Hierarchical Neural Circuit Theory of Normalization and Inter-areal Communication

  1. Asit Pal
  2. Shivang Rawat
  3. David J Heeger
  4. Stefano Martiniani

  • Curated by eLife

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    eLife Assessment

    This important study extends a model of cortical normalization (ORGaNICs) to interacting cortical areas and shows that communication through coherence and communication subspaces can arise from a single set of dynamics. The evidence is convincing, showing analytically that contrast-dependent gamma dynamics and a low-dimensional inter-areal communication subspace arise from one parameter set. The work will interest neuroscientists and theorists concerned with inter-areal communication, cortical oscillations, and divisive normalization.

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Abstract

The primate brain exhibits a hierarchical, modular architecture with conserved microcircuits executing canonical computations across reciprocally connected cortical areas. We present a hierarchical neural circuit theory with feedback connections that dynamically implements divisive normalization across its hierarchy. In a two-stage instantiation (V1 ↔ V2), increasing feedback from V2 to V1 amplifies responses in both areas. We analytically derive power spectra (V1) and coherence spectra (V1-V2) as functions of frequency (f), and compare them with experimental observations: peaks in both spectra shift to higher frequencies with increased stimulus contrast, and power decays as 1/f4 at high frequencies. The closed-form spectra are validated against direct stochastic simulation of the full nonlinear circuit across contrasts. The theory further predicts distinctive spectral signatures of feedback and input gain modulation. Crucially, the theory offers a unified view of inter-areal communication, defined as the ability to linearly predict neural activity in one brain area from neural activity in another, with emergent features consistent with empirical observations of both communication subspaces and inter-areal coherence. It admits a low-dimensional communication subspace, where inter-areal communication is lower-dimensional than within-area communication. It further predicts that: i) increasing feedback strength enhances inter-areal communication and diminishes within-area communication, without altering the subspace dimensionality; ii) communication is strongest, and the subspace dimensionality lowest, at the frequencies at which V1-V2 coherence peaks; iii) normalization reduces the subspace dimensionality. Finally, a three-area (V1 ↔ V4 and V1 ↔ V5) instantiation of the theory demonstrates that differential feedback from higher to lower cortical areas dictates their dynamic functional connectivity. Altogether, our theory provides a robust and analytically tractable framework for generating experimentally-testable predictions about normalization, inter-areal communication, and functional connectivity.

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  1. Version published to 10.7554/elife.111297.2 on eLife
  2. Version published to 10.7554/elife.111297 on eLife
  3. eLife

    eLife Assessment

    This important study extends a model of cortical normalization (ORGaNICs) to interacting cortical areas and shows that communication through coherence and communication subspaces can arise from a single set of dynamics. The evidence is convincing, showing analytically that contrast-dependent gamma dynamics and a low-dimensional inter-areal communication subspace arise from one parameter set. The work will interest neuroscientists and theorists concerned with inter-areal communication, cortical oscillations, and divisive normalization.

  4. eLife

    Reviewer #1 (Public review):

    [Editors' note: this version has been assessed by the Reviewing Editor without further input from the original reviewers. The authors have addressed the comments raised in the previous round of review.]

    In this paper, Pal and colleagues propose a mechanistic unification of two influential accounts of inter-areal communication: communication through coherence and communication subspaces. A major strength of the paper is that it does not treat coherence and communication subspaces as independent phenomena, as typically done, but instead derives both from the same circuit with divisive normalization. In this framework, noise-driven fluctuations around the normalized fixed point determine covariance and cross-power structure (which, in retrospect, makes so much sense to be related). Then, they show how these determine linear prediction performance and the effective dimensionality of the communication subspace. They also show (however not very visually, see recommendation below for a figure) how divisive normalization is crucial to shape inter-areal coherence and the dimensionality of communication.

    I found this conceptual contribution potentially very influential, but somewhat obscured by the technical complexity of the model. The central intuition (I think) is that recurrent normalization can organize cross-area fluctuations, both frequency-specific correlations and cross-covariances. Took me a while to grasp this insight, mostly because I was stuck with the model details. Note that I have some experience with network dynamics, but not with this particular model.

  5. eLife

    Reviewer #2 (Public review):

    Summary:

    The authors extend the ORGaNICs framework (a recurrent circuit that dynamically implements divisive normalization) to connected cortical areas with explicit top-down feedback. Because the network has a known analytical fixed point that coincides with (or closely approximates) the normalization equation, the authors can linearize about that fixed point and derive closed-form expressions for the power spectral density, inter-areal coherence, and communication subspaces. Using a two-area instantiation (V1 & V2) with a single fixed parameter set and no data fitting, they show the model reproduces: (i) contrast-response functions with steeper slope V2; (ii) gamma-band power and coherence peaks that shift to higher frequency with contrast; and (iii) a low-dimensional inter-areal communication subspace that is lower-dimensional than the within-area subspace. They derive parallel predictions of what happens by changing model parameters: feedback gain enhances inter-areal and suppresses within-area communication, and normalization is necessary for both the oscillatory dynamics and the reduced subspace dimensionality. A three-area extension (V1&V4, V1&V5/MT) is used to argue that differential top-down feedback can dynamically route functional connectivity.

    Strengths:

    (1) Analytical tractability: Deriving power spectra, coherence, and communication-subspace structure in closed form from a known fixed point is genuinely valuable.

    (2) Conceptual unification: Framing coherence and communication subspaces as arising from the same normalization-driven dynamics is an elegant and useful contribution.

    (3) Breadth from few assumptions: A large range of phenomena (contrast gain, gamma dynamics) emerges from normalization-based model assumptions.

    (4) Biological grounding: The mapping of model variables onto identified cell types connects the abstract computation to known cortical microcircuitry.

    (5) The prediction that input-gain versus feedback-gain modulation produce distinct spectral signatures gives experimentalists a clear way to test the framework.

  6. eLife

    Reviewer #3 (Public review):

    Summary

    The work of Pal and colleagues considers a hierarchical and multi-population version of the "oscillatory recurrent gated neural integrator circuits" (ORGaNICs) model, showing through analytics that the model captures multiple relevant experimental results: first of all, its oscillatory dynamics produce a profile with high resemblance to experimental results, both in terms of decay of power at high frequency and in terms of shifting peak as a function of stimulus contrast. Second, inter-areal communication subspace dimensionality is lower than within-area dimensionality. The authors then proceed to further characterize the model's response properties as a function of input and feedback gain. In particular, they find that frequencies transmitted with higher strength also carry more information, that changing gain modifies the dimensionality of communication subspaces, and that these properties can be used in a three-layer model, where an upstream area can select which downstream area to communicate to, based on the strength of feedback gain.

    Strengths

    This work demonstrates that a single-circuit model with normalization properties can capture both the oscillatory dynamics and the inter-areal communication properties measured in cortical circuits, matching multiple experimental results. The full analytical tractability of the model is highly advantageous, allowing for easier exploration of parameters, replicability, and effective interpretations of results compared to purely numerical approaches.

