A trace theorem in kinetic theory

AbstractA mathematical assumption made in a recent paper by Cercignani on the traces of the solutions of initial-boundary value problems for the Boltzmann Equation is shown here to be true in the case of a domain whose boundary is a Lyapunoff surface with purely diffusing boundary conditions

Asymmetry of parietal interhemispheric connections in humans

Visuospatial abilities are preferentially mediated by the right hemisphere. Although this asymmetry of function is thought to be due to an unbalanced interaction between cerebral hemispheres, the underlying neurophysiological substrate is still largely unknown. Here, using a method of trifocal transcranial magnetic stimulation, we show that the right, but not left, human posterior parietal cortex exerts a strong inhibitory activity over the contralateral homologous area by a short-latency connection. We also clarify, using diffusion-tensor magnetic resonance imaging, that such an interaction is mediated by direct transcallosal projections located in the posterior corpus callosum. We argue that this anatomo-functional network may represent a possible neurophysiological basis for the ongoing functional asymmetry between parietal cortices, and that its damage could contribute to the clinical manifestations of neglect

Strategic lesions in the anterior thalamic radiation and apathy in early Alzheimer's disease

BACKGROUND Behavioural disorders and psychological symptoms of Dementia (BPSD) are commonly observed in Alzheimer's disease (AD), and strongly contribute to increasing patients' disability. Using voxel-lesion-symptom mapping (VLSM), we investigated the impact of white matter lesions (WMLs) on the severity of BPSD in patients with amnestic mild cognitive impairment (a-MCI). METHODS Thirty-one a-MCI patients (with a conversion rate to AD of 32% at 2 year follow-up) and 26 healthy controls underwent magnetic resonance imaging (MRI) examination at 3T, including T2-weighted and fluid-attenuated-inversion-recovery images, and T1-weighted volumes. In the patient group, BPSD was assessed using the Neuropsychiatric Inventory-12. After quantitative definition of WMLs, their distribution was investigated, without an a priori anatomical hypothesis, against patients' behavioural symptoms. Unbiased regional grey matter volumetrics was also used to assess the contribution of grey matter atrophy to BPSD. RESULTS Apathy, irritability, depression/dysphoria, anxiety and agitation were shown to be the most common symptoms in the patient sample. Despite a more widespread anatomical distribution, a-MCI patients did not differ from controls in WML volumes. VLSM revealed a strict association between the presence of lesions in the anterior thalamic radiations (ATRs) and the severity of apathy. Regional grey matter atrophy did not account for any BPSD. CONCLUSIONS This study indicates that damage to the ATRs is strategic for the occurrence of apathy in patients with a-MCI. Disconnection between the prefrontal cortex and the mediodorsal and anterior thalamic nuclei might represent the pathophysiological substrate for apathy, which is one of the most common psychopathological symptoms observed in dementia

Applicability of the High Field Model: A Preliminary Numerical Study

In a companion presentation, we have discussed the theory of a mesoscopic/ macroscopic model, which can be viewed as an augmented drift-diffusion model. Here, we describe how that model is used. The device we consider for this presentation is the one dimensional GaAs n+−n−n+ structure of length 0.8μm. First, a full Hydro- Dynamic (HD) model, proven reliable when compared with Monte Carlo simulations, is used to simulate the device via the ENO finite difference method. As applied to the full device, the new model is not necessarily superior to traditional Drift-Diffusion (DD). Indeed, when we plot the quantity η= μ0E/kT0/m, where μ0 is the mobility constant and E=−ϕ′ is the electric field, we verify that the high field assumption η › 1, required for the high field model, is satisfied only in an interval given approximately by [0.2, 0.5]. When we run both the DD model and the new high field model in this restricted interval, with boundary conditions of concentration n and potential ϕ provided by the HD results, we demonstrate that the new model outperforms the DD model. This indicates that the high field and DD models should be used only in parts of the device, connected by a transition kinetic regime. This will be a domain decomposition issue involving interface conditions and adequate numerical methods

On the master equation approach to kinetic theory: linear and nonlinear Fokker--Planck equations

We discuss the relationship between kinetic equations of the Fokker-Planck type (two linear and one non-linear) and the Kolmogorov (a.k.a. master) equations of certain N-body diffusion processes, in the context of Kac's "propagation of chaos" limit. The linear Fokker-Planck equations are well-known, but here they are derived as a limit N->infty of a simple linear diffusion equation on (3N-C)-dimensional N-velocity spheres of radius sqrt(N) (with C=1 or 4 depending on whether the system conserves energy only or energy and momentum). In this case, a spectral gap separating the zero eigenvalue from the positive spectrum of the Laplacian remains as N->infty,so that the exponential approach to equilibrium of the master evolution is passed on to the limiting Fokker-Planck evolution in R^3. The non-linear Fokker-Planck equation is known as Landau's equation in the plasma physics literature. Its N-particle master equation, originally introduced (in the 1950s) by Balescu and Prigogine (BP), is studied here on the (3N-4)-dimensional N-velocity sphere. It is shown that the BP master equation represents a superposition of diffusion processes on certain two-dimensional sub-manifolds of R^{3N} determined by the conservation laws for two-particle collisions. The initial value problem for the BP master equation is proved to be well-posed and its solutions are shown to decay exponentially fast to equilibrium. However, the first non-zero eigenvalue of the BP operator is shown to vanish in the limit N->infty. This indicates that the exponentially fast approach to equilibrium may not be passed from the finite-N master equation on to Landau's nonlinear kinetic equation.Comment: 20 pages; based on talk at the 18th ICTT Conference. Some typos and a few minor technical fixes. Modified title slightl

