Brain Network-Informed Optimization

• Brain network-informed optimization is the integration of neurobiological connectivity into optimization frameworks to tailor control inputs, graph construction, and model fitting.
• It employs diverse models—including linear control, nonlinear dynamics, and graph learning—with metrics like controllability and communicability to enhance brain state transitions.
• This framework underpins advances in brain stimulation, BCI performance, and whole-brain model tuning by embedding network topology into objectives and constraints.

Brain network-informed optimization is the use of structural, functional, or effective brain network organization to define, constrain, or guide an optimization problem. In the literature, the optimized object may be a control input that steers large-scale activity, a minimal influencer set in a network-of-networks, a subject-specific graph structure, a whole-brain model parameterization, or a decoding policy for a brain-computer interface. Taken together, these studies describe a common research program in which white-matter architecture, mesoscale organization, dynamic connectivity, and cortical hierarchy are treated as determinants of achievable transitions, required energy, robustness to perturbation, and out-of-sample generalization rather than as descriptive correlates alone (Betzel et al., 2016, Stiso et al., 2018, Morone et al., 2017, Shahrzad et al., 11 Feb 2026).

1. Problem classes and formal scope

The field comprises several recurring optimization classes. In optimal control work, the brain is modeled as a networked dynamical system and the optimization variable is an input signal or stimulation policy. In graph-construction work, the optimization variable is the network itself: either a sparsified adjacency, a learned subject-specific structure, or a graph representation. In generative and whole-brain modeling, the optimization variable is a parameter vector governing wiring rules, neural-mass dynamics, or evolutionary search. In BCI and diagnostic settings, optimization targets decoding accuracy, information transfer rate, robustness to nonstationarity, or cross-dataset transfer while using network descriptors as features, priors, or constraints (Betzel et al., 2016, Fallani et al., 2018, Chen et al., 20 Sep 2025, Fallani et al., 2016).

Family	Optimized quantity	Representative formulation
State-transition control	Input energy or tracking error	LTI/LQG control on structural networks
Influence/percolation	Minimal influencer set	Optimal percolation via non-backtracking eigenvalue minimization
Graph construction	Sparse or individualized adjacency	ECO thresholding; FC/EC structure learning
Representation learning	Task-specific graph encoder depth or embedding	DRL-guided GNNs; self-supervised graph encoders
Whole-brain model fitting	Biophysical or generative parameters	ABNG, DMF curricula, Bayesian optimization

A unifying technical feature is that the optimization is not performed on an abstract graph alone. The graph carries neurobiological semantics: streamline-derived white-matter coupling, dynamic functional coupling, effective connectivity, network-of-networks control links, or resting-state network assignments. This shifts the role of topology from a summary statistic to part of the feasible set and cost geometry.

2. Linear control formulations for brain-state transitions

A central formulation models large-scale brain activity as a continuous-time linear time-invariant system,

$$\dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{u}(t),$$

where $\mathbf{x}(t)\in\mathbb{R}^n$ is the brain state, $\mathbf{A}$ is the structural coupling matrix, $\mathbf{B}$ selects controlled nodes, and $\mathbf{u}(t)$ is the input. In the 129-parcel connectome study, regions were assigned to eight canonical intrinsic connectivity networks—Visual, Somatomotor, Default Mode, Limbic, Salience/Ventral Attention, Dorsal Attention, Control, and Subcortical—and initial and target states were binary activity patterns over those systems. The canonical minimum-energy objective is

$J=\int_0^T \mathbf{u}(t)^\top \mathbf{u}(t)\,dt,$

with node-wise energies

$E_i=\int_0^T u_i(t)^2\,dt.$

For $x(0)=x_0$ and $x(T)=x_T$, the controllability Gramian

$$\mathbf{W}(T)=\int_0^T e^{\mathbf{A}t}\mathbf{B}\mathbf{B}^\top e^{\mathbf{A}^\top t}\,dt$$

yields the minimum-energy input

$\mathbf{x}(t)\in\mathbb{R}^n$0

and the minimum energy

$\mathbf{x}(t)\in\mathbb{R}^n$1

The same study also used a tracking-regularized objective,

$\mathbf{x}(t)\in\mathbb{R}^n$2

and set $\mathbf{x}(t)\in\mathbb{R}^n$3 (Betzel et al., 2016).

