arXiv subjects

David Holcman

Publications and source records attributed to David Holcman.

At least 19 recordsLinked to original sources

arXiv2026

Accelerated Simulation Algorithms for Extreme First-Passage Problems with General Emission Profiles

Fastest arrival events, where the first among many diffusing particles reaches a target, are central in triggering signal initiation in molecular stochastic systems. Classical approaches to simulate such events rely on full trajectory generation of all particles, leading to prohibitive computational costs in the large particle number regime. In this work, we present a general simulation framework for efficiently generating order statistics of arrival times by exploiting asymptotic first-passage distributions. This framework applies to diffusion processes in bounded domains with localized absorbing targets, for which short-time first-passage asymptotics are available, such as Brownian motion in dimensions one, two, and three. Starting with the case of instantaneous emission, we derive and implement a recursive inverse transform algorithm to simulate the first $k$ arrivals without tracking particle trajectories. We extend this algorithm to time-dependent emission profiles via an iterative approach, enabling the simulation of extreme statistics in systems with temporal injection, ranging from rapid to prolonged emission. Additionally, we provide asymptotic estimates of the mean fastest arrival time. To conclude, the present acceleration algorithm which bypasses Brownian simulations of trajectories can be used for spatial reaction networks, rare event detection, or diffusion-controlled activation.

math.PR↗

arXiv2025

Forecasting Excessive Anesthesia Depth Using EEG {\alpha}-Spindle Dynamics and Machine Learning

Objectives. Accurately predicting transitions to anesthetic drugs overdosage is a critical challenge in general anesthesia as it requires the identification of EEG indicators relevant for anticipating the evolution of the depth of anesthesia. Methods. In this study, we introduce a real-time, data-driven framework based on alpha spindle dynamics extracted from frontal EEG recordings. Using Empirical Mode Decomposition, we segment transient alpha spindle events and extract statistical features such as amplitude, duration, frequency, and suppression intervals. We apply these features to train a Light Gradient Boosting Machine, LGBM, classifier on a clinical EEG dataset spanning induction, maintenance, and emergence phases of general anesthesia. Results. Our model accurately classifies anesthesia phases with over 80 percent accuracy and anticipates the onset of isoelectric suppression, a marker of anesthetic drugs overdosage, with 96 percent accuracy up to 90 seconds in advance. Conclusion. The spindle-based metrics provides a non-invasive, interpretable, and predictive approach. This real-time method can be used to forecast unintentional anesthetic drugs overdosage, enabling proactive anesthesia management based solely on EEG signals. Significance. This new method is the first to provide a way to prevent too deep anesthesia and its consequence for the well-being of patients after the recovery from anesthesia.

q-bio.NC↗

arXiv2025

Weakly Supervised Segmentation and Classification of Alpha-Synuclein Aggregates in Brightfield Midbrain Images

Parkinson's disease (PD) is a neurodegenerative disorder associated with the accumulation of misfolded alpha-synuclein aggregates, forming Lewy bodies and neuritic shape used for pathology diagnostics. Automatic analysis of immunohistochemistry histopathological images with Deep Learning provides a promising tool for better understanding the spatial organization of these aggregates. In this study, we develop an automated image processing pipeline to segment and classify these aggregates in whole-slide images (WSIs) of midbrain tissue from PD and incidental Lewy Body Disease (iLBD) cases based on weakly supervised segmentation, robust to immunohistochemical labelling variability, with a ResNet50 classifier. Our approach allows to differentiate between major aggregate morphologies, including Lewy bodies and neurites with a balanced accuracy of $80\%$. This framework paves the way for large-scale characterization of the spatial distribution and heterogeneity of alpha-synuclein aggregates in brightfield immunohistochemical tissue, and for investigating their poorly understood relationships with surrounding cells such as microglia and astrocytes.

eess.IV↗

arXiv2025

Assessing One-Dimensional Cluster Stability by Extreme-Point Trimming

We develop a probabilistic method for assessing the tail behavior and geometric stability of one-dimensional n i.i.d. samples by tracking how their span contracts when the most extreme points are trimmed. Central to our approach is the diameter-shrinkage ratio, that quantifies the relative reduction in data range as extreme points are successively removed. We derive analytical expressions, including finite-sample corrections, for the expected shrinkage under both the uniform and Gaussian hypotheses, and establish that these curves remain distinct even for moderate number of removal. We construct an elementary decision rule that assigns a sample to whichever theoretical shrinkage profile it most closely follows. This test achieves higher classification accuracy than the classical likelihood-ratio test in small-sample or noisy regimes, while preserving asymptotic consistency for large n. We further integrate our criterion into a clustering pipeline (e.g. DBSCAN), demonstrating its ability to validate one-dimensional clusters without any density estimation or parameter tuning. This work thus provides both theoretical insight and practical tools for robust distributional inference and cluster stability analysis.

