fMRI GLM & Convolution Studio

Construct cognitive paradigm timelines, convolve stimulus events with the canonical Hemodynamic Response Function (HRF) to build the Design Matrix (X), inject noise and drift, and solve the General Linear Model (Y = Xβ + ε) in real time.

Experimental Design:

Stimulus Paradigm u(t) & Convolved BOLD Regressors: Task A (Orange) vs. Task B (Blue) × Canonical HRF

Voxel Time Series Y(t) (Noisy Dots) & Overlaid OLS Model Fit Ŷ = Xβ̂ (Purple Line)

True β_A vs. Estimated β̂_A

2.00 → 1.98

True β_B vs. Estimated β̂_B

1.00 → 1.04

Contrast c^Tβ̂

+0.94

t-statistic

t = 4.82

Significance (p-value)

p < 0.0001

Variance Inflation (VIF)

1.02

Statistical Contrast c:

Model: Y = β_AX_A + β_BX_B + β_driftX_drift + β₀

True Task A Effect Size (β_A): 2.0%

True Task B Effect Size (β_B): 1.0%

Thermal Noise Level (σ_noise): 0.8%

Scanner Baseline Drift: 1.5%

Core Concepts of the fMRI General Linear Model (GLM):

1. Convolution with the HRF: Neural events induce metabolic activity that triggers a sluggish, delayed blood-oxygenation response peaking 5–6 seconds later, followed by an undershoot. We convolve our stimulus timing vectors u(t) with the canonical double-gamma Hemodynamic Response Function h(t) to predict what the BOLD response should look like: X_i(t) = (u_i ∗ h)(t).

2. Detection Power vs. Estimation Efficiency: Block designs create massive cumulative signal swings by stringing events together, yielding very high statistical detection power (large t-statistics). However, they cannot isolate the shape of single events. Jittered event-related designs separate events with variable intervals, allowing the HRF to be estimated accurately (high estimation efficiency) at the cost of reduced overall raw signal amplitude.

3. Multicollinearity & Variance Inflation (VIF): When two conditions always occur in rapid succession (e.g. in the Collinear preset where Task B immediately follows Task A), their convolved regressors become heavily correlated (r → 1). In the GLM, high collinearity blows up the covariance matrix (X^TX)⁻¹, inflating the standard error of parameter estimates (β̂) and destroying our statistical ability to distinguish Task A from Task B!

4. Nuisance Regressors: Real MRI scans suffer from low-frequency hardware drifts and heating. By including a linear drift column in the design matrix X, the GLM separates baseline drifts from true task-related neural activations.