In this paper, we study the effects of a recent drop in star formation rate (SFR) on the spectra of epoch of reionization (EoR) galaxies, and the resulting degeneracy with the spectral features produced by extreme Lyman continuum leakage. In order to study these effects in the wavelength range relevant for the upcoming James Webb Space Telescope (JWST), we utilize synthetic spectra of simulated EoR galaxies from cosmological simulations together with synthetic spectra of partially quenched mock galaxies. We find that rapid declines in the SFR of EoR galaxies could seriously affect the applicability of methods that utilize the equivalent width of Balmer lines and the ultraviolet spectral slope to assess the escape fraction of EoR galaxies. In order to determine if the aforementioned degeneracy can be avoided by using the overall shape of the spectrum, we generate mock NIRCam observations and utilize a classification algorithm to identify galaxies that have undergone quenching. We find that while there are problematic cases, JWST/NIRCam or NIRSpec should be able to reliably identify galaxies with redshifts z similar to 7 that have experienced a significant decrease in the SFR (by a factor of 10-100) in the past 50-100 Myr with a success rate greater than or similar to 85 per cent. We also find that uncertainties in the dust-reddening effects on EoR galaxies significantly affect the performance of the results of the classification algorithm. We argue that studies that aim to characterize the dust extinction law most representative in the EoR would be extremely useful.

Deep Brain Stimulation (DBS) consists of sending mild electric stimuli to the brain via a chronically implanted lead. The therapy is used to alleviate the symptoms of different neurological diseases, such as Parkinson's Disease. However, its underlying biological mechanism is currently unknown. DBS patients undergo a lengthy trial-and-error procedure in order to tune the stimuli so that the treatment achieves maximal therapeutic benefits while limiting side effects that are often present with large stimulation values.

The present licentiate thesis deals with mathematical modeling for DBS, extending it towards optimization. Mathematical modeling is motivated by the difficulty of obtaining in vivo measurements from the brain, especially in humans. It is expected to facilitate the optimization of the stimuli delivered to the brain and be instrumental in evaluating the performance of novel lead designs. Both topics are discussed in this thesis.

First, an analysis of numerical accuracy is presented in order to verify the DBS models utilized in this study. Then a performance comparison between a state-of-the-art lead and a novel field-steering lead using clinical settings is provided. Afterwards, optimization schemes using intersection of volumes and electric field control are described, together with some simplification tools, in order to speed up the computations involved in the modeling.

Deep Brain Stimulation (DBS) is an established therapy that is predominantly  utilized in treating the symptoms of neurodegenerative diseases such as Parkinson's Disease and Essential Tremor, crippling diseases like Chronic Pain and Epilepsy, and psychiatric diseases such as Schizophrenia and Depression. Due to its invasive nature, DBS is considered as a last resort therapy.DBS is performed by transmitting electric pulses through an electrode implanted in the brain of the patient.

The stimulation is driven by a battery-powered Implanted Pulse Generator. The brain is a very delicate and complex organ and, therefore, accurate positioning the electrode is vital. To achieve a satisfactory therapeutical result, the stimulation targets a certain predefined brain structure that depends on the disease.

The effect of DBS depends on the individual, the chosen stimulating contact(s), and the pulse parameters, i.e. amplitude, frequency, width, and shape. Tuning these parameters to the best effect is currently done by a lengthy trial-and-error process. Insufficient stimulation does not properly alleviate the symptoms of the disease, while overstimulation or stimulation off target is prone to side effects.

This work envisions assisting physicians in DBS therapy by utilizing model-based estimation and optimization, maximizing stimulation of the target and minimizing stimulation in potentially problematic areas of the brain. This work focuses on amplitude and contact selection. Because of inter-patient differences, individualized models based on clinical imaging have to be created. Alternatively, semi-individualized models can be designed using atlases that save time but potentially introduce inaccuracies. Other optimization  applications to DBS are proposed in the thesis, e.g. fault alleviation and electrode design.

Electrical properties of the brain can change over time and alter the stimulation spread. A system identification approach has been proposed to quantify these changes.

