arXiv subjects

Benjamin Lindner

Publications and source records attributed to Benjamin Lindner.

At least 19 recordsLinked to original sources

arXiv2026

Impact of heavy-tailed synaptic strength distributions on self-sustained activity in networks of spiking neurons

We analyze states of stationary activity in randomly coupled quadratic integrate-and-fire neurons using stochastic mean-field theory. Specifically, we consider the two cases of Gaussian random coupling and Cauchy random coupling, which are representative of systems with light- or with heavy-tailed synaptic strength distributions. For both, Gaussian and Cauchy coupling, bistability between a low activity and a high activity state of self-sustained firing is possible in excitable neurons. In the system with Cauchy coupling we find analytically a directed percolation threshold, i.e., above a critical value of the synaptic strength, activity percolates through the whole network starting from a few spiking units only. The existence of the directed percolation threshold is in agreement with previous numerical results in the literature for integrate-and-fire neurons with heavy-tailed synaptic strength distribution. However, we have found that the transition can be continuous or discontinuous, depending on the excitatory-inhibitory imbalance in the network. Networks with Gaussian coupling and networks with Cauchy coupling and additional additive noise lack the percolation transition in the thermodynamic limit.

cond-mat.dis-nn↗

arXiv2026

Consistent thermodynamics reconstructed from transitions between nonequilibrium steady-states

Constructing a thermodynamic framework for nonequilibrium systems remains a major challenge, as quantities such as temperature and free energy often become ambiguous when inferred solely from steady-state properties. Here we take a transformation-based approach and experimentally examine transitions between nonequilibrium steady states (NESS). Using an optically trapped microparticle driven by a tunable correlated stochastic force, we generate active-like steady states with controllable noise statistics. By abruptly changing the trap stiffness, we measure the stochastic work, heat, and entropy produced during NESS-to-NESS transformations. We identify a state-dependent effective temperature that restores the second law for these transitions, enabling the definition of a generalized work that incorporates the consequence of the nonequilibrium fluctuations. With this quantity, we derive and experimentally verify a Crooks-like fluctuation relation linking work distributions to a nonequilibrium free-energy difference defined through the effective temperature. Finally, we establish a fluctuation-response relation for the positional variance following stiffness changes. We demonstrate that this relation is key to distinguishing systems that can be described by a unique effective temperature (i.e., those under equilibrium or white-noise conditions) from those under colored-noise, where an equilibrium-like response cannot be restored. These results delineate the applicability and limits of effective-temperature thermodynamics in driven systems.

cond-mat.stat-mech↗

arXiv2025

Stochastic synaptic dynamics under learning

Learning is based on synaptic plasticity, which affects and is driven by neural activity. Because pre- and postsynaptic spiking activity is shaped by randomness, the synaptic weights follow a stochastic process, requiring a probabilistic framework to capture the noisy synaptic dynamics. We consider a paradigmatic supervised learning example: a presynaptic neural population impinging in a sequence of episodes on a recurrent network of integrate-and-fire neurons through synapses undergoing spike-timing-dependent plasticity (STDP) with additive potentiation and multiplicative depression. We first analytically compute the drift- and diffusion coefficients for a single synapse within a single episode (microscopic dynamics), mapping the true jump process to a Langevin and the associated Fokker-Planck equations. Leveraging new analytical tools, we include spike-time--resolving cross-correlations between pre- and postsynaptic spikes, which corrects substantial deviations seen in standard theories purely based on firing rates. We then apply this microdynamical description to the network setup in which hetero-associations are trained over one-shot episodes into a feed-forward matrix of STDP synapses connecting to neurons of the recurrent network (macroscopic dynamics). By mapping statistically distinct synaptic populations to instances of the single-synapse process above, we self-consistently determine the joint neural and synaptic dynamics and, ultimately, the time course of memory degradation and the memory capacity. We demonstrate that specifically in the relevant case of sparse coding, our theory can quantitatively capture memory capacities which are strongly overestimated if spike-time--resolving cross-correlations are ignored. [...]

