arXiv subjects

Thibaud Taillefumier

Publications and source records attributed to Thibaud Taillefumier.

16 recordsLinked to original sources

arXiv2026

A functional limit theorem for additive functionals

We study a general limiting framework for the convergence of sequences of additive functionals of diffusions to Lévy subordinators, and provide explicit sufficient conditions that both ensure convergence and characterize the law of the limit. As an application, we identify a novel limiting regime for Wright-Fisher and Feller diffusions in the reflecting case and describe the corresponding limiting subordinator. This work is motivated by, and has applications in, neuroscience, where reflected diffusions are used to parametrize synchrony in doubly-stochastic models of spiking activity.

math.PR↗

arXiv2025

Physical blowups via buffered time change in a mean-field neural network

Idealized networks of integrate-and-fire neurons with impulse-like interactions obey McKean-Vlasov diffusion equations in the mean-field limit. These equations are prone to blowups: for a strong enough interaction coupling, the mean-field rate of interaction diverges in finite time with a finite fraction of neurons spiking simultaneously, thereby marking a macroscopic synchronous event. Characterizing these blowup singularities analytically is the key to understanding the emergence and persistence of spiking synchrony in mean-field neural models. Such a treatment is possible via time change techniques for a Poissonian variation of the classically considered integrate-and-fire dynamics. However, just as for shock solutions for nonlinear conservation laws, there are several admissible blowup solutions to the corresponding McKean-Vlasov equations. Because of this ambiguity, it is unclear which notion of blowup solutions shall be adopted. Here, we unambiguously define physical blowup dynamics as solutions to a fixed-point problem bearing on the time change associated to the McKean-Vlasov equation. To justify the physicality of this formulation, we introduce a buffering mechanism that regularizes blowup dynamics, while satisfying the same conservation principles as the finite-dimensional particle-system dynamics. We then show that physical blowup dynamics are recovered from these regularized solutions in the limit of vanishing buffering. Our approach also shows that these physical solutions are unique, globally defined, and avoid the so-called eternal blowup phenomenon, as long as the neurons exhibit nonzero, distributed refractory periods, a reasonable modeling assumption.

math.AP↗

arXiv2025

Subthreshold moment analysis of neuronal populations driven by synchronous synaptic inputs

Even when driven by the same stimulus, neuronal responses are well-known to exhibit a striking level of spiking variability. In-vivo electrophysiological recordings also reveal a surprisingly large degree of variability at the subthreshold level. In prior work, we considered biophysically relevant neuronal models to account for the observed magnitude of membrane voltage fluctuations. We found that accounting for these fluctuations requires weak but nonzero synchrony in the spiking activity, in amount that are consistent with experimentally measured spiking correlations. Here we investigate whether such synchrony can explain additional statistical features of the measured neural activity, including neuronal voltage covariability and voltage skewness. Addressing this question involves conducting a generalized moment analysis of conductance-based neurons in response to input drives modeled as correlated jump processes. Technically, we perform such an analysis using fixed-point techniques from queuing theory that are applicable in the stationary regime of activity. We found that weak but nonzero synchrony can consistently explain the experimentally reported voltage covariance and skewness. This confirms the role of synchrony as a primary driver of cortical variability and supports that physiological neural activity emerges as a population-level phenomenon, especially in the spontaneous regime.

q-bio.NC↗

arXiv2023

Exact analysis of the subthreshold variability for conductance-based neuronal models with synchronous synaptic inputs

The spiking activity of neocortical neurons exhibits a striking level of variability, even when these networks are driven by identical stimuli. The approximately Poisson firing of neurons has led to the hypothesis that these neural networks operate in the asynchronous state. In the asynchronous state neurons fire independently from one another, so that the probability that a neuron experience synchronous synaptic inputs is exceedingly low. While the models of asynchronous neurons lead to observed spiking variability, it is not clear whether the asynchronous state can also account for the level of subthreshold membrane potential variability. We propose a new analytical framework to rigorously quantify the subthreshold variability of a single conductance-based neuron in response to synaptic inputs with prescribed degrees of synchrony. Technically we leverage the theory of exchangeability to model input synchrony via jump-process-based synaptic drives; we then perform a moment analysis of the stationary response of a neuronal model with all-or-none conductances that neglects post-spiking reset. As a result, we produce exact, interpretable closed forms for the first two stationary moments of the membrane voltage, with explicit dependence on the input synaptic numbers, strengths, and synchrony. For biophysically relevant parameters, we find that the asynchronous regime only yields realistic subthreshold variability (voltage variance $\simeq 4-9\mathrm{mV^2}$) when driven by a restricted number of large synapses, compatible with strong thalamic drive. By contrast, we find that achieving realistic subthreshold variability with dense cortico-cortical inputs requires including weak but nonzero input synchrony, consistent with measured pairwise spiking correlations.

