Séminaire Interfaces des mathématiques et systèmes complexes

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Vendredi 7 Juin 2019 à 11 heures en salle de Conférences du LJAD

Invité Franck (Lyon)

" Title "




Archives vidéos (2009-2011)


Archives séminaires (2011-2018)


Lundi  5  Novembre 2018 à 11 h (salle de conf du LJAD)

Boris Gutkin (Paris/Moscou)

'Intrinsic and Synaptic Properties Determine Macroscopic Phase Response
Curves and Coherence States of Inter-communicating Gamma Oscillatory
Neural Circuits'

Abstract: Macroscopic oscillations of different brain regions show
multiple phase relations that are persistent across time [5]. Such phase
locking is believed to be implicated in a number of cognitive functions
and is key to the so-called Communication Through Coherence (CTC) theory
for neural information transfer [7]. The question is then to identify
the biophysical neuronal and synaptic properties that permit such motifs
to arise.

We investigate the dynamical emergence of phase locking within two
bidirectionally delayed-coupled spiking circuits with emergent gamma
band oscillations. Internally the circuits consist of excitatory and
inhibitory cells coupled synaptically in an all-to-all fashion and
parameterized to give either the pyramidal-interneuron (PING) or
interneuron gamma (ING) rhythm activity to emerge. Multiple circuits can
also be intercoupled together with reciprocal synaptic connections with
variable delays and targeting excitatory and/or inhibitory neurons.

Using mean-field approach combined with an exact reduction method [3,8],
we break down the description of each spiking network into a low
dimensional nonlinear system. The adjoint method can be applied to
derive semi-analytical expressions for the macroscopic infinitesimal
phase resetting-curves (mPRCs) [2,4]. Using the mPRCs we determine how
the phase of the global oscillation responds to incoming perturbations.
In fact, we find that depending on wether PING or ING is expressed and
wether the excitatory of inhibitory neurons are perturbed, the mPRC can
be either class I (purely positive) or class II (by-phasic). Hence we
show analytically how incoming excitation can either promote spiking
(advancing the phase) or retard the oscillation.

We further study the dynamical emergence of macroscopic phase locking of
two bidirectionally delayed-coupled spiking networks within the
framework of weakly coupled oscillators [1,9]. Using the  weak coupling
ansatz we  abbreviate the bidirectionally coupled circuits description
into a phase equation [1,9]. An analysis of the phase equation shows
that the delay is a necessary condition to get a symmetry breaking. We
find that a whole host of phase-locking relationships can exist,
depending on the coupling strength and delay. Our analysis further
allows us to track how the perturbation of on the interacting circuits
is transferred to its coupled partner. We find that such transfer is
phase-locking state dependent. We show that this transfer of signals can
be directional in the symmetrically coupled network.

[1] Ashwin, P., Coombes, S., Nicks, R.: Mathematical frameworks for
oscillatory network dynamics in neuroscience. The Journal of
Mathematical Neuroscience 6(1), 2 (2016).

[2] Brown, E., Moehlis, J., Holmes, P.: On the phase reduction and
response dynamics of neural oscillator populations. Neural Computation
16(4), 673{715 (2004).

[3] Deco, G., Jirsa, V.K., Robinson, P.A., Breakspear, M., Friston, K.:
The dynamic brain: From spiking neurons to neural masses and cortical
fields. PLoS Comput Biol 4(8), 1{35 (2008). DOI
10.1371/journal.pcbi.1000092. Neural Computation 25(12), 3207{3234

[4] Dumont, G., Ermentrout, G.B., Gutkin, B.: Macroscopic
phase-resetting curves for spiking neural networks. Phys. Rev. E 96,
042,311 (2017).

[5] Fries, P.: Neuronal gamma-band synchronization as a fundamental
process in cortical computation. Annual review of neuroscience 32,
209–224 (2009)

[7] Maris, E., Fries, P., van Ede, F.: Diverse phase relations among
neuronal rhythms and their potential function. Trends in Neurosciences
39(2), 86{99. DOI 10.1016/j.tins.2015.12.004.

[8] Montbrio, E., Pazo, D., Roxin, A.: Macroscopic description for
networks of spiking neurons. Phys. Rev. X 5, 021,028 (2015).

[9] Nakao, H.: Phase reduction approach to synchronisation of nonlinear
oscillators. Contemporary Physics 57(2), 188{214 (2016).


