arXiv subjects

Sebastian Wieczorek

Publications and source records attributed to Sebastian Wieczorek.

At least 19 recordsLinked to original sources

arXiv2026

Classifying absence seizure generation mechanisms: A critical transitions framework

Understanding how the brain switches from normal activity to an epileptic seizure is essential for improving seizure therapy, yet the underlying seizure generation mechanisms remain largely unknown. In particular, while seizure onset has been described as a critical transition (CT), there is no consensus on whether (i) bifurcation-induced, (ii) noise-induced, or (iii) bifurcation/noise-induced CTs are responsible. To clarify this, we develop a versatile CT-classification framework that can be applied to seizures in both animals and humans. First, we identify a canonical mathematical model which displays CTs that closely resemble voltage recordings of real seizures and can be of the three types mentioned above. We then identify distinctive properties of each CT-type in the model's output and use them to train a machine learning CT-type classifier. Finally, we apply the model-trained classifier to voltage recordings from epileptic rodents which consist of thousands of real absence seizures. We find that the largest proportion of analysed seizures are classified as noise-induced CTs. In other words, our results on absence seizures in rodents are in contrast to the conventional view that seizures are predominantly bifurcation-induced, and indicate that different CT mechanisms may dominate different seizure types.

math.DS↗

arXiv2026

Detecting seizure onset and offset times using human intelligence: A critical-transitions-based approach

Most existing seizure detection algorithms require extensive pre-processing of the data and rely on heuristic or currently unexplainable machine learning approaches. These approaches often struggle with balancing detection sensitivity and specificity in the presence of variable seizure morphologies, interictal epileptiform discharges, and artefacts. Here, we consider an alternative approach: our seizure detection algorithm, which is based on the concept of critical transitions and overcomes the aforementioned limitations. Specifically, we perform a receiver-operating-characteristic analysis to quantify the performance of our algorithm in terms of its agreement with expert annotations of seizure onset and offset times in the voltage recordings of seizure activity in epileptic rodents with different seizure morphologies. We demonstrate how performance depends on algorithm parameters and varies across different rodent recording sessions. We determine the optimal set of algorithm parameters for each recording session, with near expert-level performance achieved in most cases. Finally, we derive a single general set of algorithm parameters applicable across all recording sessions. The algorithm maintains its high performance in this general setting, demonstrating its versatility, robustness across varying seizure morphologies, and potential to complement machine learning algorithms.

math.DS↗

arXiv2026

Singular basins in multiscale systems: tunneling between stable states

Real-world systems often evolve on different timescales and possess multiple coexisting stable states. Whether or not a system returns to a given stable state after being perturbed away from it depends on the shape and extent of its basin of attraction. We show that basins of attraction in multiscale systems can exhibit special geometric properties in the form of singular funnels. Although singular funnels are narrow, they can extend to different regions of the phase space and, unexpectedly, impact the system's resilience to perturbations. Consequently, singular funnels may prevent common dimensionality reductions in the limit of large timescale separation, such as the quasi-static approximation, adiabatic elimination and time-averaging of the fast variables. We refer to basins of attraction with singular funnels as singular basins. We show that singular basins are universal and occur robustly in a range of multiscale systems: the normal form of a pitchfork bifurcation with a slowly adapting parameter, an adaptive active rotator, and an adaptive network of phase rotators.

math.DS↗

arXiv2026

Rate-Induced Tipping in a Non-Uniformly Moving Habitat and Determination of the Critical Rate

A habitat that is moving due to environmental change may result in tipping to extinction if the rate at which it moves is too great. We use a scalar reaction-diffusion equation with a non-autonomous reaction term, representing a spatially localized habitat moving from one asymptotic location to another, as a context for studying this phenomenon. The movement is characterized by displacement $d$ and rate parameter $r$. The system admits three steady states in both asymptotic habitat locations: a stable extinction state $u_0^*=0$, an unstable pulse (so-called edge state) $u_1^*(x)>0$, which gives rise to the Allee effect, and a stable pulse (populated base state) $u_2^*(x)>u_1^*(x)$, which corresponds to a thriving population at its carrying capacity. Numerical simulations for a specific model identify a critical displacement $d^*$ and, for $d > d^*$, demonstrate the existence of a \textit{critical rate} $r_c(d)$ at which rate-induced tipping occurs: for $r> r_c$ an initially thriving population becomes extinct due to habitat movement being too rapid. We provide analytical results for two limiting cases. For $r\ll 1$, solutions track the moving base state with error $O(r)$. For $r\gg 1$, solutions converge to the extinction state provided $d$ is sufficiently large. For $d$ too small, no tipping occurs regardless of $r$. Numerical simulations complement and extend these analytical results. At the critical rate $r=r_c(d)$, we identify a pulse-to-pulse heteroclinic connection between the base state at the past asymptotic location and the edge state at the future asymptotic location of the habitat. We also establish the uniqueness of this critical rate and non-degeneracy of the heteroclinic connection as $r$ varies.

