2023Activity reportProject-TeamMATHNEURO

6.1.1 The Gauss-Galerkin approximation method in nonlinear filtering

Participants: Fabien Campillo.

This is a translation in English of an article published by Fabien Campillo in 1986.

We study an approximation method for the one-dimensional nonlinear filtering problem, with discrete time and continuous time observation. We first present the method applied to the Fokker-Planck equation. The convergence of the approximation is established. We finally present a numerical example..

This work is available as 37.

6.2.1 A neural field model for ignition and propagation of cortical spreading depression

Participants: Emre Baspinar, Daniele Avitabile [VU Amsterdam and external collaborator of MathNeuro], Mathieu Desroches, Massimo Mantegazza [Institute of Molecular and Cellular Pharmacology (IPMC) and Inserm].

We propose a new neural field model for migraine-related cortical spreading depression (CSD). The model follows the Wilson-Cowan-Amari formalism. It is based on an excitatory-inhibitory neuron population pair which is coupled to a potassium concentration variable. This model is spatially extended to a cortical layer. Therefore, it can model both the ignition and propagation of CSD. It controls the propagation speed via connection weights and contribution weight of each population activity to the potassium accumulation in the extracellular matrix. The simulation results regarding the propagation speed are in coherence with the experiment results provided in Chever et al. (2021).

This work is available as 33.

6.3.1 From integrator to resonator neurons: A multiple-timescale scenario

Participants: Guillaume Girier [BCAM, Spain], Mathieu Desroches, Serafim Rodrigues [BCAM, Spain, and external collaborator of MathNeuro].

This work has been partially done during stays of Serafim Rodrigues and Guillaume Girier in MathNeuro.

Neuronal excitability manifests itself through a number of key markers of the dynamics and it allows to classify neurons into different groups with identifiable voltage responses to input currents. In particular, two main types of excitability can be defined based on experimental observations, and their underlying mathematical models can be distinguished through separate bifurcation scenarios. Related to these two main types of excitable neural membranes, and associated models, is the distinction between integrator and resonator neurons. One important difference between integrator and resonator neurons, and their associated model representations, is the presence in resonators, as opposed to integrators, of subthreshold oscillations following spikes. Switches between one neural category and the other can be observed and/or created experimentally, and reproduced in models mostly through changes of the bifurcation structure. In the present work, we propose a new scenario of switch between integrator and resonator neurons based upon multiple-timescale dynamics and the possibility to force an integrator neuron with a specific time-dependent slowly-varying current. The key dynamical object organizing this switch is a so-called folded-saddle singularity. We also showcase the reverse switch via a folded-node singularity and propose an experimental protocol to test our theoretical predictions.

This work has been published in Nonlinear Dynamics and is available as 26.

6.3.2 Entry-exit functions in fast-slow systems with intersecting eigenvalues

Participants: Panagiotis Kaklamanos [The University of Edinburgh, UK], Christian Kuehn [TU Munich, Germany], Nikola Popovic [The University of Edinburgh, UK], Mattia Sensi.

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 neighbourhoods 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 was published in Journal of Dynamics and Differential Equations and is available as 27.

6.4.1 Observing hidden neuronal states in experiments

Participants: Dmitry Amakhin [Sechenov Institute of Evolutionary Physiology and Biochemistry of RAS, Russia], Anton Chizhov, Guillaume Girier [BCAM, Spain], Mathieu Desroches, Jan Sieber [University of Exeter, UK], Serafim Rodrigues [BCAM, Spain, and external collaborator of MathNeuro].

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

We construct systematically experimental steady-state bifurcation diagrams for entorhinal cortex neurons. A slowly ramped voltage-clamp electrophysiology protocol serves as closed-loop feedback controlled experiment for the subsequent current-clamp open-loop protocol on the same cell. In this way, the voltage-clamped experiment determines dynamically stable and unstable (hidden) steady states of the current-clamp experiment. The transitions between observable steady states and observable spiking states in the current-clamp experiment reveal stability and bifurcations of the steady states, completing the steady-state bifurcation diagram.

