arxiv: 2605.12126 · v1 · submitted 2026-05-12 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

Leggett--Garg Tests in Neural Dynamics: Probing Non-Diffusive Stochastic Structure in Single Neurons

Partha Ghose

Authors on Pith no claims yet

Pith reviewed 2026-05-13 04:31 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Leggett-Garg inequalitiesneural dynamicsstochastic processesKac persistent random walkTelegrapher equationnon-diffusive transporttemporal correlationssingle neurons

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The pith

Persistent stochastic dynamics in single neurons can violate Leggett-Garg inequalities unlike diffusive models.

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

This paper proposes applying Leggett-Garg tests to time series data from individual neurons to determine if their activity follows simple diffusive motion or more persistent stochastic processes. Purely diffusive dynamics always satisfies the inequalities, but models with finite-velocity persistence can generate oscillatory correlations that violate them. A reader would care because this provides an experimental route to identify memory and non-Markovian features in neural firing using only classical probability bounds, offering insight into the structure of brain dynamics without appeals to quantum effects.

Core claim

The paper establishes that while diffusive stochastic processes satisfy Leggett-Garg inequalities, Kac-type persistent stochastic processes can produce temporal correlations that violate these inequalities. This violation serves as a signature of non-diffusive structure, persistence, and contextual temporal behavior in single-neuron dynamics, derived through connections to the Telegrapher's equation via analytic continuation.

What carries the argument

The Leggett-Garg inequality applied as a temporal correlation test to stochastic models of neuron membrane potential or spiking activity, contrasting Wiener processes with Kac finite-velocity processes.

Load-bearing premise

Single neuron activity can be recorded and processed in a way that reveals the intrinsic temporal correlations without interference from external noise or network interactions.

What would settle it

Measuring a sequence of neuron states at three distinct times and computing the Leggett-Garg correlator to check if it exceeds the classical bound of 1.

read the original abstract

We propose an experimental programme to test Leggett--Garg-type temporal correlations in single-neuron dynamics. The goal is to distinguish between diffusive (Wiener/cable-equation) models and non-diffusive persistent stochastic models based on Kac-type finite-velocity processes leading to the Telegrapher's equation. We show that while purely diffusive dynamics satisfies Leggett--Garg inequalities, persistent stochastic dynamics can produce oscillatory temporal correlations capable of violating these inequalities. The Leggett--Garg inequality may be viewed as a temporal analogue of Bell-type constraints. In the present context, however, violation is interpreted conservatively not as evidence of microscopic quantum coherence, but as evidence against a simple trajectory-based diffusive description. The resulting temporal correlations indicate persistence, memory, and contextual temporal structure mathematically analogous to that encountered in quantum systems. Using the analytic continuation connecting Kac processes to Dirac-like envelope equations, we argue that finite-velocity persistent stochastic transport provides a natural mechanism for such non-diffusive temporal correlations. These tests therefore offer a possible experimental probe of contextual and non-Markovian structure in neural dynamics without requiring claims of microscopic quantum coherence in the brain.

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

The paper proposes Leggett-Garg tests as a way to flag persistent non-diffusive dynamics in single neurons, but the central claim rests on an unshown mapping from analytic continuation to explicit correlation functions.

read the letter

The main takeaway is that this is a conceptual proposal rather than a completed analysis. It suggests applying Leggett-Garg inequalities to single-neuron recordings to separate standard diffusive cable models from Kac-type finite-velocity persistent processes, noting that the latter can generate oscillatory correlations capable of violating the bounds while the former cannot. The framing stays careful by treating any violation as evidence against simple trajectory diffusion, not as proof of quantum effects in the brain, and it points to the analytic continuation link between Kac processes and Dirac-like equations as the source of the oscillatory structure. That connection is already known in the stochastic-process literature, so the paper is mainly transplanting it into a neuroscience setting as an experimental programme. This is useful for modelers who want a temporal-correlation probe for memory and non-Markovian behavior without invoking quantum coherence claims. The distinction itself is standard: diffusive correlations decay and satisfy the inequality under macrorealism, while persistent ones can oscillate if the propagator allows it. The paper does not add new derivations or data, which keeps the contribution modest but honest. The soft spots are real and central. No explicit two- or three-time correlation functions are computed for a dichotomic observable such as thresholded membrane potential in the telegraph process, so it is not shown that the analytic continuation actually produces |C12 + C23 − C13| > 1 on accessible timescales without extra damping or boundary terms that would restore the bound. The abstract also gives no error analysis, no simulation results, and no concrete protocol for isolating the predicted oscillations amid measurement noise or network inputs. The weakest assumption is therefore that clean enough recordings exist to test the distinction at all. This work is aimed at computational neuroscientists and stochastic-process researchers who already think about cable versus telegrapher models. A reader looking for new empirical tests or sharpened model discrimination might find the idea worth following up, but experimentalists will need the missing calculations and feasibility checks. I would send it to peer review. The core idea is coherent on its own terms and could be strengthened by referees asking for the explicit correlations and a workable measurement scheme, even though the present version is preliminary.

