VisualPDE: rapid interactive simulations of partial differential equations

Computing has revolutionised the study of complex nonlinear systems, both by allowing us to solve previously intractable models and through the ability to visualise solutions in different ways. Using ubiquitous computing infrastructure, we provide a means to go one step further in using computers to understand complex models through instantaneous and interactive exploration. This ubiquitous infrastructure has enormous potential in education, outreach and research. Here, we present VisualPDE, an online, interactive solver for a broad class of 1D and 2D partial differential equation (PDE) systems. Abstract dynamical systems concepts such as symmetry-breaking instabilities, subcritical bifurcations and the role of initial data in multistable nonlinear models become much more intuitive when you can play with these models yourself, and immediately answer questions about how the system responds to changes in parameters, initial conditions, boundary conditions or even spatiotemporal forcing. Importantly, VisualPDE is freely available, open source and highly customisable. We give several examples in teaching, research and knowledge exchange, providing high-level discussions of how it may be employed in different settings. This includes designing web-based course materials structured around interactive simulations, or easily crafting specific simulations that can be shared with students or collaborators via a simple URL. We envisage VisualPDE becoming an invaluable resource for teaching and research in mathematical biology and beyond. We also hope that it inspires other efforts to make mathematics more interactive and accessible.Comment: 19 pages, 7 figures. This is a companion paper to the website https://visualpde.com

A chromatic transient visual evoked potential based encoding/decoding approach for brain-computer interface

This paper presents a new encoding/decoding approach to brain-computer interface (BCI) based on chromatic transient visual evoked potential (CTVEP). The proposed CTVEP-based encoding/decoding approach is designed to provide a safer and more comfortable stimulation method than the conventional VEP-based stimulation methods for BCI without loss of efficiency. For this purpose, low-frequency isoluminant chromatic stimuli are time-encoded to serve as different input commands for BCI control, and the superior comfortableness of the proposed stimulation method is validated by a survey. A combination of diversified signal processing techniques are further employed to decode the information from CTVEP. Based on experimental results, a properly designed configuration of the CTVEP-based stimulation method and a tailored signal processing framework are developed. It is demonstrated that high performance (at information transfer rate: 58.0 bits/min, accuracy: 94.9%, false alarm rate: 1.3%) for BCI can be achieved by means of the CTVEP-based encoding/decoding approach. It turns out that to achieve such good performance, only simple signal processing algorithms with very low computational complexity are required, which makes the method suitable for the development of a practical BCI system. A preliminary prototype of such a system has been implemented with demonstrated applicability. © 2011 IEEE.published_or_final_versio