A closely related stimulation framework used structural connectivity from diffusion MRI and ECoG-derived regional power states to study direct electrical stimulation. There the state was defined over $\mathbf{x}(t)\in\mathbb{R}^n$4 cortical and subcortical regions, the dynamics were again modeled by $\mathbf{x}(t)\in\mathbb{R}^n$5, and target states were subject-specific “good memory” states defined as the average of the top $\mathbf{x}(t)\in\mathbb{R}^n$6 of encoding epochs with highest predicted probability of successful memory encoding, with range $\mathbf{x}(t)\in\mathbb{R}^n$7–$\mathbf{x}(t)\in\mathbb{R}^n$8. The targeted control objective balanced state-tracking error and quadratic input energy,

$\mathbf{x}(t)\in\mathbb{R}^n$9

while open-loop stimulation predictions used a constant input

$\mathbf{A}$0

This formulation explicitly linked structural architecture to stimulation-induced state transitions and to energy required for approach to memory-relevant states (Stiso et al., 2018).

A newer extension replaces terminal-state transfer by stochastic trajectory tracking. In that work, brain dynamics were modeled as a multivariate Ornstein–Uhlenbeck process,

$\mathbf{A}$1

and the objective was to synchronize unhealthy dynamics to healthy target dynamics under noise:

$\mathbf{A}$2

The optimal input had feedback-plus-feedforward form,

$\mathbf{A}$3

with gains from coupled Riccati equations. This formulation was motivated by the observation that brain stimulation aims to normalize ongoing dynamics rather than to reach a single point in state space (Dong et al., 15 Jan 2025).

3. Topological determinants of optimization cost

The primary empirical result of early network-control work is that transition cost is strongly shaped by topology. In the 129-parcel structural connectome study, when all nodes were actuated, class energies obeyed the ordering $\mathbf{A}$4. A one-way ANOVA on log energies across tasks and participants gave $\mathbf{A}$5, $\mathbf{A}$6, and pairwise tests yielded $\mathbf{A}$7, $\mathbf{A}$8. Strength–energy correlations differed by task role: median $\mathbf{A}$9 (IQR $\mathbf{B}$0) for initial nodes, median $\mathbf{B}$1 (IQR $\mathbf{B}$2) for bulk nodes, and median $\mathbf{B}$3 (IQR $\mathbf{B}$4) for target nodes. Communicability predicted compensation under simulated suppression: averaged across all tasks, the compensation matrix correlated with weighted communicability at $\mathbf{B}$5 $\mathbf{B}$6 across participants. Rich-club organization was also decisive. Statistical rich-club presence was maximal at degree threshold $\mathbf{B}$7, optimal target states concentrated activity in hub regions, and rewiring only edges among rich-club members to minimize internal density increased energies significantly, with max $\mathbf{B}$8 (Betzel et al., 2016).

In stimulation-to-memory optimization, analogous conclusions were obtained with different topological predictors. Empirical white-matter topology predicted post-stimulation state transitions better than topological or spatial nulls: for signed maximum correlation between predicted and observed post-stimulation states, a one-way ANOVA gave $\mathbf{B}$9, $\mathbf{u}(t)$0. Energy to reach memory states increased with Frobenius distance between initial and target matrices, with linear mixed-effects slope $\mathbf{u}(t)$1 and $\mathbf{u}(t)$2, and decreased with initial memory probability, with $\mathbf{u}(t)$3 and $\mathbf{u}(t)$4. At the global level, determinant ratio explained additional variance in energy beyond network type, with $\mathbf{u}(t)$5, $\mathbf{u}(t)$6. At the regional level, persistent modal controllability, but not transient modal controllability, predicted lower energy in broadband analyses, with persistent $\mathbf{u}(t)$7, $\mathbf{u}(t)$8 (Stiso et al., 2018).

