Bulletin of Mathematical Biology

1Although resources are typically distributed continuously in space, species distributions often organize into discrete clusters. In his seminal paper [36], Turing demonstrated that such clusters can spontaneously arise in population densities, even when populations evolve in environments with continuously varying conditions. This phenomenon is known as Turing instability. In this work, we focus on two models grounded in population dynamics: a one-dimensional model based on the nonlocal Fisher-KPP equation, and a two-dimensional model involving an environmental gradient. We show that phenotypic clusters (sometimes referred to as "species") emerge in these models. We prove that they do not emerge because of Turing instability, but because of stochasticity, and that they disappear when stochasticity is reduced. First, for both models, we start our simulations with initial populations uniformly distributed in the state space. We show that phenotypic clusters quickly emerge and that the distances between them depend on the population size, that is, on the degree of stochasticity. Next, we start from already clearly defined phenotypic clusters. We identify three regimes in the connection between population size, the initial distances between clusters, and the distances between clusters at equilibrium. Last, on the two-dimensional model, we relax the hypothesis of complete clonality by varying the effective recombination rate, explore its effect on phenotypic clustering, and show that phenotypic clustering decays drastically with slight recombination.

Minimum viable populations (MVPs) are population levels large enough to surmount risk from demographic, environmental, and genetic stochasticity. MVPs are estimated by biologists to guide conservation practices. However, MVPs are generally estimated for a target population without regard for how they interact with intra- and inter-species population dynamics in the broader ecological community. Thus, how and why population dynamics interact with MVPs imposed by conservation biologists remain unclear. When MVPs are imposed on a continuous population model, traditional analyses fail to capture the range of possible outcomes those MVPs create. Here, we describe viability space decomposition (VSD) as a mathematical tool to systematically analyze the potential crossing of MVPs during population dynamics. We demonstrate that different extinction and survival outcomes can be recovered from a model with imposed MVPs using three VSD concepts in junction with a traditional phase portrait: mortality manifolds which separate conditions that lead to different existential outcomes, ordering manifolds which determine the order of extinction events for multiple populations, and collapse manifolds which determine the survival or extinction of one species given the loss of another. We employ these methods with a standard consumer-resource model, and the methods can be scaled to systems with more species. VSD is a useful tool for conservation biologists and community ecologists concerned with boundary crossing problems in any dynamical system.

Ordinary differential equation models of biochemical reactions are often formulated as stoichiometric systems in which the dynamics arise from a collection of interacting processes. A central challenge is that the functional form of each process is rarely known a priori and may be difficult to infer from data. We propose biochemically informed neural ordinary differential equations (BINODEs), a neural-ODE framework that retains the stoichiometric structure of mechanistic models while representing individual processes by neural networks. In BINODEs, the outputs of neural network processes (NNPs) are mapped to state derivatives through a linear layer analogous to a stoichiometric matrix. This architecture allows biological side information, such as process-specific inputs, sign constraints, and monotonicity assumptions, to be built directly into the model. We characterize the approximation properties of NNPs for several standard biochemical rate laws and show that the proposed framework recovers both trajectories and process-level structure in Monod, Lotka-Volterra, pharmacokinetic, and ultradian endocrine models. These results suggest that BINODEs offer a useful compromise between mechanistic interpretability and data-driven flexibility for modeling partially known biochemical or biological dynamical systems.

