A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics

To describe the collective behavior of large ensembles of neurons in neuronal network, a kinetic theory description was developed in [13, 12], where a macroscopic representation of the network dynamics was directly derived from the microscopic dynamics of individual neurons, which are modeled by conductance-based, linear, integrate-and-fire point neurons. A diffusion approximation then led to a nonlinear Fokker-Planck equation for the probability density function of neuronal membrane potentials and synaptic conductances. In this work, we propose a deterministic numerical scheme for a Fokker-Planck model of an excitatory-only network. Our numerical solver allows us to obtain the time evolution of probability distribution functions, and thus, the evolution of all possible macroscopic quantities that are given by suitable moments of the probability density function. We show that this deterministic scheme is capable of capturing the bistability of stationary states observed in Monte Carlo simulations. Moreover, the transient behavior of the firing rates computed from the Fokker-Planck equation is analyzed in this bistable situation, where a bifurcation scenario, of asynchronous convergence towards stationary states, periodic synchronous solutions or damped oscillatory convergence towards stationary states, can be uncovered by increasing the strength of the excitatory coupling. Finally, the computation of moments of the probability distribution allows us to validate the applicability of a moment closure assumption used in [13] to further simplify the kinetic theory.

Identification of neuronal network properties from the spectral analysis of calcium imaging signals in neuronal cultures

Neuronal networks in vitro are prominent systems to study the development of connections in living neuronal networks and the interplay between connectivity, activity and function. These cultured networks show a rich spontaneous activity that evolves concurrently with the connectivity of the underlying network. In this work we monitor the development of neuronal cultures, and record their activity using calcium fluorescence imaging. We use spectral analysis to characterize global dynamical and structural traits of the neuronal cultures. We first observe that the power spectrum can be used as a signature of the state of the network, for instance when inhibition is active or silent, as well as a measure of the network's connectivity strength. Second, the power spectrum identifies prominent developmental changes in the network such as GABAA switch. And third, the analysis of the spatial distribution of the spectral density, in experiments with a controlled disintegration of the network through CNQX, an AMPA-glutamate receptor antagonist in excitatory neurons, reveals the existence of communities of strongly connected, highly active neurons that display synchronous oscillations. Our work illustrates the interest of spectral analysis for the study of in vitro networks, and its potential use as a network-state indicator, for instance to compare healthy and diseased neuronal networks.

Identification of neuronal network properties from the spectral analysis of calcium imaging signals in neuronal cultures

Neuronal networks in vitro are prominent systems to study the development of connections in living neuronal networks and the interplay between connectivity, activity and function. These cultured networks show a rich spontaneous activity that evolves concurrently with the connectivity of the underlying network. In this work we monitor the development of neuronal cultures, and record their activity using calcium fluorescence imaging. We use spectral analysis to characterize global dynamical and structural traits of the neuronal cultures. We first observe that the power spectrum can be used as a signature of the state of the network, for instance when inhibition is active or silent, as well as a measure of the network's connectivity strength. Second, the power spectrum identifies prominent developmental changes in the network such as GABAA switch. And third, the analysis of the spatial distribution of the spectral density, in experiments with a controlled disintegration of the network through CNQX, an AMPA-glutamate receptor antagonist in excitatory neurons, reveals the existence of communities of strongly connected, highly active neurons that display synchronous oscillations. Our work illustrates the interest of spectral analysis for the study of in vitro networks, and its potential use as a network-state indicator, for instance to compare healthy and diseased neuronal networks.

Urban population density functions : the case of the Barcelona region

L'anàlisi de la densitat urbana és utilitzada per examinar la distribució espacial de la població dins de les àrees urbanes, i és força útil per planificar els serveis públics. En aquest article, s'estudien setze formes funcionals clàssiques de la relació existent entre la densitat i la distancia en la regió metropolitana de Barcelona i els seus onze subcentres.

Cubic spline population density functions and subcentre delimitation. The case of Barcelona

The presence of subcentres cannot be captured by an exponential function. Cubic spline functions seem more appropriate to depict the polycentricity pattern of modern urban systems. Using data from Barcelona Metropolitan Region, two possible population subcentre delimitation procedures are discussed. One, taking an estimated derivative equal to zero, the other, a density gradient equal to zero. It is argued that, in using a cubic spline function, a delimitation strategy based on derivatives is more appropriate than one based on gradients because the estimated density can be negative in sections with very low densities and few observations, leading to sudden changes in estimated gradients. It is also argued that using as a criteria for subcentre delimitation a second derivative with value zero allow us to capture a more restricted subcentre area than using as a criteria a first derivative zero. This methodology can also be used for intermediate ring delimitation.

The density of injective endomorphisms of a free group

"Vegeu el resum a l'inici del document del fitxer adjunt."

On the density distribution across space: a probabilistic approach

This paper aims at providing a Bayesian parametric framework to tackle the accessibility problem across space in urban theory. Adopting continuous variables in a probabilistic setting we are able to associate with the distribution density to the Kendall's tau index and replicate the general issues related to the role of proximity in a more general context. In addition, by referring to the Beta and Gamma distribution, we are able to introduce a differentiation feature in each spatial unit without incurring in any a-priori definition of territorial units. We are also providing an empirical application of our theoretical setting to study the density distribution of the population across Massachusetts.

