33. Dynamic Connectivity

Dynamic Connectivity¶

Overview¶

For most of its history, functional connectivity analysis summarized an entire scanning session with a single correlation matrix — static functional connectivity. Over the past decade the field has increasingly asked a different question: does connectivity change over the course of a scan, on time scales of seconds to minutes? In dynamic functional connectivity (DFC) analyses, a central goal is to identify brain states — connectivity patterns that recur across time — among which individuals transition during the scan. States have been observed reliably across groups and individuals, and state-based measures such as the average time spent in each state (dwell time) and the number of transitions between states vary with individual differences such as age and disease status. Findings like these suggest that a single static correlation matrix may be too simplistic to fully capture the interplay between brain regions — though, as we will see, this conclusion has not gone unchallenged.

There are two primary analytic paths, both starting from multivariate time series extracted from atlas regions or ICA components (Chapter 31). In the first, time-resolved connectivity is estimated — for example, a correlation matrix at each time point — and a clustering algorithm groups those matrices into a fixed set of states. In the second, a state-space model such as a hidden Markov model (HMM) estimates the states directly from the time series. Either way, state properties such as dwell time and transition frequency are then extracted and related to stimuli, behavior, or individual differences.

The workhorse estimator of time-resolved connectivity is the sliding window: compute the connectivity metric over a fixed-length window of the time series (say, 30 TRs), then slide the window one step at a time,

The approach is flexible — besides Pearson correlations, common metrics include partial correlations, coherence, mutual information, and multiplication of temporal derivatives — but it has well-known problems. The window length is fixed arbitrarily; all data outside the window are ignored; and windowing smooths over abrupt changes in connectivity. There is a fundamental tradeoff: longer windows give more stable estimates but blur dynamic changes; shorter windows track change but produce noisy, imprecise estimates. A useful rule of thumb is to set the window length to the reciprocal of the lowest frequency (largest wavelength) present in the preprocessed signal.

The tradeoff has an elegant frequency-domain interpretation: applying a moving window is equivalent to convolving the time-resolved connectivity with a fixed window, which acts approximately as a low-pass filter. The Fourier transform of a rectangular window is a sinc function whose main lobe has width $1/w$ — so a sliding-window analysis inherently restricts attention to connectivity fluctuations slower than $1/w$, and the sinc side-lobes introduce Gibbs ringing (“spectral leakage”) that distorts frequency content and lowers signal-to-noise ratio. Tapered windows reduce spectral leakage, but in simulations built from real resting-state data they were less sensitive to sharp state transitions than rectangular windows — so if abrupt state changes are expected, rectangular windows remain worth considering.

A different strategy is to model the time-varying correlation directly. Dynamic Conditional Correlation (DCC) is an extension of multivariate GARCH models from econometrics, and proceeds in two steps: first, a GARCH model estimates the time-varying variance of each series and produces standardized residuals; second, a dynamic correlation model describes the time-varying correlation matrix of those residuals as a function of past correlations and past errors. A useful intuition for the second step is the exponentially weighted recursion

Recent samples receive weight $(1-\lambda)$ and older evidence decays geometrically — $\lambda$ acts as a “forgetting factor” that plays the same role as the gain in a Kalman filter. Crucially, DCC estimates all of its parameters from the data by maximum likelihood, so no window length (or $\lambda$) needs to be chosen a priori, and the GARCH step removes time-varying noise variance that would otherwise contaminate the correlations. DCC has been shown to be less susceptible than sliding windows to noise-induced temporal variability, at the price of higher computational cost. Other alternatives include wavelet transform coherence (estimating coherence and phase lag as a function of both time and frequency), coactivation patterns (CAPs; clustering individual fMRI volumes directly), and instantaneous phase synchronization (IPS) based on the Hilbert transform — though IPS requires narrow-band filtering first, which itself acts as an implicit windowing operation.