Mercredi  7  Novembre 2018 à 11 h (salle III du LJAD)

Yohann Scribano  (Montpellier)

"Quantum trajectories approach for low-temperature molecular reactive processes"

Based on the de Broglie-Bohm formulation of quantum mechanics [1], quantum trajectories have seen a growing
interest in the chemical dynamics community over the past ten years or so [2]. As the name suggests, quantum
trajectories (QTs),are able to capture quantum dynamical effects—tunneling, in particular—which are known or
suspected to be important for many processes of current interest: tunneling effect in reactive processes (in gas or
condensed phases), non-adiabatic effects, quantum coherence, proton transfers in biochemistry, . . . .
Originally considered an interpretative tool for quantum mechanics, QTs have been recently rediscovered as a
computational method for doing quantum reaction dynamics [2]. Traditionally, analytic quantum trajectories, often
used as an interpretative tool, were extracted from conventional wave packets. In the more recent synthetic
quantum trajectory methods, both the trajectories and the wave function are computed on the fly, each affecting the
propagation of the other [2].
We recently investigate the most recent formulation of QTs proposed by Bill Poirier and coworkers [3, 4],
which regards the trajectory ensemble itself as the fundamental quantum state entity, rather than the wavefunction.
Similarly to classical trajectories, QTs obey Hamiltonian equations of motion, albeit special ones. The resultant
quantum trajectory simulation scheme so obtained is identical to a classical trajectory simulation, apart from the
addition of one extra “quantum” coordinate. Using standard techniques to integrate the equations of motion,
quantum trajectory simulations is indeed capable of providing accurate quantum dynamical information, but with
the same ease-use and computational effort as classical trajectory simulation. In this QTs formulation, a 4th-order
Newtonian-like ordinary differential equation (ODE) was derived that describes 1D stationary scattering states
exactly, solely in terms of quantum trajectories. The concept of those QTs will be presented and illustrated by our
application for a 1D Eckart barrier system [5] as well as its application in a capture model of the cold and ultra-cold
Li + CaH reaction [6]. Some perpectives will also presented on the way to perform quantum-classical trajectories
simulations for chemical reaction involving many degree of freedom (high dimensional reaction dynamics).
[1] D. Bohm, Phys.Rev. 85, 85.
[2] R. E. Wyatt, Quantum Dynamics with Trajectories (Springer, New York, 2005).
[3] B. Poirier, Chem. Phys. 370, 4 (2010).
[4] J. Schiff and B. Poirier. J. Chem. Phys., 136, 031102 (2012).
[5] G. Parlant, Y.-C. Ou, K. Park, and B. Poirier, Comp. Theor. Chem. 990, 3 (2012).
[6] Y. Scribano, G. Parlant, and B. Poirier, J. Chem. Phys. 149, 021101 (2018).

Vendredi 30 novembre 2018 à 11 heures en salle de Conférences du LJAD

Daniele Avitabile (Nottingham)

"Analysing waves and bumps with interfacial dynamics: spatio-temporal canards and
localised spatiotemporal chaos in neural networks"

We will discuss level-set based approaches to study the existence and
bifurcation structure of spatio-temporal patterns in biological neural
networks. Using this framework, which extends previous ideas in the
study of neural field models, we study the first example of canards in
an infinite-dimensional dynamical system, and we give a novel
characterisation of localised structures, informally called “bumps”,
supported by spiking neural networks.

We will initially consider a spatially-extended network with heterogeneous
synaptic kernel. Interfacial methods allow for the explicit
construction of a bifurcation equation for localised steady states. When
the model is subject to slow variations in the control parameters,
a new type of coherent structure emerges: the structure displays a
spatially-localised pattern, undergoing a slow-fast modulation at the
core. Using interfacial dynamics and geometric singular perturbation
theory, we show that these patterns follow an invariant repelling slow
manifold, hence we name them "spatio-temporal canards". We classify
spatio-temporal canards and give conditions for the existence of
folded-saddle and folded-node canards. We also find that these
structures are robust to changes in the synaptic connectivity and firing
rate. The theory correctly predicts the existence of spatio-temporal
canards with octa
hedral symmetries in a neural field model posed on a spherical domain. 
We will then discuss how the insight gained with interfacial dynamics may
be used to perform coarse-grained bifurcation analysis on neural
networks, even in models where the network does not evolve according to
an integro-differential equation. As an example I will consider a
well-known event-driven network of spiking neurons, proposed by Laing
and Chow. In this setting, we construct numerically travelling waves
whose profiles possess an arbitrary number of spikes. An open question
is the origin of the travelling waves, which have been conjectured to
form via a destabilisation of a bump solution. We provide numerical
evidence that this mechanism is not in place, by showing that
disconnected branches of travelling waves with countably many spikes
exist, and terminate at grazing points; the grazing points correspond to
travelling waves with an increasing number of spikes, a well-defined
width, and decreasing propagation speed. We interpret the so called
“bumps” and “meandering bumps”, supported by this model as localised
states of spatiotemporal chaos, whereby the dynamics visits a large
number of unstable localised travelling wave solutions. 
This is joint work with Mathieu Desroches, Edgar Knobloch, Joshua Davis and Kyle Wedgwood.