Cancer stem cells (CSCs) exhibit increased resistance to radiotherapy, contributing to tumor recurrence and progression. While CSCs are known for their intrinsic resistance, the role of their spatial organization remains poorly understood. We extend a computational model of tumorsphere growth to investigate how the spatial distribution of CSCs influences radiation response. The model explicitly tracks cell lineages and spatial positions, revealing a preferential accumulation of CSCs in the spheroid interior. Because radiosensitivity increases with oxygen availability, and oxygen levels are lowest in the tumor core, this spatial organization confers a protective advantage to the CSC population. We find that this effect is negligible in small, well-oxygenated tumorspheres but becomes pronounced as growth leads to the emergence of hypoxic regions. To isolate the role of spatial structure, we compare these results with control simulations in which CSC positions are randomly reassigned. In these synthetic configurations, CSC survival under irradiation is markedly reduced, demonstrating that spatial localization is a key determinant of radioresistance. This effect persists even after the onset of central necrosis, suggesting that the "spatial niche" of CSCs is a critical target for improving therapeutic outcomes. Author SummaryCancer stem cells are known to survive radiotherapy better than other cancer cells, often leading to tumor recurrence. While this resistance is usually attributed to intrinsic biological differences between cells, in this study we show that their physical location within the tumor plays a critical and previously underestimated role. We developed a three-dimensional computer model that simulates the growth of a tumorsphere from a single cancer stem cell. Because oxygen levels influence how sensitive cells are to radiation, our model tracks the position of each cell and calculates the oxygen distribution. We found that cancer stem cells naturally accumulate in the poorly oxygenated spheroid core, where radiation is less effective. To confirm that this location directly causes their survival advantage, we performed a "digital experiment": We artificially redistributed the same cells randomly throughout the tumorsphere before applying simulated radiation. In this random configuration, cancer stem cell survival dropped significantly. Our results show that radioresistance is not only an intrinsic cell property, but also a consequence of the spatial structure of the tumor. This finding suggests that future therapies could be improved by targeting not only the stem cells themselves, but also the protective hypoxic niches where they reside.

A growing number of studies indicate the possible involvement of astrocytes in triggering or modulating neurovascular coupling (NVC), i.e. the local dilation of blood vessels in the brain in response to neuronal activity. Astrocytes possess specialized subcellular compartments, named endfeet, that surround arterioles and capillaries, ideally positioned to mediate NVC. Various vasodilators have been shown to contribute to NVC, such as epoxyeicosatrienoic acid (EET), nitric oxide (NO), or prostaglandin E2 (PGE2), but the precise mechanisms underlying NVC and their variability remain to be fully elucidated. In particular, the involvement of astrocytes in this process is controversial. Recent translatome and proteomics data reveal that astrocytes and in particular endfeet are enriched in the proteins of the PGE2 pathway. However, how the latter could contribute to NVC remains to be characterized. Here, we develop a computational model of astrocyte-mediated NVC that recapitulates these findings and describes Ca2+ and PGE2 signaling in astrocytes, NO release by neurons, and arteriole diameter dynamics using ordinary differential equations. The model successfully reproduces the dynamics of arteriole diameter change during hyperemia from in vivo neocortical recordings in awake mice. Our simulations suggest that the astrocyte PGE2 pathway could be responsible for the late response of NVC at the arteriolar level. We further observe that PIP2-derived diacylglycerol plays a major role in driving arteriole diameter dynamics in our model, while phosphatidic acid-derived diacylglycerol, which is calcium-dependent, mainly acts as an amplifier of this response. Finally, a spatial implementation of the model using a simplified astrocyte geometry suggests that NVC is more efficient when synaptic stimulation occurs at the endfoot level rather than at other astrocytic compartments. Overall, this computational study suggests a partial role for astrocyte-mediated PGE2 release in NVC and points to astrocyte perivascular processes as sub-compartments that are ideally positioned and equipped to mediate NVC. Author summaryIn the brain, the local blood flow is regulated to meet neuronal energy demand by modulating the dilation of neighboring blood vessels. The mechanisms driving this process, known as neurovascular coupling (NVC), remain debated and are likely to differ depending on the physiological context. Recent evidence points to astrocytes, a cell type possessing specialized protrusions called "endfeet", that envelop the entire brain vascular tree. Contacts between synapses and endfeet have recently been reported, positioning the latter as ideal mediators of NVC. Here, we developed a computational model that simulates the signaling between neurons, astrocytes, and blood vessels. Our model successfully reproduces experimental recordings of blood vessels dilation in the brains of awake mice. Our simulations suggest that a specific signaling pathway in astrocytes, involving a molecule called prostaglandin E2, is a key driver of the late phase of NVC, occurring a few seconds after neuronal activity. Furthermore, our model indicates that the location of the stimulated synapses matters: signals sent to the astrocyte endfeet are particularly effective at controlling blood flow. This work helps clarify the active role of astrocytes in brain blood flow regulation, a process critical for healthy brain function.

