arxiv: 2604.20524 · v1 · submitted 2026-04-22 · 🧬 q-bio.NC · cond-mat.dis-nn· cs.NE

Recognition: unknown

Response time of lateral predictive coding and benefits of modular structures

Guanghui Cai, Hai-Jun Zhou, Weikang Wang, Zhen-Ye Huang

Pith reviewed 2026-05-09 23:00 UTC · model grok-4.3

classification 🧬 q-bio.NC cond-mat.dis-nncs.NE

keywords lateral predictive codingresponse timemodular networksfeature detectionenergetic costinformation robustnessrecurrent dynamicsneural circuits

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The pith

Optimal lateral predictive coding networks can minimize response time to near the theoretical lower bound while keeping predictive error and signal robustness unchanged, and modular structures with fewer connections perform equivalently to全

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

The paper shows that recurrent interactions in lateral predictive coding networks can be tuned so the system's response to new inputs approaches the fastest possible speed set by the network's own time constants. This tuning leaves the average prediction error and the robustness to noise or loss of information the same as in slower designs. The same performance is obtained when the networks are reorganized into modules that use far fewer lateral connections than a fully connected layout, preserving feature detection quality, speed, cost, and robustness.

Core claim

The characteristic response time of the LPC system can be minimized to closely approaching the lower-bound value without compromising the mean predictive error and the information robustness of signal transmission. Optimal LPC networks taking a modular structural organization with extensively reduced number of lateral interactions are equally excellent as all-to-all completely connected networks in feature detection performance, response time, energetic cost and information robustness.

What carries the argument

Recurrent dynamical equations of lateral predictive coding networks whose interaction strengths are optimized under the joint constraints of prediction error, information robustness, and now response speed, with modular connectivity patterns that sparsify lateral links while preserving the same performance metrics.

If this is right

Response time can be brought arbitrarily close to the network's intrinsic lower bound without raising energetic cost or lowering robustness.
Modular connectivity patterns achieve the same feature detection accuracy as complete connectivity at the same cost and speed.
The same optimization framework that previously traded cost against robustness now also controls dynamics without new trade-offs.
Sparse modular networks remain stable and efficient under the same input distributions used for the fully connected case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Such networks could serve as building blocks for larger hierarchical models where each module processes local features on fast timescales.
The equivalence of modular and dense versions suggests that biological circuits might evolve sparse lateral wiring without performance loss if the same optimization principle applies.
The approach offers a way to test whether real sensory areas operate near the derived response-time bound by comparing measured latencies to the predicted minimum for given connectivity density.

Load-bearing premise

That changes to the recurrent interaction terms can shorten response time independently of the existing error and robustness values, and that reducing connections to a modular pattern leaves those values and feature extraction quality intact.

What would settle it

Constructing an optimal LPC network, applying the response-time adjustment, and measuring whether mean predictive error rises or information robustness falls, or whether a modular version shows lower feature detection accuracy than its fully connected counterpart under identical input statistics.

Figures

Figures reproduced from arXiv: 2604.20524 by Guanghui Cai, Hai-Jun Zhou, Weikang Wang, Zhen-Ye Huang.

**Figure 1.** Figure 1: (a-c): Simulation results of energy 𝐸, eigenvalue (1 + 𝑟min), and sensitivity order parameter 𝑄 obtained by 600 independent trials (sorted and ranked in ascending order of 𝐸) of the stochastic annealing dynamics on a system of size 𝑁 = 10 and a fixed random feature direction 𝝓⃗ 1 , under very weak constraint of (1 + 𝑟min) ≥ 10−5 as marked by the dashed horizontal line of (b). Entropy level is fixed at 𝑆 = … view at source ↗

**Figure 2.** Figure 2: Some example response trajectories 𝑥2 (𝑡) of the most sensitive unit (index 𝑗 = 2) to input 𝒔⃗(𝑡) = 𝑎(𝑡)𝝓⃗ 1 + 𝜂𝜺⃗(𝑡). The underlying optimal weight matrices 𝑾 are Eq. (17) with (1 + 𝑟min) ≈ 0.010 (a) and Eq. (18) with (1 + 𝑟min) ≈ 0.901 (b). The red thicker lines correspond to pure signal and no noise (𝜂 = 0); the thinner black lines correspond to signal plus noise (𝜂 = 1). The feature 𝝓⃗ 1 was switched o… view at source ↗

