MixRx uses Large Language Models (LLMs) to classify drug combination interactions as Additive, Synergistic, or Antagonistic, given a multi-drug patient history. We evaluate the performance of 4 models, GPT-2, Mistral Instruct 2.0, and the fine-tuned counterparts. Our results showed a potential for such an application, with the Mistral Instruct 2.0 Fine-Tuned model providing an average accuracy score on standard and perturbed datasets of 81.5%. This paper aims to further develop an upcoming area of research that evaluates if LLMs can be used for biological prediction tasks.

Metagenomic disease prediction commonly relies on species abundance tables derived from large, incomplete reference catalogs, constraining resolution and discarding valuable information contained in DNA reads. To overcome these limitations, we introduce MetagenBERT, a Transformer based framework that produces end to end metagenome embeddings directly from raw DNA sequences, without taxonomic or functional annotations. Reads are embedded using foundational genomic language models (DNABERT2 and the microbiome specialized DNABERTMS), then aggregated through a scalable clustering strategy based on FAISS accelerated KMeans. Each metagenome is represented as a cluster abundance vector summarizing the distribution of its embedded reads. We evaluate this approach on five benchmark gut microbiome datasets (Cirrhosis, T2D, Obesity, IBD, CRC). MetagenBERT achieves competitive or superior AUC performance relative to species abundance baselines across most tasks. Concatenating both representations further improves prediction, demonstrating complementarity between taxonomic and embedding derived signals. Clustering remains robust when applied to as little as 10% of reads, highlighting substantial redundancy in metagenomes and enabling major computational gains. We additionally introduce MetagenBERT Glob Mcardis, a cross cohort variant trained on the large, phenotypically diverse MetaCardis cohort and transferred to other datasets, retaining predictive signal including for unseen phenotypes, indicating the feasibility of a foundation model for metagenome representation. Robustness analyses (PERMANOVA, PERMDISP, entropy) show consistent separation of different states across subsamples. Overall, MetagenBERT provides a scalable, annotation free representation of metagenomes pointing toward future phenotype aware generalization across heterogeneous cohorts and sequencing technologies.

Building on the phenomenological and microscopic models reviewed in Part I, this second part focuses on network-level mechanisms that generate emergent temperature response curves. We review deterministic models in which temperature modulates the kinetics of coupled biochemical reactions, as well as stochastic frameworks, such as Markov chains, that capture more complex multi-step processes. These approaches show how Arrhenius-like temperature dependence at the level of individual reactions is transformed into non-Arrhenius scaling, thermal limits, and temperature compensation at the system level. Together, network-level models provide a mechanistic bridge between empirical temperature response curves and the molecular organization of biological systems, giving us predictive insights into robustness, perturbations, and evolutionary constraints.

Functional-structural plant models (FSPM) replicate plants' responses to their environment and are useful for predicting behavior in a changing climate. However, they rely on detailed measurements of traits, which are difficult to collect consistently across scales, often limiting model parameterization and thorough evaluation, and thereby reducing confidence in model predictions. Here, we provided a comprehensive dataset of structural and biophysical measurements from four oil palm plants (Elaeis guinnensis) grown under multiple controlled environmental scenarios, including varying CO2 concentrations, light, temperature and humidity conditions. The dataset included detailed reconstructions of the three-dimensional plant structures derived from terrestrial LiDAR point clouds, and enabled the parametrization of biophysical processes at the leaf scale such as photosynthesis and stomatal conductance, as well as the collection of plant-scale measurements (gas exchange measurements of CO2 and H20), which can be compared with FSPM simulations. The tree-dimensional reconstructions effectively represented the architecture of the plants and showed strong correlation with the measured total leaf area. Hence, future comparisons between simulated and observed physiological traits could be used to evaluate the quality of the physiological formalisms independently. By bridging the scales from individual leaves to the entire plant, this database allows modellers to both calibrate their biophysical models at a fine spatial resolution and evaluate their predictive accuracy at the plant scale. The provided data will facilitate benchmarking of biophysical models, help identify sources of model uncertainty, and ultimately enhance model predictions, which can be applied in various fields, from cognitive studies to decision support applications.

