arXiv subjects

Ernest Montbrió

Publications and source records attributed to Ernest Montbrió.

At least 19 recordsLinked to original sources

arXiv2026

Synchronization modes in bipartite oscillator networks

Collective oscillations in neuronal systems often arise from interactions between excitatory and inhibitory populations rather than from recurrent coupling within a single ensemble. Motivated by the coexistence of strongly and partially synchronized regimes in such systems, we study the Kuramoto Sakaguchi model on a bipartite network. Despite its minimal structure, the model exhibits rich collective dynamics, including both continuous and discontinuous transitions from full synchrony to partial synchrony (PS). In the PS regime, global oscillations fail to entrain one of the two populations, whose oscillators display quasiperiodic dynamics with an average frequency that can significantly deviate from that of the global field, as observed in neuronal networks. We show that this PS state constitutes an example of self-organized quasiperiodicity, arising here in the canonical Kuramoto Sakaguchi model despite its purely linear global coupling.

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arXiv2026

Low-dimensional model for adaptive networks of spiking neurons

We investigate a large ensemble of Quadratic Integrate-and-Fire (QIF) neurons with heterogeneous input currents and adaptation variables. Our analysis reveals that for a specific class of adaptation, termed quadratic spike-frequency adaptation (QSFA), the high-dimensional system can be exactly reduced to a low-dimensional system of ordinary differential equations, which describes the dynamics of three mean-field variables: the population's firing rate, the mean membrane potential, and a mean adaptation variable. The resulting low-dimensional firing rate equations (FRE) uncover a key generic feature of heterogeneous networks with spike frequency adaptation: Both the center and the width of the distribution of the neurons' firing frequencies are reduced, and this largely promotes the emergence of collective synchronization in the network. Our findings are further supported by the bifurcation analysis of the FRE, which accurately captures the collective dynamics of the spiking neuron network, including phenomena such as collective oscillations, bursting, and macroscopic chaos.

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arXiv2024

Heterogeneous populations of quadratic integrate-and-fire neurons: on the generality of Lorentzian distributions

Over the last decade, next-generation neural mass models have become increasingly prominent in mathematical neuroscience. These models link microscopic dynamics with low-dimensional systems of so-called firing rate equations that exactly capture the collective dynamics of large populations of heterogeneous quadratic integrate-and-fire (QIF) neurons. A particularly tractable type of heterogeneity is the distribution of the QIF neurons' excitability parameters, or inputs, according to a Lorentzian. While other distributions -- such as those approximating Gaussian or uniform distributions -- admit to exact mean-field reductions, they result in more complex firing rate equations that are challenging to analyze, and it remains unclear whether they produce comparable collective dynamics. Here, we first demonstrate why Lorentzian heterogeneity is analytically favorable and, second, identify when it leads to qualitatively different collective dynamics compared to other types of heterogeneity. A stationary mean-field approach enables us to derive explicit expressions for the distributions of the neurons' firing rates and voltages in macroscopic stationary states with arbitrary heterogeneities. We also explicate the exclusive relationship between Lorentzian distributed inputs and Lorentzian distributed voltages, whose width happens to coincide with the population firing rate. A dynamic mean-field approach for unimodal heterogeneities further allows us to comprehensively analyze and compare collective dynamics. We find that different types of heterogeneity typically yield qualitatively similar dynamics. However, when gap junction coupling is present, Lorentzian heterogeneity induces nonuniversal behavior, obscuring a diversity-induced transition to synchrony.

q-bio.NC↗

arXiv2024

Exact low-dimensional description for fast neural oscillations with low firing rates

Recently, low-dimensional models of neuronal activity have been exactly derived for large networks of deterministic, Quadratic Integrate-and-Fire (QIF) neurons. Such firing rate models (FRM) describe the emergence of fast collective oscillations (>30~Hz) via the frequency-locking of a subset of neurons to the global oscillation frequency. However, the suitability of such models to describe realistic neuronal states is seriously challenged by fact that during episodes of fast collective oscillations, neuronal discharges are often very irregular and have low firing rates compared to the global oscillation frequency. Here we extend the theory to derive exact FRM for QIF neurons to include noise, and show that networks of stochastic neurons displaying irregular discharges at low firing rates during episodes of fast oscillations, are governed by exactly the same evolution equations as deterministic networks. Our results reconcile two traditionally confronted views on neuronal synchronization, and upgrade the applicability of exact FRM to describe a broad range of biologically realistic neuronal states.