This work has been submitted for publication and is available as 32.

6.4.2 Single-compartment model of a pyramidal neuron, fitted to recordings with current and conductance injection

Participants: Anton Chizhov, Dmitry Amakhin [Sechenov Institute of Evolutionary Physiology and Biochemistry of RAS, Russia], A. Erdem Sagtekin [Istanbul Technical University, Turkey], Mathieu Desroches.

For single neuron models, reproducing characteristics of neuronal activity such as the firing rate, amplitude of spikes, and threshold potentials as functions of both synaptic current and conductance is a challenging task. In the present work, we measure these characteristics of regular spiking cortical neurons using the dynamic patch-clamp technique, compare the data with predictions from the standard Hodgkin-Huxley and Izhikevich models, and propose a relatively simple five-dimensional dynamical system model, based on threshold criteria. The model contains a single sodium channel with slow inactivation, fast activation and moderate deactivation, as well as, two fast repolarizing and slow shunting potassium channels. The model quantitatively reproduces characteristics of steady-state activity that are typical for a cortical pyramidal neuron, namely firing rate not exceeding 30 Hz; critical values of the stimulating current and conductance which induce the depolarization block not exceeding 80 mV and 3 times the resting input conductance, respectively; extremum of hyperpolarization close to the midpoint between spikes. The analysis of the model reveals that the spiking regime appears through a saddle-node-on-invariant-circle bifurcation, and the depolarization block is reached through a saddle-node bifurcation of cycles. The model can be used for realistic network simulations, and it can also be implemented within the so-called mean-field, refractory density framework.

This work was published in Biological Cybernetics and is available as 24.

6.4.3 Complex excitability and "flipping" of granule cells: an experimental and computational study

Participants: Joanna Danielewicz [BCAM, Bilbao], Guillaume Girier [BCAM, Bilbao], Anton Chizhov, Mathieu Desroches, Juan Manuel Encinas [Achucarro Center for Neuroscience, Spain], Serafim Rodrigues [BCAM, Spain, and external collaborator of MathNeuro].

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

In response to prolonged depolarizing current steps, different classes of neurons display specific firing characteristics (i.e. excitability class), such as a regular train of action potentials with more or less adaptation, delayed responses, or bursting. In general, one or more specific ionic transmembrane currents underlie the different firing patterns. Here we sought to investigate the influence of artificial sodium-like (Na channels) and slow potassium-like (KM channels) voltage-gated channels conductances on firing patterns and transition to depolarization block (DB) in Dentate Gyrus granule cells with dynamic clamp - a computer-controlled real-time closed-loop electrophysiological technique, which allows to couple mathematical models simulated in a computer with biological cells. Our findings indicate that the addition of extra Na/KM channels significantly affects the firing rate of low frequency cells, but not in high frequency cells. Moreover, we have observed that 44 percent of recorded cells exhibited what we have called a “flipping” behavior. This means that these cells were able to overcome the DB and generate trains of action potentials at higher current injections steps. We have developed a unified mathematical model of flipping cells to explain this phenomenon. Based on our computational model, we conclude that the appearance of flipping is linked to the number of states for the sodium channel of the model.

This work was submitted for publication and is available as 35.

6.4.4 Idealized multiple-timescale model of cortical spreading depolarization initiation and pre-epileptic hyperexcitability caused by NaV1.1/SCN1A mutations

Participants: Louisiane Lemaire [Humbolt University, Berlin, Germany], Mathieu Desroches, Martin Krupa, Massimo Mantegazza [Institute of Molecular and Cellular Pharmacology (IPMC) and Inserm].

