Séminaire Interfaces des mathématiques et systèmes complexes

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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 octahedral 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.



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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).