Vendredi 11 janvier 2019 à 11 heures en salle 2 du LJAD

Giovanni Manfredi (Strasbourg)

"Self-gravitating systems: quantum and classical"

In recent years, there has been a renewal of interest in the set of nonlinear equations known as the
Schrödinger–Newton equations (SNEs). These consist of the ordinary Schrödinger equation endowed with
a gravitational potential that is obtained self-consistently from Poisson’s equation. The source term in
Poisson’s equation is provided by a matter density
ρ ( r , t )= | ψ |^2 that is proportional to the probability
density as given by the wave function ψ (r , t) .
The SNEs were originally put forward by Diosi [1] and Penrose [2] as a fundamental modification of
quantum mechanics for massy objects. The SNE equations may also represent the weak-field limit of
semiclassical gravity [3]. Finally, in an astrophysical context, they were used to study self-gravitating
objects (boson stars) [4] or to describe dark matter by means of a scalar field [5].
Here, I will first revisit the SNEs using a variational method [6]. With this approach, one can arrive at a
single ODE that describes the evolution of the width of the mass density. This method reproduces all the
results for the ground state and linear dynamics derived previously. In addition, it is not restricted to linear
theory and can be used to investigate nonlinear oscillations or the long-time dynamics.
Second, I will present a general-relativistic extension of the SNEs. The strategy adopted here is to first
replace Poisson’s equation with the equations of gravitoelectromagnetism (GEM). These are a linearized
approximation of the Einstein equations, which is formally almost identical to Maxwell’s equations of
ordinary electromagnetism. Next, we derive the nonrelativistic limit of the GEM equations using the
methods developed in [6]. The resulting equations can be coupled to the Schrödinger equation to obtain a
new set of SNEs augmented by a gravitomagnetic field. Such modified SNEs, though still Galilei covariant,
incorporate all three fundamental constants (G, h, c) and can be conveniently expressed in Planck’s units.
Finally, I will discuss some recent results on a class of idealized 1D models for cosmological applications.
Most existing results rely on N-body simulations, which solve the equations of motion of many particles
interacting via Newton’s force. N-body simulations show the formation of a hierarchical structure [7] that
suggests a fractal distribution of mass. The analyses are robust for high-density regions, but noisy in the
low-density ones. To gain a better insight, we propose to use a continuous probability distribution in the
phase space, which evolves according to the Vlasov-Poisson equations. As the entire phase space is
covered with a uniform mesh, regions of high and low density are sampled with equal precision and the
numerical noise remains low. Here, I will present the results of these Vlasov-Poisson simulations [8].
[1] Diosi, L.: Phys. Lett. A 105, 199 (1984)
[2] Penrose, R.: Gen. Relativ. Gravit. 28, 581 (1996)
[3] Carlip, S.: Class. Quantum Gravit. 25, 154010 (2008)
[4] Schunck, F.E., Mielke, E.W.: Class. Quantum Gravit. 20, R301 (2003)
[5] Guzman, F.S., Urena-Lopez, L.A.: Phys. Rev. D 68, 024023 (2003)
[6] Manfredi, G.: Eur. J. Phys. 34, 859 (2013)
[7] B. N. Miller and J.-L. Rouet. J. Stat. Mech. P12028 (2010); Phys. Rev. E 82, 066203 (2010).
[8] G. Manfredi, J.-L. Rouet, B. Miller, and Y. Shiozawa, Phys. Rev. E 93, 042211 (2016).