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arXiv2022

Kuramoto model for populations of quadratic integrate-and-fire neurons with chemical and electrical coupling

We derive the Kuramoto model (KM) corresponding to a population of weakly coupled, nearly identical quadratic integrate-and-fire (QIF) neurons with both electrical and chemical coupling. The ratio of chemical to electrical coupling determines the phase lag of the characteristic sine coupling function of the KM and critically determines the synchronization properties of the network. We apply our results to investigate chimera states in two coupled populations of identical QIF neurons. We find that the presence of both electrical and chemical coupling is a necessary condition for chimera states to exist. Finally, we numerically demonstrate that chimera states gradually disappear as coupling strengths cease to be weak.

nlin.AO↗

arXiv2020

Exact mean-field theory explains the dual role of electrical synapses in collective synchronization

Electrical synapses play a major role in setting up neuronal synchronization, but the precise mechanisms whereby these synapses contribute to synchrony are subtle and remain elusive. To investigate these mechanisms mean-field theories for quadratic integrate-and-fire neurons with electrical synapses have been recently put forward. Still, the validity of these theories is controversial since they assume that the neurons produce unrealistic, symmetric spikes, ignoring the well-known impact of spike shape on synchronization. Here we show that the assumption of symmetric spikes can be relaxed in such theories. The resulting mean-field equations reveal a dual role of electrical synapses: First, they equalize membrane potentials favoring the emergence of synchrony. Second, electrical synapses act as "virtual chemical synapses", which can be either excitatory or inhibitory depending upon the spike shape. Our results offer a precise mathematical explanation of the intricate effect of electrical synapses in collective synchronization. This reconciles previous theoretical and numerical works, and confirms the suitability of recent low-dimensional mean-field theories to investigate electrically coupled neuronal networks.

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arXiv2019

An exact firing rate model reveals the differential effects of chemical versus electrical synapses in spiking networks

Chemical and electrical synapses shape the dynamics of neuronal networks. Numerous theoretical studies have investigated how each of these types of synapses contributes to the generation of neuronal oscillations, but their combined effect is less understood. This limitation is further magnified by the impossibility of traditional neuronal mean-field models ---also known as firing rate models, or firing rate equations--- to account for electrical synapses. Here we introduce a novel firing rate model that exactly describes the mean field dynamics of heterogeneous populations of quadratic integrate-and-fire (QIF) neurons with both chemical and electrical synapses. The mathematical analysis of the firing rate model reveals a well-established bifurcation scenario for networks with chemical synapses, characterized by a codimension-2 Cusp point and persistent states for strong recurrent excitatory coupling. The inclusion of electrical coupling generally implies neuronal synchrony by virtue of a supercritical Hopf bifurcation. This transforms the Cusp scenario into a bifurcation scenario characterized by three codimension-2 points (Cusp, Takens-Bogdanov, and Saddle-Node Separatrix Loop), which greatly reduces the possibility for persistent states. This is generic for heterogeneous QIF networks with both chemical and electrical coupling. Our results agree with several numerical studies on the dynamics of large networks of heterogeneous spiking neurons with electrical and chemical coupling.

nlin.AO↗

arXiv2018

Dynamics of a large system of spiking neurons with synaptic delay

We analyze a large system of heterogeneous quadratic integrate-and-fire (QIF) neurons with time delayed, all-to-all synaptic coupling. The model is exactly reduced to a system of firing rate equations that is exploited to investigate the existence, stability and bifurcations of fully synchronous, partially synchronous, and incoherent states. In conjunction with this analysis we perform extensive numerical simulations of the original network of QIF neurons, and determine the relation between the macroscopic and microscopic states for partially synchronous states. The results are summarized in two phase diagrams, for homogeneous and heterogeneous populations, which are obtained analytically to a large extent. For excitatory coupling, the phase diagram is remarkably similar to that of the Kuramoto model with time delays, although here the stability boundaries extend to regions in parameter space where the neurons are not self-sustained oscillators. In contrast, the structure of the boundaries for inhibitory coupling is different, and already for homogeneous networks unveils the presence of various partially synchronized states not present in the Kuramoto model: Collective chaos, quasiperiodic partial synchronization (QPS), and a novel state which we call modulated-QPS (M-QPS). In the presence of heterogeneity partially synchronized states reminiscent to collective chaos, QPS and M-QPS persist. In addition, the presence of heterogeneity greatly amplifies the differences between the incoherence stability boundaries of excitation and inhibition. Finally, we compare our results with those of a traditional (Wilson Cowan-type) firing rate model with time delays. The oscillatory instabilities of the traditional firing rate model qualitatively agree with our results only for the case of inhibitory coupling with strong heterogeneity.