A major challenge in neuroscience is to predict how currents in nanodomains affect voltage and ionic concentrations. Cable and Rall theory provide analytic current-voltage relations by neglecting concentration gradients, and the impact of concentration gradients is usually studied numerically with the Poisson-Nernst-Planck (PNP) model. A precise quantitative understanding of the combined dynamics remains limited because analytic current-voltage-concentration relations are missing. In this work we derive such relations using a novel approach based on cross-diffusion equations. For narrow cylindrical domains, we derive time-dependent and steady-state expressions that explicitly show how currents affect voltage and ionic concentrations. We find that the influx of only one ion can significantly change the concentrations of all the other ions even if no channels for these ions are present. After a current injection we compute a biphasic voltage transient where the small-time asymptotic corresponds to the steady-state solution of the cable equation. We show that the accuracy of cable theory prediction for the voltage depends on how the current is distributed among the various ions. Finally, we develop an iterative method to accurately compute steady-state profiles for voltage and concentrations using first-order results by subdividing a cylinder into small segments.

Cooperative and competitive interactions among individuals harvesting resources can shape environmental states, such as prey abundance. In turn, environmental conditions feed back to influence strategic interactions. Eco-evolutionary game theory studies how these feedbacks shape the co-evolution of behavior and environment. Existing models typically assume deterministic, noise-free environmental dynamics. However, real environments are inherently stochastic, for example due to finite resources, and noise can qualitatively alter social outcomes. Here, we incorporate stochastic environmental dynamics into eco-evolutionary game theory. When environmental change is slow relative to strategy updates, we show that behavior reflects a mixture of the games associated with low and high environmental states, often yielding outcomes qualitatively distinct from deterministic predictions. In particular, environmental stochasticity can eliminate bistability and enforce dominance of a single behavior. When environmental dynamics are faster, populations have less opportunity to track fluctuations, and behavior converges toward strategies that are optimal on average. Stochasticity can even causes persistent oscillations in the tragedy of commons, in regimes where classical models predict stability. Our framework provides a tractable approach for analyzing social behavior linked to environmental dynamics how noise shapes long-term eco-evolutionary outcomes.

Multicellular organisms regularly encounter microbes, which are, however, only rarely pathogenic. Our understanding of this phenomenon is currently restricted due to lacking theory on evolutionary transitions between non-pathogenic and pathogenic microbial lifestyles. Here we addressed this gap by investigating a mathematical model of host-microbe interactions that is based on the danger theory of immunology, which states that danger signals related to host tissue damage play a key role in activating immune responses. We formally implemented this idea by assuming that immune activation increases with costs that microbes cause to their host, and we compared this to scenarios in which immune activation depends only on the presence or load of infecting microbes. Our model analysis revealed that cost-based - but not presence or load-based - immune activation favours the evolution of avirulence and associated non-pathogenic microbial lifestyles. Based on our results, we propose the danger hypothesis of virulence evolution which states that evolution towards avirulence and intermediate virulence are both possible - depending on whether hosts can accurately assess costs generated by microbes. The idea that basic host immune responses can select for avirulence offers a new explanation for why most microbes are not pathogenic to a given host.