Vendredi 25 janvier 2019 à 14 heures en salle 2 du LJAD

Marco Martins Afonso (Centro de Matemática da Universidade do Porto)


We study how an imposed fluid flow - laminar or turbulent - modifies
the transport properties of inertial particles (e.g. aerosols,
droplets or bubbles), namely their terminal velocity, effective
diffusivity, and concentration following a point-source emission.
Such quantities are investigated by means of analytical and
numerical computations, as functions of the control parameters of
both flow and particle; i.e., density ratio, inertia, Brownian
diffusivity, gravity (or other external forces), turbulence
intensity, compressibility degree, space dimension, and
geometric\temporal properties.

The complex interplay between these parameters leads to the
following conclusion of interest in the realm of applications: any
attempt to model dispersion and sedimentation processes - or
equivalently the wind-driven surface transport of floaters -
cannot avoid taking into account the full details of the flow
field and of the inertial particle.


Mardi 26 mars 2019 à 14 heures en salle de conférence du LJAD

Prof. Katsuhiko Sato (Hokkaido Univ.)

"Left-right asymmetric cell intercalation drives directional collective cell movement in epithelial morphogenesis"

During early development and wound healing, epithelial cells (cohesive cells)
that form a monolayer sheet sometimes move collectively in a definite direction in the sheet; that is, the
cohesive cells move unidirectionally while maintaining the attachments with adjacent cells. This phenomenon is
called collective migration of epithelial cells [1], and is considered to be an essential factor for
morphological changes of multicellular organisms [2]. While the molecular mechanisms underlying this
phenomenon are becoming understood, much of the mechanical mechanisms remain still unclear. In the present
talk, I will provide one possible mechanism for collective migration from a theoretical point of view [3]: If
the cell boundaries contract depending on their orientation (generalized planar polarity), and spatial
inhomogeneity about the cell properties, such as strength of cell adhesion, exists, then the cohesive cells
collectively move in the direction perpendicular to that of spatial inhomogeneity, by repeating rearrangement
of neighbor relationships. I will demonstrate this scenario by using the vertex model, and reproduce the
behaviors of typical collective cell migration such as that seen in zebrafish lateral line primordium. I also
provide experimental evidence for this type of movement by investigating a phenomenon in development of fly,
where monolayer epithelial sheets move 360 degrees clockwise around the genital disc [4].
1. P. Friedl and D. Gilmour, (2009) Nat. Rev. Mol. Cell Biol. 10, 445-457.
2. J. Davies, Mechanisms of Morphogenesis (Academic Press, Second Edition, 2013).
3. K. Sato, T. Hiraiwa, and T. Shibata, Phys. Rev. Lett. 115, 188102 (2015).
4. K. Sato, T. Hiraiwa, E. Maekawa, A. Isomura, T. Shibata, and E. Kuranaga, Nat. Commun. 6, 10074 (2015).

Vendredi 5 avril 2019 à 14 heures en salle 1 du LJAD

Giorgio Krstulovic (Laboratoire Lagrange, OCA)


"Non-linear wave interaction and cascades in thin elastic plates"


In fluid turbulence, energy is transferred from one scale to another by
an energy cascade that depends only on the energy dissipation rate.
Remarkably, a similar phenomenon takes place in thin elastic plates. In
the limit where the non-linearity in the equations governing the
vibrations of a plate is weak, the theory of wave turbulence (WT) can be
safely applied. Over the last ten years this cascade has been
extensively studied theoretically and numerically. This system has
revealed itself as a useful prototype to study wave interactions and

In this seminar, I will start by explaining how the theory of weak wave
turbulence applies to this problem, what are the corresponding
theoretical predictions and how well experimental results confirm such
predictions. Then, I will show that for this system, it is possible to
derive the analogous of the 4/5-law of hydrodynamic turbulence, which
provides an exact result concerning the energy transfers. Moreover, I
will show than in the fully non-linear regime, elastic deformations
present large “eddies” together with a myriad of small “crumpling
eddies”, such that folds, developable cones, and more complex stretching
structures, in close analogy with swirls, vortices and other structures
in hydrodynamic turbulence. This deformations lead to the same
Kolmogorov spectrum observed in hydrodynamic turbulence. To finish, if
time allows it, I will discuss the non-dispersive limit and explain what
are the mathematical difficulties of applying WT theory in this case.