Stochastic neural field equations: A rigorous footing

We extend the theory of neural fields which has been developed in a deterministic framework by considering the influence spatio-temporal noise. The outstanding problem that we here address is the development of a theory that gives rigorous meaning to stochastic neural field equations, and conditions ensuring that they are well-posed. Previous investigations in the field of computational and mathematical neuroscience have been numerical for the most part. Such questions have been considered for a long time in the theory of stochastic partial differential equations, where at least two different approaches have been developed, each having its advantages and disadvantages. It turns out that both approaches have also been used in computational and mathematical neuroscience, but with much less emphasis on the underlying theory. We present a review of two existing theories and show how they can be used to put the theory of stochastic neural fields on a rigorous footing. We also provide general conditions on the parameters of the stochastic neural field equations under which we guarantee that these equations are well-posed. In so doing we relate each approach to previous work in computational and mathematical neuroscience. We hope this will provide a reference that will pave the way for future studies (both theoretical and applied) of these equations, where basic questions of existence and uniqueness will no longer be a cause for concern

Global solvability of a networked integrate-and-fire model of McKean-Vlasov type

We here investigate the well-posedness of a networked integrate-and-fire model describing an infinite population of neurons which interact with one another through their common statistical distribution. The interaction is of the self-excitatory type as, at any time, the potential of a neuron increases when some of the others fire: precisely, the kick it receives is proportional to the instantaneous proportion of firing neurons at the same time. From a mathematical point of view, the coefficient of proportionality, denoted by $\alpha$, is of great importance as the resulting system is known to blow-up for large values of $\alpha$. In the current paper, we focus on the complementary regime and prove that existence and uniqueness hold for all time when $\alpha$ is small enough.Comment: Published at http://dx.doi.org/10.1214/14-AAP1044 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Synthesis of conformationally restrained peptides

The synthesis of an artificial amino acid residue, bearing two a-amino acid centres, is detailed. The residue has been designed to act as a conformational restraint when incorporated into peptides. The intended target structural motif is the a-helix, and the restraint takes the form of a macrocycle in a central position in the peptide chain, which is intended to nucleate helix formation. The synthesis has been achieved by the use of two different asymmetric methodologies. Details of the final synthetic route to the residue are included, as well as details of several other synthetic routes which proved unsuccessful. The final route involves the use of an octanoic acid derivative. This is initially reacted with a chiral lithiated pyrazine cyanocuprate complex to generate the R-chiral centre, followed by the introduction of the S-chiral centre using an asymmetric azidation methodology. These reactions have been employed in sequence to give optimum yield and efficiency. The sequence of reaction followed also simplifies the differentiation of the two chiral centres, giving the molecule in a form suitable for solid phase peptide synthesis. The attempted syntheses of peptides bearing this residue is also detailed. This process has been performed by standard Fmoc methodology, using the triply orthogonal allyl based protecting group, cleaved by palladium catalysis, to allow selective reaction to form the macrocycle. This loop is arranged in an i-(i+4) substitution pattern, suggested in the literature to be the most effective spacing for performing this task. Other sections of this thesis describe the general background to helical structure stabilisation, the asymmetric synthesis of amino acids and the solid phase synthesis of peptides

Mean-field limit of a stochastic particle system smoothly interacting through threshold hitting-times and applications to neural networks with dendritic component

In this article we study the convergence of a stochastic particle system that interacts through threshold hitting times towards a novel equation of McKean-Vlasov type. The particle system is motivated by an original model for the behavior of a network of neurons, in which a classical noisy integrate-and-fire model is coupled with a cable equation to describe the dendritic structure of each neuron

