How are the networks expressed through a resting recording?
Several learned patterns can contribute at once, and their relative contributions change through a recording. Read the cohort distribution as a comparison of component sizes, then use individual time series to see the variability behind those averages. These resting recordings have no events that identify a particular cognitive state.
Explore the figures
Choose a participant and network, then use the time slider to align network expression with the ROI signals. In the model-relationship figure, arrows and signs describe fitted predictive coefficients, not transition probabilities or causal influences.
How do brain networks contribute to resting dynamics?
Resting-state fMRI asks how activity across brain regions is organized when there is no imposed task. Rather than tying signal changes to a stimulus, this analysis asks which spatial patterns contribute to the recording, how those contributions differ across people, and how the fitted model links them during prediction. The data comprise 96 NYU control participants, represented by 100 cortical ROIs and a shared rank-7 model.
The model describes overlapping network patterns whose signed contributions can reinforce or oppose one another. Each network has a spatial coefficient matrix and a continuous expression over time. Their weighted sum, together with a constant component, represents the model’s predicted regional second moments: expected products of pairs of ROI signals given the preceding history. Because the inputs are min–max scaled rather than mean-centered, these products are not conventional covariance or Pearson connectivity.
Model, timing and interpretation
In compact notation, M(t) = W₀ + Σ zₖ(t)Wₖ, where M(t) represents a predicted regional second-moment matrix, Wₖ is a network’s spatial coefficient matrix, and zₖ(t) is its signed expression. Each subject has calibrated coefficient matrices in the same cohort-fitted predictive basis. Multiple networks can contribute simultaneously; reversing an expression reverses that component’s contribution. The coordinate sign is not an excitation/inhibition label.
Expressions are projected from history ending one sample before the paired observation target. Each prepared sample spans 3 s, and only coefficient-calibration rows are displayed. Times start at the prepared run origin. Inputs retain their existing filtering, temporal resampling and per-ROI min–max scaling.
An HMM describes probabilities of transitions between discrete states. These continuous networks have no unique HMM probability interpretation. Normalized absolute coefficients describe relative weight allocation only. The full model update depends on expressions, observations and a predicted second-moment correction; an ROI-averaged coefficient matrix is a descriptive summary of that update.
Subject consistency · · ·
How large are network contributions, and how much do subjects differ?
Which network patterns carry the largest typical contribution, and is the ordering similar across people? For network k, contribution magnitude is mean |zₖ(t)| × ||Wₖ||₂, using unique off-diagonal edges. It combines how strongly a coordinate is expressed with the size of its spatial pattern, without cancelling positive and negative expression.
Interpretation. Larger values indicate a larger fitted second-moment component, not a more active brain system or a percentage of explained variance. Overlap between subject distributions means the group ordering need not hold for every person. The shared model basis also constrains these comparisons; variation can reflect biology, recording quality and estimation error.
How do networks interact in time?
Does the predictive model describe networks as largely separate coordinates, or does it link their contributions? The matrix and graph show signed source-to-target coefficients from the transition tensor, averaged uniformly over its ROI observation dimension. Self-links describe same-network terms; off-diagonal links describe cross-network terms. A directed arrow belongs to the model’s predictive representation, not an inferred anatomical pathway.
Model transition coefficients
Directed model relationships
Interpretation. Positive and negative coefficients allow model contributions to reinforce or counteract one another. The effect of a term depends on the source expression and observation-dependent update, so a negative arrow does not always suppress its target. The sign and size of a cohort coefficient also need not characterize every subject; select a relationship to inspect that variation. These summaries alone do not establish physiological persistence, causal coupling or transfer probabilities.
What transition operator is being shown?
Cohort model is the ROI-averaged transition tensor from the fitted common model. Selected subject uses that subject’s saved operator. Both use the same latent basis and target-row/source-column convention.
This is a descriptive coefficient summary of the observation-conditioned operator. The full state update also depends on the current state, current observation and the pseudoinverse of the model’s predicted second-moment matrix. Therefore, this plot is not a standalone linear state-transition equation. No lag parameter or matrix powers are applied here.
The signed coefficients are not transfer probabilities, correlations, causal effects or evidence of neural inhibition. They can exceed one and do not sum to one. Recovering these continuous-network coefficients is different from fitting a discrete HMM. The constant bias is retained in the exported 8 × 8 operators but excluded from the learned-network display; mean-like network 1 is optional.
· ·
How do network contributions unfold within a person?
The cohort summaries compress each recording into a few numbers. The linked views instead ask when a selected network contributes, which ROI signals accompany it, and where its weighted spatial pattern lies. The common time cursor aligns those questions within one subject; resting recordings are not synchronized or averaged across participants.
Regional BOLD signals: the input to the model
Which network patterns are expressed, and when?
