Gap junctions and emergent rhythms

Gap junction coupling is ubiquitous in the brain, particularly between the dendritic trees of inhibitory interneurons. Such direct non-synaptic interaction allows for direct electrical communication between cells. Unlike spike-time driven synaptic neural network models, which are event based, any model with gap junctions must necessarily involve a single neuron model that can represent the shape of an action potential. Indeed, not only do neurons communicating via gaps feel super-threshold spikes, but they also experience, and respond to, sub-threshold voltage signals. In this chapter we show that the so-called absolute integrate-and-fire model is ideally suited to such studies. At the single neuron level voltage traces for the model may be obtained in closed form, and are shown to mimic those of fast-spiking inhibitory neurons. Interestingly in the presence of a slow spike adaptation current the model is shown to support periodic bursting oscillations. For both tonic and bursting modes the phase response curve can be calculated in closed form. At the network level we focus on global gap junction coupling and show how to analyze the asynchronous firing state in large networks. Importantly, we are able to determine the emergence of non-trivial network rhythms due to strong coupling instabilities. To illustrate the use of our theoretical techniques (particularly the phase-density formalism used to determine stability) we focus on a spike adaptation induced transition from asynchronous tonic activity to synchronous bursting in a gap-junction coupled network

Identification and Adaptive Control Methods for Some Stochastic Systems

This dissertation is focused on the identification and adaptive control of some stochastic systems. Initially a survey of some adaptive control problems for both discrete and continuous time stochastic systems is provided. Discrete time branching processes are described and some results on parameter estimation and adaptive control for these processes are reviewed. Then continuous time branching processes are introduced and the main results in this dissertation concerning estimation and adaptive control are given. The family of estimators is shown to be strongly consistent and the optimal rate of convergence of this family of estimators is obtained. Furthermore some other asymptotic properties of these estimators are verified. An adaptive control is given that posses self-tuning property. It is shown that it does not achieve the optimal asymptotic cost for the known system. Finally some computational methods and simulations are given for a variety of stochastic differential equations driven by a Brownian motion or an arbitrary fractional Brownian motion and computational properties of the parameter estimates for the branching processes are given

Directed Abelian sandpile with multiple downward neighbors

We study the directed Abelian sandpile model on a square lattice, with $K$ downward neighbors per site, $K > 2$. The $K=3$ case is solved exactly, which extends the earlier known solution for the $K=2$ case. For $K>2$, the avalanche clusters can have holes and side-branches and are thus qualitatively different from the $K=2$ case where avalance clusters are compact. However, we find that the critical exponents for $K>2$ are identical with those for the $K=2$ case, and the large scale structure of the avalanches for $K>2$ tends to the $K=2$ case.Comment: Accepted for publication in PR

Archaeology and Identity In the 19th Century Northern Cape Frontier: the Xhosa of the Pramberg

This dissertation examines the identity of the Xhosa communities that settled in the frontier zone of the Northern Cape during the first half of the 19th century. It does this through the archaeology, and ethnographic and historical accounts. The concept of a baseline Nguni identity in the Eastern Cape is examined with an emphasis on settlement, mobility and cultural interaction. The historical background and a brief history of the of the Xhosa in the Northern Cape will be detailed, focusing on the Pramberg community. The archaeology of three Xhosa sites in the Pramberg will be described and analysed, and then contrasted and compared with the ethnographic and historical evidence. The result of this comparison is a discussion of the identity change and continuity of the Pramberg Xhosa in the context of the cultural milieu of the frontier and the appropriation of land by the expanding Cape colony

A New Context Affording for Regulation: The Case of Musical Play

The present study set out to investigate theoretical speculations that regulation and musical play, an initial manifestation of musicality, are directly linked. This study aimed to explore the potential for regulation to occur during musical play and investigate the nature of the regulatory behaviours. Thirty-six children, aged 6 and 8, were observed during musical play sessions. These observations were analysed, using a coding framework, to identify and code regulatory behaviours as to the type of regulation, its social nature and the direction of activity. The data were subjected to quantitative analysis. The findings suggest that regulatory behaviours occurred during musical play. During musical play tasks, cognitive monitoring and emotional/motivational monitoring behaviours were the most prevalent, significantly more opportunities were provided for socially-shared regulation compared to self- or co-regulation, and the children more often directed their activity towards fundamental, rather than superficial aspects of tasks. The results can inform theory and practice