stat.ML↗

arXiv2024

Voltage laws in nanodomains revealed by asymptotics and simulations of electro-diffusion equations

Characterizing the local voltage distribution within nanophysiological domains, driven by ionic currents through membrane channels, is crucial for studying cellular activity in modern biophysics, yet it presents significant experimental and theoretical challenges. Theoretically, the complexity arises from the difficulty of solving electro-diffusion equations in three-dimensional domains. Currently, there are no methods available for obtaining asymptotic computations or approximated solutions of nonlinear equations, and numerically, it is challenging to explore solutions across both small and large spatial scales. In this work, we develop a method to solve the Poisson-Nernst-Planck equations with ionic currents entering and exiting through two narrow, circular window channels located on the boundary. The inflow through the first window is composed of a single cation, while the outflow maintains a constant ionic density satisfying local electro-neutrality conditions. Employing regular expansions and Green's function representations, we derive the ionic profiles and voltage drops in both small and large charge regimes. We explore how local surface curvature and window channels size influence voltage dynamics and validate our theoretical predictions through numerical simulations, assessing the accuracy of our asymptotic computations. These novel relationships between current, voltage, concentrations and geometry can enhance the characterization of physiological behaviors of nanodomains.

cond-mat.soft↗

arXiv2024

Voltage mapping in subcellular nanodomains using electro-diffusion modeling

Voltage distribution in sub-cellular micro-domains such as neuronal synapses, small protrusions or dendritic spines regulates the opening and closing of ionic channels, energy production and thus cellular homeostasis and excitability. Yet how voltage changes at such a small scale in vivo remains challenging due to the experimental diffraction limit, large signal fluctuations and the still limited resolution of fast voltage indicators. Here, we study the voltage distribution in nano-compartments using a computational approach based on the Poisson-Nernst-Planck equations for the electro-diffusion motion of ions, where inward and outward fluxes are generated between channels. We report a current-voltage (I-V) logarithmic relationship generalizing Nernst law that reveals how the local membrane curvature modulates the voltage. We further find that an influx current penetrating a cellular electrolyte can lead to perturbations from tens to hundreds of nanometers deep depending on the local channels organization. Finally, we show that the neck resistance of dendritic spines can be completely shunted by the transporters located on the head boundary, facilitating ionic flow. To conclude, we propose that voltage is regulated at a subcellular level by channels organization, membrane curvature and narrow passages.

q-bio.SC↗

arXiv2023

Single particle algorithms to reveal cellular nanodomain organization

Formation, maintenance and physiology of high-density protein-enriched organized nanodomains, first observed in electron microscopy images, remains challenging to investigate due to their small sizes. However, these regions regulate molecular trafficking, assembly and sorting required for higher cell functions, such as communication or plastic changes. Over the past ten years, super-resolution single-particle trajectories (SPTs) have been used to sample these sub-cellular environments at a nanometer resolution for both membrane and soluble proteins. We present here data analysis developments and algorithms that convert high-throughput molecular trajectories into maps of molecular density, diffusion and local drift organization. These approaches transform intrinsic trajectory properties into statistics of the underlying cellular organization. The automatic identification of large numbers of high-density regions allows quantifying their boundary location and organization, their stability over time and their ability to transiently retain molecules. To conclude recent automated algorithms can now be used to extract biophysical parameters of sub-cellular nanodomains over a large amount of trajectories.

q-bio.QM↗

arXiv2023

Modeling ionic flow between small targets: insights from diffusion and electro-diffusion theory