The heatmap summarizes all seven history-derived expressions; the line plot isolates the selected network. Red and blue mark opposite coordinate directions. Several networks can be expressed at once, and a zero crossing reverses that network’s weighted pattern rather than marking entry into a new discrete state.
Where is the selected contribution expressed?
The atlas displays zₖ(t)Wₖ: the subject’s fixed spatial pattern multiplied by its expression at the chosen time. This connects a coordinate trajectory to the ROIs it weights. Changes shown here arise from that scalar expression; they are not independent estimates of changing edges.
A sustained contribution or a changing one?
A network can contribute strongly throughout a recording while changing little, or contribute in shorter excursions. Typical magnitude measures distance from zero; fluctuation measures spread around that subject’s usual expression. Their distinction matters when separating a persistent signal component from temporal departures.
Definitions and interpretation
For a network with edge vector W and expression z(t), contribution is z(t)W. Typical magnitude = mean |z| × ||W||₂; temporal fluctuation = SD(z) × ||W||₂. Only unique off-diagonal edges enter these metrics. Their product form is unchanged by reciprocal rescaling of a coordinate and its network matrix.
A large, nearly constant component can have substantial magnitude and little fluctuation. Network 1 is mean-like but remains selectable. The separate explicit constant bias is not shown. Components can reinforce or cancel, so magnitudes do not add to explained BOLD variance. Saved expression is history-projected, not a probability or independent HMM state.
Each subject has its own fixed network matrices within the cohort-fitted basis. Recordings are already filtered and temporally resampled; adjacent samples are dependent. No event labels, cognitive states, occupancy or causal transitions are inferred. Resting timelines are not averaged across people. Visual QC and verified censoring remain incomplete.
What can be concluded?
The fitted recordings are represented by overlapping spatial patterns with different typical contribution sizes and subject-to-subject variability. The operator summarizes signed links in their predictive representation, while individual trajectories locate those contributions within each recording. Together these views describe how a low-dimensional model organizes resting BOLD signals.
They do not establish discrete cognitive states, physiological excitation or inhibition, or causal communication between systems. Functional-network identity requires spatial comparison with reference maps; reliability and generalization require subject-level checks and held-out prediction. Motion, physiological signals, preprocessing and hemodynamic timing remain alternative explanations for part of the observed variation.
More about the measures and interpretation
The figures move from the cohort to an individual recording: first comparing typical contribution sizes, then inspecting predictive relationships between networks, and finally linking their expression to the regional signals and anatomy. The goal is a compact description of resting BOLD dynamics, not an assignment of cognitive states. Spontaneous BOLD signals can have organized spatial and temporal relationships, but this framework does not identify the participant’s thoughts or establish neural causation. Neuroscience background: time-resolved resting networks.
Schaefer 100 cortical atlas · rank 7 · 96 NYU controls · shared predictive basis · min–max inputs. Results describe the fitted recordings; held-out predictive validation, motion sensitivity and verified censoring remain necessary before stronger physiological conclusions.
Large diamonds = cohort means; error bars = ±1 between-subject sample standard deviation. Small dots show every subject; gold outline marks the current selection. These bars show subject-to-subject variation, not a confidence interval, uncertainty of the mean, or temporal fluctuation within a recording. Some spread can reflect estimation variability. Click a subject dot to inspect its recording. Network 1 is hidden here by default.
Columns are source latent networks; rows are target latent networks. Red = positive coefficient; blue = negative. Self-links are same-network coefficients. Raw coefficients use fixed cohort-wide scales for colors, arrow widths and graph cutoffs. Including mean-like network 1 selects a second fixed raw scale. Relative weights use a fixed ±1 signed-share scale; changing subject never rescales either view. The graph shows up to 30 strongest qualifying links; the matrix retains every relationship. Click a cell or arrow for its summary. Network 1 follows the group checkbox above.
Gray traces show the 100 prepared ROI signals; teal highlights one region. These filtered, resampled, per-ROI min–max values are the model inputs at the observation targets. Similar rises and falls can suggest shared signal structure, but visual alignment alone cannot attribute it to neural coupling. Per-ROI scaling also prevents interpreting these heights as absolute BOLD amplitude differences between regions.
Click the heatmap to select a network and time. Color and line scales are shared across subjects; mean-like network 1 can exceed the non-mean display range. Expression size alone is not comparable contribution strength without the corresponding spatial matrix.
Solid red = positive weighted edge; solid blue = negative. Node size reflects total incident absolute contribution using all edges. Gray nodes have zero contribution. The fixed scale comes from non-mean networks across all subjects; network 1 may saturate it. Edge pruning affects this view only.
Upper curve: |zₖ(t)| × ||Wₖ||₂, the size of the selected component. Lower curve: [zₖ(t) − mean(zₖ)] × ||Wₖ||₂, its signed departure from its usual level. A flat upper curve would indicate a sustained contribution; excursions in the lower curve indicate temporal departures. Neither establishes a task response or neural influence.