Hippocampal adult neurogenesis (HANG) is a highly regulated process where neural stem cells progress through distinct stages--from Type 1 radial glia-like cells to mature neurons--via a complex series of proliferative and differentiative divisions. While recent in vivo imaging has provided valuable insights to cellular processes, the exact relationship between individual cell-fate decisions and long-term population stability remains difficult to quantify empirically. In this study, we utilized an agent-based (AB) model to simulate the stochastic dynamics of the hippocampal neurogenic niche. Our results demonstrate that while individual progenitor lineages exhibit high variability and probabilistic division symmetries (proliferative symmetric, asymmetric, and differentiative symmetric), the system achieves deterministic stability as the initial progenitor density increases. We found that the T1 progenitor pool follows a negative exponential decay profile, with its longevity primarily dictated by the differentiation rate (d,0). Critically, the terminal output of immature neurons (CIN,t) was non-linearly coupled to the proliferative capacity of transit-amplifying cells (pp,0); even marginal increases in symmetric proliferative divisions resulted in an exponential expansion of the neuronal pool. These findings suggest that the homeostatic maintenance of the hippocampal niche is governed by a kinetic tuning of division probabilities, providing a theoretical bridge between single-cell stochasticity and robust tissue-level output.

The ecological and evolutionary dynamics of populations, including viral populations, are known to be jointly shaped by deterministic and stochastic processes. While the impact of stochastic processes has been rigorously explored for viral dynamics at the level of the host population, most dynamic models for acutely-infecting respiratory viral pathogens at the within-host scale remain deterministic in their formulation. While this may be reasonable for identifying key processes shaping their within-host viral population dynamics, recent studies indicate that stochastic processes need to be invoked for understanding patterns of within-host viral evolution. Specifically, several studies have shown that viral allele frequencies can change dramatically over the time course of days in acute infections. Here, we use stochastic dynamic models to explore the role of environmental noise in shaping observed patterns of virus evolution in acute respiratory virus infections. We summarize ways in which environmental stochasticity can be biologically realized in these acute viral infections and describe within-host models that can be implemented to jointly yield viral population dynamics and evolutionary dynamics. We further develop a statistical approach to estimate the extent of environmental noise from observed within-host allele frequency changes. We test this approach on simulated data and apply it to existing influenza A virus and SARS-CoV-2 within-host data. With these applications, we show that environmental stochasticity can parsimoniously reproduce key features of empirically observed allele frequency changes without needing to invoke demographic stochasticity or to adopt Wright-Fisher model formulations with a constant effective population size. Finally, we show that purifying selection and positive selection can both still contribute to within-host viral evolution in the context of a noisy environment, providing theoretical support for studies that have found purifying and positive selection in acutely-infecting respiratory virus populations.

In the dynamic interplay between hosts and pathogens, hosts may produce a defense compound that acts as a toxin to deter pathogen attack. Conversely, pathogens may evolve to produce a counter-defense enzyme, neutralizing the hosts toxin. This evolutionary arms race incurs costs for both parties, prompting adaptations and strategic shifts. We conceptualize this interaction as an asymmetric game, with hosts and pathogens as players, and their potential responses - defense, counter-defense, or inaction - as their strategic options. In this scenario, if the pathogens counter-defense enzyme is entirely effective, then the hosts toxin is rendered obsolete. However, should the host cease toxin production, the pathogens enzyme becomes redundant, ironically reinstating the toxins utility. This interaction leads to potential red-queen cycles in defense and counter-defense strategies under certain conditions, or a balanced, optimal production of toxin and enzymes by hosts and parasites, respectively. To explore this, we introduce a game-theoretical model incorporating replicator dynamics to examine temporal shifts in strategy from active (counter-)defense to non-(counter-)defense and back. In addition, we analyze compromise strategies and interpret them as bet-hedging-like. We provide a deterministic illustration of how partial defense and counter-defense generate a fitness-buffering structure in unpredictable environments and increase the geometric mean fitness of the population. In conclusion, our analysis supports the notion of continuous periodic adjustments in strategies, notably in the levels of defensive and counter-defensive measures.

