## Section: New Results

### Modeling

#### Simulation of electrical circuits as nonsmooth dynamical systems

Participants : Vincent Acary, Olivier Bonnefon, Bernard Brogliato.

DC-DC converters are usually difficult to simulate with classical tools like SPICE because of the highly nonlinear behaviour of some components and the frequent occurrence of intrinsically generated switching events.

The simulation of such circuits modelled as nonsmooth systems has been successfully achieved with a clear advantage over several SPICE simulators and a simulator belonging to the hybrid modelling approach [31] .

#### Simulation of spiking neuronal networks

Participant : Arnaud Tonnelier.

The numerical simulation of neural networks requires special attention to reproduce accurately the firing times of spiking neurons, while allowing efficient simulation of large networks. Event-driven strategies have become increasingly popular since they allow the simulation of spiking neural networks exactly, with a computational cost similar to classical time-stepping schemes. Previous works were limited to linear integrate-and-fire neurons. In [42] we extend event-driven schemes to a class of nonlinear integrate-and-fire models. Results are presented for the quadratic integrate-and-fire model with exponential synaptic currents.

The development of an event-driven simulation algorithm has to be done
case by case.
In [28] we propose a generic technique,
*voltage-stepping schemes* , that is based on a discretization of
the voltage state-space of individual neurons.
The new simulation strategy defines a local event-driven method inducing
an implicit activity-dependent time stepping scheme.
Long time-steps are used when the neuron is slowly varying, whereas small
time-steps are used in periods of intense activity.
Our method is illustrated on nonlinear integrate-and-fire models. The efficiency of voltage-stepping schemes for the numerical simulation of spiking neural networks
has been assessed in [36] .

#### Neural dynamics

Participant : Arnaud Tonnelier.

The quadratic integrate-and-fire model (QIF) with adaptation is commonly used as an elementary neuronal model that reproduces the main characteristics of real neurons. In [29] , we introduce a QIF neuron with a nonlinear adaptive current. This model reproduces the neuron-computational features of real neurons and is analytically tractable. It is shown that, under a constant current input, chaotic firing is possible. In contrast to previous studies, the neuron is not sinusoidally forced. We show that the spike-triggered adaptation is a key parameter to understand how chaos is generated.

Precise spatiotemporal sequences of spikes are observed in many neural systems and are thought to be involved in the neural processing of sensory stimuli. In [26] we examine the capability of spiking neural networks to propagate stably spatiotemporal sequences of spikes. We derive some analytical results for the wave speed and show that the stability of simple waves is determined by the Schur criteria. The transmission of a sequence of several spikes is related to the existence of stable composite waves, i.e. the existence of stable spatiotemporal periodic traveling waves. We show that the stability of composite waves is related to the roots of a system of multivariate polynomials.

#### Modeling and simulation of mechanical rods

Participant : Florence Bertails.

In Bertails's PhD thesis, a new dynamic model for an elastic rod was presented: the Super-Helix model, which stands for one of the most promising models for simulating non-stretchable rods that can bend and twist. However, this model suffers from a quadratic complexity in the number of discrete elements, which, in the context of interactive applications, makes it limited to a few number of degrees of freedom – or equivalently to a low number of variations in curvature along the mean curve.

In our recent work [13] , we overcome this limitation by proposing a new, recursive scheme for the dynamics of a Super-Helix, inspired by the popular algorithm of Featherstone for serial multibody chains. Similarly to Featherstone's algorithm, we exploit the recursive kinematics of a Super-Helix to propagate element inertias from the free end to the fixed end of the rod, while the dynamics is solved within a second pass traversing the rod in the reverse way. Besides the gain in linear complexity, which allows us to simulate a rod of complex shape much faster than the original approach, our algorithm makes it straightforward to simulate tree-like structures of Super-Helices, which turns out to be particularly useful for animating trees and plants realistically, under large displacements. We are now looking at modeling contact and friction of thin rods with rigid objects.

#### Multiple impacts modelling

Participant : Bernard Brogliato.

The so-called Darboux-Keller approach for modelling simple impacts, is extended to the case of multiple impacts in [22] and [27] , following the impact law introduced in [38] [44] . A distributing law that accounts for the elasticity law is found, and combined with Stronge's energetic coefficient. Careful comparisions are made with experimental results found elsewhere in the physics and mechanical engineering literature on granular media, which show the validity of the model. Coulomb's friction into the model in [22] and [27] .

#### Hair dynamics

Participants : Florence Bertails-Descoubes, Florent Cadoux, Vincent Acary.

In the case of hair dynamics, we have shown in [14] that optimization-based methods such as the approach by Alart and Curnier [35] , little known in the computer graphics literature, outperform classical schemes of computer graphics (typically, penalty-based approaches) for solving the frictious contact problem, both in terms of realism and robustness.