q-bio.NC↗

arXiv2025

Probing the limits of effective temperature consistency in actively driven systems

We investigate the thermodynamic properties of a single inertial probe driven into a nonequilibrium steady-state by random collisions with self-propelled active walkers. The probe and walkers are confined within a gravitational harmonic potential. We evaluate the robustness of the effective temperature concept in this active system by comparing values of distinct, independently motivated definitions: a generalized fluctuation-dissipation relation, a kinetic temperature, and a work fluctuation relation. Our experiments reveal that, under specific conditions, these independent measurements yield a remarkably consistent effective temperature over a wide range of system configurations. Furthermore, we also identify regimes where this consistency breaks down, which delineates the fundamental limits of extending equilibrium-like thermodynamic concepts to athermal, actively driven systems.

cond-mat.stat-mech↗

arXiv2025

$Q$-functions, synchronization, and Arnold tongues for coupled stochastic oscillators

Phase reduction is an effective theoretical and numerical tool for studying synchronization of coupled deterministic oscillators. Stochastic oscillators require new definitions of asymptotic phase. The $Q$-function, i.e. the slowest decaying complex mode of the stochastic Koopman operator (SKO), was proposed as a means of phase reduction for stochastic oscillators. In this paper, we show that the $Q$-function approach also leads to a novel definition of ``synchronization" for coupled stochastic oscillators. A system of coupled oscillators in the synchronous regime may be viewed as a single (higher-dimensional) oscillator. Therefore, we investigate the relation between the $Q$-functions of the uncoupled oscillators and the higher-dimensional $Q$-function for the coupled system. We propose a definition of synchronization between coupled stochastic oscillators in terms of the eigenvalue spectrum of Kolmogorov's backward operator (the generator of the Markov process, or the SKO) of the higher dimensional coupled system. We observe a novel type of bifurcation reflecting (i) the relationship between the leading eigenvalues of the SKO for the coupled system and (ii) qualitative changes in the cross-spectral density of the coupled oscillators. Using our proposed definition, we observe synchronization domains for symmetrically-coupled stochastic oscillators that are analogous to Arnold tongues for coupled deterministic oscillators.

math.PR↗

arXiv2025

Effects of inertia on the asynchronous state of a disordered Kuramoto model

We investigate the role of inertia in the asynchronous state of a disordered Kuramoto model. We extend an iterative simulation scheme to the case of the Kuramoto model with inertia in order to determine the self-consistent fluctuation statistics, specifically, the power spectra of network noise and single oscillators. Comparison with network simulations demonstrates that this works well whenever the system is in an asynchronous state. We also find an unexpected effect when varying the degree of inertia: the correlation time of the oscillators becomes minimal at an intermediate mass of the oscillators; correspondingly, the power spectra appear flatter and thus more similar to white noise around the same value of mass. We also find a similar effect for the Lyapunov spectra of the oscillators when the mass is varied.

nlin.CD↗

arXiv2025

Fluctuation-response relations and response-response relations for membrane voltage and spike train of stochastic integrate-and-fire neurons

Neurons display spontaneous spiking (in the absence of stimulus signals) as well as a characteristic response to time-dependent external stimuli. In a simple but important class of stochastic neuron models, the integrate-and-fire model with Gaussian current noise, both aspects can mathematically be related via fluctuation-response relations (FRRs) as has been shown recently. Here we extend the class of FRRs to include the susceptibilities of the membrane voltage and subthreshold voltage nonlinearity as well as the power spectrum of the membrane voltage. For a simple but often considered IF model, the leaky IF model with white Gaussian noise, we exploit the FRRs and derive explicit expressions for the power spectrum and susceptibility of the subthreshold membrane voltage. We also put forward a relation between the response functions of the spike train and membrane voltage, a response-response relation (RRR) that holds true for a more general setting than considered in most parts of the paper. For the generalized IF model with an adaptation current and colored Gaussian noise we derive an FRR and an RRR. We briefly discuss useful applications of the derived FRRs and RRRs.

