arXiv subjects

Jonathan Touboul

Publications and source records attributed to Jonathan Touboul.

At least 19 recordsLinked to original sources

arXiv2026

Dynamics and stochastic resonance in a mathematical model of bistable phosphorylation and nuclear size control

Robust oscillations play crucial roles in a wide variety of biological processes and are often generated by deterministic mechanisms. However, stochastic fluctuations often generate complex perturbations of these deterministic oscillations, potentially strengthening or weakening their robustness. In this paper, we study bistable phosphorylation as a mechanism for robust oscillation. We present a simple nucleocytoplasmic transport and cell growth model where cargo proteins undergo bistable phosphorylation prior to nuclear import. We perform a detailed bifurcation analysis to examine the system's dynamical behavior. We then introduce additive noise into the model and study the stochastic resonance behavior and robustness of oscillations under noise. Our results show that, depending on the phosphorylation threshold, time-scale parameters, and nucleocytoplasmic transport rate, bistable phosphorylation may generate oscillations via Hopf bifurcations; moreover, stochastic resonance and Bautin bifurcations enhance the robustness of the oscillations.

math.DS↗

arXiv2025

Dynamic threshold curves and response precision in forced excitable systems

We investigate here various properties of the responses of excitable systems subject to periodic forcing and noise. While the properties of intrinsic oscillators, subject to added periodic signals, are well understood, much less is known about the factors that determine the response precision of excitable units, intrinsically at rest, when activated by periodic forcing and stochastic noise. One motivation for considering this issue comes from the behavior of auditory neurons. These neurons reportedly have the ability to fire spikes in a precise range of phases in response to incoming sound waves, a behavior for which the mechanism is unknown. To account for such a response precision, we introduce the notion of dynamic threshold curve (DTC), which estimates at each time the effective likelihood that noise will subsequently generate a spike. The DTC effectively summarizes, in a single curve, a representation of the response precision of an excitable model, as we demonstrate by showing that the distribution of spike times produced in this setting is well captured by the first passage time of a simple, Gaussian stochastic process to the distance to the DTC. This result shows that peaks and troughs of the DTC, but also their slopes, convey fine information about spike timing in response to noise. In particular, it explains properties of Type 2 and Type 3 excitable cells studied previously and provides a framework to predict the DTC properties necessary to support the response precision of auditory neurons, as we illustrate in a well-established auditory neuron model.

q-bio.NC↗

arXiv2025

Balanced Dynamics in Strongly Coupled Networks

Many mathematical models of interacting agents assume that individual interactions scale down in proportion to the network size, ensuring that the combined input received from the network does not diverge. In theoretical neuroscience, Sompolinsky and Van Vreeswijk proposed in 1996 that, should these scalings be violated (and under appropriate conditions), the system may not diverge but rather approach a balanced state where the inputs to each neuron compensate each other (in neuroscience, where inhibitory currents compensate the excitatory ones). We come back to this observation and formulate here a mathematical conjecture for the occurrence of such behaviors in general stochastic systems of interacting agents. From a mathematical viewpoint, this conjecture can be viewed as a double-limit problem in the space of probability measures, which we discuss in detail, as it provides several possible mathematical avenues for proving this result. We provide some numerical and theoretical explorations of the conjecture in classical models of neuronal networks. Moreover, we provide a complete proof of an asymptotic result consistent with one of the double-limit problems in a one-dimensional model with separable coupling inspired by models of chemically-coupled neurons. This proof relies on asymptotic methods, and particularly desingularization techniques used in some PDEs, that we apply here to the mean-field limit of the network as the coupling is made to diverge. From the applications viewpoint, this theory provides an alternative, minimalistic explanation for the widely observed balance of excitation and inhibition in the cerebral cortex not requiring the assumption of the existence of complex regulatory mechanisms.

math.AP↗

arXiv2025

3D pattern formation of a protein-membrane suspension

Many essential cellular processes, including cell division and the establishment of cell polarity during embryogenesis, are regulated by pattern-forming proteins. These proteins often need to bind to a substrate, such as the cell membrane, onto which they interact and form two-dimensional (2D) patterns. It is unclear how the membrane's continuity and dimensionality impact pattern formation. Here, we address this gap using the MinDE system, a prototypical example of pattern-forming membrane proteins. We show that when the lipid substrate is fragmented into submicrometer-sized diffusive liposomes, ATP-driven protein-protein interactions generate three-dimensional (3D) spatially extended patterns, despite the complete loss of membrane continuity. Remarkably, these 3D patterns emerge at scales four orders of magnitude larger than the individual liposomes. By systematically varying protein concentration, liposome size, and density, we observed and characterized a variety of 3D dynamical patterns not seen on continuous 2D membranes, including traveling waves, dynamical spirals, and a coexistence phase. Simulations and linear stability analysis of a coarse-grained model revealed that the physical properties of the dispersed membrane effectively rescale both the protein-membrane binding rates and diffusion, two key parameters governing pattern formation and wavelength selection. These findings highlight the robustness of Min's pattern-forming ability, suggesting that protein-membrane suspensions could serve as an adaptable template for studying out-of-equilibrium self-organization in 3D, beyond in vivo contexts.

