arXiv subjects

Rodolphe Sepulchre

Publications and source records attributed to Rodolphe Sepulchre.

At least 91 records · Page 5Linked to original sources

arXiv2018

Dominance analysis of linear complementarity systems

The paper extends the concepts of dominance and p-dissipativity to the non-smooth family of linear complementarity systems. Dominance generalizes incremental stability whereas p-dissipativity generalizes incremental passivity. The generalization aims at an interconnection theory for the design and analysis of switching and oscillatory systems. The approach is illustrated by a detailed study of classical electrical circuits that switch and oscillate.

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arXiv2018

Differentially passive circuits that switch and oscillate

The concept of passivity is central to analyze circuits as interconnections of passive components. We illustrate that when used differentially, the same concept leads to an interconnection theory for electrical circuits that switch and oscillate as interconnections of passive components with operational amplifiers (op-amps). The approach builds on recent results on dominance and p-passivity aimed at generalizing dissipativity theory to the analysis of non-equilibrium nonlinear systems. Our paper shows how those results apply to basic and well-known nonlinear circuit architectures. They illustrate the potential of dissipativity theory to design and analyze switching and oscillating circuits quantitatively, very much like their linear counterparts.

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arXiv2018

Conal Distances Between Rational Spectral Densities

The paper generalizes Thompson and Hilbert metric to the space of spectral densities. The resulting complete metric space has the differentiable structure of a Finsler manifold with explicit geodesics. The resulting distances are filtering invariant, can be computed efficiently, and admit geodesic paths that preserve rationality; these are properties of fundamental importance in many engineering applications.

math.OC↗

arXiv2018

Cellular switches orchestrate rhythmic circuits

Small inhibitory neuronal circuits have long been identified as key neuronal motifs to generate and modulate the coexisting rhythms of various motor functions. Our paper highlights the role of a cellular switching mechanism to orchestrate such circuits. The cellular switch makes the circuits reconfigurable, robust, adaptable, and externally controllable. Without this cellular mechanism, the circuits rhythms entirely rely on specific tunings of the synaptic connectivity, which makes them rigid, fragile, and difficult to control externally. We illustrate those properties on the much studied architecture of a small network controlling both the pyloric and gastric rhythms of crabs. The cellular switch is provided by a slow negative conductance often neglected in mathematical modeling of central pattern generators. We propose that this conductance is simple to model and key to computational studies of rhythmic circuit neuromodulation.

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arXiv2018

Robust and tunable bursting requires slow positive feedback

We highlight that the robustness and tunability of a bursting model critically relies on currents that provide slow positive feedback to the membrane potential. Such currents have the ability of making the total conductance of the circuit negative in a time scale that is termed slow because intermediate between the fast time scale of the spike upstroke and the ultraslow time scale of even slower adaptation currents. We discuss how such currents can be assessed either in voltage-clamp experiments or in computational models. We show that, while frequent in the literature, mathematical and computational models of bursting that lack the slow negative conductance are fragile and rigid. Our results suggest that modeling the slow negative conductance of cellular models is important when studying the neuromodulation of rhythmic circuits at any broader scale.

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arXiv2017

Robust modulation of integrate-and-fire models

By controlling the state of neuronal populations, neuromodulators ultimately affect behaviour. A key neuromodulation mechanism is the alteration of neuronal excitability via the modulation of ion channel expression. This type of neuromodulation is normally studied via conductance-based models, but those models are computationally challenging for large-scale network simulations needed in population studies. This paper studies the modulation properties of the Multi-Quadratic Integrate-and-Fire (MQIF) model, a generalisation of the classical Quadratic Integrate-and-Fire (QIF) model. The model is shown to combine the computational economy of integrate-and-fire modelling and the physiological interpretability of conductance-based modelling. It is therefore a good candidate for affordable computational studies of neuromodulation in large networks.

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arXiv2017

A dissipativity theorem for p-dominant systems

We revisit the classical dissipativity theorem of linear-quadratic theory in a generalized framework where the quadratic storage is negative definite in a p-dimensional subspace and positive definite in a complementary subspace. The classical theory assumes p = 0 and provides an inter- connection theory for stability analysis, i.e. convergence to a zero dimensional attractor. The generalized theory is shown to provide an interconnection theory for p-dominance analysis, i.e. convergence to a p-dimensional dominant subspace. In turn, this property is the differential characterization of a generalized contraction property for nonlinear systems. The proposed generalization opens a novel avenue for the analysis of interconnected systems with low-dimensional attractors.

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arXiv2017

Excitable behaviors

This chapter revisits the concept of excitability, a basic system property of neurons. The focus is on excitable systems regarded as behaviors rather than dynamical systems. By this we mean open systems modulated by specific interconnection properties rather than closed systems classified by their parameter ranges. Modeling, analysis, and synthesis questions can be formulated in the classical language of circuit theory. The input-output characterization of excitability is in terms of the local sensitivity of the current-voltage relationship. It suggests the formulation of novel questions for non-linear system theory, inspired by questions from experimental neurophysiology.

