A model for transcutaneous current stimulation: simulations and experiments.
ABSTRACT Complex nerve models have been developed for describing the generation of action potentials in humans. Such nerve models have primarily been used to model implantable electrical stimulation systems, where the stimulation electrodes are close to the nerve (near-field). To address if these nerve models can also be used to model transcutaneous electrical stimulation (TES) (far-field), we have developed a TES model that comprises a volume conductor and different previously published non-linear nerve models. The volume conductor models the resistive and capacitive properties of electrodes, electrode-skin interface, skin, fat, muscle, and bone. The non-linear nerve models were used to conclude from the potential field within the volume conductor on nerve activation. A comparison of simulated and experimentally measured chronaxie values (a measure for the excitability of nerves) and muscle twitch forces on human volunteers allowed us to conclude that some of the published nerve models can be used in TES models. The presented TES model provides a first step to more extensive model implementations for TES in which e.g., multi-array electrode configurations can be tested.
- SourceAvailable from: Matthew A Schiefer[show abstract] [hide abstract]
ABSTRACT: Functional electrical stimulation (FES) can restore limb movements through electrically initiated, coordinated contractions of paralyzed muscles. The peripheral nerve is an attractive site for stimulation using cuff electrodes. Many applications will require the electrode to selectively activate many smaller populations of axons within a common nerve trunk. The purpose of this study is to computationally model the performance of a flat interface nerve electrode (FINE) on the proximal femoral nerve for standing and stepping applications. Simulations investigated multiple FINE configurations to determine the optimal number and locations of contacts for the maximum muscular selectivity. Realistic finite element method (FEM) models were developed from digitized cross sections from cadaver femoral nerve specimens. Electrical potentials were calculated and interpolated voltages were applied to a double-cable axon model. Model output was analyzed to determine selectivity and estimate joint moments with a musculoskeletal model. Simulations indicated that a 22-contact FINE will produce the greatest selectivity. Simulations predicted that an eight-contact FINE can be expected to selectively stimulate each of the six muscles innervated by the proximal femoral nerve, producing a sufficient knee extension moment for the sit-to-stand transition and contributing 60% of the hip flexion moment needed during gait. We conclude that, whereas more contacts produce greater selectivity, eight channels are sufficient for standing and stepping with an FES system using a FINE on the common femoral nerve.IEEE Transactions on Neural Systems and Rehabilitation Engineering 05/2008; 16(2):195-204. · 3.26 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: This computer modelling study on motor cortex stimulation (MCS) introduced a motor cortex model, developed to calculate the imposed electrical potential field characteristics and the initial response of simple fibre models to stimulation of the precentral gyrus by an epidural electrode, as applied in the treatment of chronic, intractable pain. The model consisted of two parts: a three-dimensional volume conductor based on tissue conductivities and human anatomical data, in which the stimulation-induced potential field was computed, and myelinated nerve fibre models allowing the calculation of their response to this field. A simple afferent fibre branch and three simple efferent fibres leaving the cortex at different positions in the precentral gyrus were implemented. It was shown that the thickness of the cerebrospinal fluid (CSF) layer between the dura mater and the cortex below the stimulating electrode substantially affected the distribution of the electrical potential field in the precentral gyrus and thus the threshold stimulus for motor responses and the therapeutic stimulation amplitude. When the CSF thickness was increased from 0 to 2.5 mm, the load impedance decreased by 28%, and the stimulation amplitude increased by 6.6 V for each millimetre of CSF. Owing to the large anode-cathode distance (10 mm centre-to-centre) in MCS, the cathodal fields in mono- and bipolar stimulation were almost identical. Calculation of activating functions and fibre responses showed that only nerve fibres with a directional component parallel to the electrode surface were excitable by a cathode, whereas fibres perpendicular to the electrode surface were excitable under an anode.Medical & Biological Engineering & Computing 06/2005; 43(3):335-43. · 1.79 Impact Factor
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ABSTRACT: Correspondence between the nerve composition and the functional characteristics of its fiber populations is not always evident. To investigate such correspondence and to give a systematic picture of the morphology of the rat hind limb nerves, extensive morphometric study was performed on the sciatic nerve, its founding dorsal and ventral spinal roots, and its major branches. Nerve histology was examined in semithin sections via microscopic image analysis. Variation in the density of myelinated fibers, fiber interspace, and nerve cross-sectional area was studied in individual roots and nerves. In the dorsal roots, fiber numbers and cross-sectional areas were directly linearly proportional to the spinal root level number. Constituent fiber populations were identified using multicomponent lognormal models, and an optimal model for every nerve or root was selected by using an information theoretic approach. For the dorsal and ventral roots and the sciatic and peroneal nerves, optimal fiber population models consisted of three components, whereas, for the tibial and sural nerves, two components were optimal. Functional identities of the revealed fiber populations were established by using calculations of corresponding conduction velocities according to Arbuthnott et al. (J. Physiol.  308:125-157) and anatomical considerations. It is anticipated that morphological parameters established in this study would advance the development of neural prostheses in humans. The proximodistal correspondences among the fiber populations of different nerves were established by parametric statistical comparisons. The proposed approach provides a conceptual framework for understanding the comparative anatomy of the peripheral nerves and spinal roots and can be further applied in other species.The Journal of Comparative Neurology 08/2007; 503(1):85-100. · 3.66 Impact Factor
Med Biol Eng Comput manuscript No.
