Computational analyses on the structure-function relation in ion channels

Ion channels are protein molecules, embedded in the lipid bilayer of the cell membranes. They act as powerful sensing elements switching chemicalphysical stimuli into ion-fluxes. At a glance, ion channels are water-filled pores, which can open and close in response to different stimuli (gating), and one once open select the permeating ion species (selectivity). They play a crucial role in several physiological functions, like nerve transmission, muscular contraction, and secretion. Besides, ion channels can be used in technological applications for different purpose (sensing of organic molecules, DNA sequencing). As a result, there is remarkable interest in understanding the molecular determinants of the channel functioning. Nowadays, both the functional and the structural characteristics of ion channels can be experimentally solved. The purpose of this thesis was to investigate the structure-function relation in ion channels, by computational techniques. Most of the analyses focused on the mechanisms of ion conduction, and the numerical methodologies to compute the channel conductance. The standard techniques for atomistic simulation of complex molecular systems (Molecular Dynamics) cannot be routinely used to calculate ion fluxes in membrane channels, because of the high computational resources needed. The main step forward of the PhD research activity was the development of a computational algorithm for the calculation of ion fluxes in protein channels. The algorithm - based on the electrodiffusion theory - is computational inexpensive, and was used for an extensive analysis on the molecular determinants of the channel conductance. The first record of ion-fluxes through a single protein channel dates back to 1976, and since then measuring the single channel conductance has become a standard experimental procedure. Chapter 1 introduces ion channels, and the experimental techniques used to measure the channel currents. The abundance of functional data (channel currents) does not match with an equal abundance of structural data. The bacterial potassium channel KcsA was the first selective ion channels to be experimentally solved (1998), and after KcsA the structures of four different potassium channels were revealed. These experimental data inspired a new era in ion channel modeling. Once the atomic structures of channels are known, it is possible to define mathematical models based on physical descriptions of the molecular systems. These physically based models can provide an atomic description of ion channel functioning, and predict the effect of structural changes. Chapter 2 introduces the computation methods used throughout the thesis to model ion channels functioning at the atomic level. In Chapter 3 and Chapter 4 the ion conduction through potassium channels is analyzed, by an approach based on the Poisson-Nernst-Planck electrodiffusion theory. In the electrodiffusion theory ion conduction is modeled by the drift-diffusion equations, thus describing the ion distributions by continuum functions. The numerical solver of the Poisson- Nernst-Planck equations was tested in the KcsA potassium channel (Chapter 3), and then used to analyze how the atomic structure of the intracellular vestibule of potassium channels affects the conductance (Chapter 4). As a major result, a correlation between the channel conductance and the potassium concentration in the intracellular vestibule emerged. The atomic structure of the channel modulates the potassium concentration in the vestibule, thus its conductance. This mechanism explains the phenotype of the BK potassium channels, a sub-family of potassium channels with high single channel conductance. The functional role of the intracellular vestibule is also the subject of Chapter 5, where the affinity of the potassium channels hEag1 (involved in tumour-cell proliferation) and hErg (important in the cardiac cycle) for several pharmaceutical drugs was compared. Both experimental measurements and molecular modeling were used in order to identify differences in the blocking mechanism of the two channels, which could be exploited in the synthesis of selective blockers. The experimental data pointed out the different role of residue mutations in the blockage of hEag1 and hErg, and the molecular modeling provided a possible explanation based on different binding sites in the intracellular vestibule. Modeling ion channels at the molecular levels relates the functioning of a channel to its atomic structure (Chapters 3-5), and can also be useful to predict the structure of ion channels (Chapter 6-7). In Chapter 6 the structure of the KcsA potassium channel depleted from potassium ions is analyzed by molecular dynamics simulations. Recently, a surprisingly high osmotic permeability of the KcsA channel was experimentally measured. All the available crystallographic structure of KcsA refers to a channel occupied by potassium ions. To conduct water molecules potassium ions must be expelled from KcsA. The structure of the potassium-depleted KcsA channel and the mechanism of water permeation are still unknown, and have been investigated by numerical simulations. Molecular dynamics of KcsA identified a possible atomic structure of the potassium-depleted KcsA channel, and a mechanism for water permeation. The depletion from potassium ions is an extreme situation for potassium channels, unlikely in physiological conditions. However, the simulation of such an extreme condition could help to identify the structural conformations, so the functional states, accessible to potassium ion channels. The last chapter of the thesis deals with the atomic structure of the !- Hemolysin channel. !-Hemolysin is the major determinant of the Staphylococcus Aureus toxicity, and is also the prototype channel for a possible usage in technological applications. The atomic structure of !- Hemolysin was revealed by X-Ray crystallography, but several experimental evidences suggest the presence of an alternative atomic structure. This alternative structure was predicted, combining experimental measurements of single channel currents and numerical simulations. This thesis is organized in two parts, in the first part an overview on ion channels and on the numerical methods adopted throughout the thesis is provided, while the second part describes the research projects tackled in the course of the PhD programme. The aim of the research activity was to relate the functional characteristics of ion channels to their atomic structure. In presenting the different research projects, the role of numerical simulations to analyze the structure-function relation in ion channels is highlighted.