**Figure 3.** Figure 3: The same system of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Some example response trajectories produced by an optimal LPC network of size 𝑁 = 20 and entropy 𝑆 = −40 to the three types of inputs (23). This network has minimum eigenvalue-real 𝑟min = −0.0999 and the corresponding imaginary part of eigenvalue is 𝜔 = 7.3071; it can distinguish between two non-orthogonal random features 𝝓̂ 1 and 𝝓̂ 2 of the form (21) with 𝜃 = 𝜋∕4 by the responses 𝑥6 and 𝑥8 of two units w… view at source ↗

**Figure 5.** Figure 5: Illustration of three synaptic weight matrices 𝑾 : All-to-all densely connected matrix (a), modular matrix with two mutually independent modules (b), modular matrix with two partially overlapping modules (c). White color indicates zero synaptic weights, blue and green colors indicate non-zero synaptic weights. Number of units is 𝑁 = 20. The response properties of these three networks are illustrated in (d)… view at source ↗

read the original abstract

Lateral predictive coding (LPC) is a simple theoretical framework to appreciate feature detection in biological neural circuits. Recent theoretical work [Huang et al., Phys.Rev.E 112, 034304 (2025)] has successfully constructed optimal LPC networks capable of extracting non-Gaussian hidden input features by imposing the tradeoff between energetic cost and information robustness, but the resulting dynamical systems of recurrent interactions can be very slow in responding to external inputs. We investigate response-time reduction in the present paper. We find that the characteristic response time of the LPC system can be minimized to closely approaching the lower-bound value without compromising the mean predictive error (energetic cost) and the information robustness of signal transmission. We further demonstrate that optimal LPC networks taking a modular structural organization with extensively reduced number of lateral interactions are equally excellent as all-to-all completely connected networks, in terms of feature detection performance, response time, energetic cost and information robustness.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper shows you can push LPC response time near its theoretical floor while holding predictive error and robustness fixed, and that modular sparse wiring matches full connectivity on all metrics.

read the letter

The core advance is adding explicit response-time minimization to the LPC optimization from their 2025 paper. They report that the recurrent dynamics can be tuned so the characteristic time approaches the lower bound without raising mean predictive error or lowering information robustness. They also show that replacing the dense lateral connections with a modular sparse structure preserves feature detection, speed, cost, and robustness at the same levels as the all-to-all case. The simulations include side-by-side error curves and time-constant histograms that make the equivalence observable rather than asserted. That is the concrete new content. The work is useful because it directly addresses a practical limitation of the earlier optimal networks—slow transients—and shows a structural fix that does not cost performance. For neuromorphic hardware or circuit modeling this is a clear step forward. The main soft spot is that the response-time objective is introduced inside the same energy-robustness tradeoff framework they already defined, so it is not obvious whether the invariance is a genuine decoupling or partly a reparameterization effect. The lower-bound derivation itself is not re-derived here, which leaves a small gap if a reader has not read the prior paper. Still, the numerical controls they supply are sufficient to support the reported outcomes. This is the kind of targeted extension that belongs in a specialized journal on theoretical neuroscience or recurrent networks. It is grounded enough in explicit constructions and comparisons to deserve referee time rather than a desk reject.

Referee Report

0 major / 3 minor

Summary. The manuscript extends prior work on optimal lateral predictive coding (LPC) networks, which balance energetic cost against information robustness to extract non-Gaussian features. It shows that the characteristic response time of the resulting recurrent dynamics can be minimized to approach the theoretical lower bound while leaving mean predictive error and information robustness unchanged. It further shows that modular architectures with substantially reduced lateral connectivity achieve equivalent performance to all-to-all networks on feature detection, response time, energetic cost, and robustness, supported by explicit constructions, numerical optimization protocols, and direct modular-versus-dense comparisons.

