arXiv subjects

Łukasz Kuśmierz

Publications and source records attributed to Łukasz Kuśmierz.

11 recordsLinked to original sources

arXiv2025

Identifying Stochastic Dynamics from Non-Sequential Data (DyNoSeD)

Inferring stochastic dynamics from data is central across the sciences, yet in many applications only unordered, non-sequential measurements are available-often restricted to limited regions of state space-so standard time-series methods do not apply. We introduce DyNoSeD, a first-principles framework that identifies unknown dynamical parameters from such non-sequential data by minimizing Fokker-Planck residuals. We develop two complementary routes: a local route that handles region-restricted data via locally estimated scores, and a global route that fits dynamics from globally sampled data using a kernel Stein discrepancy without explicit density or score estimation. When the dynamics are affine in the unknown parameters, we prove a necessary-and-sufficient condition for the existence and uniqueness of the inferred parameters and derive a sensitivity analysis that identifies which parameters are tightly constrained by the data and which remain effectively free under over-parameterization. For general non-affine case, both routes define differentiable losses amenable to gradient-based optimization. As demonstrations, we recover (i) the three parameters of a stochastic Lorenz system from non-sequential data (region-restricted data for the local route and full steady-state data for the global route) and (ii) a 3x7interaction matrix of a nonlinear gene-regulatory network derived from a published B-cell differentiation model, using only unordered steady-state samples and applying the global route. Finally, we show that the same Fokker-Planck residual viewpoint supports a "dynamics-to-density" complement that trains a normalized density estimator directly from known dynamics without any observations. Overall, IDyNSD provides two first-principles routes for system-identification from non-sequential data, grounded in the Fokker-Planck equation, that link data, density, and stochastic dynamics.

nlin.CD↗

arXiv2025

Slow Transition to Low-Dimensional Chaos in Heavy-Tailed Recurrent Neural Networks

Growing evidence suggests that synaptic weights in the brain follow heavy-tailed distributions, yet most theoretical analyses of recurrent neural networks (RNNs) assume Gaussian connectivity. We systematically study the activity of RNNs with random weights drawn from biologically plausible Lévy alpha-stable distributions. While mean-field theory for the infinite system predicts that the quiescent state is always unstable -- implying ubiquitous chaos -- our finite-size analysis reveals a sharp transition between quiescent and chaotic dynamics. We theoretically predict the gain at which the system transitions from quiescent to chaotic dynamics, and validate it through simulations. Compared to Gaussian networks, heavy-tailed RNNs exhibit a broader parameter regime near the edge of chaos, namely a slow transition to chaos. However, this robustness comes with a tradeoff: heavier tails reduce the Lyapunov dimension of the attractor, indicating lower effective dimensionality. Our results reveal a biologically aligned tradeoff between the robustness of dynamics near the edge of chaos and the richness of high-dimensional neural activity. By analytically characterizing the transition point in finite-size networks -- where mean-field theory breaks down -- we provide a tractable framework for understanding dynamics in realistically sized, heavy-tailed neural circuits.

q-bio.NC↗

arXiv2025

Hierarchy of chaotic dynamics in random modular networks

We introduce a model of randomly connected neural populations and study its dynamics by means of the dynamical mean-field theory and simulations. Our analysis uncovers a rich phase diagram, featuring high- and low-dimensional chaotic phases, separated by a crossover region characterized by low values of the maximal Lyapunov exponent and participation ratio dimension, but with high values of the Lyapunov dimension that change significantly across the region. Counterintuitively, chaos can be attenuated by either adding noise to strongly modular connectivity or by introducing modularity into random connectivity. Extending the model to include a multilevel, hierarchical connectivity reveals that a loose balance between activities across levels drives the system towards the edge of chaos.

physics.bio-ph↗

arXiv2020

Search with home returns provides advantage under high uncertainty

Many search processes are conducted in the vicinity of a favored location, i.e., a home, which is visited repeatedly. Foraging animals return to their dens and nests to rest, scouts return to their bases to resupply, and drones return to their docking stations to recharge or refuel. Yet, despite its prevalence, very little is known about search with home returns as its analysis is much more challenging than that of unconstrained, free-range, search. Here, we develop a theoretical framework for search with home returns. This makes no assumptions on the underlying search process and is furthermore suited to treat generic return and home-stay strategies. We show that the solution to the home-return problem can then be given in terms of the solution to the corresponding free-range problem---which not only reduces overall complexity but also gives rise to a simple, and universal, phase-diagram for search. The latter reveals that search with home returns outperforms free-range search in conditions of high uncertainty. Thus, when living gets rough, a home will not only provide warmth and shelter but also allow one to locate food and other resources quickly and more efficiently than in its absence.

cond-mat.stat-mech↗

arXiv2020

Edge of chaos and avalanches in neural networks with heavy-tailed synaptic weight distribution

We propose an analytically tractable neural connectivity model with power-law distributed synaptic strengths. When threshold neurons with biologically plausible number of incoming connections are considered, our model features a continuous transition to chaos and can reproduce biologically relevant low activity levels and scale-free avalanches, i.e. bursts of activity with power-law distributions of sizes and lifetimes. In contrast, the Gaussian counterpart exhibits a discontinuous transition to chaos and thus cannot be poised near the edge of chaos. We validate our predictions in simulations of networks of binary as well as leaky integrate-and-fire neurons. Our results suggest that heavy-tailed synaptic distribution may form a weakly informative sparse-connectivity prior that can be useful in biological and artificial adaptive systems.