cond-mat.soft↗

arXiv2021

The Symmetry Basis of Pattern Formation in Reaction-Diffusion Networks

In networks of nonlinear oscillators, symmetries place hard constraints on the system that can be exploited to predict universal dynamical features and steady-states, providing a rare generic organizing principle for far-from-equilibrium systems. However, the robustness of this class of theories to symmetry-disrupting imperfections is untested. Here, we develop a model experimental reaction-diffusion network of chemical oscillators to test applications of this theory in the context of self-organizing systems relevant to biology and soft robotics. The network is a ring of 4 identical microreactors containing the oscillatory Belousov-Zhabotinsky reaction coupled to nearest neighbors via diffusion. Assuming perfect symmetry, theory predicts 4 categories of stable spatiotemporal phase-locked periodic states and 4 categories of invariant manifolds that guide and structure transitions between phase-locked states. In our experiments, we observed the predicted symmetry-derived synchronous clustered transients that occur when the dynamical trajectories coincide with invariant manifolds. However, we observe only 3 of the 4 phase-locked states that are predicted for the idealized homogeneous system. Quantitative agreement between experiment and numerical simulations is found by accounting for the small amount of experimentally determined heterogeneity. This work demonstrates that a surprising degree of the network's dynamics are constrained by symmetry in spite of the breakdown of the assumption of homogeneity and raises the question of why heterogeneity destabilizes some symmetry predicted states, but not others.

nlin.PS↗

arXiv2019

The hipster effect: When anticonformists all look the same

In such different domains as neurosciences, spin glasses, social science, economics and finance, large ensemble of interacting individuals following (mainstream) or opposing (hipsters) to the majority are ubiquitous. In these systems, interactions generally occur after specific delays associated to transport, transmission or integration of information. We investigate here the impact of anti-conformism combined to delays in the emergent dynamics of large populations of mainstreams and hipsters. To this purpose, we introduce a class of simple statistical systems of interacting agents composed of (i) mainstreams and anti-conformists in the presence of (ii) delays, possibly heterogeneous, in the transmission of information. In this simple model, each agent can be in one of two states, and can change state in continuous time with a rate depending on the state of others in the past. We express the thermodynamic limit of these systems as the number of agents diverge, and investigate the solutions of the limit equation, with a particular focus on synchronized oscillations induced by delayed interactions. We show that when hipsters are too slow in detecting the trends, they will consistently make the same choice, and realizing this too late, they will switch, all together to another state where they remain alike. Similar synchronizations arise when the impact of mainstreams on hipsters choices (and reciprocally) dominate the impact of other hipsters choices, and we show that these may emerge only when the randomness in the hipsters decisions is sufficiently large. Beyond the choice of the best suit to wear this winter, this study may have important implications in understanding synchronization of nerve cells, investment strategies in finance, or emergent dynamics in social science, domains in which delays of communication and the geometry of information accessibility are prominent.

cond-mat.dis-nn↗

arXiv2017

Mean-field equations for stochastic firing-rate neural fields with delays: Derivation and noise-induced transitions

In this manuscript we analyze the collective behavior of mean-field limits of large-scale, spatially extended stochastic neuronal networks with delays. Rigorously, the asymptotic regime of such systems is characterized by a very intricate stochastic delayed integro-differential McKean-Vlasov equation that remain impenetrable, leaving the stochastic collective dynamics of such networks poorly understood. In order to study these macroscopic dynamics, we analyze networks of firing-rate neurons, i.e. with linear intrinsic dynamics and sigmoidal interactions. In that case, we prove that the solution of the mean-field equation is Gaussian, hence characterized by its two first moments, and that these two quantities satisfy a set of coupled delayed integro-differential equations. These equations are similar to usual neural field equations, and incorporate noise levels as a parameter, allowing analysis of noise-induced transitions. We identify through bifurcation analysis several qualitative transitions due to noise in the mean-field limit. In particular, stabilization of spatially homogeneous solutions, synchronized oscillations, bumps, chaotic dynamics, wave or bump splitting are exhibited and arise from static or dynamic Turing-Hopf bifurcations. These surprising phenomena allow further exploring the role of noise in the nervous system.