Referee Report

3 major / 1 minor

Summary. The manuscript proposes an experimental programme to apply Leggett-Garg (LG) inequalities to single-neuron dynamics as a means to distinguish diffusive (Wiener/cable-equation) stochastic models from persistent (Kac-type finite-velocity) models leading to the Telegrapher's equation. It claims that purely diffusive processes satisfy the LG inequalities while persistent dynamics generate oscillatory temporal correlations capable of violating them, derived via analytic continuation connecting Kac processes to Dirac-like envelope equations. Violation is interpreted as evidence of persistence, memory, and contextual temporal structure in neural dynamics without requiring microscopic quantum coherence.

Significance. If the theoretical distinction can be made rigorous and the proposed tests implemented with suitable controls, the work would offer a novel classical probe for non-Markovian features in neural stochastic processes, potentially bridging stochastic-process theory and experimental neuroscience. The conservative interpretation of LG violation as a signature of persistence rather than quantum effects is a strength, but the manuscript remains at the level of a high-level proposal without supporting calculations, simulations, or protocols, so its immediate impact is limited.

major comments (3)

The central theoretical claim—that analytic continuation from Kac processes to Dirac-like equations produces explicit two- and three-time correlation functions C(t_i, t_j) = ⟨Q(t_i)Q(t_j)⟩ for a dichotomic observable Q (such as sign of velocity or thresholded membrane potential) that violate the LG combination |C12 + C23 − C13| > 1—remains implicit. No explicit propagator or correlation expressions are supplied to confirm that the envelope equations preserve a classical correlation structure capable of exceeding the bound on accessible timescales.
The manuscript provides no concrete experimental protocol, including the operational definition of the dichotomic observable Q from neural recordings, the selection of measurement times t1, t2, t3, controls for measurement noise, or isolation from network effects. Without these, it is impossible to assess whether real single-neuron data can cleanly isolate the predicted oscillatory correlations.
No error analysis, Monte-Carlo simulations of the persistent process under realistic noise levels, or power calculations for detecting LG violation are included. This leaves the feasibility of the proposed programme unquantified.

minor comments (1)

The abstract would benefit from a one-sentence statement of the specific form of the Leggett-Garg inequality employed (e.g., the standard three-time combination).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive comments on our manuscript. We address each major comment point by point below and will revise the manuscript to incorporate the suggested clarifications and additions.

read point-by-point responses

Referee: The central theoretical claim—that analytic continuation from Kac processes to Dirac-like equations produces explicit two- and three-time correlation functions C(t_i, t_j) = ⟨Q(t_i)Q(t_j)⟩ for a dichotomic observable Q (such as sign of velocity or thresholded membrane potential) that violate the LG combination |C12 + C23 − C13| > 1—remains implicit. No explicit propagator or correlation expressions are supplied to confirm that the envelope equations preserve a classical correlation structure capable of exceeding the bound on accessible timescales.

Authors: We agree that explicit expressions would strengthen the central claim. In the revised manuscript we will derive and present the explicit two- and three-time correlation functions C(t_i, t_j) obtained from the analytic continuation of the Kac process to the Dirac-like envelope equations. These expressions will confirm that the persistent dynamics generate oscillatory correlations capable of violating |C12 + C23 − C13| ≤ 1 on timescales relevant to single-neuron dynamics while preserving a fully classical correlation structure. This addition will make the theoretical distinction rigorous and address the implicit nature of the original presentation. revision: yes
Referee: The manuscript provides no concrete experimental protocol, including the operational definition of the dichotomic observable Q from neural recordings, the selection of measurement times t1, t2, t3, controls for measurement noise, or isolation from network effects. Without these, it is impossible to assess whether real single-neuron data can cleanly isolate the predicted oscillatory correlations.

Authors: We acknowledge that a detailed protocol is required for the proposal to be actionable. In the revision we will add a dedicated section that specifies: (i) the operational definition of the dichotomic observable Q (sign of thresholded membrane-potential fluctuations or an analogous velocity-like variable in the stochastic model); (ii) concrete choices for the measurement times t1, t2, t3 informed by typical neuronal membrane time constants and the persistence time of the Telegrapher’s equation; and (iii) experimental controls including pharmacological or voltage-clamp isolation of single-neuron dynamics, noise-filtering procedures, and checks for network contamination. These additions will allow readers to evaluate the feasibility of isolating the predicted correlations in real recordings. revision: yes
Referee: No error analysis, Monte-Carlo simulations of the persistent process under realistic noise levels, or power calculations for detecting LG violation are included. This leaves the feasibility of the proposed programme unquantified.

