Computational modeling of diffusion in the cerebellum

Toward a cerebello-thalamo-cortical computational model of spinocerebellar ataxia

Computational neural network modelling is an emerging approach for optimization of drug treatment of neurological disorders and fine-tuning of rehabilitation strategies. In the current study, we constructed a cerebello-thalamo-cortical computational neural network model to simulate a mouse model of cerebellar ataxia (pcd^5J mice) by manipulating cerebellar bursts through reduction of GABAergic inhibitory input. Cerebellar output neurons were projected to the thalamus and bidirectionally connected with the cortical network. Our results showed that reduction of inhibitory input in the cerebellum orchestrated the cortical local field potential (LFP) dynamics to generate specific motor outputs of oscillations of the theta, alpha, and beta bands in the computational model as well as in mouse motor cortical neurons. The therapeutic potential of deep brain stimulation (DBS) was tested in the computational model by increasing the sensory input to restore cortical output. Ataxia mice showed normalization of the motor cortex LFP after cerebellum DBS. We provide a novel approach to computational modelling to investigate the effect of DBS by mimicking cerebellar ataxia involving degeneration of Purkinje cells. Simulated neural activity coincides with findings from neural recordings of ataxia mice. Our computational model could thus represent cerebellar pathologies and provide insight into how to improve disease symptoms by restoring neuronal electrophysiological properties using DBS.

Long-term microglial phase-specific dynamics during single vessel occlusion and recanalization

Vascular occlusion leading to brain dysfunctions is usually considered evoking microglia-induced inflammation response. However, it remains unclear how microglia interact with blood vessels in the development of vascular occlusion-related brain disorders. Here, we illuminate long-term spatiotemporal dynamics of microglia during single vessel occlusion and recanalization. Microglia display remarkable response characteristics in different phases, including acute reaction, rapid diffusion, transition and chronic effect. Fibrinogen-induced microglial cluster promotes major histocompatibility complex II (MHCII) expression. Microglial soma represents a unique filament-shape migration and has slower motility compared to the immediate reaction of processes to occlusion. We capture proliferative microglia redistribute territory. Microglial cluster resolves gradually and microglia recover to resting state both in the morphology and function in the chronic effect phase. Therefore, our study offers a comprehensive analysis of spatiotemporal dynamics of microglia and potential mechanisms to both vessel occlusion and recanalization. Microglial phase-specific response suggests the morphological feature-oriented phased intervention would be an attractive option for vascular occlusion-related diseases treatments.

The spatiotemporal dynamics of the microglial inflammatory response to single vessel occlusion and recanalization are analysed, revealing four different response phases.

Interior Regularity Estimates for a Degenerate Elliptic Equation with Mixed Boundary Conditions

The Marchaud fractional derivative can be obtained as a Dirichlet-to–Neumann map via an extension problem to the upper half space. In this paper we prove interior Schauder regularity estimates for a degenerate elliptic equation with mixed Dirichlet–Neumann boundary conditions. The degenerate elliptic equation arises from the Bernardis–Reyes–Stinga–Torrea extension of the Dirichlet problem for the Marchaud fractional derivative.

A Simple Mathematical Model Inspired by the Purkinje Cells: From Delayed Travelling Waves to Fractional Diffusion

Recently, several experiments have demonstrated the existence of fractional diffusion in the neuronal transmission occurring in the Purkinje cells, whose malfunctioning is known to be related to the lack of voluntary coordination and the appearance of tremors. Also, a classical mathematical feature is that (fractional) parabolic equations possess smoothing effects, in contrast with the case of hyperbolic equations, which typically exhibit shocks and discontinuities. In this paper, we show how a simple toy-model of a highly ramified structure, somehow inspired by that of the Purkinje cells, may produce a fractional diffusion via the superposition of travelling waves that solve a hyperbolic equation. This could suggest that the high ramification of the Purkinje cells might have provided an evolutionary advantage of "smoothing" the transmission of signals and avoiding shock propagations (at the price of slowing a bit such transmission). Although an experimental confirmation of the possibility of such evolutionary advantage goes well beyond the goals of this paper, we think that it is intriguing, as a mathematical counterpart, to consider the time fractional diffusion as arising from the superposition of delayed travelling waves in highly ramified transmission media. The case of a travelling concave parabola with sufficiently small curvature is explicitly computed. The new link that we propose between time fractional diffusion and hyperbolic equation also provides a novelty with respect to the usual paradigm relating time fractional diffusion with parabolic equations in the limit. This paper is written in such a way as to be of interest to both biologists and mathematician alike. In order to accomplish this aim, both complete explanations of the objects considered and detailed lists of references are provided.