same time, the eigenvector-attractor correspondence also explains, why descriptive decomposition approaches, like PCA or ICA are so effective at capturing the dynamics of the system, at the first place.

As presented here, the Hopfield model is excellent in its simplicity and power, and it seems suited to tackle the structure-function relationship with the power of going further to explain task-evoked and clinical data. The work could be strengthened if that was taken into consideration. As such the model would not suffer from circularity problems and it would be possible to claim its mechanistic properties. Furthermore, as mentioned above, in the current setup, the connectivity matrix is based on statistical properties of functional activity amongst regions, and as such it is difficult to talk about a certain mechanism. This contention has for example been addressed in the Cole et al. 2016 paper with the use of a biophysical model linking structure and function, thus strengthening the mechanistic claim of the work.

We agree that investigating how the structural connectome constraints macro-scale dynamics is a crucial next step. Linking our results with the theoretical framework of self-orthogonalizing attractor networks provides a principled approach to this, as the “self-orthogonalizing” learning rule in the accompanying theoretical work provides the opportunity to fit attractor networks with structural constraints to functional data, shedding light on the plastic processes which maintain the observed approximate orthogonality even in the presence of these structural constraints. We have revised the manuscript to discuss this promising direction (lines xx), and also clarify that our phenomenological approach is inherently limited in its ability to answer mechanistic questions at the level of biophysical details (lines xx).

Recommendations for the authors: please note that you control which revisions to undertake from the public reviews and recommendations for the authors.

Reviewer #1 (Recommendations For The Authors):

(1) The statistical analyses are poorly described throughout the manuscript. The authors should provide more details on the statistical methods used for each comparison, as well as the corresponding statistics and degrees of freedom, rather than solely reporting p-values.

We thank the reviewer for pointing this out. We have revised the manuscript to include the specific test statistics, degrees of freedom, and precise p-values for all reported analyses to ensure full transparency and replicability. Additionally, we have improved the description of the statistical methods used in the manuscript (lines xx) and added two overview tables (Tables 1 and 2) that summarize the methodological approaches and the corresponding null models.

todo: fix statistics, revise methods

(2) The convergence procedure is not clearly explained in the manuscript. Is this an optimization procedure to minimize energy? If so, the authors should provide more details about the optimizer used.

We apologize for the lack of clarity. The convergence is not an optimization procedure per se, in a sense that it does not involve any external optimizer. It is simply the repeated (deterministic) application of the the same update rule also known from Hopfield networks or Boltzmann machines. However, as detailed in the accompanying theoretical paper, this update rule (or inference rule) inherently solves and optimization problem: it performs gradient descent on the free energy landscape of the network. As such, it is guaranteed to converge to a local free energy minimum in the deterministic case. We have clarified this process in the Results and Methods sections (lines xx), including the concrete links to free energy minimization.

todo: revise methods

(3) In Figure 2G, the beta values range from 0.035 to 0.06, but they are reported as 0.4 in the main text and the Supplementary Figure. Please clarify this discrepancy.

We are grateful to the reviewer for spotting this typo. The correct value for β\beta is 0.04, as reported in the Methods section. We have corrected this inconsistency in the revised manuscript and Supplementary Figure 5.

todo: correct supplementary figure

(4) Line 174: What type of null model was used to evaluate the impact of the beta values? The authors did not provide details on this anywhere in the manuscript.

We apologize for this omission. The null model is based on permuting the connectome weights while retaining the matrix symmetry, which destroys the specific topological structure but preserves the overall weight distribution. We have now clarified this at multiple places in the revised manuscript (lines xx), and added new overview tables (Tables 1 and 2) to summarize the methodological approaches and the corresponding null models.

(5) Figure 3B: It appears that the authors only demonstrate the reproducibility of the “internal” attractor across different datasets. What about other states?

This is an excellent point. We now visualize all attractor states in Figure 3B (note that these essentially consist of two symmetric pairs).

(6) Figure 3: What does “empirical” represent in Figure 3? Is it PCA? If the “empirical” method, which is a much simpler method, can achieve results similar to those of the fcHNN in terms of state occupancy, distribution, and activity flow, what are the benefits of the proposed method? Furthermore, the authors claim that the explanatory power of the fcHNN is higher than that of the empirical model and shows significant differences. However, from my perspective, this difference is not substantial (37.0% vs. 39.9%). What does this signify, particularly in comparison to PCA?

This is a crucial point that is now a central theme of our revised manuscript. The reviewer is correct that the “empirical” method is PCA. PCA - by identifying variance-heavy orthogonal directions - aims to explain the highest amount of variance possible in the data (with the assumption of Gaussian conditionals). While empirical attractors are closely aligned to the PCs (i.e. eigenvectors of the inverse covariance matrix), the alignment is only approximate. We basically take advantage of this small “gap”, to quantify, weather attractor states are a better fit to the unseen data than the PCs. Obviously, due to the otherwise strong PC-attractor correspondence, this is expected to be only a small improvement. However, it is an important piece of evidence for the validity of our framework, as it shows that attractors are not just a complementary, perhaps “noisier” variety of the PCs, but a “substrate” that generalizes better to unseen data than the PCs themselves. We have revised the manuscript to clarify this point (lines xx).

Reviewer #2 (Recommendations For The Authors):

For clarity, it might be useful to define and use consistently certain key terms. Connectome often refers to structural (anatomical) connectivity unless defined specifically this should be considered, in Figure 1B title for example Brain state often refers to different conditions ie autism, neurotypical, sleep, etc... see for review Kringelbach et al. 2020, Cell Reports. When referring to attractors of brain activity they might be called substates.

We thank the reviewer for these helpful suggestions. We have carefully revised the manuscript to ensure our terminology is precise and consistent. We now explicitly refer to the “functional connectome” (including the title) and avoid using the too general term “brain state”. Furthemore, we discuss the relationship between the attractors and the general phenomenon of brain states in light of the new theoretical framework.

todo: revise term

In Figure 2 some terms are not defined. Noise is sigma in the text but elpsilon in the figure. Only in methods, the link becomes clear. Perhaps define epsilon in the caption for clarity. The same applies to μ in the methods. It is only described above in the methods, I suggest repeating the epsilon definition for clarity

We appreciate this feedback and apologize for the inconsistency. We have revised all figures and the Methods section to ensure that all mathematical symbols (including ε, σ, and μ) are clearly and consistently defined upon their first appearance and in all figure captions.

todo: revise terms

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