q-bio.NC↗

arXiv2024

The stabilizing role of multiplicative noise in non-confining potentials

We provide a simple framework for the study of parametric (multiplicative) noise, making use of scale parameters. We show that for a large class of stochastic differential equations increasing the multiplicative noise intensity surprisingly causes the mass of the stationary probability distribution to become increasingly concentrated around the minima of the multiplicative noise term, whilst under quite general conditions exhibiting a kind of intermittent burst like jumps between these minima. If the multiplicative noise term has one zero this causes on-off intermittency. Our framework relies on first term expansions, which become more accurate for larger noise intensities. In this work we show that the full width half maximum in addition to the maximum is appropriate for quantifying the stationary probability distribution (instead of the mean and variance, which are often undefined). We define a corresponding new kind of weak sense stationarity. We consider a double well potential as an example of application, demonstrating relevance to tipping points in noisy systems.

cond-mat.stat-mech↗

arXiv2024

Furutsu-Novikov--like cross-correlation--response relations for systems driven by shot noise

We consider a dynamic system that is driven by an intensity-modulated Poisson process with intensity $\Lambda(t)=\lambda(t)+\varepsilon\nu(t)$. We derive an exact relation between the input-output cross-correlation in the spontaneous state ($\varepsilon=0$) and the linear response to the modulation ($\varepsilon>0$). If $\varepsilon$ is sufficiently small, linear response theory captures the full response. The relation can be regarded as a variant of the Furutsu-Novikov theorem for the case of shot noise. As we show, the relation is still valid in the presence of additional independent noise. Furthermore, we derive an extension to Cox-process input, which provides an instance of colored shot noise. We discuss applications to particle detection and to neuroscience. Using the new relation, we obtain a fluctuation-response-relation for a leaky integrate-and-fire neuron. We also show how the new relation can be used in a remote control problem in a recurrent neural network. The relations are numerically tested for both stationary and non-stationary dynamics. Lastly, extensions to marked Poisson processes and to higher-order statistics are presented.

cond-mat.dis-nn↗

arXiv2024

Phase-Amplitude Description of Stochastic Oscillators: A Parameterization Method Approach

The parameterization method (PM) provides a broad theoretical and numerical foundation for computing invariant manifolds of dynamical systems. PM implements a change of variables in order to represent trajectories of a system of ordinary differential equations ``as simply as possible." In this paper we pursue a similar goal for stochastic oscillator systems. For planar nonlinear stochastic systems that are ``robustly oscillatory", we find a change of variables through which the dynamics are as simple as possible $\textit{in the mean}$. We prove existence and uniqueness of a deterministic vector field, the trajectories of which capture the local mean behavior of the stochastic oscillator. We illustrate the construction of such an ``effective vector field" for several examples, including a limit cycle oscillator perturbed by noise, an excitable system derived from a spiking neuron model, and a spiral sink with noise forcing (2D Ornstein-Uhlenbeck process). The latter examples comprise contingent oscillators that would not sustain rhythmic activity without noise forcing. Finally, we exploit the simplicity of the dynamics after the change of variables to obtain the effective diffusion constant of the resulting phase variable, and the stationary variance of the resulting amplitude (isostable) variable.

math.DS↗

arXiv2024

Self-consistent autocorrelation of a disordered Kuramoto model in the asynchronous state

