A behavior-environment information loop drives sensory navigation

Damon A. Clark; Kevin S. Chen; Matthew P. Leighton; Thierry Emonet

REVIEW 4 major objections 6 minor 1 cited by

A behavior-environment information loop drives sensory navigation

T0 review · 4 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A two-way information loop between sensing and action predicts navigation performance.

desk verdict The minimal-model derivation is the real contribution; the 'reliably predicts' claim in Fig. 2 is not yet substantiated because the Phi-coefficient mapping for experimental data is never specified. read the letter →

arxiv 2607.26295 v1 pith:TITMDE2N submitted 2026-07-28 physics.bio-ph cond-mat.stat-mechq-bio.NC

Kevin S. Chen , Matthew P. Leighton , Damon A. Clark , Thierry Emonet This is my paper · ORCID

classification physics.bio-phcond-mat.stat-mechq-bio.NC MSC 94A1760J27 PACS 87.10.-e05.40.-a

keywords transferentropynavigationactivesensingsensory-motorfeedbackbacterialchemotaxisolfactorybipartiteMarkovmodelreinforcementlearning

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that navigation performance is governed by a closed loop of information: sensory inputs must reduce uncertainty about the next action (the reactive flow), and actions must reduce uncertainty about the next sensory state (the active flow). Using a minimal four-state Markov model of run-and-tumble behavior in a gradient, the authors derive a single predictor Φ that combines the reactive flow, the active flow, and their geometric mean, and show that it tracks the chemotaxis index across parameter regimes. They then estimate the same two flows directly from measured trajectories of bacteria, worms, and flies, and from a reinforcement-learning agent, and report that Φ orders navigation efficiency in every system. The payoff is a model-free way to dissect strategy as well as performance: it exposes distinct active-versus-reactive balances that yield equal navigation, spatially localized strategies in fly plume tracking, and the learned policies of an RL agent.

What carries the argument

The central object is the pair of one-step transfer-entropy rates computed from binarized behavioral and sensory time series: the reactive flow T_s→a, which measures how much the current sensory state reduces uncertainty about the next action given the current action, and the active flow T_a→s, which measures how much the current action reduces uncertainty about the next sensory state given the current sensory state. These flows are evaluated on a minimal continuous-time bipartite Markov model with four states—run/tumble actions and up/down gradient sensations—whose transition rates are set by sensory gain and signal decorrelation. Expanding both the transfer-entropy rates and the chemotaxis

What would settle it

Hold out a set of trajectory ensembles or an engineered agent that was not used to set any model parameter; estimate the full-history transfer-entropy rates rather than the one-step approximation, and compare Φ with performance. If the ranking breaks, or if any navigator succeeds with zero reactive flow and positive active flow, the central claim is falsified. The paper's own Appendix B predicts that one-step and full-history rates differ by more than 10% only outside the weak-coupling regime, so strongly persistent environments or strong-control animals are the natural test bed.

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Extended reading notes

Core claim

The central discovery is a scaling law. To leading order in weak coupling, navigation performance Δ is proportional to Φ = A√(T_s→a) + B√(T_s→a T_a→s) − C T_s→a, where T_s→a is the rate at which the sensory history reduces uncertainty about the next action and T_a→s is the rate at which actions reduce uncertainty about the next sensory state. The coefficients A, B, C are positive and depend only on the minimal model's baseline rates, not on the control parameters. The formula implies an asymmetry: when the reactive flow vanishes every term vanishes, so navigation is impossible, whereas a purely reactive strategy can still navigate when the active flow vanishes. The paper verifies Φ against n

Load-bearing premise

The load-bearing premise is that the one-step, binary-coarse-grained transfer entropies—evaluated with model coefficients A, B, C whose parameter values the paper does not state for the experimental datasets—equal the full-history causal information flow; if that equality fails, the cross-species Φ correlations are in-sample fits rather than predictions.

