Giovanni Naldi

Axonal Na+ Channels Ensure Fast Spike Activation and Back-Propagation in Cerebellar Granule Cells

In most neurons, Na+ channels in the axon are complemented by others localized in the soma and dendrites to ensure spike back-propagation. However, cerebellar granule cells are neurons with simplified architecture in which the dendrites are short and unbranched and a single thin ascending axon travels toward the molecular layer before bifurcating into parallel fibers. Here we show that in cerebellar granule cells, Na+ channels are enriched in the axon, especially in the hillock, but almost absent from soma and dendrites. The impact of this channel distribution on neuronal electroresponsiveness was investigated by multi-compartmental modeling. Numerical simulations indicated that granule cells have a compact electrotonic structure allowing excitatory postsynaptic potentials to diffuse with little attenuation from dendrites to axon. The spike arose almost simultaneously along the whole axonal ascending branch and invaded the hillock the activation of which promoted spike back-propagation with marginal delay (<200 micros) and attenuation (<20 mV) into the somato-dendritic compartment. These properties allow granule cells to perform sub-millisecond coincidence detection of pre- and postsynaptic activity and to rapidly activate Purkinje cells contacted by the axonal ascending branch.

Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences

Economic modelling and financial markets.- Agent-based models of economic interactions.- On kinetic asset exchange models and beyond: microeconomic formulation,trade network, and all that.- Microscopic and kinetic models in financial markets.- A mathematical theory for wealth distribution.- Tolstoys dream and the quest for statistical equilibrium in economics and the social sciences.- Social modelling and opinion formation.- New perspectives in the equilibrium statistical mechanics approach to social and economic sciences.- Kinetic modelling of complex socio-economic systems.- Mathematics and physics applications in sociodynamics simulation: the case of opinion formation and diffusion.- Global dynamics in adaptive models of collective choice with social influence.- Modelling opinion formation by means of kinetic equations.- Human behavior and swarming.- On the modelling of vehicular traffic and crowds by kinetic theory of active particles.- Particle, kinetic, and hydrodynamic models of swarming.- Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints.- Statistical physics and modern human warfare.- Diffusive and nondiffusive population models.

Numerical Schemes for Hyperbolic Systems of Conservation Laws with Stiff Diffusive Relaxation

Hyperbolic systems of conservation laws often have diffusive relaxation terms that lead to a small relaxation limit governed by reduced systems of a parabolic or hyperbolic type. In such systems the understanding of basic wave pattern is difficult to achieve, and standard high resolution methods fail to describe the right asymptotic behavior unless the small relaxation rate is numerically resolved. We develop high resolution underresolved numerical schemes that possess the discrete analogue of the continuous asymptotic limit, which are thus able to approximate the equilibrium system with high order accuracy even if the limiting equations may change type.

Local Field Potential Modeling Predicts Dense Activation in Cerebellar Granule Cells Clusters under LTP and LTD Control

Local field-potentials (LFPs) are generated by neuronal ensembles and contain information about the activity of single neurons. Here, the LFPs of the cerebellar granular layer and their changes during long-term synaptic plasticity (LTP and LTD) were recorded in response to punctate facial stimulation in the rat in vivo. The LFP comprised a trigeminal (T) and a cortical (C) wave. T and C, which derived from independent granule cell clusters, co-varied during LTP and LTD. To extract information about the underlying cellular activities, the LFP was reconstructed using a repetitive convolution (ReConv) of the extracellular potential generated by a detailed multicompartmental model of the granule cell. The mossy fiber input patterns were determined using a Blind Source Separation (BSS) algorithm. The major component of the LFP was generated by the granule cell spike Na+ current, which caused a powerful sink in the axon initial segment with the source located in the soma and dendrites. Reproducing the LFP changes observed during LTP and LTD required modifications in both release probability and intrinsic excitability at the mossy fiber-granule cells relay. Synaptic plasticity and Golgi cell feed-forward inhibition proved critical for controlling the percentage of active granule cells, which was 11% in standard conditions but ranged from 3% during LTD to 21% during LTP and raised over 50% when inhibition was reduced. The emerging picture is that of independent (but neighboring) trigeminal and cortical channels, in which synaptic plasticity and feed-forward inhibition effectively regulate the number of discharging granule cells and emitted spikes generating “dense” activity clusters in the cerebellar granular layer.