math.AP↗

arXiv2026

The Headless Firm: How AI Reshapes Enterprise Boundaries

The boundary of the firm is determined by coordination cost. We argue that agentic AI induces a structural change in how coordination costs scale: in prior modular systems, integration cost grew with interaction topology (O(n^2) in the number of components); in protocol-mediated agentic systems, integration cost collapses to O(n) while verification scales with task throughput rather than interaction count. This shift selects for a specific organizational equilibrium -- the Headless Firm -- structured as an hourglass: a personalized generative interface at the top, a standardized protocol waist in the middle, and a competitive market of micro-specialized execution agents at the bottom. We formalize this claim as a coordination cost model with two falsifiable empirical predictions: (1) the marginal cost of adding an execution provider should be approximately constant in a mature hourglass ecosystem; (2) the ratio of total coordination cost to task throughput should remain stable as ecosystem size grows. We derive conditions for hourglass stability versus re-centralization and analyze implications for firm size distributions, labor markets, and software economics. The analysis predicts a domain-conditional Great Unbundling: in high knowledge-velocity domains, firm size distributions shift mass from large integrated incumbents toward micro-specialized agents and thin protocol orchestrators.

cs.GT↗

arXiv2025

Cancer model with moving extinction threshold reproduces real cancer data

We propose a simple dynamic model of cancer development that captures carcinogenesis and subsequent cancer progression. A central idea of the model is to include the immune system as an extinction threshold, similar to the strong Allee effect in population biology. We first identify the limitations of commonly used Allee effect models in reproducing typical cancer progression. We then address these limitations by deriving a new model that incorporates: (i) random mutations of stem cells at a rate that increases with age and (ii) immune response whose strength may also vary over time. Our model accurately reproduces a wide range of real-world cancer data: the typical age-specific cumulative risk of most human cancers, the progression of breast cancer in mice, and the unusual age-specific cumulative risk of breast cancer in women. In the last case, we use a moving extinction threshold to reflect the different immune response at different phases of the menstrual cycle and menopausal treatment. This provides new insights into the effects of hormone replacement therapy and menstrual cycle length. This moving threshold approach can be applied to a variety of other cancer scenarios where the immune response or other important factors may vary over time.

q-bio.QM↗

arXiv2025

Pace in Concert with Phase: Rate-induced Phase-tipping in Birhythmic Oscillators

We study rate-induced phase-tipping (RP-tipping) between two stable limit cycles of a birhythmic oscillator. We say that such an oscillator RP-tips when a time variation of an input parameter preserves the bistability of the limit cycles but induces transitions from one stable limit cycle to the other, causing abrupt changes in the amplitude and frequency of the oscillations. Crucially, these transitions occur when: the rate of change of the input is in a certain interval bounded by critical rate(s), and the system is in certain phases of the cycle. We focus on two illustrative examples: the birhythmic van der Pol oscillator and the birhythmic Decroly-Goldbeter glycolysis model, each subjected to monotone and non-monotone shifts in their input parameters. We explain RP-tipping in terms of properties of the autonomous frozen system, including the phase of a cycle and partial basin instability along the parameter path traced by the changing input. We show that RP-tipping can occur as an irreversible one-way transition or as a series of transitions between the stable limit cycles. Finally, we present RP-tipping diagrams showing combinations of the rate and magnitude of parameter shifts and the phase of the oscillation that give rise to this genuine non-autonomous instability.