Editorial extensions

If this is right

Navigation performance is controlled by the geometric-mean coupling of reactive and active information flows; adding the √(T_s→a T_a→s) term improves prediction over reactive-flow-only in all datasets tested, as measured by positive ΔBIC.
Dissipation alone is a poor performance predictor: entropy production rate correlates weakly with chemotaxis in the minimal model, so the information loop, not thermodynamic cost, sets efficiency.
A purely reactive strategy with no action-to-sensory feedback can still navigate, but a purely active strategy with no sensory-to-action flow cannot; the loop is directional, not symmetric.
Equal navigation performance can arise from different active/reactive balances, so performance metrics alone underdetermine strategy; the two flows add an organizing dimension that separates distinct bacterial phenotypes and spatial fly strategies.
In learned behavior, optimal active flow is intermediate: in the two-armed bandit task, increasing action reliability raises reactive flow but lowers active flow, and in persistent environments, performance peaks at intermediate active flow.

Reading between the lines

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

Inference (editorial): the framework suggests a design rule for artificial navigators: maximize Φ rather than raw information gain, preferring control policies that raise the geometric mean of the two flows; interventions that only inflate action-driven information could degrade performance in reliable environments.
Inference (editorial): because coarse-graining makes the estimated flows lower bounds, the reported cross-system agreement is likely conservative; a finer action-state partition or longer recordings should strengthen the relation if the missing behavioral detail is navigation-relevant.
Inference (editorial): the same predictor could be applied to human sensorimotor tasks or closed-loop prosthetics, where both sensory signals and actions can be recorded; a testable extension is whether Φ ranks skill acquisition over learning or separates experts from novices.
Inference (editorial): the non-monotonic active-flow result implies that organisms navigating long-correlation environments should not be selected for maximal active sampling; measuring active flow across species with different plume statistics could reveal an optimum.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The minimal-model derivation is the real contribution; the 'reliably predicts' claim in Fig. 2 is not yet substantiated.

What the paper does well: the bipartite Markov setup and the perturbative expansion are clean, and the SI proof that weakly coupled bipartite systems give Delta ~ A sqrt(Ts->a) + B sqrt(Ts->a Ta->s) - C Ts->a is a genuine general result. The geometric-mean coupling term is new compared to Mattingly et al., and the strategy decomposition in Figs. 3-4 is suggestive and worth taking seriously. The authors are also honest about the single-step vs full-history issue and bound it in the regime where they derive Phi.

The soft spot is load-bearing: A, B, and C in Eq. 12 are explicit functions of lambda and gamma_u, the minimal-model parameters. For the experimental datasets, the paper never states how lambda and gamma_u are chosen. If they are fitted per dataset--and Table I's BIC/RMSE comparison implies a regression framework--then the R^2 values in Fig. 2 are in-sample fits, not predictions. The abstract's 'reliably predicts' overstates what the data show. The authors need to specify a fixed, a priori mapping from trajectory statistics to (lambda, gamma_u), or validate the coefficients out-of-sample, or at minimum report sensitivity of Phi to those parameters.

A couple of smaller items: the removal of low-transition ensembles is post-hoc and should be justified more carefully; and the weak-coupling regime for fly plume tracking is not obviously satisfied, though the authors do discuss non-stationary environments.

Who this is for: researchers in biological navigation and information-theoretic analyses of behavior. The theory alone justifies reading it. The experimental application needs one more round of rigor. I would send it to peer review--the core idea and derivation are solid, but the data analysis has an unresolved calibration step that should be nailed down before it becomes a widely cited result.

Referee Report

4 major / 6 minor

Summary. The paper introduces a minimal bipartite Markov model of navigation with two actions (run/tumble) and two sensory states (up/down), derives a leading-order relationship between navigational performance and bidirectional transfer entropy rates, Δ ∝ A√Ṫ_{s→a} + B√(Ṫ_{s→a}Ṫ_{a→s}) − CṪ_{s→a} ≡ Φ (Eq. 12), and validates it numerically and against experimental trajectories of E. coli, C. elegans, Drosophila, and a reinforcement-learning agent. The authors further use the decomposition into reactive and active information flows to characterize behavioral strategies, spatial dependence in fly plume tracking, and learning dynamics in a two-armed bandit task.