NaV1.1 (SCN1A) is a voltage-gated sodium channel mainly expressed in GABAergic neurons. Loss of function mutations of NaV1.1 lead to epileptic disorders, while gain of function mutations cause a migraine in which cortical spreading depolarizations (CSDs) are involved. It is still debated how these opposite effects initiate two different manifestations of neuronal hyperactivity: epileptic seizures and CSD. To investigate this question, we previously built a conductance-based model of two neurons (GABAergic and pyramidal), with dynamic ion concentrations 20. When implementing either NaV1.1 migraine or epileptogenic mutations, ion concentration modifications acted as slow processes driving the system to the corresponding pathological firing regime. However, the large dimensionality of the model complicated the exploitation of its implicit multi-timescale structure. Here, we substantially simplify our biophysical model to a minimal version more suitable for bifurcation analysis. The explicit timescale separation allows us to apply slow-fast theory, where slow variables are treated as parameters in the fast singular limit. In this setting, we reproduce both pathological transitions as dynamic bifurcations in the full system. In the epilepsy condition, we shift the spike-terminating bifurcation to lower inputs for the GABAergic neuron, to model an increased susceptibility to depolarization block. The resulting failure of synaptic inhibition triggers hyperactivity of the pyramidal neuron. In the migraine scenario, spiking-induced release of potassium leads to the abrupt increase of the extracellular potassium concentration. This causes a dynamic spike-terminating bifurcation of both neurons, which we interpret as CSD initiation.

This work was published in Journal of Mathematical Biology and is available as 28.

6.5.1 Slow-fast dynamics in a neurotransmitter release model: delayed response to a time-dependent input signal

Participants: Mattia Sensi, Mathieu Desroches, Serafim Rodrigues [BCAM, Spain, and external collaborator of MathNeuro].

We propose a generalization of the neurotransmitter release model proposed in Rodrigues et al. (PNAS, 2016). We increase the complexity of the underlying slow-fast system by considering a degree-four polynomial as parametrization of the critical manifold. We focus on the possible transient and asymptotic dynamics, exploiting the so-called entry-exit function to describe slow parts of the dynamics. We provide extensive numerical simulations, complemented by numerical bifurcation analysis.

This work was published in Physica D and is available as 30.

6.5.2 Synchronization in STDP-driven memristive neural networks with time-varying topology

Participants: Marius Yamakou [University of Erlangen-Nuremberg, Germany], Mathieu Desroches, Serafim Rodrigues [BCAM, Spain, and external collaborator of MathNeuro].

Synchronization is a widespread phenomenon in the brain. Despite numerous studies, the specific parameter configurations of the synaptic network structure and learning rules needed to achieve robust and enduring synchronization in neurons driven by spike-timing-dependent plasticity (STDP) and temporal networks subject to homeostatic structural plasticity (HSP) rules remain unclear. Here, we bridge this gap by determining the configurations required to achieve high and stable degrees of complete synchronization (CS) and phase synchronization (PS) in time-varying small-world and random neural networks driven by STDP and HSP. In particular, we found that decreasing P (which enhances the strengthening effect of STDP on the average synaptic weight) and increasing F (which speeds up the swapping rate of synapses between neurons) always lead to higher and more stable degrees of CS and PS in small-world and random networks, provided that the network parameters such as the synaptic time delay ${\tau }_c$, the average degree $\langle k\rangle$, and the rewiring probability $\beta$ have some appropriate values. When ${\tau }_c$, $\langle k\rangle$, and $\beta$ are not fixed at these appropriate values, the degree and stability of CS and PS may increase or decrease when F increases, depending on the network topology. It is also found that the time delay ${\tau }_c$ can induce intermittent CS and PS whose occurrence is independent F . Our results could have applications in designing neuromorphic circuits for optimal information processing and transmission via synchronization phenomena.

This work was published in Journal of Biological Physics and is available as 31.

6.6.1 Viruses - a major cause of amyloid deposition in the brain

Participant: Tamas Fülöp [Université de Sherbrooke, Canada], Charles Ramassamy [INRS-IAF, Canada], Simon Lévesque [Centre hospitalo-universitaire de Sherbrooke, Canada], Eric Frost [Université de Sherbrooke, Canada], Benoît Laurent [University of Sherbrooke, Canada], Guy Lacombe [University of Sherbrooke, Canada], Abdelouahed Khalil [University of Sherbrooke, Canada], Anis Larbi [University of Sherbrooke, Canada], Katsuiku Hirokawa [Tokyo Medical and Dental University, Japan], Mathieu Desroches, Serafim Rodrigues [BCAM, Spain, and external collaborator of MathNeuro].

