The hippocampus is a brain structure involved in episodic memory and spatial cognition. Neuronal activity within the hippocampus exhibits intricate temporal patterning, including oscillatory and sequential dynamics, which are believed to underlie these cognitive processes. In individual cells, a temporal activity pattern called phase precession occurs which leads to the organisation of neuronal populations into sequences. These sequences are hypothesised to form a substrate for episodic memory and the representation of spatial trajectories during navigation. In this thesis, I present a novel theory of the phenomenological properties of these neuronal activity sequences. In particular, I propose that the sequential organisation of population activity is governed by the independent phase precession of each cell. By comparison of models in which cells are independent and models in which cells exhibit coordinated activity against experimental data, I provide empirical evidence to support this hypothesis. Further, I show how independent coding affords a vast capacity for the generation of sequential activity patterns across distinct environments, allowing the representation of episodes and spatial experiences across a large number of contexts. This theory is then extended to account for grid cells, whose activity patterns form a hexagonal lattice over external space. By analysing simple forms of phase coding in populations of grid cells, I show how previously undocumented constraints on phase coding in two dimensional environments are imposed by the symmetries of grid cell firing fields. To overcome these constraints, I propose a more complex phenomenological model which can account for phase precession in both place cells and grid cells in two dimensional environments. Using insights from this theory, I then propose a biophysical circuit mechanism for hippocampal sequences. I show that this biophysical circuit model can account for the proposed phenomenological coding properties and provide experimentally testable predictions which can distinguish this model from existing models of phase precession. Finally, I outline a scheme by which this biophysical mechanism can implement supervised learning using spike time dependent plasticity in order to learn associations between events occurring on behavioural timescales. The models presented in this thesis challenge previous theories of hippocampal circuit function and suggest a much higher degree of flexibility and capacity for the generation of sequences than previously believed. This flexibility may underlie our ability to represent spatial experiences and store episodic memories across a seemingly unlimited number of distinct contexts.

This thesis studies non-simple Jacobians and non-simple abelian varieties. The moti- vation of the study is a construction which gives a distinguished genus 4 curve in the linear system of a (1, 3)-polarised surface. The main theorem characterises such curves as hyperelliptic genus 4 curves whose Jacobian contains a (1, 3)-polarised surface. This leads to investigating the locus of non-simple principally polarised abelian g- folds. The main theorem of this part shows that the irreducible components of this locus are Is~, defined as the locus of principally polarised g-folds having an abelian subvariety with induced polarisation of type d. = (d1, ... , dk), where k ≤ g/2 Moreover, there are theorems which characterise the Jacobians of curves that are etale double covers or double covers branched in two points. There is also a detailed computation showing that, for p > 1 an odd number, the hyperelliptic locus meets IS4(l,p) transversely in the Siegel upper half space

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People with schizophrenia are known to have difficulties with episodic memory (EM). The purpose of this investigation was to examine the relationship between theta-power and: i) behavioural measures of EM performance, ii) event- related potential (ERP) indices of recollection and, iii) measures of schizophrenia symptomatology. In doing so, the aim was to gain a better understanding of the basic neural mechanisms that contribute to successful EM performance, and how these may differ for people with schizophrenia. The present investigation adopted an endophenotypic approach and collected measures of schizotypy from student participants to minimise patient factors that can confound interpretations. Fifty- four participants were asked to complete a reality-monitoring exclusion EM paradigm whilst electroencephalogram (EEG) data were collected. Measures of theta-power and ERPs were time-locked to words presented during the retrieval phase. There was a significant positive correlation between theta-power over Fz between 600-1000ms post-stimulus presentation and estimates of recollection in the imagine condition as well as a significant negative correlation between these measures of theta-power for perceive items and ERP indices of recollection for imagine items. There was also a significant positive correlation between measures of frontal theta-power in the imagine condition and negative schizotypy. The epoch employed means it is likely these measures of theta- power reflect processes contributing to the content-specific retrieval of imagined items, and post-retrieval processes acting in service of differentiating imagined items in EM. Results are discussed in terms of suggestions for interventions and directions for future research.