Significance. If the central claim holds, the paper offers a data-driven, trajectory-based metric for navigation performance that could unify across biological and artificial systems. The strengths are substantial: the SI contains a general perturbative proof of the Φ scaling (SI §I), the minimal-model numerics match the analytic expansion, the one-step versus full-history transfer entropy discrepancy is quantified in a relevant regime (Appendix B), and the code and data are made available. The geometric-mean coupling term is a nontrivial and original contribution. However, the experimental application currently has a load-bearing gap concerning how the minimal-model parameters λ and γu are chosen, which affects whether the reported correlations are predictions or in-sample fits.

major comments (4)

[Section C, Eq. (12), Fig. 2] The coefficients A, B, C in Φ depend on λ and γu, the minimal-model parameters. The manuscript never states how λ and γu are set for the E. coli, C. elegans, Drosophila, or RL data. If they are adjusted per dataset, the R² values (0.51, 0.23, 0.30) and the Table I ΔBIC compare in-sample fits of a three-parameter curve to the same ensembles used to compute Ṫ, not predictions. The authors must either (i) specify a fixed, a priori method for choosing λ and γu from independent data, (ii) show the Φ-performance correlation is stable over a wide range of λ,γu, or (iii) reframe the experimental section as a descriptive regression rather than a prediction. Without one of these, the 'reliably predicts' claim in the abstract and §C is not supported.
[SI §I, Eq. (12)] The general proof of the Φ scaling assumes a performance metric Δ = Σ q(s,a)p(s,a) with an antisymmetric score function q(s,a) = −q(−s,a) and an inverse symmetry on the stimulus states. This is stated for the chemotaxis index, but the paper applies Φ to datasets with different performance metrics: accumulated reward for the RL agent, fraction of trajectories locating the source for flies, and normalized concentration change for worms. It is not shown that these metrics can be represented by an antisymmetric score function over the same binary state space. Without this, the proof does not extend to the cross-system claim. Please state explicitly which score function is used for each dataset, or provide a separate argument/test for the non-chemotaxis metrics.
[Section G and Discussion] The paper asserts that the experimental datasets are 'most closely related' to the weak-coupling regime α≈1, γu≈γd in which Eq. (12) was derived, but gives no evidence. For fly plume tracking, the environment is intermittent and non-stationary; for worms in a Gaussian landscape, the gradient is not linear; for the RL agent, the reward dynamics are non-stationary during learning. If the experimental systems are not in the perturbative regime, the leading-order form may fail. The authors should test the regime assumption, e.g., compare full-history and one-step TE rates in the SI for parameters estimated from data, or show that Φ's predictive power is robust away from α≈1. Appendix B quantifies history dependence only for the minimal model.
[Table I] The model comparison in Table I is performed on the same ensembles used to calibrate the predictor. Even if λ and γu are fixed, the BIC comparison and RMSE improvement do not by themselves demonstrate predictive power unless there is a train/test split or cross-validation. The table supports the claim that the geometric-mean term improves the in-sample fit, but not that Φ predicts navigation performance out of sample. Please clarify whether any held-out data were used.

minor comments (6)

[Table I heading] The heading 'without (wo, only p ˙Ts→a) the geometric mean term p ˙Ts→a ˙Ts→a' appears to contain a typo; the reduced predictor should be √Ṫ_{s→a} and the geometric mean should be √(Ṫ_{s→a}Ṫ_{a→s}).
[Fig. 2] The R² values are reported as point estimates. Given the finite-size corrections in Appendix E, confidence intervals or a cross-validation procedure should be reported to assess the stability of the correlations.
[Appendix D] The sentence 'We confirm the consistency of these state estimates through delay embedding' is unsupported. Supplementary Figure 2 shows an entropy-rate drop but not explicitly the consistency of the state estimates across datasets. Please provide the supporting analysis or temper the claim.
[Section C] For the RL agent, the text says 'the sensory signal corresponds to the reward signal.' This conflates the reward (performance outcome) with the sensory input that drives the policy. Clarify whether the reward signal is the sensory state or whether a distinct observation is used.
[General notation] The notation {x}^t_{-∞} in SI §II is used before it is defined in the main text. Define the history notation before first use in Eq. (B1).
[Abstract/§C] The phrase '>5000hour-length of E. coli trajectories' is awkward; suggest '>5000 h of trajectory data'.

