^{1}

^{1,2}

^{3}

^{1}

^{2}

^{3}

Transcranial direct current stimulation (tDCS) continues to demonstrate success as a medical intervention for neurodegenerative diseases, psychological conditions, and traumatic brain injury recovery. One aspect of tDCS still not fully comprehended is the influence of the tDCS electric field on neural functionality. To address this issue, we present a mathematical, multiscale model that couples tDCS administration to neuron electrodynamics. We demonstrate the model’s validity and medical applicability with computational simulations using an idealized two-dimensional domain and then an MRI-derived, three-dimensional human head geometry possessing inhomogeneous and anisotropic tissue conductivities. We exemplify the capabilities of these simulations with real-world tDCS electrode configurations and treatment parameters and compare the model’s predictions to those attained from medical research studies. The model is implemented using efficient numerical strategies and solution techniques to allow the use of fine computational grids needed by the medical community.

Transcranial direct current stimulation (tDCS) is a medical procedure that delivers electrical stimulation to the brain through electrodes positioned on the scalp. The electrodes deliver an electrical current on the order of 1-2 mA and produce an electric field in the patient’s cerebral cavity that alters neuron excitability. A common use of this treatment is to assist neurons in firing action potentials (APs) by increasing their resting membrane potential. Current biomedical research continues to demonstrate the benefits of tDCS as a medical treatment. Recognition memory in Alzheimer disease patients has improved [

Current components of tDCS include the Laplace equation [

Computational simulations of tDCS that utilize (

Beyond the tDCS modeling and simulation field, the computational neuroscience community possesses a large collection of biologically inspired, mathematical models of neural-level dynamics. The Hodgkin-Huxley model, for example, emulates voltage-gated ion channel functionality [

A multiscale model that couples tDCS and cellular level functionality would enable researchers to simulate the impact that tDCS has on neurons. For instance, correlations between tDCS and ion channel functionality, action potential behavior, and neurotransmitter dynamics could be studied. In addition, patient-specific electrode configurations and treatment parameters could be optimized based on neuron behavior. Furthermore, a multiscale model would provide a bridge between the tDCS numerical simulation field and the computational neuroscience field, thereby enabling tDCS simulations access to the sophisticated, physiologically based cellular and subcellular models of the neuroscience community.

Several researchers have investigated the influence of extracellular electrical fields on the transmembrane voltage of individual and small groups of cells [

In this paper, we combine these modeling strategies to produce a multiscale model of tDCS. We begin by coupling the bidomain model partial differential equations (PDEs) with boundary conditions that model tDCS treatments. We validate the model against several test cases on two different geometries. First, we simulate the model on an idealized two-dimensional domain, which provides a basic environment for visualizing and investigating electric potential and electric field characteristics. Then, we utilize an MRI-derived, three-dimensional human head geometry that possesses inhomogeneous and anisotropic tissue conductivities. In this setting, we examine the electric potential, electric current, electric field, and transmembrane voltage results produced from real-world tDCS electrode configurations. In both geometries, five distinct tissue types are used: skin, skull, cerebrospinal fluid (CSF), and the grey matter (GM) and white matter (WM) portions of the brain. Further, we detail the numerical methods and solution techniques that we implemented to enable reasonable simulation execution times.

To our knowledge, this paper presents the first multiscale tDCS model and simulations. We hope that the modeling and computational approaches presented in this paper help to expand tDCS simulation capabilities and further our understanding of tDCS impacts at the cellular level.

This section presents details of the model, numerics, and computational simulations used in this paper. First, an overview of the bidomain model is provided, as well as a description of the adaptation of this model for tDCS. Then, the numerical methods used to implement the multiscale tDCS model are described. Next, an overview of computational tools that we utilized is presented. Finally, the numerical experiments that were performed are described.

Modeling each cell in the brain and head is not computationally feasible; the bidomain model is based on a volume averaging approach, where the value at a point in a tissue is treated as an average over a minuscule, multicellular region around the point [

The bidomain model is given by the following system of partial differential equations for points in the brain,

Equations (

Extracellular electric field continuity at the interface between the brain and extracerebral domain is preserved by requiring that

To make the bidomain model suitable for tDCS applications, two specific areas need to be addressed. First, the boundary conditions on the scalp must model tDCS administration. Second, cellular models that emulate neuron electrodynamics are necessary. The result of this adaptation is our multiscale tDCS model.

