Functional Networks

Large language models exhibit a modular internal organization that mirrors well-studied functional networks of the human brain, but how this organization forms during training is unknown: prior work has characterized finished models, not the formation process. We track formation step by step: we train a Pythia-410M model from scratch (two trajectories, bf16 and fp32) and run attribution patching at every step, alongside probes for gradient norms, effective updates, weight norms, and first-order loss decomposition across 14 tasks in four cognitive domains. Three findings. First, the modular map is pre-carved: before any learning, the dominant task pair already overlaps at ~3.6x the attribution substrate (a task-independent baseline), and its layer-0 concentration is an architecture-level constant on this model family. Second, the partition locks in through two sharp jumps whose amplitudes do not track the learning-rate schedule (the second reaching 20.4 sigma quiet-window / 6.2 sigma global), accompanied by gradient-level relative deprivation--winners receive 2.25->2.73x the loser's gradient supply, 9.5-11.5 standard deviations below a random control--that does not propagate to updates or weights. Third, deviation from the substrate appears only in the domain being learned, consistent with the hypothesis that modularity tracks learning. We close by separating the feature-level account we can defend from the mechanistic questions we cannot, and we pre-register the scale-threshold hypothesis behind our ongoing 2.8B experiments.

Identifying individual learning styles optimizes pedagogical efficacy. While traditional questionnaires are structured, behavioral tracking methods require prolonged interaction log accumulation. To overcome these temporal constraints, this paper proposes an objective Electroencephalography (EEG) approach evaluating Phase Locking Value (PLV) connectivity against localized features across the Active-Reflective (AR) and Verbal-Visual (VV) Felder-Silverman dimensions. EEG signals were recorded from 28 participants during Raven's Advanced Progressive Matrices tasks. Support Vector Machine classification used Leave-One-Subject-Out Cross-Validation (LOSO-CV) alongside a 70:30 intra-subject split. The VV dimension achieved 70.00% subject-level accuracy driven by distinct fronto-occipital polarization. Conversely, the AR dimension yielded lower cross-subject generalizability (55.56%) due to overlapping executive networks and a "Systematic Neural Inversion" phenomenon, where stable individual connectivity signatures operated diametrically opposed to global boundaries (up to 20-0 voting margins). Ultimately, these outcomes demonstrate that rigid "one-size-fits-all" classifiers are bounded by biological diversity, emphasizing the need for future adaptive feature transformation techniques to bridge the cross-subject generalization gap.

Functional Magnetic Resonance Imaging ( fMRI ) data are often pooled into collaborative multi-site consortia, as deep learning models for analyses require large datasets to generalize well. While Federated Learning (FL) offers a privacy-preserving paradigm for collaborative training, standard approaches continue to struggle with statistical heterogeneity. In particular, site differences pose a key challenge in multi-site data settings. Additionally, existing FL approaches for fMRI rely on static Functional Connectivity ( FC), omitting dynamic information in brain networks. To address this, we propose FedDOSE, a novel framework that explicitly decomposes site differences for analysis of dynamic FC (dFC). FedDOSE introduces a Modularity-Guided Tucker Decomposition block to encode high-dimensional dFC tensors and capture modular-level spatio-temporal patterns efficiently. Class-specific prototypes are generated across all sites and subsequently aligned at the global level by using a combination of Optimal Transport (OT) barycenter formulation and Procrustes analysis. Extensive experiments for diagnosing Autism Spectrum Disorder (ASD) and Attention-Deficit Hyperactivity Disorder (ADHD) on three multi-site resting-state fMRI datasets: ABIDE-I, ABIDE-II, and ADHD-200, demonstrate that FedDOSE outperforms state-of-the-art methods in ASD and ADHD detection. Our results highlight its effectiveness in learning robust representations from multi-site datasets for reliable analysis.