Damage precipitating cognitive decline in Alzheimer's disease (AD) begins long before behavioural alterations become clinically apparent. At this prodromal stage, communication between networks of neurons connecting different brain regions starts to break down; setting in motion a chain of events leading to clinical AD. A significant challenge facing Alzheimer's researchers today is finding a cheap, easy-to-perform test capable of detecting prodromal AD. Such a test would afford significant benefits to patients, including a chance of early intervention. Perhaps, more importantly, it would also aid development and testing of novel therapies aimed at combating AD before it causes irreversible damage. Since oscillations in electrical field activity are important for facilitating connectivity across the brain and have been seen to alter in AD, this work studied how oscillations and regional connectivity are affected in the AD brain. Specifically, local field oscillations were recorded from the hippocampus and prelimbic cortex (regions implicated in memory formation and maintenance) in a double transgenic AD model - the TASTPM mouse. Here, periods of predominant theta activity were assessed both spontaneously, under urethane anaesthesia and following electrical induction through dorsal periaqueductal gray (dPAG) stimulation. From these recordings, spectral power and connectivity between regions was assessed using both a traditional measure of functional connectivity (inter-region correlation) and through a novel information theoretic approach measuring effective connectivity (transfer entropy).Perhaps the most prominent finding from this study was the observation that young TASTPM mice, at an age prior to overt cognitive decline or plaque deposition, showed significant alterations in measures of both functional and effective connectivity. This suggests that such measures may be used as biomarkers predictive of prodromal AD and, as such, may be used to aid development of drugs targeted towards treatment of prodromal AD.This study also uncovered a number of interesting observations concerning hippocampal/prelimbic connectivity. Firstly, although spectral power and inter-regional correlation peaked at ∼ 3Hz, information flow between these structures was strongest at ∼6Hz. This suggests that low and high-band theta activity may fulfil separate functions. Secondly, at theta frequencies, information flowed predominantly from the prelimbic cortex to the hippocampus. However, during lower frequency activity, information flowed predominantly in the opposite direction. Suggesting that separate frequency bands may be important for routing information flow between these structures. Finally, the strength of information transfer was seen to oscillate at approximately double the frequency of its carrier signal, perhaps suggesting locking of information transfer to certain phases of an underlying oscillation. Therefore, oscillations may structure information transfer by temporal windowing and frequency-locked routing; processes which can be studied using measures of effective connectivity such as transfer entropy.

Neste trabalho, com base em representações por matrizes de duas linhas para alguns tipos de partição (algumas já conhecidas e outras novas), identificamos propriedades sugeridas por classificá-las de acordo com a soma dos elementos de sua segunda linha. Esta soma sempre fornece alguma propriedade da partição relacionada. Se considerarmos versões sem sinal de algumas funções Mock Theta, seu termo geral pode ser interpretado como função geradora para algum tipo de partição com restrições. Para retornar aos coeficientes originais, é possível definir um peso para cada matriz e depois somá-las para contá-los. Uma representação análoga para essas partições nos permite observar propriedades sobre elas, novamente por meio de uma classificação referente à soma dos seu elementos da segunda linha. Esta seriação é feita por meio de tabelas criadas pelo software matemático Maple, as quais nos sugerem padrões e identidades relacionadas com outros tipos de partições conhecidas e, muitas vezes, encontrando uma fórmula fechada para contá-las. Tendo as conjecturas obtidas, elas são provadas por meio de bijeções entre conjuntos ou por contagem. / In this work, based on representations by matrices of two lines for some kind of partition (some already known and other new ones), we identify properties suggested by classifying them according to the sum of its second line. This sum always provides some properties of the related partition. If we consider unsigned versions of some Mock Theta Functions, its general term can be interpreted as generating function for some kind of partition with restrictions. To come back to the original coefficients, you can set a weight for each array and so add them to evaluate the coefficients. An analogous representation for partitions allows us to observe properties, again by classificating them according to the sum of its elements on the second row. This classification is made by means of tables created by mathematical software Maple, which suggest patterns, identities related to other known types of partitions and often, finding a closed formula to count them. Having established conjectured identities, all are proved by bijections between sets or counting methods.

O objetivo principal deste trabalho e descrever os números inteiros que podem ser representados nas formas 9x2+16y2+36z2+16yz+4xz+8xy e 9x2+17y2+ 32z2 - 8yz + 8xz + 6xy. Para isso, utilizamos uma série de resultados envolvendo funções theta, como a identidade do produto triplo de Jacobi e equações modulares. / The main goal of this work is to describe the integers which can be written in the forms 9x2 + 16y2 + 36z2 + 16yz + 4xz + 8xy and 9x2 + 17y2 + 32z2 - 8yz + 8xz + 6xy. To do so, we use a series of results concerning theta functions, such as the Jacobi triple product identity and modular equations.

