A Variational Latent Equilibrium for Learning in Neuronal Circuits

A recurring pattern in neural computation is the reintroduction of dynamical and biological structure into models originally simplified for scalable optimization. Early feedforward networks reduced biological neurons to threshold or rate-like summation units, an abstraction compatible with global-gradient training at scale. Since then, forward computation has diversified: modern architectures carry recurrent state, retrieve from long contexts through attention and associative memory, compress histories through structured state-space dynamics, evolve in continuous time, settle to implicit equilibria, and communicate through sparse spikes. Training has diversified less. Scalable learning remains concentrated around backpropagation, backpropagation through time, adjoint methods, implicit differentiation, and surrogate-gradient variants. We call this asymmetry the forward-backward disconnect and develop a taxonomy spanning neural model families along three coupled axes: state-dynamics structure, credit-assignment mechanism, and biological grounding. Forward and learning grounding are treated separately, and the unit of analysis is the architecture-learning configuration rather than the architecture name alone. Across static, recurrent, attention-based, state-space, continuous-time, implicit, spiking, biologically plausible, and neuromorphic families, forward dynamics have diversified while the highest demonstrated scales remain concentrated in global or closely gradient-derived error-propagation mechanisms. Closing this disconnect requires better alignment among state dynamics, credit assignment, and computational substrate.

The brain uses discrete spikes for dynamic computation, yet, how neural microcircuits (NMCs) solve temporal credit assignment using local spike timing remains a fundamental open question. Dominant spiking neural network (SNN) approaches circumvent this by approximating backpropagation through surrogate gradients, decoupling learning from biological spike timing. Here, we reformulate temporal credit assignment as a state separation problem: extracting task-required components induced by historical perturbations directly from the current neural state. This enables an online feedback learning framework for NMCs through a gradient tunneling (GT) algorithm and the lead-lag expansion technique that derives credit assignment from local synaptic spike timing, while remaining compatible with ANN-SNN hybrid architectures. Experimentally, GT-trained NMCs excel at long-timescale evidence integration and noise-robust memory retention, and perform comparably to leading SNN online learning methods on real-world benchmarks with far fewer parameters. The proposed framework addresses the two-decade-old NMC feedback learning problem and suggests a computationally plausible explanation for the brain's learning mechanisms.

Biological and neuromorphic recurrent neural networks (RNNs) are subject to spatial and temporal locality constraints on the information that can plausibly be used during learning. A common strategy to satisfy these constraints is to modify gradient descent by neglecting non-local terms to varying degrees, as in random feedback local online (RFLO) learning and truncated backpropagation through time (tBPTT). However, the learning dynamics of these algorithms, and how they compare with BPTT, remain poorly understood. We apply dynamical systems theory to data-aligned linear RNNs -- whose dynamics can be separated into orthogonal modes -- to compare stationary solutions, stability properties, and convergence rates, finding qualitatively distinct behaviour for RFLO versus BPTT and one-step tBPTT. We further observe that the solutions learned by RFLO are restricted to low-rank perturbations of initial parameters, a result which holds beyond the data-aligned setting. Our work provides analytical insight into how locality constraints shape learning dynamics, with implications for neuroscientific models of learning and alternative optimization approaches for RNNs.