Cross-site identification of major depressive disorder (MDD) from resting-state functional magnetic resonance imaging (rs-fMRI) is hindered by inter-site distribution shifts and heterogeneous functional connectivity (FC) views. These views capture complementary neural relationships but exhibit distinct site biases and graph topologies, complicating alignment without sacrificing disease-relevant information or cross-view consistency. Existing studies largely treat multi-view connectome learning and cross-site adaptation separately. To the best of our knowledge, few studies have jointly modeled multiple FC views under multi-source unsupervised domain adaptation for cross-site rs-fMRI-based MDD classification. We construct Pearson correlation, sparse representation, and Granger causality graphs, each encoded by a view-specific graph attention network. Dual-stream adaptive fusion explicitly integrates pairwise cross-view interactions, followed by lightweight hyperbolic residual encoding for curvature-aware representation refinement. Class-wise Cauchy--Schwarz alignment reduces inter-source and source-target discrepancies, complemented by adversarial learning, information maximization, and confidence-aware pseudo-labeling. Across seven unlabeled target domains, our framework achieves 73.60% mean accuracy and 71.90% AUC, demonstrating effective generalization under heterogeneous acquisition conditions. These results highlight the effectiveness of unified heterogeneous-view modeling, curvature-aware refinement, and multi-source domain adaptation for cross-site MDD identification.The source code is at https://github.com/OPUS-Lightphenexx/MM-HyperGDA

Accurate identification of Alzheimers disease (AD) using resting-state functional magnetic resonance imaging (rs-fMRI) remains challenging due to the high dimensionality, noise, and complex inter-regional dependencies inherent in functional brain connectivity, which limit the effectiveness of traditional approaches based on handcrafted connectivity features or conventional machine learning models. In this work, we present an attention-based deep learning framework for Alzheimers disease classification that operates directly on rs-fMRI functional connectivity matrices by treating brain regions as tokens and employing a Transformer-inspired self-attention mechanism to model long-range and global functional dependencies across distributed brain networks. The proposed framework learns discriminative functional representations without reliance on manual feature engineering and is evaluated on a longitudinal cohort from the Alzheimers Disease Neuroimaging Initiative (ADNI) comprising cognitively normal and Alzheimers disease subjects with multiple visits. A subject-wise evaluation protocol is adopted to prevent information leakage across visits, and class-weighted optimization is incorporated to address mild class imbalance. Experimental results for binary AD versus cognitively normal classification demonstrate that the proposed attention- based rs-fMRI model achieves an accuracy of 88.95% and a ROC-AUC of 0.90, along with a favorable precision-recall balance, highlighting the effectiveness of self-attention-driven functional connectivity modeling as a robust and interpretable approach for Alzheimers disease detection using resting-state fMRI.

Electroencephalographic (EEG) abnormalities arise from dynamic changes in neural synchrony across spatial and temporal scales, yet many computational approaches reduce these dynamics to static features. We present a Spatial Multi-Expert Graph Transformer that models each EEG recording as a sequence of dynamic functional connectivity graphs. Time-resolved connectivity is estimated using the weighted Phase Lag Index (wPLI), and hierarchical graph encoding aggregates information from electrode to regional and global levels. A multi-expert transformer architecture enables subtype-aware reasoning, with a gating mechanism adaptively fusing expert outputs for global abnormality prediction. Experiments on the TUAB dataset show competitive abnormal EEG detection performance and demonstrate the potential of dynamic graph modeling with adaptive expert fusion for interpretable, subtype-aware spatial--temporal analysis.

This study examines how deficiencies in one brain connectome modality propagate to the other, using the Krakencoder as a simulation framework. Structural and functional connectomes from 702 healthy participants in the Human Connectome Project were analyzed, with the impact of each of the Yeo-7 functional networks assessed separately. Seven scenarios were considered, each involving the removal of a single network while the remaining networks were preserved. The resulting perturbations in cross-modal predictions were quantified using three complementary metrics: KL divergence on eigenvalue spectra, Frobenius norm, and Wasserstein distance. In addition, the persistence of sex-specific information within the predicted connectomes was evaluated. Across all metrics and both prediction directions, the Default Mode Network produced the largest perturbations, whereas the Somatomotor network yielded the smallest. Sex differences in network-level perturbation signatures were subtle, with the best result being an accuracy of 66.09% from connectomes predicted under network-removal conditions. In contrast, connectomes predicted from intact inputs achieved substantially higher sex classification accuracy, reaching up to 84.76%. These findings confirm that full predicted connectomes retain considerably more sex-discriminative information than perturbation-derived signatures alone.

