The Geometry and Dimensionality of Brain-wide Activity

Joint Public Reviews:

Summary:

The authors examine the eigenvalue spectrum of the covariance matrix of neural recordings in the whole-brain larval zebrafish during hunting and spontaneous behavior. They find that the spectrum is approximately power law, and, more importantly, exhibits scale-invariance under random subsampling of neurons. This property is not exhibited by conventional models of covariance spectra, motivating the introduction of the Euclidean random matrix model. The authors show that this tractable model captures the scale invariance they observe. They also examine the effects of subsampling based on anatomical location or functional relationships. Finally, they briefly discuss the benefit of neural codes which can be subsampled without significant loss of information.

Strengths:

With large-scale neural recordings becoming increasingly common, neuroscientists are faced with the question: how should we analyze them? To address that question, this paper proposes the Euclidean random matrix model, which embeds neurons randomly in an abstract feature space. This model is analytically tractable and matches two nontrivial features of the covariance matrix: approximate power law scaling, and invariance under subsampling. It thus introduces an important conceptual and technical advance for understanding large-scale simultaneously recorded neural activity.

Weaknesses:

The downside of using summary statistics is that they can be hard to interpret. Often the finding of scale invariance, and approximate power law behavior, points to something interesting. But here caution is in order: for instance, most critical phenomena in neural activity have been explained by relatively simple models that have very little to do with computation (Aitchison et al., PLoS CB 12:e1005110, 2016; Morrell et al., eLife 12, RP89337, 2014). Whether the same holds for the properties found here remains an open question.

Author response:

We are grateful for the thorough and constructive feedback provided on our manuscript.

Regarding the main concern about power law behavior and scale invariance, we would like to clarify that our study does not aim to establish criticality. Instead, we focus on describing and understanding a specific scale-invariant property: the collapsed eigenspectra in neural activity under random sampling. Indeed, we tested Morrell et al.’s latent-variable model (eLife 12, RP89337, 2024, [1]), where a slowly varying latent factor drives population activity. Although it produces a seemingly power-law-like spectrum, random sampling does not replicate the strict spectral collapse observed in our data (second row in Author response image 1). This highlights that simply adding latent factors does not fully recapitulate the scale invariance we measure, suggesting richer or more intricate processes may be involved in real neural recordings.

Author response image 1.

Morrell et al.’s latent variable model [1, 2]. A-D: Functional sampled (RSap) eigenspectral of the Morrell et al. model. E-H: Random sampled (RSap) eigenspectra of the same model. Briefly, in Morrell et al.’s latent variable model [1, 2], neural activity is driven by Nf latent fields and a place fields. The latent fields are modeled as Ornstein-Uhlenbeck processes with a time constant τ . The parameters ϵ and η control the mean and variance of individual neurons’ firing rates, respectively. The following are the parameter values used. A,E: Using the same parameters as in [1]: N_f = 10, ϵ = −2.67, η = 6, τ = 0.1. Half of the cells are also coupled to the place field. B,C,D,F,G,H: Using parameters from [2]: N_f = 5, ϵ = −3, η = 4. There is no place field. The time constant τ = 0.1, 1, 10 for B,F, C,G, and D,H, respectively.

We decided to make 5 key revisions.

• As mentioned, we have evaluated the latent variable model proposed by Morrell et al. and found that they fail to reproduce the scale-invariant eigenspectra observed in our data; these results will be presented in the Discussion section and supported by a new Supplementary Figure.

• We will include a discussion on the findings of Manley et al. (2024, [2]) regarding the issue of saturating dimensionality in the Discussion section, highlighting the methodological differences and their implications.

• We will add a new mathematical derivation in the Methods section, elucidating the bounded dimensionality using the spectral properties of our model.

• We will elaborate in the Discussion section to further emphasize the robustness of our findings by demonstrating their consistency across diverse datasets and experimental techniques.

• We will incorporate a brief discussion on the implications for neural coding. In particular, Fisher information can become unbounded when the slope of the power-law rank plot is less than one, as highlighted in the recent work by Moosavi et al. (bioRxiv 2024.08.23.608710, Aug, 2024 [3]) in the Discussion section.

We believe these revisions will address the concerns raised by you and collectively strengthen our manuscript to provide a more comprehensive and robust understanding of the geometry and dimensionality of brain-wide activity.

References

(1) M. C. Morrell, A. J. Sederberg, I. Nemenman, Latent dynamical variables produce signatures of spatiotemporal criticality in large biological systems. Physical Review Letters 126, 118302 (2021).

(2) M. C. Morrell, I. Nemenman, A. Sederberg, Neural criticality from effective latent variables. eLife 12, RP89337 (2024).