On the surface of the head, there are three separate boundary conditions needed to model tDCS. First, current delivered via tDCS anode electrodes is implemented by the nonhomogeneous Neumann boundary condition

Simulating single neuron transmembrane voltage dynamics was accomplished with the FitzHugh-Nagumo (FHN) model [

FitzHugh-Nagumo model action potential response.

The PDE system (

The multiscale tDCS model was solved with a Godunov operator splitting scheme [

Solve the ODE system:

Solve the PDE system:

The result is numerical solutions of

The ODE system in step

The linear system (

Several of the multiscale tDCS numerical simulations are performed on a three-dimensional grid derived from human MRI data. The SimNIBS software package [

Portions of the computational grid used in multiscale tDCS simulations.

Head boundary of mesh

WM and GM portion of mesh

Finite element solutions were performed with Diffpack. An anisotropic conductivity tensor field for the brain region of the MRI-derived mesh is generated by SimNIBS and stored in a Matlab [

Numerical experiments were contrived to examine the following four properties:

action potential conduction velocity;

tDCS electric potential;

tDCS electric current and field;

tDCS-induced transmembrane voltage increase.

Experiments were run with a global time-step

The multiscale tDCS model was assessed and validated against several two- and three-dimensional numerical experiments. The following subsections describe each of these.

Figure

Two-dimensional geometry used in multiscale tDCS simulations; the gray scale illustrates the electrical conductivity of the different tissue types.

Isotropic extracellular conductivities were assigned to different tissues: skin = 0.465, skull = 0.010, CSF = 1.654, GM = 0.276, and WM = 0.126, each with units S/m [

Individual two-dimensional experiments are described in the following paragraphs.

Action potential conduction: an AP centered at (

tDCS electric potential and field: models of tDCS based on the Laplace equation (

These comparisons are performed using two simulations. First, tDCS was simulated with the anode and cathode electrodes positioned at (

This numerical experiment was repeated with a different electrode configuration, placing the anode electrode at (

Three-dimensional experiments were conducted on an MRI-derived volume mesh. Figure

Three separate electrode montages were selected for the three-dimensional simulations (see Table

Multiscale tDCS three-dimensional numerical experiment electrode configurations, specified using the international 10-20 system.

Anode | Cathode(s) | Target region | |
---|---|---|---|

Montage 1 | C3 | C4 | Motor cortex (ipsilateral to anode) |

Montage 2 | C3 | Fp2 | Motor cortex (ipsilateral to anode) |

Montage 3 | Forehead symmetric | Mastoids (both) | Motor cortex; STN and SN |

In all montages, the anode electric current magnitude was set to 1.0 mA (see Section ^{2} [

Again, the Laplace equation (

Transmembrane voltage results for the AP numerical experiment described in Section

Action potential conduction in two-dimensional geometry.

Transmembrane voltage at time

Transmembrane voltage at time

Figure

Electric potential (

Electric potential at time

Electric potential time-course plots at five points in the domain

These electric potential results are of the same order of magnitude as those reported by Szmurło et al. [

This numerical experiment confirms that the selected parameter set produces biologically reasonable action potential results. Conduction speeds are appropriate and the electric potential resulting from an AP is consistent with previous research reports. In the following sections, the multiscale model is validated when tDCS is administered.

Electric potential (

Laplace-based model

tDCS multiscale model;

tDCS multiscale model;

Figure

Electric field results for the first tDCS electrode configuration: anode at (

Laplace-based model

Multiscale model;

Electric potential (

Laplace-based model

Multiscale model;

Electric field results for the second tDCS electrode configuration: anode at (

Laplace-based model

Multiscale model;

These two experiments demonstrate that the multiscale tDCS model can accurately compute electric potentials and fields when tDCS is administered. In the next section these validations are continued. In addition, the ability of the multiscale model to accurately identify regions of the brain that are electrically excited by tDCS is also demonstrated.