Bioimaging of animal embryogenesis: mathematical methods and computational algorithms

Some fundamental biological processes such as embryonic development have been preserved during evolution and are common to species belonging to different phylogenetic positions, but are nowadays largely unknown. The understanding of cell morphodynamics leading to the formation of organized spatial distribution of cells such as tissues and organs can be achieved through the reconstruction of cells shape and position during the development of a live animal embryo. We design in this work a chain of image processing methods to automatically segment and track cells nuclei and membranes during the development of a zebrafish embryo, which has been largely validates as model organism to understand vertebrate development, gene function and healingrepair mechanisms in vertebrates. The embryo is previously labeled through the ubiquitous expression of fluorescent proteins addressed to cells nuclei and membranes, and temporal sequences of volumetric images are acquired with laser scanning microscopy. Cells position is detected by processing nuclei images either through the generalized form of the Hough transform or identifying nuclei position with local maxima after a smoothing preprocessing step. Membranes and nuclei shapes are reconstructed by using PDEs based variational techniques such as the Subjective Surfaces and the Chan Vese method. Cells tracking is performed by combining informations previously detected on cells shape and position with biological regularization constraints. Our results are manually validated and reconstruct the formation of zebrafish brain at 7-8 somite stage with all the cells tracked starting from late sphere stage with less than 2% error for at least 6 hours. Our reconstruction opens the way to a systematic investigation of cellular behaviors, of clonal origin and clonal complexity of brain organs, as well as the contribution of cell proliferation modes and cell movements to the formation of local patterns and morphogenetic fields.

