We revisit the controversial "abscopal" effect in the context of Personalized Ultra-Fractionated Stereotactic Adaptive Radiotherapy (PULSAR). By allowing long interval between fractions, PULSAR may enhance systemic immune activation and increase the likelihood of abscopal responses compared with conventional daily fractionation. To quantify treatment-induced effects, we introduce an interaction-picture transformation adapted from quantum mechanics, which separates intrinsic tumor growth from radiation and immune-mediated perturbations. In this preliminary study, we tested this method to two preclinical bilateral tumor models (4T1 and MC38). Our model provides a quantitative measure of the interaction strength between primary and secondary tumors at the individual level, capturing dynamics over time rather than relying solely on cohort averages. This approach frames the abscopal effect as a continuous, stochastic phenomenon rather than a binary response. The framework is flexible for future studies, particularly in concurrent radiation and immunotherapy with PULSAR, where different radiation doses and fractionation schedules can be compared, and immune checkpoint inhibitors (ICIs) can be incorporated to further enhance systemic anti-tumor immunity. The framework can also help us make cross-study comparison of abscopal effects and standardizes the reporting of abscopal magnitude beyond simple statistical significance.

Mass-action networks are special cases of chemical reaction networks. For these systems, we argue that conserved quantities are dual to internal cycles. We introduce maximal invariant polyhedral supports, and we conjecture that there is a duality relation between preclusters and maximal invariant polyhedral supports. Given the close relation between maximal invariant polyhedral supports and siphons, we also conjecture that siphons and preclusters are dual objects.

Phase-amplitude coupling (PAC), a form of cross-frequency interaction, has been implicated in various cognitive functions and, by extension, in neural communication and information integration. Accurately detecting and characterising PAC is essential for understanding its role in processes such as memory and attention. However, this remains a significant challenge. Most existing methods rely on variations in the temporal profile to detect PAC, but they often suffer from key limitations, most notably, their sensitivity to filter bandwidth selection and their susceptibility to detecting spurious couplings. Previous studies have suggested that approaches grounded in the actual generative dynamics of PAC may offer improved accuracy. In this study, we adopt a dynamical systems perspective and propose a novel method for PAC detection and characterisation based on nonlinear system identification. This approach involves identifying a nonlinear dynamical model that captures the temporal dynamics underlying PAC. The resulting generative model enables noise-free simulation of estimated PAC signals, facilitating detailed analysis of modulation strength and the low-frequency phase at which the high-frequency bursts occur. The proposed method accounts for harmonic-induced spurious couplings through empirically derived criteria and remains robust to high noise levels and variations in slow-frequency power, offering an accurate and interpretable framework for PAC analysis. The performance of the proposed approach is illustrated using several simulated examples and a real case using local field potentials (LFP) data. The results are compared with several popular methods.

Classical reaction-diffusion models of the 14th-century Black Death fail to explain the rapid genetic radiation of \textit{Yersinia pestis} and the anomalous emergence of vast, untouched geographic safe zones, such as Central Europe. In this work, we resolve these historical anomalies by embedding macroscopic pathogen dynamics within a non-Abelian gauge theory. Utilizing the Doi-Peliti formalism, we map the stochastic master equation of a multi-strain epidemic into a covariant classical field theory. We introduce an $SU(N)$ environmental gauge field, $\mathbf{A}_\mu$, which actively couples geographic displacement to phenotypic mutation, treating evolutionary drift as a spatial transport phenomenon. We demonstrate via linear stability analysis that this covariant advection drives a Differential Flow (Turing-Hopf) instability, spontaneously breaking spatial symmetry to generate traveling waves of mutation. Furthermore, by extending the pathogen multiplet to the large-$N$ ('t Hooft) continuum limit, we prove that historical safe zones are not statistical outliers nor the result of perfect quarantine, but are mathematically necessary topological voids. In this continuous limit, the destructive interference of the mutating wavefronts analytically resolves into a stable, isotropic macroscopic node governed by a zeroth-order Bessel function ($J_0$), precisely mapping onto the historical survival of Poland and Bohemia.

