{"id":"4a635228-013a-469a-a6b6-efd22fef0ec1","arxiv_id":"2607.29537","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The inverse controllability Gramian ('affordance landscape') of a network reveals its cheapest computations, with landscape heterogeneity tracking the sensory-association axis and increasing over training in artificial networks.","lead":"This paper repackages a standard control-theory quantity — the minimum input energy needed to drive a network between activity states — into a 'computational affordance landscape' of a network. Applied across scales, it finds the cheapest computation in a fly navigation circuit is updating direction, that human sensory networks have more varied cost landscapes than association networks, and that training artificial networks widens the cost spread.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The human connectome and RNN analyses compute the landscape from the raw weight matrix W rather than the linearized Jacobian, so the reported heterogeneity trends may reflect weight magnitude, not computational cost.","rationale":"The reader's weakest_assumption identifies exactly the same issue: W in the empirical sections is not the Jacobian of the dynamics. I agree with this assessment. The theoretical development is careful: Eq. (3) requires the Jacobian. The ring-attractor application follows this correctly (Eq. 24), making the fly result credible. However, the human and RNN applications skip the linearization step. For human connectomes, the structural connectivity matrix is a static, non-negative graph; without an equilibrium and decay term, the 'cost' it induces is not the minimal control energy of any neural dynamics. The IQR is not invariant to scaling, so the S-A axis correlation could reflect differences in edge-weight distributions or node counts (despite regression). For RNNs, using W_rec instead of the Jacobian conflates weight magnitude with landscape shape: training increases weights, which inflates the IQR. The paper's claim that learning 'sculpts the landscape' therefore lacks support unless the Jacobian is used or weights are normalized. These are not minor technicalities; they bear directly on the two empirical results beyond the ring attractor. The framework itself may still be valid, and the paper could be revised to address the issue, so a conditional recommendation is appropriate. The reader's CONDITIONAL verdict remains appropriate; my stress-test does not change it.","tokens_in":16953,"tokens_out":6901,"duration_ms":73612,"concrete_test":"Recompute the HCP analysis using a stable linearized dynamics, e.g., W' = W - I or W' = -(I - W) (and also test a normalized Laplacian), and re-evaluate the Spearman correlation between IQR and S-A axis. For the RNNs, at each checkpoint compute the Jacobian J = -I + diag(tanh'(W_rec r*)) W_rec (with r* the network's fixed point or a typical state) and compute the IQR of the Gramian eigenvalues from J; also compute the IQR from W_rec/||W_rec|| to separate scale effects. If the S-A correlation loses significance (|r_S| < 0.5, p > 0.05) or the IQR training curve ceases to be monotonic, the reported results are artifacts of using raw weights.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (3) defines the affordance landscape from the Jacobian W of the linearized dynamics about an equilibrium. The HCP and RNN sections do not use a Jacobian: for human connectomes, W is the raw structural connectivity matrix (streamline counts normalized by gray-matter volume), with no equilibrium or decay term; for RNNs, the landscape is computed from the raw recurrent weight matrix W_rec, although Eq. (10) implies the local Jacobian is J = -I + diag(f')W_rec (evaluated at a fixed point). Because the Gramian eigenvalues and their IQR are scale-sensitive, the reported correlations and training trends could be driven by differences in overall connectivity strength rather than by the shape of a genuine computational landscape. For example, if training simply increases the norm of W_rec, the IQR of the Gramian eigenvalues will increase even if the normalized structure is unchanged. Similarly, sensory and association networks differ in connection weights and sizes; without a principled linearization, the S-A correlation (r_S=-0.64) may be a property of the raw matrices, not of the network's computational affordances. The ring-attractor analysis (Eq. 24) correctly linearizes and is therefore not affected. This matters because the human and RNN results are two of the three main empirical supports for the framework; the paper should either justify W as the relevant linearization or recompute the landscapes with a dynamical model.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a control-theoretic framework for relating network structure to computational function. A computation is defined as an input-driven transition between activity states, and its minimum energy cost is expressed through the controllability Gramian (Eq. 3); the inverse Gramian defines a 'computational affordance landscape.' The framework is applied in three settings: a Drosophila ring-attractor model, where the cheapest mode is the derivative of the bump with respect to heading and the optimal input matches known shift-neuron anatomy; human HCP structural connectomes, where the IQR of the affordance spectrum correlates with the sensorimotor-association axis (r_S=-0.64, p=0.006); and trained RNNs, where landscape heterogeneity increases over training. The authors argue the