Authors: The original manuscript was framed as a high-level theoretical proposal. To quantify feasibility we will include, in a new subsection, Monte-Carlo simulations of the Kac-type persistent process with added realistic measurement noise, an error analysis for the estimated correlation functions, and statistical power calculations indicating the number of trials needed to detect LG violations at a chosen significance level. These numerical results will provide concrete guidance on the experimental resources required. revision: yes

Circularity Check

0 steps flagged

No circularity: standard stochastic distinctions applied to LG inequalities

full rationale

The paper's central distinction—that purely diffusive Wiener/cable dynamics satisfies Leggett-Garg inequalities while Kac-type persistent processes can produce oscillatory correlations capable of violating them—rests on the established correlation structures of these processes (exponentially decaying for Markovian diffusion versus damped oscillatory for finite-velocity telegraph processes) together with the definition of the LG combination. No equations in the provided text reduce a claimed prediction or uniqueness result to a fitted parameter, self-definition, or load-bearing self-citation; the analytic-continuation argument is invoked only to motivate the envelope equations whose correlation properties are already known from the stochastic-process literature. The derivation therefore remains self-contained against external benchmarks and introduces no circular step.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two standard mathematical facts about stochastic processes and one interpretive stance; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)

standard math Purely diffusive (Wiener/cable-equation) dynamics satisfies Leggett-Garg inequalities
Standard result for Markovian diffusive processes; invoked in the abstract to contrast with the persistent case.
domain assumption Persistent stochastic dynamics based on Kac-type finite-velocity processes can produce oscillatory correlations that violate LG inequalities
Derived via analytic continuation to the Telegrapher's equation; presented as a known property of such processes.

pith-pipeline@v0.9.0 · 5500 in / 1395 out tokens · 33718 ms · 2026-05-13T04:31:56.056819+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

while purely diffusive dynamics satisfies Leggett-Garg inequalities

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

Quantum-like brain: Interference of minds,

A. Khrennikov, “Quantum-like brain: Interference of minds,”Biosystems84, 225–241 (2006)

work page 2006
[2]

J. R. Busemeyer and P. D. Bruza,Quantum Models of Cognition and Decision, Cam- bridge University Press (2012)

work page 2012
[3]

Orchestrated reduction of quantum coherence in brain microtubules: A model for consciousness,

R. Penrose and S. Hameroff, “Orchestrated reduction of quantum coherence in brain microtubules: A model for consciousness,”Math. Comput. Simul.40, 453–480 (1996)

work page 1996
[4]

Quantum mechanics, interference, and the brain,

J. A. de Barros and P. Suppes, “Quantum mechanics, interference, and the brain,”J. Math. Psychol.53, 306-313 (2009)

work page 2009
[5]

Importance of quantum decoherence in brain processes,

M. Tegmark, “Importance of quantum decoherence in brain processes,”Phys. Rev. E61, 4194–4206 (2000)

work page 2000
[6]

Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?

A. J. Leggett and A. Garg, “Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?”Phys. Rev. Lett.54, 857 (1985)

work page 1985
[7]

On the Einstein-Podolsky-Rosen paradox,

J. S. Bell, “On the Einstein-Podolsky-Rosen paradox,”Physics1, 195–200 (1964). 10

work page 1964
[8]

Risken,The Fokker–Planck Equation: Methods of Solution and Applications, 2nd ed., Springer (1989)

H. Risken,The Fokker–Planck Equation: Methods of Solution and Applications, 2nd ed., Springer (1989)

work page 1989
[9]

N. G. van Kampen,Stochastic Processes in Physics and Chemistry, Elsevier (1992)

work page 1992
[10]

Dayan and L

P. Dayan and L. F. Abbott,Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, MIT Press (2001)

work page 2001
[11]

A stochastic model related to the telegrapher’s equation,

M. Kac, “A stochastic model related to the telegrapher’s equation,”Rocky Mountain J. Math.4, 497-510 (1974)

work page 1974
[12]

Classical World Arising out of Quantum Physics under the Restriction of Coarse-Grained Measurements,

J. Kofler and C. Brukner, “Classical World Arising out of Quantum Physics under the Restriction of Coarse-Grained Measurements,”Phys. Rev. Lett.99, 180403 (2007)

work page 2007
[13]

Violation of a Leggett–Garg inequality with ideal non-invasive measurements,

G. C. Knee, S. Simmons, E. M. Gauger, J. J. L. Morton, H. Riemann, N. V. Abrosimov, P. Becker, H.-J. Pohl, K. M. Itoh, M. L. W. Thewalt, G. A. D. Briggs, and S. C. Benjamin, “Violation of a Leggett–Garg inequality with ideal non-invasive measurements,”Nat. Commun.3, 606 (2012)

work page 2012
[14]

Addressing the clumsiness loophole in a Leggett–Garg test of macrorealism,

M. M. Wilde and A. Mizel, “Addressing the clumsiness loophole in a Leggett–Garg test of macrorealism,”Foundations of Physics42, 256–265 (2012)

work page 2012
[15]

Relativistic Extension of the Analogy between Quantum Mechanics and Brownian Motion,

B. Gaveau, T. Jacobson, M. Kac, and L. S. Schulman, “Relativistic Extension of the Analogy between Quantum Mechanics and Brownian Motion,”Phys. Rev. Lett.53, 419 (1984)

work page 1984
[16]

The FitzHugh–Nagumo equations and quantum noise,

P. Ghose and D. A. Pinotsis, “The FitzHugh–Nagumo equations and quantum noise,” Computational and Structural Biotechnology Journal30, 12–20 (2025). 11

work page 2025