q-bio.NC↗

arXiv2022

Global solutions with infinitely many blowups in a mean-field neural network

We recently introduced idealized mean-field models for networks of integrate-and-fire neurons with impulse-like interactions -- the so-called delayed Poissonian mean-field models. Such models are prone to blowups: for a strong enough interaction coupling, the mean-field rate of interaction diverges in finite time with a finite fraction of neurons spiking simultaneously. Due to the reset mechanism of integrate-and-fire neurons, these blowups can happen repeatedly, at least in principle. A benefit of considering Poissonian mean-field models is that one can resolve blowups analytically by mapping the original singular dynamics onto uniformly regular dynamics via a time change. Resolving a blowup then amounts to solving the fixed-point problem that implicitly defines the time change, which can be done consistently for a single blowup and for nonzero delays. Here we extend this time-change analysis in two ways: First, we exhibit the existence and uniqueness of explosive solutions with a countable infinity of blowups in the large interaction regime. Second, we show that these delayed solutions specify "physical" explosive solutions in the limit of vanishing delays, which in turn can be explicitly constructed. The first result relies on the fact that blowups are self-sustaining but nonoverlapping in the time-changed picture. The second result follows from the continuity of blowups in the time-changed picture and incidentally implies the existence of periodic solutions. These results are useful to study the emergence of synchrony in neural network models.

math.PR↗

arXiv2022

Characterization of blowups via time change in a mean-field neural network

Idealized networks of integrate-and-fire neurons with impulse-like interactions obey McKean-Vlasov diffusion equations in the mean-field limit. These equations are prone to blowups: for a strong enough interaction coupling, the mean-field rate of interaction diverges in finite time with a finite fraction of neurons spiking simultaneously, thereby marking a macroscopic synchronous event. Characterizing these blowup singularities analytically is the key to understanding the emergence and persistence of spiking synchrony in mean-field neural models. However, such a resolution is hindered by the first-passage nature of the mean-field interaction in classically considered dynamics. Here, we introduce a delayed Poissonian variation of the classical integrate-and-fire dynamics for which blowups are analytically well defined in the mean-field limit. Albeit fundamentally nonlinear, we show that this delayed Poissonian dynamics can be transformed into a noninteracting linear dynamics via a deterministic time change. We specify this time change as the solution of a nonlinear, delayed integral equation via renewal analysis of first-passage problems. This formulation also reveals that the fraction of simultaneously spiking neurons can be determined via a self-consistent, probability-conservation principle about the time-changed linear dynamics. We utilize the proposed framework in a companion paper to show analytically the existence of singular mean-field dynamics with sustained synchrony for large enough interaction coupling.

math.PR↗

arXiv2022

Metastable spiking networks in the replica-mean-field limit

Characterizing metastable neural dynamics in finite-size spiking networks remains a daunting challenge. We propose to address this challenge in the recently introduced replica-mean-field (RMF) limit. In this limit, networks are made of infinitely many replicas of the finite network of interest, but with randomized interactions across replicas. Such randomization renders certain excitatory networks fully tractable at the cost of neglecting activity correlations, but with explicit dependence on the finite size of the neural constituents. However, metastable dynamics typically unfold in networks with mixed inhibition and excitation. Here, we extend the RMF computational framework to point-process-based neural network models with exponential stochastic intensities, allowing for mixed excitation and inhibition. Within this setting, we show that metastable finite-size networks admit multistable RMF limits, which are fully characterized by stationary firing rates. Technically, these stationary rates are determined as the solutions of a set of delayed differential equations under certain regularity conditions that any physical solutions shall satisfy. We solve this original problem by combining the resolvent formalism and singular-perturbation theory. Importantly, we find that these rates specify probabilistic pseudo-equilibria which accurately capture the neural variability observed in the original finite-size network. We also discuss the emergence of metastability as a stochastic bifurcation, which can be interpreted as a static phase transition in the RMF limits. In turn, we expect to leverage the static picture of RMF limits to infer purely dynamical features of metastable finite-size networks, such as the transition rates between pseudo-equilibria.

