Non-Markovian stochastic processes and their applications: from anomalous diffusion to time series analysis

This work provides a forward step in the study and comprehension of the relationships between stochastic processes and a certain class of integral-partial differential equation, which can be used in order to model anomalous diffusion and transport in statistical physics. In the first part, we brought the reader through the fundamental notions of probability and stochastic processes, stochastic integration and stochastic differential equations as well. In particular, within the study of H-sssi processes, we focused on fractional Brownian motion (fBm) and its discrete-time increment process, the fractional Gaussian noise (fGn), which provide examples of non-Markovian Gaussian processes. The fGn, together with stationary FARIMA processes, is widely used in the modeling and estimation of long-memory, or long-range dependence (LRD). Time series manifesting long-range dependence, are often observed in nature especially in physics, meteorology, climatology, but also in hydrology, geophysics, economy and many others. We deepely studied LRD, giving many real data examples, providing statistical analysis and introducing parametric methods of estimation. Then, we introduced the theory of fractional integrals and derivatives, which indeed turns out to be very appropriate for studying and modeling systems with long-memory properties. After having introduced the basics concepts, we provided many examples and applications. For instance, we investigated the relaxation equation with distributed order time-fractional derivatives, which describes models characterized by a strong memory component and can be used to model relaxation in complex systems, which deviates from the classical exponential Debye pattern. Then, we focused in the study of generalizations of the standard diffusion equation, by passing through the preliminary study of the fractional forward drift equation. Such generalizations have been obtained by using fractional integrals and derivatives of distributed orders. In order to find a connection between the anomalous diffusion described by these equations and the long-range dependence, we introduced and studied the generalized grey Brownian motion (ggBm), which is actually a parametric class of H-sssi processes, which have indeed marginal probability density function evolving in time according to a partial integro-differential equation of fractional type. The ggBm is of course Non-Markovian. All around the work, we have remarked many times that, starting from a master equation of a probability density function f(x,t), it is always possible to define an equivalence class of stochastic processes with the same marginal density function f(x,t). All these processes provide suitable stochastic models for the starting equation. Studying the ggBm, we just focused on a subclass made up of processes with stationary increments. The ggBm has been defined canonically in the so called grey noise space. However, we have been able to provide a characterization notwithstanding the underline probability space. We also pointed out that that the generalized grey Brownian motion is a direct generalization of a Gaussian process and in particular it generalizes Brownain motion and fractional Brownain motion as well. Finally, we introduced and analyzed a more general class of diffusion type equations related to certain non-Markovian stochastic processes. We started from the forward drift equation, which have been made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation has been interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time-evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the same memory kernel K(t). We developed several applications and derived the exact solutions. Moreover, we considered different stochastic models for the given equations, providing path simulations.

Model predictive control in thermal management of multiprocessor systems-on-chip

MultiProcessor Systems-on-Chip (MPSoC) are the core of nowadays and next generation computing platforms. Their relevance in the global market continuously increase, occupying an important role both in everydaylife products (e.g. smartphones, tablets, laptops, cars) and in strategical market sectors as aviation, defense, robotics, medicine. Despite of the incredible performance improvements in the recent years processors manufacturers have had to deal with issues, commonly called “Walls”, that have hindered the processors development. After the famous “Power Wall”, that limited the maximum frequency of a single core and marked the birth of the modern multiprocessors system-on-chip, the “Thermal Wall” and the “Utilization Wall” are the actual key limiter for performance improvements. The former concerns the damaging effects of the high temperature on the chip caused by the large power densities dissipation, whereas the second refers to the impossibility of fully exploiting the computing power of the processor due to the limitations on power and temperature budgets. In this thesis we faced these challenges by developing efficient and reliable solutions able to maximize performance while limiting the maximum temperature below a fixed critical threshold and saving energy. This has been possible by exploiting the Model Predictive Controller (MPC) paradigm that solves an optimization problem subject to constraints in order to find the optimal control decisions for the future interval. A fully-distributedMPC-based thermal controller with a far lower complexity respect to a centralized one has been developed. The control feasibility and interesting properties for the simplification of the control design has been proved by studying a partial differential equation thermal model. Finally, the controller has been efficiently included in more complex control schemes able to minimize energy consumption and deal with mixed-criticalities tasks

Neurogeometry of stereo vision

This work aims to develop a neurogeometric model of stereo vision, based on cortical architectures involved in the problem of 3D perception and neural mechanisms generated by retinal disparities. First, we provide a sub-Riemannian geometry for stereo vision, inspired by the work on the stereo problem by Zucker (2006), and using sub-Riemannian tools introduced by Citti-Sarti (2006) for monocular vision. We present a mathematical interpretation of the neural mechanisms underlying the behavior of binocular cells, that integrate monocular inputs. The natural compatibility between stereo geometry and neurophysiological models shows that these binocular cells are sensitive to position and orientation. Therefore, we model their action in the space R3xS2 equipped with a sub-Riemannian metric. Integral curves of the sub-Riemannian structure model neural connectivity and can be related to the 3D analog of the psychophysical association fields for the 3D process of regular contour formation. Then, we identify 3D perceptual units in the visual scene: they emerge as a consequence of the random cortico-cortical connection of binocular cells. Considering an opportune stochastic version of the integral curves, we generate a family of kernels. These kernels represent the probability of interaction between binocular cells, and they are implemented as facilitation patterns to define the evolution in time of neural population activity at a point. This activity is usually modeled through a mean field equation: steady stable solutions lead to consider the associated eigenvalue problem. We show that three-dimensional perceptual units naturally arise from the discrete version of the eigenvalue problem associated to the integro-differential equation of the population activity.