nlin.AO↗

arXiv2018

The Winfree model with heterogeneous phase-response curves: Analytical results

We study an extension of the Winfree model of coupled phase oscillators in which both natural frequencies and phase-response curves (PRCs) are heterogeneous. In the first part of the paper we resort to averaging and derive an approximate model, in which the oscillators are coupled through their phase differences. Remarkably, this simplified model is the 'Kuramoto model with distributed shear' (2011 Phys. Rev. Lett. 106 254101). We find that above a critical level of PRC heterogeneity the incoherent state is always stable. In the second part of the paper we perform the analysis of the full model for Lorentzian heterogeneities, resorting to the Ott-Antonsen ansatz. The critical level of PRC heterogeneity obtained within the averaging approximation has a different manifestation in the full model depending on the sign of the center of the distribution of PRCs.

nlin.AO↗

arXiv2018

Kuramoto model for excitation-inhibition-based oscillations

The Kuramoto model (KM) is a theoretical paradigm for investigating the emergence of rhythmic activity in large populations of oscillators. A remarkable example of rhythmogenesis is the feedback loop between excitatory (E) and inhibitory (I) cells in large neuronal networks. Yet, although the EI-feedback mechanism plays a central role in the generation of brain oscillations, it remains unexplored whether the KM has enough biological realism to describe it. Here we derive a two-population KM that fully accounts for the onset of EI-based neuronal rhythms and that, as the original KM, is analytically solvable to a large extent. Our results provide a powerful theoretical tool for the analysis of large-scale neuronal oscillations.

nlin.AO↗

arXiv2018

Firing rate equations require a spike synchrony mechanism to correctly describe fast oscillations in inhibitory networks

Recurrently coupled networks of inhibitory neurons robustly generate oscillations in the gamma band. Nonetheless, the corresponding Wilson-Cowan type firing rate equation for such an inhibitory population does not generate such oscillations without an explicit time delay. We show that this discrepancy is due to a voltage-dependent spike-synchronization mechanism inherent in networks of spiking neurons which is not captured by standard firing rate equations. Here we investigate an exact low-dimensional description for a network of heterogeneous canonical type-I inhibitory neurons which includes the sub-threshold dynamics crucial for generating synchronous states. In the limit of slow synaptic kinetics the spike-synchrony mechanism is suppressed and the standard Wilson-Cowan equations are formally recovered as long as external inputs are also slow. However, even in this limit synchronous spiking can be elicited by inputs which fluctuate on a time-scale of the membrane time-constant of the neurons. Our meanfield equations therefore represent an extension of the standard Wilson-Cowan equations in which spike synchrony is also correctly described.

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arXiv2017

Synchrony-induced modes of oscillation of a neural field model

We investigate the modes of oscillation of heterogeneous ring-networks of quadratic integrate-and-fire neurons with non-local, space-dependent coupling. Perturbations of the equilibrium state with a particular wave number produce transient standing waves with a specific frequency, analogous to those in a tense string. In the neuronal network, the equilibrium corresponds to a spatially homogeneous, asynchronous state. Perturbations of this state excite the network's oscillatory modes, which reflect the interplay of episodes of synchronous spiking with the excitatory-inhibitory spatial interactions. In the thermodynamic limit, an exact low-dimensional neural field model describing the macroscopic dynamics of the network is derived. This allows us to obtain formulas for the Turing eigenvalues of the spatially-homogeneous state, and hence to obtain its stability boundary. We find that the frequency of each Turing mode depends on the corresponding Fourier coefficient of the synaptic pattern of connectivity. The decay rate instead, is identical for all oscillation modes as a consequence of the heterogeneity-induced desynchronization of the neurons. Finally, we numerically compute the spectrum of spatially-inhomogeneous solutions branching from the Turing bifurcation, showing that similar oscillatory modes operate in neural bump states, and are maintained away from onset.

q-bio.NC↗

arXiv2017

Synchronization scenarios in the Winfree model of coupled oscillators

The emergence of collective synchronization was reproduced long ago by Winfree in a classical model consisting of an ensemble of pulse-coupled phase oscillators. By means of the Ott-Antonsen ansatz, we derive an exact low-dimensional representation which is exhaustively investigated for a variety of pulse types and phase response curves (PRCs). Two structurally different synchronization scenarios are found, which are linked via the mutation of a Bogdanov-Takens point. From our results, we infer a general rule of thumb relating pulse shape and PRC offset with each scenario. Finally, we compare the exact synchronization threshold with the prediction of the averaging approximation given by the Kuramoto-Sakaguchi model. At the leading order, the discrepancy appears to behave as an odd function of the PRC offset.

