MathNeuro focuses on the applications of multi-scale dynamics to neuroscience. This involves the modelling and analysis of systems with multiple time scales and space scales, as well as stochastic effects. We look both at single-cell models, microcircuits and large networks. In terms of neuroscience, we are mainly interested in questions related to synaptic plasticity and neuronal excitability, in particular in the context of pathological states such as epileptic seizures and neurodegenerative diseases such as Alzheimer.

Our work is quite mathematical but we make heavy use of computers for numerical experiments and simulations. We have close ties with several top groups in biological neuroscience. We are pursuing the idea that the "unreasonable effectiveness of mathematics" can be brought, as it has been in physics, to bear on neuroscience.

Modeling such assemblies of neurons and simulating their behavior involves putting together a mixture of the most recent results in neurophysiology with such advanced mathematical methods as dynamical systems theory, bifurcation theory, probability theory, stochastic calculus, theoretical physics and statistics, as well as the use of simulation tools.

We conduct research in the following main areas:

The study of neural networks is certainly motivated by the long term goal to understand how brain is working. But, beyond the comprehension of brain or even of simpler neural systems in less evolved animals, there is also the desire to exhibit general mechanisms or principles at work in the nervous system. One possible strategy is to propose mathematical models of neural activity, at different space and time scales, depending on the type of phenomena under consideration. However, beyond the mere proposal of new models, which can rapidly result in a plethora, there is also a need to understand some fundamental keys ruling the behaviour of neural networks, and, from this, to extract new ideas that can be tested in real experiments. Therefore, there is a need to make a thorough analysis of these models. An efficient approach, developed in our team, consists of analysing neural networks as dynamical systems. This allows to address several issues. A first, natural issue is to ask about the (generic) dynamics exhibited by the system when control parameters vary. This naturally leads to analyse the bifurcations 59 60 occurring in the network and which phenomenological parameters control these bifurcations. Another issue concerns the interplay between the neuron dynamics and the synaptic network structure.

Modeling neural activity at scales integrating the effect of thousands of neurons is of central importance for several reasons. First, most imaging techniques are not able to measure individual neuron activity (microscopic scale), but are instead measuring mesoscopic effects resulting from the activity of several hundreds to several hundreds of thousands of neurons. Second, anatomical data recorded in the cortex reveal the existence of structures, such as the cortical columns, with a diameter of about 50

Our group is developing mathematical methods allowing to produce dynamic mean-field equations from the physiological characteristics of neural structure (neurons type, synapse type and anatomical connectivity between neurons populations). These methods use tools from advanced probability theory such as the theory of Large Deviations 48 and the study of interacting diffusions 3.

Neural fields are a phenomenological way of describing the activity of population of neurons by delayed integro-differential equations. This continuous approximation turns out to be very useful to model large brain areas such as those involved in visual perception. The mathematical properties of these equations and their solutions are still imperfectly known, in particular in the presence of delays, different time scales and noise.

Our group is developing mathematical and numerical methods for analysing these equations. These methods are based upon techniques from mathematical functional analysis, bifurcation theory 18, 62, equivariant bifurcation analysis, delay equations, and stochastic partial differential equations. We have been able to characterize the solutions of these neural fields equations and their bifurcations, apply and expand the theory to account for such perceptual phenomena as edge, texture 42, and motion perception. We have also developed a theory of the delayed neural fields equations, in particular in the case of constant delays and propagation delays that must be taken into account when attempting to model large size cortical areas 61. This theory is based on center manifold and normal forms ideas 17.

Neuronal rhythms typically display many different timescales, therefore it is important to incorporate this slow-fast aspect in models. We are interested in this modeling paradigm where slow-fast point models, using Ordinary Differential Equations (ODEs), are investigated in terms of their bifurcation structure and the patterns of oscillatory solutions that they can produce. To insight into the dynamics of such systems, we use a mix of theoretical techniques — such as geometric desingularisation and centre manifold reduction 52 — and numerical methods such as pseudo-arclength continuation 46. We are interested in families of complex oscillations generated by both mathematical and biophysical models of neurons. In particular, so-called mixed-mode oscillations (MMOs)12, 44, 51, which represent an alternation between subthreshold and spiking behaviour, and bursting oscillations45, 50, also corresponding to experimentally observed behaviour 43; see Figure 1. We are working on extending these results to spatio-temporal neural models 2.