(will be inserted by the editor)
Andreas Kuhn · Thierry Keller · Marc Lawrence ·
A model for current regulated transcutaneous electrical
stimulation: Simulations and experiments
Received: / Accepted:
Abstract Complex nerve models have been developed for describing the generation of action potentials in
humans. Such nerve models have primarily been used to model implantable electrical stimulation systems,
where the stimulation electrodes are close to the nerve (near-field). To address if these nerve models can also
be used to model transcutaneous electrical stimulation (TES) (far-field), we have developed a TES model
that comprises a volume conductor and different previously published non-linear nerve models. The volume
conductor models the resistive and capacitive properties of electrodes, electrode-skin interface, skin, fat, mus-
Andreas Kuhn · Thierry Keller · Marc Lawrence · Manfred Morari
Automatic Control Laboratory, ETH Zurich
Tel.: +41 44 632 6571
Fax: +41 44 632 1211
Andreas Kuhn · Thierry Keller · Marc Lawrence
Sensory-Motor Systems Laboratory, ETH Zurich
Number of words: 4923
Number of words (abstract): 167
cle, and bone. The non-linear nerve models were used to conclude from the potential field within the volume
conductor on nerve activation. A comparison of simulated and experimentally measured chronaxie values (a
measure for the excitability of nerves) and muscle twitch forces on human volunteers allowed us to conclude
that some of the published nerve models can be used in TES models. The presented TES model provides a
first step to more extensive model implementations for TES in which e.g., multi-array electrode configurations
can be tested.
Keywords Transcutaneous electrical stimulation · finite element model · active nerve model · capacitive
Transcutaneous electrical stimulation (TES) can be used to artificially activate nerve and muscle fibers by
applying electrical current pulses between pairs of electrodes placed on the skin surface. The applied current
flows through the skin and underlying tissues (bulk tissues) where a spatiotemporal potential field is generated
bundles that lie within the bulk tissues experience activation and can generate action potentials (APs) due to
the electrically induced potential field. These APs travel along the axons to the muscle where a contraction
of the muscle is generated. For single stimulation pulses the generated twitch force is increased when the
pulse amplitude or the pulse duration (PD) is increased  because additional axons are recruited in the nerve
Two-step models have been proposed to describe nerve activation in TES . The first step describes
the electrical potential field within the electrodes, the electrode-skin interfaces, and the bulk tissues (volume
conductor). Analytical models [32, 39] finite difference models , and finite element (FE) models  were
used to calculate the potential field in the volume conductor. The second step describes the complex behavior
of the axons’ transmembrane potential (TP), which depends upon the spatiotemporal potential field along
the axon . Several two step models were proposed to describe TES [43, 51, 27]. However, these models
exclusively employ static models (i.e. neglecting capacitive effects) to describe the volume conductor, and
linear nerve models to describe nerve activation. Up to now non-linear nerve models, which can describe more
facets of nerve activation , were mainly used for implantable systems [31, 49, 47], epidural stimulation
, or motor cortex stimulation , where the exciting electrodes are small and close to the nerve (near-
field). To address if these nerve models can also be used to model TES (far-field), we have developed a
TES model that comprises a volume conductor and different non-linear nerve models. Such a model that
describes TES from the applied stimulation current pulse to nerve recruitment is useful for the development
and enhancement of new stimulation technology. For example, the irregular potential fields that are delivered
with multi-channel array electrodes [8, 30] can be described using such models. These irregular potential
fields produced with multi-channel array electrodes can be varied spatially and temporally and require time
varying solutions to describe nerve activation appropriately. In this paper a suitable axon model to be used in
such a TES model is identified and verified with experiments.
