A Variational Latent Equilibrium for Learning in Cortex

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.

💡 Research Summary

The paper introduces a novel theoretical framework called Variational Latent Equilibrium (VLE) that aims to reconcile the powerful learning capabilities of the brain with the constraints of biological plausibility. Traditional deep learning for temporal tasks relies on back‑propagation through time (BPTT), which requires non‑local error signals and weight transport that are difficult to implement in real neural tissue. The authors start from an energy‑based formulation, defining a total energy E(t) = ½∑ₙ eₙ²(t) + β C(t), where eₙ denotes a local neuron‑specific error, C is the global cost, and β is a small nudging term applied only at output neurons. By applying the calculus of variations to the integral of this energy, they obtain Euler‑Lagrange equations that describe the continuous‑time dynamics of each neuron.

These dynamics naturally decompose into three compartments: a somatic voltage uᵢ, an input compartment that integrates feed‑forward signals, and an error compartment that stores the locally computed error eᵢ. The resulting differential equation τₘ · ẋᵢ = –xᵢ + Σⱼ Wᵢⱼ φⱼ( ûʳⱼ ) + eᵢ captures both the low‑pass filtering performed by membrane integration and a prospective, forward‑looking response to changing inputs.

To handle temporal processing, the authors define four elementary operators: forward discount, forward look‑ahead, backward low‑pass, and backward look‑ahead. These operators correspond to biologically observed phenomena—membrane low‑pass filtering and neurons’ ability to anticipate future inputs. By substituting the forward discount operator with its forward look‑ahead approximation and the backward low‑pass with a causal low‑pass filter, the error signals become temporally local, eliminating the causality violation inherent in the standard adjoint method (AM).

Spatial locality, or the “weight‑transport problem,” is addressed by introducing a separate set of backward weights Bᵢⱼ. Rather than assuming Bᵢⱼ = Wⱼᵢ, the framework learns Bᵢⱼ through a local gradient descent on an auxiliary energy E_B = ‖Wⱼᵢ · ĥεᵣⱼ – Bᵢⱼ · ĥεₘⱼ‖². This learning rule, ˙Bᵢⱼ ∝ –∂E_B/∂Bᵢⱼ, ensures that the backward pathway dynamically aligns with the forward pathway, achieving the same error propagation as AM while remaining fully local in both space and time. The approach subsumes earlier ideas such as Feedback Alignment (FA) and Phaseless Alignment Learning (PAL) but extends them to deep, multilayer networks where simple random backward weights are insufficient.

The authors demonstrate the theory on two experimental setups. First, a minimal two‑neuron “student‑teacher” chain shows that when backward weights are either learned or set to the transpose of forward weights, the loss rapidly declines and the output matches the target signal; with fixed random backward weights, learning fails. Second, multilayer recurrent networks trained with VLE achieve loss curves nearly identical to those obtained with exact BPTT, while requiring only local information at each synapse and neuron.

Special cases reveal connections to existing models. When the membrane and prospective time constants are equal (τₘ = τᵣ), the forward look‑ahead and backward low‑pass operators cancel, reducing VLE to the previously proposed Latent Equilibrium (LE) model, which already respects causality because errors depend only on the instantaneous state. In the general case, VLE can be viewed as a principled extension of the Generalized Latent Equilibrium (GLE) model, adding a biologically plausible mechanism for learning backward weights.

Overall, the paper provides a rigorous, physics‑inspired derivation of a learning rule that is mathematically equivalent to the continuous‑time adjoint method yet respects the spatial and temporal locality required by real cortical circuits. By framing learning as an energy minimization problem and by decomposing neuronal dynamics into three interacting compartments, VLE offers a concrete blueprint for both neuroscientific investigations (e.g., targeted electrophysiology or optogenetics) and the design of low‑power neuromorphic hardware that can implement deep temporal learning without the prohibitive memory and computation costs of BPTT. Future work is suggested to extend VLE to spiking neurons, integrate spike‑timing‑dependent plasticity, and scale the framework to large‑scale brain models.