Excitability refers to the all-or-none property of neurons 47, 49. That is, the ability to respond nonlinearly to an input with a dramatic change of response from “none” — no response except a small perturbation that returns to equilibrium — to “all” — large response with the generation of an action potential or spike before the neuron returns to equilibrium. The return to equilibrium may also be an oscillatory motion of small amplitude; in this case, one speaks of resonator neurons as opposed to integrator neurons. The combination of a spike followed by subthreshold oscillations is then often referred to as mixed-mode oscillations (MMOs) 44. Slow-fast ODE models of dimension at least three are well capable of reproducing such complex neural oscillations. Part of our research expertise is to analyse the possible transitions between different complex oscillatory patterns of this sort upon input change and, in mathematical terms, this corresponds to understanding the bifurcation structure of the model. In particular, we also study possible combinations of different scenarios of complex oscillations and their relevance to revisit unexplained experimental data, e.g. in the context of bursting oscillations 45. In all case, the role of noise 40 is important and we take it into consideration, either as a modulator of the underlying deterministic dynamics or as a trigger of potential threshold crossings. Furthermore, the shape of time series of this sort with a given oscillatory pattern can be analysed within the mathematical framework of dynamic bifurcations; see the section on slow-fast dynamics in Neuronal Models. The main example of abnormal neuronal excitability is hyperexcitability and it is important to understand the biological factors which lead to such excess of excitability and to identify (both in detailed biophysical models and reduced phenomenological ones) the mathematical structures leading to these anomalies. Hyperexcitability is one important trigger for pathological brain states related to various diseases such as chronic migraine 56, epilepsy 58 or even Alzheimer's Disease 53. A central central axis of research within our group is to revisit models of such pathological scenarios, in relation with a combination of advanced mathematical tools and in partnership with biological labs.

Neural networks show amazing abilities to evolve and adapt, and to store and process information. These capabilities are mainly conditioned by plasticity mechanisms, and especially synaptic plasticity, inducing a mutual coupling between network structure and neuron dynamics. Synaptic plasticity occurs at many levels of organization and time scales in the nervous system 39. It is of course involved in memory and learning mechanisms, but it also alters excitability of brain areas and regulates behavioral states (e.g., transition between sleep and wakeful activity). Therefore, understanding the effects of synaptic plasticity on neurons dynamics is a crucial challenge.

Our group is developing mathematical and numerical methods to analyse this mutual interaction. On the one hand, we have shown that plasticity mechanisms 10, 16, Hebbian-like or STDP, have strong effects on neuron dynamics complexity, such as dynamics complexity reduction, and spike statistics.

The processes by which memories are formed and stored in the brain are multiple and not yet fully understood. What is hypothesised so far is that memory formation is related to the activation of certain groups of neurons in the brain. Then, one important mechanism to store various memories is to associate certain groups of memory items with one another, which then corresponds to the joint activation of certain neurons within different subgroup of a given population. In this framework, plasticity is key to encode the storage of chains of memory items. Yet, there is no general mathematical framework to model the mechanism(s) behind these associative memory processes. We are aiming at developing such a framework using our expertise in multi-scale modelling, by combining the concepts of heteroclinic dynamics, slow-fast dynamics and stochastic dynamics.

The general objective that we wish to pursue in this project is to investigate non-equilibrium phenomena pertinent to storage and retrieval of sequences of learned items. In previous work by team members 9, 1, 14, it was shown that with a suitable formulation, heteroclinic dynamics combined with slow-fast analysis in neural field systems can play an organizing role in such processes, making the model accessible to a thorough mathematical analysis. Multiple choice in cognitive processes require a certain flexibility in the neural network, which has recently been investigated in the submitted paper 28.

Our goal is to contribute to identify general processes under which cognitive functions can be organized in the brain.