Collective improvisation in dance provides a rich natural laboratory for studying emergent coordination in coupled neuro-motor systems. Here, we investigate how training shapes spontaneous synchronization patterns in both movement and brain signals during collaborative performance. Using a dual-recording protocol integrating 3D motion capture and hyperscanning EEG, participants engaged in free, interaction-driven, and rule-based improvisation before and after a program of generative dance, grounded in cellular-automata. Motor behavior was modeled through a time-resolved {\alpha}-exponent derived from Movement Element Decomposition scaling between mean velocity and displacement, revealing fluctuations in energetic strategies and degrees of freedom. Synchronization events were quantified using Motif Synchronization (biomechanical data) and multilayer Time-Varying Graphs (neural data), enabling the detection of nontrivial lead-lag dependencies beyond zero-lag entrainment. Results indicate that training produced an intriguing dissociation: inter-brain synchronization increased, particularly within the frontal lobe, while interpersonal motor synchrony decreased. This opposite trend suggests that enhanced participatory sense-making fosters neural alignment while simultaneously expanding individual motor explorations, thereby reducing coupling in movement. Our findings position collaborative improvisation as a complex dynamical regime in which togetherness emerges not from identical motor outputs but from shared neural intentionality distributed across multilayer interaction networks, exemplifying the coupling-decoupling paradox, whereby increasing inter-brain synchrony supports the exploration of broader and mutually divergent motor trajectories. These results highlight the nonlinear nature of social coordination, offering new avenues for modeling creative joint action in human systems.

As intelligent systems are developed across diverse substrates - from machine learning models and neuromorphic hardware to in vitro neural cultures - understanding what gives a system agency has become increasingly important. Existing definitions, however, tend to rely on top-down descriptions that are difficult to quantify. We propose a bottom-up framework grounded in a system's information-processing order: the extent to which its transformation of input evolves over time. We identify three orders of information processing. Class I systems are reactive and memoryless, mapping inputs directly to outputs. Class II systems incorporate internal states that provide memory but follow fixed transformation rules. Class III systems are adaptive; their transformation rules themselves change as a function of prior activity. While not sufficient on their own, these dynamics represent necessary informational conditions for genuine agency. This hierarchy offers a measurable, substrate-independent way to identify the informational precursors of agency. We illustrate the framework with neurophysiological and computational examples, including thermostats and receptor-like memristors, and discuss its implications for the ethical and functional evaluation of systems that may exhibit agency.

Fitness consequence of dispersal depends on property of the entire landscape, which patches are available and what are the cost of moving. These are information that are not available locally when an organism make the decision to disperse. This poses a problem to the organism, where it is unclear how an adaptive decision can be made. This also poses a problem to the scientist, since in order to study the adaptiveness of dispersal, we need information of the entire landscape. For theorist, this is through making a series of assumption about either the landscape or the organism, and for empiricists, this means a large amount of measurements needs to be made across a large area. In this paper, we propose a link between local demographic process, which an organism can have access to, to the fitness consequence of dispersal. This meant local environmental cue can be used for the decision on dispersal, and hence allow the evolution of plastic dispersal strategy. We will then show that using this approach, evolution of dispersal on complex landscape can be modelled with relative ease, and to show that accidental dispersal in one patch can drive the evolution of adaptive dispersal in another.

Cooperation in large groups and one-shot interactions is often hindered by freeloading. Punishment can enforce cooperation, but it is usually regarded as wasteful because the costs of punishing offset its benefits. Here, we analyze an evolutionary game model that integrates upstream and downstream reciprocity with costly punishment: integrated strong reciprocity (ISR). We demonstrate that ISR admits a stable mixed equilibrium of ISR and unconditional defection (ALLD), and that costly punishment can become productive: When sufficiently efficient, it raises collective welfare above the no-punishment baseline. ALLD players persist as evolutionary shields, preventing invasion by unconditional cooperation (ALLC) or alternative conditional strategies (e.g., antisocial punishment). At the same time, the mixed equilibrium of ISR and ALLD remains robust under modest complexity costs that destabilize other symmetric cooperative systems.

We present a data-driven framework to characterize large-scale brain dynamical states directly from correlation matrices at the single-subject level. By treating correlation thresholding as a percolation-like probe of connectivity, the approach tracks multiple cluster- and network-level observables and identifies a characteristic percolation threshold, rc, at which these signatures converge. We use $r_c$ as an operational and physically interpretable descriptor of large-scale brain dynamical state. Applied to resting-state fMRI data from a large cohort of healthy individuals (N = 996), the method yields stable, subject-specific estimates that covary systematically with established dynamical indicators such as temporal autocorrelations. Numerical simulations of a whole-brain model with a known critical regime further show that $r_c$ tracks changes in collective dynamics under controlled variations of excitability. By replacing arbitrary threshold selection with a criterion intrinsic to correlation structure, the r-spectra provides a physically grounded approach for comparing brain dynamical states across individuals.