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arXiv2017

LTI Stochastic Processes: a Behavioral Perspective

This paper revisits the definition of linear time-invariant (LTI) stochastic process within a behavioral systems framework. Building on [Willems, 2013], we derive a canonical representation of an LTI stochastic process and a physically grounded notion of interconnection between independent stochastic processes. We use this framework to analyze the invariance properties enjoyed by distances between spectral densities of LTI processes.

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arXiv2016

Path-complete positivity of switching systems

The notion of path-complete positivity is introduced as a way to generalize the property of positivity from one LTI system to a family of switched LTI systems whose switching rule is constrained by a finite automaton. The generalization builds upon the analogy between stability and positivity, the former referring to the contraction of a norm, the latter referring to the contraction of a cone (or, equivalently, a projective norm). We motivate and investigate the potential of path-positivity and we propose an algorithm for the automatic verification of positivity.

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arXiv2016

Scaled stochastic gradient descent for low-rank matrix completion

The paper looks at a scaled variant of the stochastic gradient descent algorithm for the matrix completion problem. Specifically, we propose a novel matrix-scaling of the partial derivatives that acts as an efficient preconditioning for the standard stochastic gradient descent algorithm. This proposed matrix-scaling provides a trade-off between local and global second order information. It also resolves the issue of scale invariance that exists in matrix factorization models. The overall computational complexity is linear with the number of known entries, thereby extending to a large-scale setup. Numerical comparisons show that the proposed algorithm competes favorably with state-of-the-art algorithms on various different benchmarks.

cs.LG↗

arXiv2016

A three-scale model of spatio-temporal bursting

We study spatio-temporal bursting in a three-scale reaction diffusion equation organized by the winged cusp singularity. For large time-scale separation the model exhibits traveling bursts, whereas for large space-scale separation the model exhibits standing bursts. Both behaviors exhibit a common singular skeleton, whose geometry is fully determined by persistent bifurcation diagrams of the winged cusp. The modulation of spatio-temporal bursting in such a model naturally translates into paths in the universal unfolding of the winged-cusp.

math.DS↗

arXiv2016

A spectral characterization of nonlinear normal modes

This paper explores the relationship that exists between nonlinear normal modes (NNMs) defined as invariant manifolds in phase space and the spectral expansion of the Koopman operator. Specifically, we demonstrate that NNMs correspond to zero level sets of specific eigenfunctions of the Koopman operator. Thanks to this direct connection, a new, global parametrization of the invariant manifolds is established. Unlike the classical parametrization using a pair of state-space variables, this parametrization remains valid whenever the invariant manifold undergoes folding, which extends the computation of NNMs to regimes of greater energy. The proposed ideas are illustrated using a two-degree-of-freedom system with cubic nonlinearity.

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arXiv2016

Riemannian preconditioning

This paper exploits a basic connection between sequential quadratic programming and Riemannian gradient optimization to address the general question of selecting a metric in Riemannian optimization, in particular when the Riemannian structure is sought on a quotient manifold. The proposed method is shown to be particularly insightful and efficient in quadratic optimization with orthogonality and/or rank constraints, which covers most current applications of Riemannian optimization in matrix manifolds.

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arXiv2015

Differential positivity characterizes one-dimensional normally hyperbolic attractors

The paper shows that normally hyperbolic one-dimensional compact attractors of smooth dynamical systems are characterized by differential positivity, that is, the pointwise infinitesimal contraction of a smooth cone field. The result is analog to the characterization of zero-dimensional hyperbolic attractors by differential stability, which is the pointwise infinitesimal contraction of a Riemannian metric.

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arXiv2015

On the Projective Geometry of Kalman Filter

Convergence of the Kalman filter is best analyzed by studying the contraction of the Riccati map in the space of positive definite (covariance) matrices. In this paper, we explore how this contraction property relates to a more fundamental non-expansiveness property of filtering maps in the space of probability distributions endowed with the Hilbert metric. This is viewed as a preliminary step towards improving the convergence analysis of filtering algorithms over general graphical models.

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arXiv2015

Factor Analysis of Moving Average Processes

The paper considers an extension of factor analysis to moving average processes. The problem is formulated as a rank minimization of a suitable spectral density. It is shown that it can be adequately approximated via a trace norm convex relaxation.

math.OC↗

arXiv2015

Sparse plus low-rank autoregressive identification in neuroimaging time series

This paper considers the problem of identifying multivariate autoregressive (AR) sparse plus low-rank graphical models. Based on the corresponding problem formulation recently presented, we use the alternating direction method of multipliers (ADMM) to efficiently solve it and scale it to sizes encountered in neuroimaging applications. We apply this decomposition on synthetic and real neuroimaging datasets with a specific focus on the information encoded in the low-rank structure of our model. In particular, we illustrate that this information captures the spatio-temporal structure of the original data, generalizing classical component analysis approaches.

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