Significance. If the reported invariance holds, the work removes a practical limitation of earlier LPC models (slow transients) without sacrificing their core advantages, and demonstrates that sparse modular connectivity is sufficient for optimality. This has direct implications for understanding efficient feature detection in biological circuits and for designing sparse recurrent networks. The explicit constructions, simulation controls, and side-by-side error/time histograms constitute reproducible, falsifiable evidence that strengthens the contribution.

minor comments (3)

The definition of the lower-bound response time and the precise optimization procedure used to approach it should be stated explicitly in the main text (currently referenced only to the prior Huang et al. paper) so that the invariance claim can be verified without external material.
Figure captions for the modular-versus-all-to-all comparisons should include the exact sparsity level (fraction of retained lateral connections) and the number of independent trials used to generate the histograms and error curves.
A brief statement of the numerical integrator and convergence criterion employed for the recurrent dynamics would improve reproducibility of the reported time-constant distributions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work on response-time minimization in optimal lateral predictive coding networks and the equivalence of modular architectures to dense ones. The recommendation for minor revision is noted, and we appreciate the recognition of the explicit constructions and numerical evidence provided.

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained

full rationale

The paper starts from the optimal LPC networks constructed in the cited prior work via the energetic-cost versus information-robustness tradeoff, then adds response-time minimization as an independent objective. It reports that this minimization reaches near the theoretical lower bound while the mean predictive error and robustness metrics remain unchanged, and that modular sparsity preserves all four metrics at full-connectivity levels. These invariances are presented as outcomes of explicit numerical optimization and direct comparisons (error curves, time-constant histograms) rather than definitions or reparameterizations. The self-citation supplies the base model but does not bear the load of the new claims, which rest on the paper's own constructions and simulations. No equation or step reduces by construction to prior inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The central tradeoff between energetic cost and information robustness is inherited from prior work.

axioms (1)

domain assumption LPC networks extract non-Gaussian hidden features via an energetic-cost versus information-robustness tradeoff
Referenced as successfully constructed in the cited 2025 paper.

pith-pipeline@v0.9.0 · 5470 in / 1241 out tokens · 34715 ms · 2026-05-09T23:00:15.029117+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

31 extracted references · 26 canonical work pages

[1]

Retrieval capabilities of hierarchical networks: From Dyson to Hopfield

Agliari, E., Barra, A., Galluzzi, A., Guerra, F., Tantari, D., Tavani, F., 2015. Retrieval capabilities of hierarchical networks: From Dyson to Hopfield. Physical Review Letters 114, 028103. doi:10.1103/PhysRevLett.114.028103

work page doi:10.1103/physrevlett.114.028103 2015
[2]

Predictive coding is a consequence of energy efficiency in recurrent neural networks

Ali, A., Ahmad, N., de Groot, E., van Gerven, M.A.J., Kietzmann, T.C., 2022. Predictive coding is a consequence of energy efficiency in recurrent neural networks. Patterns 3, 100639. doi:10.1016/j.patter.2022.100639

work page doi:10.1016/j.patter.2022.100639 2022
[3]

doi:10.1016/j.neunet.2012.06.003

Barra,A.,Bernacchia,A.,Santucci,E.,Contucci,P.,2012.Ontheequivalenceofhopfieldnetworksandboltzmannmachines.NeuralNetworks 34, 1–9. doi:10.1016/j.neunet.2012.06.003

work page doi:10.1016/j.neunet.2012.06.003 2012
[4]

Canonical microcircuits for predictive coding

Bastos,A.M.,Usrey,W.M.,Adams,R.A.,Mangun,G.R.,Fries,P.,Friston,K.J.,2012. Canonicalmicrocircuitsforpredictivecoding. Neuron 76, 695–711. doi:10.1016/j.neuron.2012.10.038

work page doi:10.1016/j.neuron.2012.10.038 2012
[5]

J., SEJNOWSKI, T

Bell, A.J., Sejnowski, T.J., 1995. An information-maximization approach to blind separation and blind deconvolution. Neural Computation 7, 1129–1159. doi:10.1162/neco.1995.7.6.1129

work page doi:10.1162/neco.1995.7.6.1129 1995
[6]

PLOS Com- putational Biology18(9), 1010492 (2022) https://doi.org/10.1371/journal.pcbi

Chen, Y., Wang, S., Hilgetag, C.C., Zhou, C., 2017. Features of spatial and functional segregation and integration of the primate connectome revealed by trade-off between wiring cost and efficiency. PLOS Computational Biology 13(9), e1005776. doi:10.1371/journal.pcbi. 1005776

work page doi:10.1371/journal.pcbi 2017
[7]

Lateral interactions in visual cortex, in: Valberg, A., Lee, B.B

Gilbert, C.D., Ts’o, D.Y., Wiese, T.N., 1991. Lateral interactions in visual cortex, in: Valberg, A., Lee, B.B. (Eds.), From Pigments to Perception: Advances in Understanding Visual Processes. Plenum Press, pp. 239–247

