Neural population geometry: An approach for understanding biological and artificial neural networks

Abstract

Advances in experimental neuroscience have transformed our ability to explore the structure and function of neural circuits. At the same time, advances in machine learning have unleashed the remarkable computational power of artificial neural networks (ANNs). While these two fields have different tools and applications, they present a similar challenge: namely, understanding how information is embedded and processed through high-dimensional representations to solve complex tasks. One approach to addressing this challenge is to utilize mathematical and computational tools to analyze the geometry of these high-dimensional representations, i.e., neural population geometry. We review examples of geometrical approaches providing insight into the function of biological and artificial neural networks: representation untangling in perception, a geometric theory of classification capacity, disentanglement, and abstraction in cognitive systems, topological representations underlying cognitive maps, dynamic untangling in motor systems, and a dynamical approach to cognition. Together, these findings illustrate an exciting trend at the intersection of machine learning, neuroscience, and geometry, in which neural population geometry provides a useful population-level mechanistic descriptor underlying task implementation. Importantly, geometric descriptions are applicable across sensory modalities, brain regions, network architectures, and timescales. Thus, neural population geometry has the potential to unify our understanding of structure and function in biological and artificial neural networks, bridging the gap between single neurons, population activities, and behavior.

Figure 1

(a) Representation straightening for invariant object recognition (b) Temporal straightening for temporal natural video sequences. (c) Geometry of Abstraction. Representations encoding abstraction (i.e., cross-conditional generalization) show geometry where coding directions can be rotated or translated between conditions, known as parallelism (Right). (d) Neural manifolds arise as a result of stimulus variability. Population responses to two object classes (dog vs. cat) in the presence of the stimulus variability (orientation) give rise to two object manifolds. Invariant object recognition becomes the problem of classifying between two object manifolds. Axes represent the firing rates of neurons. (e) Manifold capacity is high if object manifolds are well separated and low when object manifolds are entangled in neural state space. Part (a) adapted from Ref. [7]. Part (b) adapted from Ref. [30]. Part (c) adapted from Ref. [31]. Part (e) adapted from Ref. [32].

Figure 2

(a–b) Manifold discovery methods. (a) Spine Parameterization for Unsupervised Decoding (SPUD). (b) Manifold Inference for Neural Dynamics (MIND) (c–d) Population dynamics as cognition. (c) (Left) Temporal trajectories during macaque cycling task in M1 and (Right) SMA. (d) Dorsomedial Frontal Cortex (DMFC) response profiles during Bayesian computation. Part (a) adapted from Ref. [46]. Part (b) adapted from Ref. [10]. Part (c) adapted from Ref. [51]. Part (d) adapted from Ref. [8].