Construction of direction selectivity in V1: from simple to complex cells

Despite detailed knowledge about the anatomy and physiology of the primary visual cortex (V1), the immense number of feed-forward and recurrent connections onto a given V1 neuron make it difficult to understand how the physiological details relate to a given neuron’s functional properties. Here, we focus on a well-known functional property of many V1 complex cells: phase-invariant direction selectivity (DS). While the energy model explains its construction at the conceptual level, it remains unclear how the mathematical operations described in this model are implemented by cortical circuits. To understand how DS of complex cells is constructed in cortex, we apply a nonlinear modeling framework to extracellular data from macaque V1. We use a modification of spike-triggered covariance (STC) analysis to identify multiple biologically plausible "spatiotemporal features" that either excite or suppress a cell. We demonstrate that these features represent the true inputs to the neuron more accurately, and the resulting nonlinear model compactly describes how these inputs are combined to result in the functional properties of the cell. In a population of 59 neurons, we find that both simple and complex V1 cells are selective to combinations of excitatory and suppressive motion features. Because the strength of DS and simple/complex classification is well predicted by our models, we can use simulations with inputs matching thalamic and simple cells to assess how individual model components contribute to these measures. Our results unify experimental observations regarding the construction of DS from thalamic feed-forward inputs to V1: based on the differences between excitatory and inhibitory inputs, they suggest a connectivity diagram for simple and complex cells that sheds light on the mechanism underlying the DS of cortical cells. More generally, they illustrate how stage-wise nonlinear combination of multiple features gives rise to the processing of more abstract visual information

Generalized gradient flow structure of internal energy driven phase field systems

In this paper we introduce a general abstract formulation of a variational thermomechanical model, by means of a unified derivation via a generalization of the principle of virtual powers for all the variables of the system, including the thermal one. In particular, choosing as thermal variable the entropy of the system, and as driving functional the internal energy, we get a gradient flow structure (in a suitable abstract setting) for the whole nonlinear PDE system. We prove a global in time existence of (weak) solutions result for the Cauchy problem associated to the abstract PDE system as well as uniqueness in case of suitable smoothness assumptions on the functionals

Microformal geometry and homotopy algebras

We extend the category of (super)manifolds and their smooth mappings by introducing a notion of microformal or "thick" morphisms. They are formal canonical relations of a special form, constructed with the help of formal power expansions in cotangent directions. The result is a formal category so that its composition law is also specified by a formal power series. A microformal morphism acts on functions by an operation of pullback, which is in general a nonlinear transformation. More precisely, it is a formal mapping of formal manifolds of even functions (bosonic fields), which has the property that its derivative for every function is a ring homomorphism. This suggests an abstract notion of a "nonlinear algebra homomorphism" and the corresponding extension of the classical "algebraic-functional" duality. There is a parallel fermionic version. The obtained formalism provides a general construction of $L_{\infty}$-morphisms for functions on homotopy Poisson ($P_{\infty}$-) or homotopy Schouten ($S_{\infty}$-) manifolds as pullbacks by Poisson microformal morphisms. We also show that the notion of the adjoint can be generalized to nonlinear operators as a microformal morphism. By applying this to $L_{\infty}$-algebroids, we show that an $L_{\infty}$-morphism of $L_{\infty}$-algebroids induces an $L_{\infty}$-morphism of the "homotopy Lie--Poisson" brackets for functions on the dual vector bundles. We apply this construction to higher Koszul brackets on differential forms and to triangular $L_{\infty}$-bialgebroids. We also develop a quantum version (for the bosonic case), whose relation with the classical version is like that of the Schr\"odinger equation with the Hamilton--Jacobi equation. We show that the nonlinear pullbacks by microformal morphisms are the limits at $\hbar\to 0$ of certain "quantum pullbacks", which are defined as special form Fourier integral operators.Comment: LaTeX 2e. 47 p. Some editing of the expositio

A new approach to quantitative propagation of chaos for drift, diffusion and jump processes

This paper is devoted the the study of the mean field limit for many-particle systems undergoing jump, drift or diffusion processes, as well as combinations of them. The main results are quantitative estimates on the decay of fluctuations around the deterministic limit and of correlations between particles, as the number of particles goes to infinity. To this end we introduce a general functional framework which reduces this question to the one of proving a purely functional estimate on some abstract generator operators (consistency estimate) together with fine stability estimates on the flow of the limiting nonlinear equation (stability estimates). Then we apply this method to a Boltzmann collision jump process (for Maxwell molecules), to a McKean-Vlasov drift-diffusion process and to an inelastic Boltzmann collision jump process with (stochastic) thermal bath. To our knowledge, our approach yields the first such quantitative results for a combination of jump and diffusion processes.Comment: v2 (55 pages): many improvements on the presentation, v3: correction of a few typos, to appear In Probability Theory and Related Field