Phage display is a powerful laboratory technique used to study the interactions between proteins and other molecules, whether other proteins, peptides, DNA or RNA. The under-utilisation of this data in conjunction with deep learning models for protein design may be attributed to; high experimental noise levels; the complex nature of data pre-processing; and difficulty interpreting these experimental results. In this work, we propose a novel approach utilising a Bayesian Neural Network within a training loop, in order to simulate the phage display experiment and its associated noise. Our goal is to investigate how understanding the experimental noise and model uncertainty can enable the reliable application of such models to reliably interpret phage display experiments. We validate our approach using actual binding affinity measurements instead of relying solely on proxy values derived from 'held-out' phage display rounds.

A hallmark of aging is loss of information in gene regulatory networks. These networks are tightly connected, raising the question of whether information could be restored by perturbing single genes. We develop a simple theoretical framework for information transmission in gene regulatory networks that describes the information gained or lost when a gene is "knocked in" (exogenously expressed). Applying the framework to gene expression data from muscle cells in young and old mice, we find that single knock-ins can restore network information by up to 10%. Our work advances the study of information flow in networks and identifies potential gene targets for rejuvenation.

Efficient navigation in swarms often relies on the emergence of decentralized approaches that minimize traversal time or energy. Stigmergy, where agents modify a shared environment that then modifies their behavior, is a classic mechanism that can encode this strategy. We develop a theoretical framework for stigmergic transport by casting it as a stochastic optimal control problem: agents (collectively) lay and (individually) follow trails while minimizing expected traversal time. Simulations and analysis reveal two emergent behaviors: path straightening in homogeneous environments and path refraction at material interfaces, both consistent with experimental observations of insect trails. While reminiscent of Fermat's principle, our results show how local, noisy agent+field interactions can give rise to geodesic trajectories in heterogeneous environments, without centralized coordination or global knowledge, relying instead on an embodied slow fast dynamical mechanism.

Differential gene expression (DGE) analysis is foundational to transcriptomic research, yet tool selection can substantially influence results. This study presents a comprehensive comparison of two widely used DGE tools, edgeR and DESeq2, using real and semi-simulated bulk RNA-Seq datasets spanning viral, bacterial, and fibrotic conditions. We evaluated tool performance across three key dimensions: (1) sensitivity to sample size and robustness to outliers; (2) classification performance of uniquely identified gene sets within the discovery dataset; and (3) generalizability of tool-specific gene sets across independent studies. First, both tools showed similar responses to simulated outliers, with Jaccard similarity between the DEG sets from perturbed and original (unperturbed) data decreasing as more outliers were added. Second, classification models trained on tool-specific genes showed that edgeR achieved higher F1 scores in 9 of 13 contrasts and more frequently reached perfect or near-perfect precision. Dolan-More performance profiles further indicated that edgeR maintained performance closer to optimal across a greater proportion of datasets. Third, in cross-study validation using four independent SARS-CoV-2 datasets, gene sets uniquely identified by edgeR yielded higher AUC, precision, and recall in classifying samples from held-out datasets. This pattern was consistent across folds, with some test cases achieving perfect separation using edgeR-specific genes. In contrast, DESeq2-specific genes showed lower and more variable performance across studies. Overall, our findings highlight that while DESeq2 may identify more DEGs even under stringent significance conditions, edgeR yields more robust and generalizable gene sets for downstream classification and cross-study replication, which underscores key trade-offs in tool selection for transcriptomic analyses.

Personal health analytics systems face a persistent cold-start dilemma: users expect meaningful insights early in data collection, while conventional statistical inference requires data volumes that often exceed engagement horizons. Existing approaches either delay inference until fixed statistical thresholds are met -- leading to user disengagement -- or surface heuristic insights without formal uncertainty quantification, risking false confidence. We propose a progressive Bayesian confidence architecture that formalizes early-stage inference through phased interpretation of posterior uncertainty. Drawing on Bayesian updating and epistemic strategies from financial risk modeling under sparse observations, we map posterior contraction to interpretable tiers of insight, ranging from exploratory directional evidence to robust associative inference. We demonstrate the framework's performance through controlled experimentation with synthetic N-of-1 health data, showing that calibrated early insights can be generated within 5--7 days while maintaining explicit epistemic humility. Compared to fixed-threshold baselines requiring 30+ days of data, the proposed approach yields earlier directional signals (mean: 5.3 vs 31.7 days, p<0.001) while controlling false discovery rates below 6% (5.9% at day 30) despite 26-day earlier detection, compared to 0% FDR for fixed-threshold baselines that delay insights by 30 days. In addition, we show strong uncertainty calibration (76% credible interval coverage for ground-truth correlations at day 90). This work contributes a methodological framework for uncertainty-aware early inference in personalized health analytics that bridges the gap between user engagement requirements and statistical rigor.