Mathematical models of cognitive processes

The research activity carried out during the PhD course was focused on the development of mathematical models of some cognitive processes and their validation by means of data present in literature, with a double aim: i) to achieve a better interpretation and explanation of the great amount of data obtained on these processes from different methodologies (electrophysiological recordings on animals, neuropsychological, psychophysical and neuroimaging studies in humans), ii) to exploit model predictions and results to guide future research and experiments. In particular, the research activity has been focused on two different projects: 1) the first one concerns the development of neural oscillators networks, in order to investigate the mechanisms of synchronization of the neural oscillatory activity during cognitive processes, such as object recognition, memory, language, attention; 2) the second one concerns the mathematical modelling of multisensory integration processes (e.g. visual-acoustic), which occur in several cortical and subcortical regions (in particular in a subcortical structure named Superior Colliculus (SC)), and which are fundamental for orienting motor and attentive responses to external world stimuli. This activity has been realized in collaboration with the Center for Studies and Researches in Cognitive Neuroscience of the University of Bologna (in Cesena) and the Department of Neurobiology and Anatomy of the Wake Forest University School of Medicine (NC, USA). PART 1. Objects representation in a number of cognitive functions, like perception and recognition, foresees distribute processes in different cortical areas. One of the main neurophysiological question concerns how the correlation between these disparate areas is realized, in order to succeed in grouping together the characteristics of the same object (binding problem) and in maintaining segregated the properties belonging to different objects simultaneously present (segmentation problem). Different theories have been proposed to address these questions (Barlow, 1972). One of the most influential theory is the so called “assembly coding”, postulated by Singer (2003), according to which 1) an object is well described by a few fundamental properties, processing in different and distributed cortical areas; 2) the recognition of the object would be realized by means of the simultaneously activation of the cortical areas representing its different features; 3) groups of properties belonging to different objects would be kept separated in the time domain. In Chapter 1.1 and in Chapter 1.2 we present two neural network models for object recognition, based on the “assembly coding” hypothesis. These models are networks of Wilson-Cowan oscillators which exploit: i) two high-level “Gestalt Rules” (the similarity and previous knowledge rules), to realize the functional link between elements of different cortical areas representing properties of the same object (binding problem); 2) the synchronization of the neural oscillatory activity in the γ-band (30-100Hz), to segregate in time the representations of different objects simultaneously present (segmentation problem). These models are able to recognize and reconstruct multiple simultaneous external objects, even in difficult case (some wrong or lacking features, shared features, superimposed noise). In Chapter 1.3 the previous models are extended to realize a semantic memory, in which sensory-motor representations of objects are linked with words. To this aim, the network, previously developed, devoted to the representation of objects as a collection of sensory-motor features, is reciprocally linked with a second network devoted to the representation of words (lexical network) Synapses linking the two networks are trained via a time-dependent Hebbian rule, during a training period in which individual objects are presented together with the corresponding words. Simulation results demonstrate that, during the retrieval phase, the network can deal with the simultaneous presence of objects (from sensory-motor inputs) and words (from linguistic inputs), can correctly associate objects with words and segment objects even in the presence of incomplete information. Moreover, the network can realize some semantic links among words representing objects with some shared features. These results support the idea that semantic memory can be described as an integrated process, whose content is retrieved by the co-activation of different multimodal regions. In perspective, extended versions of this model may be used to test conceptual theories, and to provide a quantitative assessment of existing data (for instance concerning patients with neural deficits). PART 2. The ability of the brain to integrate information from different sensory channels is fundamental to perception of the external world (Stein et al, 1993). It is well documented that a number of extraprimary areas have neurons capable of such a task; one of the best known of these is the superior colliculus (SC). This midbrain structure receives auditory, visual and somatosensory inputs from different subcortical and cortical areas, and is involved in the control of orientation to external events (Wallace et al, 1993). SC neurons respond to each of these sensory inputs separately, but is also capable of integrating them (Stein et al, 1993) so that the response to the combined multisensory stimuli is greater than that to the individual component stimuli (enhancement). This enhancement is proportionately greater if the modality-specific paired stimuli are weaker (the principle of inverse effectiveness). Several studies have shown that the capability of SC neurons to engage in multisensory integration requires inputs from cortex; primarily the anterior ectosylvian sulcus (AES), but also the rostral lateral suprasylvian sulcus (rLS). If these cortical inputs are deactivated the response of SC neurons to cross-modal stimulation is no different from that evoked by the most effective of its individual component stimuli (Jiang et al 2001). This phenomenon can be better understood through mathematical models. The use of mathematical models and neural networks can place the mass of data that has been accumulated about this phenomenon and its underlying circuitry into a coherent theoretical structure. In Chapter 2.1 a simple neural network model of this structure is presented; this model is able to reproduce a large number of SC behaviours like multisensory enhancement, multisensory and unisensory depression, inverse effectiveness. In Chapter 2.2 this model was improved by incorporating more neurophysiological knowledge about the neural circuitry underlying SC multisensory integration, in order to suggest possible physiological mechanisms through which it is effected. This endeavour was realized in collaboration with Professor B.E. Stein and Doctor B. Rowland during the 6 months-period spent at the Department of Neurobiology and Anatomy of the Wake Forest University School of Medicine (NC, USA), within the Marco Polo Project. The model includes four distinct unisensory areas that are devoted to a topological representation of external stimuli. Two of them represent subregions of the AES (i.e., FAES, an auditory area, and AEV, a visual area) and send descending inputs to the ipsilateral SC; the other two represent subcortical areas (one auditory and one visual) projecting ascending inputs to the same SC. Different competitive mechanisms, realized by means of population of interneurons, are used in the model to reproduce the different behaviour of SC neurons in conditions of cortical activation and deactivation. The model, with a single set of parameters, is able to mimic the behaviour of SC multisensory neurons in response to very different stimulus conditions (multisensory enhancement, inverse effectiveness, within- and cross-modal suppression of spatially disparate stimuli), with cortex functional and cortex deactivated, and with a particular type of membrane receptors (NMDA receptors) active or inhibited. All these results agree with the data reported in Jiang et al. (2001) and in Binns and Salt (1996). The model suggests that non-linearities in neural responses and synaptic (excitatory and inhibitory) connections can explain the fundamental aspects of multisensory integration, and provides a biologically plausible hypothesis about the underlying circuitry.

