Principles and open questions in functional brain network reconstruction

Abstract

Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network representation involves often covert theoretical assumptions and methodological choices which affect the way networks are reconstructed from experimental data, and ultimately the resulting network properties and their interpretation. Here, we review some fundamental conceptual underpinnings and technical issues associated with brain network reconstruction, and discuss how their mutual influence concurs in clarifying the organization of brain function.

Keywords: brain dynamics; brain topology; edges; functional imaging; functional networks; nodes; resting state; structure-function relationship; temporal networks.

FIGURE 1

An n‐dimensional manifold M can be described locally by the n‐dimensional real space ℝ ⁿ. A local chart (φ, U) is an open subset of the manifold U ⊆ M together with a one to one map φ: U→ℝ ⁿ from this subset to an open set of the Euclidean space. The piecewise one‐to‐one mapping to the Euclidean space allows generalizing Euclidean space properties onto manifolds. A transition map between two open subsets of ℝ ⁿ provides a way to compare two charts of an atlas

FIGURE 2

Genuinely functional activity results from a complex relation between the structure ${\mathcal{S}}_{\mathrm{\Phi }}$ of the neurophysiological space Φ and the structure ${\mathcal{S}}_{\mathrm{\Psi }}$ of the abstract space Ψ of cognitive functions made observable by performance measures Γ (see text above). Thus subdivisions in one space are used to define subdivisions in the other

FIGURE 3

Manifold M and the corresponding tangent space T _p*M at point p. At each point p of M, there is one and only one tangent vector, and a scalar product can be defined in the associated tangent space T*M. If M is the space of positive definite matrices, T*M is identified with the Euclidean space of symmetric matrices. The M ↦ T*M homomorphism allows replacing the Riemannian metric in M with the Euclidean metric in T*M, and treating the projected connectivity matrices in the tangent space as Euclidean objects

FIGURE 4

A general recipe for post hoc iterative network reconstruction parameter update. Instead of fixing parameters (e.g., edge density, which topological metric to extract) a priori, the proposed methodology involves reconstructing networks using a large set of different parameters; the corresponding discrimination power is then evaluated using a classification problem, and the combination yielding the clearest difference between two groups of patients is chosen as the most informative one. See Zanin et al. (2012) for details