| Popis: |
Computational approaches are becoming increasingly important in neuroscience, where complex, nonlinear systems modelling neural activity across multiple spatial and temporal scales are the norm. This paper considers collocation techniques for solving neural field models, which typically take the form of a partial integro-dfferential equation. In particular, we investigate and compare the convergence properties of linear and quadratic collocation on both regular grids and more general meshes not fixed to the regular Cartesian grid points. For regular grids we perform a comparative analysis against more standard techniques, in which the convolution integral is computed either by using Fourier based methods or via the trapezoidal rule. Perhaps surprisingly, we find that on regular, periodic meshes, linear collocation displays better convergence properties than quadratic collocation, and is in fact comparable with the spectral convergence displayed by both the Fourier based and trapezoidal techniques. However, for more general meshes we obtain superior convergence of the\ud convolution integral using higher order methods, as expected. |