Development of Clinical Decision Support Systems based on Mathematical Models of Physiological Systems

In the last years of research, I focused my studies on different physiological problems. Together with my supervisors, I developed/improved different mathematical models in order to create valid tools useful for a better understanding of important clinical issues. The aim of all this work is to develop tools for learning and understanding cardiac and cerebrovascular physiology as well as pathology, generating research questions and developing clinical decision support systems useful for intensive care unit patients. I. ICP-model Designed for Medical Education We developed a comprehensive cerebral blood flow and intracranial pressure model to simulate and study the complex interactions in cerebrovascular dynamics caused by multiple simultaneous alterations, including normal and abnormal functional states of auto-regulation of the brain. Individual published equations (derived from prior animal and human studies) were implemented into a comprehensive simulation program. Included in the normal physiological modelling was: intracranial pressure, cerebral blood flow, blood pressure, and carbon dioxide (CO2) partial pressure. We also added external and pathological perturbations, such as head up position and intracranial haemorrhage. The model performed clinically realistically given inputs of published traumatized patients, and cases encountered by clinicians. The pulsatile nature of the output graphics was easy for clinicians to interpret. The manoeuvres simulated include changes of basic physiological inputs (e.g. blood pressure, central venous pressure, CO2 tension, head up position, and respiratory effects on vascular pressures) as well as pathological inputs (e.g. acute intracranial bleeding, and obstruction of cerebrospinal outflow). Based on the results, we believe the model would be useful to teach complex relationships of brain haemodynamics and study clinical research questions such as the optimal head-up position, the effects of intracranial haemorrhage on cerebral haemodynamics, as well as the best CO2 concentration to reach the optimal compromise between intracranial pressure and perfusion. We believe this model would be useful for both beginners and advanced learners. It could be used by practicing clinicians to model individual patients (entering the effects of needed clinical manipulations, and then running the model to test for optimal combinations of therapeutic manoeuvres). II. A Heterogeneous Cerebrovascular Mathematical Model Cerebrovascular pathologies are extremely complex, due to the multitude of factors acting simultaneously on cerebral haemodynamics. In this work, the mathematical model of cerebral haemodynamics and intracranial pressure dynamics, described in the point I, is extended to account for heterogeneity in cerebral blood flow. The model includes the Circle of Willis, six regional districts independently regulated by autoregulation and CO2 reactivity, distal cortical anastomoses, venous circulation, the cerebrospinal fluid circulation, and the intracranial pressure-volume relationship. Results agree with data in the literature and highlight the existence of a monotonic relationship between transient hyperemic response and the autoregulation gain. During unilateral internal carotid artery stenosis, local blood flow regulation is progressively lost in the ipsilateral territory with the presence of a steal phenomenon, while the anterior communicating artery plays the major role to redistribute the available blood flow. Conversely, distal collateral circulation plays a major role during unilateral occlusion of the middle cerebral artery. In conclusion, the model is able to reproduce several different pathological conditions characterized by heterogeneity in cerebrovascular haemodynamics and can not only explain generalized results in terms of physiological mechanisms involved, but also, by individualizing parameters, may represent a valuable tool to help with difficult clinical decisions. III. Effect of Cushing Response on Systemic Arterial Pressure. During cerebral hypoxic conditions, the sympathetic system causes an increase in arterial pressure (Cushing response), creating a link between the cerebral and the systemic circulation. This work investigates the complex relationships among cerebrovascular dynamics, intracranial pressure, Cushing response, and short-term systemic regulation, during plateau waves, by means of an original mathematical model. The model incorporates the pulsating heart, the pulmonary circulation and the systemic circulation, with an accurate description of the cerebral circulation and the intracranial pressure dynamics (same model as in the first paragraph). Various regulatory mechanisms are included: cerebral autoregulation, local blood flow control by oxygen (O2) and/or CO2 changes, sympathetic and vagal regulation of cardiovascular parameters by several reflex mechanisms (chemoreceptors, lung-stretch receptors, baroreceptors). The Cushing response has been described assuming a dramatic increase in sympathetic activity to vessels during a fall in brain O2 delivery. With this assumption, the model is able to simulate the cardiovascular effects experimentally observed when intracranial pressure is artificially elevated and maintained at constant level (arterial pressure increase and bradicardia). According to the model, these effects arise from the interaction between the Cushing response and the baroreflex response (secondary to arterial pressure increase). Then, patients with severe head injury have been simulated by reducing intracranial compliance and cerebrospinal fluid reabsorption. With these changes, oscillations with plateau waves developed. In these conditions, model results indicate that the Cushing response may have both positive effects, reducing the duration of the plateau phase via an increase in cerebral perfusion pressure, and negative effects, increasing the intracranial pressure plateau level, with a risk of greater compression of the cerebral vessels. This model may be of value to assist clinicians in finding the balance between clinical benefits of the Cushing response and its shortcomings. IV. Comprehensive Cardiopulmonary Simulation Model for the Analysis of Hypercapnic Respiratory Failure We developed a new comprehensive cardiopulmonary model that takes into account the mutual interactions between the cardiovascular and the respiratory systems along with their short-term regulatory mechanisms. The model includes the heart, systemic and pulmonary circulations, lung mechanics, gas exchange and transport equations, and cardio-ventilatory control. Results show good agreement with published patient data in case of normoxic and hyperoxic hypercapnia simulations. In particular, simulations predict a moderate increase in mean systemic arterial pressure and heart rate, with almost no change in cardiac output, paralleled by a relevant increase in minute ventilation, tidal volume and respiratory rate. The model can represent a valid tool for clinical practice and medical research, providing an alternative way to experience-based clinical decisions. In conclusion, models are not only capable of summarizing current knowledge, but also identifying missing knowledge. In the former case they can serve as training aids for teaching the operation of complex systems, especially if the model can be used to demonstrate the outcome of experiments. In the latter case they generate experiments to be performed to gather the missing data.

