Amari-Type Neural Fields: Dynamics & Thermodynamics

• Amari-type neural fields are continuous mean-field models using nonlinear integro-differential equations to capture spatiotemporal neural activity.
• They integrate nonlocal synaptic interactions and stochastic noise to analyze equilibrium, pattern formation, and thermodynamic irreversibility.
• Spectral decomposition and Fourier analysis quantify energy dissipation, guiding model design for accurate representation of brain dynamics.

Amari-type neural fields are a class of continuous, spatially extended mean-field models describing the collective activity of neural tissue. Formulated as nonlinear integro-differential equations, these models capture the spatiotemporal evolution of neural activation in terms of local decay, nonlocal synaptic interactions, and, in stochastic generalizations, noise-driven fluctuations. The Amari framework is foundational in mathematical neuroscience, providing a tractable description of population-level brain activity and supporting rigorous analysis of pattern formation, control, non-equilibrium thermodynamics, and neural computation. The following sections present the principles, mathematical structures, dynamical features, thermodynamic implications, and control-theoretic aspects of Amari-type neural fields, as elucidated in current research including (Lucente et al., 18 Oct 2025).

1. Mathematical Formulation of Amari Neural Field Equations

The canonical Amari neural field equation models the coarse-grained activity field $u(\mathbf{x}, t)$ over a spatial domain $\Omega \subset \mathbb{R}^d$:

$$\frac{\partial u(\mathbf{x}, t)}{\partial t} = -\frac{1}{\tau}u(\mathbf{x}, t) + \frac{1}{\tau}\int_{\Omega} w(\mathbf{x}, \mathbf{y}) f[u(\mathbf{y}, t)]\, d\mathbf{y} + \xi(\mathbf{x}, t)$$

where:

• $\tau$ is the characteristic relaxation timescale,
• $w(\mathbf{x}, \mathbf{y})$ is the synaptic connectivity kernel, often spatially homogeneous or symmetric,
• $f[\cdot]$ is the nonlinear activation (firing rate) function, typically of sigmoidal form,
• $\xi(\mathbf{x}, t)$ is a Gaussian, possibly spatially correlated, stochastic noise term:

$$\langle \xi(\mathbf{x}, t) \xi(\mathbf{y}, t') \rangle = \gamma(\mathbf{x}, \mathbf{y})\delta(t-t')$$

with $\gamma(\mathbf{x}, \mathbf{y})$ positive-definite.

Linearization around a homogeneous steady state $u_0$ yields $\eta(\mathbf{x}, t) = u(\mathbf{x}, t) - u_0$ governed by

$$\partial_t \eta(\mathbf{x}, t) = -\frac{1}{\tau}\eta(\mathbf{x}, t) + \frac{f'[u_0]}{\tau} \int_{\Omega} w(\mathbf{x}, \mathbf{y}) \eta(\mathbf{y}, t)\, d\mathbf{y} + \xi(\mathbf{x}, t)$$

which is the basis for analyzing small fluctuations, spectral properties, and thermodynamic structure.

2. Stochastic Dynamics and Noise Structure

When stochasticity is included, the dynamics are described by linear (or nonlinear) stochastic integro-differential equations driven by spatially correlated noise. The form and properties of the noise kernel $\gamma(\mathbf{x}, \mathbf{y})$—whether delta-correlated (white), colored, or asymmetric—critically determine the system's thermodynamic behavior. Operator notation,

$$\dot{\eta}(\mathbf{x}, t) = \Lambda[\eta](\mathbf{x}, t) + \xi(\mathbf{x}, t)$$

where $\Lambda$ is a linear integral operator, streamlines the spectral analysis and the determination of equilibrium or non-equilibrium stationary states.

Equilibrium is achieved only if the deterministic linear operator $\Lambda$ and the noise covariance $\gamma$ satisfy a generalized detailed balance condition,

$$\Lambda^T \circ \Gamma^{-1} = \Gamma^{-1}\circ \Lambda$$

where $\Gamma$ is the operator representation of $\gamma$.

3. Thermodynamic Irreversibility and Entropy Production

A fundamental insight is that neural field dynamics are generically irreversible, remaining out of equilibrium except in special parameter regimes. Entropy production quantifies this irreversibility by measuring the rate at which the system's dynamics break time-reversal symmetry. The trajectory-level entropy production over $[0, t]$ is expressed via

$$\Sigma(t) = \log\left( \frac{\mathcal{P}[\Omega_0^t]}{\mathcal{P}[\overline{\Omega}_0^t]} \right)$$

where $\mathcal{P}[\Omega_0^t]$ and $\mathcal{P}[\overline{\Omega}_0^t]$ denote the probabilities of forward and backward paths, respectively. In the stationary regime, the mean entropy production rate is

$$\sigma = \operatorname{Tr}\left( [\Lambda^T \circ \Gamma^{-1} - \Gamma^{-1}\circ\Lambda] \circ \Lambda \circ C \right)$$

with $C$ the stationary covariance of $\eta$, solution to the associated Lyapunov equation.

