Fundamentals of Functional Connectivity and Gradients
Principle and practice of mapping the brain's functional architecture into a low-dimensional space.
I. Functional Connectivity
At the level of basic measurements, neuroimaging data can be said to consist typically of a set of signals (usually time series) at each of a collection of pixels (in two dimensions) or voxels (in three dimensions). Building from such data, various forms of higher-level data representations are employed in neuroimaging. Since brain regions cooperate within large-scale functional networks to support specific cognitive processes, increasingly in recent years, there has emerged a substantial interest in network-based representations.
There are three main types of brain connectivity:
Connectivity Type
Description
Modality
Functional Connectivity
Investigates the undirected statistical dependence between separate brain regions characterized by similar temporal dynamics. It does not imply direct structural connection.
fMRI, EEG, MEG
Effective Connectivity
Refers to the directed, or causal influence of one region over another.
fMRI, EEG, MEG, TMS
Structural Connectivity
Investigates the physical connections (e.g., axons, white matter) between brain regions.
DTI, sMRI
Functional connectivities are inferred when signals from two separate regions fluctuate in synchrony. Basically, computing correlation. Signals can be from any modalities. For example:
fMRI: Functional magnetic resonance imaging (fMRI) provides an indirect measure of neuronal activity by evaluating changes in blood oxygenation over different areas of the brain. Good spatial resolution, low temporal resolution.
II. Functional Gradients: Low-Dimensional Representation
1. The Curse of Dimensionality
The Problem: Functional connectivity analysis generates a $p \times p$ matrix, where $p$ is the number of brain regions. If $p=400$, each region is defined by a complex 400-dimensional connectivity vector. This high-dimensional space is noisy, difficult to visualize, and computationally expensive to model.
The Solution: Dimensionality reduction techniques (e.g., PCA, t-SNE, Diffusion Mapping) are applied to extract the “dominant axes” of variance. These axes represent the underlying organizational principles of the brain.
2. Why called “Gradient”?
A Functional Gradient is analogous to a Principal Component (PC) score, serving as a continuous coordinate system for the brain. Rather than categorizing regions into discrete clusters, a gradient positions each region along a continuous spectrum determined by the similarity of its connectivity profile.
3. Why use Functional Gradients?
It is low-dimensional but encodes the brain activity sufficiently well. For example, gradients is shown to distinguish between primary sensory/motor regions (e.g., Visual Cortex) that process immediate external stimuli and association regions (e.g., Default Mode Network) involved in abstract, internal cognition like memory and planning.
III. Data Acquisition Workflow
This workflow moves beyond simple “Region A connects to Region B” analysis to map the macro-scale hierarchy of the brain.
Construct Connectivity Matrix:
Input: Time-series data for $p$ regions.
Operation: Pearson correlation between all pairs.
Result:Symmetric, Dense Matrix.
Row-wise Thresholding:
Operation: For each row, keep only the top 10% strongest connections.
Result:Sparse, Asymmetric Matrix. (Asymmetry arises because being in A’s top 10% doesn’t guarantee A is in B’s top 10%).
Purpose: Filter out noise and weak correlations.
Affinity Matrix Calculation:
Operation: Compute Cosine Similarity between the sparse rows (connectivity profiles).
Question: “Do Region A and Region B have similar ‘friends’ (connectivity patterns)?”
Result:Symmetric Affinity Matrix. represents contextual similarity, not direct correlation.
Dimensionality Reduction:
Operation: Apply Diffusion Map Embedding.
Result:Principal Gradients. Each region gets a coordinate/score along these axes.