math.DS↗

arXiv2024

Canard cascading in networks with adaptive mean-field coupling

Canard cascading (CC) is observed in dynamical networks with global adaptive coupling. It is a fast-slow phenomenon characterized by a recurrent sequence of fast transitions between distinct and slowly evolving quasi-stationary states. In this letter, we uncover the dynamical mechanisms behind CC, using an illustrative example of globally and adaptively coupled semiconductor lasers, where CC represents sequential switching on and off the lasers. Firstly, we show that CC is a robust and truly adaptive network effect that is scalable with network size and does not occur without adaptation. Secondly, we uncover multiple saddle slow manifolds (unstable quasi-stationary states) linked by heteroclinic orbits (fast transitions) in the phase space of the system. This allows us to identify CC with a novel heteroclinic canard orbit that organises different unstable quasi-stationary states into an intricate fast-slow limit cycle. Although individual quasi-stationary states are unstable (saddles), the CC cycle as a whole is attractive and robust to parameter changes.

nlin.AO↗

arXiv2023

Rate-Induced Tipping in Heterogeneous Reaction-Diffusion Systems: An Invariant Manifold Framework and Geographically Shifting Ecosystems

We propose a framework to study tipping points in reaction-diffusion equations (RDEs) in one spatial dimension, where the reaction term decays in space (asymptotically homogeneous) and varies linearly with time (nonautonomous) due to an external input. A compactification of the moving-frame coordinate together with Lin's method to construct heteroclinic orbits along intersections of stable and unstable invariant manifolds allows us to (i) obtain multiple coexisting pulse and front solutions for the RDE by computing heteroclinic orbits connecting equilibria at negative and positive infinity in the compactified moving-frame ordinary differential equation, (ii) detect tipping points as dangerous bifurcations of such heteroclinic orbits, and (iii) obtain tipping diagrams by numerical continuation of such bifurcations. We apply our framework to an illustrative model of a habitat patch that features an Allee effect in population growth and is geographically shrinking or shifting due to human activity or climate change. Thus, we identify two classes of tipping points to extinction: bifurcation-induced tipping (B-tipping) when the shrinking habitat falls below some critical length and rate-induced tipping (R-tipping) when the shifting habitat exceeds some critical speed. We explore two-parameter R-tipping diagrams to understand how the critical speed depends on the size of the habitat patch and the dispersal rate of the population, uncover parameter regions where the shifting population survives, and relate these regions to the invasion speed in an infinite homogeneous habitat. Furthermore, we contrast the tipping instabilities with gradual transitions to extinction found for logistic population growth without the Allee effect.

math.DS↗

arXiv2023

Rate-Induced Tipping to Metastable Zombie Fires

Zombie fires in peatlands disappear from the surface, smoulder underground during the winter, and `come back to life' in the spring. They can release hundreds of megatonnes of carbon into the atmosphere per year and are believed to be caused by surface wildfires. Here, we propose rate-induced tipping (R-tipping) to a subsurface hot metastable state in bioactive peat soils as a main cause of Zombie fires. Our hypothesis is based on a conceptual soil-carbon model subjected to realistic changes in weather and climate patterns, including global warming scenarios and summer heatwaves. Mathematically speaking, R-tipping to the hot metastable state is a nonautonomous instability, due to crossing an elusive quasithreshold, in a multiple-timescale dynamical system. To explain this instability, we provide a framework combining a special compactification technique with concepts from geometric singular perturbation theory. This framework allows us to reduce an R-tipping problem due to crossing a quasithreshold to a heteroclinic orbit problem in a singular limit. We identify generic cases of tracking-tipping transitions via: (i) unfolding of a codimension-two heteroclinic folded saddle-node type-I singularity for global warming, and (ii) analysis of a codimension-one saddle-to-saddle hetroclinic orbit for summer heatwaves, in turn revealing new types of excitability quasithresholds.

math.DS↗

arXiv2023

Rate-Induced Tipping: Thresholds, Edge States and Connecting Orbits

Rate-induced tipping (R-tipping) occurs when time-variation of input parameters of a dynamical system interacts with system timescales to give genuine nonautonomous instabilities. Such instabilities appear as the input varies at some critical rates and cannot, in general, be understood in terms of autonomous bifurcations in the frozen system with a fixed-in-time input. This paper develops an accessible mathematical framework for R-tipping in multidimensional nonautonomous dynamical systems with an autonomous future limit. We focus on R-tipping via loss of tracking of base attractors that are equilibria in the frozen system, due to crossing what we call regular thresholds. These thresholds are associated with regular edge states: compact hyperbolic invariant sets with one unstable direction and orientable stable manifold, that lie on a basin boundary in the frozen system. We define R-tipping and critical rates for the nonautonomous system in terms of special solutions that limit to a compact invariant set of the future limit system that is not an attractor. We focus on the case when the limit set is a regular edge state, which we call the regular R-tipping edge state that anchors the associated regular R-tipping threshold at infinity. We introduce the concept of edge tails to rigorously classify R-tipping into reversible, irreversible, and degenerate cases. The main idea is to compactify the problem and use regular edge states of the future limit system to analyse R-tipping in the nonautonomous system. This allows us to give sufficient conditions for the occurrence of R-tipping in terms of easily testable properties of the frozen system and input variation, and necessary and sufficient conditions for the occurrence of reversible and irreversible R-tipping in terms of computationally verifiable (heteroclinic) connections to regular R-tipping edge states in the compactified system.

