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Interpreting neural computations by examining intrinsic and embedding dimensionality of neural activity
Mehrdad Jazayeri, Srdjan Ostojic
TL;DR
Rapidly expanding neural recordings require clearer ways to interpret population dimensionality, but studies use differing definitions. This review distinguishes intrinsic and embedding dimensionality, relates them to encoded latent variables and information processing, and proposes network models for testing computational hypotheses.
Problem
Neural recordings contain redundant population activity, while dimensionality studies lack unified definitions and terminology for relating that activity to task-relevant variables.
Method
The review distinguishes intrinsic and embedding dimensionality, synthesizes experimental studies, and discusses network models for testing computational and normative hypotheses.
Results
Intrinsic dimensionality reflects latent variables encoded in collective activity, whereas embedding dimensionality reflects how neural circuits process and communicate that information.
Takeaways & Limitations
Contrasting intrinsic and embedding dimensionality across brain areas and behavioral settings can generate hypotheses about neural computations and experimentally testable predictions.
Takeaways & Limitations
Accurately estimating intrinsic and embedding dimensions from noisy neural recordings remains an active research problem.
Abstract
from arXiv · showhide
The ongoing exponential rise in recording capacity calls for new approaches for analysing and interpreting neural data. Effective dimensionality has emerged as an important property of neural activity across populations of neurons, yet different studies rely on different definitions and interpretations of this quantity. Here we focus on intrinsic and embedding dimensionality, and discuss how they might reveal computational principles from data. Reviewing recent works, we propose that the intrinsic dimensionality reflects information about the latent variables encoded in collective activity, while embedding dimensionality reveals the manner in which this information is processed. We conclude by highlighting the role of network models as an ideal substrate for testing more specifically various hypotheses on the computational principles reflected through intrinsic and embedding dimensionality.
Highlights
The review presents intrinsic dimensionality as reflecting the information encoded in collective neural activity, embedding dimensionality as describing how circuits process that information, and network models as a testing ground for these roles.
- Intrinsic dimensionality reflects the nature of information encoded in collective neural activity.
- Embedding dimensionality describes how neural circuits process information.
- Network models provide a testing ground for the computational roles of intrinsic and embedding dimensionality.
Addresses
The authors are affiliated with institutions in the United States and France, including MIT, INSERM U960, and École Normale Supérieure–PSL Research University.
- One affiliation is the McGovern Institute for Brain Research and Department of Brain & Cognitive Sciences at MIT.
- The MIT affiliation is located in Cambridge, Massachusetts 02139, USA.
- Another affiliation is Laboratoire de Neurosciences Cognitives, INSERM U960, at École Normale Supérieure–PSL Research University in Paris.
Introduction
The review addresses how redundant population activity represents task-relevant variables amid rapidly expanding neural recordings, distinguishing intrinsic and embedding dimensionality as guides to neural computations.
- Recording capacity is rising exponentially, while recorded populations often contain far more neurons than task-relevant behavioral variables.
- This mismatch makes population signals highly redundant and raises questions about how task-relevant variables are represented.
- Figure 1 illustrates nonlinear population organization in visual, frontal motor-planning, and head-direction activity.
Dimensionality of Neural Activity
The review distinguishes embedding dimensionality, which captures structured activity under a linear representation, from intrinsic dimensionality, which counts independent latent variables; estimating both from noisy neural data remains difficult.
- Ambient dimensionality is the number of recorded neurons and describes all possible neural states, not the structure of neural activity.
- Embedding dimensionality is the number of Euclidean dimensions required to capture a chosen fraction of structured activity, often using PCA.A common explained-variance threshold is 80%, although criteria differ across studies.
- Intrinsic dimension is the number of independent variables needed to describe neural activity, unlike embedding dimension, which depends on its representation in state space.A ring can have one intrinsic dimension while being embedded in any number of Euclidean dimensions.
- Recent nonlinear analyses have linked intrinsic dimensions to latent variables in neural representations, including navigation-related variables.One study found hippocampal and entorhinal activity constrained to a manifold with intrinsic dimensionality of approximately 3, with two dimensions corresponding to navigational space.
- Estimating intrinsic and embedding dimensions from noisy neural recordings is challenging, and most methods appear to overestimate them.
- The review focuses on how latent variables and their embedding might relate to task variables and underlying computations.
Box 1: Representing neural data in terms of manifolds
Neural population activity can be viewed as lying on manifolds within the full neural state space. Intrinsic dimensionality counts the variables needed to describe the manifold, whereas embedding dimensionality captures how it occupies ambient space; estimating either from noisy recordings remains difficult.
- Neural activity is assumed to be restricted to one or more manifolds within the full state space.
- Intrinsic dimension is the minimum number of continuous variables needed to parametrize a manifold.
- Embedding dimension measures how many dimensions a manifold explores within the ambient Euclidean space.
- A ring has intrinsic dimension one, while its embedding dimension increases when it is warped through higher-dimensional space.
- Estimating intrinsic and embedding dimensions from noisy neural recordings remains an active research problem.
Computational factors influencing intrinsic dimensionality
Intrinsic dimensionality reflects the information represented in collective neural activity, including stimuli, movements, and latent variables. Its expected value and interpretation vary across brain areas and task complexity, while naturalistic dependencies and unknown latent variables complicate estimation.
- Intrinsic dimensionality is shaped by incoming stimuli, ongoing movements, and latent variables related to prior experiences and future expectations.
- Natural stimuli may require higher intrinsic dimensionality as their spatial or temporal dependencies become more complex.
- Motor-cortex population dynamics may provide a basis set for outgoing muscle-like commands and carry upcoming-movement information.
- Estimating intrinsic dimensionality is difficult because long-term dependencies can make it very high and cognitive tasks involve unknown latent variables.
Computational factors influencing embedding dimensionality
Embedding dimensionality concerns how latent variables are processed and communicated in neural state space. Candidate principles include linear readout, separable or overlapping subspaces, and constrained high-dimensional representations, but their usefulness depends on the computational setting.
- Embedding determines how stimuli, movements, and latent variables are processed and communicated through collective neural activity.
- Linear decoders test whether desired information can be extracted by projecting population activity along specific directions.
- Different task variables may occupy orthogonal subspaces, allowing separate decoders to extract them without interference.
- Higher-dimensional embeddings can facilitate noninterfering extraction of task variables, but unconstrained dimensionality can impair generalization.
- Linear decodability is less clearly appropriate for higher brain areas, where controllability may better constrain latent-variable embeddings.
Network models as tools for testing hypotheses about intrinsic and embedding dimensionality
Neural network models offer a substrate for testing hypotheses about how intrinsic and embedding dimensionality support computation. Modeling and model-reduction approaches can connect task behavior, neural activity, and latent-variable organization.
- Network models can instantiate computational hypotheses and generate testable predictions about neural dimensionality.
- Comparing network models with brain activity can narrow hypotheses, while analyzing candidate models can reveal computational mechanisms.
- Systematic characterization of intrinsic and embedding dimensionality in network models remains an open issue.
- Recurrent dynamics are a candidate mechanism for expanding intrinsic dimensionality with internal latent variables beyond immediate sensory inputs.
- Model-reduction techniques can reverse-engineer large recurrent networks to study how latent variables interact and are embedded to implement tasks.
Conclusion
The review argues that intrinsic and embedding dimensionality can help extract computational principles from neural data. Testing this proposal requires combining nonlinear analyses of dense recordings with computational models.
- Intrinsic and embedding dimensionality are proposed as fruitful concepts for extracting computational principles from neural data.