Month: August 2016

Transcranial Electric Stimulation

Although this blog is dedicated to TMS I think that it is worthwhile to examine briefly the electric field space that TES (Transcranial Electric Stimulation) is capable of producing. TES also goes by the names tDCS (Transcranial Direct Current Stimulation) and tACS (Transcranial Alternating Current Stimulation). The common feature of these methods is that an electric field is established within the brain tissue by the application of an electric current at the scalp surface using electrodes.

For direct comparison to the TMS description given in a previous blog post we will again assume a spherically symmetric conductor. We will assume a simple sphere with a single homogeneous isotropic conductivity throughout its volume. If a model consisting of multiple concentric spherical shells of differing conductivity (see [1] for example) is used instead then the three principal results given at the end of this post still hold. Consult the previous blog post for definitions of symbols and notation if needed. Our primary goal here is to establish that TMS and TES electric fields are orthogonal and as such are geometrically very different fields.

As with the previous blog post we again assume that the frequencies involved are such that the quasi-static approximation holds. As discussed in the previous blog post, under the quasi-static approximation the condition \nabla \cdot {\bf J} = 0 applies everywhere. By using this condition, along with Ohm’s Law, and constructing an infinitesimal Gaussian pillbox at the surface it is a simple matter to show that the following boundary condition must hold at the surface:

\sigma {\bf E}(a,\theta,\phi,t) \cdot {\bf n} =  {\bf J} (a,\theta,\phi,t) \cdot  {\bf n} \qquad \qquad \qquad \qquad \qquad \qquad \qquad  (1)

where \sigma is the conductivity of the spherical conductor, {\bf n} is a unit normal vector at the surface of the conductor,  and {\bf J} is the applied current density at the surface of the conductor and is assumed to be known in TES.

Since in TES, unlike TMS, there is no magnetic induction capable of producing a significant electric field then the electric field is given by {\bf E} = - \nabla \Phi . And since, as with TMS, no net charge density exists within the spherical conductor then according to Maxwell’s Laws \nabla \cdot {\bf E} = 0 . Together these two equations yield the Laplace Equation for the interior of the spherical conductor:

\nabla^2 \Phi({\bf r},t) = 0 \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad  \qquad \qquad \qquad (2)

The solution of the Laplace equation is written in terms scalar spherical harmonics as:

\Phi({\bf r},t) = \sum\limits_{j=0}^{\infty} \sum\limits_{m =-j}^{m=j} [A_{jm}(t) r^{j} + B_{jm}(t) r^{-(j+1)} ] Y_{jm}(\theta,\phi)  \qquad \qquad \; \;\; (3)

and since the scalar potential at the center of the sphere must be finite we can set B_{jm} = 0 so that

\Phi({\bf r},t) = \sum\limits_{j=0}^{\infty} \sum\limits_{m =-j}^{m=j} A_{jm}(t) r^{j} Y_{jm}(\theta,\phi)  \qquad \qquad \qquad \qquad \qquad \qquad (4)

Using {\bf E} = - \nabla \Phi we may then write the electric field within the conductor as (see [2] for vector spherical harmonics):

{\bf E}({\bf r},t)  = - \sum\limits_{j=0}^{\infty} \sum\limits_{m =-j}^{m=j} A_{jm}(t) \nabla \left[r^{j} \; Y_{jm}(\theta,\phi) \right]  \qquad \qquad \qquad \qquad \; (5)

{\bf E}({\bf r},t) = - \sum\limits_{j=0}^{\infty} \sum\limits_{m =-j}^{m=j} A_{jm}(t) \sqrt{j(2j+1)} \; r^{j-1} \; {\bf Y}_{jm}^{j-1}(\theta,\phi)  \qquad \qquad \!\!\! (6)

By applying the boundary condition given by equation (1) we may obtain the coefficients A_{jm} in terms of the applied current normal to the spherical surface:

A_{jm} = - \frac{1}{\sigma j a^{j-1}}  \int_0^{2\pi} \! \int_0^{\pi} [{\bf J} (a,\theta,\phi,t) \cdot  {\bf n}] \; Y^*_{jm}(\theta,\phi) \; d\phi \; d\theta \sin \theta \qquad  \;\; (7)

where we have made use of the orthogonality relationship for scalar spherical harmonics  (see [2]) .

Equations (6) and (7) provide the following three principle messages of this blog post:

  1. The TES electric field is a linear superposition of the fields due to the vector spherical harmonics {\bf Y}_{jm}^{j-1}(\theta,\phi) as given by Equation (6).
  2. The coefficients in this linear superposition are determined by the applied current density at the scalp as given by Equation (7).
  3. Since the TMS field is a superposition of fields due to the vector spherical harmonics {\bf Y}_{jm}^{j}(\theta,\phi) then the TES electric field and the TMS electric field are geometrically very different. In fact they are orthogonal under the dot inner product as well as the usual inner product used in vector spherical harmonic analysis, that is: \int_0^{2\pi}  \int_0^{\pi} {\bf Y}_{jm}^j (\theta,\phi) \cdot {\bf Y}_{jm}^{j-1 *}(\theta,\phi)  \; d\phi \; d\theta \sin \theta  = 0 .

Notice from the following definitions of the vector spherical harmonics  [2] :

{\bf Y}_{jm}^{j}(\theta,\phi) = - {\bf e}_{\theta}  \frac{m}{\sqrt{j(j+1)} } \frac{Y_{jm}(\theta,\phi)}{\sin \theta} -  {\bf e}_{\phi}  \frac{i}{\sqrt{j(j+1)} } \frac{\partial Y_{jm}(\theta,\phi)}{\partial \theta}

and

{\bf Y}_{jm}^{j-1}(\theta,\phi) = \sqrt{\frac{j}{2j+1}} \left( {\bf e}_r  Y_{jm}(\theta,\phi) + {\bf e}_{\theta}  \frac{i}{j} \frac{\partial Y_{jm}(\theta,\phi)}{\partial \theta}  +  {\bf e}_{\phi}  \frac{m}{j} \frac{Y_{jm}(\theta,\phi)}{\sin \theta} \right)

that the electric field due to TES has a component normal to the surface of the spherical conductor whereas the TMS electric field has no such component.

Of course for real heads and brains the assumption of a spherically symmetric conductor does not strictly hold. Nevertheless the above analysis provides good intuition with respect to the relationship between the electric fields established in TES and TMS systems.

Before leaving this topic I should also note that the electric field magnitudes that can be established safely using TES and TMS are very different. TMS fields are typically near 100 V/m whereas TES fields are typically less than 1.0 V/m. The primary safety issue for TES concerns the magnitude of the current established in the head. Applied currents capable of generating fields above 1.0 V/m in TES can produce currents within the head that cause unpleasant sensations and burns. Also note that the typical electric field strength used in TMS is capable of directly causing an action potential in neurites whereas the typical electric field strength used in TES is too weak to directly cause an action potential. The mechanism of TES effects are thought to influence neuronal activity by other means.

References:

[1] J. P. Dmochowski, M. Bikson and L. C. Parra, The point spread function of the human head and its implications for transcranial current stimulation, Phys. Med. Biol., Vol 57, 2012, 6459–6477.

[2] D. A. Varshalovich, A. N. Moskalev and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988, 208-229.