## Section: New Results

### Mean field theory and stochastic processes

#### Mean-field limit of interacting 2D nonlinear stochastic spiking neurons

Participants : Benjamin Aymard, Fabien Campillo, Romain Veltz.

In this work, we propose a nonlinear stochastic model of a network of stochastic spiking neurons. We heuristically derive the mean-field limit of this system. We then design a Monte Carlo method for the simulation of the microscopic system, and a finite volume method (based on an upwind implicit scheme) for the mean-field model. The finite volume method respects numerical versions of the two main properties of the mean-field model, conservation and positivity, leading to existence and uniqueness of a numerical solution. As the size of the network tends to infinity, we numerically observe propagation of chaos and convergence from an individual description to a mean-field description. Numerical evidences for the existence of a Hopf bifurcation (synonym of synchronised activity) for a sufficiently high value of connectivity, are provided.

This work has been submitted for publication and is available as [38].

#### Stochastic modeling for biotechnologies Anaerobic model AM2b

Participants : Fabien Campillo, Mohsen Chebbi [ENIT, University of Tunis, Tunisia] , Salwa Toumi [INSAT, University of Carthage, Tunisia] .

The model AM2b is conventionally represented by a system of differential equations. However, this model is valid only in a large population context and our objective is to establish several stochastic models at different scales. At a microscopic scale, we propose a pure jump stochastic model that can be simulated exactly. But in most situations this exact simulation is not feasible, and we propose approximate simulation methods of Poisson type and of diffusive type. The diffusive type simulation method can be seen as a discretization of a stochastic differential equation. Finally, we formally present a result of law of large numbers and of functional central limit theorem which demonstrates the convergence of these stochastic models towards the initial deterministic models.

This work has been published in ARIMA and is available as [22].

#### Cross frequency coupling in next generation inhibitory neural mass models

Participants : Andrea Ceni [University of Exeter, UK] , Simona Olmi, Alessandro Torcini [Institute of Complex Systems, Florence, Italy] , David Angulo-Garcia [Polytechnic University of Cartagena, Colombia] .

Coupling among neural rhythms is one of the most important mechanisms at the basis of cognitive processes in the brain. In this study we consider a neural mass model, rigorously obtained from the microscopic dynamics of an inhibitory spiking network with exponential synapses, able to autonomously generate collective oscillations (COs). These oscillations emerge via a super-critical Hopf bifurcation, and their frequencies are controlled by the synaptic time scale, the synaptic coupling and the excitability of the neural population. Furthermore, we show that two inhibitory populations in a master-slave configuration with different synaptic time scales can display various collective dynamical regimes: namely, damped oscillations towards a stable focus, periodic and quasi-periodic oscillations, and chaos. Finally, when bidirectionally coupled the two inhibitory populations can exhibit different types of theta-gamma cross-frequency couplings (CFCs): namely, phase-phase and phase-amplitude CFC. The coupling between theta and gamma COs is enhanced in presence of a external theta forcing, reminiscent of the type of modulation induced in Hippocampal and Cortex circuits via optogenetic drive.

This work has been submitted for publication and is available as [40].

#### Conductance-Based Refractory Density Approach for a Population of Bursting Neurons

Participants : Anton Chizhov [IOFFE Institute, St Petersburg, Russia] , Fabien Campillo, Mathieu Desroches, Antoni Guillamon [Polytechnic University of Catalonia, Barcelona, Spain] , Serafim Rodrigues [Ikerbasque & MCEN team, Basque Center for Applied Mathematics, Spain] .

