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Neural population geometry: An approach for understanding biological and artificial neural networks
SueYeon Chung, L. F. Abbott
TL;DR
Biological and artificial neural networks both require explanations of how high-dimensional representations encode and process information for complex tasks. This review uses neural population geometry to synthesize geometric analyses across perception, classification, cognition, topology, motor control, and inference, finding that these descriptions apply across modalities, brain regions, architectures, and timescales.
Problem
Mixed selectivity, complex-task variability, and the high dimensionality of neural and artificial networks create a need for population-level tools that reveal representational mechanisms.
Method
The review synthesizes geometric approaches linking neural population representations to task implementation across biological and artificial networks.
Results
Geometric analyses provide insight into perception, classification capacity, cognitive abstraction, cognitive-map topology, motor control, and Bayesian inference.
Takeaways & Limitations
Neural population geometry offers a population-level mechanistic description applicable across sensory modalities, brain regions, network architectures, and timescales.
Takeaways & Limitations
Formal connections between representational geometry and encoded task information remain to be developed for a larger array of tasks.
Abstract
from arXiv · showhide
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, populations and behavior.
1 Center for Theoretical Neuroscience, Columbia University, New York City, United States
Neural population geometry analyzes how neural representations are configured to understand task implementation in biological and artificial networks. The review highlights applications spanning perception, classification, cognition, cognitive maps, motor systems, and future theory.
- Neural population geometry analyzes geometric properties of neural populations as a population-level link between neural responses and task implementation.
- The reviewed applications include perceptual untangling, manifold classification capacity, cognitive abstraction, cognitive-map topology, motor untangling, and dynamical cognition.
- Future work includes developing geometric measures, connecting representational geometry to neuronal biophysics, and extending theory to more tasks.
Introduction
Neural circuits and artificial networks encode information in high-dimensional activity patterns that often form lower-dimensional geometric structures. Neural population geometry provides tools for studying these representations across complex tasks, while the review surveys applications from perception and classification to cognition, topology, and motor control.
- Neural activity forms manifold-like representations in a high-dimensional neural state space, with coordinates corresponding to individual neurons or network units.Examples include lines, surfaces, trajectories, subspaces, and point clouds.
- Mixed selectivity and task variability make simplistic tuning-based analyses inadequate for large neural populations involved in complex tasks.Population-level geometric analysis is presented as an approach suited to these challenges.
- Artificial neural networks provide a testbed for developing population-level tools that can reveal mechanisms in high-dimensional neural and artificial networks.The shared interpretability challenge motivates geometric approaches across both systems.
- The review surveys geometric analyses of categorization, perceptual untangling, cognitive abstraction, topological maps, motor control, and Bayesian inference.
- A neural manifold broadly denotes a geometric structure or lower-dimensional subspace in population activity, although noise and sparse sampling can make real data non-manifolds mathematically.
- Neural population geometry refers to the configuration of neural manifolds within ambient neural state space.
The Geometry of Perception and Decision Making
Neural population geometry explains perception and decision-making by characterizing how representations become separable, straightened, abstract, and topologically structured. These geometric analyses extend from points and trajectories to stimulus-induced manifolds and reveal task-related structure in biological and artificial networks.
- Perceptual untangling: Linear separability provides a geometric account of invariant recognition by transforming object representations into linearly separable forms.The untangling hypothesis proposes that visual processing separates object categories in neural state space.
- Temporal straightening: Natural video sequences, but not artificial video sequences, produce straighter neural response trajectories in neural network models and human perceptual space.Temporal straightening is assessed by measuring trajectory curvature.
- Abstraction: Disentangled representations support context switches through rotations or translations of a dividing surface while retaining information about other variables.Prefrontal and hippocampal recordings, together with task-trained networks, quantify this geometry using a parallelism score.
- Extensions from points to manifolds: Stimulus variability turns object responses into manifolds, making invariant recognition a problem of classifying between object manifolds rather than isolated points.Object manifolds preserve identity across variations such as orientation, pose, position, and scale.
- Classification capacity: Manifold capacity measures both linear separability and the maximum number of object classes that can be linearly read out from a representation.Capacity is higher for smaller manifold dimensions and radii, while dimensionality and radius can trade off to produce similar capacities.
