Discrete Spherical Harmonic Transforms: Numerical Preconditioning and Optimization

Blais, J. A. R.

doi:10.1007/978-3-540-69387-1_74

Cited by 6 publications

(4 citation statements)

References 11 publications

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“… where is the domain of sampled points. Modern implementations of these expansions use fast Fourier transforms over latitude bands, among other methods, to improve computational efficiency [25, 24].…”

Section: Methodsmentioning

confidence: 99%

Spherical harmonics representation of the steady-state membrane potential shift induced by tDCS in realistic neuron models

Galan-Gadea¹,

Salvador²,

Bartolomei

et al. 2022

Preprint

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ObjectiveWe provide a systematic framework for the quantification of the effect of externally applied weak electric fields on realistic neuron compartment models as captured by physiologically relevant quantities such as the membrane potential or transmembrane current as a function of the orientation of the field.ApproachWe define a response function as the steady-state change of the membrane potential induced by a canonical external field of 1 V/m as a function of its orientation. We estimate the function values through simulations employing reconstructions of the rat somatosensory cortex from the Blue Brain Project. The response of different cell types is simulated using the NEURON simulation environment. We represent and analyze the angular response as an expansion in spherical harmonics.Main resultsWe report membrane perturbation values comparable to those in the literature, extend them to different cell types, and provide their profiles as spherical harmonic coefficients. We show that in resting condition, responses are dominated by their dipole terms (ℓ = 1), in agreement with experimental findings and compartment theory. Indeed, we show analytically that for a passive cell only the dipole term is nonzero. However, for states other than resting, other terms, while small, appear relevant. In particular, we show how ℓ = 0 and ℓ = 2 terms can modify the function to induce asymmetries in the response.SignificanceThis work provides a practical framework for the realistic representation of the effects of weak electric fields on different neuron types and their main regions—an important milestone for the development of micro- and mesoscale models and the optimization of brain stimulation solutions.

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Section: Methodsmentioning

confidence: 99%

Spherical harmonics representation of the steady-state membrane potential shift induced by tDCS in realistic neuron models

Galan-Gadea¹,

Salvador²,

Bartolomei

et al. 2022

Preprint

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show abstract

“…The spectral coefficient is obtained through projection of the spherical harmonics using orthogonality as -----------------1 x 2 -( ) The spherical harmonics spectrum of the Gaussian bell, given as grid point values on the spherical domain, can be obtained either by the Fourier-series method or the Gaussian quadrature (Sneeuw and Bun, 1996;Rod Blais, 2008;Wittwer et al, 2008;Cheong et al, 2012). The projection of Legendre functions is performed by the integral formula on the global domain for the Fourier-series method, while it is done by weighted sum of gridpoint values on Gaussian grids for the Gaussian quadrature method (Swarztrauber, 1993;Cheong et al, 2012).…”

Section: Spectral Methods With Gaussian Quadrature and Fourier-series Methodsmentioning

confidence: 99%

Spherical Harmonics Power-spectrum of Global Geopotential Field of Gaussian-bell Type

Cheong¹,

Kong²

2013

Journal of the Korean earth science society

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Spherical harmonics power spectrum of the geopotential field of Gaussian-bell type on the sphere was investigated using integral formula that is associated with Legendre polynomials. The geopotential field of Gaussian-bell type is defined as a function of sine of angular distance from the bell's center in order to guarantee the continuity on the global domain. Since the integral-formula associated with the Legendre polynomials was represented with infinite series of polynomial, an estimation method was developed to make the procedure computationally efficient while preserving the accuracy. The spherical harmonics power spectrum was shown to vary significantly depending on the scale parameter of the Gaussian bell. Due to the accurate procedure of the new method, the power (degree variance) spanning over orders that were far higher than machine roundoff was well explored. When the scale parameter (or width) of the Gaussian bell is large, the spectrum drops sharply with the total wavenumber. On the other hand, in case of small scale parameter the spectrum tends to be flat, showing very slow decaying with the total wavenumber. The accuracy of the new method was compared with theoretical values for various scale parameters. The new method was found advantageous over discrete numerical methods, such as Gaussian quadrature and Fourier method, in that it can produce the power spectrum with accuracy and computational efficiency for all range of total wavenumber. The results of present study help to determine the allowable maximum scale parameter of the geopotential field when a Gaussian-bell type is adopted as a localized function.

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“…where D is the domain of sampled points. Modern implementations of these expansions use fast Fourier transforms over latitude bands, among other methods, to improve computational efficiency [24,25].…”

Section: Spherical Harmonics-based Representationmentioning

confidence: 99%

Spherical harmonics representation of the steady-state membrane potential shift induced by tDCS in realistic neuron models

Galan-Gadea¹,

Salvador²,

Bartolomei³

et al. 2023

J. Neural Eng.

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Objective: We provide a systematic framework for quantifying the effect of externally applied weak electric fields on realistic neuron compartment models as captured by physiologically relevant quantities such as the membrane potential or transmembrane current as a function of the orientation of the field. Approach: We define a response function as the steady-state change of the membrane potential induced by a canonical external field of 1 V/m as a function of its orientation. We estimate the function values through simulations employing reconstructions of the rat somatosensory cortex from the Blue Brain Project. The response of different cell types is simulated using the NEURON simulation environment. We represent and analyze the angular response as an expansion in spherical harmonics. Main results: We report membrane perturbation values comparable to those in the literature, extend them to different cell types, and provide their profiles as spherical harmonic coefficients. We show that at rest, responses are dominated by their dipole terms (ℓ = 1), in agreement with experimental findings and compartment theory. Indeed, we show analytically that for a passive cell, only the dipole term is nonzero. However, while minor, other terms are relevant for states different from resting. In particular, we show how ℓ = 0 and ℓ = 2 terms can modify the function to induce asymmetries in the response. Significance: This work provides a practical framework for the representation of the effects of weak electric fields on different neuron types and their main regions—an important milestone for developing micro- and mesoscale models and optimizing brain stimulation solutions.

show abstract

Discrete Spherical Harmonic Transforms: Numerical Preconditioning and Optimization

Cited by 6 publications

References 11 publications

Spherical harmonics representation of the steady-state membrane potential shift induced by tDCS in realistic neuron models

Spherical harmonics representation of the steady-state membrane potential shift induced by tDCS in realistic neuron models

Spherical Harmonics Power-spectrum of Global Geopotential Field of Gaussian-bell Type

Spherical harmonics representation of the steady-state membrane potential shift induced by tDCS in realistic neuron models

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