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Energy landscape analysis of neuroimaging data
Takahiro Ezaki, Takamitsu Watanabe, Masayuki Ohzeki, Naoki Masuda
TL;DR
The paper addresses how energy landscape analysis can be adapted and evaluated for neuroimaging data, where pairwise maximum entropy methods remain relatively new. It reviews inference methods, develops new technical components, and applies them to resting-state fMRI, finding that reliable fitting is limited to relatively small systems and requires sufficient data length.
Problem
Energy landscape methods have been used for electrophysiological data, but their use for neuroimaging data is still in its infancy.
Method
The paper reviews pairwise maximum entropy methods, introduces inference algorithms and a Dijkstra-like method, and applies them to resting-state fMRI data.
Results
Reliable pairwise maximum entropy estimation was achieved only up to N ≈5, while longer data with tmax = 750–4500 could accommodate N ≈8.
Takeaways & Limitations
Energy landscape analysis provides a data-driven framework for studying resting-state fMRI dynamics while making fit accuracy dependent on brain-system size and data length.
Takeaways & Limitations
The method cannot currently be applied to relatively large brain systems.
Abstract
from arXiv · showhide
Computational neuroscience models have been used for understanding neural dynamics in the brain and how they may be altered when physiological or other conditions change. We review and develop a data-driven approach to neuroimaging data called the energy landscape analysis. The methods are rooted in statistical physics theory, in particular the Ising model, also known as the (pairwise) maximum entropy model and Boltzmann machine. The methods have been applied to fitting electrophysiological data in neuroscience for a decade, but their use in neuroimaging data is still in its infancy. We first review the methods and discuss some algorithms and technical aspects. Then, we apply the methods to functional magnetic resonance imaging data recorded from healthy individuals to inspect the relationship between the accuracy of fitting, the size of the brain system to be analyzed, and the data length.
I. INTRODUCTION
The paper develops energy landscape analysis as a data-driven Ising or pairwise maximum entropy approach for studying brain dynamics in neuroimaging data. It reviews the pipeline and applies it to resting-state fMRI to examine fitting accuracy, brain-system size, and data length.
- A. Energy landscape analysis: Energy landscape analysis maps multivariate brain dynamics onto an energy landscape inferred from neural data, using the Ising model or pairwise maximum entropy model.Activity patterns occupy basins, with stochastic fluctuations allowing movement between them.
- A. Energy landscape analysis: The data-driven approach infers Ising-model parameters from the observed data, distinguishing it from earlier approaches used to emulate fMRI signals.The model can be constructed from multivariate time series recorded at specified regions of interest.
- I. INTRODUCTION: The study reviews methods and technical details, introduces alternative inference algorithms and a Dijkstra-like technique, and validates them on resting-state fMRI.The analysis uses publicly shared data from healthy human participants.
- I. INTRODUCTION: The application examines how fitting accuracy relates to the number of regions of interest and the length of fMRI data.This links model-validation requirements to the size and duration of the analyzed brain system.
- A. Pairwise maximum entropy model: The pipeline specifies a brain network, binarizes ROI signals into active or inactive states, estimates pattern frequencies, and fits a Boltzmann distribution.The resulting activity pattern is an N-dimensional binary vector with 2^N possible states.
- A. Pairwise maximum entropy model: The fitted energy assigns lower visitation frequency to high-energy activity patterns, while h_i represents baseline ROI activity and J_ij represents pairwise ROI interaction.Parameters are chosen so the model matches empirical first- and second-order moments.
B. Algorithms to estimate the pairwise MEM
This section reviews likelihood-based algorithms for estimating pairwise maximum entropy parameters. Maximum likelihood is principled but becomes computationally costly as the number of ROIs grows.
- B. Algorithms to estimate the pairwise MEM: Maximum likelihood estimation updates h and J by gradient ascent using differences between empirical and model means and correlations.The likelihood is concave in the parameters, supporting the maximum likelihood estimator.
- B. Algorithms to estimate the pairwise MEM: Iterative scaling provides an alternative update rule that replaces moment differences with signed logarithms of empirical-to-model ratios.The method updates both single-ROI and pairwise interaction parameters.
- B. Algorithms to estimate the pairwise MEM: Maximum likelihood is computationally costly for large N because each update requires evaluating all 2^N activity patterns.The paper supplies Matlab code for calculating the estimator on arbitrary multivariate time-series data.
2. Pseudo-likelihood maximization
Pseudo-likelihood maximization approximates the full likelihood by conditioning on one spin at a time, avoiding calculations over all activity patterns. Minimum probability flow instead minimizes short-time probability flow from observed patterns and has complementary computational trade-offs.
