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Continuous Energy Landscape Model for Analyzing Brain State Transitions

Triet M. Tran, Seyed Majid Razavi, Dee H. Wu, Sina Khanmohammadi

TL;DR

This work tackles the information loss inherent in binary brain-state discretization by introducing a continuous energy landscape framework learned from fMRI data. The authors derive a continuous energy function $E(\mathbf{x}) = \tfrac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^T \mathbf{S}^{-1}(\mathbf{x}-\boldsymbol{\mu}) - \mathbf{h}^T\mathbf{x}$ with a positive-definite precision matrix $\mathbf{S}$ learned via Graph Convolutional Networks, enabling exact, scalable energy computations without binarization. Through simulations (SLDS and Kuramoto) and real rs-fMRI data from brain tumor patients, the continuous model yields higher likelihoods, better basin-recovery metrics, and stronger prediction of post-surgical cognitive outcomes (e.g., working memory, executive function, and reaction time) compared to traditional discrete Ising-based models. The findings suggest that preserving full signal fidelity in energy landscape analyses improves understanding of neural dynamics and enhances biomarker potential for clinical decision-making, while also outlining avenues for incorporating directionality and structural connectivity in future work.

Abstract

Energy landscape models characterize neural dynamics by assigning energy values to each brain state that reflect their stability or probability of occurrence. The conventional energy landscape models rely on binary brain state representation, where each region is considered either active or inactive based on some signal threshold. However, this binarization leads to significant information loss and an exponential increase in the number of possible brain states, making the calculation of energy values infeasible for large numbers of brain regions. To overcome these limitations, we propose a novel continuous energy landscape framework that employs Graph Neural Networks (GNNs) to learn a continuous precision matrix directly from functional MRI (fMRI) signals, preserving the full range of signal values during energy landscape computation. We validated our approach using both synthetic data and real-world fMRI datasets from brain tumor patients. Our results on synthetic data generated from a switching linear dynamical system (SLDS) and a Kuramoto model show that the continuous energy model achieved higher likelihood and more accurate recovery of basin geometry, state occupancy, and transition dynamics than conventional binary energy landscape models. In addition, results from the fMRI dataset indicate a 0.27 increase in AUC for predicting working memory and executive function, along with a 0.35 improvement in explained variance (R2) for predicting reaction time. These findings highlight the advantages of utilizing the full signal values in energy landscape models for capturing neuronal dynamics, with strong implications for diagnosing and monitoring neurological disorders.

Continuous Energy Landscape Model for Analyzing Brain State Transitions

TL;DR

This work tackles the information loss inherent in binary brain-state discretization by introducing a continuous energy landscape framework learned from fMRI data. The authors derive a continuous energy function with a positive-definite precision matrix learned via Graph Convolutional Networks, enabling exact, scalable energy computations without binarization. Through simulations (SLDS and Kuramoto) and real rs-fMRI data from brain tumor patients, the continuous model yields higher likelihoods, better basin-recovery metrics, and stronger prediction of post-surgical cognitive outcomes (e.g., working memory, executive function, and reaction time) compared to traditional discrete Ising-based models. The findings suggest that preserving full signal fidelity in energy landscape analyses improves understanding of neural dynamics and enhances biomarker potential for clinical decision-making, while also outlining avenues for incorporating directionality and structural connectivity in future work.

Abstract

Energy landscape models characterize neural dynamics by assigning energy values to each brain state that reflect their stability or probability of occurrence. The conventional energy landscape models rely on binary brain state representation, where each region is considered either active or inactive based on some signal threshold. However, this binarization leads to significant information loss and an exponential increase in the number of possible brain states, making the calculation of energy values infeasible for large numbers of brain regions. To overcome these limitations, we propose a novel continuous energy landscape framework that employs Graph Neural Networks (GNNs) to learn a continuous precision matrix directly from functional MRI (fMRI) signals, preserving the full range of signal values during energy landscape computation. We validated our approach using both synthetic data and real-world fMRI datasets from brain tumor patients. Our results on synthetic data generated from a switching linear dynamical system (SLDS) and a Kuramoto model show that the continuous energy model achieved higher likelihood and more accurate recovery of basin geometry, state occupancy, and transition dynamics than conventional binary energy landscape models. In addition, results from the fMRI dataset indicate a 0.27 increase in AUC for predicting working memory and executive function, along with a 0.35 improvement in explained variance (R2) for predicting reaction time. These findings highlight the advantages of utilizing the full signal values in energy landscape models for capturing neuronal dynamics, with strong implications for diagnosing and monitoring neurological disorders.
Paper Structure (39 sections, 1 theorem, 56 equations, 3 figures, 4 tables)