Homeostasis is widely observed in biological systems and refers to their ability to maintain an output quantity approximately constant despite variations in external disturbances. Mathematically, homeostasis can be formulated through an input-output function mapping an external parameter to an output variable. Infinitesimal homeostasis occurs at isolated points where the derivative of this input-output function vanishes, allowing tools from singularity theory and combinatorial matrix theory to characterize homeostatic mechanisms in terms of network topology. However, the required combinatorial enumeration becomes increasingly intractable as network size grows, and the reliance on advanced graph-theoretic concepts limits accessibility and practical use in biological applications. To overcome these limitations, we develop a Python-based algorithm that automates the identification of homeostasis subnetworks and their associated homeostasis conditions directly from network topology. Given an input-output network specified solely by its connectivity structure and designated input and output nodes, the algorithm identifies the relevant graph-theoretical structures and enumerates all homeostatic mechanisms. We demonstrate its applicability across a range of biological examples, including small and large networks, networks with single or multiple input nodes or parameters, and cases where input and output coincide. This wide applicability stems from our extension of the theoretical framework from single-input-single-output networks to networks with multiple input nodes through an augmented single-input-node representation. The resulting computational framework provides a scalable and systematic approach to classifying homeostatic mechanisms in complex biological networks, facilitating the application of advanced mathematical theory to a broad range of biological systems.

Biological learning achieves temporal credit assignment despite sparse and imprecise feedback, often relying on neuromodulatory signals acting over space and time. Here, we introduce a learning mechanism in which error information diffuses locally through the network, similar to volume transmission of neuromodulators. This distributed modulation allows neurons to learn even in the absence of direct feedback, using the local concentration of the diffusing credit signal. Applied to recurrent spiking neural networks with sparse feedback connectivity, diffusive credit signaling improves learning across three benchmark tasks. Using eligibility propagation as a baseline learning mechanism, we show how diffusion-based modulation can provide a plausible mechanism for credit assignment in sparsely connected neural circuits.

We introduce a filtration-based framework for studying when and why adding taxa improves phylodynamic inference, by constructing a natural ordering of observed tips and applying sequential Bayesian analysis to the resulting filtration. We decompose the expected variance reduction on taxa addition into learning, mismatch, and covariance components, classify estimands into learning classes based on the pathwise behaviour of the mismatch, and show that for absorbing estimands an oracle who knows the latent absorption status obtains event-wise learning guarantees unavailable to the analyst. The gap between oracle and analyst is irreducible assumptions that are likely to hold for many real phylodynamic estimands, establishing a fundamental limit on what sequence data alone can reveal about the latent genealogy.

We present a compact dynamical mean-field theory (DMFT) for large networks of coupled phase oscillators whose phases live on the circle $S^1$ and interact with both coherent mean-field coupling and quenched randomness. Starting from wrapped Langevin dynamics, we build a path-integral representation that keeps the $2\pi$-periodicity of the phases explicit. After averaging over the disorder in the thermodynamic limit, this construction reduces to a single-oscillator stochastic equation driven by a deterministic mean field and a self-consistent colored Gaussian noise, whose covariance is fixed by a circular two-time correlator. In the limit of vanishing disorder, the formalism reproduces the Ott--Antonsen reduction and recovers standard Kuramoto and theta-neuron neural-mass equations. The same framework accommodates arbitrary $2\pi$-periodic coupling functions, including those obtained from infinitesimal phase response curves (iPRCs) of biophysical neuron models. As an example, we show that for adaptive exponential integrate-and-fire neurons, inserting an iPRC-fitted coupling into the compact DMFT yields quantitative predictions for synchronization thresholds, providing a direct route from single-neuron phase response data to network-level mean-field predictions for arbitrary phase-reducible oscillators.

Due to its ability to summarise 'real-time' epidemic behaviour, the time-dependent reproduction number, Rt, is a useful metric for tracking pathogen transmission and quantifying the effects of interventions during infectious disease outbreaks. The predominant models underlying inferred Rt trajectories are renewal equations, their success owing in part to the relatively few assumptions they require. One necessary assumption is the generation time distribution, which summarises the time periods between infections in infector-infectee transmission pairs. This distribution is typically assumed to be the same across all members of a population. In reality, however, it may vary systematically between population groups. In this study, we consider two Rt inference frameworks based on renewal equation models: one for a single, homogeneous group and another accounting for a structured population. We compare the estimates of Rt generated by the two models and investigate, both analytically and through simulations, under which conditions the conclusions drawn from these modelling paradigms differ. We also demonstrate a methodology for selecting the generation time for the one-group model that correctly encapsulates variations between different population groups; this allows us to use a renewal framework for a one-group model to infer Rt when, in fact, the population is structured. Finally, we use real epidemic data to demonstrate that practical Rt estimates can differ depending on whether the underlying model is the one-group model or the multi-group model. Our results motivate the need for rigorous collection of detailed epidemic data and consideration of differences between population groups to improve the accuracy of Rt estimates that are used to guide public health policy responses.

Background: Pleuroparenchymal fibroelastosis (PPFE) is an upper lobe predominant fibrotic lung abnormality associated with increased mortality in established interstitial lung disease. However, the clinical significance of radiologic PPFE progression in lung cancer screening populations remains unclear. We investigated whether longitudinal change in PPFE quantified on low dose CT independently associates with mortality and respiratory morbidity. Methods: We analysed longitudinal low-dose CT scans and clinical data from two lung cancer screening studies: the National Lung Screening Trial (NLST; n=7980) and the SUMMIT study (n=8561). An automated algorithm quantified PPFE volume on baseline and follow up scans. Annualised change in PPFE (dPPFE) was derived and dichotomised using a distribution based threshold to define progressive PPFE. Associations between dPPFE and mortality were evaluated using Cox proportional hazards models adjusted for demographic and clinical variables. In the SUMMIT cohort, dPPFE was also examined in relation to clinical outcomes. Findings: dPPFE independently associated with mortality in both cohorts (NLST: HR 1.25, 95% CI 1.01-1.56, p=0.042; SUMMIT: HR 3.14, 95% CI 1.66-5.97, p<0.001). Kaplan-Meier curves showed reduced survival among participants with progressive PPFE in both cohorts. In SUMMIT, dPPFE was associated with higher respiratory admissions (IRR 2.79, p<0.001), increased antibiotic and steroid use (IRR 1.55, p=0.010), and a trend towards higher mMRC scores (OR 1.40, p=0.055). Interpretation: Radiologic PPFE progression independently associates with mortality across two large lung cancer screening cohorts and with adverse clinical outcomes. Quantitative assessment of PPFE progression may provide a clinically relevant imaging biomarker for identifying individuals at increased respiratory risk within screening programmes.

Brains remain unrivaled in their ability to recognize and generate complex spatiotemporal patterns. While AI is able to reproduce some of these capabilities, deep learning algorithms remain largely at odds with our current understanding of brain circuitry and dynamics. This is prominently the case for backpropagation through time (BPTT), the go-to algorithm for learning complex temporal dependencies. In this work we propose a general formalism to approximate BPTT in a controlled, biologically plausible manner. Our approach builds on, unifies and extends several previous approaches to local, time-continuous, phase-free spatiotemporal credit assignment based on principles of energy conservation and extremal action. Our starting point is a prospective energy function of neuronal states, from which we calculate real-time error dynamics for time-continuous neuronal networks. In the general case, this provides a simple and straightforward derivation of the adjoint method result for neuronal networks, the time-continuous equivalent to BPTT. With a few modifications, we can turn this into a fully local (in space and time) set of equations for neuron and synapse dynamics. Our theory provides a rigorous framework for spatiotemporal deep learning in the brain, while simultaneously suggesting a blueprint for physical circuits capable of carrying out these computations. These results reframe and extend the recently proposed Generalized Latent Equilibrium (GLE) model.

Animal brains exhibit remarkable efficiency in perception and action, while being robust to both external and internal perturbations. The means by which brains accomplish this remains, for now, poorly understood, hindering our understanding of animal and human cognition, as well as our own implementation of efficient algorithms for control of dynamical systems.A potential candidate for a robust mechanism of state estimation and action computation is the free energy principle, but existing implementations of this principle have largely relied on conventional, biologically implausible approaches without spikes. We propose a novel, efficient, and robust spiking control framework with realistic biological characteristics. The resulting networks function as free energy constrainers, in which neurons only fire if they reduce the free energy of their internal representation. The networks offer efficient operation through highly sparse activity while matching performance with other similar spiking frameworks, and have high resilience against both external (e.g. sensory noise or collisions) and internal perturbations (e.g. synaptic noise and delays or neuron silencing) that such a network would be faced with when deployed by either an organism or an engineer. Overall, our work provides a novel mathematical account for spiking control through constraining free energy, providing both better insight into how brain networks might leverage their spiking substrate and a new route for implementing efficient control algorithms in neuromorphic hardware.

We present a mathematical model of the curvature blindness illusion in which sinusoids appear as angular zigzags when drawn with alternating contrast polarity against a gray background. The model identifies two complementary mechanisms, both operating in V1. First, polarity channel separation: simple cells are selective for contrast polarity, and lateral connections link only same polarity neurons; where the line switches from darker than background to lighter than background at each peak and trough, the encoding population changes and the lateral chain is broken, segmenting the contour into half-wavelength pieces. Second, orientation channel fragmentation: at moderate contrast, the active orientation window is narrow, and within each half-wavelength segment no single orientation channel spans the full range of edge normals; the inflection point at the center of each segment anchors a locally straight percept. Together, the two mechanisms produce a zigzag: polarity breaks supply the corners, and fragmentation straightens the segments between them.

Protein function is executed at the molecular surface, where shape and chemistry act together to govern interaction. Yet most comparison methods treat these aspects separately, privileging either global fold or local descriptors and missing their coupled organization. Here we introduce IFACE (Intrinsic Field-Aligned Coupled Embedding), a correspondence-based framework that aligns protein surfaces through probabilistic coupling of intrinsic geometry with spatially distributed chemical fields. From this alignment, we derive a joint geometric--chemical distance that integrates structural and physicochemical discrepancies within a single formulation. Across diverse proteins, this distance separates conformational variability from true structural divergence more effectively than fold-based similarity measures. Applied to the cytochrome P450 family, it reveals coherent family-level organization and identifies conserved buried catalytic pockets despite the complex topology. By linking interpretable surface correspondences with a unified distance, IFACE establishes a principled basis for comparing protein interfaces and detecting functionally related interaction patches across proteins.

Under-ice blooms in the Arctic can develop rapidly under conditions where conventional early warning signals based on critical slowing down fail due to strong noise or limited observational records. We analyze noise-induced transitions in a temperature phytoplankton stochastic differential equation model exhibiting bistability between background and bloom states. The committor function defines a stochastic separatrix as its 1/2-isocommittor, and the normal width of the associated transition layer yields a geometric indicator via arc-length averaging. Under systematic variation of noise intensity, this indicator scales linearly with noise strength, while the logarithm of the mean first passage time follows the Freidlin-Wentzell asymptotic law. Eliminating the noise parameter produces an affine scaling between the logarithmic transition time and the inverse square of the geometric indicator. The relation is robust under variations in discretization, neighborhood definition, and diffusion structure, and holds in the weak noise regime where the transition-layer width scales linearly with noise strength. Unlike variance or lag-one autocorrelation, the geometric indicator remains well defined when rapid transitions preclude reliable time-series estimation. These results provide a geometrically interpretable precursor of bloom onset that may support model-based ecological monitoring in high-variability Arctic systems.

Genomic language models (GLMs) have emerged as powerful tools for learning representations of DNA sequences, enabling advances in variant prediction, regulatory element identification, and cross-task transfer learning. However, as these models are increasingly trained or fine-tuned on sensitive genomic cohorts, they risk memorizing specific sequences from their training data, raising serious concerns around privacy, data leakage, and regulatory compliance. Despite growing awareness of memorization risks in general-purpose language models, little systematic evaluation exists for these risks in the genomic domain, where data exhibit unique properties such as a fixed nucleotide alphabet, strong biological structure, and individual identifiability. We present a comprehensive, multi-vector privacy evaluation framework designed to quantify memorization risks in GLMs. Our approach integrates three complementary risk assessment methodologies: perplexity-based detection, canary sequence extraction, and membership inference. These are combined into a unified evaluation pipeline that produces a worst-case memorization risk score. To enable controlled evaluation, we plant canary sequences at varying repetition rates into both synthetic and real genomic datasets, allowing precise quantification of how repetition and training dynamics influence memorization. We evaluate our framework across multiple GLM architectures, examining the relationship between sequence repetition, model capacity, and memorization risk. Our results establish that GLMs exhibit measurable memorization and that the degree of memorization varies across architectures and training regimes. These findings reveal that no single attack vector captures the full scope of memorization risk, underscoring the need for multi-vector privacy auditing as a standard practice for genomic AI systems.