    The work also makes a useful conceptual link between normalization, coherence-based communication, and subspace-based communication. In particular, it shows how both phenomena can emerge from the same circuit dynamics, where normalization is a key factor.

    Interestingly, the model is also extended to multiple areas, showing how attention (in the form of changes in feedback gain) can synchronize the activity of a downstream area with one of two upstream areas, thus effectively selecting which area to communicate with.

    In general, this is an interesting computational framework and a useful starting point for future modeling work. A particular strength is that it connects normalization, oscillatory dynamics, coherence, and communication subspaces within one analytically tractable model, making it possible to generate mechanistic hypotheses about when inter-areal communication should be stronger, lower-dimensional, or preferentially routed through feedback.

  7. eLife

    Author response:

    The following is the authors’ response to the original reviews.

    eLife Assessment:

    This important study extends a model of cortical normalization (ORGaNICs) to interacting cortical areas and shows that communication through coherence and communication subspaces can arise from a single set of dynamics. The evidence is solid, showing analytically that contrast-dependent gamma dynamics and a low-dimensional inter-areal communication subspace arise from one parameter set, though the comparisons to data remain qualitative and the analytics rest on a linearization that is not checked against numerical simulation. The work will interest neuroscientists and theorists concerned with inter-areal communication, cortical oscillations, and divisive normalization.

    We thank the editors and reviewers for this assessment. We have addressed both stated qualifications. On the linearization, we now validate the closed-form power and coherence spectra against direct numerical simulation of the full nonlinear stochastic circuit at every contrast (new SI Fig. S6 and SI Sec. II). On the comparison to data, in addition to the unfitted comparison of Fig. 2, we now provide a quantitative comparison in which the model is fitted to the experimental power and coherence spectra, with residuals reported at every contrast (new SI Fig. S7 and SI Sec. III).

    Public Reviews:

    Reviewer #1 (Public review):

    In this paper, Pal and colleagues propose a mechanistic unification of two influential accounts of inter-areal communication: communication through coherence and communication subspaces. A major strength of the paper is that it does not treat coherence and communication subspaces as independent phenomena, as typically done, but instead derives both from the same circuit with divisive normalization. In this framework, noise-driven fluctuations around the normalized fixed point determine covariance and cross-power structure (which, in retrospect, makes so much sense to be related). Then, they show how these determine linear prediction performance and the effective dimensionality of the communication subspace. They also show (however not very visually, see recommendation below for a figure) how divisive normalization is crucial to shape inter-areal coherence and the dimensionality of communication.

    I found this conceptual contribution potentially very influential, but somewhat obscured by the technical complexity of the model. The central intuition (I think) is that recurrent normalization can organize cross-area fluctuations, both frequency-specific correlations and cross-covariances. Took me a while to grasp this insight, mostly because I was stuck with the model details. Note that I have some experience with network dynamics, but not with this particular model.

    We thank the reviewer for this positive assessment. We agree that recurrent normalization shapes the cross-spectral density of the noise-driven fluctuations about the normalized fixed point, and that power, coherence, and the communication subspace can therefore be viewed as complementary readouts of the same underlying dynamics. To make this central connection explicit from the outset, we have added a new paragraph, “The core mechanism in brief” (Results, end of “Theory”), which states this mechanism before the main results are presented. The dependence of coherence and communication-subspace dimensionality on normalization is shown in SI Figs. S4e–h and S5, which are now cited wherever the role of normalization is discussed. In addition, the new Discussion subsection “Experimental predictions of the model” describes how normalization could be manipulated experimentally and the readouts expected.

    Reviewer #2 (Public review):

    Summary:

    The authors extend the ORGaNICs framework (a recurrent circuit that dynamically implements divisive normalization) to connected cortical areas with explicit top-down feedback. Because the network has a known analytical fixed point that coincides with (or closely approximates) the normalization equation, the authors can linearize about that fixed point and derive closed-form expressions for the power spectral density, inter-areal coherence, and communication subspaces. Using a two-area instantiation (V1 & V2) with a single fixed parameter set and no data fitting, they show the model reproduces: (i) contrast-response functions with steeper slope V2; (ii) gamma-band power and coherence peaks that shift to higher frequency with contrast; and (iii) a low-dimensional inter-areal communication subspace that is lower-dimensional than the within-area subspace. They derive parallel predictions of what happens by changing model parameters: feedback gain enhances inter-areal and suppresses within-area communication, and normalization is necessary for both the oscillatory dynamics and the reduced subspace dimensionality. A three-area extension (V1&V4, V1&V5/MT) is used to argue that differential top-down feedback can dynamically route functional connectivity.

    We thank the reviewer for this summary.

    Strengths:

    (1) Analytical tractability: Deriving power spectra, coherence, and communication-subspace structure in closed form from a known fixed point is genuinely valuable.

    (2) Conceptual unification: Framing coherence and communication subspaces as arising from the same normalization-driven dynamics is an elegant and useful contribution.

    (3) Breadth from few assumptions: A large range of phenomena (contrast gain, gamma dynamics) emerges from normalization-based model assumptions.

    (4) Biological grounding: The mapping of model variables onto identified cell types connects the abstract computation to known cortical microcircuitry.

    (5) The prediction that input-gain versus feedback-gain modulation produce distinct spectral signatures gives experimentalists a clear way to test the framework.

    We thank the reviewer for the positive assessment.

    Weaknesses:

    (1) Comparisons are qualitative, not quantitative: The theory/experiment panels are visual side-by-side comparisons. There is no quantitative goodness-of-fit for any predictions.

    We thank the reviewer for this comment. We agree, and have added a quantitative comparison. New SI Fig. S7 and SI Sec. III fit the model parameters to the experimental power and coherence spectra and report the residuals at every contrast.

    (2) The simulations use τ ≈ 1ms for all cell types, which the authors acknowledge is unrealistically short; realistic values would shift the gamma peaks to lower frequencies.

    We thank the reviewer for this comment. We agree. The Limitations (Discussion) state that more realistic values would shift the gamma-band oscillations to lower frequencies, and that when all time constants were released as free parameters in the fit to the experimental spectra (SI Fig. S7) they converged to values between 0.2 and 3ms. The time constants are now described consistently as effective population time constants rather than as intrinsic single-neuron time constants (Limitations).

    (3) Divisive normalization is a special case and is recovered exactly only for the identity recurrent matrix (self-normalization). Some statements that the circuit implements divisive normalization exactly need softening.

    We thank the reviewer for this comment. We agree, and have softened the statements throughout. The main text now states that exact recovery of Eq. 2 requires W11 = I and β = γ = 1, that for the centre-surround W11 used in all reported results the circuit closely approximates normalization (to within 0.6%), and that the fixed point is obtained numerically, at first mention (Results, “Theory”), in the Methods (“The Model” and the new subsection “Numerical computation of fixed points”) and in the Discussion (“Normalization”).

    (4) The element-wise (multiplicative) interaction in the modulator dynamics is not tied to a specific cellular mechanism.