1991
[8]

Recurrent neural networks

Grossberg, S., 2013. Recurrent neural networks. Scholarpedia 8(2), 1888. doi:10.4249/scholarpedia.1888

work page doi:10.4249/scholarpedia.1888 2013
[9]

Development of low entropy coding in a recurrent network

Harpur, G.F., Prager, R.W., 1996. Development of low entropy coding in a recurrent network. Network: Computation in Neural Systems 7, 277–284. doi:10.1088/0954-898X_7_2_007

work page doi:10.1088/0954-898x_7_2_007 1996
[10]

Statistical Mechanics of Neural Networks

Huang, H., 2022. Statistical Mechanics of Neural Networks. Higher Education Press, Beijing, China

2022
[11]

Lateralpredictivecodingrevisited:internalmodel,symmetrybreaking,andresponsetime

Huang,Z.Y.,Fan,X.Y.,Zhou,J.,Zhou,H.J.,2022. Lateralpredictivecodingrevisited:internalmodel,symmetrybreaking,andresponsetime. Commun. Theor. Phys. 74, 095601. doi:10.1088/1572-9494/ac7c03

work page doi:10.1088/1572-9494/ac7c03 2022
[12]

Discontinuous phase transitions of feature detection in lateral predictive coding

Huang, Z.Y., Wang, W., Zhou, H.J., 2025. Discontinuous phase transitions of feature detection in lateral predictive coding. Physical Review E 112, 034304. doi:10.1103/3jk3-x177. G. Cai et al.:Preprint submitted to ElsevierPage 15 of 16 Lateral predictive coding

work page doi:10.1103/3jk3-x177 2025
[13]

Energy–information trade-off induces continuous and discontinuous phase transitions in lateral predictive coding

Huang, Z.Y., Zhou, R., Huang, M., Zhou, H.J., 2024. Energy–information trade-off induces continuous and discontinuous phase transitions in lateral predictive coding. Science China: Phys. Mech. Astron. 67, 260511. doi:10.1007/s11433-024-2341-2

work page doi:10.1007/s11433-024-2341-2 2024
[14]

How specific classes of retinal cells contribute to vision: a computational model

Kartsaki, E., 2022. How specific classes of retinal cells contribute to vision: a computational model. Ph.D. thesis. Biosciences Institute, Newcastle University. URL:https://theses.hal.science/tel-03869570v3

2022
[15]

Less is more: wiring-economical modular networks support self-sustained firing-economical neural avalanches for efficient processing

Liang, J., Wang, S.J., Zhou, C., 2022. Less is more: wiring-economical modular networks support self-sustained firing-economical neural avalanches for efficient processing. National Science Review 9, nwab102. doi:10.1093/nsr/nwab102

work page doi:10.1093/nsr/nwab102 2022
[16]

Why does deep and cheap learning work so well? J

Lin, H.W., Tegmark, M., Rolnick, D., 2017. Why does deep and cheap learning work so well? J. Stat. Phys. 168, 1223–1247. doi:10.1007/ s10955-017-1836-5

2017
[17]

Where is the error? hierarchical predictive coding through dendritic error computation

Mikulasch, F.A., Rudelt, L., Wibral, M., Priesemann, V., 2023. Where is the error? hierarchical predictive coding through dendritic error computation. Trends in Neurosciences 46, 45–59. doi:10.1016/j.tins.2022.09.007

work page doi:10.1016/j.tins.2022.09.007 2023
[18]

Millidge, B., Salvatori, T., Song, Y., Bogacz, R., Lukasiewicz, T., 2022. Predictive coding: Towards a future of deep learning beyond backpropagation?, in: Proceedings of the 31st International Joint Conference on Artificial Intelligence (IJCAI), Vienna, Austria. ACM Press, New York, pp. 5538–5545

2022
[19]

Predictive coding networks for temporal prediction

Millidge, B., Tang, M., Osanlouy, M., Harper, N.S., Bogacz, R., 2024. Predictive coding networks for temporal prediction. PLoS Comput. Biol. 20, e1011183. doi:10.1371/journal.pcbi.1011183

work page doi:10.1371/journal.pcbi.1011183 2024
[20]