The Kuramoto model has provided deep insights into synchronization phenomena and remains an important paradigm to study the dynamics of coupled oscillators. Yet, despite its success, the asynchronous regime in the Kuramoto model has received limited attention. Here, we adapt and enhance the mean-field approach originally proposed by Stiller and Radons [Phys. Rev. E 58 (1998)] to study the asynchronous state in the Kuramoto model with a finite number of oscillators and with disordered connectivity. By employing an iterative stochastic mean field (IMF) approximation, the complex N-oscillator system can effectively be reduced to a one-dimensional dynamics, both for homogeneous and heterogeneous networks. This method allows us to investigate the power spectra of individual oscillators as well as of the multiplicative "network noise" in the Kuramoto model in the asynchronous regime. By taking into account the finite system size and disorder in the connectivity, our findings become relevant for the dynamics of coupled oscillators that appear in the context of biological or technical systems.

math-ph↗

arXiv2024

The noise intensity of a Markov chain

Stochastic transitions between discrete microscopic states play an important role in many physical and biological systems. Often, these transitions lead to fluctuations on a macroscopic scale. A classic example from neuroscience is the stochastic opening and closing of ion channels and the resulting fluctuations in membrane current. When the microscopic transitions are fast, the macroscopic fluctuations are nearly uncorrelated and can be fully characterized by their mean and noise intensity. We show how, for an arbitrary Markov chain, the noise intensity can be determined from an algebraic equation, based on the transition rate matrix. We demonstrate the validity of the theory using an analytically tractable two-state Markovian dichotomous noise, an eight-state model for a Calcium channel subunit (De Young-Keizer model), and Markov models of the voltage-gated Sodium and Potassium channels as they appear in a stochastic version of the Hodgkin-Huxley model.

cond-mat.stat-mech↗

arXiv2024

Generalized dynamical phase reduction for stochastic oscillators

Phase reduction is an important tool for studying coupled and driven oscillators. The question of how to generalize phase reduction to stochastic oscillators remains actively debated. In this work, we propose a method to derive a self-contained stochastic phase equation of the form $\mathop{}\!\mathrm{d} \phi = a(\phi)\mathop{}\!\mathrm{d} t + \sqrt{2D(\phi)}\,\mathop{}\!\mathrm{d} W(t)$ that is valid not only for noise-perturbed limit cycles, but also for noise-induced oscillations. We show that our reduction captures the asymptotic statistics of qualitatively different stochastic oscillators, and use it to infer their phase-response properties.

math-ph↗

arXiv2024

The fluctuation-dissipation relation holds for a macroscopic tracer in an active bath

The fluctuation-dissipation relation (FDR) links thermal fluctuations and dissipation at thermal equilibrium through temperature. Extending it beyond equilibrium conditions in pursuit of broadening thermodynamics is often feasible, albeit with system-dependent specific conditions. We demonstrate experimentally that a generalized FDR holds for a harmonically trapped tracer colliding with self-propelled walkers. The generalized FDR remains valid across a large spectrum of active fluctuation frequencies, extending from underdamped to critically damped dynamics, which we attribute to a single primary channel for energy input and dissipation in our system.

cond-mat.stat-mech↗

arXiv2023

Detecting a periodic signal by a population of spiking neurons in the weakly nonlinear response regime

Motivated by experimental observations, we investigate a variant of the cocktail party problem: the detection of a weak periodic stimulus in the presence of fluctuations and another periodic stimulus which is stronger than the periodic signal to be detected. Specifically, we study the response of a population of stochastic leaky integrate-and-fire (LIF) neurons to two periodic signals and focus in particular on the question, whether the presence of one of the stimuli can be detected from the population activity. As a detection criterion, we use a simple threshold-crossing of the population activity over a certain time window. We show by means of the receiver operating characteristics (ROC) that the detectability depends only weakly on the time window of observation but rather strongly on the stimulus amplitude. Counterintuitively, the detection of the weak periodic signal can be facilitated by the presence of a strong periodic input current depending on the frequencies of the two signals and on the dynamical regime in which the neurons operate. Beside numerical simulations of the model we present an analytical approximation for the ROC curve that is based on the weakly nonlinear-response theory for a stochastic LIF neuron. We discuss the validity of this approximation as well as the relevance of our results for a detection problem in weakly electric fish.