    We thank the reviewer for this comment. We agree: The Limitations (Discussion) state that, while weighted sums and gain control may be implemented via balanced excitatory–inhibitory mechanisms, we can only speculate about the cellular mechanisms for the element-wise products. We speculate that the element-wise product between presynaptic firing rates in Eq. 1b might be mediated by coincident spike detection (assuming that each of the spike trains is Poisson and that the synaptic current is proportional to the probability of coincident spikes within a short time window (Srinivasan & Bernard, 1976)), and that the element-wise product between postsynaptic membrane potential and presynaptic firing rate in Eq. 1c might be mediated by NMDA receptors (Polsky, Mel & Schiller, 2004; Silver, 2010) (see Discussion, last paragraph of “Biological Plausibility”).

    Reviewer #3 (Public review):

    Summary:

    The work of Pal and colleagues considers a hierarchical and multi-population version of the “oscillatory recurrent gated neural integrator circuits” (ORGaNICs) model, showing through analytics that the model captures multiple relevant experimental results: first of all, its oscillatory dynamics produce a profile with high resemblance to experimental results, both in terms of decay of power at high frequency and in terms of shifting peak as a function of stimulus contrast. Second, inter-areal communication subspace dimensionality is lower than within-area dimensionality. The authors then proceed to further characterize the model’s response properties as a function of input and feedback gain. In particular, they find that frequencies transmitted with higher strength also carry more information, that changing gain modifies the dimensionality of communication subspaces, and that these properties can be used in a three-layer model, where an upstream area can select which downstream area to communicate to, based on the strength of feedback gain.

    We thank the reviewer for the summary.

    Strengths:

    This work demonstrates that a single-circuit model with normalization properties can capture both the oscillatory dynamics and the inter-areal communication properties measured in cortical circuits, matching multiple experimental results. The full analytical tractability of the model is highly advantageous, allowing for easier exploration of parameters, replicability, and effective interpretations of results compared to purely numerical approaches.

    The work also makes a useful conceptual link between normalization, coherence-based communication, and subspace-based communication. In particular, it shows how both phenomena can emerge from the same circuit dynamics, where normalization is a key factor.

    Interestingly, the model is also extended to multiple areas, showing how attention (in the form of changes in feedback gain) can synchronize the activity of a downstream area with one of two upstream areas, thus effectively selecting which area to communicate with.

    In general, this is an interesting computational framework and a useful starting point for future modeling work. A particular strength is that it connects normalization, oscillatory dynamics, coherence, and communication subspaces within one analytically tractable model, making it possible to generate mechanistic hypotheses about when inter-areal communication should be stronger, lower-dimensional, or preferentially routed through feedback.

    We thank the reviewer for the positive assessment.

    Weaknesses:

    Although I see the analytic approach as a strength, at the same time I regard the lack of any numerical comparison as a big weakness. Circuit simulations would not only confirm the correctness of the analytics, but also offer further insights on the error margins and on the regimes where the analytics are valid. This is because, to my understanding, the analytics are based on a linear approximation around the operating regime, which means deviations might be expected, especially for high gain levels in the input, or in the feedforward and feedback pathways.

    We thank the reviewer for this comment. We agree that a numerical check of the linearization is required, and have added one. New SI Fig. S6 and SI Sec. II integrate the full nonlinear stochastic system (SI Eqs. S31–S44, with the firing-rate noise and the filtered synaptic noise of SI Sec. VI) with the Euler–Maruyama scheme (time step 10 µs; 100 independent trials of 16s at each contrast) and compare the V1 power spectra and V1–V2 coherence spectra estimated from the simulated trajectories (Welch’s method) with the closed-form expressions at all eight contrasts of Fig. 2. At the baseline parameters, the root-mean-square deviation is below 0.017 for the normalised power and below 0.018 for the coherence, and the maximum absolute deviation is below 0.072, at every contrast; residuals are shown in the lower panels of Fig. S6. The Methods (“Power and coherence spectra”) now describe this validation. On the regime of validity: the linearization holds for small fluctuations about the fixed point, as stated in the Methods, and the simulations bear this out. When the amplitudes of the noise sources are increased, the deviation between the simulated and the analytical spectra grows, and it does so first at the lowest contrasts. This is because at low contrast < 10% the activities lie near zero, and the noise fluctuations can drive them negative, where the rectification in the model clips them to zero; the linearized dynamics do not capture this clipping. Note that changing the strength of the noise does not alter the analytical results, since the plotted spectra are all normalized. The deviations do not, however, grow with the gains: changing the input or feedback gain changes the fixed point, and the linearization about the new fixed point remains valid, because its accuracy is controlled by the strength of the fluctuations relative to the fixed point, not by the gains themselves. This characterization of the validity regime is now stated in SI Sec. II. All results reported in the paper use the baseline parameters, for which the linearization is accurate at every contrast and for all reported gain values. The stable ranges of the gains are given in SI Table S2.

    Another problem is that the analytically tractable model seems to rely on effective connectivity weights that break Dale’s law. Numerical simulations with explicitly modeled excitatory and inhibitory units might give insights into effects due, e.g., to the additional transmission delays mentioned in the Discussion.

    We thank the reviewer for this comment. We agree, and have made the distinction between effective and anatomical connectivity explicit. The Results (“Theory”) and the Methods (“The Model”) now state, where the matrices are introduced, that the recurrent, feedforward and feedback matrices are effective (net) connectivity, with negative entries standing for disynaptic inhibition through interneurons that are not modelled explicitly. The Limitations (Discussion) now state that the circuit as written does not respect Dale’s law at the level of individual connections, that the posited inhibitory interneurons would introduce additional delays, and that examining how inter-areal synchronization depends on feedback delay is an important direction for future work.

    Another weakness is the use of the term “predictions” to indicate features of the model dynamics that are purely described in the context of the model parameters. Although the model’s response properties may certainly lead to predictions, I think the term requires a better contextualization in terms of neurophysiology and experimental neuroscience. The Discussion draws very interesting and valuable bridges between neuron morphology, interneuron types, and model parameters. But it seems it’s left to the reader to backtrack and figure out which biological mechanisms or experimental manipulations should correspond to changes in input or feedback gain, and how these should be distinguished from possible changes in feedforward gain.

    We thank the reviewer for this comment. Each prediction is now grounded in a candidate experimental manipulation and a measurable readout in a new Discussion subsection, “Experimental predictions of the model”, which covers each gain parameter, the frequency-resolved subspace, the routing result and normalization, and states explicitly that the proposed manipulations of the gains are speculative and rest on the hypothesized cell-type mapping. The V2 input gain is addressed in our response to the next point.

    Relatedly, the manuscript places substantial emphasis on modulation of feedback gain, but does not comparably explore modulation of the feedforward gain, β2, which regulates the V1-to-V2 drive.

    This seems important because changes in feedforward gain could also influence communication subspace dimensionality and oscillatory dynamics. Therefore, predictions related to top-down feedback modulations should be taken with a grain of salt.