The project underlying MathNeuro revolves around pillars of neuronal behaviour –excitability, plasticity, memory– in link with the initiation and propagation of pathological brain states in diseases such as cortical spreading depression (in link with certain forms of migraine with aura), epileptic seizures and Alzheimer’s Disease. Our work on memory processes can also potentially be applied to studying mental disorders such as schizophrenia or obsessive disorder troubles.

The team decided to invest significantly in software development in collaboration with the SED (Service d’Expérimentation et de Développement) of Sophia Antipolis and the team led by Serafim Rodrigues at BCAM in Bilbao, see Section 6.

Until now the team has only developed small ad-hoc simulation tools associated with articles. This year we have decided to implement a systematic and ambitious development strategy and to move to tested, reliable, documented and disseminable tools.

In collaboration with Joanna Danielewicz, Research Technician (biologist) in MCEN at BCAM, Bilbao.

MathNeuro wishes to call upon the support and competence of the SED of Sophia Antipolis Méditerrannée to develop a collaborative project around an electrophysiology laboratory currently set up in Bilbao, which has already benefited from the support of Inria Sophia Antipolis Méditerrannée.

The goal of this project is to develop a software environment that will play an important role within MathNeuro and will be the focus of our collaboration with the MCEN lab (BCAM Bilbao) directed by Serafim Rodrigues.

The MCEN experimental neurophysiological dynamic-clamp device 57, 41 to interact with: an artificial neuron, an in vitro neuron, an in vivo neuron, to which we also want to add numerical simulations (neuron in silico). Beyond the activity of a single neuron, this device can also monitor the activity of a single ion channel or the activity of several neurons. This device is not only a measuring device, it also allows to inject a current into the neuron and thus to control the nerve cell 55.

Fabien Campillo has recently updated and made available four demonstration programs: three simulation of individual based models (IBM forest, IBM cellulose, IBM clonal) as well as a set of particle filtering demonstration software (SMC demos); see the gitlab repositories and the dedicated web page.

These tools are not recent, they are written in Matlab and we wish to update them in Python. The IBM models are Markov models now very developed and analyzed in life sciences. SMC Demos has met with some success and has participated in the diffusion of particle filtering tools.

This work was partially carried out during an invitation of Daniele Avitabile within the MathNeuro team.

Bump attractors are wandering localised patterns observed in in vivo experiments of spatially-extended neurobiological networks. They are important for the brain's navigational system and specific memory tasks. A bump attractor is characterised by a core in which neurons fire frequently, while those away from the core do not fire. We uncover a relationship between bump attractors and travelling waves in a classical network of excitable, leaky integrate-and-fire neurons. This relationship bears strong similarities to the one between complex spatiotemporal patterns and waves at the onset of pipe turbulence. Waves in the spiking network are determined by a firing set, that is, the collection of times at which neurons reach a threshold and fire as the wave propagates. We define and study analytical properties of the voltage mapping, an operator transforming a solution's firing set into its spatiotemporal profile. This operator allows us to construct localised travelling waves with an arbitrary number of spikes at the core, and to study their linear stability. A homogeneous "laminar" state exists in the network, and it is linearly stable for all values of the principal control parameter. Sufficiently wide disturbances to the homogeneous state elicit the bump attractor. We show that one can construct waves with a seemingly arbitrary number of spikes at the core; the higher the number of spikes, the slower the wave, and the more its profile resembles a stationary bump. As in the fluid-dynamical analogy, such waves coexist with the homogeneous state, are unstable, and the solution branches to which they belong are disconnected from the laminar state; we provide evidence that the dynamics of the bump attractor displays echoes of the unstable waves, which form its building blocks.

This work has been accepted for publication in SIAM Review and is available as 19.

This work was partially achieved during a stay of Daniele Avitabile in MathNeuro.

We show that during the transition from and to elliptic burstings both classical and mixed-type torus canards appear in a Wilson-Cowan type neuronal network model, as well as in its corresponding mean-field framework. We show numerically the overlap between the network and mean-field dynamics. We comment on that mixed-type torus canards result from the nonsymmetric fast subsystem. This nonsymmetricity provides the dynamics linked to an associated canonical form presented in 38, and it occurs mainly due to the three-timescale nature of the Wilson-Cowan type models.