- 2. Pseudo-likelihood maximization: Pseudo-likelihood maximization approximates the likelihood with products of single-spin conditional probabilities while holding the other spins fixed.Its updates use conditional means and correlations rather than the full model expectations.
- 2. Pseudo-likelihood maximization: The pseudo-likelihood estimator approaches the maximum likelihood estimator while circumventing calculation of full model moments.This addresses the computational burden of likelihood maximization.
- 3. Minimum probability flow: Minimum probability flow fits h and J by minimizing probability flow from activity patterns observed in the data during an infinitesimal relaxation.The method initializes the distribution with the empirical distribution and uses transitions between patterns differing at one ROI.
- 3. Minimum probability flow: Minimum probability flow is ineffective when N is small and tmax is large, but efficient in computation time and memory when N is large or tmax is small.Its usefulness therefore depends on the relationship between system size and data length.
C. Accuracy indices
The paper evaluates pairwise maximum entropy fits by comparing model distributions with empirical data and by measuring how much pairwise correlations explain total dependence. These indices distinguish pairwise explanatory power from higher-order structure.
- C. Accuracy indices: The pairwise maximum entropy model uses N + N(N − 1)/2 parameters while a fully specified empirical distribution requires 2^N − 1 parameters.The model imposes agreement with empirical first- and second-order moments but may misrepresent higher-order correlations.
- C. Accuracy indices: The section introduces these measures because previous studies used them to quantify the accuracy with which MEMs fit empirical data.They assess fit beyond the first two moments matched during model construction.
- C. Accuracy indices: The ratio I2/IN measures the fraction of total empirical correlation contribution accounted for by pairwise correlations.A value of 1 indicates that pairwise correlations account for all correlations, whereas 0 indicates no information from pairwise correlations.
- C. Accuracy indices: The index r compares how closely the pairwise and independent MEM distributions approximate the empirical distribution.For the maximum likelihood estimator, r equals I2/IN.
D. Disconnectivity graph and energy landscape
The disconnectivity graph organizes activity patterns by local minima, attractive basins, and the energy barriers separating them. It links lower energy with more frequent patterns and higher barriers with less frequent switching.
- D. Disconnectivity graph and energy landscape: A leaf’s vertical position encodes activity-pattern energy and frequency, with lower positions indicating higher frequency.The endpoint energy specifies the pattern’s frequency of appearance.
- D. Disconnectivity graph and energy landscape: A high energy barrier between two local minima implies a low switching frequency between their activity patterns.The dynamics must pass through a high-energy pattern to move along the connecting path.
- D. Disconnectivity graph and energy landscape: The disconnectivity graph represents local minima as leaves and groups pairs according to the highest energy along paths connecting them.Local minima are patterns with lower energy than all neighbors; the graph’s branching structure describes energy barriers between them.
- D. Disconnectivity graph and energy landscape: The landscape contains 2^N activity patterns, with each pattern connected to N neighbors differing at one ROI.The graph is built from the complete activity-pattern state space and its one-ROI neighborhoods.
- D. Disconnectivity graph and energy landscape: Each activity pattern belongs to an attractive basin determined by repeatedly moving to the neighboring pattern with the smallest energy.This procedure continues until a local minimum is reached, which defines the basin assignment.
E. 0/1 versus 1/-1
The pairwise maximum-entropy model can represent activity with either 0/1 or −1/1 variables, which are mathematically equivalent through a one-to-one transformation. The −1/1 convention preserves spin-state symmetry, while 0/1 representation is useful for expressing neuronal firing.
- E. 0/1 versus 1/-1: The Ising model usually uses σ_i ∈ {−1,1}, whereas neuronal spike and fMRI analyses commonly use either representation.The −1/1 convention respects symmetry between spin states and supports analytical calculations using (σ_i)^2 = 1.
- E. 0/1 versus 1/-1: The 0/1 representation expresses neuronal firing activity directly through multiplication of binary variables.This convention provides a simple form for representing instantaneous firing rates.
- E. 0/1 versus 1/-1: Iterative scaling does not generally work with σ_i ∈ {−1,1} because the empirical-to-model moment ratio can be problematic.The affected ratio is the logarithm of ⟨σ_i⟩_empirical and ⟨σ_i⟩_model.
- E. 0/1 versus 1/-1: The two activity representations are mathematically equivalent through the transformation 2˜σ_i − 1 = σ_i.The mapping relates ˜σ_i ∈ {0,1} to σ_i ∈ {−1,1}.
III. RESULTS
The methods were applied to resting-state fMRI from healthy adults across three brain networks, comparing likelihood, pseudo-likelihood, and minimum probability flow estimation. Pseudo-likelihood closely matched likelihood, whereas minimum probability flow showed larger deviations and distinct accuracy-index behavior.