    We thank the reviewer for this comment. We agree, and have added the V2 input gain to every analysis in which feedback and V1 input gain are varied. New SI Fig. S8 varies β2 over the same range as β1 (0.5–1.5) and reports, side by side with feedback and V1 input gain, the contrast-response functions of V1 and V2, the V1 power spectra and V1–V2 coherence spectra at 3% and 50% contrast, and the inter- and within-area prediction performance at 100% contrast. V2 input gain has a signature distinct from both feedback and V1 input gain:

    - Increasing β2 leaves V1 responses unchanged and enhances only V2 responses (Fig. S8c), whereas feedback and V1 input gain enhance both areas.

    - At 3% contrast, β2 produces neither the alpha peak in V1 power nor the beta-band coherence peak that appear at low feedback gain (Fig. S8f,l).

    - At 50% contrast, increasing β2 shifts the gamma peak in power and coherence to higher frequencies and broadens it, like feedback gain, but also reduces its amplitude (Fig. S8i,o).

    - Increasing β2 enhances inter-areal communication, like feedback gain, but leaves within-area communication approximately unchanged (Fig. S8r), whereas feedback gain reduces within-area communication and V1 input gain reduces both.

    These results are reported in the Results (“Contrast response functions”, “Inter-areal coherence”, “Communication subspaces”; the “V1 power spectra” section refers the reader to SI Fig. S8 for the V2 input gain), in the Discussion (opening summary, “Role of feedback” and “Experimental predictions of the model”).

    Last but not least, the model dynamics are split among multiple elements and nonlinear interactions, reaching a level of complexity far higher than the other ORGaNICs formulations present in the literature. The authors derive these dynamics in the supplementary material, as a dynamical system that converges to a fixed-point solution that includes “exact divisive normalization”. I wonder, however, if there could be simpler solutions that also produce normalization, either approximate or in a different form than the one proposed by the authors. Note also that the designation of “excitatory neurons” is misleading: despite the presence of two explicitly inhibitory populations, the “excitatory” units also interact with negative effective weights both recurrently and in the inter-areal interactions, thus breaking Dale’s law.

    We thank the reviewer for the comment. The complexity is, moreover, not unstructured: as now stated in the Results (“Theory”), each population performs a defined operation — principal cells y sum the gain-modulated input, recurrent and feedback drives, u accumulates the normalization signal, a implements the division, and q provides the sign inversion required by the feedback term in the dynamics of a, and it is through these populations that normalization is implemented dynamically, and simultaneously across the reciprocally coupled areas, rather than imposed as a static input–output relation. Whether a materially simpler circuit could produce normalization, approximately or in a different form, while retaining analytical tractability is an open question that we do not resolve here; we would regard such a circuit as a valuable simplification of the present framework. On terminology: we agree, and have changed it. The y population is now called “principal cells” (or “principal neurons”) throughout the main text, figure captions and SI; “excitatory” is reserved for the anatomically excitatory populations in the hypothesised cell-type mapping (Discussion, “Biological plausibility”); the distinction between effective and anatomical weights is stated where the matrices are introduced (Results, “Theory”); and the Limitations state that the principal cells interact through mixed-sign effective weights both recurrently and across areas.

    Recommendations for the authors:

    Reviewer #1 (Recommendations for the authors):

    First of all, my sincere apologies for my long delay. It has been quite a hectic period.

    If I could suggest one improvement, it would be to add an early conceptual figure and/or minimal toy model that illustrates the core mechanism before introducing the full circuit. For the illustration, I would try to get inspiration from illustrations of the original Semedo et al paper. Such a figure could show how divisive normalization shapes the covariance/cross-power structure, how this affects coherence, and how the resulting eigen spectrum gives rise to a low-dimensional communication subspace. Starting from this stripped-down mechanism would make the novel insight more transparent, after which the more biologically motivated model would be easier to appreciate.

    We thank the reviewer for this suggestion. We agree that the core mechanism should be stated before the full circuit is introduced, and have done so in the text rather than with a new figure. A new paragraph, “The core mechanism in brief”, closes the “Theory” section (Results), before any result is presented. It follows the progression the reviewer outlines: the noise-driven fluctuations about the normalized fixed point are characterized by the cross-spectral density matrix (Eq. 6); the power spectra are its diagonal entries, the coherence its normalised off-diagonal entries (Eq. 7), and the communication subspace follows from its frequency-resolved covariance through reduced rank regression, so that power, coherence and the communication subspace are complementary readouts of the same underlying dynamics. How normalization shapes this matrix, by creating a resonance and by suppressing weakly driven neurons, which lowers its rank, is stated in the Discussion (“Unified view of inter-areal communication”), and removing normalization removes all three effects (SI Fig. S4e–h). The full circuit is then introduced as the implementation of this mechanism. We have not added a schematic figure, because the corresponding panels already exist: the with/without-normalization comparison (SI Figs. S4 and S5).

    Minor comments/suggestions:

    The authors forgot to cite other work modelling communication subspaces. Citing them could improve impact. Disclaimer: I am among the authors of one of them.

    https://www.nature.com/articles/s41467-023-42519-5

    https://www.science.org/doi/full/10.1126/sciadv.adl6120

    https://www.pnas.org/doi/10.1073/pnas.2404039122

    https://www.biorxiv.org/content/10.64898/2026.05.18.725829v1.abstract

    We thank the reviewer for these references. We have added all four (Barbosa et al. 2023; Gozel & Doiron 2024; Clark & Beiran 2025; Mastrogiuseppe et al. 2026) in the Introduction, where communication subspaces are introduced.

    It could improve the impact if the authors could flesh out how they would test ‘experimentally testable predictions’ (e.g., lines 162, 329).

    We thank the reviewer for this suggestion. We agree, and have revised accordingly. Each prediction is now paired with a candidate manipulation and a readout in a new Discussion subsection, “Experimental predictions of the model”.

    Figure 2 c-d and e-f could make the message stronger if a correlation between model and data is shown: how good is the prediction?

    We thank the reviewer for this suggestion. We agree that a quantitative measure of agreement is needed, and have provided it as a fit rather than a correlation coefficient. New SI Fig. S7 and SI Sec. III fit the model to the experimental power and coherence spectra of Roberts et al. (2013) at all measured contrasts and show the residuals. The Fig. 2 caption and the Results (“V1 power spectra”, “Inter-areal coherence”) now cite this comparison.

    Figure 4c is very interesting and deserves more spotlight (could even be tested with available data?)

    We thank the reviewer for this suggestion. We agree, and have given this result more prominence and specified the test. It is stated in the Abstract in its own right. The Methods (“Frequency decomposition of the communication subspace”) now end with a paragraph that states how the analysis may be applied to experimental recordings: from the cross-spectral density matrix of simultaneously recorded V1 and V2 populations, its real part at each frequency is the frequency-specific covariance, whose source, target and cross blocks give the communication subspace, the prediction performance and the dimensionality at that frequency in closed form (Eqs. 20 and 22); no band-pass filtering or per-band regression is needed. The resulting test, that the dimensionality should be smallest in the band of peak V1–V2 coherence and that this band should shift to higher frequency with contrast, is stated at the end of that Methods paragraph and in the Discussion (“Experimental predictions of the model”). We have not performed this analysis, which requires the raw simultaneous recordings.