The Feller diffusion is studied as the limit of a coalescent point process in which the density of the node height distribution is skewed towards zero. Using a unified approach, a number of recent results pertaining to scaling limits of branching processes are reinterpreted as properties of the Feller diffusion arising from this limit. The notion of Bernoulli sampling of a finite population is extended to the diffusion limit to cover finite Poisson-distributed samples drawn from infinite continuum populations. We show that the coalescent tree of a Poisson-sampled Feller diffusion corresponds to a coalescent point process with a node height distribution taking the same algebraic form as that of a Bernoulli-sampled birth-death process. By adapting methods for analysing k-sampled birth-death processes, in which the sample size is pre-specified, we develop methods for studying the coalescent properties of the k-sampled Feller diffusion.

Biomolecular condensates govern essential cellular processes yet elude description by traditional equilibrium models. This roadmap, distilled from structured discussions at a workshop and reflecting the consensus of its participants, clarifies key concepts for researchers, funding bodies, and journals. After unifying terminology that often separates disciplines, we outline the core physics of condensate formation, review their biological roles, and identify outstanding challenges in nonequilibrium theory, multiscale simulation, and quantitative in-cell measurements. We close with a forward-looking outlook to guide coordinated efforts toward predictive, experimentally anchored understanding and control of biomolecular condensates.

The trade-off between predictive accuracy and data availability makes it difficult to predict protein--protein binding affinity accurately. The lack of experimentally resolved protein structures limits the performance of structure-based machine learning models, which generally outperform sequence-based methods. In order to overcome this constraint, we suggest a regression framework based on knowledge distillation that uses protein structural data during training and only needs sequence data during inference. The suggested method uses binding affinity labels and intermediate feature representations to jointly supervise the training of a sequence-based student network under the guidance of a structure-informed teacher network. Leave-One-Complex-Out (LOCO) cross-validation was used to assess the framework on a non-redundant protein--protein binding affinity benchmark dataset. A maximum Pearson correlation coefficient (P_r) of 0.375 and an RMSE of 2.712 kcal/mol were obtained by sequence-only baseline models, whereas a P_r of 0.512 and an RMSE of 2.445 kcal/mol were obtained by structure-based models. With a P_r of 0.481 and an RMSE of 2.488 kcal/mol, the distillation-based student model greatly enhanced sequence-only performance. Improved agreement and decreased bias were further confirmed by thorough error analyses. With the potential to close the performance gap between sequence-based and structure-based models as larger datasets become available, these findings show that knowledge distillation is an efficient method for transferring structural knowledge to sequence-based predictors. The source code for running inference with the proposed distillation-based binding affinity predictor can be accessed at this https URL.

Many biological networks grow by elongation of filaments that can branch and fuse -- typical examples include fungal mycelium or slime mold. These networks must simultaneously perform multiple tasks such as transport, exploration, and robustness under finite resources. Yet, how such multi-task architectures emerge from local growth processes remains poorly understood. Here, we introduce a minimal model of spatial network morphogenesis based solely on stochastic branching, fusion, and stopping, during elongation. Despite the absence of global optimization or feedback, the model generates a broad morphospace from tree-like, to loopy, as well as hybrid architectures. By quantifying multiple functional objectives, we show that (i) these synthetic structures occupy similar regions of performance space than evolved empirical fungal networks, and (ii) that their Pareto front of optimal trade-offs lies close to that of these same fungal networks. Our results show that biological architectures approaching multi-objective optimality can arise from simple local growth rules, and identify branching and fusion as fundamental ingredients shaping the architecture of living transport networks.