framework unifies structure-function relationships across scales.","tokens_in":17260,"tokens_out":4806,"duration_ms":52169,"significance":"If the central claims hold, the framework is valuable: it provides a single quantitative object—the affordance landscape—that can be computed from network structure and compared to biological function across very different scales. The mathematical derivation of C_min is standard and correct, and the ring-attractor analysis is a clean demonstration that a continuous attractor's marginal mode is the cheapest to drive. The paper also ships openly available data and code, which strengthens reproducibility. The main significance would be the cross-scale unification, but this depends on the empirical sections being computed from the correct linearized dynamics rather than from raw weight matrices. The human and RNN results, in particular, are the broadest claims and currently rest on unstated modeling choices; those need to be fixed before the paper's central message is fully convincing.","major_comments":[{"comment":"The Gramian in Eq. (3) is defined from the Jacobian W of the linearized dynamics about an equilibrium. In the HCP analysis, W is taken to be the raw structural connectivity matrix (SIFT2-weighted streamline counts normalized by gray-matter volume), with no equilibrium, decay term, or input Jacobian specified. In the RNN analysis, the affordance landscape is computed from the raw recurrent weight matrix W_rec, although Eq. (10) implies a local Jacobian J = (-I + diag(f') W_rec)/tau at a fixed point (and something different along task trajectories). The Gramian eigenvalues and their IQR are not invariant to replacing J by W: a scale change in W alters e^{Ws} and hence the eigenvalue spread. The reported r_S=-0.64 and the training-time increase in Fig. 5c could therefore reflect raw connectivity strength or weight norm rather than the cost of actual computations. Because these are two of th","section":"Methods, 'Human Connectome Project (HCP) data' and 'Artificial recurrent neural networks'; Eq. (3)"},{"comment":"The choices B=I and T=1 are adopted globally without sensitivity analysis. Eq. (5) explicitly depends on T, and the eigenvalue spread of the Gramian depends on B. For the HCP data, T=1 has no stated time unit, so the Gramian is not dimensionless and the IQR comparisons across networks may depend on the arbitrary choice of T. The ring-attractor result is likely robust, but the HCP heterogeneity ranking and the RNN training trend could change with T. Please provide a sensitivity sweep over T (and over B if the all-nodes-input assumption is relaxed), or give a principled argument for why the results are independent of these choices for the matrices studied.","section":"Methods, 'Computing the affordance landscape'"},{"comment":"The identification of bump rotation as the cheapest mode follows mathematically from the continuous family of fixed points: differentiating Eq. (22) with respect to theta shows that d h*/d theta is in the kernel of J, and since the mode cost decreases with the eigenvalue, the zero mode is automatically the cheapest. Thus the agreement with the known function of the circuit is a consistency check of the model's defining symmetry, not an independent empirical prediction. The paper should state this explicitly. The genuinely nontrivial outcome is the optimal-input profile in Fig. 3f-g, which matches the known shift-neuron anatomy and does not follow from the zero mode alone. Reframing the ring-attractor section in this way would strengthen, not weaken, the paper.","section":"Supplementary Information Sec. 2, Eq. (26); Fig. 3d-e"}],"minor_comments":[{"comment":"The same symbol W is used for the Jacobian matrix and for the controllability Gramian, which is confusing. Please use a distinct notation, e.g., A for the Jacobian and \\mathcal{W} for the Gramian.","section":"Eq. (3) and Methods, Eq. (8)"},{"comment":"The statistical unit for the Spearman correlation is unclear. The figure caption says 'Each point represents an individual subject' but then reports a single r_S and p. If the analysis uses network-level medians, the sample size is 17; if it pools all subjects, the points are not independent. Please clarify the exact test, the n used, and whether the p-value is corrected for the number of networks.","section":"Fig. 4c and 'Human Connectome Project (HCP) data'"},{"comment":"It would help to state explicitly whether each Yeo network is treated as an induced submatrix of the full 400x400 structural connectivity matrix, and if so, how inter-network connections and inputs from other networks are modeled. This is a modeling choice that affects the interpretation of the network-specific affordance landscape.","section":"Methods, 'Human Connectome Project (HCP) data'"},{"comment":"The weight noise is described as 'small' but sigma=0.5 is not negligible compared to J_E=10 and J_I=-9. A sentence justifying this magnitude, or a brief robustness check, would be useful.","section":"Methods, 'Ring attractor network'"}],"recommendation":"major_revision","confidential_remarks":"The derivation and the ring-attractor demonstration are sound, and the data/code availability is a strength. The main risk is that the human connectome and RNN results are artifacts of analyzing raw weight matrices rather than the linearized dynamics that define the affordance landscape. I do not think this is grounds for rejection, because the issue is fixable by reanalysis or explicit justification. However, the