This work is available as 34.

From the action potentials of neurons and cardiac cells to the amplification of calcium signals in oocytes, excitability is a hallmark of many biological signalling processes. In recent years, excitability in single cells has been related to multiple-timescale dynamics through canards, special solutions which determine the effective thresholds of the all-or-none responses. However, the emergence of excitability in large populations remains an open problem. We show that the mechanism of excitability in large networks and mean-field descriptions of coupled quadratic integrate-and-fire (QIF) cells mirrors that of the individual components. We initially exploit the Ott-Antonsen ansatz to derive low-dimensional dynamics for the coupled network and use it to describe the structure of canards via slow periodic forcing. We demonstrate that the thresholds for onset and offset of population firing can be found in the same way as those of the single cell. We combine theoretical analysis and numerical computations to develop a novel and comprehensive framework for excitability in large populations, applicable not only to models amenable to Ott-Antonsen reduction, but also to networks without a closed-form mean-field limit, in particular sparse networks.

This work has been published in PLoS Computational Biology and is available as 20.

This work was partially achieved during a stay of Daniele Avitabile in MathNeuro.

We report a detailed analysis on the emergence of bursting in a recently developed neural mass model that takes short-term synaptic plasticity into account. The one being used here is particularly important, as it represents an exact meanfield limit of synaptically coupled quadratic integrate & fire neurons, a canonical model for type I excitability. In absence of synaptic dynamics, a periodic external current with a slow frequency

This work was published in Nonlinear Dynamics and is available as 32.

This work was partially carried out during an invitation of the author within the MathNeuro team.

We propose a numerical analysis framework for the study of the numerical approximation of nonlinear integro-differential equations for the evolution of neuronal activity. Neural fields are often simulated heuristically and, in spite of their popularity in mathematical neuroscience, their numerical analysis is not yet fully established. We introduce generic projection methods for neural fields, and derive a priori error bounds for these schemes. We extend an existing framework for stationary integral equations to the time-dependent case, which is relevant for neuroscience applications. We find that the convergence rate of a projection scheme for a neural field is determined to a great extent by the convergence rate of the projection operator. This abstract analysis, which unifies the treatment of collocation and Galerkin schemes, is carried out in operator form, without resorting to quadrature rules for the integral term, which are introduced only at a later stage, and whose choice is enslaved by the choice of the projector.

This work has been accepted for publication in SIAM Journal on Numerical Analysis and is available as 21.

This work has been partially done during a stay of Serafim Rodrigues in MathNeuro.

Bursting is one of the fundamental rhythms that excitable cells can generate either in response to incoming stimuli or intrinsically. It has been a topic of intense research in computational biology for several decades. The classification of bursting oscillations in excitable systems has been the subject of active research since the early 1980s and is still ongoing. As a by-product, it establishes analytical and numerical foundations for studying complex temporal behaviors in multiple timescale models of cellular activity. In this review, we first present the seminal works of Rinzel and Izhikevich in classifying bursting patterns of excitable systems. We recall a complementary mathematical classification approach by Bertram and colleagues, and then by Golubitsky and colleagues, which, together with the Rinzel-Izhikevich proposals, provide the state-of-the-art foundations to these classifications. Beyond classical approaches, we review a recent bursting example that falls outside the previous classification systems. Generalizing this example leads us to propose an extended classification, which requires the analysis of both fast and slow subsystems of an underlying slow-fast model and allows the dissection of a larger class of bursters. Namely, we provide a general framework for bursting systems with both subthreshold and superthreshold oscillations. A new class of bursters with at least 2 slow variables is then added, which we denote folded-node bursters , to convey the idea that the bursts are initiated or annihilated via a folded-node singularity. Key to this mechanism are so-called canard or duck orbits, organizing the underpinning excitability structure. We describe the 2 main families of folded-node bursters, depending upon the phase (active/spiking or silent/nonspiking) of the bursting cycle during which folded-node dynamics occurs. We classify both families and give examples of minimal systems displaying these novel bursting patterns. Finally, we provide a biophysical example by reinterpreting a generic conductance-based episodic burster as a folded-node burster, showing that the associated framework can explain its subthreshold oscillations over a larger parameter region than the fast subsystem approach.