Development of functional MRI protocols in the study of neurodegenerative diseases: MRI study in patients with REM sleep behavior disorder, idiopathic Parkinson's disease and multisystem atrophy

In the central nervous system, iron in several proteins is involved in many important processes: oxygen transportation, oxidative phosphorylation, mitochondrial respiration, myelin production, the synthesis and metabolism of neurotransmitters. Abnormal iron homoeostasis can induce cellular damage through hydroxyl radical production, which can cause the oxidation, modification of lipids, proteins, carbohydrates, and DNA, lead to neurotoxicity. Moreover increased levels of iron are harmful and iron accumulations are typical hallmarks of brain ageing and several neurodegenerative disorders particularly PD. Numerous studies on post mortem tissue report on an increased amount of total iron in the substantia nigra in patients with PD also supported by large body of in vivo findings from Magnetic Resonance Imaging (MRI) studies. The importance and approaches for in vivo brain iron assessment using multiparametric MRI is increased over last years. Quantitative MRI may provide useful biomarkers for brain integrity assessment in iron-related neurodegeneration. Particularly, a prominent change in iron- sensitive T2* MRI contrast within the sub areas of the SN overlapping with nigrosome 1 were shown to be a hallmark of Parkinson's Disease with high diagnostic accuracy. Moreover, differential diagnosis between Parkinson's Disease (PD) and atypical parkinsonian syndromes (APS) remains challenging, mainly in the early phases of the disease. Advanced brain MR imaging enables to detect the pathological changes of nigral and extranigral structures at the onset of clinical manifestations and during the course of the disease. The Nigrosome-1 (N1) is a substructure of the healthy Substantia Nigra pars compacta enriched by dopaminergic neurons; their loss in Parkinson’s disease and atypical parkinsonian syndromes is related to the iron accumulation. N1 changes are supportive MR biomarkers for diagnosis of these neurodegenerative disorders, but its detection is hard with conventional sequences, also using high field (3T) scanner. Quantitative susceptibility mapping (QSM), an iron-sensitive technique, enables the direct detection of Neurodegeneration

Quantum integrability as a new method for exact results on N=2 supersymmetric gauge theories and black holes

In this thesis I show a triple new connection we found between quantum integrability, N=2 supersymmetric gauge theories and black holes perturbation theory. I use the approach of the ODE/IM correspondence between Ordinary Differential Equations (ODE) and Integrable Models (IM), first to connect basic integrability functions - the Baxter’s Q, T and Y functions - to the gauge theory periods. This fundamental identification allows several new results for both theories, for example: an exact non linear integral equation (Thermodynamic Bethe Ansatz, TBA) for the gauge periods; an interpretation of the integrability functional relations as new exact R-symmetry relations for the periods; new formulas for the local integrals of motion in terms of gauge periods. This I develop in all details at least for the SU(2) gauge theory with Nf=0,1,2 matter flavours. Still through to the ODE/IM correspondence, I connect the mathematically precise definition of quasinormal modes of black holes (having an important role in gravitational waves’ obervations) with quantization conditions on the Q, Y functions. In this way I also give a mathematical explanation of the recently found connection between quasinormal modes and N=2 supersymmetric gauge theories. Moreover, it follows a new simple and effective method to numerically compute the quasinormal modes - the TBA - which I compare with other standard methods. The spacetimes for which I show these in all details are in the simplest Nf=0 case the D3 brane in the Nf=1,2 case a generalization of extremal Reissner-Nordström (charged) black holes. Then I begin treating also the Nf=3,4 theories and argue on how our integrability-gauge-gravity correspondence can generalize to other types of black holes in either asymptotically flat (Nf=3) or Anti-de-Sitter (Nf=4) spacetime. Finally I begin to show the extension to a 4-fold correspondence with also Conformal Field Theory (CFT), through the renowned AdS/CFT correspondence.

Decoding Agc1 deficiency: bioinformatics approaches to unveil the functional implications of Slc25a12 on the transcriptome and epigenome

In the brain, mutations in SLC25A12 gene encoding AGC1 cause an ultra-rare genetic disease reported as a developmental and epileptic encephalopathy associated with global cerebral hypomyelination. Symptoms of the disease include diffused hypomyelination, arrested psychomotor development, severe hypotonia, seizures and are common to other neurological and developmental disorders. Amongst the biological components believed to be most affected by AGC1 deficiency are oligodendrocytes, glial cells responsible for myelination. Recent studies (Poeta et al, 2022) have also shown how altered levels of transcription factors and epigenetic modifications greatly affect proliferation and differentiation in oligodendrocyte precursor cells (OPCs). In this study we explore the transcriptomic landscape of Agc1 in two different system models: OPCs silenced for Agc1 and iPSCs from human patients differentiated to neural progenitors. Analyses range from differential expression analysis, alternative splicing, master regulator analysis. ATAC-seq results on OPCs were integrated with results from RNA-Seq to assess the activity of a TF based on the accessibility data from its putative targets, which allows to integrate RNA-Seq data to infer their role as either activators or repressors. All the findings for this model were also integrated with early data from iPSCs RNA-seq results, looking for possible commonalities between the two different system models, among which we find a downregulation in genes encoding for SREBP, a transcription factor regulating fatty acids biosynthesis, a key process for myelination which could explain the hypomyelinated state of patients. We also find that in both systems cells tend to form more neurites, likely losing their ability to differentiate, considering their progenitor state. We also report several alterations in the chromatin state of cells lacking Agc1, which confirms the hypothesis for which Agc1 is not a disease restricted only to metabolic alterations in the cells, but there is a profound shift of the regulatory state of these cells.