A phylogenetic network represents evolutionary relationships involving hybridization, gene flow, or admixture. While the full network may not be identifiable from genomic data under common coalescent models, its tree of blobs, depicting only the tree-like portions of the network structure, is. We introduce ECToBlob (Edge Contraction for Tree of Blobs), a new statistically-consistent algorithm to estimate the tree of blobs from quartet concordance factors. Starting from a resolved tree, ECToBlob successively contracts edges which statistical tests indicate do not belong in the tree of blobs, due to reticulate or polytomous signal. We show that ASTRAL provides a valid starting tree under common assumptions, in that, asymptotically in the number of loci, trees optimizing ASTRALs criterion refine the tree of blobs. We describe several algorithm variants, differing in how evidence from multiple tests are combined to determine if the edge should be contracted, and provide software implementations. Relevance to Life SciencesHybridization, gene flow, or admixture are now recognized as important aspects of evolutionary history, but their genomic signal is confounded with that from a coalescent process, creating substantial challenges for inferring phylogenetic networks. The networks tree of blobs identifies areas where reticulation occurred, separated by tree-like branching. ECToBlob quickly estimates the tree of blobs using quartet concordance factors from gene trees, and provides a measure of statistical support for its result. Performance is illustrated through simulation and on empirical data, using an implementation in the R package MSCquartets. While the presence of a blob may be all that can be inferred in some cases, in others ECToBlob offers a robust and principled way to focus further analyses on more local reticulate structure. Mathematical ContentThis work makes contributions to mathematical phylogenetics in optimization, combinatorics, and statistics. We show that any tree maximizing quartet support (the criterion underlying ASTRAL) is a refinement of the networks tree of blobs under the coalescent model. Second, we give a concise proof that whether a network has a cut-edge corresponding to a given split is determined by information in certain subcollections of its 4-taxon subnetworks (quarnets). Finally, we propose valid statistical approaches for combining p-values across multiple quarnet hypothesis tests, proving that their use with specific decreasing test levels leads to statistically consistent inference as the number of loci grows. MSC codes05C90, 60J95, 62-04, 62F07, 92D15

Predicting the outcome of species or pathogen strain competition is a fundamental aim in both community ecology and infectious disease dynamics. Recent work revealed major challenges in predicting strain co-circulation from ecological coexistence theory due to overcompensatory competition among pathogens for susceptible resources, which can prevent the re-invasion of other competing strains. This resource overcompensation is ubiquitous across host-pathogen systems, but not apparent in simple Lotka-Volterra competition system, highlighting fundamental differences between pathogen strain and species competition. To address this gap, we begin by deriving classical models of pathogen strain and species competition from a resource-consumer model. This generalization illustrates that the relative time scale between resource and consumer dynamics limits the degree of resource overcompensation and therefore dictates the outcome of stochastic competition. Moreover, by introducing a mathematical framework for quantifying pairwise and higher-order terms from general competition systems, we show that a simple, ecological competition model can accurately predict the equilibrium dynamics of strain competition. A case study of rotavirus strain competition reveals that the ability to predict the outcome of strain competition from ecological theory depends on the underlying cross immunity structure. This work synthesizes coexistence theory across two fields by providing a unifying framework for predicting the outcome of complex ecological competition.

Most organisms become increasingly likely to die as they grow older, a phenomenon known as demographic senescence. Despite this statement saying nothing about the particular shape of a mortality curve, the Gompertz-Makeham law remains remarkably accurate in a broad range of species. We develop a general mathematical framework in which individuals are modelled as comprising a fixed number of interacting intra-organismal sub-systems, each characterised by stochastic failure and repair rates, such that the number of failed sub-systems follows a birth-death process. In many organisms, failure begets failure because sub-systems are typically interdependent. We use diffusion approximations to demonstrate that this interdependence generically produces Gompertz-Makeham mortality curves. Since individuals who die can no longer age, observed cohorts become increasingly composed of lucky individuals that avoided death by (stochastically) taking paths in failure space associated with lower mortality despite no intrinsic differences in their quality. Selective disappearance of unlucky individuals generates deviations from Gompertz-Makeham predictions at advanced ages, producing a late-life mortality plateau. We show that while these deviations must always exist, they may often be too small to detect, either because the failure accumulation process is stereotyped or because detection requires unreasonably large cohort sizes. Our work establishes Gompertz-Makeham curves arising from stochastic failure accumulation as a null expectation in organisms with many interdependent sub-systems.