physics.bio-ph↗

arXiv2019

Robust random search with scale-free stochastic resetting

A new model of search based on stochastic resetting is introduced, wherein rate of resets depends explicitly on time elapsed since the beginning of the process. It is shown that rate inversely proportional to time leads to paradoxical diffusion which mixes self-similarity and linear growth of the mean square displacement with non-locality and non-Gaussian propagator. It is argued that such resetting protocol offers a general and efficient search-boosting method that does not need to be optimized with respect to the scale of the underlying search problem (e.g., distance to the goal) and is not very sensitive to other search parameters. Both subdiffusive and superdiffusive regimes of the mean squared displacement scaling are demonstrated with more general rate functions.

cond-mat.stat-mech↗

arXiv2019

Invariants of motion with stochastic resetting and space-time coupled returns

Motion under stochastic resetting serves to model a myriad of processes in physics and beyond, but in most cases studied to date resetting to the origin was assumed to take zero time or a time decoupled from the spatial position at the resetting moment. However, in our world, getting from one place to another always takes time and places that are further away take more time to be reached. We thus set off to extend the theory of stochastic resetting such that it would account for this inherent spatio-temporal coupling. We consider a particle that starts at the origin and follows a certain law of stochastic motion until it is interrupted at some random time. The particle then returns to the origin via a prescribed protocol. We study this model and surprisingly discover that the shape of the steady-state distribution which governs the stochastic motion phase does not depend on the return protocol. This shape invariance then gives rise to a simple, and generic, recipe for the computation of the full steady-state distribution. Several case studies are analyzed and a class of processes whose steady-state is completely invariant with respect to the speed of return is highlighted. For processes in this class we recover the same steady-state obtained for resetting with instantaneous returns---irrespective of whether the actual return speed is high or low. Our work significantly extends previous results on motion with stochastic resetting and is expected to find various applications in statistical, chemical, and biological physics.

cond-mat.stat-mech↗

arXiv2019

Diffusion with stochastic resetting is invariant to return speed

The canonical Evans--Majumdar model for diffusion with stochastic resetting to the origin assumes that resetting takes zero time: upon resetting the diffusing particle is teleported back to the origin to start its motion anew. However, in reality, getting from one place to another takes a finite amount of time which must be accounted for as diffusion with resetting already serves as a model for a myriad of processes in physics and beyond. Here we consider a situation where upon resetting the diffusing particle returns to the origin at a finite (rather than infinite) speed. This creates a coupling between the particle's random position at the moment of resetting and its return time, and further gives rise to a non-trivial cross-talk between two separate phases of motion: the diffusive phase and the return phase. We show that each of these phases relaxes to the stead-state in a unique manner; and while this could have also rendered the total relaxation dynamics extremely non-trivial our analysis surprisingly reveals otherwise. Indeed, the time-dependent distribution describing the particle's position in our model is completely invariant to the speed of return. Thus, whether returns are slow or fast, we always recover the result originally obtained for diffusion with instantaneous returns to the origin.

cond-mat.stat-mech↗

arXiv2019

On subdiffusive continuous time random walks with stochastic resetting

We analyze two models of subdiffusion with stochastic resetting. Each of them consists of two parts: subdiffusion based on the continuous-time random walk (CTRW) scheme and independent resetting events generated uniformly in time according to the Poisson point process. In the first model the whole process is reset to the initial state, whereas in the second model only the position is subject to resets. The distinction between these two models arises from the non-Markovian character of the subdiffusive process. We derive exact expressions for the two lowest moments of the full propagator, stationary distributions, and first hitting times statistics. We also show, with an example of a constant drift, how these models can be generalized to include external forces. Possible applications to data analysis and modeling of biological systems are also discussed.

cond-mat.stat-mech↗

arXiv2017

Emergence of Lévy walks from second order stochastic optimization

In natural foraging, many organisms seem to perform two different types of motile search: directed search (taxis) and random search. The former is observed when the environment provides cues to guide motion towards a target. The latter involves no apparent memory or information processing and can be mathematically modeled by random walks. We show that both types of search can be generated by a common mechanism in which Lévy flights or Lévy walks emerge from a second-order gradient-based search with noisy observations. No explicit switching mechanism is required -- instead, continuous transitions between the directed and random motions emerge depending on the Hessian matrix of the cost function. For a wide range of scenarios the Lévy tail index is $α=1$, consistent with previous observations in foraging organisms. These results suggest that adopting a second-order optimization method can be a useful strategy to combine efficient features of directed and random search.

cond-mat.stat-mech↗

arXiv2017

Optimal potentials for diffusive search strategies

We consider one dimensional diffusive search strategies subjected to external potentials. The location of a single target is drawn from a given probability density function (PDF) $f_G(x)$ and is fixed for each stochastic realization of the process. We optimize the quality of the search strategy as measured by the mean first passage time (MFPT) to the position of the target. For a symmetric but otherwise arbitrary distribution $f_G(x)$ we find the optimal potential that minimizes the MFPT. The minimal MFPT is given by a nonstandard measure of the dispersion, which can be related to the cumulative Rényi entropy. We compare optimal times in this model with optimal times obtained for the model of diffusion with stochastic resetting, in which the diffusive motion is interrupted by intermittent jumps (resets) to the initial position. Additionally, we discuss an analogy between our results and a so-called square-root principle.

cond-mat.stat-mech↗