Maximum principle, mean value operators and quasi boundedness in non-euclidean settings

This work deals with some classes of linear second order partial differential operators with non-negative characteristic form and underlying non- Euclidean structures. These structures are determined by families of locally Lipschitz-continuous vector fields in RN, generating metric spaces of Carnot- Carath´eodory type. The Carnot-Carath´eodory metric related to a family {Xj}j=1,...,m is the control distance obtained by minimizing the time needed to go from two points along piecewise trajectories of vector fields. We are mainly interested in the causes in which a Sobolev-type inequality holds with respect to the X-gradient, and/or the X-control distance is Doubling with respect to the Lebesgue measure in RN. This study is divided into three parts (each corresponding to a chapter), and the subject of each one is a class of operators that includes the class of the subsequent one. In the first chapter, after recalling “X-ellipticity” and related concepts introduced by Kogoj and Lanconelli in [KL00], we show a Maximum Principle for linear second order differential operators for which we only assume a Sobolev-type inequality together with a lower terms summability. Adding some crucial hypotheses on measure and on vector fields (Doubling property and Poincar´e inequality), we will be able to obtain some Liouville-type results. This chapter is based on the paper [GL03] by Guti´errez and Lanconelli. In the second chapter we treat some ultraparabolic equations on Lie groups. In this case RN is the support of a Lie group, and moreover we require that vector fields satisfy left invariance. After recalling some results of Cinti [Cin07] about this class of operators and associated potential theory, we prove a scalar convexity for mean-value operators of L-subharmonic functions, where L is our differential operator. In the third chapter we prove a necessary and sufficient condition of regularity, for boundary points, for Dirichlet problem on an open subset of RN related to sub-Laplacian. On a Carnot group we give the essential background for this type of operator, and introduce the notion of “quasi-boundedness”. Then we show the strict relationship between this notion, the fundamental solution of the given operator, and the regularity of the boundary points.

Methods for the estimation of the cortical activity and connectivity during cognitive tasks in humans

The research activity characterizing the present thesis was mainly centered on the design, development and validation of methodologies for the estimation of stationary and time-varying connectivity between different regions of the human brain during specific complex cognitive tasks. Such activity involved two main aspects: i) the development of a stable, consistent and reproducible procedure for functional connectivity estimation with a high impact on neuroscience field and ii) its application to real data from healthy volunteers eliciting specific cognitive processes (attention and memory). In particular the methodological issues addressed in the present thesis consisted in finding out an approach to be applied in neuroscience field able to: i) include all the cerebral sources in connectivity estimation process; ii) to accurately describe the temporal evolution of connectivity networks; iii) to assess the significance of connectivity patterns; iv) to consistently describe relevant properties of brain networks. The advancement provided in this thesis allowed finding out quantifiable descriptors of cognitive processes during a high resolution EEG experiment involving subjects performing complex cognitive tasks.

Multimodal (EEG-fMRI) functional connectivity study of levodopa effect in Parkinson’s disease

Aim: To assess if the intake of levodopa in patients with Parkinson’s Disease (PD) changes cerebral connectivity, as revealed by simultaneous recording of hemodynamic (functional MRI, or fMRI) and electric (electroencephalogram, EEG) signals. Particularly, we hypothesize that the strongest changes in FC will involve the motor network, which is the most impaired in PD. Methods: Eight patients with diagnosis of PD “probable”, therapy with levodopa exclusively, normal cognitive and affective status, were included. Exclusion criteria were: moderate-severe rest tremor, levodopa induced dyskinesia, evidence of gray or white matter abnormalities on structural MRI. Scalp EEG (64 channels) were acquired inside the scanner (1.5 Tesla) before and after the intake of levodopa. fMRI functional connectivity was computed from four regions of interest: right and left supplementary motor area (SMA) and right and left precentral gyrus (primary motor cortex). Weighted partial directed coherence (w-PDC) was computed in the inverse space after the removal of EEG gradient and cardioballistic artifacts. Results and discussion: fMRI group analysis shows that the intake of levodopa increases hemodynamic functional connectivity among the SMAs / primary motor cortex and: sensory-motor network itself, attention network and default mode network. w-PDC analysis shows that EEG connectivity among regions of the motor network has the tendency to decrease after the intake the levodopa; furthermore, regions belonging to the DMN have the tendency to increase their outflow toward the rest of the brain. These findings, even if in a small sample of patients, suggest that other resting state physiological functional networks, beyond the motor one, are affected in patients with PD. The behavioral and cognitive tasks corresponding to the affected networks could benefit from the intake of levodopa.

Advanced neuroimaging methodologies to improve connectivity detection in normal and abnormal language brain networks

The language connectome was in-vivo investigated using multimodal non-invasive quantitative MRI. In PPA patients (n=18) recruited by the IRCCS ISNB, Bologna, cortical thickness measures showed a predominant reduction on the left hemisphere (p<0.005) with respect to matched healthy controls (HC) (n=18), and an accuracy of 86.1% in discrimination from Alzheimer’s disease patients (n=18). The left temporal and para-hippocampal gyri significantly correlated (p<0.01) with language fluency. In PPA patients (n=31) recruited by the Northwestern University Chicago, DTI measures were longitudinally evaluated (2-years follow-up) under the supervision of Prof. M. Catani, King’s College London. Significant differences with matched HC (n=27) were found, tract-localized at baseline and widespread in the follow-up. Language assessment scores correlated with arcuate (AF) and uncinate (UF) fasciculi DTI measures. In left-ischemic stroke patients (n=16) recruited by the NatBrainLab, King’s College London, language recovery was longitudinally evaluated (6-months follow-up). Using arterial spin labelling imaging a significant correlation (p<0.01) between language recovery and cerebral blood flow asymmetry, was found in the middle cerebral artery perfusion, towards the right. In HC (n=29) recruited by the DIBINEM Functional MR Unit, University of Bologna, an along-tract algorithm was developed suitable for different tractography methods, using the Laplacian operator. A higher left superior temporal gyrus and precentral operculum AF connectivity was found (Talozzi L et al., 2018), and lateralized UF projections towards the left dorsal orbital cortex. In HC (n=50) recruited in the Human Connectome Project, a new tractography-driven approach was developed for left association fibres, using a principal component analysis. The first component discriminated cortical areas typically connected by the AF, suggesting a good discrimination of cortical areas sharing a similar connectivity pattern. The evaluation of morphological, microstructural and metabolic measures could be used as in-vivo biomarkers to monitor language impairment related to neurodegeneration or as surrogate of cognitive rehabilitation/interventional treatment efficacy.