Genome characterization through a mathematical model of the genetic code: an analysis of the whole chromosome 1 of A. thaliana

The objective of this work is to characterize the genome of the chromosome 1 of A.thaliana, a small flowering plants used as a model organism in studies of biology and genetics, on the basis of a recent mathematical model of the genetic code. I analyze and compare different portions of the genome: genes, exons, coding sequences (CDS), introns, long introns, intergenes, untranslated regions (UTR) and regulatory sequences. In order to accomplish the task, I transformed nucleotide sequences into binary sequences based on the definition of the three different dichotomic classes. The descriptive analysis of binary strings indicate the presence of regularities in each portion of the genome considered. In particular, there are remarkable differences between coding sequences (CDS and exons) and non-coding sequences, suggesting that the frame is important only for coding sequences and that dichotomic classes can be useful to recognize them. Then, I assessed the existence of short-range dependence between binary sequences computed on the basis of the different dichotomic classes. I used three different measures of dependence: the well-known chi-squared test and two indices derived from the concept of entropy i.e. Mutual Information (MI) and Sρ, a normalized version of the “Bhattacharya Hellinger Matusita distance”. The results show that there is a significant short-range dependence structure only for the coding sequences whose existence is a clue of an underlying error detection and correction mechanism. No doubt, further studies are needed in order to assess how the information carried by dichotomic classes could discriminate between coding and noncoding sequence and, therefore, contribute to unveil the role of the mathematical structure in error detection and correction mechanisms. Still, I have shown the potential of the approach presented for understanding the management of genetic information.

Analysis and mathematical modeling of wave-structure interaction

The aim of this thesis, included within the THESEUS project, is the development of a mathematical model 2DV two-phase, based on the existing code IH-2VOF developed by the University of Cantabria, able to represent together the overtopping phenomenon and the sediment transport. Several numerical simulations were carried out in order to analyze the flow characteristics on a dike crest. The results show that the seaward/landward slope does not affect the evolution of the flow depth and velocity over the dike crest whereas the most important parameter is the relative submergence. Wave heights decrease and flow velocities increase while waves travel over the crest. In particular, by increasing the submergence, the wave height decay and the increase of the velocity are less marked. Besides, an appropriate curve able to fit the variation of the wave height/velocity over the dike crest were found. Both for the wave height and for the wave velocity different fitting coefficients were determined on the basis of the submergence and of the significant wave height. An equation describing the trend of the dimensionless coefficient c_h for the wave height was derived. These conclusions could be taken into consideration for the design criteria and the upgrade of the structures. In the second part of the thesis, new equations for the representation of the sediment transport in the IH-2VOF model were introduced in order to represent beach erosion while waves run-up and overtop the sea banks during storms. The new model allows to calculate sediment fluxes in the water column together with the sediment concentration. Moreover it is possible to model the bed profile evolution. Different tests were performed under low-intensity regular waves with an homogeneous layer of sand on the bottom of a channel in order to analyze the erosion-deposition patterns and verify the model results.