Affective and cognitive disorders manifest as distributed, time-varying brain network dynamics across regions, channels, and time, challenging robust representation learning from EEG/sEEG for clinical diagnosis. We propose RECTOR (Masked Region-Channel-Temporal Modeling), an end-to-end self-supervised framework that unifies joint region-channel-temporal representation learning beyond fixed anatomical priors. At its core, RECTOR-SA is a hierarchical, block-sparse self-attention induced by Adaptive Functional Partitioning that evolves region structures from static anatomical definitions to adaptive functional regions. The self-supervision is driven by Masked Topology and Representation Learning, which jointly optimizes three complementary objectives: Masked Predictive Modeling, Topological Structure Modeling, and Cross-View Consistency. Across diverse benchmarks, RECTOR sets a new state-of-the-art in EEG emotion recognition and sEEG task-engagement classification. Crucially, its strong robustness to missing channels and cross-montage generalization underscores its potential for large-scale pre-training on heterogeneous EEG/sEEG, providing interpretable insights at both region and channel levels.

How the wiring and functional organization of cortex shape recurrent computation remains a central question in both neuroscience and machine learning. Here, we leverage data released through the Machine Intelligence from Cortical Networks (MICrONS) program--a functional connectomics resource spanning multiple areas of mouse visual cortex, in which dense calcium imaging is co-registered with high-resolution electron microscopy reconstruction from the same animal--to build biologically grounded recurrent neural networks. Using neuronal spatial coordinates, anatomical connectivity, and function-derived relationships from nearly 12,000 coregistered excitatory neurons, we initialize recurrent weights and impose communication-aware spatial constraints during learning. Across three cognitive decision-making tasks, networks constrained by cortical structure and function consistently outperform baseline and partially constrained models. Functional weight initialization provides the largest gain, while real spatial embedding yields robust additional improvements across conditions. These biologically grounded networks also develop low-entropy, modular, and small-world organization, and retain strong performance even when recurrence is restricted to positive weights. Together, our results show that the machinery of cortex--its geometry, wiring, and functional structure--can be harnessed as a powerful inductive basis for building recurrent networks that learn more effectively while converging toward key organizational principles of biological computation.

Decoding emotional states from neural signals has been typically framed as a discrete, single-label classification task based on emotionally stable stimuli, a formulation that oversimplifies the continuous, fluid, and co-occurring nature of human affect. This study reconceptualizes emotion decoding by adopting a multi-target regression framework to track multiple overlapping emotional dimensions as continuous trajectories over time. Leveraging the robust generalization capabilities of Large Language Models (LLMs), we extracted fine-grained, continuous sentiment profiles from a naturalistic auditory narrative, Alice in Wonderland, to serve as scalable proxies for subjective affect from human fMRI dataset. Departing from standard classification paradigms or mass-univariate subtractive contrasts that filter out network dynamics, we leverage regularized and kernel-based machine learning algorithms as continuous estimators to track the magnitude of macroscale neural state variations. We demonstrate that models trained on temporal snapshots of Dynamic Functional Connectivity (DFC) significantly outperform static region-of-interest (ROI) amplitude representations, effectively capturing continuous emotional trajectories under rapidly fluctuating narrative input. Furthermore, by implementing graph-theoretical Explainable AI (XAI) techniques, we deconstruct the underlying predictive features to reveal highly interpretable, emotion-specific topological configurations. Collectively, these results highlight the utility of LLM-automated annotation in affective neuroscience and provide compelling empirical evidence for psychological constructionist frameworks, demonstrating that dynamic, distributed network interactions offer superior explanatory power over strictly locationist accounts of emotion.

Understanding complex interactions between brain regions is critical for early neurodegenerative disease classification such as Alzheimer's Disease (AD) and Parkinson's Disease (PD). While graph-based models are widely used to analyze brain networks, most existing approaches primarily focus on pairwise interactions between directly connected nodes, limiting their ability to capture higher-order dependencies across multiple regions. Although hypergraph-based methods have been proposed to model higher-order relations, many rely on predefined hyperedges or restrict learning to hyperedge weights, reducing flexibility and limiting their capacity to capture multi-resolution structural patterns. In this regard, we introduce an adaptive multi-scale hyperedge learning framework, i.e., MuHL, which constructs hierarchical node features and dynamically learns high-order interactions through continuous hyperedge construction over multi-resolution graph signals. Extensive experiments on multiple brain network benchmarks demonstrate that MuHL consistently improves disease classification performance across different stages, and further identifies key regions of interest (ROIs) and their group-wise interactions from the learned hyperedges that are associated with disease progression, highlighting its potential as a powerful tool for brain network analysis in neurodegenerative disorders.

