[Paper Review] A Variational Latent Equilibrium for Learning in Neuronal Circuits

Simon D. Brandt, Paul Haider+3 more|arXiv (Cornell University)|Mar 10, 2026

Neural dynamics and brain function0 citations

TL;DR

Introduces a variational latent equilibrium framework to approximate backpropagation through time in a biologically plausible, local manner and derives error propagation and learning rules from an energy-based formulation.

ABSTRACT

Brains remain unrivaled in their ability to recognize and generate complex spatiotemporal patterns. While AI is able to reproduce some of these capabilities, deep learning algorithms remain largely at odds with our current understanding of brain circuitry and dynamics. This is prominently the case for backpropagation through time (BPTT), the go-to algorithm for learning complex temporal dependencies. In this work we propose a general formalism to approximate BPTT in a controlled, biologically plausible manner. Our approach builds on, unifies and extends several previous approaches to local, time-continuous, phase-free spatiotemporal credit assignment based on principles of energy conservation and extremal action. Our starting point is a prospective energy function of neuronal states, from which we calculate real-time error dynamics for time-continuous neuronal networks. In the general case, this provides a simple and straightforward derivation of the adjoint method result for neuronal networks, the time-continuous equivalent to BPTT. With a few modifications, we can turn this into a fully local (in space and time) set of equations for neuron and synapse dynamics. Our theory provides a rigorous framework for spatiotemporal deep learning in the brain, while simultaneously suggesting a blueprint for physical circuits capable of carrying out these computations. These results reframe and extend the recently proposed Generalized Latent Equilibrium (GLE) model.

Motivation & Objective

• Motivate learning in neuronal circuits under biological and physical constraints through a constrained optimization lens.
• Develop a first-principles variational framework that unifies and extends LE and GLE within a local, time-continuous setting.
• Derive neuron- and synapse-specific dynamics and local plasticity rules from an energy functional.
• Show how the framework recovers adjoint method results and provides a biologically plausible path to spatiotemporal credit assignment.

Proposed method

• Postulate a network energy E(t) = 1/2 sum_i e_i^2(t) + beta C(t) and minimize integrated energy to align with the target cost C.
• Define local mismatch errors e_i from neuron states and synaptic inputs as e_i = u_i^m(bar) - sum_j W_ij phi_j(u^r_j_bar).
• Derive gradient-based learning rules dot{W}_{ij} = e_i r_j and dot{theta}_i ~ -∂E/∂θ_i indicating local plasticity.
• Show that the Euler-Lagrange equations yield error dynamics equivalent to the adjoint method for the considered system.
• Discuss prospects for implementing these dynamics in fully local (in space and time) brain-like circuits.
• Relate the framework to Generalized Latent Equilibrium (GLE) and neuronal variants such as LE and NLA.

Experimental results

Research questions

• RQ1 How can constrained optimization be solved in a biologically plausible, time-continuous manner for spatiotemporal tasks?
• RQ2 Can a first-principles energy-based formulation reproduce adjoint-based learning results (AM) while remaining local in space and time?
• RQ3 What are the explicit error propagation and plasticity rules that implement spatiotemporal credit assignment in neuronal circuits?
• RQ4 How does the variational latent equilibrium framework relate to and extend GLE, LE, and related models?
• RQ5 What approximations are needed for biological plausibility, and how can distortions be corrected within this framework?

Key findings

• A variational energy framework yields error dynamics equivalent to the adjoint method for spatiotemporal learning.
• Local plasticity rules dot{W}_{ij} = e_i r_j are derived from local neuron energies, enabling biologically plausible learning.
• The derivation recovers AM results while simplifying and clarifying the underlying principles.
• The framework unifies and extends LE and GLE, providing a path to fully local, time-continuous spatiotemporal credit assignment.
• Potential applications demonstrated in learning to reproduce complex temporal behavior, with a blueprint for physical neural circuits.