Under stochastic tracking control, the key intrinsic predictor was average controllability. Tracking energy was significantly negatively correlated with intrinsic average controllability across cohorts. For $\mathbf{u}(t)$9 ROIs, correlations ranged from $J=\int_0^T \mathbf{u}(t)^\top \mathbf{u}(t)\,dt,$0 to $J=\int_0^T \mathbf{u}(t)^\top \mathbf{u}(t)\,dt,$1; for $J=\int_0^T \mathbf{u}(t)^\top \mathbf{u}(t)\,dt,$2 ROIs, they ranged from $J=\int_0^T \mathbf{u}(t)^\top \mathbf{u}(t)\,dt,$3 to $J=\int_0^T \mathbf{u}(t)^\top \mathbf{u}(t)\,dt,$4. By contrast, modal controllability showed no significant correlation with tracking energy, and deterministic state-approach energy related predominantly to target-state magnitude rather than to intrinsic controllability. This dissociation is methodologically important because it separates topology-driven ease of trajectory synchronization from mere size of the target displacement (Dong et al., 15 Jan 2025).

4. Beyond terminal-state transfer: percolation, robust control, and nonlinear steering

Not all brain network-informed optimization is framed as quadratic control. In a robust network-of-networks model of brain activation, the optimization objective is to identify the minimal set of essential nodes whose activation can broadcast to the whole-brain NoN, or equivalently whose inactivation optimally fragments the giant activated component. The problem is expressed through the largest eigenvalue $J=\int_0^T \mathbf{u}(t)^\top \mathbf{u}(t)\,dt,$5 of a modified non-backtracking matrix, and the minimal influencer set satisfies

$J=\int_0^T \mathbf{u}(t)^\top \mathbf{u}(t)\,dt,$6

For the empirical 3-NoN derived from dual-task fMRI, random-input robustness was high, with $J=\int_0^T \mathbf{u}(t)^\top \mathbf{u}(t)\,dt,$7, whereas Collective Influence optimization yielded $J=\int_0^T \mathbf{u}(t)^\top \mathbf{u}(t)\,dt,$8. The resulting Neural Collective Influence map concentrated high-CI influencers in anterior cingulate cortex, extended into posterior parietal cortex, and was rare in V1/V2. This framework differs fundamentally from LTI control because it optimizes activation integration and percolation robustness rather than Gramian-mediated transfer energy (Morone et al., 2017).

Optimal nonlinear control of a whole-brain FitzHugh–Nagumo network gives a second departure from linear state-transfer thinking. There the cost functional penalized deviations from a desired network dynamic, control energy, and spatially non-sparse inputs,

$J=\int_0^T \mathbf{u}(t)^\top \mathbf{u}(t)\,dt,$9

and was applied to attractor switching and synchronization tasks. The phase diagram was organized by background input $E_i=\int_0^T u_i(t)^2\,dt.$0 and global coupling $E_i=\int_0^T u_i(t)^2\,dt.$1. At low bifurcation, high-degree hubs received the largest control energies and remained controlled under high sparsity; at high bifurcation, low-degree nodes received the largest energies. In synchronization tasks, the alignment phase at one operating point showed a positive correlation between node energy and degree, $E_i=\int_0^T u_i(t)^2\,dt.$2, $E_i=\int_0^T u_i(t)^2\,dt.$3, whereas another showed a negative correlation, $E_i=\int_0^T u_i(t)^2\,dt.$4, $E_i=\int_0^T u_i(t)^2\,dt.$5. The general conclusion was explicit: intuitions from linear controllability, solely based on connectome features, do not generally carry over to nonlinear systems, and node roles depend critically on task and state-space location (Chouzouris et al., 2021).