Mild cognitive impairment (MCI) and subjective cognitive decline (SCD) are closely associated with the early Alzheimer's disease continuum, where accurate and explainable diagnosis is important for early risk assessment and intervention. Existing connectome-based deep learning models can improve classification performance but often provide limited insight into disease-related functional and structural connectivity changes. This paper proposes an atlas-knowledge-guided Generative Counterfactual Attention-guided Network (GCAN) for explainable cognitive decline diagnosis using multimodal brain connectomes. GCAN formulates diagnosis as a source-to-target counterfactual generation problem, where target-label connectomes are generated from source-label inputs and their differences are used to construct counterfactual attention maps. To preserve connectome topology, an Atlas-aware Bidirectional Transformer (AABT) performs network-level token encoding and decoding under brain-atlas constraints. The framework is further extended from functional connectivity (FC) to joint functional and structural connectivity (SC) modeling, enabling counterfactual analysis of complementary functional reorganization and structural topology changes. Experiments on hospital-collected and ADNI datasets show that GCAN achieves competitive performance across HC vs. SCD, HC vs. MCI, and SCD vs. MCI classification tasks. Visualization, circular connectome analysis, CAM-based comparison, ablation studies, and confidence interval analysis further support the interpretability and reliability of the proposed framework. Modality-specific FC and SC pre-trained classifiers are used to provide target-state priors for counterfactual generation while being separated from the downstream diagnostic classifier to prevent data leakage.

Electroencephalogram (EEG)-based emotion recognition is an important affective computing task, and recent EEG foundation models provide useful generic representations for downstream adaptation. However, under the fine-tuning setting, three limitations remain prominent: insufficient modeling of multi-scale emotional dynamics, inadequate exploitation of inter-channel functional connectivity, and the limited expressive power of simple linear classification heads. To address these issues, this paper proposes a new EEG emotion recognition method, termed MSCGC-KAN, which introduces a structured task head composed of multi-scale causal graph convolution and Kolmogorov--Arnold feature mapping. Built on a pre-trained CBraMod backbone, MSCGC-KAN enhances downstream adaptation by jointly strengthening multi-scale temporal modeling, learnable inter-channel connectivity modeling, and nonlinear discriminative mapping within a compact task-specific head. This design preserves the representation advantage of the foundation model while making the classifier more sensitive to emotion-related spatiotemporal patterns. Extensive experiments are conducted on the public FACED and SEED-VII datasets. The proposed method achieves a balanced accuracy of 60.66%, a Cohen's Kappa of 0.5525, and a weighted F1-score of 60.40% on FACED, and obtains 33.27%, 0.2223, and 33.64%, respectively, on SEED-VII. Compared with the CBraMod+Linear baseline, the balanced accuracy is improved by 5.91 and 2.03 percentage points on the two datasets, respectively. These results indicate that structured task-head design is an effective way to improve EEG emotion recognition when fine-tuning pre-trained EEG models.

Major depressive disorder (MDD) is a common neuropsychiatric condition whose accurate diagnosis from resting-state functional magnetic resonance imaging (rs-fMRI) remains difficult. Dynamic functional connectivity (DFC) captures time-varying interactions among brain regions and provides rich spatio-temporal information, yet current DFC-based methods face three limitations: sliding-window Pearson correlation yields noisy estimates sensitive to window length and motion artifacts; correlation-derived node features do not fully exploit frequency-domain properties of blood-oxygen-level-dependent (BOLD) signals; and most spatio-temporal graph models handle spatial structure and temporal dynamics in separate stages, restricting their ability to represent coupled brain network evolution. To overcome these issues, we reformulate DFC learning as joint spatio-temporal graph representation learning under a Hawkes-process-inspired temporal dependency prior and propose HWSTCL, a two-stage framework built on a reliability-refined joint spatio-temporal graph with a kernel-weighted pretraining objective. Within each temporal window, BOLD signals are encoded as spectral node descriptors and functional edges are refined by an exponential distance-decay prior that down-weights less reliable long-range connections. The joint graph is then formed by linking each region to itself across future windows through a Hawkes-inspired exponential kernel, allowing spatial and temporal information to be propagated together during message passing. A kernel-weighted contrastive objective further promotes temporal consistency for each region across windows while reducing redundant similarity between different regions. Experiments on a benchmark rs-fMRI dataset show that HWSTCL outperforms recent baselines and yields coherent spatio-temporal representations for MDD diagnosis.

