A Balance Equation Determines a Switch in Neuronal Excitability

in PLoS Computational Biology (in press)

We use the qualitative insight of a planar neuronal phase portrait to detect an excitability switch in arbitrary conductance-based models from a simple mathematical condition. The condition expresses a ... [more ▼]

We use the qualitative insight of a planar neuronal phase portrait to detect an excitability switch in arbitrary conductance-based models from a simple mathematical condition. The condition expresses a balance between ion channels that provide a negative feedback at resting potential (restorative channels) and those that provide a positive feedback at resting potential (regenerative channels). Geometrically, the condition imposes a transcritical bifurcation that rules the switch of excitability through the variation of a single physiological parameter. Our analysis of six di erent published conductance based models always nds the transcritical bifurcation and the associated switch in excitability, which suggests that the mathematical predictions have a physiological rel- evance and that a same regulatory mechanism is potentially involved in the excitability and signaling of many neurons. [less ▲]

Kick synchronization versus diffusive synchronization

in Proceedings of the 51st IEEE Conference on Decision and Control (2012, December)

The paper provides an introductory discussion about two fundamental models of oscillator synchronization: the (continuous-time) diffusive model, that dominates the mathematical literature on ... [more ▼]

The paper provides an introductory discussion about two fundamental models of oscillator synchronization: the (continuous-time) diffusive model, that dominates the mathematical literature on synchronization, and the (hybrid) kick model, that accounts for most popular examples of synchronization, but for which only few theoretical results exist. The paper stresses fundamental differences between the two models, such as the different contraction measures underlying the analysis, as well as important analogies that can be drawn in the limit of weak coupling. [less ▲]

A novel phase portrait for neuronal excitability

in PLoS ONE (2012), 7(8),

Fifty years ago, FitzHugh introduced a phase portrait that became famous for a twofold reason: it captured in a physiological way the qualitative behavior of Hodgkin-Huxley model and it revealed the power ... [more ▼]

Fifty years ago, FitzHugh introduced a phase portrait that became famous for a twofold reason: it captured in a physiological way the qualitative behavior of Hodgkin-Huxley model and it revealed the power of simple dynamical models to unfold complex firing patterns. To date, in spite of the enormous progresses in qualitative and quantitative neural modeling, this phase portrait has remained a core picture of neuronal excitability. Yet, a major difference between the neurophysiology of 1961 and of 2011 is the recognition of the prominent role of calcium channels in firing mechanisms. We show that including this extra current in Hodgkin-Huxley dynamics leads to a revision of FitzHugh-Nagumo phase portrait that affects in a fundamental way the reduced modeling of neural excitability. The revisited model considerably enlarges the modeling power of the original one. In particular, it captures essential electrophysiological signatures that otherwise require non-physiological alteration or considerable complexification of the classical model. As a basic illustration, the new model is shown to highlight a core dynamical mechanism by which calcium channels control the two distinct firing modes of thalamocortical neurons. [less ▲]

Decision-making in noisy bistable models : a local analysis for non local predictions

Poster (2012, May)

Anisotropy preserving interpolation of diffusion tensors

Conference (2012, March)

Metrics for oscillator models: an input-to-phase approach

Conference (2012, March)

Fixed-rank matrix factorizations and Riemannian low-rank optimization

E-print/Working paper (2012)

Motivated by the problem of learning a linear regression model whose parameter is a large fixed-rank non-symmetric matrix, we consider the optimization of a smooth cost function defined on the set of ... [more ▼]

Motivated by the problem of learning a linear regression model whose parameter is a large fixed-rank non-symmetric matrix, we consider the optimization of a smooth cost function defined on the set of fixed-rank matrices. We adopt the geometric optimization framework of optimization on Riemannian matrix manifolds. We study the underlying geometries of several well-known fixed-rank matrix factorizations and then exploit the Riemannian geometry of the search space in the design of a class of gradient descent and trust-region algorithms. The proposed algorithms generalize our previous results on fixed-rank symmetric positive semidefinite matrices, apply to a broad range of applications, scale to high-dimensional problems and confer a geometric basis to recent contributions on the learning of fixed-rank non-symmetric matrices. We make connections with existing algorithms in the context of low-rank matrix completion and discuss relative usefulness of the proposed framework. Numerical experiments suggest that the proposed algorithms compete with the state-of-the-art and that manifold optimization offers an effective and versatile framework for the design of machine learning algorithms that learn a fixed-rank matrix. [less ▲]

Low-rank optimization with trace norm penalty

E-print/Working paper (2012)

The paper addresses the problem of low-rank trace norm minimization. We propose an algorithm that alternates between fixed-rank optimization and rank-one updates. The fixed-rank optimization is ... [more ▼]

The paper addresses the problem of low-rank trace norm minimization. We propose an algorithm that alternates between fixed-rank optimization and rank-one updates. The fixed-rank optimization is characterized by an efficient factorization that makes the trace norm differentiable in the search space and the computation of duality gap numerically tractable. The search space is nonlinear but is equipped with a particular Riemannian structure that leads to efficient computations. We present a second-order trust-region algorithm with a guaranteed quadratic rate of convergence. Overall, the proposed optimization scheme converges super-linearly to the global solution while still maintaining complexity that is linear in the number of rows of the matrix. To compute a set of solutions efficiently for a grid of regularization parameters we propose a predictor-corrector approach on the quotient manifold that outperforms the naive warm-restart approach. The performance of the proposed algorithm is illustrated on problems of low-rank matrix completion and multivariate linear regression. [less ▲]

Low-rank optimization for distance matrix completion

in Proceedings of the 50th IEEE Conference on Decision and Control (2011, December)

This paper addresses the problem of low-rank distance matrix completion. This problem amounts to recover the missing entries of a distance matrix when the dimension of the data embedding space is possibly ... [more ▼]

This paper addresses the problem of low-rank distance matrix completion. This problem amounts to recover the missing entries of a distance matrix when the dimension of the data embedding space is possibly unknown but small compared to the number of considered data points. The focus is on high-dimensional problems. We recast the considered problem into an optimization problem over the set of low-rank positive semidefinite matrices and propose two efficient algorithms for low-rank distance matrix completion. In addition, we propose a strategy to determine the dimension of the embedding space. The resulting algorithms scale to high-dimensional problems and monotonically converge to a global solution of the problem. Finally, numerical experiments illustrate the good performance of the proposed algorithms on benchmarks. [less ▲]

Sensitivity analysis of phase response curves

Conference (2011, May 23)

The Phase Response Curve (PRC) has proven a useful tool for the reduction of complex oscillator models to one-dimensional phase models. We introduce the sensitivity analysis of this important mathematical ... [more ▼]

The Phase Response Curve (PRC) has proven a useful tool for the reduction of complex oscillator models to one-dimensional phase models. We introduce the sensitivity analysis of this important mathematical object and its numerical implementation. As an application, we study simple biochemical models of circadian oscillators and discuss how sensitivity analysis helps drawing connections between the state-space model of the oscillator and its phase response curve. [less ▲]