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.

The Libor Market Model: from theory to calibration

This thesis is focused on the financial model for interest rates called the LIBOR Market Model. In the appendixes, we provide the necessary mathematical theory. In the inner chapters, firstly, we define the main interest rates and financial instruments concerning with the interest rate models, then, we set the LIBOR market model, demonstrate its existence, derive the dynamics of forward LIBOR rates and justify the pricing of caps according to the Black’s formula. Then, we also present the Swap Market Model, which models the forward swap rates instead of the LIBOR ones. Even this model is justified by a theoretical demonstration and the resulting formula to price the swaptions coincides with the Black’s one. However, the two models are not compatible from a theoretical point. Therefore, we derive various analytical approximating formulae to price the swaptions in the LIBOR market model and we explain how to perform a Monte Carlo simulation. Finally, we present the calibration of the LIBOR market model to the markets of both caps and swaptions, together with various examples of application to the historical correlation matrix and the cascade calibration of the forward volatilities to the matrix of implied swaption volatilities provided by the market.

N-3 fatty acids and cardiovascular prevention

In this study we elucidate the role of polyunsaturated fatty acids (PUFAs) in the prevention of cardiovascular diseases, focusing the attention on their role in the modulation of acyl composition of cell lipids and of gene expression. Regarding this latter mechanism, the effectiveness of PUFAs as activators of two transcriptional factors, SREBPs and PPARs, have been considered. Two different model system have been used: primary cultures of neonatal rat cardiomyocytes and an human hepatoma cell line (HepG2). Cells have been supplemented with different PUFAs at physiological concentration, and special attention has been devoted to the main n-3 PUFAs, EPA and DHA. PUFAs influence on global gene expression in cardiomyocytes has been evaluated using microarray technique. Furthermore, since it is not fully elucidated which transcription factors are involved in this modulation in the heart, expression and activation of the three different PPAR isoforms have been investigated. Hepatocytes have been used as experimental model system in the evaluation of PUFAs effect on SREBP activity. SREBPs are considered the main regulator of cholesterol and triglyceride synthesis, which occur mainly in the liver. In both experimental models the modification of cell lipid fatty acid composition subsequent to PUFAs supplementation has been evaluated, and related to the effects observed at molecular level. The global vision given by the obtained results may be important for addressing new researches and be useful to educators and policy makers in setting recommendations for reaching optimal health through good nutrition.

Classical limit of the Nelson model

Since the development of quantum mechanics it has been natural to analyze the connection between classical and quantum mechanical descriptions of physical systems. In particular one should expect that in some sense when quantum mechanical effects becomes negligible the system will behave like it is dictated by classical mechanics. One famous relation between classical and quantum theory is due to Ehrenfest. This result was later developed and put on firm mathematical foundations by Hepp. He proved that matrix elements of bounded functions of quantum observables between suitable coherents states (that depend on Planck's constant h) converge to classical values evolving according to the expected classical equations when h goes to zero. His results were later generalized by Ginibre and Velo to bosonic systems with infinite degrees of freedom and scattering theory. In this thesis we study the classical limit of Nelson model, that describes non relativistic particles, whose evolution is dictated by Schrödinger equation, interacting with a scalar relativistic field, whose evolution is dictated by Klein-Gordon equation, by means of a Yukawa-type potential. The classical limit is a mean field and weak coupling limit. We proved that the transition amplitude of a creation or annihilation operator, between suitable coherent states, converges in the classical limit to the solution of the system of differential equations that describes the classical evolution of the theory. The quantum evolution operator converges to the evolution operator of fluctuations around the classical solution. Transition amplitudes of normal ordered products of creation and annihilation operators between coherent states converge to suitable products of the classical solutions. Transition amplitudes of normal ordered products of creation and annihilation operators between fixed particle states converge to an average of products of classical solutions, corresponding to different initial conditions.

Regulation of wake-sleep states and state-dependent cardiovascular function in diet-induced obesity rats

Obesity often predisposes to coronary heart disease, heart failure, and sudden death. Also, several studies suggest a reciprocal enhancing interaction between obesity and sleep curtailment. Aim of the present study was to go deeper in the understanding of sleep and cardiovascular regulation in an animal model of diet-induced obesity (DIO). According to this, Wake-Sleep (W-S) regulation, and W-S dependent regulation of cardiovascular and metabolic/thermoregulatory function was studied in DIO rats, under normal laboratory conditions and during sleep deprivation and the following recovery period, enhancing either wake or sleep, respectively. After 8 weeks of the delivery of a hypercaloric (HC) diet, treated animals were heavier than those fed a normocaloric (NC) diet (NC: 441 ±17g; HC: 557±17g). HC rats slept more than NC ones during the activity period (Dark) of the normal 12h:12h light-dark (LD) cycle (Wake: 67.3±1.2% and 57.2 ±1.6%; NREM sleep (NREMS): 26.8±1.0% and 34.0±1.4%; REM sleep (REMS): 5.7±0. 6% and 8.6±0.7%; for NC and HC, respectively; p<0.05 for all). HC rats were hypertensive throughout the W-S states, as shown by the mean arterial blood pressure values across the 24-h period (Wake: 90.0±5.3 and 97.3±1.3; NREMS: 85.1±5.5 and 92.2±1.2; REMS: 87.2±4.5 and 96.5±1.1, mmHg for NC and HC, respectively; p<0.05 for all). Also, HC rats appeared to be slightly bradycardic compared to NC ones (Wake: 359.8±9.3 and 352.4±7.7; NREMS: 332.5±10.1 and 328.9±5.4; REMS: 338.5±9.3 and 334.4±5.8; bpm for NC and HC, respectively; p<0.05 for Wake). In HC animals, sleep regulation was not apparently altered during the sleep rebound observed in the recovery period following sleep deprivation, although REMS rebound appeared to be quicker in NC animals. In conclusion, these results indicate that in the rat obesity interfere with W-S and cardiovascular regulation and that DIO rats are suitable for further studies aimed at a better understanding of obesity comorbidities.