Circularity Check

1 steps flagged · score 6.0 of 10

Fig. 2's cross-system 'prediction' is in-sample: the coefficients of Φ depend on λ and γu, whose values for experimental datasets are never stated, while Table I's BIC/RMSE comparison indicates a fit to the same ensembles.

fitted input called prediction [Section B, Eq. (12); Section C 'Bidirectional information flow predicts navigation performance...'; Table I]
"'∆ ∝ A√Ṫ_{s→a} + B√(Ṫ_{s→a}Ṫ_{a→s}) − CṪ_{s→a} ≡ Φ, (12) where A=..., B=..., C=... are positive functions of λ and γu but not α or γd.' ... 'Model comparison of the information theoretic predictor. Positive ΔBIC (Bayesian Information Criterion) indicate support for the full predictor Φ. ... We compare predictors with (w) and without (wo, only √Ṫ_{s→a}) the geometric mean term √(Ṫ_{s→a}Ṫ_{a→s}). RMSE improvement is defined as the percentile increase 100 (1 − RMSE_w/RMSE_wo).'"

Φ is not a parameter-free observable derived from data: its coefficients are explicit functions of the minimal-model rates λ and γu. When Φ is applied to bacteria, worms, flies, and RL agents, the paper never states how λ and γu are set for those datasets. The only quantitative evidence reported is Table I's BIC/RMSE comparison between full and reduced predictors on the same trajectory ensembles, which is a regression fit. Thus the R² values in Fig. 2 are in-sample correlations of a fitted combination of transfer entropies, not out-of-sample predictions of a fixed derived metric. The central claim that Φ 'reliably predicts navigation efficiency' reduces to calibrating its coefficients on the data it is then said to predict.

full rationale

The minimal-model derivation itself is self-contained: Eq. (12) follows from the stated perturbative expansions (Eqs. 10–11) and from the SI's general low-information expansion; the Fig. 1 phase diagram checks the model's own prediction. I find the principal circularity in the experimental application. The coefficients A, B, C in Eq. (12) are functions of λ and γu, but the text gives no method for fixing these parameters for bacterial, worm, fly, or RL trajectories. Table I reports positive ΔBIC and RMSE improvement for a full model versus a reduced model, which implies the predictor was fit to the same ensembles plotted in Fig. 2. Consequently the reported R² for experimental data are in-sample fit quality measures rather than predictions of a parameter-free information-theoretic formula. I do not count the one-step transfer entropy approximation or the weak-coupling assumption as circular, because they are stated approximations (Appendix B, Section G) rather than steps that reuse the predicted quantity. Self-citations to [5,32] are not load-bearing here: the present SI re-derives the needed scaling, and those prior results are external publications. Therefore the score is 6: one central 'prediction' partially reduces to a fitted regression, while the functional form and measured transfer entropies retain independent content.

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

The central theoretical derivation rests on standard Markov/entropy machinery plus the weak-coupling expansion; the fragile additions are the transfer of minimal-model coefficients to data and the coarse-graining choices. No new physical entities are postulated.

free parameters (4)

A,B,C coefficients for experimental Φ (via λ and γu) = unspecified; possibly fitted per dataset
Eq. (12) defines A,B,C as functions of λ and γu; experimental datasets provide no measurement of these, and the text never states the values used in Fig. 2.
Embedding time window K = chosen per dataset (e.g., ~2 s for bacteria)
Appendix D selects K by the point where steady-state entropy rate drops; data-dependent preprocessing choice.
Number of behavioral states N = 2
Appendix D clusters delay-embedded kinematics into exactly N=2 classes for all datasets; no sensitivity analysis shown.
Sensory binarization threshold = not specified numerically
Sensory signals are binarized into up/down or odor/no-odor (flies) before TE estimation; threshold choice affects all downstream estimates.