math.DS↗

arXiv2017

Wild oscillations in a nonlinear neuron model with resets: (II) Mixed-mode oscillations

This work continues the analysis of complex dynamics in a class of bidimensional nonlinear hybrid dynamical systems with resets modeling neuronal voltage dynamics with adaptation and spike emission. We show that these models can generically display a form of mixed-mode oscillations (MMOs), which are trajectories featuring an alternation of small oscillations with spikes or bursts (multiple consecutive spikes). The mechanism by which these are generated relies fundamentally on the hybrid structure of the flow: invariant manifolds of the continuous dynamics govern small oscillations, while discrete resets govern the emission of spikes or bursts, contrasting with classical MMO mechanisms in ordinary differential equations involving more than three dimensions and generally relying on a timescale separation. The decomposition of mechanisms reveals the geometrical origin of MMOs, allowing a relatively simple classification of points on the reset manifold associated to specific numbers of small oscillations. We show that the MMO pattern can be described through the study of orbits of a discrete adaptation map, which is singular as it features discrete discontinuities with unbounded left- and right-derivatives. We study orbits of the map via rotation theory for discontinuous circle maps and elucidate in detail complex behaviors arising in the case where MMOs display at most one small oscillation between each consecutive pair of spikes.

math.DS↗

arXiv2017

Large deviations for randomly connected neural networks: I. Spatially extended systems

In a series of two papers, we investigate the large deviations and asymptotic behavior of stochastic models of brain neural networks with random interaction coefficients. In this first paper, we take into account the spatial structure of the brain and consider (i) the presence of interaction delays that depend on the distance between cells and (ii) Gaussian random interaction amplitude whose mean and variance depend on the neurons positions and scale as the inverse of the network size. We show that the empirical measure satisfies a large-deviation principle with good rate function reaching its minimum at a unique spatially extended probability measure. This result implies averaged convergence of the empirical measure and propagation of chaos. The limit is characterized through complex non-Markovian implicit equation in which the network interaction term is replaced by a non-local Gaussian process whose statistics depend on the solution over the whole neural field.

math.PR↗

arXiv2017

Large deviations for randomly connected neural networks: II. State-dependent interactions

This work continues the analysis of large deviations for randomly connected neural networks models of the brain. The originality of the model relies on the fact that the directed impact of one particle onto another depends on the state of both particles, and (ii) have random Gaussian amplitude with mean and variance scaling as the inverse of the network size. Similarly to the spatially extended case, we show that under sufficient regularity assumptions, the empirical measure satisfies a large-deviation principle with good rate function achieving its minimum at a unique probability measure, implying in particular its convergence in both averaged and quenched cases, as well as a propagation of chaos property (in the averaged case only). The class of model we consider notably includes a stochastic version of Kuramoto model with random connections.

math.PR↗

arXiv2017

Power-law statistics and universal scaling in the absence of criticality

Critical states are sometimes identified experimentally through power-law statistics or universal scaling functions. We show here that such features naturally emerge from networks in self-sustained irregular regimes away from criticality. In these regimes, statistical physics theory of large interacting systems predict a regime where the nodes have independent and identically distributed dynamics. We thus investigated the statistics of a system in which units are replaced by independent stochastic surrogates, and found the same power-law statistics, indicating that these are not sufficient to establish criticality. We rather suggest that these are universal features of large-scale networks when considered macroscopically. These results put caution on the interpretation of scaling laws found in nature.

q-bio.NC↗

arXiv2016

Propagation of chaos in neural fields

We consider the problem of the limit of bio-inspired spatially extended neuronal networks including an infinite number of neuronal types (space locations), with space-dependent propagation delays modeling neural fields. The propagation of chaos property is proved in this setting under mild assumptions on the neuronal dynamics, valid for most models used in neuroscience, in a mesoscopic limit, the neural-field limit, in which we can resolve the quite fine structure of the neuron's activity in space and where averaging effects occur. The mean-field equations obtained are of a new type: they take the form of well-posed infinite-dimensional delayed integro-differential equations with a nonlocal mean-field term and a singular spatio-temporal Brownian motion. We also show how these intricate equations can be used in practice to uncover mathematically the precise mesoscopic dynamics of the neural field in a particular model where the mean-field equations exactly reduce to deterministic nonlinear delayed integro-differential equations. These results have several theoretical implications in neuroscience we review in the discussion.