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

Clinically, Alzheimer's disease (AD) is a syndrome with a spectrum of various cognitive disorders. There is a complete dissociation between the pathology and the clinical presentation. Therefore, we need a disruptive new approach to be able to prevent and treat AD. In this review, we extensively discuss the evidence why the amyloid beta is not the pathological cause of AD which makes therefore the amyloid hypothesis not sustainable anymore. We review the experimental evidence underlying the role of microbes, especially that of viruses, as a trigger/cause for the production of amyloid beta leading to the establishment of a chronic neuroinflammation as the mediator manifesting decades later by AD as a clinical spectrum. In this context, the emergence and consequences of the infection/antimicrobial protection hypothesis are described. The epidemiological and clinical data supporting this hypothesis are also analyzed. For decades, we have known that viruses are involved in the pathogenesis of AD. This discovery was ignored and discarded for a long time. Now we should accept this fact, which is not a hypothesis anymore, and stimulate the research community to come up with new ideas, new treatments, and new concepts.

This work was published in Expert Review of Neurotherapeutics and is available as 25.

6.6.2 Topological Data Analysis of Human Brain Data

Participant: Ufuk Cem Birbiri [Université Côte d'Azur and MathNeuro].

This work is a research report written by Ufuk Cem Birbiri out of the report of his master 2 internship done under the supervision of Mathieu Desroches in 2022. This project has also involved Serafim Rodrigues (BCAM, Spain) and Fernando Santos (University of Amsterdam and Institute for Advanced Study, The Netherlands). It is available as 34. The work finds its place within our research line on using advanced data-scientific methods, such as Topological Data Analysis, to study biomarkers of aging.

6.7.1 The one step fixed-lag particle smoother as a strategy to improve the prediction step of particle filtering

Participant: Samuel Nyobe [University of Yaoundé, Cameroon], Fabien Campillo, Serge Moto [University of Yaoundé, Cameroon], Vivien Rossi [Cirad and University of Yaoundé, Cameroon].

Sequential Monte Carlo methods have been a major breakthrough in the field of numerical signal processing for stochastic dynamical state-space systems with partial and noisy observations. However, these methods still present certain weaknesses. One of the most fundamental is the degeneracy of the filter due to the impoverishment of the particles: the prediction step allows the particles to explore the state-space and can lead to the impoverishment of the particles if this exploration is poorly conducted or when it conflicts with the following observation that will be used in the evaluation of the likelihood of each particle. In this article, in order to improve this last step within the framework of the classic bootstrap particle filter, we propose a simple approximation of the one step fixed- lag smoother. At each time iteration, we propose to perform additional simulations during the prediction step in order to improve the likelihood of the selected particles.

This work was published in ARIMA and is available as  52.

7.1.1 Visits of international scientists

7.1.2 Visits to international teams

7.1.3 H2020 projects

7.2.1 ANR projects

8.1.1 Scientific events: selection

• Mathieu Desroches
  was co-organiser of the two-part Mini-Symposium Multiple-Timescale Dynamics with a View Towards Biological Applications, at the SIAM Conference on Applications of Dynamical Systems (DS23), Portland (USA), 14-18 May 2023.