The Scottish Churches and the Organ in the Nineteenth Century

"The use of organs in the public worship of God is contrary to the law of the land, and to the law and constitution of our Established Church". This was the opinion of the Presbytery of Glasgow when it condemned the use of an organ in St. Andrew's Church, Glasgow, in 1807, the first use of a musical instrument in public worship by any presbyterian congregation in Scotland. So began the controversy in Scotland about what came to be known as "the organ question", a controversy which continued through the nineteenth century and into the twentieth. A presbyterian congregation attempted to use an organ in 1829, but in 1850 church organs in Scotland were still confined to episcopal and Roman catholic places of worship and to a few chapels of small minority sects. By then organs were widely used among nonconformists in the north of England. During the fifties, some Scottish independent congregations and two English presbyterian congregations followed their example, and two Scottish presbyterian congregations attempted to do so but were prevented by their Church courts. From 1863 onwards, instruments began to appear in churches of the establishment. By 1866 the General Assembly of the Church of Scotland had made it clear that it saw no objection in principle to the use of organs, and the U. P. Church and the Free Church permitted their use in 1872 and 1883 respectively. Once introduced, instrumental music proved popular, and the Church of Scotland, in which there had been none before 1863, found in 1906 that instruments were used by about nine-tenths of its congregations. The "continuing" Free Church, a small minority, maintained its opposition, and ordered the removal of an organ in 1 908. The organ controversy in Scotland has been virtually ignored by church historians and organ historians alike. This thesis is therefore largely concerned with establishing the facts. It examines the arguments used in the debate and traces the history of the adoption of instru- mental music in each of the major Scottish denominations. It relates instrumental music to other innovations which were transforming Scottish worship; it examines the initiation and implementation of organ projects by congregations, the nature of the instruments, how space was found for them in church buildings, how they were used in worship, and how organists were found to play them. Finally, it assesses briefly the various factors which contributed to the general desire for instrumental music in Scottish worship. Attention is drawn to an extensive pamphlet literature, most items of which have remained unnoticed since the era of their publication

The typewriter trade in Scotland, from the 1870s to 1920s

This thesis explores the typewriter trade in Scotland from the 1870s to the 1920s. It analyses the businesses and individuals involved in the marketing, sale and use of writing machines, revealing the processes by which typewriters went from little known novelties in the mid-1870s to essential technologies for commercial and professional work by the early twentieth century. Alongside conventional archival and print-based sources, this thesis makes use of typewriters held at National Museums Scotland and the Glasgow Museum Resource Centre. These collections shed light on the leading figures in Scotland’s typewriter trade, while strengthening our understanding of the reasons why typewriters were designed, advertised, sold and used in the way that they were. Throughout the variety and diversity of businesses involved in the commercialisation of typewriters is revealed, demonstrating that in addition to the buying and selling of writing machines, Scottish businesses profited from producing typewritten transcriptions on demand; providing typing tuition; selling typewriter supplies; repairing typewriters; and dealing in second-hand machines. The focus on these customer facing businesses constitutes an entirely fresh approach to the history of typewriters. To date, scholars interested in the historical significance of these technologies have concentrated on either manufacturing and technical developments or on the expansion of typing as an area of employment. However, there has been hardly any analysis of the businesses that mediated between manufacturers on the one side and users on the other, in Scotland or anywhere else. The lacuna in the historiography has implied that the businesses which sold typewriters and typewriter services played a trivial role in commercialisation. In reality, the businessmen and women in Scotland’s trade were active agents in the sale and promotion of typewriters. Through advertising, exhibitions, lectures, canvassing, typing classes, sales and a whole host of other promotional methods, they introduced typewriters to the Scottish public and demonstrated the potential that these devices had for streamlining office work and transforming the production of written documentation."Regarding funding, I am extremely grateful to the Arts & Humanities Research Council which has funded this project through the excellent Collaborative Doctoral Partnership scheme (grant number AH/R002711). In addition, the University of St Andrews and National Museums Scotland have also made significant financial contributions to the project, including valuable funding extensions in the wake of the Covid pandemic." -- Acknowledgement

Early premiership of Lord Liverpool 1812-15 : palma non sine pulvere

Robert Banks Jenkinson, 2nd Earl of Liverpool KG (1770-1828), was First Lord of the Treasury and Prime Minister for almost fifteen years in the early nineteenth century. He survived in the premiership for longer than all but two of his predecessors and longer than all of his successors, at least so far. Liverpool is, however, one the most overlooked and underestimated of Prime Ministers. Norman Gash's book is the first and so far only modem biography of Liverpool. This study, however, is less than three hundred pages in length, is based on only seven of the hundreds of volumes of the Liverpool Papers in the custody of the British Library and is far from exhaustive in its use of printed sources. There is evidently considerable scope and need for further research on the subject of Liverpool's life and career especially during the period of his premiership and based on a trawl through all the manuscript sources now available. This doctoral dissertation seeks to examine Liverpool during his early premiership between 1812 and 1815, one of the least studied but most significant periods of both Liverpool's life and career, and his administration, from the point Liverpool succeeded to the highest office to the resettlement of Europe after the long war with France. The opening section aims to place Liverpool in his historical context. There is a particular emphasis here on an analysis of the political system that Liverpool was required to master. Liverpool's early life and career before he rose to the premiership is the focus of the next section. The main body of the thesis is divided into two parts. One part examines Liverpool during his early premiership in a mainly chronological style and is concerned almost entirely with the issues of war and peace, and the other part seeks to examine a number of major themes that are most satisfactorily looked at in isolation from the main narrative. For example, one chapter covers the premier's relationship with the monarchy