leggett-garg-neural-dynamics

name

leggett-garg-neural-dynamics

description

Leggett-Garg inequality testing methodology for neural dynamics. Proposes experimental tests to distinguish diffusive vs non-diffusive stochastic structure in single neurons using temporal correlations. Connects Kac processes to Telegrapher equation and Dirac-like envelope equations. Activation: Leggett-Garg inequality, neural dynamics, Telegrapher equation, persistent stochastic process, non-diffusive neuron, quantum-inspired neuroscience, temporal correlations.

Leggett-Garg Tests in Neural Dynamics

Methodology for probing non-diffusive stochastic structure in single-neuron dynamics using Leggett-Garg-type temporal correlation tests, based on Ghose (2026), arXiv:2605.12126.

Core Concept

The Leggett-Garg inequality (LGI) is a temporal analogue of Bell inequalities. This methodology proposes using LGI tests on single-neuron dynamics to distinguish between:

1. Diffusive models (Wiener process / cable equation): Standard neural dynamics models where membrane potential evolves as Brownian motion. These ALWAYS satisfy LGIs.
2. Non-diffusive persistent models (Kac-type finite-velocity processes): Models with memory and persistence that lead to the Telegrapher's equation. These CAN violate LGIs.

Key Theoretical Framework

Telegrapher's Equation vs Cable Equation

The cable equation (standard neural model): $$\frac{\partial V}{\partial t} = D \frac{\partial^2 V}{\partial x^2}$$

The Telegrapher's equation (persistent stochastic model): $$\frac{1}{v^2} \frac{\partial^2 V}{\partial t^2} + \frac{1}{D} \frac{\partial V}{\partial t} = \frac{\partial^2 V}{\partial x^2}$$

Key difference: Telegrapher's equation includes a second-order time derivative, capturing finite-velocity transport with memory effects.

Leggett-Garg Inequality

For measurements at three times $t_1 < t_2 < t_3$, with dichotomic observable $Q(t) = \pm 1$:

$$K = C_{12} + C_{23} - C_{13} \leq 1$$

where $C_{ij} = \langle Q(t_i)Q(t_j) \rangle$ is the two-time correlation function.

Violation condition: $K > 1$ indicates non-diffusive, persistent temporal structure.

Analytic Continuation: Kac → Dirac

• Kac processes (persistent random walks with finite velocity) are analytically continued to Dirac-like envelope equations
• This provides a mathematical bridge between classical persistent stochastic transport and quantum-like temporal correlations
• Violation is interpreted conservatively: NOT evidence of quantum coherence, but evidence against simple diffusive description

Experimental Protocol

Step 1: Measurement Setup

• Record single-neuron membrane potential at high temporal resolution
• Define dichotomic observable: $Q(t) = +1$ if $V(t) > V_{threshold}$, $Q(t) = -1$ otherwise
• Choose appropriate threshold (e.g., resting potential or mean firing threshold)

Step 2: Correlation Measurement

• Measure two-time correlation functions $C(t_1, t_2)$ for multiple time pairs
• Use non-invasive or weakly-invasive measurement to minimize disturbance
• Ensure measurement independence (no-signaling in time)

Step 3: LGI Computation

• Compute $K = C(t_1,t_2) + C(t_2,t_3) - C(t_1,t_3)$ for triplets of measurement times
• Test for violation: $K > 1$
• Optimize time intervals to maximize potential violation

Step 4: Interpretation

• If $K \leq 1$: Consistent with diffusive (cable equation) dynamics
• If $K > 1$: Evidence of persistent, non-Markovian temporal structure
• Oscillatory temporal correlations are the key signature of violation

Key Findings

• Purely diffusive dynamics always satisfies LGIs
• Persistent stochastic dynamics can produce oscillatory temporal correlations capable of violating LGIs
• Violation indicates: persistence, memory, and contextual temporal structure
• These features are mathematically analogous to quantum systems, but the interpretation is conservative (no quantum coherence claim)

Applications

• Probing non-Markovian structure in neural dynamics
• Distinguishing competing models of single-neuron behavior
• Understanding memory effects in neural processing
• Bridge between neuroscience and quantum foundations (conceptual, not ontological)
• Experimental probe of contextual temporal structure without requiring quantum brain claims

Pitfalls

1. Conservative interpretation: LGI violation does NOT imply quantum coherence in neurons. It indicates non-diffusive stochastic structure only.
2. Measurement invasiveness: Strong measurements can artificially destroy temporal correlations. Use weak or non-invasive techniques.
3. Time resolution: Telegrapher's equation effects operate at very short timescales. Sub-millisecond recording resolution needed.
4. No-signaling assumption: LGI tests require measurement independence. Ensure measurements at $t_i$ do not influence future dynamics beyond the natural evolution.
5. Alternative classical explanations: Persistent stochastic processes are classical — distinguish from quantum carefully.

Mathematical Tools

• Kac process: Random walk with finite velocity, alternating direction at Poisson-distributed times
• Telegrapher's equation: Derived from Kac process in the diffusion limit with finite velocity
• Dirac envelope equation: Obtained via analytic continuation from Telegrapher's equation
• Temporal correlation functions: $C(t_1, t_2) = \langle Q(t_1)Q(t_2) \rangle$

Related Skills

• neural-dynamics-universal-translator: Cross-model neural dynamics alignment
• neural-critical-dynamics-theory: Critical dynamics in neural systems
• stochastic-physical-neural-networks: Physical neural networks with stochastic dynamics
• quantum-neuroscience-patterns: Umbrella skill for quantum-neuroscience research patterns
• orchid-kuramoto-quantum-consensus: Bio-inspired Kuramoto quantum consensus protocol