We investigate how internal representations emerge across hierarchical processing systems by introducing a neuroscience-inspired framework for analyzing deep spiking neural networks (SNN) through the lens of functional connectivity. Drawing on concepts from systems neuroscience and information theory, we form the first-order functionally-connected (1FC) group of a neuron based on its statistically significant pairwise correlations with neurons from the previous layer of a trained SNN architecture. We then track its response properties during inference under various conditions. Our analysis shows that several principles of functional connectivity previously observed in biological cortex are preserved in spiking ResNet architectures. These 1FC ensembles display interesting properties: their aggregate cofiring reliably predicts downstream neuronal responses through a robust, ReLU-like input-output relationship, whose gain scales systematically with ensemble size. Reliable encoding of the presented class emerges only during high 1FC cofiring events, which themselves occur infrequently, indicating that informative representations are concentrated in rare but highly coordinated activity patterns. Under uniform random noise or adversarial perturbations, these response profiles are disrupted, particularly in early and intermediate layers. This enables a targeted high-resolution interrogation at specific nodes and pathways. We showed that the functional connectivity structure is shaped by learning and this structure breaks under weight permutation. These establish 1FC ensembles as a functionally meaningful substrate for input encoding and information transfer, with potential implications in designing targeted fine-grained diagnostics on the information flow.

Visual image reconstruction from functional Magnetic Resonance Imaging (fMRI) is a fundamental task in brain decoding, providing a crucial pathway for understanding human perceptual mechanisms and developing advanced brain-computer interfaces (BCIs). However, most current methods simply flatten fMRI signals from localized visual cortices into one-dimensional (1D) vectors, mapping them directly into latent spaces such as that of Contrastive Language-Image Pre-training (CLIP). This paradigm not only disrupts the inherent network topology of the brain-leading to limited neuroscientific interpretability-but also overlooks the synergistic contributions of other distributed functional networks in processing high-level visual semantics. To address these limitations, we propose FPED, a Functional-Network Prior-Guided Mixture of Experts (MoE) framework for interpretable brain decoding. FPED explicitly models different functional brain networks as specialized experts and employs adaptive routing to capture their complementary contributions to visual semantic understanding. Unlike conventional homogeneous decoding paradigms, our framework incorporates neurobiologically grounded priors to enable structured and interpretable network-level representation learning. Experimental results demonstrate that FPED achieves highly competitive semantic reconstruction performance with only 0.68B parameters. The learned routing dynamics reveal biologically meaningful correspondence between functional brain networks and modality-specific semantic processing, providing transparent neuroscientific interpretability. This suggests that brain network-aware expert modeling is a promising direction for bridging neural decoding and biologically inspired artificial intelligence.

Masked autoencoders (MAEs) have recently shown promise for self-supervised representation learning of resting-state brain functional connectivity (FC). However, a fundamental question remains unresolved: how should FC matrices be tokenized to align with the intrinsic modular organization of large-scale brain networks? Existing approaches typically adopt region-centric or graph-based schemes that treat FC as structurally homogeneous elements and overlook the large-scale network brain organization. We introduce NERVE (Network-Aware Representations of Brain Functional Connectivity via Bilinear Tokenization), a self-supervised learning framework that redefines FC tokenization by partitioning FC matrices into patches of intra- and inter-network connectivity blocks. Unlike image-based MAE, where fixed-size patches share a common tokenizer, FC patches defined by network pairs are heterogeneous in size and correspond to distinct functional roles. To resolve this problem, NERVE embeds FC patches through a novel structured bilinear factorization. This formulation preserves network identity and reduces parameter complexity from quadratic to linear scaling in the number of networks. We evaluate NERVE across three large-scale developmental cohorts (ABCD, PNC, and CCNP) for behavior and psychopathology prediction. Compared to structurally agnostic MAE variants and graph-based self-supervised baselines, the proposed network-aware formulation yields more stable and transferable representations, particularly in cross-cohort evaluation. Ablation studies confirm that the proposed bilinear network embedding and anatomically grounded parcellation are critical for performance. These findings highlight the importance of incorporating domain-specific structural priors into self-supervised learning for functional connectomics. Code is available at: https://github.com/leomlck/NERVE.