The conductance-based refractory density (CBRD) approach is a parsimonious mathematical-computational framework for modelling interacting populations of regular spiking neurons, which, however, has not been yet extended for a population of bursting neurons. The canonical CBRD method allows to describe the firing activity of a statistical ensemble of uncoupled Hodgkin-Huxley-like neurons (differentiated by noise) and has demonstrated its validity against experimental data. The present manuscript generalises the CBRD for a population of bursting neurons; however, in this pilot computational study, we consider the simplest setting in which each individual neuron is governed by a piecewise linear bursting dynamics. The resulting population model makes use of slow-fast analysis, which leads to a novel methodology that combines CBRD with the theory of multiple timescale dynamics. The main prospect is that it opens novel avenues for mathematical explorations, as well as, the derivation of more sophisticated population activity from Hodgkin-Huxley-like bursting neurons, which will allow to capture the activity of synchronised bursting activity in hyper-excitable brain states (e.g. onset of epilepsy).

This work has been published in Bulletin of Mathematical Biology and is available as [23].

#### Long time behavior of a mean-field model of interacting neurons

Participants : Quentin Cormier [Inria Tosca] , Étienne Tanré [Inria Tosca] , Romain Veltz.

We study the long time behavior of the solution to some McKean-Vlasov stochastic differential equation (SDE) driven by a Poisson process. In neuroscience, this SDE models the asymptotic dynamics of the membrane potential of a spiking neuron in a large network. We prove that for a small enough interaction parameter, any solution converges to the unique (in this case) invariant measure. To this aim, we first obtain global bounds on the jump rate and derive a Volterra type integral equation satisfied by this rate. We then replace temporary the interaction part of the equation by a deterministic external quantity (we call it the external current). For constant current, we obtain the convergence to the invariant measure. Using a perturbation method, we extend this result to more general external currents. Finally, we prove the result for the non-linear McKean-Vlasov equation.

This work has been published in Stochastic Processes and their Applications and is available as [24].

#### Effective low-dimensional dynamics of a mean-field coupled network of slow-fast spiking lasers

Participants : Axel Dolcemascolo [INPHYNI, Nice] , Alexandre Miazek [INPHYNI, Nice] , Romain Veltz, Francesco Marino [National Institute of Optics, Italy] , Stéphane Barland [INPHYNI, Nice] .

Low dimensional dynamics of large networks is the focus of many theoretical works, but controlled laboratory experiments are comparatively very few. Here, we discuss experimental observations on a mean-field coupled network of hundreds of semiconductor lasers, which collectively display effectively low-dimensional mixed mode oscillations and chaotic spiking typical of slow-fast systems. We demonstrate that such a reduced dimensionality originates from the slow-fast nature of the system and of the existence of a critical manifold of the network where most of the dynamics takes place. Experimental measurement of the bifurcation parameter for different network sizes corroborate the theory.

This work has been submitted for publication and is available as [42].

#### The mean-field limit of a network of Hopfield neurons with correlated synaptic weights

Participants : Olivier Faugeras, James Maclaurin [NJIT, USA] , Étienne Tanré [Inria Tosca] .

We study the asymptotic behaviour for asymmetric neuronal dynamics in a network of Hopfield neurons. The randomness in the network is modelled by random couplings which are centered Gaussian correlated random variables. We prove that the annealed law of the empirical measure satisfies a large deviation principle without any condition on time. We prove that the good rate function of this large deviation principle achieves its minimum value at a unique Gaussian measure which is not Markovian. This implies almost sure convergence of the empirical measure under the quenched law. We prove that the limit equations are expressed as an infinite countable set of linear non Markovian SDEs.

This work has been submitted for publication and is available as [43].

#### Asymptotic behaviour of a network of neurons with random linear interactions

Participants : Olivier Faugeras, Émilie Soret, Étienne Tanré [Inria Tosca] .

We study the asymptotic behaviour for asymmetric neuronal dynamics in a network of linear Hopfield neurons. The randomness in the network is modelled by random couplings which are centered i.i.d. random variables with finite moments of all orders. We prove that if the initial condition of the network is a set of i.i.d random variables with finite moments of all orders and independent of the synaptic weights, each component of the limit system is described as the sum of the corresponding coordinate of the initial condition with a centered Gaussian process whose covariance function can be described in terms of a modified Bessel function. This process is not Markovian. The convergence is in law almost surely w.r.t. the random weights. Our method is essentially based on the CLT and the method of moments.