math.DS↗

arXiv2023

Loxodromes in Open Multi-Section Lasers

We introduce a formalism to efficiently calculate lasing modes and optical power flow in multi-section lasers with open boundaries. The formalism is underpinned by a projection of the complex-valued electric field and its spatial derivative onto a suitably extended complex $\mathcal{Z}$-plane, to reduce the order of the problem and simplify analysis. In a single-section laser, we show that a laser mode is a loxodrome on the extended complex $\mathcal{Z}$-plane. In a multi-section laser, we obtain loxodromes for individual sections of the laser. Then, a multi-section mode is constructed by continuously concatenating individual loxodromes from each section using the open boundary conditions. A natural visualization of this construction is given by stereographic projection of the extended complex $\mathcal{Z}$-plane onto the Riemann sphere. Our formalism simplifies analysis of lasing modes in open multi-section lasers and provides new insight into the mode geometry and degeneracy.

physics.optics↗

arXiv2023

Stochastic Resonance in Climate Reddening Increases the Risk of Cyclic Ecosystem Extinction via Phase-tipping

Human activity is leading to changes in the mean and variability of climatic parameters in most locations around the world. The changing mean has received considerable attention from scientists and climate policy makers. However, recent work indicates that the changing variability, that is, the amplitude and the temporal autocorrelation of deviations from the mean, may have greater and more imminent impact on ecosystems. In this paper, we demonstrate that changes in climate variability alone could drive cyclic predator-prey ecosystems to extinction via so-called phase tipping (P-tipping), a new type of instability that occurs only from certain phases of the predator-prey cycle. We construct a mathematical model of a variable climate and couple it to two self-oscillating paradigmatic predator-prey models. Most importantly, we combine realistic parameter values for the Canada lynx and snowshoe hare with actual climate data from the boreal forest. In this way, we demonstrate that critically important species in the boreal forest have increased likelihood of P-tipping to extinction under predicted changes in climate variability, and are most vulnerable during stages of the cycle when the predator population is near its maximum. Furthermore, our analysis reveals that stochastic resonance is the underlying mechanism for the increased likelihood of P-tipping to extinction.

q-bio.PE↗

arXiv2022

Rate-induced tipping can trigger plankton blooms

Plankton blooms are complex nonlinear phenomena whose occurrence can be described by the two-timescale (fast-slow) phytoplankton-zooplankton model intrpduced by Truscott and Brindley 1994. In their work, they observed that a sufficiently fast rise of the water temperature causes a critical transition from a low phytoplankton concentration to a single outburst: a so-called plankton bloom. However, the dynamical mechanism responsible for the observed transition has not been identified to the present day. Using techniques from geometric singular perturbation theory, we uncover the formerly overlooked rate-sensitive quasithreshold which is given by special trajectories called canards. The transition from low to high concentrations occurs when this rate-sensitive quasithreshold moves past the current state of the plankton system at some narrow critical range of warming rates. In this way, we identify rate-induced tipping as the underlying dynamical mechanism. Our findings explain the previously reported transitions to a single plankton bloom, and allow us to predict a new type of transition to a sequence of blooms for higher rates of warming. This could provide a possible mechanism of the observed increased frequency of harmful algal blooms.

nlin.PS↗

arXiv2021

Phase tipping: How cyclic ecosystems respond to contemporary climate

We identify the phase of a cycle as a new critical factor for tipping points (critical transitions) in cyclic systems subject to time-varying external conditions. As an example, we consider how contemporary climate variability induces tipping from a predator-prey cycle to extinction in two paradigmatic predator-prey models with an Allee effect. Our analysis of these examples uncovers a counter-intuitive behaviour, which we call phase tipping or P-tipping, where tipping to extinction occurs only from certain phases of the cycle. To explain this behaviour, we combine global dynamics with set theory and introduce the concept of partial basin instability for attracting limit cycles. This concept provides a general framework to analyse and identify easily testable criteria for the occurrence of phase tipping in externally forced systems, and can be extended to more complicated attractors.