This work has been published in PLoS Computational Biology and is available as 23.

Mixed affective states in bipolar disorder (BD) is a common psychiatric condition that occurs when symptoms of the two opposite poles coexist during an episode of mania or depression. A four-dimensional model by A. Goldbeter [27, 28] rests upon the notion that manic and depressive symptoms are produced by two competing and auto-inhibited neural networks. Some of the rich dynamics that this model can produce, include complex rhythms formed by both small-amplitude (subthreshold) and large-amplitude (suprathreshold) oscillations and could correspond to mixed bipolar states. These rhythms are commonly referred to as mixed mode oscillations (MMOs) and they have already been studied in many different contexts [7, 50]. In order to accurately explain these dynamics one has to apply a mathematical apparatus that makes full use of the timescale separation between variables. Here we apply the framework of multiple-timescale dynamics to the model of BD in order to understand the mathematical mechanisms underpinning the observed dynamics of changing mood. We show that the observed complex oscillations can be understood as MMOs due to a so-called folded-node singularity. Moreover, we explore the bifurcation structure of the system and we provide possible biological interpretations of our findings. Finally, we show the robustness of the MMOs regime to stochastic noise and we propose a minimal three- dimensional model which, with the addition of noise, exhibits similar yet purely noise-driven dynamics. The broader significance of this work is to introduce mathematical tools that could be used to analyse and potentially control future, more biologically grounded models of BD.

This work was published in Cognitive Neurodynamics and is available as 29.

The phenomenon of slow passage through a Hopf bifurcation is ubiquitous in multiple-timescale dynamical systems, where a slowly varying quantity replacing a static parameter induces the solutions of the resulting slow–fast system to feel the effect of the Hopf bifurcation with a delay. This phenomenon is well understood in the context of smooth slow–fast dynamical systems; in the present work, we study it for the first time in piecewise linear (PWL) slow–fast systems. This special class of systems is indeed known to reproduce all features of their smooth counterpart while being more amenable to quantitative analysis and offering some level of simplification, in particular, through the existence of canonical (linear) slow manifolds. We provide conditions for a PWL slow–fast system to exhibit a slow passage through a Hopf-like bifurcation, in link with possible connections between canonical attracting and repelling slow manifolds. In doing so, we fully describe the so-called way-in/way-out function. Finally, we investigate this slow passage effect in the Doi–Kumagai model, a neuronal PWL model exhibiting elliptic bursting oscillations.

The classical Lanchester's model is shortly reviewed and analysed, with particular attention to the critical issues that intrinsically arise from the mathematical formalization of the problem. We then generalize a particular version of such a model describing the dynamics of warfare when three or more armies are involved in the conflict. Several numerical simulations are provided.

This work has been submitted for publication and is available as 35.

We study delayed loss of stability in a class of fast-slow systems with two fast variables and one slow one, where the linearization of the fast vector field along a one-dimensional critical manifold has two real eigenvalues which intersect before the accumulated contraction and expansion are balanced along any individual eigendirection. That interplay between eigenvalues and eigendirections renders the use of known entry-exit relations unsuitable for calculating the point at which trajectories exit neighborhoods of the given manifold. We illustrate the various qualitative scenarios that are possible in the class of systems considered here, and we propose novel formulae for the entry-exit functions that underlie the phenomenon of delayed loss of stability therein.

This work has been submitted for publication and is available as 36.

This work was partially done during a stay of Serafim Rodrigues in MathNeuro.

We propose a minimalist phenomenological model for the 'interfacial water' phenomenon that occurs near hydrophilic polymeric surfaces. We achieve this by combining a Ginzburg–Landau approach with Maxwell's equations which leads us to a well-posed model providing a macroscopic interpretation of experimental observations. From the derived governing equations, we estimate the unknown parameters using experimental measurements from the literature. The resulting profiles of the polarization and electric potential show exponential decay near the surface, in qualitative agreement with experiments. Furthermore, the model's quantitative prediction of the electric potential at the hydrophilic surface is in excellent agreement with experiments. The proposed model is a first step towards a more complete parsimonious macroscopic model that will, for example, help to elucidate the effects of interfacial water on cells (e.g. neuronal excitability, the effects of infrared neural stimulation or the effects of drugs mediated by interfacial water.