q-bio.NC↗

arXiv2021

Replica-Mean-Field Limits of Fragmentation-Interaction-Aggregation Processes

Network dynamics with point-process-based interactions are of paramount modeling interest. Unfortunately, most relevant dynamics involve complex graphs of interactions for which an exact computational treatment is impossible. To circumvent this difficulty, the replica-mean-field approach focuses on randomly interacting replicas of the networks of interest. In the limit of an infinite number of replicas, these networks become analytically tractable under the so-called "Poisson Hypothesis". However, in most applications, this hypothesis is only conjectured. Here, we establish the Poisson Hypothesis for a general class of discrete-time, point-process-based dynamics, that we propose to call fragmentation-interaction-aggregation processes, and which are introduced in the present paper. These processes feature a network of nodes, each endowed with a state governing their random activation. Each activation triggers the fragmentation of the activated node state and the transmission of interaction signals to downstream nodes. In turn, the signals received by nodes are aggregated to their state. Our main contribution is a proof of the Poisson Hypothesis for the replica-mean-field version of any network in this class. The proof is obtained by establishing the propagation of asymptotic independence for state variables in the limit of an infinite number of replicas. Discrete time Galves-Löcherbach neural networks are used as a basic instance and illustration of our analysis.

math.PR↗

arXiv2020

The Pair-Replica-Mean-Field Limit for Intensity-based Neural Networks

Replica-mean-field models have been proposed to decipher the activity of neural networks via a multiply-and-conquer approach. In this approach, one considers limit networks made of infinitely many replicas with the same basic neural structure as that of the network of interest, but exchanging spikes in a randomized manner. The key point is that these replica-mean-field networks are tractable versions that retain important features of the finite structure of interest. To date, the replica framework has been discussed for first-order models, whereby elementary replica constituents are single neurons with independent Poisson inputs. Here, we extend this replica framework to allow elementary replica constituents to be composite objects, namely, pairs of neurons. As they include pairwise interactions, these pair-replica models exhibit non-trivial dependencies in their stationary dynamics, which cannot be captured by first-order replica models. Our contributions are two-fold: $(i)$ We analytically characterize the stationary dynamics of a pair of intensity-based neurons with independent Poisson input. This analysis involves the reduction of a boundary-value problem related to a two-dimensional transport equation to a system of Fredholm integral equations---a result of independent interest. $(ii)$ We analyze the set of consistency equations determining the full network dynamics of certain replica limits. These limits are those for which replica constituents, be they single neurons or pairs of neurons, form a partition of the network of interest. Both analyses are numerically validated by computing input/output transfer functions for neuronal pairs and by computing the correlation structure of certain pair-dominated network dynamics.

math.PR↗

arXiv2019

Replica-mean-field limits for intensity-based neural networks

Neural computations emerge from myriads of neuronal interactions occurring in intricate spiking networks. Due to the inherent complexity of neural models, relating the spiking activity of a network to its structure requires simplifying assumptions, such as considering models in the thermodynamic mean-field limit. In the thermodynamic mean-field limit, an infinite number of neurons interact via vanishingly small interactions, thereby erasing the finite size of interactions. To better capture the finite-size effects of interactions, we propose to analyze the activity of neural networks in the replica-mean-field limit. Replica-mean-field models are made of infinitely many replicas which interact according to the same basic structure as that of the finite network of interest. Here, we analytically characterize the stationary dynamics of an intensity-based neural network with spiking reset and heterogeneous excitatory synapses in the replica-mean-field limit. Specifically, we functionally characterize the stationary dynamics of these limit networks via ordinary differential equations derived from the Poisson Hypothesis of queuing theory. We then reduce this functional characterization to a system of self-consistency equations specifying the stationary neuronal firing rates. Of general applicability, our approach combines rate-conservation principles from point-process theory and analytical considerations from generating-function methods. We validate our approach by demonstrating numerically that replica-mean-field models better capture the dynamics of neural networks with large, sparse connections than their thermodynamic counterparts. Finally, we explain that improved performance by analyzing the neuronal rate-transfer functions, which saturate due to finite-size effects in the replica-mean-field limit.

math.DS↗

arXiv2016

Bacterial cartels at steady supply

Metagenomics has revealed hundreds of bacterial species in almost all microbiota. In a few well-studied cases, bacterial communities have been observed to coordinate their metabolic fluxes. In principle, bacteria can divide tasks to reap the benefits of specialization, as in human economies. However, the benefits and stability of an economy of bacterial specialists are far from obvious. Here, we physically model the population dynamics of bacteria that compete for steadily supplied resources. Importantly, we explicitly model the metabolic fluxes yielding cellular biomass production under the constraint of a limited enzyme budget. In our framework, we find that population dynamics generally leads to the coexistence of different metabolic types, which satisfy an extended competitive exclusion principle (even allowing for adaptive mutation). We establish that these consortia act as cartels, whereby population dynamics pins down resource concentrations at values for which no other strategy can invade. Finally, we propose that at steady supply, cartels of competing strategies automatically yield maximum biomass, thereby achieving a collective optimum.