- Intrinsic geometry: Dimensionality-reduction methods characterize neural manifolds, but standard nonlinear methods can fail when the underlying topology is complex.SPUD uses persistent homology to discover non-trivial ring structure, while MIND defines distances through transition probabilities to infer task-relevant intrinsic dimensions.
The Geometry of Movement and Cognition
Geometric analyses reveal how neural population trajectories support movement generation, sequencing, and cognition. Across motor and cognitive tasks, trajectory shape and curvature expose computational structure.
- Dynamic untangling of internally generated activities: A tangling index distinguishes internally generated motor trajectories by detecting when recorded population trajectories cross or nearly cross.Tangling is lower in primary motor cortex than in primary sensory cortex or muscle activity during cycling, supporting a role in movement generation.
- Dynamic untangling of internally generated activities: Helical trajectories in supplementary motor area represent the sequence of cycling movements, a pattern also reproduced by recurrent networks tracking cycle number.Motor cortex activity repeated across cycles, whereas SMA activity followed a helical trajectory.
- Population dynamics as cognition: Recurrent-network dynamics link fixed points to memory, line attractors to integration, and limit cycles to neuronal oscillation patterns.These motifs form part of a broader program connecting network dynamics to cognition.
- Population dynamics as cognition: In macaque frontal cortex, experience warps population representations during time reproduction, while recurrent-network curvature supports an underlying Bayesian computation.The warped map incorporates prior statistics into the mapping from sensory representation to motor output.
Conclusion
The review presents neural population geometry as a richer descriptor of computation and a possible bridge across biological and artificial systems. It also identifies unresolved links between geometry, task information, and neuronal biophysics.
- Conclusion: The review identifies outstanding opportunities and challenges for future research using neural population geometry.These opportunities are framed at the intersection of neuroscience and artificial intelligence.
- Conclusion: Neural population geometry can distinguish representations with equal task capacity but different geometric configurations, providing a more accurate population-level descriptor than simple task probes.Dimensionality captures task information and representational redundancy, while invariant object classification capacity also depends on manifold radius.
- Conclusion: The growing range of tasks and brain regions with structured geometry requires theories connecting representational geometry formally to encoded task information.This is identified as a future theoretical challenge.
- Conclusion: Future work should relate population geometry to neuronal biophysics, including how cell types, connectivity, activation profiles, and sparsity constrain task-encoding geometry.Examples link trajectory curvature to encoded priors and suggest heterogeneity can support beneficial geometric changes.
- Conclusion: Because geometric descriptions generalize across task modalities, brain regions, and timescales, the approach may unify structure and function across biological and artificial neural networks.The proposed scope spans brain regions and computational levels.
Highlighted papers
Highlighted studies use geometry and topology to connect neural population structure with classification, abstraction, Bayesian behavior, motor robustness, and cognitive maps. Together, they show how specialized geometric measures expose task-relevant organization.
- Classification and geometry: Manifold capacity formalizes the number of linearly separable category manifolds per neuron as a function of manifold geometry.Relevant geometric properties include manifold dimension and manifold radius.
- Bayesian computation: Prior-reflecting behavior is associated with neural-manifold geometry, including trajectory curvature warped by context-dependent prior statistics.The approach targets population-level neural mechanisms underlying Bayesian behavior.
- Motor systems: Motor-cortex population trajectories show lower tangling than EMG muscle activity, suggesting reduced dependence on input dynamics and greater noise robustness.Tangling is used as a geometric measure for dynamic trajectories.
- Cognitive maps: SPUD uses persistent homology to identify intrinsic dimensionality and topology, decoding mouse head direction from a one-dimensional ring structure.The ring persisted during sleep despite the lack of sensory input.
- Cognitive maps: MIND defines distances between nearby behavioral states using transition probabilities to identify dimensions relevant for topological task maps.It was applied to foraging and sound-manipulation tasks involving transitions between behavioral states.
- Abstraction: Parallelism score measures whether coding directions align across training conditions and relates this geometry to cross-condition linear-readout generalization.Such representations were observed in dorsolateral prefrontal cortex, anterior cingulate cortex, and hippocampus.