This work was published in Journal of Physics: Condensed Matter and is available as 24.

An important function of the brain is to predict which stimulus is likely to occur based on the perceived cues. The present research studied the branching behavior of a computational network model of populations of excitatory and inhibitory neurons, both analytically and through simulations. Results show how synaptic efficacy, retroactive inhibition and short-term synaptic depression determine the dynamics of selection between different branches predicting sequences of stimuli of different probabilities. Further results show that changes in the probability of the different predictions depend on variations of neuronal gain. Such variations allow the network to optimize the probability of its predictions to changing probabilities of the sequences without changing synaptic efficacy.

This work was published in Journal of Computational Neuroscience and is available as 28.

In collaboration with Anna Song (Imperial College London, The Francis Crick Institute) and Romain Veltz (on secondment to Inria).

We study a simplified model of the representation of colors in the primate primary cortical visual area V1. The model is described by an initial value problem related to a Hammerstein equation. The solutions to this problem represent the variation of the activity of populations of neurons in V1 as a function of space and color. The two space variables describe the spatial extent of the cortex while the two color variables describe the hue and the saturation represented at every location in the cortex. We prove the well-posedness of the initial value problem. We focus on its stationary and periodic in space solutions. We show that the model equation is equivariant with respect to the direct product G of the group of the Euclidean transformations of the planar lattice determined by the spatial periodicity and the group of color transformations, isomorphic to O(2), and study the equivariant bifurcations of its stationary solutions when some parameters in the model vary. Their variations may be caused by the consumption of drugs and the bifurcated solutions may represent visual hallucinations in space and color. Some of the bifurcated solutions can be determined by applying the Equivariant Branching Lemma (EBL) by determining the axial subgroups of G. These define bifurcated solutions which are invariant under the action of the corresponding axial subgroup. We compute analytically these solutions and illustrate them as color images. Using advanced methods of numerical bifurcation analysis we then explore the persistence and stability of these solutions when varying some parameters in the model. We conjecture that we can rely on the EBL to predict the existence of patterns that survive in large parameter domains but not to predict their stability. On our way we discover the existence of spatially localized stable patterns through the phenomenon of "snaking".

This work has been published in Compte Rendus Mathématique and is available as 25.

This work was partially achieved during a stay of Serafim Rodrigues in MathNeuro.

In the brain, spiking patterns live in a high-dimensional space of neurons and time. Thus, determining the intrinsic structure of this space presents a theoretical and experimental challenge. To address it, we introduce a new framework for applying topological data analysis (TDA) to spike train data and use it to determine the geometry of spiking patterns in the visual cortex. Key to our approach is a parametrized family of distances based on the timing of spikes that quantifies the dissimilarity between neuronal responses. We applied TDA to visually driven single-unit and multiple single-unit spiking activity in macaque V1 and V2. TDA across timescales reveals a common geometry for spiking patterns in V1 and V2 which, among simple models, is most similar to that of a low-dimensional space endowed with Euclidean or hyperbolic geometry with modest curvature. Remarkably, the inferred geometry depends on timescale and is clearest for the timescales that are important for encoding contrast, orientation and spatial correlations.

This work was published in Journal of the Royal Society Interface and is available as 27.

This work was notably done in collaboration with Tamás Fülöp (Centre de recherche sur le vieillissement, CSSS-UIGS,
Université de Sherbrooke). This work was partially done during a stay of Serafim Rodrigues in MathNeuro.