q-bio.PE↗

arXiv2015

Optimal census by quorum sensing

Quorum sensing is the regulation of gene expression in response to changes in cell density. To measure their cell density, bacterial populations produce and detect diffusible molecules called autoinducers. Individual bacteria internally represent the external concentration of autoinducers via the level of monitor proteins. In turn, these monitor proteins typically regulate both their own production and the production of autoinducers, thereby establishing internal and external feedbacks. Here, we ask whether feedbacks can increase the information available to cells about their local density. We quantify available information as the mutual information between the abundance of a monitor protein and the local cell density for biologically relevant models of quorum sensing. Using variational methods, we demonstrate that feedbacks can increase information transmission, allowing bacteria to resolve up to two additional ranges of cell density when compared with bistable quorum-sensing systems. Our analysis is relevant to multi-agent systems that track an external driver implicitly via an endogenously generated signal.

q-bio.MN↗

arXiv2012

A phase transition in the first passage of a Brownian process through a fluctuating boundary: implications for neural coding

Finding the first time a fluctuating quantity reaches a given boundary is a deceptively simple-looking problem of vast practical importance in physics, biology, chemistry, neuroscience, economics and industry. Problems in which the bound to be traversed is itself a fluctuating function of time include widely studied settings in neural coding, such as neuronal integrators with irregular inputs and internal noise. We show that the probability p(t) that a Gauss-Markov process will first exceed the boundary at time t suffers a phase transition as a function of the roughness of the boundary, as measured by its Hölder exponent H, with critical value Hc = 1/2. For smoother boundaries, H > 1/2, the probability density is a continuous func- tion of time. For rougher boundaries, H < 1/2, the probability is concentrated on a Cantor-like set of zero measure: the probability density becomes divergent, almost everywhere either zero or infin- ity. The critical point Hc = 1/2 corresponds to a widely-studied case in the theory of neural coding, where the external input integrated by a model neuron is a white-noise process, such as uncorrelated but precisely balanced excitatory and inhibitory inputs. We argue this transition corresponds to a sharp boundary between rate codes, in which the neural firing probability varies smoothly, and temporal codes, in which the neuron fires at sharply-defined times regardless of the intensity of internal noise.

q-bio.NC↗

arXiv2010

Multi-Resolution Schauder Approach to Multidimensional Gauss-Markov Processes

The study of multidimensional stochastic processes involves complex computations in intricate functional spaces. In particular, the diffusion processes, which include the practically important Gauss-Markov processes, are ordinarily defined through the theory of stochastic integration. Here, inspired by the Lévy-Cieselski construction of the Wiener process, we propose an alternative representation of multidimensional Gauss-Markov processes as expansions on well-chosen Schauder bases, with independent random coefficients of normal law with zero mean and unitary variance. We thereby offer a natural multi-resolution description of Gauss-Markov processes as limits of the finite-dimensional partial sums of the expansion, that are strongly almost-surely convergent. Moreover, such finite-dimensional random processes constitute an optimal approximation of the process, in the sense of minimizing the associated Dirichlet energy under interpolating constraints. This approach allows simpler treatment in many applied and theoretical fields and we provide a short overview of applications we are currently developing.

math.PR↗

arXiv2008

A Discrete Construction for Gaussian Markov Processes

In the Lévy construction of Brownian motion, a Haar-derived basis of functions is used to form a finite-dimensional process $W^{N}$ and to define the Wiener process as the almost sure path-wise limit of $W^{N}$ when $N$ tends to infinity. We generalize such a construction to the class of centered Gaussian Markov processes $X$ which can be written $X_{t} = g(t) \cdot \int_{0}^{t} f(t) dW_{t}$ with $f$ and $g$ being continuous functions. We build the finite-dimensional process $X^{N}$ so that it gives an exact representation of the conditional expectation of $X$ with respect to the filtration generated by ${\lbrace X_{k/2^{N}}\rbrace}$ for $0 \leq k \leq 2^{N}$. Moreover, we prove that the process $X^{N}$ converges in distribution toward $X$.

math.PR↗

arXiv2007

A Haar-like Construction for the Ornstein Uhlenbeck Process

The classical Haar construction of Brownian motion uses a binary tree of triangular wedge-shaped functions. This basis has compactness properties which make it especially suited for certain classes of numerical algorithms. We present a similar basis for the Ornstein-Uhlenbeck process, in which the basis elements approach asymptotically the Haar functions as the index increases, and preserve the following properties of the Haar basis: all basis elements have compact support on an open interval with dyadic rational endpoints; these intervals are nested and become smaller for larger indices of the basis element, and for any dyadic rational, only a finite number of basis elements is nonzero at that number. Thus the expansion in our basis, when evaluated at a dyadic rational, terminates in a finite number of steps. We prove the covariance formulae for our expansion and discuss its statistical interpretation and connections to asymptotic scale invariance.

math.PR↗