Neuroscience & Brain Science
Operations Research in Computational Neuroscience
From partitioning the Human Connectome Project's 360-region network into functional communities, to solving the ill-posed inverse problem of EEG/MEG source localisation with sparsity-promoting regularisation, to designing optimal fMRI stimulus sequences that maximise statistical power — three operations research problems at the frontier of brain science.
The Human Connectome Project (HCP) maps structural and functional connectivity across roughly 360 cortical regions using diffusion MRI tractography and resting-state fMRI correlations. The resulting weighted graph encodes how strongly each pair of regions co-activates or is physically connected. Identifying functional communities — groups of regions that are more densely connected to each other than to the rest of the network — reveals the brain's modular architecture: default mode, salience, executive control, and sensorimotor networks.
Problem type: Modularity maximisation (Newman-Girvan Q). Partition N nodes of a weighted graph into communities to maximise the fraction of within-community edge weight minus the expected fraction under a null model. This is NP-hard (Brandes et al., 2007). The Louvain algorithm provides a fast, greedy multi-level heuristic with near-optimal results.
// A = adjacency matrix (weighted)
// k_i = degree of node i, m = total edge weight
// γ = resolution parameter (default 1.0)
// δ(c_i, c_j) = 1 if same community
// NP-hard — Louvain greedy heuristic
Connectome Solver
- Brandes, U., et al. (2007). On modularity clustering. IEEE Transactions on Knowledge and Data Engineering, 20(2), 172-188. Published
- Blondel, V.D., Guillaume, J.-L., Lambiotte, R., & Lefebvre, E. (2008). Fast unfolding of communities in large networks. Journal of Statistical Mechanics, P10008. Published
- Crimi, A., et al. (2023). Structurally constrained effective brain connectivity. Scientific Reports, 13, 7260. Published
- Bassett, D.S. & Sporns, O. (2017). Network neuroscience. Nature Neuroscience, 20, 353-364. Published
Electroencephalography (EEG) records electrical potentials from 64–256 scalp electrodes, while magnetoencephalography (MEG) measures the magnetic field from ~300 sensors. Both modalities observe a linear mixture of thousands of cortical sources. Estimating the underlying source activity from surface measurements is a classic underdetermined inverse problem: there are far more unknowns (source amplitudes) than measurements. Regularisation is required to obtain a unique, stable solution — with different penalty terms yielding fundamentally different neurophysiological interpretations.
Problem type: Regularised least-squares inverse problem. Reconstruct cortical source amplitudes x from sensor measurements b via the lead-field matrix A. L1 penalty (LASSO) promotes focal, sparse solutions; L2 penalty (MNE) yields smooth, distributed solutions; Elastic Net blends both.
// A = lead-field matrix (sensors × sources)
// b = measured sensor data
// λ1 = 0: MNE (L2 only, smooth)
// λ2 = 0: LASSO (L1 only, sparse)
// Both > 0: Elastic Net
Source Localisation Solver
- Pascual-Marqui, R.D. (1994). Low resolution electromagnetic tomography (LORETA). International Journal of Psychophysiology, 18(1), 49-65. Operational
- Gramfort, A., et al. (2013). MNE software for processing MEG and EEG data. Frontiers in Neuroscience, 7, 267. Operational
In event-related fMRI, the experimenter presents a sequence of stimuli (e.g., faces, houses, words) and measures the blood-oxygen-level-dependent (BOLD) response. Each stimulus type produces a haemodynamic response function (HRF) that peaks ~5 seconds post-stimulus and lasts ~25 seconds. The design matrix X is constructed by convolving the stimulus sequence with the HRF. The order, timing, and proportion of stimulus types critically determine the statistical power of the estimated contrasts. Optimal design maximises estimation efficiency while balancing detection power and psychological constraints.
Problem type: Optimal experimental design (D-optimality). Choose a stimulus sequence of length T with K types to maximise the determinant (or trace inverse) of the information matrix (XᵀX), where X is the convolution of the design with the HRF kernel.
where X = S * h // convolution with HRF
s.t. St ∈ {1, ..., K} ∀ t // stimulus type at each time
// h(t) = canonical HRF kernel
// Combinatorial over K^T sequences
fMRI Design Solver
- Dale, A.M. (1999). Optimal experimental design for event-related fMRI. Human Brain Mapping, 8(2-3), 109-114. Published
- Wager, T.D. & Nichols, T.E. (2003). Optimization of experimental design in fMRI: A general framework using a genetic algorithm. NeuroImage, 18(2), 293-309. Published