On the use of pseudo-spectral method in model reduction and simulation of active dendrites

Abstract

Most of dendrites in the central nervous system are now known to have active channels. These active dendrites play important roles not only in signal summation but also in computation. For the simulation of these important active dendrites, the compartment model based on the finite volume or finite difference discretization was mainly adopted. In this paper, we employ the Chebychev pseudo-spectral method well developed in computational physics, and demonstrate that it can achieve a higher precision with the same number of equations than the compartment model. Moreover, it is also shown that the Chebychev pseudo-spectral method converges faster to attain a given precision. Hence, for the simulations of active dendrites, the Chebychev pseudo-spectral method can be an attractive alternative to the compartment model since it leads to a low order model with higher precision or converges faster for a given precision.

Introduction

Dendrites are known to integrate various input signals coming from other neurons. In the past, most of computational dendritic studies focused on passive dendrites due to their simplicity. However, recently, most of dendrites in the central nervous system are shown to have active channels. The presence of active channels makes it possible that the signals from distal synapses can be transmitted to soma by reducing the signal attenuation [1]. Moreover, some computational and experimental studies of the neurons with active dendrites suggest that active dendrites may be very important for computation. Indeed, Mel and his coworkers [2], [3], [4] proposed that nonlinear computations are possible in each side branch and the entire active dendrite forms a two layer network of nonlinear elements. Thanks to these stimulating results, active dendrites started to receive a great deal of attention recently. For the computational studies of active dendrites, detailed simulations are necessary that are computationally very demanding.

Both passive and active dendrites are described by the cable equations that are partial differential equations (PDEs). For the numerical solutions of the cable equations, a spatial discretization scheme is often adopted to get a system of approximate ordinary differential equations (ODEs). For this, the compartment model based on the finite volume or finite difference scheme was settled as a standard technique. For instance, the existing neuron simulators such as NEURON [5] and GENESIS [6] are based exclusively on the compartment model. However, the finite volume method is not the best computational scheme in most cases. Indeed, it was popular simply because it is just easy to use.

It is well known in computational physics that the spectral method converges much faster than any other computational schemes for the problems with simple geometries [7]. Since the same precision is achieved with a smaller number of approximate ODEs in the spectral method, a higher precision will be attained with the same number of approximate ODEs. Hence, the spectral method is superior in finding a low order model with a reasonable precision as well. Indeed, local basis functions used in the finite volume or finite difference method are designed to approximate the local variation of solution and, thus, a good approximation of the global spatial variation of solution is hardly achieved with a small number of local basis functions. In contrast, global basis functions employed in the spectral method are able to capture the global spatial features of solution with relatively small number of global basis functions.

To see whether the spectral methods can be powerful in computational neuroscience as well, the third author and his colleagues investigated the application of the spectral method (well known in computational physics) to passive dendrites and myelinated axons [8], [9]. A passive dendrite or a myelinated part of a myelinated axon is a linear cable. Since eigenfunctions can be found for the linear cables, eigenfunction expansion is adopted to propose computationally very efficient Galerkin type spectral methods for these cables [8], [9]. For this, the singular perturbation is shown to be important. However, active dendrites are nonlinear cables due to the presence of active channels and, thus, the eigenfunction expansion based Galerkin type techniques are hardly applicable.

In computational physics, it is the pseudo-spectral methods based on interpolations that are very popular for nonlinear problems [7], [10], [11]. Especially, when boundary conditions are not periodic, the Chebyshev polynomial based pseudo-spectral method is known to be the best choice in many cases [7], [10], [11]. Hence, to see whether spectral method well known in computational physics is also advantageous over the compartment model in the reduced model and simulation of active dendrites, we will consider the Chebyshev pseudo-spectral method in this paper. It turned out that, for the same number of ODEs, the proposed scheme achieved a more precise solution. Moreover, to achieve a given precision, the proposed scheme converged faster. Hence, the Chebyshev pseudo-spectral method is superior than the conventional compartment model approach.