Machine-learning datasets are typically characterized by measuring their size and class balance. However, there exists a richer and potentially more useful set of measures, termed S-entropy (similarity-sensitive entropy), that incorporate elements' frequencies and between-element similarities. Although these have been available in the R and Julia programming languages for other applications, they have not been as readily available in Python, which is widely used for machine learning, and are not easily applied to machine-learning-sized datasets without special coding considerations. To address these issues, we developed $\textit{sentropy}$, a Python package that calculates S-entropy and is tailored to large datasets. $\textit{sentropy}$ can calculate any of the frequency-sensitive measures of Hill's D-number framework and their similarity-sensitive counterparts. $\textit{sentropy}$ also outputs measures that compare datasets. We first briefly review S-entropy, illustrating how it incorporates elements' frequencies and elements' pairwise similarities. We then describe $\textit{sentropy}$'s key features and usage. We end with several examples - immunomics, metagenomics, computational pathology, and medical imaging - illustrating $\textit{sentropy}$'s applicability across a range of dataset types and fields.

A goal of computational studies of protein-protein interfaces (PPIs) is to predict the binding site between two monomers that form a heterodimer. The simplest version of this problem is to rigidly re-dock the bound forms of the monomers, which involves generating computational models of the heterodimer and then scoring them to determine the most native-like models. Scoring functions have been assessed previously using rank- and classification-based metrics, however, these methods are sensitive to the number and quality of models in the scoring function training set. We assess the accuracy of seven PPI scoring functions by comparing their scores to a measure of structural similarity to the x-ray crystal structure (i.e. the DockQ score) for a non-redundant set of heterodimers from the Protein Data Bank. For each heterodimer, we generate re-docked models uniformly sampled over DockQ and calculate the Spearman correlation between the PPI scores and DockQ. For some targets, the scores and DockQ are highly correlated; however, for many targets, there are weak correlations. Several physical features can explain the difference between difficult- and easy-to-score targets. For example, strong correlations exist between the score and DockQ for targets with highly intertwined monomers and many interface contacts. We also develop a new score based on only three physical features that matches or exceeds the performance of current PPI scoring functions. These results emphasize that PPI prediction can be improved by focusing on correlations between the PPI score and DockQ and incorporating more discriminating physical features into PPI scoring functions.

In recent years, the rapid advancement of large language models (LLMs) in natural language processing has sparked significant interest among researchers to understand their mechanisms and functional characteristics. Although prior studies have attempted to explain LLM functionalities by identifying and interpreting specific neurons, these efforts mostly focus on individual neuron contributions, neglecting the fact that human brain functions are realized through intricate interaction networks. Inspired by research on functional brain networks (FBNs) in the field of neuroscience, we utilize similar methodologies estabilished in FBN analysis to explore the "functional networks" within LLMs in this study. Experimental results highlight that, much like the human brain, LLMs exhibit certain functional networks that recur frequently during their operation. Further investigation reveals that these functional networks are indispensable for LLM performance. Inhibiting key functional networks severely impairs the model's capabilities. Conversely, amplifying the activity of neurons within these networks can enhance either the model's overall performance or its performance on specific tasks. This suggests that these functional networks are strongly associated with either specific tasks or the overall performance of the LLM. Code is available at this https URL.

Chemical reaction networks underpin biological and physical phenomena across scales, from microbial interactions to planetary atmosphere dynamics. Bacterial communities exhibit complex competitive interactions for resources, human organs and tissues demonstrate specialized biochemical functions, and planetary atmospheres can display diverse organic and inorganic chemical processes. Despite their complexities, comparing these networks methodically remains a challenge due to the vast underlying degrees of freedom. In biological systems, comparative genomics has been pivotal in tracing evolutionary trajectories and classifying organisms via DNA sequences. However, purely genomic classifications often fail to capture functional roles within ecological systems. Metabolic changes driven by nutrient availability highlight the need for classification schemes that integrate metabolic information. Here we introduce and apply a computational framework for a classification scheme of organisms that compares matrix representations of chemical reaction networks using the Grassmann distance, corresponding to measuring distances between the nullspaces of stoichiometric matrices. Applying this framework to human gut microbiome data confirms that metabolic distances are distinct from phylogenetic distances, underscoring the limitations of genetic information in metabolic classification. Importantly, our analysis of metabolic distances reveals functional groups of organisms enriched or depleted in specific metabolic processes and shows robustness to metabolically silent genetic perturbations. The generalizability of metabolic Grassmann distances is illustrated by application to chemical reaction networks in human tissue and planetary atmospheres, highlighting its potential for advancing functional comparisons across diverse chemical reaction systems.