This work has been submitted for publication and is available as [44].

#### On a toy network of neurons interacting through their dendrites

Participants : Nicolas Fournier [LPSM, Sorbonne Université] , Étienne Tanré [Inria Tosca] , Romain Veltz.

Consider a large number $n$ of neurons, each being connected to approximately $N$ other ones, chosen at random. When a neuron spikes, which occurs randomly at some rate depending on its electric potential, its potential is set to a minimum value vmin, and this initiates, after a small delay, two fronts on the (linear) dendrites of all the neurons to which it is connected. Fronts move at constant speed. When two fronts (on the dendrite of the same neuron) collide, they annihilate. When a front hits the soma of a neuron, its potential is increased by a small value ${w}_{n}$. Between jumps, the potentials of the neurons are assumed to drift in $[{v}_{min},\infty )$, according to some well-posed ODE. We prove the existence and uniqueness of a heuristically derived mean-field limit of the system when $n,N\to \infty $ with ${w}_{n}\simeq {N}^{-1/2}$. We make use of some recent versions of the results of Deuschel and Zeitouni [55] concerning the size of the longest increasing subsequence of an i.i.d. collection of points in the plan. We also study, in a very particular case, a slightly different model where the neurons spike when their potential reach some maximum value ${v}_{max}$, and find an explicit formula for the (heuristic) mean-field limit.

This work has been accepted for publication in Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques and is available as [27].

#### Bumps and oscillons in networks of spiking neurons

Participants : Helmut Schmidt [Max Planck Institute for Human Cognitive and Brain Science, Germany] , Daniele Avitabile [VU Amsterdam, Inria MathNeuro] .

We study localized patterns in an exact mean-field description of a spatially-extended network of quadratic integrate-and-fire (QIF) neurons. We investigate conditions for the existence and stability of localized solutions, so-called bumps, and give an analytic estimate for the parameter range where these solutions exist in parameter space, when one or more microscopic network parameters are varied. We develop Galerkin methods for the model equations, which enable numerical bifurcation analysis of stationary and time-periodic spatially-extended solutions. We study the emergence of patterns composed of multiple bumps, which are arranged in a snake-and-ladder bifurcation structure if a homogeneous or heterogeneous synaptic kernel is suitably chosen. Furthermore, we examine time-periodic, spatially-localized solutions (oscillons) in the presence of external forcing, and in autonomous, recurrently coupled excitatory and inhibitory networks. In both cases we observe period doubling cascades leading to chaotic oscillations.

This work has been submitted for publication and is available as [46].

#### Slow-fast dynamics in the mean-field limit of neural networks

Participants : Daniele Avitabile [VU Amsterdam, Inria MathNeuro] , Emre Baspinar, Mathieu Desroches, Olivier Faugeras.

In the context of the Human Brain Project (HBP, see section 5.1.1.1. below), we have recruited Emre Baspinar in December 2018 for a two-year postdoc. Within MathNeuro, Emre is working on analysing slow-fast dynamical behaviours in the mean-field limit of neural networks.

In a first project, he has been analysing the slow-fast structure in the mean-field limit of a network of FitzHugh-Nagumo neuron models; the mean-field was previously established in [3] but its slow-fast aspect had not been analysed. In particular, he has proved a persistence result of Fenichel type for slow manifolds in this mean-field limit, thus extending previous work by Berglund *et al.* [47], [48]. A manuscript is in preparation.

In a second project, he has been looking at a network of Wilson-Cowan systems whose mean-field limit is an ODE, and he has studied elliptic bursting dynamics in both the network and the limit: its slow-fast dissection, its singular limits and the role of canards. In passing, he has obtained a new characterisation of ellipting bursting via the construction of periodic limit sets using both the slow and the fast singular limits and unravelled a new singular-limit scenario giving rise to elliptic bursting via a new type of torus canard orbits. A manuscript is in preparation.