math.DS↗

arXiv2020

Tipping Phenomena and Points of No Return in Ecosystems: Beyond Classical Bifurcations

We discuss tipping phenomena (critical transitions) in nonautonomous systems using an example of a bistable ecosystem model with environmental changes represented by time-varying parameters [Scheffer et al., Ecosystems, 11 (2008), pp. 275--279]. We give simple testable criteria for the occurrence of nonautonomous tipping from the herbivore-dominating equilibrium to the plant-only equilibrium using global properties of the autonomous frozen system with fixed-in-time parameters. To begin with, we use classical autonomous bifurcation analysis to identify a codimension-three degenerate Bogdanov-Takens bifurcation: the source of a dangerous subcritical Hopf bifurcation and the organizing center for bifurcation-induced tipping (B-tipping). Then, we introduce the concept of basin instability for equilibria to identify parameter paths along which genuine nonautonomous rate-induced tipping (R-tipping) occurs without crossing any classical autonomous bifurcations. We explain nonautonomous R-tipping in terms of maximal canard trajectories and produce nonautonomous tipping diagrams in the plane of the magnitude and rate of a parameter shift to uncover intriguing R-tipping tongues and wiggling tipping-tracking bifurcation curves. Discussion of nontrivial dynamics arising from the interaction between B-tipping and R-tipping identifies "points of no return" where tipping cannot be prevented by the parameter trend reversal and "points of return tipping" where tipping is inadvertently induced by the parameter trend reversal. Our results give new insight into the sensitivity of ecosystems to the magnitudes and rates of environmental change. Finally, a comparison between "tilted" saddle-node and subcritical Hopf normal forms reveals some universal tipping properties due to basin instability, a generic dangerous bifurcation, or the combination of both.

math.DS↗

arXiv2020

Compactification for Asymptotically Autonomous Dynamical Systems: Theory, Applications and Invariant Manifolds

We develop a general compactification framework to facilitate analysis of nonlinear nonautonomous ODEs where nonautonomous terms decay asymptotically. The strategy is to compactify the problem: the phase space is augmented with a bounded but open dimension and then extended at one or both ends by gluing in flow-invariant subspaces that carry autonomous dynamics of the limit systems from infinity. We derive the weakest decay conditions possible for the compactified system to be continuously differentiable on the extended phase space. This enables us to use equilibria and other compact invariant sets of the limit systems from infinity to analyse the original nonautonomous problem in the spirit of dynamical systems theory. Specifically, we prove that solutions of interest are contained in unique invariant manifolds of saddles for the limit systems when embedded in the extended phase space. The uniqueness holds in the general case, that is even if the compactification gives rise to a centre direction and the manifolds become centre or centre-stable manifolds. A wide range of problems including pullback attractors, rate-induced critical transitions (R-tipping) and nonlinear wave solutions fit naturally into our framework.

math.DS↗

arXiv2019

When very slow is too fast -- collapse of a predator-prey system

Critical transitions or regime shifts are sudden and unexpected changes in the state of an ecosystem, that are usually associated with dangerous levels of environmental change. However, recent studies show that critical transitions can also be triggered by dangerous rates of environmental change. In contrast to classical regime shifts, such rate-induced critical transitions do not involve any obvious loss of stability, or a bifurcation, and thus cannot be explained by the linear stability analysis. In this work, we demonstrate that the well-known Rosenzweig-MacArthur predator-prey model can undergo a rate-induced critical transition in response to a continuous decline in the habitat quality, resulting in a collapse of the predator and prey populations. Rather surprisingly, the collapse occurs even if the environmental change is slower than the slowest process in the model. To explain this counterintuitive phenomenon, we combine methods from geometric singular perturbation theory with the concept of a moving equilibrium, and study critical rates of environmental change with dependence on the initial state and the system parameters. Moreover, for a fixed rate of environmental change, we determine the set of initial states that undergo a rate-induced population collapse. Our results suggest that ecosystems may be more sensitive to how fast environmental conditions change than previously assumed. In particular, unexpected critical transitions with dramatic ecological consequences can be triggered by environmental changes that (i) do not exceed any dangerous levels, and (ii) are slower than the natural timescales of the ecosystem. This poses an interesting research question whether regime shifts observed in the natural world are predominantly rate-induced or bifurcation-induced.

q-bio.PE↗