A Unified Dynamical Field Theory of Learning, Inference, and Emergence

TL;DR

This work proposes a unified dynamical field theory for learning, inference, and emergence by starting from a minimal stochastic equation and its MSRJD path-integral representation. It posits that inference corresponds to saddle-point trajectories on a learned dynamical landscape and that loop corrections around these trajectories generate emergent collective modes, organizing cognition through dynamical time scales. A central construct is the time-scale density of states (TDOS), which encodes how learning reshapes the spectrum of slow relaxation modes to support memory, stability, and context-dependent computation. By showing that classic models (Hopfield, RNNs, transformers) arise as limits, the framework unifies disparate neural architectures under a single dynamical principle and explains cognitive function as an emergent property of collective time-scale organization rather than microscopic precision. The TDOS-centric view provides experimentally testable predictions about temporal correlations and spectral content across biological and artificial systems, offering a principled route toward understanding cognition as emergent dynamical phenomena.

Abstract

Learning, inference, and emergence in biological and artificial systems are often studied within disparate theoretical frameworks, ranging from energy-based models to recurrent and attention-based architectures. Here we develop a unified dynamical field theory in which learning and inference are governed by a minimal stochastic dynamical equation admitting a Martin--Siggia--Rose--Janssen--de Dominicis formulation. Within this framework, inference corresponds to saddle-point trajectories of the associated action, while fluctuation-induced loop corrections render collective modes dynamically emergent and generate nontrivial dynamical time scales. A central result of this work is that cognitive function is controlled not by microscopic units or precise activity patterns, but by the collective organization of dynamical time scales. We introduce the \emph{time-scale density of states} (TDOS) as a compact diagnostic that characterizes the distribution of collective relaxation modes governing inference dynamics. Learning and homeostatic regulation are naturally interpreted as processes that reshape the TDOS, selectively generating slow collective modes that support stable inference, memory, and context-dependent computation despite stochasticity and structural irregularity. This framework unifies energy-based models, recurrent neural networks, transformer architectures, and biologically motivated homeostatic dynamics within a single physical description, and provides a principled route toward understanding cognition as an emergent dynamical phenomenon.

Figures (4)

• Figure 1: Conceptual overview of the unified dynamical field theory. (a) Collective neural states evolve on a learned state-space geometry, shaped by an effective potential $\Phi(x)$, a state-dependent metric $G(x)$, non-conservative reentrant flows $R(x)$, and stochastic fluctuations $\xi(t)$. (b) Inference corresponds to saddle-point trajectories of the MSRJD action, while fluctuations around these trajectories generate emergent collective modes through loop corrections. (c) Learning and homeostatic regulation reorganize the time-scale density of states (TDOS), selectively stabilizing slow collective modes that support robust inference, memory, and context-dependent cognition.
• Figure 2: Recurrent versus reentrant neural architectures. (a) Recurrent architectures update a latent hidden state sequentially in time. The recurrence operates through discrete or implicit temporal state updates, while the underlying representation space and geometry remain fixed. Temporal dependencies are captured by iterating the same state-transition rule across time steps. (b) Reentrant architectures implement internal self-referential feedback at the same representational level. Here, the system’s output is continuously fed back as a new input, dynamically reshaping the effective state-space geometry and collective dynamics. Reentry enables reflective computation, context-dependent reinterpretation, and dynamical modulation of collective modes beyond simple temporal recurrence.
• Figure 3: Effective homeostatic potential of the Fast-weights Homeostatic Reentry Network. (a) One-dimensional radial effective potential $\Phi(r)$ obtained from the exact radial reduction of the FHRN dynamics. For $\gamma\rho>1$, the potential develops a nontrivial minimum at a finite radius $r^*=\sqrt{1+(\gamma\rho-1)/\kappa}$, corresponding to a stable homeostatic shell. The origin $r=0$ is unstable, while radial fluctuations are strongly stabilized near $r=r^*$. (b) Two-dimensional visualization of the same effective potential $\Phi(y_1,y_2)=\Phi(y_1^2+y_2^2)$ in the full state space. The Mexican-hat–like geometry illustrates radial stabilization combined with a flat angular direction, providing a geometric substrate for collective computation. Reentrant dynamics generate non-conservative flows along the shell, while the potential enforces homeostatic control of activity magnitude.
• Figure 4: Time-scale density of states $\rho_\epsilon(\lambda)$ of the FHRN. The TDOS is computed by linearizing the deterministic dynamics $\dot{x} = -\nabla \Phi(x) + R(x)$ around a stable homeostatic fixed point $x^*$, forming the stability matrix $M = -\partial F / \partial x |_{x^*}$, and smoothing the real parts of its eigenvalues with a Gaussian kernel.