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.

Periodicities in the Hodgkin-Huxley model and versions of this model with stochastic input

A field of computational neuroscience develops mathematical models to describe neuronal systems. The aim is to better understand the nervous system. Historically, the integrate-and-fire model, developed by Lapique in 1907, was the first model describing a neuron. In 1952 Hodgkin and Huxley [8] described the so called Hodgkin-Huxley model in the article “A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve”. The Hodgkin-Huxley model is one of the most successful and widely-used biological neuron models. Based on experimental data from the squid giant axon, Hodgkin and Huxley developed their mathematical model as a four-dimensional system of first-order ordinary differential equations. One of these equations characterizes the membrane potential as a process in time, whereas the other three equations depict the opening and closing state of sodium and potassium ion channels. The membrane potential is proportional to the sum of ionic current flowing across the membrane and an externally applied current. For various types of external input the membrane potential behaves differently. This thesis considers the following three types of input: (i) Rinzel and Miller [15] calculated an interval of amplitudes for a constant applied current, where the membrane potential is repetitively spiking; (ii) Aihara, Matsumoto and Ikegaya [1] said that dependent on the amplitude and the frequency of a periodic applied current the membrane potential responds periodically; (iii) Izhikevich [12] stated that brief pulses of positive and negative current with different amplitudes and frequencies can lead to a periodic response of the membrane potential. In chapter 1 the Hodgkin-Huxley model is introduced according to Izhikevich [12]. Besides the definition of the model, several biological and physiological notes are made, and further concepts are described by examples. Moreover, the numerical methods to solve the equations of the Hodgkin-Huxley model are presented which were used for the computer simulations in chapter 2 and chapter 3. In chapter 2 the statements for the three different inputs (i), (ii) and (iii) will be verified, and periodic behavior for the inputs (ii) and (iii) will be investigated. In chapter 3 the inputs are embedded in an Ornstein-Uhlenbeck process to see the influence of noise on the results of chapter 2.

The porcine animal model goes through the 3Rs: development of in vitro and ex vivo system to study vascular biology

Since the publication of the book of Russell and Burch in 1959, scientific research has never stopped improving itself with regard to the important issue of animal experimentation. The European Directive 2010/63/EU “On the protection of animals used for scientific purposes” focuses mainly on the animal welfare, fixing the Russell and Burch’s 3Rs principles as the foundations of the document. In particular, the legislator clearly states the responsibility of the scientific community to improve the number of alternative methods to animal experimentation. The swine is considered a species of relevant interest for translational research and medicine due to its biological similarities with humans. The surgical community has, in fact, recognized the swine as an excellent model replicating the human cardiovascular system. There have been several wild-type and transgenic porcine models which were produced for biomedicine and translational research. Among these, the cardiovascular ones are the most represented. The continuous involvement of the porcine animal model in the biomedical research, as the continuous advances achieved using swine in translational medicine, support the need for alternative methods to animal experimentation involving pigs. The main purpose of the present work was to develop and characterize novel porcine alternative methods for cardiovascular translational biology/medicine. The work was mainly based on two different models: the first consisted in an ex vivo culture of porcine aortic cylinders and the second consisted in an in vitro culture of porcine aortic derived progenitor cells. Both the models were properly characterized and results indicated that they could be useful to the study of vascular biology. Nevertheless, both the models aim to reduce the use of experimental animals and to refine animal based-trials. In conclusion, the present research aims to be a small, but significant, contribution to the important and necessary field of study of alternative methods to animal experimentation.

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.

Analysis and numerical solution of the Peterlin viscoelastic model

Liquids and gasses form a vital part of nature. Many of these are complex fluids with non-Newtonian behaviour. We introduce a mathematical model describing the unsteady motion of an incompressible polymeric fluid. Each polymer molecule is treated as two beads connected by a spring. For the nonlinear spring force it is not possible to obtain a closed system of equations, unless we approximate the force law. The Peterlin approximation replaces the length of the spring by the length of the average spring. Consequently, the macroscopic dumbbell-based model for dilute polymer solutions is obtained. The model consists of the conservation of mass and momentum and time evolution of the symmetric positive definite conformation tensor, where the diffusive effects are taken into account. In two space dimensions we prove global in time existence of weak solutions. Assuming more regular data we show higher regularity and consequently uniqueness of the weak solution. For the Oseen-type Peterlin model we propose a linear pressure-stabilized characteristics finite element scheme. We derive the corresponding error estimates and we prove, for linear finite elements, the optimal first order accuracy. Theoretical error of the pressure-stabilized characteristic finite element scheme is confirmed by a series of numerical experiments.