physics.bio-ph↗

arXiv2023

Scattering theory of Non-Brownian active particles with social distancing

We consider deterministic self-propelled particles with anti-alignment interactions. An asymptotically exact kinetic theory for particle scattering at low densities is constructed by a non-local closure of the BBGKY-hierarchy, involving pair correlations. We show that the mean-field assumption of molecular chaos yields unphysical predictions, whereas the scattering theory shows excellent agreement with agent-based simulations. To extend the theory to high densities, a self-consistent mapping to a random-telegraph process is performed. The approach is used to derive a one-particle Langevin-equation and leads to analytical expressions for the correlations of its effective noise.

cond-mat.stat-mech↗

arXiv2023

Asymptotically exact scattering theory of the Kuramoto-Vicsek model

We consider the Kuramoto-Vicsek model of self-propelled particles with velocity-alignment interactions. Starting from the exact $N$-particle Liouville equation, a kinetic equation for the one-particle distribution function is obtained in a self-consistent manner. We show that the usual mean-field assumption of molecular chaos leads to qualitatively wrong predictions such as an infinite coefficient of self-diffusion. Going beyond mean-field and applying the refined assumption of \emph{one-sided molecular chaos} where the two-particle-correlations during binary interactions are explicitly taken into account, we analytically calculate the scattering of particles in the limit of low density and obtain explicit expressions for the dynamical noise of an effective one-particle Langevin-equation and the corresponding self-diffusion. The theory is developed in detail for anti-aligning couplings, where exact analytical results are obtainable. In this calculation, the superposition principle of traditional kinetic theory is modified to handle a system with non-Hamiltonian dynamics involving phase-space compression. The predicted theoretical expressions for the relaxation of hydrodynamic modes and the self-diffusion coefficient are in excellent, quantitative agreement with agent-based simulations. At large particle densities, a given particle is constantly approached and abandoned by different collision partners. Modeling this switching by a random telegraph process and exactly solving a self-consistent integral equation, we obtain explicit expressions for the noise correlations of the effective one-particle Langevin-equation. We also consider the effect of frozen disorder in the particle speeds and show how this can be used to calculate the exact Boltzmann collision operator for positive alignment strengths.

cond-mat.stat-mech↗

arXiv2023

A Universal Description of Stochastic Oscillators

Many systems in physics, chemistry and biology exhibit oscillations with a pronounced random component. Such stochastic oscillations can emerge via different mechanisms, for example linear dynamics of a stable focus with fluctuations, limit-cycle systems perturbed by noise, or excitable systems in which random inputs lead to a train of pulses. Despite their diverse origins, the phenomenology of random oscillations can be strikingly similar. Here we introduce a nonlinear transformation of stochastic oscillators to a new complex-valued function $Q^*_1(\textbf{x})$ that greatly simplifies and unifies the mathematical description of the oscillator's spontaneous activity, its response to an external time-dependent perturbation, and the correlation statistics of different oscillators that are weakly coupled. The function $Q^*_1(\textbf{x})$ is the eigenfunction of the Kolmogorov backward operator with the least negative (but non-vanishing) eigenvalue $\lambda_1=\mu_1+i\omega_1$. The resulting power spectrum of the complex-valued function is exactly given by a Lorentz spectrum with peak frequency $\omega_1$ and half-width $\mu_1$; its susceptibility with respect to a weak external forcing is given by a simple one-pole filter, centered around $\omega_1$; and the cross-spectrum between two coupled oscillators can be easily expressed by a combination of the spontaneous power spectra of the uncoupled systems and their susceptibilities. Our approach makes qualitatively different stochastic oscillators comparable, provides simple characteristics for the coherence of the random oscillation, and gives a framework for the description of weakly coupled oscillators.

nlin.AO↗