Recently, the Less In More Out device, a fluidically actuated soft total artificial heart was proposed. This device uses arrays of pouch motors to achieve a positive fluidic lever when pneumatically actuated against physiological hemodynamic conditions. Extensive experimental characterization demonstrated its potential; however, experiments alone cannot resolve the internal mechanical fields that govern device durability and performance. Here, we develop a computational framework to investigate intrinsic device mechanics, such as stress concentrations, strain paths, and fatigue life, and to explore targeted design modifications that improve durability and efficiency. We show that our model reproduces the nonlinear deformation and pressure-volume relationships measured experimentally under varying hemodynamic conditions. Across designs, devices with fewer pouches deliver higher stroke volumes but exhibit up to 50% higher peak von Mises stresses, which reduces their fatigue life. Our simulations further identify heat-sealed seams and buckling regions as durability-limiting features. As a proof of concept, we vary the valve support aspect ratio and relative endocardial-epicardial pouch fabric compliance, reducing the peak von Mises stress by ~10% while maintaining identical physiological outputs and improving mechanical efficiency. Overall, our framework enables detailed evaluation of stress hotspots, buckling, and fatigue life, and offers a foundation for optimizing artificial hearts and other fluidically actuated fabric-based soft robotic devices.

Allostery is a fundamental mechanism of protein regulation and is commonly interpreted as modulating enzymatic activity or product abundance. Here we show that this view is incomplete. Using a stochastic model of allosteric regulation combined with an information-theoretic analysis, we quantify the mutual information between an enzyme's regulatory state and the states of downstream signaling components. Beyond controlling steady-state production levels, allostery also regulates the timing and duration over which information is transmitted. By tuning the temporal operating regime of signaling pathways, allosteric regulation enables distinct dynamical outcomes from identical molecular components, providing a physical mechanism for temporal information flow, signaling specificity, and coordination without changes in metabolic pathways.

Computing observables from conditioned dynamics is typically computationally hard, because, although obtaining independent samples efficiently from the unconditioned dynamics is usually feasible, generally most of the samples must be discarded (in a form of importance sampling) because they do not satisfy the imposed conditions. Sampling directly from the conditioned distribution is non-trivial, as conditioning breaks the causal properties of the dynamics which ultimately renders the sampling procedure efficient. One standard way of achieving it is through a Metropolis Monte-Carlo procedure, but this procedure is normally slow and a very large number of Monte-Carlo steps is needed to obtain a small number of statistically independent samples. In this work, we propose an alternative method to produce independent samples from a conditioned distribution. The method learns the parameters of a generalized dynamical model that optimally describe the conditioned distribution in a variational sense. The outcome is an effective, unconditioned, dynamical model, from which one can trivially obtain independent samples, effectively restoring causality of the conditioned distribution. The consequences are twofold: on the one hand, it allows us to efficiently compute observables from the conditioned dynamics by simply averaging over independent samples. On the other hand, the method gives an effective unconditioned distribution which is easier to interpret. The method is flexible and can be applied virtually to any dynamics. We discuss an important application of the method, namely the problem of epidemic risk assessment from (imperfect) clinical tests, for a large family of time-continuous epidemic models endowed with a Gillespie-like sampler. We show that the method compares favorably against the state of the art, including the soft-margin approach and mean-field methods.

Neuroimaging provides essential tools for characterizing brain activity by quantifying connectivity strength between remote regions, using different modalities that capture different aspects of connectivity. Yet, decoding meaningful neural signatures must contend with modality-specific challenges, including measurement noise, spatial and temporal distortions, heterogeneous acquisition protocols, and limited sample sizes. A unifying perspective emerges when these data are expressed through symmetric positive definite (SPD)-valued representations: across neuroimaging modalities, SPD-valued representations naturally give rise to SPD matrices that capture dependencies between sensors or brain regions. Endowing the SPD space with Riemannian metrics equips it with a non-Euclidean geometric structure, enabling principled statistical modeling and machine learning on the resulting manifold. This review consolidates machine learning methodologies that operate on the SPD manifold under a unified framework termed SPD matrix learning. SPD matrix learning brings conceptual clarity across multiple modalities, establishes continuity with decades of geometric statistics in neuroimaging, and positions SPD modeling as a methodological bridge between classical analysis and emerging AI-driven paradigms. We show that (i) modeling on the SPD manifold is mathematically natural and numerically stable, preserving symmetry and positive definiteness while avoiding degeneracies inherent to Euclidean embeddings; (ii) SPD matrix learning extends a broad family of established geometric statistical tools used across neuroimaging; and (iii) SPD matrix learning integrates new-generation AI technologies, driving a new class of neuroimaging problems that were previously out of reach. Taken together, SPD matrix learning offers a principled and forward-looking framework for next-generation neuroimaging analytics.