The flow of ions through permeable channels causes voltage drop in physiological nanodomains such as synapses, dendrites and dendritic spines, and other protrusions. How the voltage changes around channels in these nanodomains has remained poorly studied. We focus this book chapter on summarizing recent efforts in computing the steady-state current, voltage and ionic concentration distributions based on the Poisson-Nernst-Planck equations as a model of electro-diffusion. We first consider the spatial distribution of an uncharged particle density and derive asymptotic formulas for the concentration difference by solving the Laplace's equation with mixed boundary conditions. We study a constant particles injection rate modeled by a Neumann flux condition at a channel represented by a small boundary target, while the injected particles can exit at one or several narrow patches. We then discuss the case of two species (positive and negative charges) and take into account motions due to both concentration and electrochemical gradients. The voltage resulting from charge interactions is calculated by solving the Poisson's equation. We show how deep an influx diffusion propagates inside a nanodomain, for populations of both uncharged and charged particles. We estimate the concentration and voltage changes in relations with geometrical parameters and quantify the impact of membrane curvature.

cond-mat.soft↗

arXiv2023

Memory of Ivan Kupka (1937-2023)

This brief text is in memory of Professor Ivan Kupka. It presents his vision, scientific life, his interest in mathematics and our join collaboration.

math.HO↗

arXiv2023

The WQN algorithm for EEG artifact removal in the absence of scale invariance

Electroencephalogram (EEG) signals reflect brain activity across different brain states, characterized by distinct frequency distributions. Through multifractal analysis tools, we investigate the scaling behaviour of different classes of EEG signals and artifacts. We show that brain states associated to sleep and general anaesthesia are not in general characterized by scale invariance. The lack of scale invariance motivates the development of artifact removal algorithms capable of operating independently at each scale. We examine here the properties of the wavelet quantile normalization algorithm, a recently introduced adaptive method for real-time correction of transient artifacts in EEG signals. We establish general results regarding the regularization properties of the WQN algorithm, showing how it can eliminate singularities introduced by artefacts, and we compare it to traditional thresholding algorithms. Furthermore, we show that the algorithm performance is independent of the wavelet basis. We finally examine its continuity and boundedness properties and illustrate its distinctive non-local action on the wavelet coefficients through pathological examples.

stat.ME↗

arXiv2023

Extreme diffusion with point-sink killing field

We study here the escape time for the fastest diffusing particle from the boundary of an interval with point-sink killing sources. Killing represents a degradation that leads to the probabilistic removal of the moving Brownian particles. We compute asymptotically the mean time it takes for the fastest particle escaping alive and obtain the extreme statistic distribution. These computations relies on an explicit expression for the time dependent flux of the Fokker-Planck equation using the time dependent Green's function and Duhamel's formula. We obtain a general formula for several point-sink killing, showing how they directly interact. The range of validity of the present formula for the mean extreme times of the fastest is evaluated with Brownian simulations. Finally, we discuss some applications to the early calcium signaling at neuronal synapses.

cond-mat.stat-mech↗

arXiv2022

Narrow escape in composite domains forming heterogeneous networks

Cellular networks are often composed of thin tubules connecting much larger node compartments. These structures serve for active or diffusion transport of proteins. Examples are glial networks in the brain, the endoplasmic reticulum in cells or dendritic spines located on dendrites. In this latter case, a large ball forming the head is connected by a narrow passage. In all cases, how the transport of molecules, ions or proteins is regulated determines the time scale of chemical reactions or signal transduction. In the present study, based on modeling diffusion in three dimensions, we compute the mean time for a Brownian particle to reach a narrow target inside such a composite network made of tubules connected to spherical nodes. We derive asymptotic formulas by solving a mixed Neumann-Dirichlet boundary value problem with small Dirichlet part. We first consider the case of a network domain organized in a 2-D lattice structure that consists of spherical ball compartments connected via narrow cylindrical passages. When there is a single target we derive a matrix equation for each Mean First Passage Time (MFPT) averaged over each spherical compartment. We then consider a composite domain consisting of a spherical head-like domain connected to a large cylinder via a narrow cylindrical neck. For Brownian particles starting within the narrow neck, we derive formulas for the MFPT to reach a target on the spherical head. When diffusing particles can be absorbed upon hitting additional absorbing boundaries of the large cylinder, we compute the probability and conditional MFPT to reach a target. We compare these formulas with numerical solutions of the mixed boundary value problem and with Brownian simulations. To conclude, the present analysis reveals that the mean arrival time, driven by diffusion in heterogeneous networks, is controlled by the target and narrow passage sizes.

