Geometric Irreversibility and Dissipation Bounds in Non-Reciprocal Neural Attractor Dynamics

Technical Summary: Geometric Irreversibility and Dissipation Bounds in Non-Reciprocal Neural Attractor Dynamics

Problem Statement
Classical attractor-network theory often models neural states using a scalar landscape (quasipotential) where stable states correspond to minima and transitions are governed by barrier heights. While useful, this equilibrium-like approximation is incomplete for neural systems that sustain metabolic energy consumption and directed recurrent interactions. Such systems often break detailed balance, generating circulating probability currents even when stationary densities appear equilibrium-like. The paper addresses three specific gaps:

1. Whether two systems with identical stationary densities and quasipotential barriers can exhibit different rare-transition geometries.
2. Whether the geometric asymmetry of forward and reverse transition paths can be quantitatively linked to steady-state entropy production.
3. Whether this path-level irreversibility propagates into thermodynamic precision bounds for neural current observables.

Methodology
The authors employ a hierarchy of models to isolate the effects of non-reciprocal drift:

• Solvable Normal Form: A two-dimensional stochastic differential equation (SDE) where the drift $F$ is decomposed into a gradient component ($-\nabla U$) and a solenoidal (rotational) component ($\chi R \nabla U$). The potential $U(x,y)$ defines a bistable landscape with two attractors and a saddle. The parameter $\chi$ controls non-reciprocity.
• Large-Deviation Theory: The Freidlin–Wentzell framework is used to compute the most probable transition paths (instantons) and action barriers. The geometric minimum-action method (gMAM) is utilized to find optimal paths in high-dimensional spaces by minimizing a geometric action functional.
• High-Dimensional Extensions: The normal form is embedded into $N$-dimensional ($N \in \{50, 100, 200, 500\}$) sparse recurrent networks with antisymmetric coupling matrices to test scalability.
• Numerical Validation:
  • Monte Carlo Simulations: Euler–Maruyama integration is used to estimate first-passage times and validate the large-deviation scaling of transition rates.
  • Thermodynamic Uncertainty Relations (TUR): The product of relative current variance and entropy production is computed to test if it satisfies the bound $Q \ge 1$.
  • Synthetic Readouts: Principal Component Analysis (PCA) is applied to synthetic population activity to illustrate how curved transition geometries might appear in neural recordings.

Key Contributions and Results

1. Decoupling of Landscape and Geometry:
   The study demonstrates that a system can possess a stationary density $\rho_{ss} \propto \exp(-U/\epsilon)$ and a quasipotential barrier identical to its equilibrium counterpart, yet sustain non-zero probability currents and irreversible switching paths. The rotational drift term preserves the fixed points and the scalar barrier height but bends the transition channels into directed loops.

2. Quantitative Triad of Irreversibility:
   The paper establishes a quantitative relationship between the solenoidal drift strength ($\chi$), path asymmetry, and dissipation:

   • Path Asymmetry ($A_{path}$): The geometric difference between forward (escape) and reverse (relaxation) paths scales linearly with non-reciprocity ($A_{path} \propto \chi$).
   • Entropy Production ($\langle \sigma \rangle$): The steady-state entropy production rate scales quadratically with non-reciprocity ($\langle \sigma \rangle \propto \chi^2$).
   • Scaling Relation: In the small-$\chi$ regime, the geometric path irreversibility serves as the square-root signature of dissipation: $A_{path}^2 \propto \epsilon \langle \sigma \rangle$.

3. Validation of Large-Deviation Theory:
   Monte Carlo simulations of first-passage times confirm that the transition rates obey the Freidlin–Wentzell scaling law $k \sim \exp(-\Delta V / \epsilon)$. The fitted barrier ($\Delta V \approx 0.2516$) matches the analytic value ($0.25$) within 1%, validating that the rare events are governed by the action barrier rather than local Gaussian fluctuations.

4. Thermodynamic Precision Bounds:
   The study verifies that time-integrated housekeeping currents in these non-reciprocal networks satisfy the Thermodynamic Uncertainty Relation (TUR). The uncertainty-cost product $Q = \frac{\text{Var}(J_T)}{\langle J_T \rangle^2} \langle \sigma \rangle T$ remains above the theoretical bound of 1. This confirms that improving the precision of neural currents requires increased dissipation, longer integration times, or architectural changes.

5. High-Dimensional Robustness:
   The path–current–dissipation structure persists in high-dimensional networks ($N=500$). The path-asymmetry index and entropy production scale consistently with system size and non-reciprocity strength, demonstrating that the variational pipeline converges in full state spaces without relying on low-dimensional projections.

Significance and Claims
The paper claims to identify a falsifiable nonequilibrium signature of attractor computation. It argues that scalar landscape descriptions are insufficient for a complete understanding of neural dynamics because they fail to predict:

• The specific geometry of rare transitions (escape vs. relaxation paths).
• The directionality of probability currents.
• The thermodynamic cost required to achieve specific levels of precision in state-space currents.

The authors emphasize that their results are derived from controlled, analytically tractable models and do not claim to measure total biological metabolic costs. Instead, the work provides a framework for analyzing resolved coarse-grained dynamics. The primary contribution is the demonstration that two systems with identical occupancy landscapes can differ fundamentally in their transition routes and thermodynamic efficiency. This suggests that for neural or neuromorphic systems, state stability, transition geometry, and dissipation are distinct but coupled properties that must be analyzed together to understand the physical implementation of computation. The work proposes that future empirical studies should test whether observed transition ensembles concentrate around predicted instanton tubes and whether non-equilibrium models outperform equilibrium surrogates in predicting direction-specific transition intermediates.