    Reviewer #2 (Recommendations for the authors):

    Major:

    (1) If possible, try to add at least one quantitative model-data comparison (variance explained) even without fitting if needed, so readers can gauge agreement beyond visual inspection.

    We thank the reviewer for this comment. We agree, and have added a quantitative comparison. New SI Fig. S7 and SI Sec. III fit the model parameters to the experimental power and coherence spectra measurements and report the residuals at every contrast.

    (2) Show how the gamma-peak predictions change under physiologically realistic time constants (e.g., 10-20 ms), so that readers get an idea.

    We thank the reviewer for this suggestion. The Limitations subsection of the Discussion now explains that if every time constant of the model is multiplied by a common factor k then every spectral feature would be shifted to a frequency k times lower with its shape preserved. The Limitations also now state that when all time constants were released as free parameters in the fit to the experimental spectra (SI Fig. S7) they converged to values between 0.2 and 3ms. The uniform 1ms values are now described throughout as effective population time constants. We decided not to add a figure because it would look identical to Figs. 2 and 3 with a relabelled frequency axis.

    (3) For the communication-subspace results, report how dimensionality estimates scale with the number of modeled/subsampled neurons, so the comparison to the experimental literature is calibrated.

    We thank the reviewer for this suggestion. We agree, and have added the analysis as new SI Fig. S9. Fig. S9 reports the dimensionality of the inter-areal (V1–V2) and within-area (V1–V1) communication subspaces (i) as a function of the number of modeled principal neurons per area, N = 9, 18, 36, 72 and 144, at the sampling fraction of the main text (30/72 of the neurons as source and as target, i.e. 4–60 neurons; the connectivity kernels and normalization pool are defined on the ring and scale with N, with no parameter re-tuned), and (ii) as a function of the number of sampled source and target neurons, 5–30, at N = 72. Each point is the mean ± SD over 200 random draws of source and target populations at 100% contrast and baseline parameters, with dimensionality defined as the rank at which prediction performance reaches 95% of its full-rank value. The dimensionality grows slowly with both quantities (roughly two-fold over the 16-fold change in N), remains far below the number of sampled neurons, and the inter-areal subspace is lower-dimensional than the within-area subspace at every population size.

    Minor:

    (1) Qualify quantitative-sounding language to reflect that comparisons are qualitative and unfitted.

    We thank the reviewer for this comment. We agree, and have revised accordingly. The unfitted comparisons are now described as qualitative: “successfully replicates key experimental observations” now reads “reproduces the main qualitative features” (Results, “V1 power spectra”), the captured trends and observations are described as qualitative in the power-spectra and coherence sections, and “align well with” now reads “qualitatively consistent with the experimental data” (Results, “Communication subspaces”). The fitted comparison of SI Fig. S7 is cited alongside the power-spectra and coherence comparisons.

    (2) Clarify in the main text the scope of the “exact normalization” claim

    We thank the reviewer for this comment. We agree, and have revised accordingly. At first mention (Results, “Theory”) the main text now states that the implementation is exact for W11 = I with β = γ = 1, that for the centre-surround W11 used in all reported results the circuit closely approximates normalization, and that the fixed point for the parameters used is obtained numerically by integrating the deterministic dynamics to steady state.

    Reviewer #3 (Recommendations for the authors):

    (1) Abstract: I think the text could be improved in terms of clarity and readability. For example, the (f) for frequency appears only after the 1/f4, and similarly for the explanation for “communication”. The term “high-coherence” is not properly defined.

    We thank the reviewer for this comment. We agree, and have revised the Abstract. Frequency and its symbol f are introduced together before 1/f4 appears; “communication” is defined at first use (“the ability to linearly predict neural activity in one brain area from neural activity in another”); “high-coherence frequencies” is replaced by “the frequencies at which V1–V2 coherence peaks”.

    (2) L50 and L56: The term “a priori constraints” seems to refer to two general features of the model. The model, however, is much more constrained in terms of parameters and architecture. So the statement that “these two constraints alone give rise to a rich set of emergent properties” is not really corroborated by the results.

    We thank the reviewer for this comment. We agree, and have revised the Introduction. It now states the two assumptions (a hierarchy with feedforward, recurrent and feedback connections; recurrently implemented normalization within each area) and then notes that a concrete model additionally requires choices about the connectivity, the normalization pool, and the noise, which are specified in the Methods and whose influence on the results is examined in SI Fig. S4 and the SI section “Robustness with respect to model parameters”. The sentence “Remarkably, these two constraints alone give rise to...” now reads “Given these choices, and a single set of parameters, the theory gives rise to...”.

    (3) L69: “competing hypotheses”, but they were called “complementary hypotheses” in L34.

    We thank the reviewer for this comment. We agree, and have made the wording consistent. Communication through coherence and communication subspaces are now described, in the Introduction and in the Discussion (“Unified view of inter-areal communication”), as hypotheses that have been treated as alternatives in the literature and that our results show to be complementary.

    (4) L132: “all neurons perform the same computation”; this comment is a bit too generic and dismissive.

    We thank the reviewer for this comment. We agree, and have rewritten the sentence as a neutral statement where it first occurred (Results, “Theory”) and removed the duplicate sentence from the Discussion (“Biological plausibility”). In Wilson–Cowan-type excitatory–inhibitory models the same form of rate equation is applied to each population, with different signs and weights, whereas here the four populations perform different roles with structurally different equations.

    (5) L134: “unique computations to each cell type” although it is clear that different cell types have different roles in the overall dynamics, it is not clear to me which is the unique computation attributed to each of them.

    We thank the reviewer for this comment. The computation performed by each cell type is now stated at that point in the text (Results, “Theory”): principal cells y sum the gain-modulated input, recurrent and feedback drives (squaring output nonlinearity); excitatory modulator cells u accumulate the normalization signal, a weighted sum of principal-cell responses (square-root output nonlinearity); inhibitory modulator cells a implement the division, i.e. the multiplicative gain control of the recurrent and feedback drives; and inhibitory cells q track principal-cell activity and provide the sign inversion required by the feedback term in the dynamics of a (SI Eqs. S21–S22). Additionally, they all have different activation nonlinearities.

    (6) L146: “positive and negative weights”: mixed signs for feedforward and feedback weights.

    We thank the reviewer for this comment. We agree, and have revised accordingly. The phrase now reads “an embedding matrix of effective weights”, followed by a new sentence stating that all weight matrices describe effective connectivity rather than anatomical synapses and that negative weights may be implemented with disynaptic inhibition through interneurons that are not modelled explicitly, with a cross-reference to the Discussion. The same statement accompanies the definition of the matrices in the Methods (“The Model”).

    (7) L160: Again mixed signs for recurrent weights.

    We thank the reviewer for this comment. We agree, see the response above.

    (8) L189: Normalization is exact, but only for parameters different from those used. I think it should be added there that the fixed point for the parameters in use is found numerically through simulation of the dynamics.