The main aim of DBS is to alleviate the symptoms of the disease and quantifying symptoms is important. The ultimate vision of this work is to design a closed-loop system that can deliver optimal stimulation to the brain while automatically adapting to changes in the brain and the severity of symptoms.

Objective: Deep brain stimulation (DBS) consists of delivering electrical stimuli to a brain target via an implanted lead to treat neurological and psychiatric conditions. Individualized stimulation is vital to ensure therapeutic results, since DBS may otherwise become ineffective or cause undesirable side effects. Since the DBS pulse generator is battery-driven, power consumption incurred by the stimulation is important. In this study, target coverage and power consumption are compared over a patient population for clinical and model-based patient-specific settings calculated by constrained optimization.

Approach: Brain models for five patients undergoing bilateral DBS were built. Mathematical optimization of activated tissue volume was utilized to calculate stimuli amplitudes, with and without specifying the volumes, where stimulation was not allowed to avoid side effects. Power consumption was estimated using measured impedance values and battery life under both clinical and optimized settings.

Results: It was observed that clinical settings were generally less aggressive than the ones suggested by unconstrained model-based optimization, especially under asymmetrical stimulation. The DBS settings satisfying the constraints were close to the clinical values.

Significance: The use of mathematical models to suggest optimal patient-specific DBS settings that observe technological and safety constraints can save time in clinical practice. It appears though that the considered safety constraints based on brain anatomy depend on the patient and further research into it is needed. This work highlights the need of specifying the brain volumes to be avoided by stimulation while optimizing the DBS amplitude, in contrast to minimizing general stimuli overspill, and applies the technique to a cohort of patients. It also stresses the importance of considering power consumption in DBS optimization, since it increases with the square of the stimuli amplitude and also critically affects battery life through pulse frequency and duty cycle.

Deep Brain Stimulation (DBS) is a well-established treatment in neurodegenerative diseases, e.g. Parkinson's Disease. It consists of delivering electrical stimuli to a target in the brain via a chronically implanted lead. To expedite the tuning of DBS stimuli to best therapeutical effect, mathematical models have been developed during recent years. The electric field produced by the stimuli in the brain for a given lead position is evaluated by numerically solving a Partial Differential Equation with the medium conductivity as a parameter. The latter is patient- and target-specific but difficult to measure in vivo. Estimating brain tissue conductivity through medical imaging is feasible but time consuming due to registration, segmentation and post-processing. On the other hand, brain atlases are readily available and processed. This study analyzes how alternations in the conductivity due to inter-patient variability or lead position uncertainties affect both the stimulation shape and the activation of a given target. Results suggest that stimulation shapes are similar, with a Dice's Coefficient between 93.2 and 98.8%, with a higher similarity at lower depths. On the other hand, activation shows a significant variation of 17 percentage points, with most of it being at deeper positions as well. It is concluded that, as long as the lead is not too deep, atlases can be used for conductivity maps with acceptable accuracy instead of fully individualized though medical imaging models.

Deep Brain Stimulation (DBS) is an established therapy for treating e.g. Parkinson's disease, essential tremor, as well as epilepsy. In DBS, chronic pulsatile electrical stimulation is administered to a certain target area of the brain through a surgically implanted lead. The stimuli parameters have to be properly tuned in order to achieve therapeutical effect that in most cases is alleviation of motor symptoms. Tuning of DBS currently is a tedious task since it is performed manually by medical personnel in a trial-and-error manner. It can be dramatically improved and expedited by means of recently developed mathematical models together with control and estimation technology. This paper presents a control engineering perspective on DBS, viewing it as a control system for minimizing the severity of the symptoms through coordinated manipulation of the stimuli parameters. The DBS model structure comprises a stimuli model, an activation model, and a symptoms model. Each of those is individualized from patient data obtained through medical imaging, electrical measurements, and objective symptom quantification. The proposed approach is illustrated by simulation and clinical data from an individualized DBS model being developed by the authors.