Unravelling the mystery of Alzheimer’s disease (AD) requires urgent resolution given the worldwide increase of the aging population. There is a growing concern that the current leading AD hypothesis, the amyloid cascade hypothesis, does not stand up to validation with respect to emerging new data. Indeed, several paradoxes are being discussed in the literature, for instance, both the deposition of the amyloid-β peptide (Aβ) and the intracellular neurofibrillary tangles could occur within the brain without any cognitive pathology. Thus, these paradoxes suggest that something more fundamental is at play in the onset of the disease and other key and related pathomechanisms must be investigated. The present study follows our previous investigations on the infectious hypothesis, which posits that some pathogens are linked to late onset AD. Our studies also build upon the finding that Aβ is a powerful antimicrobial agent, produced by neurons in response to viral infection, capable of inhibiting pathogens as observed in in vitro experiments. Herein, we ask what are the molecular mechanisms in play when Aβ neutralizes infectious pathogens? Methods: To answer this question, we probed at nanoscale lengths with FRET (Förster Resonance Energy Transfer), the interaction between Aβ peptides and glycoprotein B (responsible of virus-cell binding) within the HSV-1 virion Results: The experiments show an energy transfer between Aβ peptides and glycoprotein B when membrane is intact. No energy transfer occurs after membrane disruption or treatment with blocking antibody. Conclusion: We concluded that Aβ insert into viral membrane, close to glycoprotein B, and participate in virus neutralization.

This work have been published in Journal of Alzheimer's Disease Reports and is available as 22.

Notably with Tamás Fulop (Centre de recherche sur le vieillissement - CSSS-UIGS, Université de Sherbrooke). This work was partially achieved during a stay of Serafim Rodrigues in MathNeuro.

Organismal ageing is associated with many physiological changes, including differences in the immune system of most animals. These differences are often considered to be a key cause of age-associated diseases as well as decreased vaccine responses in humans. The most often cited vaccine failure is seasonal influenza, but, while it is usually the case that the efficiency of this vaccine is lower in older than younger adults, this is not always true, and the reasons for the differential responses are manifold. Undoubtedly, changes in the innate and adaptive immune response with ageing are associated with failure to respond to the influenza vaccine, but the cause is unclear. Moreover, recent advances in vaccine formulations and adjuvants, as well as in our understanding of immune changes with ageing, have contributed to the development of vaccines, such as those against herpes zoster and SARS-CoV-2, that can protect against serious disease in older adults just as well as in younger people. In the present article, we discuss the reasons why it is a myth that vaccines inevitably protect less well in older individuals, and that vaccines represent one of the most powerful means to protect the health and ensure the quality of life of older adults.

We have considerably revised and improved a preprint from last year. This work consists in improving the prediction step in the particle filter. This step is still delicate since if the particles explore areas of the state space that will not match the next observation via the likelihood function, the filter will lose the state track and the filter will be inoperative. We have improved this prediction step by using a simple technique, which may be close to a smoothing technique, that uses the next observation.

This work has been submitted for publication and is available as 54.

These works of Mattia Sensi, that are outside the themes of MathNeuro, were started before his stay at Inria were completed this year, and were done in collaboration in first with Stefania Ottaviano (Dipartimento di Matematica, Padova), second with
Sara Sottile (University of Trento), and in the other hand in
collaboration with Sara Sottile and Ozan Kahramanoğulları (University of Trento), and third with Massimo A. Achterberg (TU Delft).

In a first paper 37 we consider the global stability of multi-group SAIRS epidemic models with vaccination. The role of asymptomatic and symptomatic infectious individuals is explicitly considered in the transmission pattern of the disease among the groups in which the population is divided. We provide a global stability analysis for the model.

In a second paper 31, we study how network properties and epidemic parameters influence stochastic SIR dynamics on scale-free random networks. With the premise that social interactions are described by power-law distributions, we study a SIR stochastic dynamic on a static scale-free random network generated via configuration model. We verify our model with respect to deterministic considerations and provide a theoretical result on the probability of the extinction of the disease. We explore the variability in disease spread by stochastic simulations.

In a third paper 33, we study a minimal model for adaptive SIS epidemics. The interplay between disease spreading and personal risk perception is of key importance for modelling the spread of infectious diseases. We propose a planar system of ordinary differential equations (ODEs) to describe the co-evolution of a spreading phenomenon and the average link density in the personal contact network. Contrary to standard epidemic models, we assume that the contact network changes based on the current prevalence of the disease in the population. We assume that personal risk perception is described using two functional responses: one for link-breaking and one for link-creation.

HBP SGA3 project on cordis.europa.eu