    We thank the reviewer for this suggestion. We agree, and have added the statement in those terms. The passage now states that the implementation is exact only for W11 = I with β = γ = 1, that for the centre-surround W11 used in all reported results the circuit closely approximates normalization (deviates from Eq. 2 by less than 0.6%), and that the fixed point for these parameters is obtained numerically by integrating the deterministic dynamics to steady state, after which all spectra and subspaces are computed in closed form by linearising about it. The same statement is made in the Methods, in the new subsection “Numerical computation of fixed points”.

    (9) L212: There should be something more explicit to corroborate the “experimentally testable”.

    We thank the reviewer for this comment. We agree, and have revised accordingly. Each prediction is now paired with a manipulation and a readout in a new Discussion subsection, “Experimental predictions of the model”, which is cited where the predictions are stated in the Introduction and the Discussion.

    (10) L214: I think it’s important to clarify a bit more that even if both areas have input gains, the manipulation is only of the input gain in the upstream area, β1.

    We thank the reviewer for this comment. We agree, and have clarified this at that point in the text (“with the V2 input gain β2 held at its baseline value”) and in the captions of Fig. 3 and SI Figs. S1–S3, each of which now states that β2 is held at 1.0. β2 is varied separately in new SI Fig. S8.

    (11) Figure 2: Please consider using a better color scale for the power and the coherence lines. The lower three contrasts are literally indistinguishable. Same for the top two contrasts.

    We thank the reviewer for this suggestion. We have retained the palette, for the following reasons. The palette is the “Reds” sequential scheme of matplotlib, derived from the ColorBrewer scheme of the same name (Harrower & Brewer, 2003). The curves encode an ordered variable (contrast), for which a single-hue sequential palette is the recommended choice (Zeileis et al., 2020). This encoding is also colour-vision-deficiency safe: the curves are distinguished by lightness rather than by red-versus-green discrimination, and the ColorBrewer “Reds” scheme is classified as colourblind-safe on the basis of empirical testing with colour-vision-impaired viewers (Olson & Brewer, 1997). The same palette is used in the new SI Figs. S6 and S7 for consistency.

    (12) L273: Something that confuses me is how to disambiguate between input gain and input intensity, since the theory considers effects around fixed points. This equivalence is also mentioned in the discussion.

    We thank the reviewer for this comment. In the model, the V1 input gain β1 enters the dynamics only through the product β1z1 (Eq. 1a), and the input drive z1 is proportional to stimulus contrast (Methods). Therefore, V1 input gain and stimulus intensity are mathematically equivalent because they enter the dynamics only through their product (β1 is scalar). This equivalence would not generally hold for spatially or feature-selective gain modulation, where attention can differentially modulate subsets (β1 is vector) of neurons relative to the normalization pool (Reynolds & Heeger, 2009).

    (13) L375: There seems to be a lot of emphasis on modulation of the feedback gain, but related to L214, I wonder why there has been no manipulation of the feedforward gain, β2, which might be equally important in influencing the communication subspace dimensionality and in modulating the oscillatory dynamics. In terms of experimental predictions, the two might be very different, so why consider only one and ignore the other? See also statement at L388. To attribute the effect to top-down feedback, one should prove that bottom-up modulation does not produce the same effect.

    We thank the reviewer for this comment. We agree with the methodological point in the final sentence, and have acted on it. β2 is now varied across the same analyses as γ1 and β1 (new SI Fig. S8; outcomes listed in our response to Weakness 4 above). Bottom-up modulation does not produce the same effect: increasing β2 enhances inter-areal communication, as feedback gain does, but leaves within-area communication unchanged (feedback gain reduces it), leaves V1 responses unchanged, produces no low-contrast alpha or beta peaks, and reduces rather than increases the amplitude of the gamma peak. The passage at L375 (Results, “Communication subspaces”) now reports that increasing β2 enhances inter-areal communication while leaving within-area communication approximately unchanged, with a substantially smaller effect than feedback gain (SI Fig. S8r); the readouts that distinguish the two gains are described in the Discussion (“Experimental predictions of the model”).

    (14) Figure 4: See comment for Figure 2.

    As for Fig. 2, we have retained the palette; see our response to item (11). The contrast values are given in the legend, the luminance ordering is monotonic in contrast, and the same palette is used across Figs. 2 and 4 and the SI.

    (15) L440: I think here it would be more honest to say that the two measures, of coherence and inter-areal communication, are not independent. If two signals are highly correlated at a certain frequency, then the prediction power will also peak for that same frequency. Or would it be possible to have conditions where it would not be the case, showing that the two measures are in fact independent?

    We thank the reviewer for this comment. We agree, and now state the dependence explicitly. The passage at L440 (Results, “Frequency decomposition of communication subspaces”), the Discussion (“Unified view of inter-areal communication”) and the Methods (“Interpretation of the results”) now state that coherence (Eq. 7) and frequency-resolved prediction performance (Eq. 22) are functionals of the same cross-spectral density matrix, so that the coincidence of their peaks is a mathematical consequence rather than an independent empirical confirmation. The claim retained is that coherence, prediction performance and the reduction in dimensionality are all generated by normalization; removing it eliminates all three (SI Fig. S4e–h).

    (16) L446: This paragraph mentions dimensionality, but only the following paragraph defines it.

    We thank the reviewer for this comment. We agree, and have moved the definition (the number of source dimensions required to reach the full-rank prediction performance, dictated by the rank of ) so that it precedes the first use of “dimensionality” (Results, “Frequency decomposition of communication subspaces”); the sentence that previously followed has been removed.

    (17) L485: See comment for L375.

    We address this together with item (13) and the corresponding point of the public review (Weakness 4). We have added a statement regarding the V2 input gain at the end of the section “Dynamic modulation of functional connectivity through feedback” (Results), and the conclusion of that section now describes the three-area result as a demonstration that differential feedback strength can control functional connectivity.

    (18) L575: I find this part a bit too vague. Within the scope of the paper, it is not clear what the network is “processing” and what kind of “balancing” it performs.

    We thank the reviewer for this comment. In this passage, “processing” refers to the computation performed by the principal cells: each cell sums its gain-scaled input, recurrent and feedback drives (Eq. 1a). “Balancing” refers to how these drives are combined: the recurrent gain control adjusts them so that the steady-state responses follow the normalization equation (Eq. 2), exactly or approximately depending on the recurrent weights. This is the mechanistic contrast with predictive coding: predictive coding matches top-down predictions against feedforward signals so as to minimize the prediction error, whereas hierarchical ORGaNICs balances the input, recurrent and feedback drives so as to maintain normalization.

    (19) L583: Similarly to above, I find these lines very confusing. In the context of this paper, there is no notion of “predictable stimulus”, so I find this part very confusing. Also, the model always produces sustained activity by construction, since it has stable steady-state solutions across a wide range of input patterns.

