Thermodynamic Optimization of Sensory Adaptation via Game-Theoretic Path Integrals

TL;DR

This work presents a field-theoretic framework in which sensory adaptation emerges from a variational free-energy principle, formulated as a stochastic differential game between an organism and its environment, and shows that the resulting adaptive dynamics are mathematically equivalent to a class of model reference adaptive control schemes.

Abstract

Biological sensory systems, from \textit{E.~coli} chemotaxis to sensory neurons in \textit{C.~elegans}, achieve reliable adaptation over wide dynamic ranges despite operating in strongly noisy and overdamped regimes. Here, we present a field-theoretic framework in which sensory adaptation emerges from a variational free-energy principle, formulated as a stochastic differential game between an organism and its environment. Using an Onsager--Machlup path-integral formalism, we show that the resulting adaptive dynamics are mathematically equivalent to a class of model reference adaptive control schemes and can be interpreted as a dynamic renormalization of the system's Green's function. Within this framework, the phasic overshoot commonly observed in sensory responses arises naturally from an effective inertia ($m^* \approx τγ$) generated by memory-dissipation coupling, rather than from biochemical fine-tuning. Quantitative fits to experimental data across species yield $R^2 > 0.88$, and indicate that adaptive sensory processing operates within a narrow thermodynamically optimal regime bounded by signal-to-noise and stability constraints.

Figures (4)

• Figure 1: Dynamics of Adaptive Attention. (a) The neural state $x(t)$ (solid blue) tracks the step stimulus but exhibits a pronounced overshoot compared to the ideal MRAC reference $x_{\text{ref}}(t)$ (dashed red). This deviation indicates the presence of thermodynamic inertia. (b) Thermodynamic control parameters: The stiffness $k(t)$ (orange) drops transiently to facilitate rapid state transitions (Attention), while the memory $\mu(t)$ (green dashed) acts as a lagging integrator, gradually adapting to the new input level. (c) The rapid quenching of prediction error drives the restoration of stability and the recovery of stiffness.
• Figure 2: Thermodynamic breathing mode of the adaptive kernel. Heatmap of the impulse response kernel $G(t, t-\tau_{\text{lag}})$ plotted on a logarithmic scale. The vertical axis represents the causal time lag $\tau_{\text{lag}}$. (Red Zone) Attention Phase: Transient expansion of the causal horizon (increasing $\tau_{\text{lag}}$). Biologically, this corresponds to a "memory stretch" where the system lowers stiffness to integrate past signals over extended timescales. (Blue Zone) Habituation Phase: Contraction of the horizon (decreasing $\tau_{\text{lag}}$). This represents a "memory reset," restoring stiffness to prioritize recent data and filter thermal fluctuations.
• Figure 3: Thermodynamic phase diagram of sensory adaptation. The sensory susceptibility $\chi$ is plotted as a function of the stiffness parameter $k$, which acts as an effective thermodynamic control parameter. The hyperbolic constraint $\chi \propto 1/k$ reflects the fundamental trade-off between sensitivity and stability. The red-shaded region denotes the thermal noise limit ($\text{SNR} < 1$), while the blue-shaded region corresponds to a stability limit associated with oscillatory instability. To ensure a consistent comparison across different biological scales, we normalize the dynamical variables by the intrinsic friction $\gamma$. We define the dimensionless stiffness $\tilde{k} \equiv k\gamma$ and the dimensionless susceptibility $\tilde{\chi} \equiv \chi/\gamma$. In this renormalized representation, the fundamental constraint reduces to the universal form $\tilde{\chi} \tilde{k} \approx 1$. The operating points for both E. coli and C. elegans collapse onto the same thermodynamic trajectory, proving that the trade-off between sensitivity and stability is governed by a scale-invariant principle.
• Figure 4: Evidence of Thermodynamic Inertia across Biological Scales. The system simulates the response to a smoothed step input $I(t)$ (green dotted). (a) E. coli exhibits rapid adaptation with transient ringing. The short adaptation time ($\tau \approx 3.4$ s) results in a small effective mass ($m^* \approx 0.75$), allowing quick settling. (b) C. elegans shows a prolonged phasic overshoot. The longer memory timescale ($\tau \approx 11.8$ s) generates a large effective inertia ($m^* \approx 1.30$), facilitating extended temporal integration. Solid red lines represent the active adaptive model; black dashed lines represent passive relaxation without adaptation. The predicted overshoot reproduces the qualitative features of experimentally observed responses Berg1975Chalasani2007, with quantitative parameter fits provided in the Supplemental Material (Fig. S1, Table S1).