A common misconception is therefore that “optimal nodes” are fixed properties of a connectome. The available evidence supports a narrower statement: node optimality is formulation-dependent. It can be hub-centric in minimum-energy LTI switching, low-average-controllability-averse in stochastic tracking, influencer-centric in optimal percolation, or regime-specific in nonlinear oscillator networks.

5. Graph learning, BCI optimization, and network-based diagnosis

In BCI and neuroadaptive systems, optimization is often centered on decoding or feedback rather than on direct state transfer. Network-neuroscience perspectives identify persistent obstacles—accuracy, information transfer rate, nonstationarity, plasticity, fatigue, and heterogeneous “BCI illiteracy”—and propose network metrics, dynamic connectivity, multilayer models, and network control theory as ways to define robust features and normative stimulation or neurofeedback targets. Typical objectives combine accuracy or ITR with stability and fatigue terms, and network regularizers can penalize high synchronizability or preserve modular structure (Fallani et al., 2018).

A complementary review of network-based BCIs operationalizes this position with concrete graph estimators and learning objectives. It reports, for example, that time-frequency clustering-coefficient sequences classified EEG finger-tap motor imagery or execution with $E_i=\int_0^T u_i(t)^2\,dt.$6 accuracy for all subjects, that single graph metrics yielded approximately $E_i=\int_0^T u_i(t)^2\,dt.$7 accuracy in left-versus-right hand motor imagery, that network features with sparse logistic regression and SVM achieved up to $E_i=\int_0^T u_i(t)^2\,dt.$8 accuracy in lower-limb motor imagery, and that PLV-based network strength and communicability yielded intracranial cognitive BCI discrimination with SVM AUC $E_i=\int_0^T u_i(t)^2\,dt.$9 ($x(0)=x_0$0). The same review describes graph-regularized learning of the form

$x(0)=x_0$1

treating network topology as a smoothness prior on classifier weights (Gonzalez-Astudillo et al., 2020).

Adaptive graph representation learning extends the same logic. BN-GNN models each subject’s connectome as a graph with initial node features $x(0)=x_0$2 and uses deep reinforcement learning to choose, for each graph instance, the optimal number of GNN aggregation layers. The action space is $x(0)=x_0$3, the reward is a validation-performance difference,

$x(0)=x_0$4

and the learned policy is trained by DDQN. Across eight real-world brain-network datasets, BN-GNN achieved the best accuracy on all eight, with average gain approximately $x(0)=x_0$5 absolute over the best baseline, and performance peaked near $x(0)=x_0$6 (Zhao et al., 2022).

Self-supervised structure learning makes the graph itself an optimization variable. BCNSL starts from rs-fMRI time series from $x(0)=x_0$7 AAL regions, builds complementary priors from Pearson FC and transfer-entropy EC, and learns row-normalized adjacencies

$x(0)=x_0$8

with a multi-head similarity term and a graph loss

$x(0)=x_0$9

Cross-view FC/EC embeddings are aligned by a contrastive loss. In cross-dataset diagnosis, the method achieved ACC $x(T)=x_T$0, AUC $x(T)=x_T$1 on ADNI; ACC $x(T)=x_T$2, AUC $x(T)=x_T$3 on ABIDE; and ACC $x(T)=x_T$4, AUC $x(T)=x_T$5 on ADHD, with all improvements over prior methods significant at $x(T)=x_T$6 (Chen et al., 20 Sep 2025).