nlin.AO↗

arXiv2016

From quasiperiodic partial synchronization to collective chaos in populations of inhibitory neurons with delay

Collective chaos is shown to emerge, via a period-doubling cascade, from quasiperiodic partial synchronization in a population of identical inhibitory neurons with delayed global coupling. This system is thoroughly investigated by means of an exact model of the macroscopic dynamics, valid in the thermodynamic limit. The collective chaotic state is reproduced numerically with a finite population, and persists in the presence of weak heterogeneities. Finally, the relationship of the model's dynamics with fast neuronal oscillations is discussed.

nlin.CD↗

arXiv2015

Macroscopic description for networks of spiking neurons

A major goal of neuroscience, statistical physics and nonlinear dynamics is to understand how brain function arises from the collective dynamics of networks of spiking neurons. This challenge has been chiefly addressed through large-scale numerical simulations. Alternatively, researchers have formulated mean-field theories to gain insight into macroscopic states of large neuronal networks in terms of the collective firing activity of the neurons, or the firing rate. However, these theories have not succeeded in establishing an exact correspondence between the firing rate of the network and the underlying microscopic state of the spiking neurons. This has largely constrained the range of applicability of such macroscopic descriptions, particularly when trying to describe neuronal synchronization. Here we provide the derivation of a set of exact macroscopic equations for a network of spiking neurons. Our results reveal that the spike generation mechanism of individual neurons introduces an effective coupling between two biophysically relevant macroscopic quantities, the firing rate and the mean membrane potential, which together govern the evolution of the neuronal network. The resulting equations exactly describe all possible macroscopic dynamical states of the network, including states of synchronous spiking activity. Finally we show that the firing rate description is related, via a conformal map, with a low-dimensional description in terms of the Kuramoto order parameter, called Ott-Antonsen theory. We anticipate our results will be an important tool in investigating how large networks of spiking neurons self-organize in time to process and encode information in the brain.

q-bio.NC↗

arXiv2014

Low-dimensional dynamics of populations of pulse-coupled oscillators

Large communities of biological oscillators show a prevalent tendency to self-organize in time. This cooperative phenomenon inspired Winfree to formulate a mathematical model that originated the theory of macroscopic synchronization. Despite its fundamental importance, a complete mathematical analysis of the model proposed by Winfree ---consisting of a large population of all-to-all pulse-coupled oscillators--- is still missing. Here we show that the dynamics of the Winfree model evolves into the so-called Ott-Antonsen manifold. This important property allows for an exact description of this high-dimensional system in terms of a few macroscopic variables, and the full investigation of its dynamics. We find that brief pulses are capable of synchronizing heterogeneous ensembles which fail to synchronize with broad pulses, specially for certain phase response curves. Finally, to further illustrate the potential of our results, we investigate the possibility of `chimera' states in populations of identical pulse-coupled oscillators. Chimeras are self-organized states in which the symmetry of a population is broken into a synchronous and an asynchronous part. Here we derive three ordinary differential equations describing two coupled populations, and uncover a variety of chimera states, including a new class with chaotic dynamics.

nlin.AO↗

arXiv2011

Collective synchronization in the presence of reactive coupling and shear diversity

We analyze the synchronization dynamics of a model obtained from the phase reduction of the mean-field complex Ginzburg-Landau equation with heterogeneity. We present exact results that uncover the role of dissipative and reactive couplings on the synchronization transition when shears and natural frequencies are independently distributed. As it occurs in the purely dissipative case, an excess of shear diversity prevents the onset of synchronization, but this does not hold true if coupling is purely reactive. In this case the synchronization threshold turns out to depend on the mean of the shear distribution, but not on all the other distribution's moments.

nlin.SI↗

arXiv2011

The Kuramoto model with distributed shear

We uncover a solvable generalization of the Kuramoto model in which shears (or nonisochronicities) and natural frequencies are distributed and statistically dependent. We show that the strength and sign of this dependence greatly alter synchronization and yield qualitatively different phase diagrams. The Ott-Antonsen ansatz allows us to obtain analytical results for a specific family of joint distributions. We also derive, using linear stability analysis, general formulae for the stability border of incoherence.

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