    We thank the reviewer for this comment. The passage refers to a limitation of the predictive coding framework. A constant, continuously presented stimulus is fully predictable, so in predictive coding the forward-propagated activity, which represents the prediction error, should be explained away and decay toward zero; cortical neurons instead maintain sustained responses to constant stimuli, barring adaptation (Burns, Xing & Shapley, 2010). We agree that our model produces sustained responses by construction, because it has a stable fixed point for a wide range of inputs; this behaviour is closer to the observed physiology than the explaining-away account.

    (20) L601, 602: I don’t think that not producing mathematically exact normalization is a significant, critical limitation in a model. We can be pretty confident that neural circuits are not, in fact, producing mathematically exact normalization. Or, to say it more constructively, simpler models that produce approximate normalization might actually be closer to the biological reality.

    We thank the reviewer for this comment. We agree, and have rewritten the passage (Discussion, “Normalization”). It no longer presents approximate normalization as a critical limitation of prior circuit models; it instead states how the present circuit differs: the normalization equation, with an arbitrary normalization pool, is a fixed point of the recurrent dynamics; the dynamics remain stable even with strong recurrent connections; and the fixed point and the linearization about it yield the power spectra, coherence, and communication subspaces in closed form.

    (21) L653: this line states that feedback is essential for network stability. But the paragraph seems to refer to top-down feedback drive, which should not be necessary for dynamical stability in the model.

    We thank the reviewer for this comment. The line refers to the feedback drive onto the inhibitory modulator cells (a), which contributes to the local normalization signal; it is this feedback-mediated inhibition, not the top-down drive to the principal cells, that is essential for stability. We have clarified the sentence accordingly (Discussion, “Role of feedback”).

    (22) L661: I think this statement is misleading and this part should be stated more cautiously. Although plausible in general, I don’t see these results anywhere in the paper. It would require a more functional connotation of the model, which now has a very homogeneous ring structure.

    We thank the reviewer for this comment. We agree, and have stated the claim more cautiously (Discussion, “Role of feedback”): the passage now describes the generative-model regime as a hypothesis (“the network is hypothesized to behave like generative models...”) rather than as a property demonstrated by the present results.

    (23) L715, L719: I feel confused by the contrast between spatially global feature-based attention and spatially-selective attention.

    We thank the reviewer for this comment. The two forms of attention differ in the domain over which neurons are selected for gain modulation. Feature-based attention is spatially global: attending to a feature boosts the neurons selective for that feature across the entire visual field, regardless of retinotopic location (Saenz, Buracas & Boynton, 2002). Spatially selective attention instead boosts only the neurons whose receptive fields overlap the attended location, regardless of their feature preference. Both are gain modulations that can be implemented in the model through the input gain: applied as a vector rather than a scalar, β1 raises the gain of a selected subset of neurons relative to the normalization pool (Reynolds & Heeger, 2009), as noted in our response to item (12). On the ring, a β1 profile over preferred orientation implements feature-selective gain; implementing spatially selective attention in the same way would require adding a spatial (retinotopic) dimension over which β1 varies, as now noted in the Discussion (“Attention and functional connectivity”). The three-area routing result corresponds to the spatially global case, in which boosting an entire feature-selective area increases its feedback to V1 at all locations.

    (24) L729, L730: Here it is important to define exogenous and endogenous attention. If they are defined by the pathways on which they act, then the reasoning becomes circular.

    We thank the reviewer for this comment. We agree, and have defined both terms behaviourally at first use, independently of the model (Discussion, “Attention and functional connectivity”): exogenous attention as the fast, involuntary orienting driven by a salient stimulus, and endogenous attention as the slower, voluntary, goal-directed deployment of attention. The mapping onto input gain and feedback gain, respectively, is then stated as a testable hypothesis rather than as a definition, falsifiable through the divergent spectral signatures of the two gains.

    (25) Figure 7: I suggest changing the color map into a “diverging” color map centered at zero, so that negative and positive values appear visually distinct and zero has a neutral color. For example, the “seismic” or “PuOr” color maps in matplotlib would be appropriate.

    We thank the reviewer for this suggestion. We agree, and have revised Fig. 7 accordingly. It now uses the PuOr colour map, centred at zero with a shared colour bar, so that positive and negative weights in the tuning-curve, recurrent and feedback matrices are visually distinct and zero is neutral.

    References cited in this letter

    Burns, S.P., Xing, D. & Shapley, R.M. (2010). Comparisons of the dynamics of local field potential and multiunit activity signals in macaque visual cortex. Journal of Neuroscience 30, 13739–13749.

    Harrower, M. & Brewer, C.A. (2003). ColorBrewer.org: an online tool for selecting colour schemes for maps. The Cartographic Journal 40, 27–37.

    Olson, J.M. & Brewer, C.A. (1997). An evaluation of color selections to accommodate map users with color-vision impairments. Annals of the Association of American Geographers 87, 103–134. Polsky, A., Mel, B.W. & Schiller, J. (2004). Computational subunits in thin dendrites of pyramidal cells. Nature Neuroscience 7, 621–627.

    Reynolds, J.H. & Heeger, D.J. (2009). The normalization model of attention. Neuron 61, 168–185.

    Saenz, M., Buracas, G.T. & Boynton, G.M. (2002). Global effects of feature-based attention in human visual cortex. Nature Neuroscience 5, 631–632.

    Silver, R.A. (2010). Neuronal arithmetic. Nature Reviews Neuroscience 11, 474–489.

    Zeileis, A., Fisher, J.C., Hornik, K., Ihaka, R., McWhite, C.D., Murrell, P., Stauffer, R. & Wilke, C.O. (2020). colorspace: a toolbox for manipulating and assessing colors and palettes. Journal of Statistical Software 96, 1–49.

    Srinivasan, M.V. & Bernard, G.D. (1976). A proposed mechanism for multiplication of neural signals. Biological Cybernetics 21, 227–236.

  8. Version published to 10.1101/2025.07.15.664935v4 on bioRxiv
  9. eLife

    eLife Assessment

    This important study extends a model of cortical normalization (ORGaNICs) to interacting cortical areas and shows that communication through coherence and communication subspaces can arise from a single set of dynamics. The evidence is solid, showing analytically that contrast-dependent gamma dynamics and a low-dimensional inter-areal communication subspace arise from one parameter set, though the comparisons to data remain qualitative and the analytics rest on a linearization that is not checked against numerical simulation. The work will interest neuroscientists and theorists concerned with inter-areal communication, cortical oscillations, and divisive normalization.

  10. eLife

    Reviewer #1 (Public review):

    In this paper, Pal and colleagues propose a mechanistic unification of two influential accounts of inter-areal communication: communication through coherence and communication subspaces. A major strength of the paper is that it does not treat coherence and communication subspaces as independent phenomena, as typically done, but instead derives both from the same circuit with divisive normalization. In this framework, noise-driven fluctuations around the normalized fixed point determine covariance and cross-power structure (which, in retrospect, makes so much sense to be related). Then, they show how these determine linear prediction performance and the effective dimensionality of the communication subspace. They also show (however not very visually, see recommendation below for a figure) how divisive normalization is crucial to shape inter-areal coherence and the dimensionality of communication.

