A neuronal least-action principle for real-time learning in cortical circuits

Reviewer #1 (Public Review):

The manuscript considers a hierarchical network of neurons, of the type that can be found in the sensory cortex, and assumes that they aim to constantly predict sensory inputs that may change in time. The paper describes the dynamics of neurons and rules of synaptic plasticity that minimize the integral of prediction errors over time.

The manuscript describes and analyses the model in great detail, and presents multiple and diverse simulations illustrating the model's functioning. However, the manuscript could be made more accessible and easier to read. The paper may help to understand the organization of cortical neurons, their properties, as well as the function of their particular components (such as apical dendrites).

Reviewer #2 (Public Review):

Neuroscientists often state that we have no theory of the brain. The example of theoretical physics is often cited, where numerous and quite complex phenomena are explained by a compact mathematical description. Lagrangian and Hamiltonian pictures provide such powerful 'single equation'. These frameworks are referred to as 'energy', an elegant way to turn numerous differential equations into a single compact relationship between observable quantities (state variables like position and speed) and scaling constants (like the gravity constant or the Planck constant). Such energy-pictures have been used in theoretical neuroscience since the 1980s.

The manuscript "neuronal least-action principle for real-time learning in cortical circuits" by Walter Senn and collaborators describes a theoretical framework to link predictive coding, error-based learning, and neuronal dynamics. The central concept is that an energy function combining self-supervised and supervised objectives is optimized by realistic neuronal dynamics and learning rules when considering the state of a neuron as a mixture of the current membrane potential and its rate of change. As compared with previous energy functions in theoretical neuroscience, this theory captures a more extensive range of observations while satisfying normative constraints. Particularly, no theory had to my knowledge related to adaptive dynamics widely observed in the brain (referred to as prospective coding in the text, but is sometimes referred to as adaptive coding or redundancy reduction) with the dynamics of learning rules.

The manuscript first exposes the theory of two previously published papers by the same group on somato-dendritic error with apical and basal dendrites. These dynamics are then related to an energy function, whose optimum recovers the dynamics. The rest of the manuscript illustrates how features of this model fit either normative or observational constraints. Learning follows a combination of self-supervised learning (learning to predict the next step) and supervised learning (learning to predict an external signal). The credit assignment problem is solved by an apical-compartment projecting a set of interneurons with learning rules whose role is to align many weight matrices to avoid having to do multiplexing. An extensive method section and supplementary material expand on mathematical proofs and makes more explicit the mathematical relationship between different frameworks.

Experts would say that much of the article agglomerates previous theoretical papers by the same authors that have been published recently either in archival servers or in conference proceedings. A number of adaptations to previous theoretical results were necessary, so the present article is not easily reduced to a compendium of previous pre-prints. However, the manuscript is by no means easy to read as there are several inconsistencies and it lacks a single thread. Also, there remains a few thorny assumptions (unobserved details of the learning rules or soma-dendrites interactions), but the theory is likely going to be regarded as an important step towards a comprehensive theory of the brain.