8.1.2 Invited talks

• Fabien Campillo
  gave an invited talk entitled “Nonlinear Filtering in Neuroscience” at the 21st INFORMS Applied Probability Society Conference, Nancy (France), 30 June 2023.
• Anton Chizhov
  gave an invited talk, jointly with Lyle Graham (CNRS, Université de Paris) entitled “Neurons and neuronal populations: From recordings in vivo to simulations of cortical tissue”, as a Neuromod mini-course, Inria centre at Université Côte d'Azur, 11 January 2023.
• Mathieu Desroches
  gave an invited talk entitled: “Classification of bursting patterns: Mind the slow variables” at the SIAM Conference on Applications of Dynamical Systems (DS23), Portland (USA), 17 May 2023.
• Mathieu Desroches
  gave an invited seminar talk entited “Classification of bursting patterns: Review & Extension” at the Institut de Neurosciences des Systèmes, Marseille (France), 14 September 2023.

8.1.3 Leadership within the scientific community

• Fabien Campillo
  is a founding member of the African scholarly Society on Digital Sciences (ASDS).
• Mathieu Desroches
  is member of the scientific committee of the Complex Systems Academy of the UCAJEDI IDEX.

8.1.4 Scientific expertise

• Mathieu Desroches
  has been reviewing grant proposals for the Agence Nationale de la Recherche (AAPG 2023).
• Mathieu Desroches
  has been reviewing grant proposals for the Complex Systems Academy of the UCAJEDI Idex.

8.1.5 Research administration

• Fabien Campillo
  is member of the “Inria Evaluation Committee (CE)”.
• Fabien Campillo
  is member of the “Health and Safety committee (CSHCT)” of the Inria centre at Université Côte d'Azur.
• Mathieu Desroches
  is supervising the PhD seminar at the Inria centre at Université Côte d'Azur.

8.2.1 Teaching

• Master:
  Mathieu Desroches, Modèles Mathématiques et Computationnels en Neuroscience (Lectures, example classes and computer labs), 18 hours (Feb. 2023), M1 (BIM), Sorbonne Université, Paris, France.
• Master:
  Mathieu Desroches, Multiple Timescale Dynamics in Neuroscience, (Lectures, example classes and computer labs), 9 hours (Jan. 2023) and 21 hours (Nov.-Dec. 2023), M1 (Mod4NeuCog), Université Côte d'Azur, Sophia Antipolis, France.
• Masters and Engineer schools:
  With the project to write a book, Fabien Campillo proposes a set of applications of particle filtering, developed in the context of lectures given during many years in Masters and Engineer schools. See the associated web page and git repository.

8.2.2 Supervision

• PhD
  Guillaume Girier, Basque Center for Applied Mathematics (BCAM, Bilbao, Spain) is doing a PhD on “A mathematical, computational and experimental study of neuronal excitability”, co-supervised by S. Rodrigues (BCAM) and Mathieu Desroches, in co-tutelle between the University of the Basque Country and Université Côte d'Azur.
• PhD
  Jordi Penalva Vadell, University of the Balearic Islands (UIB, Palma, Spain) is doing a PhD on “Neuronal piecewise linear models reproducing bursting dynamics”, co-supervised by A. E. Teruel (UIB), C. Vich (UIB) and Mathieu Desroches.
• Master 2 internship
  Safaa Habib (Université Côte d’Azur - UCA, Nice), “Revisiting a Single-Neuron Model of Seizure-Like Events, Ictal Activity and Depolarization Block”, supervised by Mathieu Desroches, April - September 2023.

8.2.3 Juries

• Fabien Campillo
  was member of the jury and reviewer of the PhD of Jana Zaherddine, entitled “Modèles mathématiques de l’allocation dynamique des ressources dans une cellule de bactérie”, supervised by Philipe Robert (Inria Paris Research Centre) and Vincent Fromion (Inrae, Jouy-en-Josas), 19 December 2023.
• Pascal Chossat
  was member of the jury of the PhD of Maria Virginia Bolelli, entitled “Neurogeometry of stereo vision”, supervised by Giovanna Citti (University of Bologna, Italy) and Alessandro Sarti (EHESS, Paris, France), 27 March 2023.
• Mathieu Desroches
  was member of the jury and reviewer of the PhD of Lisa Blum Moyse, entitled “Computational neuroscience models at different levels of abstraction for synaptic plasticity, astrocyte modulation of synchronization and systems memory consolidation”, supervised by Hugues Berry (Inria Lyon Centre), 14 September 2023.

