Computational techniques for fluid dynamics
Computational techniques for fluid dynamics
Cable theory for dendritic neurons
Methods in neuronal modeling
Numerical methods for neuronal modeling
Methods in neuronal modeling
Approximation by the Legendre collocation method of a model problem in electrophysiology
Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
Numerical recipes in C (2nd ed.): the art of scientific computing
Numerical recipes in C (2nd ed.): the art of scientific computing
Pseudospectra of the convection-diffusion operator
SIAM Journal on Applied Mathematics
The book of GENESIS (2nd ed.): exploring realistic neural models with the GEneral NEural SImulation System
Spectral methods in MatLab
Biophysics of Computation: Information Processing in Single Neurons (Computational Neuroscience Series)
The NEURON Book
Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems
Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems
Lapicque’s 1907 paper: from frogs to integrate-and-fire
Biological Cybernetics
Adaptive numerical simulation of intracellular calcium dynamics using domain decomposition methods
Applied Numerical Mathematics
Numerical solution of calcium-mediated dendritic branch model
Journal of Computational and Applied Mathematics
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We use high-order approximation schemes for the space derivatives in the nonlinear cable equation and investigate the behavior of numerical solution errors by using exact solutions, where available, and grid convergence. The space derivatives are numerically approximated by means of differentiation matrices. Nonlinearity in the equation arises from the Hodgkin-Huxley dynamics of the gating variables for ion channels. We have investigated in particular the effects of synaptic current distribution and compared the accuracy of the spectral solutions with that of finite differencing. A flexible form for the injected current is used that can be adjusted smoothly from a very broad to a narrow peak, which furthermore leads, for the passive cable, to a simple, exact solution. We have used three distinct approaches to assess the numerical solutions: comparison with exact solutions in an unbranched passive cable, the convergence of solutions with progressive refinement of the grid in an active cable, and the simulation of spike initiation in a biophysically realistic single-neuron model. The spectral method provides good numerical solutions for passive cables comparable in accuracy to those from the second-order finite difference method and far greater accuracy in the case of a simulated system driven by inputs that are smoothly distributed in space. It provides faster convergence in active cables and in a realistic neuron model due to better approximation of propagating spikes.