Once time-resolved connectivity is in hand, the data have expanded enormously — a $T \times p$ time series ($T$ time points by $p$ regions) becomes a $p \times p \times T$ array of time-resolved connections — and the challenge becomes summarizing it. Simple summaries include the mean (essentially the static correlation) and the variance of each connection across time. The more ambitious summary is a set of brain states, most commonly estimated by k-means clustering of the time-resolved matrices. But clustering involves consequential choices. K-means assumes spherical clusters of similar size, is sensitive to outliers, and ignores temporal order entirely — permuting the time points changes nothing. Best practice includes reducing dimensionality (e.g., with PCA) before clustering sparse, high-dimensional data; running multiple centroid initializations; and choosing the number of clusters with an information criterion (AIC/BIC), a permutation or bootstrap null, or Bayesian methods. Methods that respect temporal order — change-point analysis, which partitions the multivariate series at moments of significant covariance change, and HMMs, which model the data with a fixed number of latent states (each a multivariate Gaussian defined by its mean and covariance) plus a transition-probability matrix — will almost certainly do better than order-blind clustering, and both can capture rapid state switches.

Finally, the pitfalls. Time-varying connectivity estimates rest on very few effective degrees of freedom: windows are short, fMRI noise is autocorrelated, and nuisance regression (motion, drift, physiology) removes further degrees of freedom — all of which inflate temporal noise that is easily misread as genuine connectivity change (partial correlations are hit hardest). Fluctuations in arousal and vigilance, and time-varying shifts in the BOLD mean caused by respiration, cardiac activity, transient head motion, or scanner drift, can all produce apparent dynamics. The very existence of resting-state DFC has been controversial: Laumann and colleagues argued that much of the observed “connectivity dynamics” at rest reflects sampling variability, head motion, and drowsiness, so a single static matrix may suffice; and Lindquist showed that sliding-window analysis can reintroduce signals that were previously regressed out — motion-induced signals yielded essentially the same brain states and transitions as the preprocessed data. The lesson is not that dynamics are illusory, but that rigorous statistics — explicit stationary null models, careful denoising, and confound checks — must be part of any dynamic connectivity analysis. The hands-on tutorial below makes exactly this point.

Hands-on tutorial¶

In this tutorial you will generate convincing-looking “dynamics” from data whose correlation never changes, then test them properly against a stationary null. This is the single most important exercise in dynamic connectivity: learning what pure sampling variability looks like before you interpret anything.

Step 1 — Sliding-window correlation on a static-correlation null. We simulate two time series with a constant true correlation of 0.4 and watch the windowed correlation swing anyway.

% Adapted from Lindquist's Dynamic Correlation Toolbox
% (github.com/canlab/Lindquist_Dynamic_Correlation); see sliding_window.m
rng(33);
T = 600;                                    % number of time points (TRs)
r_true = 0.4;                               % true, constant correlation
w = 30;                                     % sliding-window length (TRs)
Sigma = [1 r_true; r_true 1];              % 2x2 correlation matrix of the generating process
dat = mvnrnd([0 0], Sigma, T);              % correlation NEVER changes

rho = NaN(T, 1);                            % sliding-window correlation
for t = w:T
    c = corr(dat(t-w+1:t, :));
    rho(t) = c(1, 2);
end

figure; plot(rho); yline(r_true, '--');
xlabel('Time (TRs)'); ylabel(sprintf('Windowed correlation (w = %d)', w));
fprintf('Range: [%.2f, %.2f] despite constant true r = %.1f\n', ...
    min(rho), max(rho), r_true);

import numpy as np
import matplotlib.pyplot as plt

rng = np.random.default_rng(33)
T, r_true, w = 600, 0.4, 30    # T = time points (TRs), r_true = true correlation, w = window length
L = np.linalg.cholesky(np.array([[1, r_true], [r_true, 1]]))  # 2x2 correlation -> Cholesky factor
dat = rng.standard_normal((T, 2)) @ L.T     # correlation NEVER changes

rho = np.full(T, np.nan)                    # sliding-window correlation
for t in range(w, T + 1):
    rho[t - 1] = np.corrcoef(dat[t - w:t].T)[0, 1]

plt.plot(rho); plt.axhline(r_true, ls="--", color="k")
plt.xlabel("Time (TRs)"); plt.ylabel(f"Windowed correlation (w = {w})")
print(f"Range: [{np.nanmin(rho):.2f}, {np.nanmax(rho):.2f}] "
      f"despite constant true r = {r_true}")

Example output:

Range: [0.03, 0.78] despite constant true r = 0.4

The windowed correlation sweeps from near 0 to almost 0.8 — with no true dynamics whatsoever. Cluster matrices like these and you will get “brain states” out of pure noise.

