Nystrom Methods in the RKQ Algorithm for Initial-value Problems

We incorporate explicit Nystrom methods into the RKQ algorithm for stepwise global error control in numerical solutions of initial-value problems. The initial-value problem is transformed into an explicitly second-order problem, so as to be suitable for Nystrom integration. The Nystrom methods used are fourth-order, fifth-order and 10th-order. Two examples demonstrate the effectiveness of the algorithm.Comment: This is an extension of ideas published in J. Math. Res. (open access); see refs [1] and [2

Spatial navigation and multiscale representation by hippocampal place cells

Hippocampal lesions are known to impair success in navigation tasks. While such tasks could be solved by memorizing complete paths from a starting location to the goal, animals still perform successfully when placed in a novel starting position. We propose a navigation algorithm to solve the latter problem by exploiting two facts about hippocampal organization: (1) The size of the place fields of hippocampal place cells varies systematically along the dorsoventral axis, with dorsal place cells having smaller place fields than ventral (Kjelstrup et. al. 2008); and (2) the theta oscillation propagates as a traveling wave from dorsal to ventral hippocampus (Lubenov and Siapas, 2009). Taken together, these observations imply that the hippocampal representation of space progresses from fine- to coarse-grained within every theta cycle. 

The algorithm assumes that place cells can be activated by the animal's imagining a goal location, in addition to physically standing in the appropriate location. In the proposed algorithm, place cell activation propagates from small scale to large scale until place cells are found which respond strongly to both the physical location and the goal location. These place fields have their centers aligned roughly in the direction of the goal, providing a crude estimate of which direction the animal should step to approach the goal. Fine-grained directional information is contained in the smaller scale place fields within these large ones. Our algorithm therefore identifies a sequence of place cells, one from each scale, whose centers lie roughly along the line to the goal. 

Simulations reveal successful navigation to the goal, even around obstacles. By minimizing the number of steps the animal takes to reach the goal, we predict the organization of the optimal place field "map"; specifically the fraction of place cells which should be allocated to each spatial scale. This prediction is, in principle, experimentally testable.

The set of place fields with centers lying along a line to the goal is used to compute a step direction by maximizing the probability that those cells will be active in the next time step, given that a particular step direction is chosen.

The proposed algorithm handles navigation around obstacles by including "border cells" (Solstad et. al. 2008) which inhibit place cells in proportion to the degree of overlap between the place field and the obstacle. Furthermore, including firing rate adaptation of place cells prevents the animal from getting stuck in one spot

An Euler-type method for Volterra integro-differential equations

We describe an algorithm, based on Euler's method, for solving Volterra integro-differential equations. The algorithm approximates the relevant integral by means of the composite Trapezium Rule, using the discrete nodes of the independent variable as the required nodes for the integration variable. We have developed an error control device, using Richardson extrapolation, and we have achieved accuracy better than 1e-12 for all numerical examples considered.Comment: 11 page

Error propagation in an explicit and an implicit numerical method for Volterra integro-differential equations

We study error propagation in both an explicit and an implicit method for solving Volterra integro-differential equations. We determine the relationship between local and global errors. We derive upper bounds for the global error, and show that the global order for both methods is expected to be first-order. A few numerical examples illustrate our results.Comment: 14p, 5 fig

Stability analysis of an implicit and explicit numerical method for Volterra integro-differential equations with kernel K(x,y(t),t)

We present implicit and explicit versions of a numerical algorithm for solving a Volterra integro-differential equation. These algorithms are an extension of our previous work, and cater for a kernel of general form. We use an appropriate test equation to study the stability of both algorithms,, numerically deriving stability regions. The region for the implicit method appears to be unbounded, while the explicit has a bounded region close to the origin. We perform a few calculations to demonstrate our results.Comment: 10 pages, 1 Figur

Improving the Efficiency of Runge-Kutta Reintegration by Means of the RKGL Algorithm

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Range and Domain Partitioning in Piecewise Polynomial Approximation

Abstract: Error control in piecewise polynomial interpolation of a smooth univariate function f requires that the interval of approximation be subdivided into many subintervals, on each of which an interpolating polynomial is determined. The number of such subintervals is often over- estimated through the use of a high-order derivative of f . We report on a partitioning algorithm, in which we attempt to reduce the number of subintervals required, by imposing conditions on f and its relevant higher derivative. One of these conditions facilitates a distinction between the need for absolute or relative error control. Two examples demonstrate the effectiveness of this partitioning algorithm. Key Words: Piecewise Polynomial; Range Partitioning; Domain Partitioning; Error Contro

Synthesis of empty bacterial microcompartments, directed organelle protein incorporation, and evidence of filament-associated organelle movement

Compartmentalization is an important process, since it allows the segregation of metabolic activities and, in the era of synthetic biology, represents an important tool by which defined microenvironments can be created for specific metabolic functions. Indeed, some bacteria make specialized proteinaceous metabolic compartments called bacterial microcompartments (BMCs) or metabolosomes. Here we demonstrate that the shell of the metabolosome (representing an empty BMC) can be produced within E. coil cells by the coordinated expression of genes encoding structural proteins. A plethora of diverse structures can be generated by changing the expression profile of these genes, including the formation of large axial filaments that interfere with septation. Fusing GFP to PduC, PduD, or PduV, none of which are shell proteins, allows regiospecific targeting of the reporter group to the empty BMC. Live cell imaging provides unexpected evidence of filament-associated BMC movement within the cell in the presence of Pdu