This work focuses on the global and partial identification problem for fractional differential equations. We provide a general numerical procedure based on global and local optimization algorithms with two refinements for biological systems that ensure solution positivity and homogeneous parameter units. The method is applied to a new fractional model of Dengue outbreak called the Fractional Homogeneous Nishiura (FHN) model, calibrated using data of newly infected people in Cape Verde. We show that our identification method yields a better fit between data and model solutions than previous approaches and that our FHN model captures the dynamics of Dengue more closely than existing systems.

The basic reproduction number R0 is central to malaria epidemiology, yet it is typically treated as a static quantity derived under memoryless assumptions for mosquito demography. In natural systems, however, mosquito populations are shaped by delayed processes such as larval development and density-dependent feedback, introducing biological memory into vector dynamics. We develop a minimal delay-based framework that incorporates this memory into the Ross-Macdonald model by describing adult mosquito abundance with a retarded differential equation. This formulation induces a time-dependent transmission potential R0(t). Using complex analysis and the argument principle, we derive an explicit stability threshold [Formula], which separates stable from oscillatory transmission regimes. Near this threshold, delayed feedback produces slow relaxation times and sustained transient oscillations, implying that transmission potential may vary intrinsically even in the absence of external forcing. To account for ecological variability, we extend this deterministic condition into a probabilistic framework and define the stability probability as [Formula]. Numerical simulations and global sensitivity analysis show that recruitment and developmental delays are the primary drivers of instability, while adult mortality has a weaker stabilizing effect. These results indicate that malaria interventions may influence not only the magnitude of malaria transmission but also its dynamical stability. By linking delay dynamics, transmission theory, and uncertainty quantification, this framework provides a basis for stability-aware modeling and interpretation of malaria transmission under ecological variability. Author summaryMalaria transmission is often summarized by a single number, R0, treated as a fixed indicator of whether transmission will increase or decline. This assumes mosquito populations respond instantly to environmental conditions. In reality, mosquitoes develop through stages where larval conditions, such as crowding, nutrition, or temperature, affect adult populations only after a delay. This creates biological memory: todays mosquitoes reflect past environments. We show that this memory can fundamentally reshape transmission dynamics. When developmental delays are included, transmission potential is no longer constant but can fluctuate over time, even in stable environments. These fluctuations can persist or amplify depending on the balance between mosquito growth, mortality, and delay. As a result, variability in mosquito abundance or malaria transmission may arise from intrinsic dynamics rather than external drivers alone. Under ecological variability, stability becomes probabilistic, allowing estimation of how likely transmission is to remain stable. Interventions that reduce larval productivity or increase adult mortality may therefore both lower transmission and make it more predictable, improving interpretation and control strategies.

Hemodynamic forces play a key role in early cardiac morphogenesis, yet many computational studies assume Newtonian blood behavior. Here, we evaluate the impact of nonNewtonian shearthinning rheology on flow patterns, pressure distributions, and wall shear stress (WSS) during cardiac looping using idealized threedimensional models of the embryonic heart tube. Five geometries representing progressive looping stages, from a linear tube to an Sshaped configuration with ventricular ballooning, were analyzed under pulsatile flow using both Newtonian and powerlaw viscosity models. Across all stages, Reynolds numbers (Re {approx} 1-7) and Womersley numbers (Wo {approx} 0.3) indicated laminar, quasisteady flow consistent with embryonic conditions. Incorporating shearthinning rheology produced substantial deviations from Newtonian predictions, with peak systolic WSS differing by up to [~]40% and pressure drops by up to [~]20%. These effects were most pronounced in regions of increased curvature and geometric complexity. These findings demonstrate that nonNewtonian rheology significantly influences predicted hemodynamic environments during cardiac looping and should be incorporated into computational models aimed at understanding mechanobiological regulation of early heart development.