Step 2 — Test against a stationary null. Phase-randomized surrogates keep each series’ power spectrum and their cross-spectrum (hence the static correlation and autocorrelation) but are stationary by construction. We ask: is the observed variability of the windowed correlation larger than a static process would produce?

% Stationary surrogates: same spectra and static correlation, no dynamics
obs = std(rho, 'omitnan');                  % observed test statistic: SD of windowed r
nsur = 200;                                 % number of surrogates; use ~1,000 for publication-grade p-values
nullsd = zeros(nsur, 1);                    % null distribution of the test statistic
F = fft(dat);                               % Fourier coefficients of both series
half = 2:ceil(T/2);                        % positive-frequency bins (excluding DC)
for s = 1:nsur
    ph = zeros(T, 1);
    ph(half) = 2*pi*rand(numel(half), 1);   % same rotation for both series
    ph(T - half + 2) = -ph(half);           % conjugate symmetry
    sur = real(ifft(F .* exp(1i*ph)));
    rs = NaN(T, 1);
    for t = w:T
        c = corr(sur(t-w+1:t, :)); rs(t) = c(1, 2);
    end
    nullsd(s) = std(rs, 'omitnan');
end
p = (1 + sum(nullsd >= obs)) / (nsur + 1);
fprintf('SD of windowed r = %.3f, stationary-null p = %.3f\n', obs, p);

def phase_randomize(dat, rng):
    """Stationary surrogate with the same power- and cross-spectra."""
    F = np.fft.rfft(dat, axis=0)
    phi = rng.uniform(0, 2*np.pi, size=(F.shape[0], 1))  # same rotation
    phi[0] = 0.0                                     # keep the mean (DC) component real
    if dat.shape[0] % 2 == 0:
        phi[-1] = 0.0                                    # keep Nyquist real
    return np.fft.irfft(F * np.exp(1j*phi), n=dat.shape[0], axis=0)

def windowed_sd(d):
    rs = [np.corrcoef(d[t - w:t].T)[0, 1] for t in range(w, T + 1)]
    return np.std(rs)

obs = windowed_sd(dat)         # observed test statistic: SD of windowed r
nsur = 200                     # number of surrogates; use ~1,000 for publication-grade p-values
null = np.array([windowed_sd(phase_randomize(dat, rng)) for _ in range(nsur)])
p = (1 + np.sum(null >= obs)) / (nsur + 1)   # add-one p-value (Phipson & Smyth correction)
print(f"SD of windowed r = {obs:.3f}, stationary-null p = {p:.3f}")

Example output:

SD of windowed r = 0.151, stationary-null p = 0.537

For this static-null dataset, the test should (correctly) find nothing unusual: the observed variability is entirely typical of a stationary process. The full labs continue the arc: the window-length bias–variance tradeoff, an exponentially weighted (DCC-flavored) estimator racing sliding windows on a true regime-switching signal, and the stationary-null test applied to data with real dynamics — where it correctly rejects.

Thought questions¶

1. A colleague analyzes HCP resting-state data (TR = 0.72 s) with a 20-TR sliding window. Using the filter interpretation of windowing, what range of connectivity-fluctuation frequencies can their analysis actually resolve, and how does the rule of thumb — window length equal to the reciprocal of the lowest frequency in the signal — interact with the fact that the data were high-pass filtered at 0.01 Hz during preprocessing?

2. Laumann and colleagues argued that resting-state “dynamics” largely reflect sampling variability, head motion, and drowsiness. Design an analysis plan that could convince a skeptic that state transitions in your data are neuronal: which null models, physiological measurements, and external validations would you include, and what result pattern would change your own mind in the other direction?

3. You find that patients dwell longer in a “weakly connected” state than controls do, and the group difference is statistically significant. List at least three non-neuronal explanations for this finding, and describe how you would test or rule out each one.

4. K-means clustering ignores temporal order — permuting the time points leaves the clusters unchanged — while HMMs and change-point models explicitly use it. Describe a ground-truth scenario where the two approaches would identify very different states, and explain what each method’s assumptions contribute to the disagreement.

5. Narrow windows, autocorrelated noise, and nuisance regression all reduce the effective degrees of freedom available for each time-resolved connectivity estimate. Explain mechanistically how each one inflates apparent dynamics, and why the problem is worse for partial correlations than for simple correlations.

Quiz yourself¶

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