6. Generative, evolutionary, and hyperparameter search in whole-brain models

Another major branch of brain network-informed optimization searches parameter spaces of connectome generators and biophysical whole-brain models. In action-based generative modeling of structural connectomes, the probability of an edge is shaped jointly by geometric distance and topological actions. In the ABNG(vis) variant,

$x(T)=x_T$7

and optimization minimizes multi-objective KS distances over degree distribution, degree correlations, and clustering/transitivity. For the group representative network, the geometric-only null required $x(T)=x_T$8, whereas ABNG(vis) reduced $x(T)=x_T$9 to $$\mathbf{W}(T)=\int_0^T e^{\mathbf{A}t}\mathbf{B}\mathbf{B}^\top e^{\mathbf{A}^\top t}\,dt$$0. In the ABNG model fitted to the group representative connectome, the action probabilities emphasized homophily—SLW $$\mathbf{W}(T)=\int_0^T e^{\mathbf{A}t}\mathbf{B}\mathbf{B}^\top e^{\mathbf{A}^\top t}\,dt$$1 and SJ $$\mathbf{W}(T)=\int_0^T e^{\mathbf{A}t}\mathbf{B}\mathbf{B}^\top e^{\mathbf{A}^\top t}\,dt$$2—with a smaller PAB component of $$\mathbf{W}(T)=\int_0^T e^{\mathbf{A}t}\mathbf{B}\mathbf{B}^\top e^{\mathbf{A}^\top t}\,dt$$3. Subject-specific mean visibility $$\mathbf{W}(T)=\int_0^T e^{\mathbf{A}t}\mathbf{B}\mathbf{B}^\top e^{\mathbf{A}^\top t}\,dt$$4 was negatively correlated with general intelligence at $$\mathbf{W}(T)=\int_0^T e^{\mathbf{A}t}\mathbf{B}\mathbf{B}^\top e^{\mathbf{A}^\top t}\,dt$$5 overall, and in females at $$\mathbf{W}(T)=\int_0^T e^{\mathbf{A}t}\mathbf{B}\mathbf{B}^\top e^{\mathbf{A}^\top t}\,dt$$6 with $$\mathbf{W}(T)=\int_0^T e^{\mathbf{A}t}\mathbf{B}\mathbf{B}^\top e^{\mathbf{A}^\top t}\,dt$$7, indicating that lower geometric penalization, hence greater tolerance for long-range connections, was associated with higher cognitive ability (Arora et al., 2022).

In Dynamic Mean Field modeling, the optimization problem becomes higher-dimensional and more strongly nonconvex. A 20-parameter local-circuit block was either shared across the cortex or replicated for each of seven canonical resting-state networks, producing a 140-parameter heterogeneous model. Four search strategies were compared: flat heterogeneous optimization, HICO, reverse curriculum, and shuffled curriculum. All heterogeneous strategies improved in-sample fit relative to the homogeneous baseline, with paired $$\mathbf{W}(T)=\int_0^T e^{\mathbf{A}t}\mathbf{B}\mathbf{B}^\top e^{\mathbf{A}^\top t}\,dt$$8-test $$\mathbf{W}(T)=\int_0^T e^{\mathbf{A}t}\mathbf{B}\mathbf{B}^\top e^{\mathbf{A}^\top t}\,dt$$9 values of $\mathbf{x}(t)\in\mathbb{R}^n$00 for flat heterogeneous, $\mathbf{x}(t)\in\mathbb{R}^n$01 for HICO, and $\mathbf{x}(t)\in\mathbb{R}^n$02 for reverse curriculum. Yet only curricular approaches generalized under leave-one-out averaging; homogeneous and flat heterogeneous solutions frequently collapsed to zero fitness after averaging, whereas HICO and reverse maintained robust non-zero leave-one-out fitness. Most importantly, only HICO yielded parameter sets that predicted behavioral abilities reliably (Shahrzad et al., 11 Feb 2026).

