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Identifying Feedforward and Feedback Controllable Subspaces of Neural Population Dynamics

Ankit Kumar, Loren M. Frank, Kristofer E. Bouchard

TL;DR

The paper treats neural population dynamics as controllable linear systems and introduces FCCA, a data-driven method to identify feedback-controllable subspaces by estimating LQG costs from second-order statistics. It shows that PCA captures feedforward controllability while FCCA targets subspaces requiring feasible, low-dimensional feedback controllers; non-normal dynamics under Dale's Law drive these subspaces apart, and FCCA subspaces better predict behavior across diverse neural recordings. Together, these results provide a control-theoretic lens on neural dynamics and suggest targeting feedback-controllable subspaces could improve brain-machine interface performance. The approach offers a practical, scalable way to analyze large-scale population data from a control perspective, with potential extensions to nonlinear controllability and inverse control frameworks.

Abstract

There is overwhelming evidence that cognition, perception, and action rely on feedback control. However, if and how neural population dynamics are amenable to different control strategies is poorly understood, in large part because machine learning methods to directly assess controllability in neural population dynamics are lacking. To address this gap, we developed a novel dimensionality reduction method, Feedback Controllability Components Analysis (FCCA), that identifies subspaces of linear dynamical systems that are most feedback controllable based on a new measure of feedback controllability. We further show that PCA identifies subspaces of linear dynamical systems that maximize a measure of feedforward controllability. As such, FCCA and PCA are data-driven methods to identify subspaces of neural population data (approximated as linear dynamical systems) that are most feedback and feedforward controllable respectively, and are thus natural contrasts for hypothesis testing. We developed new theory that proves that non-normality of underlying dynamics determines the divergence between FCCA and PCA solutions, and confirmed this in numerical simulations. Applying FCCA to diverse neural population recordings, we find that feedback controllable dynamics are geometrically distinct from PCA subspaces and are better predictors of animal behavior. Our methods provide a novel approach towards analyzing neural population dynamics from a control theoretic perspective, and indicate that feedback controllable subspaces are important for behavior.

Identifying Feedforward and Feedback Controllable Subspaces of Neural Population Dynamics

TL;DR

The paper treats neural population dynamics as controllable linear systems and introduces FCCA, a data-driven method to identify feedback-controllable subspaces by estimating LQG costs from second-order statistics. It shows that PCA captures feedforward controllability while FCCA targets subspaces requiring feasible, low-dimensional feedback controllers; non-normal dynamics under Dale's Law drive these subspaces apart, and FCCA subspaces better predict behavior across diverse neural recordings. Together, these results provide a control-theoretic lens on neural dynamics and suggest targeting feedback-controllable subspaces could improve brain-machine interface performance. The approach offers a practical, scalable way to analyze large-scale population data from a control perspective, with potential extensions to nonlinear controllability and inverse control frameworks.

Abstract

There is overwhelming evidence that cognition, perception, and action rely on feedback control. However, if and how neural population dynamics are amenable to different control strategies is poorly understood, in large part because machine learning methods to directly assess controllability in neural population dynamics are lacking. To address this gap, we developed a novel dimensionality reduction method, Feedback Controllability Components Analysis (FCCA), that identifies subspaces of linear dynamical systems that are most feedback controllable based on a new measure of feedback controllability. We further show that PCA identifies subspaces of linear dynamical systems that maximize a measure of feedforward controllability. As such, FCCA and PCA are data-driven methods to identify subspaces of neural population data (approximated as linear dynamical systems) that are most feedback and feedforward controllable respectively, and are thus natural contrasts for hypothesis testing. We developed new theory that proves that non-normality of underlying dynamics determines the divergence between FCCA and PCA solutions, and confirmed this in numerical simulations. Applying FCCA to diverse neural population recordings, we find that feedback controllable dynamics are geometrically distinct from PCA subspaces and are better predictors of animal behavior. Our methods provide a novel approach towards analyzing neural population dynamics from a control theoretic perspective, and indicate that feedback controllable subspaces are important for behavior.
Paper Structure (14 sections, 7 theorems, 36 equations, 4 figures, 1 table)

This paper contains 14 sections, 7 theorems, 36 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Controllable subspaces of linear dynamical systems
  3. Principal Components Analysis Eigenvalues Measure Feedforward Controllability
  4. Linear Quadratic Gaussian Singular Values Measure Feedback Controllability
  5. The Feedback Controllability Components Analysis Method.
  6. Control-Theoretic Intuition for FCCA
  7. PCA and FCCA subspaces diverge in non-normal dynamical systems
  8. FCCA subspaces are better predictors of behavior than PCA subspaces
  9. Discussion
  10. Appendix
  11. Details of neural datasets
  12. Proof of Theorem 2
  13. FFC Objective Function
  14. Supplementary Figures

Key Result

Theorem 1

Let $(A, B, C)$ be a minimal realization of $G(s)$. Then, the eigenvalues of $QP$ are similarity invariant. Further, these eigenvalues are real and strictly positive. If $\mu_1^2 \geq \mu_2^2 \geq \mu_N^2 > 0$ denote the eigenvalues of $QP$ in decreasing order, then there exists a state space transf The realization $(\tilde{A}, \tilde{B}, \tilde{C})$ will be called the closed-loop balanced realiza

Figures (4)

  • Figure 1: In principle, a controller of dimension as large as the neural state space may be required to effectively regulate dynamics within a FBC subspace ($H_1(s)$). However, subspaces optimized to minimize either the rank, or more practically, the trace of $PQ$ will require controllers of lower dimensionality to achieve near-optimal performance ($H_2(s)$).
  • Figure 2: (Black) Average subspace angles between $d=2$ FCCA and PCA projections applied to Dale's law constrained linear dynamical systems (LDS) as a function of non-normality. Spread indicates standard deviation over 20 random generations of A and 10 random initializations of FCCA. (Blue) Subspace angles between $d=2$ FCCA and PCA projections applied to firing rates derived from spiking activity driven by Dale's Law constrained LDS. Spread around both curves indicates standard deviation taken over 20 random generations of $A$ matrices and 10 random initializations of FCCA.
  • Figure 3: (a) Average subspace angles between FCCA and PCA at $d=2$ across recording sessions (median $\pm IQR$ indicated). (b) Five-fold cross-validated position prediction $r^2$ as a function of projection dimension between for FCCA (red) and PCA (black) and without dimensionality reduction (dashed blue). Mean $\pm$ standard error across folds and recording sessions indicated. (inset) Total area under the curve (AUC) of decoding performance averaged over folds for PCA and FCCA within each recording session (** : $p < 10^{-2}, n=8$, Wilcoxon signed rank test)
  • Figure 4: Full range of variation in cross-validated position $r^2$ from projected FCCA activity relative to the median cross-validated $r^2$. Spread is taken across 20 initializations of FCCA and across all recording sessions

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Theorem 2
  • Lemma 1
  • Proposition 1
  • Lemma 2
  • Proposition 2
  • Lemma 3