assumptions (4)

domain assumption Behavioral and sensory dynamics are conditionally independent given current state and follow a bipartite Markov chain with no simultaneous transitions.
Needed for Eq. (5), the TE decomposition, and formulas (7)-(8); latent internal states or memory would break it (Discussion acknowledges ignoring latent states).
domain assumption One-step transfer entropy rates are adequate proxies for full-history transfer entropy rates in the systems analyzed.
Used in Eq. (6) and all empirical estimates; validated only in the minimal model (Appendix B), not in worm/fly data.
domain assumption Experimental datasets lie in the weak-control/low-persistence regime (γd−γu≪1, 1−α≪1) where the Φ expansion converges.
Needed for Eq. (12); authors assert datasets are 'most closely related' to this regime (Section G) but do not test it for worm or fly plume data.
ad hoc to paper Performance metrics for all systems can be represented by an antisymmetric score function q(s,a) with an inverse symmetry.
The SI proof of Φ's scaling assumes this structure; fly 'fraction of trajectories that found source' and worm concentration-change metrics are not obviously of this form.

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Cite this review

Pith. "Pith review of A behavior-environment information loop drives sensory navigation." pith.science (2026). https://pith.science/paper/TITMDE2N

@misc{pith2026260726295,
  author       = {Pith},
  title        = {Pith review of: A behavior-environment information loop drives sensory navigation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TITMDE2N}},
  note         = {Machine review of arXiv:2607.26295}
}

read the original abstract

As organisms navigate the environment to locate critical resources, their behavioral actions must be tightly coupled to their sensory inputs. Here, we introduce an information-theoretic framework that quantifies this coupling using transfer entropy, which measures information flow between sensory inputs and behavioral outputs. Information flow from sensory inputs to behavior defines a "reactive" component of a navigational strategy, whereas information flow from behavior to sensory inputs defines an "active" component, whereby actions shape subsequent sensory experiences. Analyzing these bidirectional information flows enables us to both predict navigational performance and dissect navigation strategies from trajectories. Using a minimal model that captures the active and reactive components, we connect macroscopic performance to microscopic information flows. We then apply the framework to experimentally measured trajectories of bacteria, worms, and flies, as well as to machine learning agents navigating sensory landscapes. Across systems, bidirectional information flow reliably predicts navigation efficiency, revealing a common behavioral-environment feedback loop. Decomposing active and reactive information flows further exposes distinct strategies underlying bacterial chemotaxis, the spatial dependency of the navigation strategy in fly olfactory navigation, and the learned policies of a reinforcement-trained agent. Together, these results establish bidirectional information flow as a unifying principle for understanding navigation in biological and artificial systems.

Figures

Figures reproduced from arXiv: 2607.26295 by the authors.

**Figure 2.** FIG. 2. Bidirectional information flow between sensing [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Bidirectional information flow reveals diverse [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. Flies navigating complex plumes are more reactive [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. Bidirectional information flow reflects constraints [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. Non-monotonic scaling of the chemotactic index, [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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Forward citations

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Navigation driven by bidirectional information transmission between sensing and actuation
physics.bio-ph 2026-07 conditional novelty 7.5 of 10

Navigation performance in shallow gradients equals a closed-form function of the feedforward and feedback transfer entropies, holding for simulated E. coli without fitted parameters.

Reference graph

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+ 2γdγu +γ d +γ u. Despite its simplicity, this minimal model can implement a range of navigation strategies. To build intuition we defineγ d =Kandγ u = 1/K, with K >1, which reduces the control strategy to one parameter sensory gainK, whereasαcontrols how long the signal remains correlated during runs. The chemotaxis performance becomes: ∆ = K−1 (K+ 1)(2...

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Reviewed August 1, 2026 · model on record in the stance chip above.

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