Functional connectivity (FC) derived from resting-state fMRI is widely used to characterize large-scale brain network alterations in neurological and psychiatric disorders. However, FC construction critically depends on the choice of brain atlas, and different parcellations may emphasize distinct organizational features, leading to heterogeneous and sometimes inconsistent representations. Existing multi-atlas approaches partially alleviate this issue but often fuse atlas-derived features or predictions at a relatively shallow level, while single-atlas disentanglement methods do not explicitly address cross-atlas heterogeneity. We propose Multi-Atlas Disentangled Connectivity LEarning (MADCLE), a multi-branch representation learning framework that jointly encodes FC matrices derived from different brain atlases. Rather than introducing a single explicitly shared latent variable across parcellations, MADCLE learns atlas-wise disease-related representations and encourages them to be cross-atlas consistent through distributional alignment. Meanwhile, covariate-related and atlas-dependent residual factors are modeled separately using covariate similarity supervision, atlas-specific reconstruction, and decorrelation constraints, thereby reducing the leakage of non-disease and parcellation-dependent information into the disease-related embeddings. Experiments on the ADNI and ADHD-200 datasets suggest that MADCLE achieves competitive or improved performance compared with single-atlas baselines, multi-atlas GNN/Transformer models, and recent multi-atlas consistency frameworks. These results support the potential value of structured disentanglement for FC-based disorder identification under heterogeneous parcellation schemes.

Understanding how biological and artificial neural networks implement computation from connectivity is a central problem in neuroscience and machine learning. In neural systems, structural and functional connectivity are known to diverge, motivating approaches that move beyond direct connections alone. Here, we show that the spatial and temporal function of recurrent neural networks (RNNs) trained on hierarchically modular tasks can be recovered by modelling the network as a graph and analysing the multi-hop pathways between input and output units. In particular, decomposing these pathways by hop length reveals how the network temporally routes information. This perspective reframes regularisation: if function is implemented through multi-hop communication, then standard penalties such as L1 regularisation, which act only on individual weights, constrain single-hop structure rather than the multi-hop pathways that support computation. Motivated by this view, we introduce resolvent-RNNs (R-RNNs), which constrain multi-hop pathways and thereby induce temporal sparsity beyond that achieved by standard L1 regularisation. Compared with L1 regularisation, R-RNNs achieve improved performance by inducing temporal sparsity that matches the task structure, even when the task signal is sparse. Moreover, R-RNNs exhibit stronger sparsity-function alignment, reflected in their increased robustness under strong regularisation. Together, our results identify multi-hop communication as a key principle linking structure to function in recurrent networks, and suggest that sparsity should be defined over functional pathways rather than individual parameters.

Several brain foundation models (FM) have recently been proposed to predict brain disorders by modelling dynamic functional connectivity (FC). While they demonstrate remarkable model performance and zero- or few-shot generalization, the salient features identified as potential biomarkers are yet to be thoroughly evaluated. We propose RE-CONFIRM, a framework for evaluating the robustness of potential biomarker candidates elucidated by deep learning (DL) models including FMs. From experiments on five large datasets of Autism Spectrum Disorder (ASD), Attention-deficit Hyperactivity Disorder (ADHD), and Alzheimer's Disease (AD), we found that although commonly used performance metrics provide an intuitive assessment of model predictions, they are insufficient for evaluating the robustness of biomarkers identified by these models. RE-CONFIRM metrics revealed that simply finetuning FMs leads to models that fail to capture regional hubs effectively, even in disorders where hubs are known to be implicated, such as ASD and ADHD. In view of this, we propose Hub-LoRA (Low-Rank Adaptation) as a fine-tuning technique that enables FMs to not only outperform customised DL models but also produce neurobiologically faithful biomarkers supported by meta-analyses. RE-CONFIRM is generalizable and can be easily applied to ascertain the robustness of DL models trained on functional MRI datasets. Code is available at: https://github.com/SCSE-Biomedical-Computing-Group/RE-CONFIRM.

To understand the training dynamics of neural networks, prior studies have considered the mean-field limit of two-layer neural networks as the width tends to infinity, establishing theoretical guarantees for its convergence under gradient flow training as well as approximation and generalization capabilities. In this work, we study the infinite-width limit of a type of three-layer neural network where the first-layer weights are randomly sampled and untrained. To rigorously define the limiting model, we extend the mean-field theory by lifting the representation of neurons from Euclidean to functional spaces. This allows us to establish the mean-field training dynamics as a functional gradient flow with a time-varying kernel that remains positive-definite under suitable assumptions, thus proving a linear-rate convergence of its training loss. Furthermore, we define novel function spaces that contain the solutions obtained through the mean-field training dynamics and prove Rademacher complexity bounds for these spaces. Notably, our analysis applies to a range of scaling choices of the model, resulting in two distinct regimes of the mean-field limit that both exhibit feature learning through training.