    I found this conceptual contribution potentially very influential, but somewhat obscured by the technical complexity of the model. The central intuition (I think) is that recurrent normalization can organize cross-area fluctuations, both frequency-specific correlations and cross-covariances. Took me a while to grasp this insight, mostly because I was stuck with the model details. Note that I have some experience with network dynamics, but not with this particular model.

  11. eLife

    Reviewer #2 (Public review):

    Summary:

    The authors extend the ORGaNICs framework (a recurrent circuit that dynamically implements divisive normalization) to connected cortical areas with explicit top-down feedback. Because the network has a known analytical fixed point that coincides with (or closely approximates) the normalization equation, the authors can linearize about that fixed point and derive closed-form expressions for the power spectral density, inter-areal coherence, and communication subspaces. Using a two-area instantiation (V1 & V2) with a single fixed parameter set and no data fitting, they show the model reproduces: (i) contrast-response functions with steeper slope V2; (ii) gamma-band power and coherence peaks that shift to higher frequency with contrast; and (iii) a low-dimensional inter-areal communication subspace that is lower-dimensional than the within-area subspace. They derive parallel predictions of what happens by changing model parameters: feedback gain enhances inter-areal and suppresses within-area communication, and normalization is necessary for both the oscillatory dynamics and the reduced subspace dimensionality. A three-area extension (V1&V4, V1&V5/MT) is used to argue that differential top-down feedback can dynamically route functional connectivity.

    Strengths:

    (1) Analytical tractability: Deriving power spectra, coherence, and communication-subspace structure in closed form from a known fixed point is genuinely valuable.

    (2) Conceptual unification: Framing coherence and communication subspaces as arising from the same normalization-driven dynamics is an elegant and useful contribution.

    (3) Breadth from few assumptions: A large range of phenomena (contrast gain, gamma dynamics) emerges from normalization-based model assumptions.

    (4) Biological grounding: The mapping of model variables onto identified cell types connects the abstract computation to known cortical microcircuitry.

    (5) The prediction that input-gain versus feedback-gain modulation produce distinct spectral signatures gives experimentalists a clear way to test the framework.

    Weaknesses:

    (1) Comparisons are qualitative, not quantitative: The theory/experiment panels are visual side-by-side comparisons. There is no quantitative goodness-of-fit for any predictions.

    (2) The simulations use τ ≈ 1 ms for all cell types, which the authors acknowledge is unrealistically short; realistic values would shift the gamma peaks to lower frequencies.

    (3) Divisive normalization is a special case and is recovered exactly only for the identity recurrent matrix (self-normalization). Some statements that the circuit implements divisive normalization exactly need softening.

    (4) The element-wise (multiplicative) interaction in the modulator dynamics is not tied to a specific cellular mechanism.

  12. eLife

    Reviewer #3 (Public review):

    Summary

    The work of Pal and colleagues considers a hierarchical and multi-population version of the "oscillatory recurrent gated neural integrator circuits" (ORGaNICs) model, showing through analytics that the model captures multiple relevant experimental results: first of all, its oscillatory dynamics produce a profile with high resemblance to experimental results, both in terms of decay of power at high frequency and in terms of shifting peak as a function of stimulus contrast. Second, inter-areal communication subspace dimensionality is lower than within-area dimensionality. The authors then proceed to further characterize the model's response properties as a function of input and feedback gain. In particular, they find that frequencies transmitted with higher strength also carry more information, that changing gain modifies the dimensionality of communication subspaces, and that these properties can be used in a three-layer model, where an upstream area can select which downstream area to communicate to, based on the strength of feedback gain.

    Strengths

    This work demonstrates that a single-circuit model with normalization properties can capture both the oscillatory dynamics and the inter-areal communication properties measured in cortical circuits, matching multiple experimental results. The full analytical tractability of the model is highly advantageous, allowing for easier exploration of parameters, replicability, and effective interpretations of results compared to purely numerical approaches.

    The work also makes a useful conceptual link between normalization, coherence-based communication, and subspace-based communication. In particular, it shows how both phenomena can emerge from the same circuit dynamics, where normalization is a key factor.

    Interestingly, the model is also extended to multiple areas, showing how attention (in the form of changes in feedback gain) can synchronize the activity of a downstream area with one of two upstream areas, thus effectively selecting which area to communicate with.

    In general, this is an interesting computational framework and a useful starting point for future modeling work. A particular strength is that it connects normalization, oscillatory dynamics, coherence, and communication subspaces within one analytically tractable model, making it possible to generate mechanistic hypotheses about when inter-areal communication should be stronger, lower-dimensional, or preferentially routed through feedback.

    Weaknesses

    Although I see the analytic approach as a strength, at the same time I regard the lack of any numerical comparison as a big weakness. Circuit simulations would not only confirm the correctness of the analytics, but also offer further insights on the error margins and on the regimes where the analytics are valid. This is because, to my understanding, the analytics are based on a linear approximation around the operating regime, which means deviations might be expected, especially for high gain levels in the input, or in the feedforward and feedback pathways.

    Another problem is that the analytically tractable model seems to rely on effective connectivity weights that break Dale's law. Numerical simulations with explicitly modeled excitatory and inhibitory units might give insights into effects due, e.g., to the additional transmission delays mentioned in the Discussion.

    Another weakness is the use of the term "predictions" to indicate features of the model dynamics that are purely described in the context of the model parameters. Although the model's response properties may certainly lead to predictions, I think the term requires a better contextualization in terms of neurophysiology and experimental neuroscience. The Discussion draws very interesting and valuable bridges between neuron morphology, interneuron types, and model parameters. But it seems it's left to the reader to backtrack and figure out which biological mechanisms or experimental manipulations should correspond to changes in input or feedback gain, and how these should be distinguished from possible changes in feedforward gain.

    Relatedly, the manuscript places substantial emphasis on modulation of feedback gain, but does not comparably explore modulation of the feedforward gain, β2, which regulates the V1-to-V2 drive. This seems important because changes in feedforward gain could also influence communication subspace dimensionality and oscillatory dynamics. Therefore, predictions related to top-down feedback modulations should be taken with a grain of salt.

    Last but not least, the model dynamics are split among multiple elements and nonlinear interactions, reaching a level of complexity far higher than the other ORGaNICs formulations present in the literature. The authors derive these dynamics in the supplementary material, as a dynamical system that converges to a fixed-point solution that includes "exact divisive normalization". I wonder, however, if there could be simpler solutions that also produce normalization, either approximate or in a different form than the one proposed by the authors. Note also that the designation of "excitatory neurons" is misleading: despite the presence of two explicitly inhibitory populations, the "excitatory" units also interact with negative effective weights both recurrently and in the inter-areal interactions, thus breaking Dale's law.

  13. Version published to 10.7554/elife.111297.1 on eLife
  14. Version published to 10.1101/2025.07.15.664935v3 on bioRxiv
  15. Version published to 10.1101/2025.07.15.664935v2 on bioRxiv
  16. Version published to 10.1101/2025.07.15.664935v1 on bioRxiv