Bayesian optimization has been used at smaller scale to tune rhythm-generating neural-mass models. In a four-population thalamic LGN model, the optimized variables were eight sigmoid parameters, $\mathbf{x}(t)\in\mathbb{R}^n$03 and $\mathbf{x}(t)\in\mathbb{R}^n$04 for $\mathbf{x}(t)\in\mathbb{R}^n$05, each bounded by $\mathbf{x}(t)\in\mathbb{R}^n$06mV and $\mathbf{x}(t)\in\mathbb{R}^n$07mV. The objective enforced target-band feasibility first and then rewarded dominant PSD peak height and band-appropriate amplitude. Using a budget of $\mathbf{x}(t)\in\mathbb{R}^n$08 BO iterations and $\mathbf{x}(t)\in\mathbb{R}^n$09 stochastic simulations per evaluation, the method identified feasible theta, alpha, and beta parameter sets. Across $\mathbf{x}(t)\in\mathbb{R}^n$10 independent stochastic trials, the theta and beta solutions satisfied band and amplitude requirements in $\mathbf{x}(t)\in\mathbb{R}^n$11 of runs, whereas the alpha solution did so in approximately $\mathbf{x}(t)\in\mathbb{R}^n$12 of runs (Kothari et al., 2021).

These studies collectively show that network-informed optimization in generative and biophysical modeling is not limited to fitting connectivity. It also serves as a mechanism for separating biologically meaningful parameterizations from merely high-fitting but fragile ones.

7. Thresholding, assumptions, and recurrent methodological tensions

A foundational preprocessing problem is that inferred brain connectivity matrices are often unrealistically dense. The Efficiency–Cost Optimization criterion addresses this by maximizing

$\mathbf{x}(t)\in\mathbb{R}^n$13

where $\mathbf{x}(t)\in\mathbb{R}^n$14 is graph density, $\mathbf{x}(t)\in\mathbb{R}^n$15 is global efficiency, and $\mathbf{x}(t)\in\mathbb{R}^n$16 is local efficiency. Analytically and numerically, the maximizer satisfies

$\mathbf{x}(t)\in\mathbb{R}^n$17

Across pooled real connectomes from micro- to macro-scales and multiple modalities, the empirical fit was $\mathbf{x}(t)\in\mathbb{R}^n$18 with adjusted $\mathbf{x}(t)\in\mathbb{R}^n$19 in the law $\mathbf{x}(t)\in\mathbb{R}^n$20. ECO also outperformed minimum spanning tree and planar maximally filtered graph in discriminating brain states, with Kruskal–Wallis $\mathbf{x}(t)\in\mathbb{R}^n$21 (Fallani et al., 2016).

Several assumptions recur across the broader literature. Linear time-invariant dynamics are routinely justified as approximations near equilibrium because they retain tractability and preserve topology-driven propagation, but the same papers note that real dynamics are nonlinear, state-dependent, and noisy. Binary state definitions and time-invariant actuator sets are simplifications, and dynamic functional connectivity estimates can be unreliable in short windows. Structure-learning methods alleviate manual threshold selection but inherit the biases of Pearson correlation, transfer entropy estimation, tractography, and parcellation. Whole-brain parameter search can overfit individuals even when fit quality is excellent, as shown by the collapse of flat DMF solutions under subject averaging (Betzel et al., 2016, Dong et al., 15 Jan 2025, Shahrzad et al., 11 Feb 2026).

A second recurrent tension concerns what network metrics can and cannot predict. In linear models, strength, communicability, rich-club organization, determinant ratio, persistent modal controllability, and average controllability all carry explanatory power for energy or compensation. In nonlinear oscillator networks, however, node roles reverse with task and operating regime, and synchronization may require collective control even when sparse switching is possible. This suggests that topology is indispensable but insufficient: it informs optimization through the chosen dynamical model and cost functional, not in isolation (Chouzouris et al., 2021).

A plausible implication is that brain network-informed optimization is best understood not as a single algorithmic doctrine but as a family of model-based strategies for embedding neurobiological structure into objectives, constraints, and search trajectories. What unifies the family is the claim—recurrently supported across control, graph learning, BCI, and whole-brain modeling—that optimization quality depends materially on respecting the organization of the brain network on which the problem is posed.