cond-mat.soft↗

arXiv2022

The WQN algorithm to adaptively correct artifacts in the EEG signal

Wavelet quantile normalization (WQN) is a nonparametric algorithm designed to efficiently remove transient artifacts from single-channel EEG in real-time clinical monitoring. Today, EEG monitoring machines suspend their output when artifacts in the signal are detected. Removing unpredictable EEG artifacts would thus allow to improve the continuity of the monitoring. We analyze the WQN algorithm which consists in transporting wavelet coefficient distributions of an artifacted epoch into a reference, uncontaminated signal distribution. We show that the algorithm regularizes the signal. To confirm that the algorithm is well suited, we study the empirical distributions of the EEG and the artifacts wavelet coefficients. We compare the WQN algorithm to the classical wavelet thresholding methods and study their effect on the distribution of the wavelet coefficients. We show that the WQN algorithm preserves the distribution while the thresholding methods can cause alterations. Finally, we show how the spectrogram computed from an EEG signal can be cleaned using the WQN algorithm.

stat.ME↗

arXiv2022

How large should be the redundant numbers of copy to make a rare event probable

The redundancy principle provides the framework to study how rare events are made possible with probability 1 in accelerated time, by making many copies of similar random searchers. But what is $n$ large? To estimate large $n$ with respect to the geometrical properties of a domain and the dynamics, we present here a criteria based on splitting probabilities between a small fraction of the exploration space associated to an activation process and other absorbing regions where trajectories can be terminated. We obtain explicit computations especially when there is a killing region located inside the domain that we compare with stochastic simulations. We present also examples of extreme trajectories with killing in dimension 2. For a large $n$, the optimal trajectories avoid penetrating inside the killing region. Finally we discuss some applications to cell biology.

cond-mat.soft↗

arXiv2022

Arrival time for the fastest among $N$ switching stochastic particles

The first arrivals among $N$ Brownian particles is ubiquitous in the life sciences, as it often trigger cellular processes from the molecular level. We study here the case where stochastic particles, which represent molecules, proteins or molecules can switch between two states inside the non-negative real line. The switching process is modeled as a two-state Markov chain and particles can only escape in state 1. We estimate the fastest arrival time by solving asymptotically the Fokker-Planck equations for three different initial distributions: Dirac-delta, uniformly distributed and long-tail decay. The derived formulas reveal that the fastest particle avoid switching when the switching rates are much smaller than the diffusion time scale, but switches twice when the diffusion is state 2 is much faster than in state 1. The present results are compared to stochastic simulations revealing the range of validity of the derived formulas.

cond-mat.stat-mech↗

arXiv2021

Zeev Schuss' legacy: an unexpected path in applied mathematics

Z. Schuss (1937-2018) was an applied mathematician, with several contributions in asymptotic, stochastic processes, PDEs, modeling and signal processing. He is well known for his original approach to the activation escape problem, based on WKB and boundary layer analysis, summarized in six books published by Springer. The text summarizes his intellectual approach in science, views and personal path across the XX century, WWII and his personal contribution to academia.

math.HO↗

arXiv2021

Modeling and asymptotic analysis of the concentration difference in a nanoregion between an influx and outflux diffusion across narrow windows

When a flux of Brownian particles is injected in a narrow window located on the surface of a bounded domain, these particles diffuse and can eventually escape through a cluster of narrow windows. At steady-state, we compute asymptotically the distribution of concentration between the different windows. The solution is obtained by solving Laplace's equation using Green's function techniques and second order asymptotic analysis, and depends on the influx amplitude, the diffusion properties as well as the geometrical organization of all the windows, such as their distances and the mean curvature. We explore the range of validity of the present asymptotic expansions using numerical simulations of the mixed boundary value problem. Finally, we introduce a length scale to estimate how deep inside a domain a local diffusion current can spread. We discuss some applications in biophysics.

math.AP↗

arXiv2021

Asymptotics for the fastest among n stochastics particles: role of an extended initial distribution and an additional drift component

We derive asymptotic formulas for the mean exit time $\barτ^{N}$ of the fastest among $N$ identical independently distributed Brownian particles to an absorbing boundary for various initial distributions (partially uniformly and exponentially distributed). Depending on the tail of the initial distribution, we report here a continuous algebraic decay law for $\barτ^{N}$, which differs from the classical Weibull or Gumbell results. We derive asymptotic formulas in dimension 1 and 2, for half-line and an interval that we compare with stochastic simulations. We also obtain formulas for an additive constant drift on the Brownian motion. Finally, we discuss some applications in cell biology where a molecular transduction pathway involves multiple steps and a long-tail initial distribution.

physics.data-an↗