Cerebellar white matter disruption in Alzheimer’s Disease patients: a Diffusion Tensor Imaging study

The cognitive role of the cerebellum has recently gained much attention, and its pivotal role in Alzheimer’s disease (AD) has now been widely recognized. Diffusion tensor imaging (DTI) has been used to evaluate the disruption of the microstructural milieu in AD, and though several white matter (WM) tracts such as corpus callosum, inferior and superior longitudinal fasciculus, cingulum, fornix, and uncinate fasciculus have been evaluated in AD, data on cerebellar WM tracts are currently lacking. We performed a tractography-based DTI reconstruction of the middle cerebellar peduncle (MCP), and the left and right superior cerebellar peduncles separately (SCPL and SCPR) and addressed the differences in fractional anisotropy (FA), axial diffusivity (Dax), radial diffusivity (RD), and mean diffusivity (MD) in the three tracts between 50 patients with AD and 25 healthy subjects. We found that AD patients showed a lower FA and a higher RD compared to healthy subjects in MCP, SCPL, and SCPR. Moreover, a higher MD was found in SCPR and SCPL and a higher Dax in SCPL. This result is important as it challenges the traditional view that WM bundles in the cerebellum are unaffected in AD and might identify new targets for therapeutic interventions

Association between a genetic variant of type-1 cannabinoid receptor and inflammatory neurodegeneration in multiple sclerosis

Genetic ablation of type-1 cannabinoid receptors (CB1Rs) exacerbates the neurodegenerative damage of experimental autoimmune encephalomyelitis, the rodent model of multiple sclerosis (MS). To address the role on CB1Rs in the pathophysiology of human MS, we first investigated the impact of AAT trinucleotide short tandem repeat polymorphism of CNR1 gene on CB1R cell expression, and secondly on the inflammatory neurodegeneration process responsible for irreversible disability in MS patients. We found that MS patients with long AAT repeats within the CNR1 gene (≥12 in both alleles) had more pronounced neuronal degeneration in response to inflammatory white matter damage both in the optic nerve and in the cortex. Optical Coherence Tomography (OCT), in fact, showed more severe alterations of the retinal nerve fiber layer (RNFL) thickness and of the macular volume (MV) after an episode of optic neuritis in MS patients carrying the long AAT genotype of CNR1. MS patients with long AAT repeats also had magnetic resonance imaging (MRI) evidence of increased gray matter damage in response to inflammatory lesions of the white matter, especially in areas with a major role in cognition. In parallel, visual abilities evaluated at the low contrast acuity test, and cognitive performances were negatively influenced by the long AAT CNR1 genotype in our sample of MS patients. Our results demonstrate the biological relevance of the (AAT)n CNR1 repeats in the inflammatory neurodegenerative damage of MS

The linear Fokker-Planck equation for the Ornstein-Uhlenbeck process as an (almost) nonlinear kinetic equation for an isolated N-particle system

It is long known that the Fokker-Planck equation with prescribed constant coefficients of diffusion and linear friction describes the ensemble average of the stochastic evolutions in velocity space of a Brownian test particle immersed in a heat bath of fixed temperature. Apparently, it is not so well known that the same partial differential equation, but now with constant coefficients which are functionals of the solution itself rather than being prescribed, describes the kinetic evolution (in the infinite particle limit) of an isolated N-particle system with certain stochastic interactions. Here we discuss in detail this recently discovered interpretation.Comment: Minor revisions and corrections (including the title

In vivo mapping of brainstem nuclei functional connectivity disruption in Alzheimer's disease

We assessed here functional connectivity changes in the locus coeruleus (LC) and ventral tegmental area (VTA) of patients with Alzheimer's disease (AD). We recruited 169 patients with either AD or amnestic mild cognitive impairment due to AD and 37 elderly controls who underwent cognitive and neuropsychiatric assessments and resting-state functional magnetic resonance imaging at 3T. Connectivity was assessed between LC and VTA and the rest of the brain. In amnestic mild cognitive impairment patients, VTA disconnection was predominant with parietal regions, while in AD patients, it involved the posterior nodes of the default-mode network. We also looked at the association between neuropsychiatric symptoms (assessed by the neuropsychiatric inventory) and VTA connectivity. Symptoms such as agitation, irritability, and disinhibition were associated with VTA connectivity with the parahippocampal gyrus and cerebellar vermis, while sleep and eating disorders were associated with VTA connectivity to the striatum and the insular cortex. This suggests a contribution of VTA degeneration to AD pathophysiology and to the occurrence of neuropsychiatric symptoms. We did not find evidence of LC disconnection, but this could be explained by the size of this nucleus, which makes it difficult to isolate. These results are consistent with animal findings and have potential implications for AD prognosis and therapies