acase@tulane.edu / Modular-type transformation formulas are the identities that are invariant under the transformation α → 1/α, and they can be represented as F (α) = F (β) where α β = 1. We derive a new transformation formula of the form F (α, z, w) = F (β, z, iw) that is a one-variable generalization of the well-known Ramanujan-Guinand identity of the form F (α, z) = F (β, z) and a two-variable generalization of Koshliakov’s formula of the form F (α) = F (β) where α β = 1. The formula is generated by first finding an integral J that is comprised of an invariance function Z and evaluating the integral to give F (α, z, w) mentioned above. The modified Bessel function K z (x) appearing in Ramanujan-Guinand identity is generalized to a new function, denoted as K z,w (x), that yields a pair of functions reciprocal in the Koshliakov kernel, which in turn yields the invariance function Z and hence the integral J and the new formula. The special function K z,w (x), first defined as the inverse Mellin transform of a product of two gamma functions and two confluent hypergeometric functions, is shown to exhibit a rich theory as evidenced by a number of integral and series representations as well as a differential-difference equation. The second topic of the thesis is 2-adic valuations of integer sequences associated with quadratic polynomials of the form x 2 +a. The sequence {n 2 +a : n ∈ Z} contains numbers divisible by any power of 2 if and only if a is of the form 4 m (8l+7). Applying this result to the sequences derived from the sums of four or fewer squares when one or more of the squares are kept constant leads to interesting results, that also points to an inherent connection with the functions r k (n) that count the number of ways to represent n as sums of k integer squares. Another class of sequences studied is the shifted sequences of the polygonal numbers given by the quadratic formula, for which the most common examples are the triangular numbers and the squares. / 1 / Aashita Kesarwani

Let V( ) denote a local system of weight on X = A2;n(C), where X is the moduli space of principle polarized abelian varieties of genus 2 over C with xed n-level structure. The inner cohomology of X with coe cients in V( ), H3 ! (X;V( )), has a Hodge ltration of weight 3. Each term of this Hodge ltration can be presented as space of cuspidal automorphic representations of genus 2. We consider the purely non-holomorphic part of H3 ! (X;V( )) denoted by H3 Ends(X;V( )). First of all we show that there is a non-zero subspace of H3 Ends(X;V( )) denoted by V (K), where K is an open compact subgroup of GSp(4;A), such that elements of V (K) are obtained by the global theta lifting of cuspidal automorphic representations of GL(2) GL(2)=Gm. This means that there is a non-zero part of H3 Ends(X;V( )) which is endoscopic. Secondly, we consider the local theta correspondence and nd an explicit answer for the level of lifted cuspidal automorphic representations to GSp(4; F) over a non-archimedean local eld F. Therefore, we can present an explicit way for nding a basis for V (K) for a xed level structure K. ii There is a part of the Hodge structure that only contributes in H(3;0) ! (X;V( )) H(0;3) ! (X;V( )). This part is endoscopic and coming from the Yoshida lift from O(4). Finally, in the case X = A2, if eendo(A2;V( )) denotes the motive corresponded to the strict endoscopic part (the part that contributes only in non-holomorphic terms of the Hodge ltration), then we have eendo(A2;V( )) = s 1+ 2+4S[ 1 2 + 2]L 2+1; (1) where = ( 1; 2) and is far from walls. Here S[k] denotes the motive corresponded to Sk, the space of cuspidal automorphic forms of weight k and trivial level, and sk = dim(Sk). ii

Let V( ) denote a local system of weight on X = A2;n(C), where X is the moduli space of principle polarized abelian varieties of genus 2 over C with xed n-level structure. The inner cohomology of X with coe cients in V( ), H3 ! (X;V( )), has a Hodge ltration of weight 3. Each term of this Hodge ltration can be presented as space of cuspidal automorphic representations of genus 2. We consider the purely non-holomorphic part of H3 ! (X;V( )) denoted by H3 Ends(X;V( )). First of all we show that there is a non-zero subspace of H3 Ends(X;V( )) denoted by V (K), where K is an open compact subgroup of GSp(4;A), such that elements of V (K) are obtained by the global theta lifting of cuspidal automorphic representations of GL(2) GL(2)=Gm. This means that there is a non-zero part of H3 Ends(X;V( )) which is endoscopic. Secondly, we consider the local theta correspondence and nd an explicit answer for the level of lifted cuspidal automorphic representations to GSp(4; F) over a non-archimedean local eld F. Therefore, we can present an explicit way for nding a basis for V (K) for a xed level structure K. ii There is a part of the Hodge structure that only contributes in H(3;0) ! (X;V( )) H(0;3) ! (X;V( )). This part is endoscopic and coming from the Yoshida lift from O(4). Finally, in the case X = A2, if eendo(A2;V( )) denotes the motive corresponded to the strict endoscopic part (the part that contributes only in non-holomorphic terms of the Hodge ltration), then we have eendo(A2;V( )) = s 1+ 2+4S[ 1 2 + 2]L 2+1; (1) where = ( 1; 2) and is far from walls. Here S[k] denotes the motive corresponded to Sk, the space of cuspidal automorphic forms of weight k and trivial level, and sk = dim(Sk). ii