Phase-space features of a reduced version of the Toda-like Hamiltonian, $\mathcal{H}(x,\,k)$, written in a form constrained by the condition $\partial^2 \mathcal{H} / \partial x \partial k = 0$, with $x$ and $k$ as canonically conjugate variables, are analyzed in terms of Wigner currents. For Wigner currents convoluted with either thermodynamic or Gaussian ensembles, the underlying Hamiltonian dynamics admits analytic corrections due to quantum distortions over the classical phase-space pattern, computed and interpreted through quantifiers of quantumness and stationarity. Notably, while emulating the Lotka-Volterra (LV) dynamics that describe ecological competition systems, the Toda-like classical dynamics allows for analytical solutions with computable periods corresponding to closed phase-space orbits of isotropic prey-predator population distributions. The essential conditions for understanding how classical and quantum evolution can coexist are provided at different scales of quantumness, driven by the associated convoluting ensemble parameter. In the case of Gaussian statistical ensembles, the exact profile of the quantum distortions over classical prey-predator phase-space trajectories is obtained non-perturbatively. Our results indicate that, besides the classical stability admitted by LV models, the Toda-like patterns also exhibit quantum stability. Therefore, this can be regarded as the first step as a predictive theoretical framework towards more robust descriptions of quantum patterns in competitive microscopic biosystems.

Temporal point processes offer a powerful framework for sampling from discrete distributions, yet they remain underutilized in existing literature. We show how to construct, for any target multivariate count distribution with downward-closed support, a multivariate temporal point process whose event-count vector in a fixed-length sliding window converges in distribution to the target as time tends to infinity. Structured as a system of potentially coupled infinite-server queues with deterministic service times, the sampler exhibits a discrete form of momentum that suppresses random-walk behaviour. The admissible families of processes permit both reversible and non-reversible dynamics. As an application, we derive a recurrent stochastic neural network whose dynamics implement sampling-based computation and exhibit some biologically plausible features, including relative refractory periods and oscillatory dynamics. The introduction of auxiliary randomness reduces the sampler to a birth-death process, establishing the latter as a degenerate case with the same limiting distribution. In simulations on 63 target distributions, our sampler always outperforms these birth-death processes and frequently outperforms Zanella processes in multivariate effective sample size, with further gains when normalized by CPU time.

The Hopfield model provides a paradigmatic framework for associative memory. Its classical implementation, based on the Hebbian learning rule, suffers from catastrophic forgetting: when one attempts storing too many patterns, the network fails to retrieve any of them. Yet, the Hebbian rule does not take into account that synaptic strength is bounded. Introducing this biologically plausible modification, known as "clipping", eliminates catastrophic forgetting; the model is now able to retrieve the most recently seen memories, eliminating older ones. Yet, its memorization capacity is much reduced with respect to the unclipped case. Here, we investigate the effects of adding a "dreaming" phase on the capacity of a clipped Hopfield model. Following a proposal by Hopfield, Feinstein and Palmer, we assume that during the dreaming phase, the model generates random patterns that are then "unlearned". We show that while clipping still removes catastrophic forgetting, alternating learning and dreaming phases improves the memorization capacity and makes the search for optimal performance more realistic from an evolutionary perspective.

Understanding peptide properties is often assumed to require modeling long-range molecular interactions, motivating the use of complex graph neural networks and pretrained transformers. Yet, whether such long-range dependencies are essential remains unclear. We investigate if simple, domain-specific molecular fingerprints can capture peptide function without these assumptions. Atomic-level representation aims to provide richer information than purely sequence-based models and better efficiency than structural ones. Across 132 datasets, including LRGB and five other peptide benchmarks, models using count-based ECFP, Topological Torsion, and RDKit fingerprints with LightGBM achieve state-of-the-art accuracy. Despite encoding only short-range molecular features, these models outperform GNNs and transformer-based approaches. Control experiments with sequence shuffling and amino acid counts confirm that fingerprints, though inherently local, suffice for robust peptide property prediction. Our results challenge the presumed necessity of long-range interaction modeling and highlight molecular fingerprints as efficient, interpretable, and computationally lightweight alternatives for peptide prediction.

Chimeric antigen receptor (CAR) T cell therapy has shown remarkable success in hematological malignancies, yet patient responses remain highly variable and the roles of CD4+ and CD8+ subsets are not fully understood. We present an extended mathematical framework of CAR-T cell dynamics that explicitly models CD4+ helper and CD8+ cytotoxic lineages and their interactions with tumor antigen burden. Building on the Kirouac et al. (2023) model of antigen-regulated memory-effector-exhaustion transitions, our system of differential equations incorporates CD4-mediated modulation of CD8+ proliferation, cytotoxicity, and memory regeneration through biologically grounded, saturating interactions. Sensitivity analyses identify effector proliferation, antigen turnover, and CD8+ expansion rates as dominant drivers of treatment outcome. Virtual patient simulations recover reported qualitative trends in CAR-T composition, including enhanced expansion and tumor clearance for defined CD4:CD8 products relative to CD8-only formulations, while also revealing inter-patient variability and time-dependent effects. To assess the practical limits of patient-level prediction under parameter uncertainty, we introduce controlled noise into key parameters and show that direct mechanistic classification rapidly degrades. We then demonstrate that a simple feed-forward neural network can partially recover predictive signal from noisy inputs, outperforming a naive baseline while remaining consistent with mechanistic sensitivities. This work positions the extended model as a hypothesis generator, and illustrates how data-driven methods can complement mechanistic modeling when parameter uncertainty constrains predictive confidence.

Subjective cognitive decline (SCD) doubles dementia risk. This study investigates how self-perceived cognitive worsening manifests in neural dynamics during naturalistic speech perception. EEG was collected from 60 cognitively normal older adults while they listened to speech varied in prosodic contexts, categorized by expressive styles (scrambled, descriptive, dialogue, exciting). Encoding models mapping three speech representations -- acoustic, subsyllabic segmentation and phonotactic features -- to the ongoing EEG signals were built. Cortical tracking strength (CTS) showed that models fitted with linguistic features outperformed acoustic ones. Crucially, a greater degree of SCD was associated with weaker CTS of (1) higher-level linguistic but not acoustic features, and (2) prosodically flat speech (scrambled and descriptive). Thus, the CTS of higher-level linguistic features while listening to prosodically flat speech may serve as a potential biomarker for early-stage cognitive decline.

Virtually every biological rate depends on temperature, yet the resulting rate-temperature relationships often deviate strongly from simple Arrhenius behavior. In this first part of a two-part review, we survey phenomenological models used to describe biological temperature responses across scales, from enzymatic reactions to organismal performance. We discuss common functional forms, including symmetric and asymmetric thermal performance curves and extensions of the Arrhenius law, and we highlight how these models define operational quantities such as optimal temperatures, thermal breadths, and thermal limits. We also discuss microscopic models for the effect of temperature, which however do not capture cooperative effects. In Part II of this review, we will discuss how system-level temperature response curves emerge from the interaction of many underlying reactions.

Vertical federated learning (VFL) enables multi-laboratory collaboration on distributed multi-omics datasets without sharing raw data, but exhibits severe instability under extreme data scarcity (P >> N) when applied generically. Here, we investigate how domain-aware design choices; specifically gradient saliency-guided feature selection with biologically motivated priors; affect the stability, interpretability, and failure modes of VFL architectures in small-sample coral stress classification (N = 13 samples, P = 90,579 features across transcriptomics, proteomics, metabolomics, and microbiome data). We benchmark REEF (Robust Expert Encoder Federation), a domain-aware VFL framework, against two baselines on the Montipora capitata thermal stress dataset: (i) a standard NVFlare-based VFL and (ii) LASER, a state-of-the-art label-aware VFL method. REEF achieves an AUROC of 0.776 +/- 0.039 after reducing dimensionality by 98.6% (90,579 to 1,300 features), substantially outperforming NVFlare VFL at chance level (AUROC 0.500 +/- 0.125, p = 0.0106, Cohen's d = 2.265) and numerically exceeding LASER (AUROC 0.557 +/- 0.191, p = 0.0995, Cohen's d = 1.068), with 3-5-fold variance reduction. An equal-weights ablation confirms that biological priors specifically contribute stability: removing priors yields statistically indistinguishable mean AUROC (p = 0.405) but 2.3x higher variance (CV 0.110 vs 0.050). Negative control experiments using permuted labels produce AUROC near or below chance (0.357 for REEF, 0.238 for NVFlare), consistent with the absence of gross data leakage. These results motivate design principles for VFL in extreme P >> N regimes, emphasizing domain-informed dimensionality reduction, stability-focused evaluation, and interpretable feature selection for scarce biological data.