math.PR↗

arXiv2016

Real eigenvalues of non-symmetric random matrices: Transitions and Universality

In the past 20 years, the study of real eigenvalues of non-symmetric real random matrices has seen important progress. Notwithstanding, central questions still remain open, such as the characterization of their asymptotic statistics and the universality thereof. In this letter we show that for a wide class of matrices, the number $k_n$ of real eigenvalues of a matrix of size $n$ is asymptotically Gaussian with mean $\bar k_n=\mathcal{O}(\sqrt{n})$ and variance $\bar k_n(2-\sqrt{2})$. Moreover, we show that the limit distribution of real eigenvalues undergoes a transition between bimodal for $k_n=o(\sqrt{n})$ to unimodal for $k_n=\mathcal{O}(n)$, with a uniform distribution at the transition. We predict theoretically these behaviours in the Ginibre ensemble using a log-gas approach, and show numerically that they hold for a wide range of random matrices with independent entries beyond the universality class of the circular law.

math-ph↗

arXiv2015

Limits and dynamics of stochastic neuronal networks with random heterogeneous delays

Realistic networks display heterogeneous transmission delays. We analyze here the limits of large stochastic multi-populations networks with stochastic coupling and random interconnection delays. We show that depending on the nature of the delays distributions, a quenched or averaged propagation of chaos takes place in these networks, and that the network equations converge towards a delayed McKean-Vlasov equation with distributed delays. Our approach is mostly fitted to neuroscience applications. We instantiate in particular a classical neuronal model, the Wilson and Cowan system, and show that the obtained limit equations have Gaussian solutions whose mean and standard deviation satisfy a closed set of coupled delay differential equations in which the distribution of delays and the noise levels appear as parameters. This allows to uncover precisely the effects of noise, delays and coupling on the dynamics of such heterogeneous networks, in particular their role in the emergence of synchronized oscillations. We show in several examples that not only the averaged delay, but also the dispersion, govern the dynamics of such networks.

math-ph↗

arXiv2015

The real Ginibre ensemble with $k = O(n)$ real eigenvalues

We consider the ensemble of Real Ginibre matrices with a positive fraction $α>0$ of real eigenvalues. We demonstrate a large deviations principle for the joint eigenvalue density of such matrices and we introduce a two phase log-gas whose stationary distribution coincides with the spectral measure of the ensemble. Using these tools we provide an asymptotic expansion for the probability $p^n_{αn}$ that an $n\times n$ Ginibre matrix has $k=αn$ real eigenvalues and we characterize the spectral measures of these matrices.

math-ph↗

arXiv2015

Complex oscillations in the delayed Fitzhugh-Nagumo equation

Motivated by the dynamics of neuronal responses, we analyze the dynamics of the Fitzhugh-Nagumo slow-fast system with delayed self-coupling. This system provides a canonical example of a canard explosion for sufficiently small delays. Beyond this regime, delays significantly enrich the dynamics, leading to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a delay-induced subcritical Bogdanov-Takens instability arising at the fold points of the S-shaped critical manifold. Underlying the transition from canard-induced to delay-induced dynamics is an abrupt switch in the nature of the Hopf bifurcation.

math.DS↗

arXiv2015

On the Dynamic Interplay between Positive and Negative Affects

Emotional disorders and psychological flourishing are the result of complex interactions between positive and negative affects that depend on external events and the subject's internal representations. Based on psychological data, we mathematically model the dynamical balance between positive and negative affects as a function of the response to external positive and negative events. This modeling allows the investigation of the relative impact of two leading forms of therapy on affect balance. The model uses a delay differential equation to analytically study the complete bifurcation diagram of the system. We compare the results of the model to psychological data on a single, recurrently depressed patient that was administered the two types of therapies considered (viz., coping-focused vs. affect-focused). The model leads to the prediction that stabilization at a normal state may rely on evaluating one's emotional state through an historical ongoing emotional state rather than in a narrow present window. The simple mathematical model proposed here offers a theoretically grounded quantitative framework for investigating the temporal process of change and parameters of resilience to relapse.

stat.AP↗

arXiv2015

Absorption properties of stochastic equations with Hölder diffusion coefficients

In this article, we address the absorption properties of a class of stochastic differ- ential equations around singular points where both the drift and diffusion functions vanish. According to the Hölder coefficient alpha of the diffusion function around the singular point, we identify different regimes. Stability of the absorbing state, large deviations for the absorption time, existence of stationary or quasi-stationary distributions are discussed. In particular, we show that quasi-stationary distributions only exist for alpha < 3/4, and for alpha in the interval (3/4, 1), no quasi-stationary distribution is found and numerical simulations tend to show that the process conditioned on not being absorbed initiates an almost sure exponential convergence towards the absorbing state (as is demonstrated to be true for alpha = 1). Applications of these results to stochastic bifurcations are discussed.

math.PR↗