International journals

• 24 articleA.Anton Chizhov, D.Dmitry Amakhin, A. E.A. Erdem Sagtekin and M.Mathieu Desroches. Single-compartment model of a pyramidal neuron, fitted to recordings with current and conductance injection.Biological Cybernetics (Modeling)September 2023HALDOIback to text
• 25 articleT.Tamas Fulop, C.Charles Ramassamy, S.Simon Lévesque, E.Eric Frost, B.Benoit Laurent, G.Guy Lacombe, A.Abedelouahed Khalil, A.Anis Larbi, K.Katsuiku Hirokawa, M.Mathieu Desroches, S.Serafim Rodrigues, K.Karine Bourgade, A.Alan Cohen and J.Jacek Witkowski. Viruses - a major cause of amyloid deposition in the brain.Expert Review of Neurotherapeutics239July 2023, 775-790HALDOIback to text
• 26 articleG.Guillaume Girier, M.Mathieu Desroches and S.Serafim Rodrigues. From integrator to resonator neurons: A multiple-timescale scenario.Nonlinear DynamicsJune 2023HALDOIback to text
• 27 articleP.Panagiotis Kaklamanos, C.Christian Kuehn, N.Nikola Popović and M.Mattia Sensi. Entry-exit functions in fast-slow systems with intersecting eigenvalues.Journal of Dynamics and Differential EquationsJune 2023HALDOIback to text
• 28 articleL.Louisiane Lemaire, M.Mathieu Desroches, M.Martin Krupa and M.Massimo Mantegazza. Idealized multiple-timescale model of cortical spreading depolarization initiation and pre-epileptic hyperexcitability caused by NaV1.1/SCN1A mutations.Journal of Mathematical Biology866June 2023, 92HALDOIback to text
• 29 articleS.Samuel Nyobe, F.Fabien Campillo, S.Serge Moto and V.Vivien Rossi. The one step fixed-lag particle smoother as a strategy to improve the prediction step of particle filtering.Revue Africaine de Recherche en Informatique et Mathématiques Appliquées39November 2023, 1-23HALDOI
• 30 articleM.Mattia Sensi, M.Mathieu Desroches and S.Serafim Rodrigues. Slow-fast dynamics in a neurotransmitter release model: delayed response to a time-dependent input signal.Physica D: Nonlinear Phenomena4552023, 133887HALDOIback to text
• 31 articleM.Marius Yamakou, M.Mathieu Desroches and S.Serafim Rodrigues. Synchronization in STDP-driven memristive neural networks with time-varying topology.Journal of Biological PhysicsAugust 2023HALDOIback to text

Reports & preprints

• 32 miscD.Dmitry Amakhin, A.Anton Chizhov, G.Guillaume Girier, M.Mathieu Desroches, J.Jan Sieber and S.Serafim Rodrigues. Observing hidden neuronal states in experiments.August 2023HALback to text
• 33 miscE.E Baspinar, D.D Avitabile, M.M Desroches and M.M Mantegazza. A neural field model for ignition and propagation of cortical spreading depression.February 2023HALback to text
• 34 reportU. C.Ufuk Cem Birbiri. Topological Data Analysis of Human Brain Data.Université Côte d'Azur, FranceAugust 2023HALback to text
• 35 miscJ.Joanna Danielewicz, G.Guillaume Girier, A.Anton Chizhov, M.Mathieu Desroches, J. M.Juan Manuel Encinas and S.Serafim Rodrigues. Complex excitability and "flipping" of granule cells: an experimental and computational study.October 2023HALback to text
• 36 miscP.Panagiotis Kaklamanos, A.Andrea Pugliese, M.Mattia Sensi and S.Sara Sottile. A geometric analysis of the SIRS model with secondary infections.2023HAL

Other scientific publications

• 37 miscF.Fabien Campillo. The Gauss-Galerkin approximation method in nonlinear filtering.February 2023HALback to text