Deep Brain Stimulation (DBS) is an established treatment, in e.g. Parkinson's Disease, whose underlying biological mechanisms are unknown. In DBS, electrical stimulation is delivered through electrodes surgically implanted into certain regions of the brain of the patient. Mathematical models aiming at a better understanding of DBS and optimization of its therapeutical effect through the simulation of the electrical field propagating in the brain tissue have been developed in the past decade. The contribution of the present study is twofold: First, an analytical approximation of the electric field produced by an emitting contact is suggested and compared to the numerical solution given by a Finite Element Method (FEM) solver. Second, the optimal stimulation settings are evaluated by fitting the field distribution to a target one to control the spread of the stimulation. Optimization results are compared to those of a geometric approach, maximizing the intersection between the target and the activated volume in the brain tissue and reducing the stimulated area beyond said target. Both methods exhibit similar performance with respect to the optimal stimuli, with the electric field control approach being faster and more versatile.

The James Webb Space Telescope (JWST) NIRSpec instrument will allow rest-frame ultraviolet/optical spectroscopy of galaxies in the epoch of reionization (EoR). Some galaxies may exhibit significant leakage of hydrogen-ionizing photons into the intergalactic medium, resulting in faint nebular emission lines. We present a machine learning framework for identifying cases of very high hydrogen-ionizing photon escape from galaxies based on the data quality expected from potential NIRSpec observations of EoR galaxies in lensed fields. We train our algorithm on mock samples of JWST/NIRSpec data for galaxies at redshifts z = 6-10. To make the samples more realistic, we combine synthetic galaxy spectra based on cosmological galaxy simulations with observational noise relevant for z greater than or similar to 6 objects of a brightness similar to EoR galaxy candidates uncovered in Frontier Fields observations of galaxy cluster Abell-2744 and MACS-J0416. We find that ionizing escape fractions (f(esc)) of galaxies brighter than m(AB,1500) approximate to 27 mag may be retrieved with mean absolute error Delta f(esc) approximate to 0.09(0.12) for 24 h (1.5 h) JWST/NIRSpec exposures at resolution R = 100. For 24 h exposure time, even fainter galaxies (m(AB,1500) < 28.5 mag) can be processed with Delta f(esc) approximate to 0.14. This framework simultaneously estimates the redshift of these galaxies with a relative error less than 0.03 for both 24 (m(AB,1500) < 28.5 mag) and 1.5 h (m(AB,1500) < 27 mag) exposure times. We also consider scenarios where just a minor fraction of galaxies attain high f(esc) and present the conditions required for detecting a subpopulation of high-f(esc) galaxies within the data set.

The James Webb Space Telescope (JWST) will observe the rest-frame ultraviolet/optical spectra of galaxies from the epoch of reionization (EoR) in unprecedented detail. While escaping into the intergalactic medium, hydrogen-ionizing (Lyman continuum; LyC) photons from the galaxies will contribute to the bluer end of the UV slope and make nebular emission lines less prominent. We present a method to constrain leakage of the LyC photons using the spectra of high redshift (z greater than or similar to 6) galaxies. We simulate JWST/NIRSpec observations of galaxies at z = 6-9 by matching the fluxes of galaxies observed in the Frontier Fields observations of galaxy cluster MACS-J0416. Our method predicts the escape fraction f(esc) with a mean absolute error Delta f(esc) approximate to 0.14. The method also predicts the redshifts of the galaxies with an error approximate to 0.0003.

Deep brain stimulation (DBS) is a an established therapy in neurological and mental disorders making use of electrical pulses chronically delivered to a certain disease-specific neural target through surgically implanted electrodes. The therapeutical effect of DBS is highly individual and depends on the target coverage by the stimuli and the amount of spill beyond it. This can be suitably formulated as an optimization problem. Since the biological mechanism underlying the DBS therapy is mainly unknown, and due to high inter-patient and intra-patient variability of the DBS effect, a pragmatic approach to the DBS programming is to consider the process as tuning of a control system for the symptoms. Such a technology assumes that the symptoms are accurately quantified. The paper summarizes the progress in the individualized DBS and presents the results of a limited clinical study making use of the proposed DBS programming approach.