The Synaptic Weight Matrix: Dynamics, Symmetry Breaking, and Disorder

A key role in simplified models of neural circuitry (Wilson and Cowan, 1972) is played by the matrix of synaptic weights, also called connectivity matrix, whose elements describe the amount of influence the firing of one neuron has on another. Biologically, this matrix evolves over time whether or not sensory inputs are present, and symmetries possessed by the internal dynamics of the network may break up spontaneously, as found in the development of the visual cortex (Hubel and Wiesel, 1977). In this thesis, a full analytical treatment is provided for the simplest case of such a biological phenomenon, a single dendritic arbor driven by correlation-based dynamics (Linsker, 1988). Borrowing methods from the theory of Schrödinger operators, a complete study of the model is performed, analytically describing the break-up of rotational symmetry that leads to the functional specialization of cells. The structure of the eigenfunctions is calculated, lower bounds are derived on the critical value of the correlation length, and explicit expressions are obtained for the long-term profile of receptive fields, i.e. the dependence of neural activity on external inputs. The emergence of a functional architecture of orientation preferences in the cortex is another crucial feature of visual information processing. This is discussed through a model consisting of large neural layers connected by an infinite number of Hebb-evolved arbors. Ohshiro and Weiliky (2006), in their study of developing ferrets, found correlation profiles of neural activity in contradiction with previous theories of the phenomenon (Miller, 1994; Wimbauer, 1998). The theory proposed herein, based upon the type of correlations they measured, leads to the emergence of three different symmetry classes. The contours of a highly structured phase diagram are traced analytically, and observables concerning the various phases are estimated in every phase by means of perturbative, asymptotic and variational methods. The proper modeling of axonal competition proves to be key to reproducing basic phenomenological features. While these models describe the long-term effect of synaptic plasticity, plasticity itself makes the connectivity matrix highly dependent on particular histories, hence its stochasticity cannot be considered perturbatively. The problem is tackled by carrying out a detailed treatment of the spectral properties of synaptic-weight matrices with an arbitrary distribution of disorder. Results include a proof of the asymptotic compactness of random spectra, calculations of the shape of supports and of the density profiles, a fresh analysis of the problem of spectral outliers, a study of the link between eigenvalue density and the pseudospectrum of the mean connectivity, and applications of these general results to a variety of biologically relevant examples. The strong non-normality of synaptic-weight matrices (biologically engineered through Dale’s law) is believed to play important functional roles in cortical operations (Murphy and Miller, 2009; Goldman, 2009). Accordingly, a comprehensive study is dedicated to its effect on the transient dynamics of large disordered networks. This is done by adapting standard field-theoretical methods (such as the summation of ladder diagrams) to the non-Hermitian case. Transient amplification of activity can be measured from the average norm squared; this is calculated explicitly for a number of cases, showing that transients are typically amplified by disorder. Explicit expressions for the power spectrum of response are derived and applied to a number of biologically plausible networks, yielding insights into the interplay between disorder and non-normality. The fluctuations of the covariance of noisy neural activity are also briefly discussed. Recent optogenetic measurements have raised questions on the link between synaptic structure and the response of disordered networks to targeted perturbations. Answering to these developments, formulae are derived that establish the operational regime of networks through their response to specific perturbations, and a minimal threshold is found to exist for counterintuitive responses of an inhibitory-stabilized circuit such as have been reported in Ozeki et al. (2016), Adesnik (2016), Kato et al. (2017). Experimental advances are also bringing to light unsuspected differences between various neuron types, which appear to translate into different roles in network function (Pfeffer et al., 2013; Tremblay et al., 2016). Accordingly, the last part of the thesis focuses on networks with an arbitrary number of neuronal types, and predictions are provided for the response of networks with a multipopulation structure to targeted input perturbations

Augmentation of Brain Function: Facts, Fiction and Controversy. Volume III: From Clinical Applications to Ethical Issues and Futuristic Ideas

The final volume in this tripartite series on Brain Augmentation is entitled “From Clinical Applications to Ethical Issues and Futuristic Ideas”. Many of the articles within this volume deal with translational efforts taking the results of experiments on laboratory animals and applying them to humans. In many cases, these interventions are intended to help people with disabilities in such a way so as to either restore or extend brain function. Traditionally, therapies in brain augmentation have included electrical and pharmacological techniques. In contrast, some of the techniques discussed in this volume add specificity by targeting select neural populations. This approach opens the door to where and how to promote the best interventions. Along the way, results have empowered the medical profession by expanding their understanding of brain function. Articles in this volume relate novel clinical solutions for a host of neurological and psychiatric conditions such as stroke, Parkinson’s disease, Huntington’s disease, epilepsy, dementia, Alzheimer’s disease, autism spectrum disorders (ASD), traumatic brain injury, and disorders of consciousness. In disease, symptoms and signs denote a departure from normal function. Brain augmentation has now been used to target both the core symptoms that provide specificity in the diagnosis of a disease, as well as other constitutional symptoms that may greatly handicap the individual. The volume provides a report on the use of repetitive transcranial magnetic stimulation (rTMS) in ASD with reported improvements of core deficits (i.e., executive functions). TMS in this regard departs from the present-day trend towards symptomatic treatment that leaves unaltered the root cause of the condition. In diseases, such as schizophrenia, brain augmentation approaches hold promise to avoid lengthy pharmacological interventions that are usually riddled with side effects or those with limiting returns as in the case of Parkinson’s disease. Brain stimulation can also be used to treat auditory verbal hallucination, visuospatial (hemispatial) neglect, and pain in patients suffering from multiple sclerosis. The brain acts as a telecommunication transceiver wherein different bandwidth of frequencies (brainwave oscillations) transmit information. Their baseline levels correlate with certain behavioral states. The proper integration of brain oscillations provides for the phenomenon of binding and central coherence. Brain augmentation may foster the normalization of brain oscillations in nervous system disorders. These techniques hold the promise of being applied remotely (under the supervision of medical personnel), thus overcoming the obstacle of travel in order to obtain healthcare. At present, traditional thinking would argue the possibility of synergism among different modalities of brain augmentation as a way of increasing their overall effectiveness and improving therapeutic selectivity. Thinking outside of the box would also provide for the implementation of brain-to-brain interfaces where techniques, proper to artificial intelligence, could allow us to surpass the limits of natural selection or enable communications between several individual brains sharing memories, or even a global brain capable of self-organization. Not all brains are created equal. Brain stimulation studies suggest large individual variability in response that may affect overall recovery/treatment, or modify desired effects of a given intervention. The subject’s age, gender, hormonal levels may affect an individual’s cortical excitability. In addition, this volume discusses the role of social interactions in the operations of augmenting technologies. Finally, augmenting methods could be applied to modulate consciousness, even though its neural mechanisms are poorly understood. Finally, this volume should be taken as a debate on social, moral and ethical issues on neurotechnologies. Brain enhancement may transform the individual into someone or something else. These techniques bypass the usual routes of accommodation to environmental exigencies that exalted our personal fortitude: learning, exercising, and diet. This will allow humans to preselect desired characteristics and realize consequent rewards without having to overcome adversity through more laborious means. The concern is that humans may be playing God, and the possibility of an expanding gap in social equity where brain enhancements may be selectively available to the wealthier individuals. These issues are discussed by a number of articles in this volume. Also discussed are the relationship between the diminishment and enhancement following the application of brain-augmenting technologies, the problem of “mind control” with BMI technologies, free will the duty to use cognitive enhancers in high-responsibility professions, determining the population of people in need of brain enhancement, informed public policy, cognitive biases, and the hype caused by the development of brain- augmenting approaches

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal