Detecting cell assemblies in large neuronal populations

Recent progress in the technology for single unit recordings has given the neuroscientific community theopportunity to record the spiking activity of large neuronal populations. At the same pace, statistical andmathematical tools were developed to deal with high-dimensional datasets typical of such recordings.A major line of research investigates the functional role of subsets of neurons with significant co-firingbehavior: the Hebbian cell assemblies. Here we review three linear methods for the detection of cellassemblies in large neuronal populations that rely on principal and independent component analysis.Based on their performance in spike train simulations, we propose a modified framework that incorpo-rates multiple features of these previous methods. We apply the new framework to actual single unitrecordings and show the existence of cell assemblies in the rat hippocampus, which typically oscillate attheta frequencies and couple to different phases of the underlying field rhythm

Components of learning and assessment in linear algebra

Linear algebra provides theory and technology that are the cornerstones of a range of cutting edge mathematical applications, from designing computer games to complex industrial problems, as well as more traditional applications in statistics and mathematical modelling. Once past introductions to matrices and vectors, the challenges of balancing theory, applications and computational work across mathematical and statistical topics and problems are considerable, particularly given the diversity of abilities and interests in typical cohorts. This paper considers two such cohorts in a second level linear algebra course in different years. The course objectives and materials were almost the same, but some changes were made in the assessment package. In addition to considering effects of these changes, the links with achievement in first year courses are analysed, together with achievement in a following computational mathematics course. Some results that may initially appear surprising provide insight into the components of student learning in linear algebra.

Constructive development of the solutions of linear equations in introductory ordinary differential equations

The solution of linear ordinary differential equations (ODEs) is commonly taught in first year undergraduate mathematics classrooms, but the understanding of the concept of a solution is not always grasped by students until much later. Recognising what it is to be a solution of a linear ODE and how to postulate such solutions, without resorting to tables of solutions, is an important skill for students to carry with them to advanced studies in mathematics. In this study we describe a teaching and learning strategy that replaces the traditional algorithmic, transmission presentation style for solving ODEs with a constructive, discovery based approach where students employ their existing skills as a framework for constructing the solutions of first and second order linear ODEs. We elaborate on how the strategy was implemented and discuss the resulting impact on a first year undergraduate class. Finally we propose further improvements to the strategy as well as suggesting other topics which could be taught in a similar manner.