Mathematical Optimization Applied to Thermal and Electrical Energy Systems

This Thesis aims at building and discussing mathematical models applications focused on Energy problems, both on the thermal and electrical side. The objective is to show how mathematical programming techniques developed within Operational Research can give useful answers in the Energy Sector, how they can provide tools to support decision making processes of Companies operating in the Energy production and distribution and how they can be successfully used to make simulations and sensitivity analyses to better understand the state of the art and convenience of a particular technology by comparing it with the available alternatives. The first part discusses the fundamental mathematical background followed by a comprehensive literature review about mathematical modelling in the Energy Sector. The second part presents mathematical models for the District Heating strategic network design and incremental network design. The objective is the selection of an optimal set of new users to be connected to an existing thermal network, maximizing revenues, minimizing infrastructure and operational costs and taking into account the main technical requirements of the real world application. Results on real and randomly generated benchmark networks are discussed with particular attention to instances characterized by big networks dimensions. The third part is devoted to the development of linear programming models for optimal battery operation in off-grid solar power schemes, with consideration of battery degradation. The key contribution of this work is the inclusion of battery degradation costs in the optimisation models. As available data on relating degradation costs to the nature of charge/discharge cycles are limited, we concentrate on investigating the sensitivity of operational patterns to the degradation cost structure. The objective is to investigate the combination of battery costs and performance at which such systems become economic. We also investigate how the system design should change when battery degradation is taken into account.

Closed-loop investigation of the dynamical properties of cardiomyocyte electrophysiology by means of Dynamic clamp and Mathematical modelling

The research field of my PhD concerns mathematical modeling and numerical simulation, applied to the cardiac electrophysiology analysis at a single cell level. This is possible thanks to the development of mathematical descriptions of single cellular components, ionic channels, pumps, exchangers and subcellular compartments. Due to the difficulties of vivo experiments on human cells, most of the measurements are acquired in vitro using animal models (e.g. guinea pig, dog, rabbit). Moreover, to study the cardiac action potential and all its features, it is necessary to acquire more specific knowledge about single ionic currents that contribute to the cardiac activity. Electrophysiological models of the heart have become very accurate in recent years giving rise to extremely complicated systems of differential equations. Although describing the behavior of cardiac cells quite well, the models are computationally demanding for numerical simulations and are very difficult to analyze from a mathematical (dynamical-systems) viewpoint. Simplified mathematical models that capture the underlying dynamics to a certain extent are therefore frequently used. The results presented in this thesis have confirmed that a close integration of computational modeling and experimental recordings in real myocytes, as performed by dynamic clamp, is a useful tool in enhancing our understanding of various components of normal cardiac electrophysiology, but also arrhythmogenic mechanisms in a pathological condition, especially when fully integrated with experimental data.

Searching and retrieving in content-based repositories of formal mathematical knowledge

In this thesis, the author presents a query language for an RDF (Resource Description Framework) database and discusses its applications in the context of the HELM project (the Hypertextual Electronic Library of Mathematics). This language aims at meeting the main requirements coming from the RDF community. in particular it includes: a human readable textual syntax and a machine-processable XML (Extensible Markup Language) syntax both for queries and for query results, a rigorously exposed formal semantics, a graph-oriented RDF data access model capable of exploring an entire RDF graph (including both RDF Models and RDF Schemata), a full set of Boolean operators to compose the query constraints, fully customizable and highly structured query results having a 4-dimensional geometry, some constructions taken from ordinary programming languages that simplify the formulation of complex queries. The HELM project aims at integrating the modern tools for the automation of formal reasoning with the most recent electronic publishing technologies, in order create and maintain a hypertextual, distributed virtual library of formal mathematical knowledge. In the spirit of the Semantic Web, the documents of this library include RDF metadata describing their structure and content in a machine-understandable form. Using the author's query engine, HELM exploits this information to implement some functionalities allowing the interactive and automatic retrieval of documents on the basis of content-aware requests that take into account the mathematical nature of these documents.