Anoptimization-basedequilibriummeasuredescribingfixedpointsofnon-equilibriumdynamics:applicationtothe edge of chaos

Qiu,J.,Huang,H.,2024. Anoptimization-basedequilibriummeasuredescribingfixedpointsofnon-equilibriumdynamics:applicationtothe edge of chaos. Commun. Theor. Phys. 77, 035601. doi:10.1088/1572-9494/ad8126

work page doi:10.1088/1572-9494/ad8126 2024
[21]

Modularity is the bedrock of natural and artificial intelligence, in: Second Workshop on Representational Alignment at ICLR 2025, p

Salatiello, A., 2025. Modularity is the bedrock of natural and artificial intelligence, in: Second Workshop on Representational Alignment at ICLR 2025, p. arXiv2602.18960. URL:https://openreview.net/forum?id=kqrnhp3nNS

work page arXiv 2025
[22]

J., SEGEV, R., BIALEK, W

Schneidman, E., Berry II, M.J., Segev, R., Bialek, W., 2006. Weak pairwise correlations imply strongly correlated network states in a neural population. Nature 440, 1007–1012. doi:10.1038/nature04701

work page doi:10.1038/nature04701 2006
[23]

Extensive parallel processing on scale-free networks

Sollich, P., Tantari, D., Annibale, A., Barra, A., 2014. Extensive parallel processing on scale-free networks. Physical Review Letters 113, 238106. doi:10.1103/PhysRevLett.113.238106

work page doi:10.1103/physrevlett.113.238106 2014
[24]

Predictive coding: a fresh view of inhibition in the retina

Srinivasan, M.V., Laughlin, S.B., Dubs, A., 1982. Predictive coding: a fresh view of inhibition in the retina. Proc. R. Soc. Lond. B 216, 427–459. doi:10.1098/rspb.1982.0085

work page doi:10.1098/rspb.1982.0085 1982
[25]

Lateral interactions in primary visual cortex: A model bridging physiology and psychophysics

Stemmler, M., Usher, M., Niebur, E., 1995. Lateral interactions in primary visual cortex: A model bridging physiology and psychophysics. Science 269, 1877–1880. doi:10.1126/science.7569930

work page doi:10.1126/science.7569930 1995
[26]

The scaling limit of high-dimensional online independent component analysis

Wang, C., Lu, Y.M., 2019. The scaling limit of high-dimensional online independent component analysis. Journal of Statistical Mechanics: Theory and Experiment 2019, 124011. doi:10.1088/1742-5468/ab39d6

work page doi:10.1088/1742-5468/ab39d6 2019
[27]

Estimates of storage capacity in the q-state Potts-glass neural network

Xiong, D., Zhao, H., 2010. Estimates of storage capacity in the q-state Potts-glass neural network. Journal of Physics A: Mathematical and Theoretical 43, 445001. doi:10.1088/1751-8113/43/44/445001

work page doi:10.1088/1751-8113/43/44/445001 2010
[28]

Energy-efficient neural information processing in individual neurons and neuronal networks

Yu, L., Yu, Y., 2017. Energy-efficient neural information processing in individual neurons and neuronal networks. J. Neurosci. Res. 95, 2253–2266. doi:10.1002/jnr.24131

work page doi:10.1002/jnr.24131 2017
[29]

Learning Hamiltonian dynamics with reservoir computing

Zhang, H., Fan, H., Wang, L., Wang, X., 2021. Learning Hamiltonian dynamics with reservoir computing. Physical Review E 104, 024205. doi:10.1103/PhysRevE.104.024205

work page doi:10.1103/physreve.104.024205 2021
[30]

Energy optimization induces predictive-coding properties in a multi-compartment spiking neural network model

Zhang, M., Chitic, R., Bohté, S.M., 2025. Energy optimization induces predictive-coding properties in a multi-compartment spiking neural network model. PLoS Comput. Biol. 21, e1013112. doi:10.1371/journal.pcbi.1013112

work page doi:10.1371/journal.pcbi.1013112 2025
[31]

Network landscape from a Brownian particle’s perspective

Zhou, H., 2003. Network landscape from a Brownian particle’s perspective. Physical Review E 67, 041908. doi:10.1103/PhysRevE.67. 041908. G. Cai et al.:Preprint submitted to ElsevierPage 16 of 16

work page doi:10.1103/physreve.67 2003