The role of fixed delays in neuronal rate models

Fixed delays in neuronal interactions arise through synaptic and dendritic processing. Previous work has shown that such delays, which play an important role in shaping the dynamics of networks of large numbers of spiking neurons with continuous synaptic kinetics, can be taken into account with a rate model through the addition of an explicit, fixed delay. Here we extend this work to account for arbitrary symmetric patterns of synaptic connectivity and generic nonlinear transfer functions. Specifically, we conduct a weakly nonlinear analysis of the dynamical states arising via primary instabilities of the stationary uniform state. In this way we determine analytically how the nature and stability of these states depend on the choice of transfer function and connectivity. While this dependence is, in general, nontrivial, we make use of the smallness of the ratio in the delay in neuronal interactions to the effective time constant of integration to arrive at two general observations of physiological relevance. These are: 1 - fast oscillations are always supercritical for realistic transfer functions. 2 - Traveling waves are preferred over standing waves given plausible patterns of local connectivity.

Sobolev mappings: Lipschitz density is not an isometric invariant of the target

"Vegeu el resum a l'inici del document del fitxer adjunt."

Gaussian estimates for the density of the non-linear stochastic heat equation in any space dimension

In this paper, we establish lower and upper Gaussian bounds for the probability density of the mild solution to the stochastic heat equation with multiplicative noise and in any space dimension. The driving perturbation is a Gaussian noise which is white in time with some spatially homogeneous covariance. These estimates are obtained using tools of the Malliavin calculus. The most challenging part is the lower bound, which is obtained by adapting a general method developed by Kohatsu-Higa to the underlying spatially homogeneous Gaussian setting. Both lower and upper estimates have the same form: a Gaussian density with a variance which is equal to that of the mild solution of the corresponding linear equation with additive noise.

High tidal volume ventilation affects neuronal activation in mechanically ventilated rats

Els malalts crítics presenten sovint seqüeles cognitives a llarg termini, l’aplicació de ventilació mecànica (VM) pot contribuir al seu desenvolupament. El principal objectiu del nostre estudi fou investigar l’efecte de dos patrons de ventilació (volum corrent elevat/baix) en l’activació neuronal (expressió de c-fos) en determinades àrees cerebrals en un model en rates. Després de 3 hores sota VM, es va trobar activació neuronal; la seva intensitat va ser superior al grup de volum corrent elevat, suggerint un efecte iatrogènic de la VM al cervell. Aquests resultats suggereixen que cal aprofundir en l’estudi del crosstalk cervell-pulmó en malalts crítics sota VM.

Cortical mapping of the neuronal circuits modulating the muscle tone. Introduction to the electrophysiological treatment of the spastic hand.

L’objectiu d’aquest estudi es investigar l’organització cortical junt amb la connectivitat còrtico-subcortical en subjectes sans, com a estudi preliminar. Els mapes corticals s’han fet per TMS navegada, i els punts motors obtinguts s’han exportant per estudi tractogràfic i anàlisi de las seves connexions. El coneixement precís de la localització de l’àrea cortical motora primària i les seves connexions es la base per ser utilitzada en estudis posteriors de la reorganització cortical i sub-cortical en pacients amb infart cerebral. Aquesta reorganització es deguda a la neuroplasticitat i pot ser influenciada per els efectes neuromoduladors de la estimulació cerebral no invasiva.

Comparing methods for dimensionality reduction when data are density functions

Functional Data Analysis (FDA) deals with samples where a whole function is observedfor each individual. A particular case of FDA is when the observed functions are densityfunctions, that are also an example of infinite dimensional compositional data. In thiswork we compare several methods for dimensionality reduction for this particular typeof data: functional principal components analysis (PCA) with or without a previousdata transformation and multidimensional scaling (MDS) for diferent inter-densitiesdistances, one of them taking into account the compositional nature of density functions. The difeerent methods are applied to both artificial and real data (householdsincome distributions)

Hilbert space on probability density functions with Aitchison geometry

Compositional data analysis motivated the introduction of a complete Euclidean structure in the simplex of D parts. This was based on the early work of J. Aitchison (1986) and completed recently when Aitchinson distance in the simplex was associated with an inner product and orthonormal bases were identified (Aitchison and others, 2002; Egozcue and others, 2003). A partition of the support of a random variable generates a composition by assigning the probability of each interval to a part of the composition. One can imagine that the partition can be refined and the probability density would represent a kind of continuous composition of probabilities in a simplex of infinitely many parts. This intuitive idea would lead to a Hilbert-space of probability densitiesby generalizing the Aitchison geometry for compositions in the simplex into the set probability densities

Automatic classification of breast density

A recent trend in digital mammography is computer-aided diagnosis systems, which are computerised tools designed to assist radiologists. Most of these systems are used for the automatic detection of abnormalities. However, recent studies have shown that their sensitivity is significantly decreased as the density of the breast increases. This dependence is method specific. In this paper we propose a new approach to the classification of mammographic images according to their breast parenchymal density. Our classification uses information extracted from segmentation results and is based on the underlying breast tissue texture. Classification performance was based on a large set of digitised mammograms. Evaluation involves different classifiers and uses a leave-one-out methodology. Results demonstrate the feasibility of estimating breast density using image processing and analysis techniques