Neural population geometry: An approach for understanding biological and artificial neural networks (2104.07059v3)

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, populations and behavior.

• The paper demonstrates that manifold structures in neural populations enable linear separability for interpreting complex perceptual data.
• It quantifies geometric transformations that encode abstract task contexts using metrics like the parallelism score.
• The study employs dimensionality reduction to uncover low-dimensional subspaces in high-dimensional state spaces, informing models for invariant recognition and movement.

Analyzing Neural Population Geometry in Biological and Artificial Neural Networks

In the paper "Neural population geometry: An approach for understanding biological and artificial neural networks," SueYeon Chung and L. F. Abbott explore the concept of neural population geometry and its application to both biological and artificial neural networks (ANNs). This research investigates the geometric properties of neural populations to understand how information is processed through high-dimensional representations. The paper explores several aspects of neural population geometry, offering insights into the function and structure of neural networks.

Key Concepts and Findings

The paper defines neural population geometry as the geometric configuration of manifold-like structures formed by the activity patterns of neurons. These structures emerge from variability in response to stimuli and internal dynamics, manifesting in high-dimensional state spaces. The paper reviews the application of geometric analysis across various domains, including perception, cognition, and motor control.

1. Perceptual Untangling: The authors explore the hypothesis that the ventral visual stream processes information to transform complex visual representations into a linearly separable form. This process is integral to simplifying tasks such as categorization and recognition, where discrimination between different stimuli can be optimized when neural activities are separated by linear hyperplanes.
2. Geometry of Abstraction: They discuss the ability of neural representations to encode abstract information, demonstrating how task context switching can be represented through geometric transformations, such as translation and rotation, within the neural state space. Measures like the parallelism score quantify disentangled representations, suggesting abstraction does not lead to the loss of crucial information.
3. Extensions from Points to Manifolds: The researchers describe how variability in neural activities results in point-cloud manifolds rather than single points in state space, reinforcing the need to model these responses as manifold structures. The theory of manifold capacity, which encompasses traits like dimensionality and radius, is applied to understanding linear separability and storage capacity in tasks involving invariant object discrimination.
4. Intrinsic Geometry and Movement: Dimensionality reduction techniques, such as PCA and manifold inference methods, are employed to reveal the lower-dimensional subspaces where neural activities reside. These techniques help uncover complex topological structures within neural populations, applicable to both sensory and cognitive domains.
5. Dynamic Untangling: In motor systems, the paper exemplifies dynamic untangling by highlighting how state-space trajectories should avoid crossings, ensuring consistent change rates in dynamic variables. This phenomenon underlines the role of motor system areas in movement generation, contrasting them with those regions purely reflecting motor effects.

Implications and Future Directions

The paper posits that neural population geometry offers a unified framework for understanding neural network functions across modalities and time scales. The geometry provides a mechanistic descriptor that can bridge the gap between neuron activity, population dynamics, and behavioral outcomes.

The authors suggest that future research should focus on enhancing geometric measures as population-level hypotheses and exploring connections between representational geometry and neuron biophysical properties. Understanding neural population geometry can lead to more accurate modeling of tasks across a broader range of neural activities, potentially improving interpretation and prediction models in computational neuroscience.

The approach encourages the integration of population geometry into studies of neural circuit functionality, offering opportunities to refine theories regarding neural representation of tasks. Continued exploration of population-level metrics, such as manifold capacity and dimensionality, will be crucial in bridging biological and artificial network analyses, facilitating advancements in both neuroscience and AI applications.