This paper contains 39 sections, 1 theorem, 56 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Discrete Energy Landscape Models
  4. Methods
  5. Continuous Energy Landscape Derivation
  6. Unimodality of the quadratic CEL
  7. Boundedness of the energy landscape
  8. Graph-Convolutional Parametrization of the Precision Matrix
  9. Training Graph Convolutional Networks (GCNs)
  10. Experiments
  11. Simulated Data
  12. Switching Linear Dynamical System (SLDS)
  13. Kuramoto Oscillator Network with Template Locking
  14. Extracting Energy Landscapes
  15. Discrete Energy Landscape
  16. ...and 24 more sections

Key Result

Lemma 1

Let $\mathbf{S}\in\mathbb R^{N\times N}$ be symmetric positive-definite matrix and let $\boldsymbol{\mu},\mathbf{h}$ be vectors in $\mathbb R^N$. Define $E(\mathbf{x})$ by eq:cel_energy_S. Then $E$ is strictly convex with unique minimizer $\mathbf{x}_*=\boldsymbol{\mu}+\mathbf{S}^{-1}\mathbf{h},$ an so $E$ is bounded below and coercive. Thus, $p(\mathbf{x})\propto\exp[-E(\mathbf{x})]$ coincides pr

Figures (3)

  • Figure 1: Continuous energy landscape modeling via graph neural networks. In Step one (Network Construction), we construct subject-specific functional networks $G=(V,E)$ by thresholding Pearson correlations from standardized fMRI time series and form the normalized weight matrix $\tilde{\mathbf{B}}$. In Step 2 (Parameter Learning), we estimate the precision matrix $\mathbf{S} = \mathbf{Z}\mathbf{Z}^\top + \epsilon \mathbf{I}$ together with the external field $\mathbf{h}(\boldsymbol{\Theta})$ by applying a graph convolutional network that produces node embeddings $\mathbf{H}(\boldsymbol{\Theta})$, which are projected via the trainable map $\mathbf{W}_Z(\boldsymbol{\Theta})$ to obtain embeddings $\mathbf{Z}$. Lastly, in the final step (Energy Construction), we utilize the learned $\mathbf{S}$, $\boldsymbol{\mu}_{\boldsymbol{\Theta}}$, and $\mathbf{h}(\boldsymbol{\Theta})$ to calculate the continuous energy values $E(\mathbf{x}_t;\boldsymbol{\Theta})$ and summarize the resulting energy landscape across time and subjects.
  • Figure 2: Switching Linear Dynamical System (SLDS): grouped performance on Basin Recovery (BR), Transition Matrix Accuracy (TMA), and State Distribution Agreement (SDA) for the discrete (DEL) and continuous (CEL) energy landscape models across the simulation grid ($N\in\{6,\ldots,14\}$, $K\in\{3,4,5\}$, $T\in\{500,1000\}$, SNR levels). The bar plots summarize results across repeats, where paired Wilcoxon signed-rank $p$-values shown at the top of each figure. Here, CEL denotes the mixture extension (CEL-Mix) used for multi-basin recovery. The CEL model provides a significantly better performance in capturing the underlying energy states and their transitions.
  • Figure 3: Kuramoto network with template locking: grouped performance on BR, TMA, and SDA for the discrete (DEL) and continuous (CEL) energy landscape models across the simulation grid. Bars summarize results across repeats; paired Wilcoxon signed-rank $p$-values are shown in-figure. Here, CEL denotes the mixture extension (CEL-Mix) used for multi-basin recovery. The CEL model provides a significantly better performance in terms of BR, but not for TMA and SDA. This was expected, as Ising-based discrete energy models are well-suited to capture the emergent bistability exhibited by Kuramoto dynamics.

Theorems & Definitions (2)

  • Lemma 1: Boundedness and Gaussianity of the quadratic landscape
  • proof