At stationarity, there is an explicit relationship between this entropy production rate and the time variation of the system's Shannon entropy:

$$S_{\text{sys}}(t) = -\int \mathcal{D}\eta~ P_t[\eta] \log P_t[\eta]$$

$$\sigma = \dot{S}_{\text{tot}} = \dot{S}_{\text{res}}$$

This formalism grounds neural field models within nonequilibrium statistical mechanics and stochastic thermodynamics, establishing entropy production as a universal observable for irreversibility.

4. Spectral Decomposition and Spatial Mode Analysis

When $w(\mathbf{x}, \mathbf{y})$ and $\gamma(\mathbf{x}, \mathbf{y})$ are translationally invariant, Fourier analysis decomposes the system into independent spatial modes:

$$\frac{\partial}{\partial t}\eta_\mathbf{k}(t) = \lambda_{\mathbf{k}} \eta_\mathbf{k}(t) + \xi_\mathbf{k}(t)$$

with $$\langle \xi_{\mathbf{k}}(t) \xi_{\mathbf{k}'}^*(t') \rangle = \gamma_\mathbf{k} \delta_{\mathbf{k}, \mathbf{k}'} \delta(t-t')$$.

The entropy production per mode is:

$$\sigma_{\mathbf{k}} = -\frac{\Im^2[\lambda_{\mathbf{k}}]}{\Re[\lambda_{\mathbf{k}}]}$$

and the total entropy production is a sum over all modes:

$$\sigma = -\sum_{\mathbf{k}} \frac{\Im^2[\lambda_{\mathbf{k}}]}{\Re[\lambda_{\mathbf{k}}]}$$

This quantifies dissipation and irreversibility across spatial scales, connecting microscopic symmetry breaking to macroscopic information dissipation and energy flow.

5. Equilibrium versus Non-equilibrium Regimes

Amari-type neural fields only admit equilibrium (time-reversible) stationary states when connectivity and noise satisfy the symmetry constraint. Standard choices—symmetric kernels and spatially white noise—guarantee equilibrium. Spatial correlations in the noise or asymmetry in the weights result in persistent entropy production and nonequilibrium steady states. Non-equilibrium behavior is thus intrinsic to the mesoscopic regime of neural population dynamics, except in finely tuned situations.

The following table summarizes the key equations involved in Amari-type neural fields with stochasticity and their thermodynamic content:

Quantity	Equation
Amari Stochastic Equation	$$\partial_t u = -\tau^{-1}u + \tau^{-1}\int w f[u] d\mathbf{y} + \xi$$
Linearized SDE	$\dot{\eta} = \Lambda[\eta] + \xi$
Noise Correlation	$$\langle \xi(\mathbf{x}, t)\xi(\mathbf{y}, t') \rangle = \gamma(\mathbf{x}, \mathbf{y}) \delta(t-t')$$
Entropy Production Rate	$$\sigma = \operatorname{Tr} \left( [\Lambda^T \Gamma^{-1} - \Gamma^{-1} \Lambda] \Lambda C \right)$$
Equilibrium Criterion	$\Lambda^T \Gamma^{-1} = \Gamma^{-1} \Lambda$
Fourier-Space Entropy Prod.	$$\sigma = -\sum_{\mathbf{k}} \frac{\Im^2[\lambda_{\mathbf{k}}]}{\Re[\lambda_{\mathbf{k}}]}$$

6. Implications for Analysis and Model Design

The explicit calculation of entropy production and its dependence on both noise and network structure delivers crucial constraints and interpretation for the construction of spatially extended neural field models. Only specific choices of stochasticity yield equilibrium stationary states; generic noise/connectivity combinations place the system out of equilibrium. This is essential for the modeling of real neural tissue, which exhibits persistent irreversibility, energy dissipation, and nontrivial information flow. The magnitude and spectral distribution of entropy production across spatial modes can inform on which scales and structural features drive non-equilibrium behavior.

7. Connections to Information Processing and Brain Dynamics

Entropy production in Amari-type neural fields provides an intrinsic measure of irreversibility and dissipative activity, independent of particular observed variables. Its spatial mode decomposition identifies which network motifs and connectivity structures contribute to nonequilibrium phenomena, directly linking macroscopic neural population dynamics to theoretical principles of information processing and thermodynamics. The connection established between stochastic neural field modeling and contemporary statistical mechanics clarifies the correct mathematical treatment of stochasticity and grounds neural field theory within a rigorous analytical paradigm (Lucente et al., 18 Oct 2025).

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Amari-type neural fields are thus a theoretically robust, analytically tractable framework for studying large-scale brain dynamics, providing precise conditions for equilibrium and non-equilibrium behavior, spectral and thermodynamic quantification of irreversibility, and a foundation for the rigorous modeling of stochastic neural systems.