Mathematical skills: intergenerational features and relationships with cognitive and linguistic abilities

This thesis aimed to investigate the cognitive underpinnings of math skills, with particular reference to cognitive, and linguistic markers, core mechanisms of number processing and environmental variables. In particular, the issue of intergenerational transmission of math skills has been deepened, comparing parents’ and children’s basic and formal math abilities. This pattern of relationships amongst these has been considered in two different age ranges, preschool and primary school children. In the first chapter, a general introduction on mathematical skills is offered, with a description of some seminal works up to recent studies and latest findings. The first chapter concludes with a review of studies about the influence of environmental variables. In particular, a review of studies about home numeracy and intergenerational transmission is examined. The first study analyzed the relationship between mathematical skills of children attending primary school and those of their mothers. The objective of this study was to understand the influence of mothers' math abilities on those of their children. In the second study, the relationship between parents’ and children numerical processing has been examined in a sample of preschool children. The goal was to understand how mathematical skills of parents were relevant for the development of the numerical skills of children, taking into account children’s cognitive and linguistic skills as well as the role of home numeracy. The third study had the objective of investigating whether the verbal and nonverbal cognitive skills presumed to underlie arithmetic are also related to reading. Primary school children were administered measures of reading and arithmetic to understand the relationships between these two abilities and testing for possible shared cognitive markers. Finally, in the general discussion a summary of main findings across the study is presented, together with clinical and theoretical implications.

The spatial association of numbers and mathematical symbols

The "SNARC effect" refers to the finding that people respond faster to small numbers with the left hand and to large numbers with the right hand. This effect is often explained by hypothesizing that numbers are represented from left to right in ascending order (Mental Number Line). However, the SNARC effect may not depend on quantitative information, but on other factors such as the order in which numbers are often represented from left to right in our culture. Four experiments were performed to test this hypothesis. In the first experiment, the concept of spatial association was extended to nonnumeric mathematical symbols: the minus and plus symbols. These symbols were presented as fixation points in a spatial compatibility paradigm. The results demonstrated an opposite influence of the two symbols on the target stimulus: the minus symbol tends to favor the target presented on the left, while the plus symbol the target presented on the right, demonstrating that spatial association can emerge in the absence of a numerical context. In the last three experiments, the relationship between quantity and order was evaluated using normal numbers and mirror numbers. Although mirror numbers denote quantity, they are not encountered in a left-to-right spatial organization. In Experiments 1 and 2, participants performed a magnitude classification task with mirror and normal numbers presented together (Experiment 1) or separately (Experiment 2). In Experiment 3, participants performed a new task in which quantity information processing was not required: the mirror judgment task. The results show that participants access the quantity of both normal and mirror numbers, but only the normal numbers are spatially organized from left to right. In addition, the physical similarity between the numbers, used as a predictor variable in the last three experiments, showed that the physical characteristics of numbers influenced participants' reaction times.

Mathematical methods to analyze and interpret calcium signals of astrocytes

Astrocytes are the most numerous glial cell type in the mammalian brain and permeate the entire CNS interacting with neurons, vasculature, and other glial cells. Astrocytes display intracellular calcium signals that encode information about local synaptic function, distributed network activity, and high-level cognitive functions. Several studies have investigated the calcium dynamics of astrocytes in sensory areas and have shown that these cells can encode sensory stimuli. Nevertheless, only recently the neuro-scientific community has focused its attention on the role and functions of astrocytes in associative areas such as the hippocampus. In our first study, we used the information theory formalism to show that astrocytes in the CA1 area of the hippocampus recorded with 2-photon fluorescence microscopy during spatial navigation encode spatial information that is complementary and synergistic to information encoded by nearby "place cell" neurons. In our second study, we investigated various computational aspects of applying the information theory formalism to astrocytic calcium data. For this reason, we generated realistic simulations of calcium signals in astrocytes to determine optimal hyperparameters and procedures of information measures and applied them to real astrocytic calcium imaging data. Calcium signals of astrocytes are characterized by complex spatiotemporal dynamics occurring in subcellular parcels of the astrocytic domain which makes studying these cells in 2-photon calcium imaging recordings difficult. However, current analytical tools which identify the astrocytic subcellular regions are time consuming and extensively rely on user-defined parameters. Here, we present Rapid Astrocytic calcium Spatio-Temporal Analysis (RASTA), a novel machine learning algorithm for spatiotemporal semantic segmentation of 2-photon calcium imaging recordings of astrocytes which operates without human intervention. We found that RASTA provided fast and accurate identification of astrocytic cell somata, processes, and cellular domains, extracting calcium signals from identified regions of interest across individual cells and populations of hundreds of astrocytes recorded in awake mice.