Krotov and Hopfield (2021) proposed a biologically plausible two-layer associative memory network with memory storage capacity exponential in the number of visible neurons. However, the capacity was only linear in the number of hidden neurons. This limitation arose from the choice of nonlinearity between the visible and hidden units, which enforced winner-take-all dynamics in the hidden layer, thereby restricting each hidden unit to encode only a single memory. We overcome this limitation by introducing a novel associative memory network with a threshold nonlinearity that enables distributed representations. In contrast to winner-take-all dynamics, where each hidden neuron is tied to an entire memory, our network allows hidden neurons to encode basic components shared across many memories. Consequently, complex patterns are represented through combinations of hidden neurons. These representations reduce redundancy and allow many correlated memories to be stored compositionally. Thus, we achieve much higher capacity: exponential in the number of hidden units, provided the number of visible units is sufficiently large relative to the number of hidden units. Exponential capacity arises because all binary states of the hidden units can become stable memory patterns. Moreover, the distributed hidden representation, which has much lower dimensionality than the visible layer, preserves class-discriminative structure, supporting efficient nonlinear decoding. These results establish a new regime for associative memory, enabling high-capacity, robust, and scalable architectures consistent with biological constraints.

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.

Neural populations exhibit latent dynamical structures that drive time-evolving spiking activities, motivating the search for models that capture both intrinsic network dynamics and external unobserved influences. In this work, we introduce LangevinFlow, a sequential Variational Auto-Encoder where the time evolution of latent variables is governed by the underdamped Langevin equation. Our approach incorporates physical priors -- such as inertia, damping, a learned potential function, and stochastic forces -- to represent both autonomous and non-autonomous processes in neural systems. Crucially, the potential function is parameterized as a network of locally coupled oscillators, biasing the model toward oscillatory and flow-like behaviors observed in biological neural populations. Our model features a recurrent encoder, a one-layer Transformer decoder, and Langevin dynamics in the latent space. Empirically, our method outperforms state-of-the-art baselines on synthetic neural populations generated by a Lorenz attractor, closely matching ground-truth firing rates. On the Neural Latents Benchmark (NLB), the model achieves superior held-out neuron likelihoods (bits per spike) and forward prediction accuracy across four challenging datasets. It also matches or surpasses alternative methods in decoding behavioral metrics such as hand velocity. Overall, this work introduces a flexible, physics-inspired, high-performing framework for modeling complex neural population dynamics and their unobserved influences.

Orientation-rich images, such as fingerprints and textures, often exhibit coherent angular directional patterns that are challenging to model using standard generative approaches based on isotropic Euclidean diffusion. Motivated by the role of phase synchronization in biological systems, we propose a score-based generative model built on periodic domains by leveraging stochastic Kuramoto dynamics in the diffusion process. In neural and physical systems, Kuramoto models capture synchronization phenomena across coupled oscillators -- a behavior that we re-purpose here as an inductive bias for structured image generation. In our framework, the forward process performs \textit{synchronization} among phase variables through globally or locally coupled oscillator interactions and attraction to a global reference phase, gradually collapsing the data into a low-entropy von Mises distribution. The reverse process then performs \textit{desynchronization}, generating diverse patterns by reversing the dynamics with a learned score function. This approach enables structured destruction during forward diffusion and a hierarchical generation process that progressively refines global coherence into fine-scale details. We implement wrapped Gaussian transition kernels and periodicity-aware networks to account for the circular geometry. Our method achieves competitive results on general image benchmarks and significantly improves generation quality on orientation-dense datasets like fingerprints and textures. Ultimately, this work demonstrates the promise of biologically inspired synchronization dynamics as structured priors in generative modeling.

Although commonly associated with limbless animals like snakes and fish, multi-legged organisms like centipedes also utilize undulatory locomotion. Whether these undulations are actively reinforced or resisted by the axial musculature remains an open question. We present a dynamical model of centipede locomotion that integrates leg-ground interactions, passive body mechanics, and active lateral musculature. By varying stepping rate, actuation, and body stiffness, we examine how locomotor strategies affect speed and an effective energetic efficiency. Coordination emerges only when body stiffness is tuned to stepping frequency: overly flexible bodies lose synchrony, while overly rigid ones move slowly and inefficiently. This leads to the prediction that centipedes utilize speed dependent active stiffness to maintain this coordination. Our results suggest that lateral muscles also have a speed dependent function, revealed by optimizing speed and an effective cost, that resists a phase lag between leg touchdowns and body curvature. Together, we find that centipedes actively modulate body mechanics to achieve rapid, efficient locomotion, highlighting how complex control can emerge from embodied physical properties rather than solely from neural computation.