Numerical schemes for kinetic equations in diffusive regimes

Abstract The diffusive scaling of many finite-velocity kinetic models leads to a small-relaxation time behavior governed by reduced systems which are parabolic in nature. Here we demonstrate that standard numerical methods for hyperbolic conservation laws with stiff relaxation fail to capture the right asymptotic behavior. We show how to design numerical schemes for the study of the diffusive limit that possess the discrete analogue of the continuous asymptotic limit. Numerical results for a model of relaxing heat flow and for a model of nonlinear diffusion are presented.

Wavelet Methods for the Numerical Solution of Boundary Value Problems on the Interval

Abstract We will describe some results on the numerical solution of differential equations on the interval, in particular with regard to the treatment of boundary conditions. We will concentrate on the elliptic case where we will test some methods specially suited to treat Dirichlets boundary conditions. Among such methods we will describe a Galerkin method based on the wavelets on the interval of [9] and a wavelet collocation method. For both methods we give some error estimates, along with some numerical results.

First‐Order Continuous Models of Opinion Formation

We study certain nonlinear continuous models of opinion formation derived from a kinetic description involving exchanges of opinion between individual agents. These models imply that the only possible final opinions are the extremal ones, and they are similar to models of pure drift in magnetization. Both analytical and numerical methods allow us to recover the final distribution of opinion between the two extremal ones.

A bistable model of cell polarity

Ultrasensitivity, as described by Goldbeter and Koshland, has been considered for a long time as a way to realize bistable switches in biological systems. It is not as well recognized that when ultrasensitivity and reinforcing feedback loops are present in a spatially distributed system such as the cell plasmamembrane, they may induce bistability and spatial separation of the system into distinct signaling phases. Here we suggest that bistability of ultrasensitive signaling pathways in a diffusive environment provides a basic mechanism to realize cell membrane polarity. Cell membrane polarization is a fundamental process implicated in several basic biological phenomena, such as differentiation, proliferation, migration and morphogenesis of unicellular and multicellular organisms. We describe a simple, solvable model of cell membrane polarization based on the coupling of membrane diffusion with bistable enzymatic dynamics. The model can reproduce a broad range of symmetry-breaking events, such as those observed in eukaryotic directional sensing, the apico-basal polarization of epithelium cells, the polarization of budding and mating yeast, and the formation of Ras nanoclusters in several cell types.

High-Order Relaxation Schemes for Nonlinear Degenerate Diffusion Problems

Several relaxation approximations to partial differential equations have been recently proposed. Examples include conservation laws, Hamilton-Jacobi equations, convection-diffusion problems, and gas dynamics problems. The present paper focuses on diffusive relaxation schemes for the numerical approximation of nonlinear parabolic equations. These schemes are based on a suitable semilinear hyperbolic system with relaxation terms. High-order methods are obtained by coupling ENO and weighted essentially nonoscillatory (WENO) schemes for space discretization with implicit-explicit (IMEX) schemes for time integration. Error estimates and a convergence analysis are developed for semidiscrete schemes with a numerical analysis for fully discrete relaxed schemes. Various numerical results in one and two dimensions illustrate the high accuracy and good properties of the proposed numerical schemes, also in the degenerate case. These schemes can be easily implemented on parallel computers and applied to more general systems of nonlinear parabolic equations in two- and three-dimensional cases.

A wavelet-based method for numerical solution of nonlinear evolution equations

Abstract We describe an adaptive algorithm for solving one-dimensional system of nonlinear partial differential equations. Different strategies are considered for the discretization in time while a multiscale collocation method is applied for the discretization in space. In particular we look at the so called Rothe method which is based on first time then space discretization. Numerical experiments are presented for a set of nonlinear problems from the literature.