Phage therapy: A software system for phage quantification and kinetic model inference.

One of the most serious problems of the modern medicine is the growing emergence of antibiotic resistance among pathogenic bacteria. In this circumstance, different and innovative approaches for treating infections caused by multidrug-resistant bacteria are imperatively required. Bacteriophage Therapy is one among the fascinating approaches to be taken into account. This consists of the use of bacteriophages, viruses that infect bacteria, in order to defeat specific bacterial pathogens. Phage therapy is not an innovative idea, indeed, it was widely used around the world in the 1930s and 1940s, in order to treat various infection diseases, and it is still used in Eastern Europe and the former Soviet Union. Nevertheless, Western scientists mostly lost interest in further use and study of phage therapy and abandoned it after the discovery and the spread of antibiotics. The advancement of scientific knowledge of the last years, together with the encouraging results from recent animal studies using phages to treat bacterial infections, and above all the urgent need for novel and effective antimicrobials, have given a prompt for additional rigorous researches in this field. In particular, in the laboratory of synthetic biology of the department of Life Sciences at the University of Warwick, a novel approach was adopted, starting from the original concept of phage therapy, in order to study a concrete alternative to antibiotics. The innovative idea of the project consists in the development of experimental methodologies, which allow to engineer a programmable synthetic phage system using a combination of directed evolution, automation and microfluidics. The main aim is to make “the therapeutics of tomorrow individualized, specific, and self-regulated” (Jaramillo, 2015). In this context, one of the most important key points is the Bacteriophage Quantification. Therefore, in this research work, a mathematical model describing complex dynamics occurring in biological systems involving continuous growth of bacteriophages, modulated by the performance of the host organisms, was implemented as algorithms into a working software using MATLAB. The developed program is able to predict different unknown concentrations of phages much faster than the classical overnight Plaque Assay. What is more, it gives a meaning and an explanation to the obtained data, making inference about the parameter set of the model, that are representative of the bacteriophage-host interaction.

A 2-dimensional computational model to analyze the effects of cellular heterogeinity on cardiac pacemaking

The mechanical action of the heart is made possible in response to electrical events that involve the cardiac cells, a property that classifies the heart tissue between the excitable tissues. At the cellular level, the electrical event is the signal that triggers the mechanical contraction, inducing a transient increase in intracellular calcium which, in turn, carries the message of contraction to the contractile proteins of the cell. The primary goal of my project was to implement in CUDA (Compute Unified Device Architecture, an hardware architecture for parallel processing created by NVIDIA) a tissue model of the rabbit sinoatrial node to evaluate the heterogeneity of its structure and how that variability influences the behavior of the cells. In particular, each cell has an intrinsic discharge frequency, thus different from that of every other cell of the tissue and it is interesting to study the process of synchronization of the cells and look at the value of the last discharge frequency if they synchronized.

High altitude, a natural research laboratory for the study of cardiovascular physiology and pathophysiology

High altitude constitutes an exciting natural laboratory for medical research. Although initially, the aim of high-altitude research was to understand the adaption of the organism to hypoxia and find treatments for altitude-related diseases, during the past decade or so, the scope of this research has broadened considerably. Two important observations led the foundation for the broadening of the scientific scope of high-altitude research. First, high-altitude pulmonary edema represents a unique model that allows studying fundamental mechanisms of pulmonary hypertension and lung edema in humans. Second, the ambient hypoxia associated with high-altitude exposure facilitates the detection of pulmonary and systemic vascular dysfunction at an early stage. Here, we will review studies that, by capitalizing on these observations, have led to the description of novel mechanisms underpinning lung edema and pulmonary hypertension and to the first direct demonstration of fetal programming of vascular dysfunction in humans.

Physical coronary arteriogenesis: a human "model" of collateral growth promotion

In patients with coronary artery disease, the size of myocardial infarction mainly determines the subsequent clinical outcome. Accordingly, it is the primary strategy to decrease cardiovascular mortality by minimizing infarct size. Promotion of collateral artery growth (arteriogenesis) is an appealing option of reducing infarct size. It has been demonstrated in experimental models that tangential fluid shear stress is the major trigger of arterial remodeling and, thus, of collateral growth. Lower-leg, high-pressure external counterpulsation triggered to occur during diastole induces a flow velocity signal and thus tangential endothelial shear stress in addition to the flow signal caused by cardiac stroke volume. We here present two cases of cardiac transplant recipients as human "models" of physical coronary arteriogenesis, providing an example of progressing and regressing clinical arteriogenesis, and review available evidence from clinical studies on other feasible forms of physical arteriogenesis.

Murine model of surgically induced acute aortic dissection type A

This study aimed at developing a murine model of surgically induced acute aortic dissection type A for investigation of the formation and progression of acute aortic dissection and to test whether this system could be used for biomarker discovery.