- III. RESULTS: Pseudo-likelihood maximization produced parameter estimates close to likelihood maximization across all networks, particularly for J.The comparison was performed for the DMN, FPN, and CON.
- III. RESULTS: Minimum probability flow was not applied to the CON because all 28 activity patterns appeared at least once, making the right-hand side of Eq. (15) zero.The condition is described as D = Ω.
- III. RESULTS: Minimum probability flow deviated more from likelihood maximization than pseudo-likelihood did, particularly for h.The comparison is reported for estimates in the DMN and FPN; it was not applied to the CON.
- III. RESULTS: For likelihood maximization, the two accuracy indices had the same value, while pseudo-likelihood indices differed slightly but were approximately similar to maximum-likelihood values.The indices are reported for each network and estimation method in Table I.
- III. RESULTS: For minimum probability flow, r was substantially smaller than for likelihood or pseudo-likelihood, whereas I2/IN exceeded those methods’ values.I2/IN exceeded unity because S1 > SN > S2 for minimum probability flow.
B. Disconnectivity graphs
Energy-landscape analysis revealed synchronized all-active and all-inactive patterns as local minima, with more local minima in the FPN than in the DMN or CON. Fitting accuracy improved with data length and was approximately governed by the number of visits per activity pattern.
- B. Disconnectivity graphs: The FPN had many more local minima than the DMN and CON, while all-active and all-inactive synchronized patterns were local minima.The reported pattern differs from an earlier dataset, and the reason for the discrepancy is unclear.
- C. Effects of the data length: Increasing the number of ROIs makes unvisited activity patterns more likely because the state space contains 2^N patterns.Unvisited patterns still receive positive probability in the maximum-entropy model, while empirical probabilities based on few visits are unreliable.
- C. Effects of the data length: Accuracy improved as data length increased, and results for the DMN, FPN, and CON roughly collapsed onto a single curve.The analysis used likelihood maximization on samples of varying length and examined r as a function of ℓ/2^N.
- C. Effects of the data length: Accuracy of 0.8 and 0.9 required each activity pattern to be visited approximately 5 and 16 times, respectively.These visit frequencies are suggested by the fitted accuracy curve.
- C. Effects of the data length: The non-overlapping-sample analysis produced results similar to those from overlapping time windows.The time series was divided into halves, quarters, eighths, and progressively shorter non-overlapping samples.
IV. DISCUSSION
The paper presents computational methods for estimating pairwise maximum-entropy models and energy landscapes from resting-state fMRI, while showing that data requirements constrain the size of analyzable brain systems. It demonstrates the methods in healthy participants and notes potential applications to altered brain dynamics, but emphasizes that results require corroboration.
- The study explained three methods for estimating pairwise maximum-entropy models and calculating energy landscapes, including pseudo-likelihood maximization, minimum probability flow, and a Dijkstra variant.
- The methods were applied to healthy-participant fMRI data to assess how much data is needed for sufficiently accurate model fitting.
- The current method remains demanding in data volume and cannot yet be applied to relatively large ROI systems or voxel-based data.
- The present results differ from the authors’ previous findings using another dataset, so the conclusions should be corroborated with different data sets.
- The methods may support studies of medical conditions because altered brain dynamics have been suggested to deform energy landscapes relative to healthy controls.
Materials and Methods
The study analyzed publicly shared resting-state fMRI data from two healthy adults using a standardized acquisition and preprocessing pipeline. Signals were extracted from 30 predefined regions of interest, producing time series for energy-landscape analysis.
- The dataset comprised resting-state fMRI from two healthy adults selected because the available data were sufficiently large for the analysis.
- The EPI acquisition used a 3T MRI with TR 0.72 s, TE 33.1 ms, 72 slices, and 2.0 mm isotropic resolution.
- Participants underwent four approximately 15-minute resting-state runs while viewing a fixed cross mark and were instructed to relax.
- The analysis extracted ROI-averaged signals from 30 regions, comprising 12 default-mode, 11 frontoparietal, and 7 remaining regions.
L hi
The paper compares fitting accuracy across brain networks and estimation algorithms, and examines how fit accuracy changes with normalized data length using overlapping and non-overlapping windows.
- L hi: The analysis compares likelihood maximization with pseudo-likelihood estimation and minimum probability flow.Figure 2 organizes comparisons across the DMN, FPN, and CON.
- L hi: Fit accuracy is plotted against data length normalized by the 2^N possible activity patterns for overlapping and non-overlapping windows.Figure 4 reports standard-deviation error bars for both sampling schemes.
- L hi: Fitting accuracy is evaluated across different brain networks and estimation algorithms.Table I summarizes accuracy for each network and algorithm.