Understanding how complex systems self-organize, exhibit emergent properties beyond their constituent elements remains a challenge across physics, biology, and cognitive science. In resource-constrained neuronal systems, existing theoretical approaches, including gauge theoretic formulations, statistical physics-inspired methods, dynamical population models, and variational principles such as the Free Energy Principle, address important aspects of this problem but do not fully specify the physical conditions and thermodynamic costs under which self-organizing behavior occurs. Here, we introduce Dynamic Resource Theory (DRT) as a general physical framework for describing self-organization under constrained resource availability. DRT formalizes complexity as a physical property of self-organizing systems arising from coupled mechanisms of resource allocation and dynamic reallocation of internal resources. This framework provides a thermodynamic and variational account of how stability is preserved while adaptive reconfiguration remains possible, consistent with stationary action and thermodynamic constraints. DRT is formulated within a gauge theoretic setting and directly incorporates the energetic costs associated with maintaining structure and enabling system-level reconfiguration. Within DRT, baseline resource allocation preserves system stability, while internal and external demands perturb the system, driving self-organization through dynamic resource reallocation across a coupled free energy landscape without assuming subsystem separability. We then develop Neural Resource Theory (NRT) and Cognitive Resource Theory (CRT) as principled specializations of DRT, illustrating how this structure is instantiated in resource constrained neuronal and cognitive systems. We conclude by discussing the broader implications of DRT for understanding how complexity, emergence, and adaptive capacity arise over time through thermodynamically permissible reallocation processes across scales.

Neuronal computation depends on the balance between excitation and inhibition, yet how this balance is implemented across the dendritic tree remains unclear. Classical views predict that inhibition should be most effective near the soma or along the path from excitation to output, but many interneuron subtypes preferentially target remote dendritic compartments. This apparent paradox is sharpened by active dendrites, where local NMDA spikes, calcium plateaus and backpropagating action potentials can make distal branches powerful contributors to somatic firing. Here we develop an analytical framework that extracts general principles of inhibition from biophysically detailed multi-compartment simulations. By reformulating the implicit voltage update of detailed neuron models as a matrix recursion, we derive exact voltage sensitivities to inhibitory synaptic perturbations. This leads to a unified {Phi}-a law: the somatic impact of inhibition factorizes into a global dendritic susceptibility term and a local synaptic perturbation term. Using this law to map inhibitory leverage and identify optimal inhibitory interventions, we show that active dendritic excitation can shift inhibitory hot zones from perisomatic regions toward distal or intermediate compartments. Across neocortical, hippocampal and striatal neuron models, the same response law explains convergent inhibitory strategies despite distinct cellular mechanisms. Our framework turns detailed numerical simulation into analytical theory, providing a general principle for how diverse dendritic inhibition controls active neurons.

Thin structures such as heart valves and aortic dissection flaps interact dynamically with blood flow in human vessels. Their flexibility and capacity for large deformations generate complex, highly transient hemodynamic patterns over the cardiac cycle. Accurately resolving these interactions remains challenging for conventional boundary-fitted fluid-structure interaction approaches. We present an immersed boundary method for simulating thin structures in incompressible flow on unstructured grids. The method couples a stabilized finite element fluid solver with a nonlinear, rotation-free shell formulation through a direct forcing immersed boundary approach. The framework supports both weak (explicit) and strong (implicit) time-coupling strategies, enabling stable simulations over a wide range of solid-to-fluid density ratios. Hydrodynamic forces acting on thin structures are computed from fluid solutions sampled on both sides of the structure, allowing accurate force reconstruction for zero-thickness shells. To our knowledge, this is the first immersed boundary formulation that couples an unstructured finite element fluid solver with a two-dimensional, rotation-free shell model to simulate interactions between thin structures and incompressible flow. Fluid-structure coupling is achieved using predefined finite element shape functions, which provide consistent projection between Eulerian and Lagrangian fields without additional interpolation procedures. The framework is validated using three-dimensional benchmark problems involving thin structures. Then, valve-like model is used to compare strong and weak coupling strategies. Finally, the method is applied to an idealized type-B aortic dissection model. The proposed approach is implemented within the open-source software CRIMSON, a finite element platform for cardiovascular simulation.