Development of predictive energy management strategies for hybrid electric vehicles supported by connectivity

Nowadays, the spreading of the air pollution crisis enhanced by greenhouse gases emission is leading to the worsening of the global warming. In this context, the transportation sector plays a vital role, since it is responsible for a large part of carbon dioxide production. In order to address these issues, the present thesis deals with the development of advanced control strategies for the energy efficiency optimization of plug-in hybrid electric vehicles (PHEVs), supported by the prediction of future working conditions of the powertrain. In particular, a Dynamic Programming algorithm has been developed for the combined optimization of vehicle energy and battery thermal management. At this aim, the battery temperature and the battery cooling circuit control signal have been considered as an additional state and control variables, respectively. Moreover, an adaptive equivalent consumption minimization strategy (A-ECMS) has been modified to handle zero-emission zones, where engine propulsion is not allowed. Navigation data represent an essential element in the achievement of these tasks. With this aim, a novel simulation and testing environment has been developed during the PhD research activity, as an effective tool to retrieve routing information from map service providers via vehicle-to-everything connectivity. Comparisons between the developed and the reference strategies are made, as well, in order to assess their impact on the vehicle energy consumption. All the activities presented in this doctoral dissertation have been carried out at the Green Mobility Research Lab} (GMRL), a research center resulting from the partnership between the University of Bologna and FEV Italia s.r.l., which represents the industrial partner of the research project.

Performance assessment of vehicular connectivity in hybrid electric vehicles for fuel and emissions reduction

Nowadays, the spreading of the air pollution crisis enhanced by greenhouse gases emission is leading to the worsening of global warming. Recently, several metropolitan cities introduced Zero-Emissions Zones where the use of the Internal Combustion Engine is forbidden to reduce localized pollutants emissions. This is particularly problematic for Plug-in Hybrid Electric Vehicles, which usually work in depleting mode. In order to address these issues, the present thesis presents a viable solution by exploiting vehicular connectivity to retrieve navigation data of the urban event along a selected route. The battery energy needed, in the form of a minimum State of Charge (SoC), is calculated by a Speed Profile Prediction algorithm and a Backward Vehicle Model. That value is then fed to both a Rule-Based Strategy, developed specifically for this application, and an Adaptive Equivalent Consumption Minimization Strategy (A-ECMS). The effectiveness of this approach has been tested with a Connected Hardware-in-the-Loop (C-HiL) on a driving cycle measured on-road, stimulating the predictions with multiple re-routings. However, even if hybrid electric vehicles have been recognized as a valid solution in response to increasingly tight regulations, the reduced engine load and the repeated engine starts and stops may reduce substantially the temperature of the exhaust after-treatment system (EATS), leading to relevant issues related to pollutant emission control. In this context, electrically heated catalysts (EHCs) represent a promising solution to ensure high pollutant conversion efficiency without affecting engine efficiency and performance. This work aims at studying the advantages provided by the introduction of a predictive EHC control function for a light-duty Diesel plug-in hybrid electric vehicle (PHEV) equipped with a Euro 7-oriented EATS. Based on the knowledge of future driving scenarios provided by vehicular connectivity, engine first start can be predicted and therefore an EATS pre-heating phase can be planned.

Assessing brain connectivity through electroencephalographic signal processing and modeling analysis

Brain functioning relies on the interaction of several neural populations connected through complex connectivity networks, enabling the transmission and integration of information. Recent advances in neuroimaging techniques, such as electroencephalography (EEG), have deepened our understanding of the reciprocal roles played by brain regions during cognitive processes. The underlying idea of this PhD research is that EEG-related functional connectivity (FC) changes in the brain may incorporate important neuromarkers of behavior and cognition, as well as brain disorders, even at subclinical levels. However, a complete understanding of the reliability of the wide range of existing connectivity estimation techniques is still lacking. The first part of this work addresses this limitation by employing Neural Mass Models (NMMs), which simulate EEG activity and offer a unique tool to study interconnected networks of brain regions in controlled conditions. NMMs were employed to test FC estimators like Transfer Entropy and Granger Causality in linear and nonlinear conditions. Results revealed that connectivity estimates reflect information transmission between brain regions, a quantity that can be significantly different from the connectivity strength, and that Granger causality outperforms the other estimators. A second objective of this thesis was to assess brain connectivity and network changes on EEG data reconstructed at the cortical level. Functional brain connectivity has been estimated through Granger Causality, in both temporal and spectral domains, with the following goals: a) detect task-dependent functional connectivity network changes, focusing on internal-external attention competition and fear conditioning and reversal; b) identify resting-state network alterations in a subclinical population with high autistic traits. Connectivity-based neuromarkers, compared to the canonical EEG analysis, can provide deeper insights into brain mechanisms and may drive future diagnostic methods and therapeutic interventions. However, further methodological studies are required to fully understand the accuracy and information captured by FC estimates, especially concerning nonlinear phenomena.