Figure

Multiscale model electric potential and current simulation results using montage 1;

Electric potential on head surface; viewing perspective is from directly above the head with the nasion facing downward

Electric current density and field stream lines from coronal cross-section taken through the anode and cathode electrode centers; viewing perspective is from the posterior

The shunting of the electric field along the scalp and skull is noticeable in Figure

Figure

Transmembrane voltage increase in sagittal cross-section through motor cortex ipsilateral to the anode; viewing perspective is from the left side with head facing left. The arrows in (a) locate the primary motor cortex.

After 1 ms of tDCS administration (Figure

Montage 2 electric potential and electric field results are presented in Figure

Multiscale model electric potential and current simulation results using montage 2;

Electric potential on head surface

Electric current density and field stream lines in a plane intersecting the anode and cathode electrode centers; head is facing towards the left

Figure

Transmembrane voltage increase in plane longitudinally through the motor cortex ipsilateral to the anode; viewing perspective is from the left posterior with the head facing towards the left. The arrows in (a) locate the primary motor cortex.

The multiscale simulation predicts an increase in transmembrane voltage in the motor cortex after 1 ms of tDCS treatment (Figure

Montages 1 and 2 possess similar transmembrane voltage trends in the motor cortex region. The simulations predict that montage 1 will, however, increase the resting membrane voltage in this region approximately 1.5 times that of montage 2. This phenomena can be explained by the fact that the electric current distribution with montage 1 is more confined to this locality due to the closer proximity of its electrodes to each other and to the motor cortex [

Figure

Multiscale model electric potential and current simulation results using montage 3;

Electric potential on head surface

Electric current density and field stream lines in a plane intersecting the anode and cathode electrode centers; head is facing towards the left

Based on the research communities’ suggestion that motor cortex stimulation enhances mobility and movement capabilities in Parkinson’s disease patients (see Section

Transmembrane voltage increase in plane through left motor cortex viewed from the back of the head. The arrows in (a) locate the primary motor cortex region.

Increases in motor cortex excitability are observable at 10 ms (Figure

Next, the increase in membrane resting potential in the subthalamic nucleus and substantia nigra regions (Figure

Transmembrane voltage increase in coronal slice through the STN and SN, viewed from the back of the head. The arrows in (a) locate the STN and SN regions.

Resting membrane voltage increases in these regions are much larger than those seen in the motor cortex, with AP sensitivity values comparable to those attained with montages 1 and 2. After 1 ms of tDCS administration, AP sensitivity increases in the STN and SN regions are observable (Figure

These three-dimensional numerical experiments further validate the multiscale model’s ability to accurately compute the electric potentials and currents generated during tDCS treatments. In addition, using an MRI-derived head geometry and anisotropic tissue conductivities, the ability of the multiscale model to identify regions in the brain that have elevated resting membrane potentials during tDCS treatments with three real-world electrode configurations has been shown.

We have presented a novel, multiscale model of tDCS that couples the mathematics of this procedure to neuronal functioning. The model has been validated against several test cases with comparisons to existing simulations and medical research results. In all of these experiments, the multiscale model accurately simulates tDCS electric potentials and electric fields. We verified the model’s ability to correctly identify those areas of the brain known to be electrically stimulated by specific, real-world tDCS electrode montages. Further, we demonstrated the model’s medical applicability with simulations on a three-dimensional head geometry, derived from MRI data, with anisotropic and inhomogeneous tissue conductivities.

To our knowledge, this paper presents the first multiscale model and simulations of tDCS, which effectively couples cellular-level functionality with tDCS treatment conditions. In addition, our simulation implementation strategies provide an intersection between the tDCS simulation and computational neuroscience communities. In the future, we plan to enhance the fidelity of our simulations with more robust, location-specific neuron models. We also plan to investigate alternative electrode configurations and the numerical methods that most efficiently execute these simulations.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to acknowledge the financial support received from Virginia Tech’s Open Access Subvention Fund and the entire inuTech team for their assistance with Diffpack.