fix is load-bearing for two of the three main empirical claims, so I recommend major revision rather than minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the things you should know. The paper renames the inverse controllability Gramian as a \"computational affordance landscape\" and shows that the cost of a state transition is D^T W^{-1} D. That math is textbook. The new content is the set of applications: a ring-attractor model of Drosophila heading, a correlation with the S-A axis in human connectomes, and an increase in eigenvalue dispersion during RNN training. The ring-attractor section is the best part. The authors linearize the actual dynamics about a stable bump, identify the zero mode, and show it corresponds to rotating the bump; the optimal input pattern matches the known shift-neuron anatomy. That is a clean, genuinely satisfying result.\n\nWhere it gets soft: the human and RNN analyses do not linearize any dynamical system. For the HCP data, W is the raw streamline-count matrix, with no equilibrium, no decay term, and no justification for why that matrix plays the role of a Jacobian. For the RNNs, Eq. (10) implies the local Jacobian is -I + diag(f') W_rec, but the paper computes the landscape from W_rec directly. The stress-test is right that the IQR of Gramian eigenvalues is scale-sensitive, so the S-A correlation and the training trend could be driven by overall connection strength or network size rather than the shape of a genuine computational landscape. The authors regress out node count but not edge weight magnitude, and they do not test sensitivity to T=1 and B=I. These are not fatal to the framework, but they are load-bearing for the human and RNN claims.\n\nAlso, the ring-attractor \"prediction\" is not an independent empirical outcome. Because the bump family is continuously parameterized by heading, the derivative of the bump is necessarily in the kernel of the Jacobian. The paper essentially shows that the linearization inherits the model's symmetry. That is worth saying in the paper; it is a consistency check, not a falsifiable prediction. The authors do not overclaim too much, but the word \"prediction\" appears.\n\nOn balance: a capable, well-written paper, with a correct mathematical core and one beautiful application. The human and RNN sections need substantial revision before the empirical claims can be trusted. A serious referee should engage with it. I would accept it for peer review, but my own verdict would be conditional on the authors either justifying W as the relevant linearization or recomputing with a proper dynamical model and sensitivity analysis.","headline":"Useful reframing of standard control theory with an elegant ring-attractor result, but the human and RNN analyses rest on unstated linearization choices that the paper needs to fix.","tokens_in":17751,"tokens_out":3048,"would_cite":false,"duration_ms":34489,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A network's cheapest computations are read off from its inverse controllability Gramian; in a fly compass circuit, the cheapest is the heading update, and the same landscape explains human cortical specialization and learning-driven change.","keywords":["computational affordance landscape","controllability Gramian","network control theory","ring attractor","head-direction circuit","sensorimotor-association axis","recurrent neural network training","structure-function relationships"],"falsifier":"Compute the affordance landscape for the same trained RNNs using the full Jacobian of Eq. (10), J = −I + W_rec diag(f'(r)), instead of W_rec alone, and check whether heterogeneity still increases monotonically across all five tasks; if the trend reverses or flattens, the 'learning sculpts the landscape' claim rests on the choice of linearization. For the fly circuit, a biophysical simulation that perturbs the bump in the predicted low-cost direction and measures the actual input energy would test whether heading rotation really is the cheapest computation.","tokens_in":16855,"feed_emoji":"🧭","tokens_out":4896,"duration_ms":52667,"temperature":0.7,"pith_summary":"This paper tries to establish that the set of activity transitions a network can perform cheaply is fully determined by its structure, and that this 'computational affordance landscape' can be computed from standard control theory. It applies the idea to a Drosophila head-direction circuit, where the cheapest mode corresponds exactly to updating the heading representation and the optimal inputs match the known shift-neuron anatomy. It then shows that in human cortical networks, the heterogeneity of this landscape tracks the sensorimotor-association axis: sensory networks have heterogeneous cost landscapes, while association networks have flat, homogeneous ones. Finally, it shows that training recurrent neural networks on a single task progressively increases landscape heterogeneity, suggesting that learning sculpts which computations a network can do cheaply. If the framework holds, it gives a parameter-free structural account of why certain computations are easy for a given neural circuit.","feed_headline":"A fly's compass update is its network's cheapest move","feed_subtitle":"Control-theoretic cost landscape also explains sensory vs association cortex and how training sculpts neural networks.","key_machinery":"The central object is the inverse controllability Gramian W^{-1}, called the computational affordance landscape. For linearized dynamics dr/dt = Wr + Bv, the minimum cost to reach a target state r_f from r_0 is C_min = (r_f − e^{WT}r_0)^T W^{-1}(r_f − e^{WT}r_0). When each node receives input independently (B = I) and W is symmetric, the cost of exciting each mode is 2λ_i/(e^{2λ_i T} − 1), so modes with eigenvalues near zero, i.e. the slowest modes, are the cheapest. In the ring attractor, marginal stability puts the bump-derivative mode in the kernel of the Jacobian, so rotation is the zero-cost direction; this is the load-bearing step that connects cheap computation to biological function.","core_discovery":"The paper's central claim is that a network's structure, through the controllability Gramian W = ∫₀^T e^{Ws} BB^T e^{W^T s} ds, defines a 'computational affordance landscape' W^{-1} that ranks every possible activity transition by the minimum input energy needed to drive it. For a model ring-attractor compass circuit, this landscape has a single ultra-low-cost mode, which is exactly the derivative of the activity bump with respect to heading; the minimum-energy inputs that rotate the bump are an increase ahead of the bump and a decrease behind it, matching the fly's known shift-neuron anatomy. In human structural connectomes, the heterogeneity of the landscape, measured by the interquartile","pith_inferences":["The paper does not say this explicitly, but the same landscape can serve as a design target: one could choose or train a network's weights so that a desired computation falls in the low-cost subspace, and neuromodulation can be viewed as an online mechanism for reshuffling the landscape.","Because the human and RNN analyses use raw structural or recurrent weights rather than full Jacobians, recomputing the landscape from the correct local linearization, including the decay term and activation derivatives, is a natural stress test; the monotone training trend may depend on that choice.","The bilateral-symmetry result suggests a general principle linking structural symmetry to energy-efficient activity patterns, which could be tested across species and network scales using the same Gramian machinery.","The sensorimotor-association correlation is a single summary statistic; a stronger test would ask whether within-network functional diversity, such as the number of distinct low-cost modes, predicts behavioral variability across individuals."],"forward_implications":["The relative ordering of computation costs follows directly from an eigendecomposition of the Gramian, so no task-specific simulation is needed to know what a circuit finds easy.","In the ring-attractor model, marginal stability is not incidental: the null direction is exactly the behavior the circuit must perform, making rotation the structurally cheapest computation.","Human sensory networks are wired so that a few activity modes are very cheap, while association networks keep costs flat across modes; the paper interprets this as specialization versus flexible, domain-general recombination.","Learning in recurrent networks progressively widens the spread of mode costs, implying that acquiring a single skill actively carves the landscape into easy and hard directions.","The oscillatory extension of the framework yields frequency-dependent cost landscapes, predicting which oscillation frequencies a circuit supports cheaply and where in the circuit those frequencies are routed."],"fun_headline_variants":["For a fly compass, updating heading is cheapest","Brain computation cost is set by network topology","Learning diversifies the cost landscape of neural networks","Sensory and association networks differ in computational cost","Control theory prices computations in brain networks"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the matrix W used to build the Gramian genuinely captures the local dynamics of the network: for human connectomes this is the raw structural connectivity matrix with no specified equilibrium, and for trained RNNs it is the raw recurrent weight matrix rather than the full Jacobian, so if those matrices misrepresent the real dynamics, the reported cost landscape may not reflect actual computational costs.","fun_headline_variants_meta":{"raw":{"variants":["For a fly compass, updating heading is cheapest","Brain computation cost is set by network topology","Learning diversifies the cost landscape of neural networks","Sensory and association networks differ in computational cost","Control theory prices computations in brain networks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001027,"raw_usage":{"total_tokens":4182,"prompt_tokens":777,"completion_tokens":3405,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":521,"completion_tokens_details":{"reasoning_tokens":3337}},"tokens_in":521,"tokens_out":3405,"duration_ms":27277,"temperature":1.0,"reasoning_tokens":3337,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T05:01:45.239928+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the affordance landscape for the same trained RNNs using the full Jacobian of Eq. (10), J = −I + W_rec diag(f'(r)), instead of W_rec alone, and check whether heterogeneity still increases monotonically across all five tasks; if the trend reverses or flattens, the 'learning sculpts the landscape' claim rests on the choice of linearization. For the fly circuit, a biophysical simulation that perturbs the bump in the predicted low-cost direction and measures the actual input energy would test whether heading rotation really is the cheapest computation.","supporting_citations":[],"review_version":1}