A method to experimentally verify electrical stimulation models is to compare simulated strength-duration
curves with experimentally obtained strength-duration data . Strength-duration curves describe the stim-
ulation current amplitude versus the PD for threshold activation. From strength-duration curves rheobase and
a few hundred milliseconds) that produces an activation. Chronaxie is the PD required for activation with an
amplitude of two times the rheobase. Experimentally obtained chronaxie values using electrodes placed close
to the excited axon (clamp experiments, animal studies and needle electrodes) are between 30µs and 150µs
[9, 3, 42]. Published non-linear nerve models were experimentally verified in this range of chronaxie values
[38, 4, 52, 42, 37]. However, chronaxie values that were obtained experimentally using surface electrodes are
longer. In humans the chronaxie values using surface electrodes were found to be between 200µs and 700µs
[48, 15, 23]. It is unclear if the short chronaxie values (30-150µs) of such non-linear nerve models that were
designed for implantable systems are increased significantly when used in a TES model to describe chronaxie
values measured with surface electrodes in TES (200-700µs).
Apart from strength-duration curves, which describe only the excitability at motor threshold (thickest
axons activated), measured force or torque versus PD curves were used to describe the excitability of nerves
(where also thinner axons are activated) . Such measurements at higher stimulation intensities provide
additionally an understanding of the nerve recruitment. These curves show either the stimulation amplitude
versus the PD at a fixed force output [54, 6, 22] or the force versus the PD at a fixed amplitude [1, 14]. The
influence of the muscle properties on the measurement can be minimized by measuring twitch forces (single
stimulation pulse) instead of tetanic forces . This has the advantage that experimentally measured twitch
forces can be directly compared with nerve recruitment obtained from nerve models . As such, we present
twitch force measurements on human volunteers that are compared with the nerve recruitment from our TES
model. This comparison enabled us to conclude, which nerve models are most suitable to be used in TES
2.1 TES model
The developed TES model comprises an FE model that describes the potential field in the volume conductor
(forearm) and an active (non-linear) nerve model that calculates nerve activation. The following two subsec-
tions introduce the two models and how they are linked together.
Finite element model
The electric scalar potential (VFE) within the arm model (volume conductor) and the electrodes was described
by Equation (1), which can be derived from Amperes’s Law. It takes into account both the resistive (σ) and the
dielectric properties (ε = ε0εr) of the tissues. The electrical potentials were calculated with the finite element
time domain (FETD) solver of the FEM package Ansys (EMAG, Ansys Inc., Canonsburg, PA).
) = 0(1)
The two stimulation electrodes were modeled as a good conducting substrate (conductive carbon rubber)
with a 1mm thick electrode-skin interface layer (hydrogel) with a size of 5cm by 5cm and a center to center
spacing of 11cm. These parameters were chosen as in the experimental setup (section 2.2). The amplitudes
and durations of the current-regulated pulses that were applied to the electrodes could be varied. The bulk
tissues were modeled with a multiple layer cylinder (Fig. 1) representing the forearm. A comparison of a
cylindrical model geometry with a more detailed geometry segmented from MRI scans, revealed that nerve
activation did not change significantly (<5%) using the more detailed geometry . Therefore, a cylindrical
(fat), 33.5mm (muscle), 6mm (cortical bone) and 6.5mm (bone marrow). The cylinder had a length of 40cm.
The FE model was verified in  with experimental measurements where the potential on the skin and the
potential in the muscle were measured and compared.
The electrical properties (resistive and capacitive) that were used for the tissues and electrodes are given
in Table 1. The anisotropy of the muscles’ resistivity and permittivity was considered using a factor of three
between the axial and longitudinal direction .
which represented nerve bundle locations that were parallel to the longitudinal axis of the arm model. The
potential at time t and position n on one nerve line is labelledVFE,n(t).
Four different active axon models (see Table 2) were combined with the FE model. These myelinated axon
models were chosen to cover different axon model structures and a wide range of chronaxie values (near-field)
 (published chronaxie values are given in Table 2). Models A and B are based on the Frankenhaeuser-
Huxley membrane  that describe sodium, potassium, and leakage membrane currents  of the nodes
of Ranvier. Model C is the CRRSS model (CRRSS stands for its authors’ names) that only incorporates
sodium and leakage currents at the nodes of Ranvier. The CRRSS membrane is similar to the Hudgkin-
Huxley membrane  but without potassium channels because they were found to be less important in the
excitation process of myelinated mammalian nerves . Model D (MRG model) incorporates a double cable
structure that does not only describe membrane currents at the nodes of Ranvier but also at the paranodal and
internodal sections .
The four introduced axon models describe the TP of a single axon with a certain diameter. However,
in humans axons are gathered in nerve bundles consisting of many axons with different diameters. Multiple
nerve bundles innervate muscles and these nerve bundles lie in different depth within the body . Therefore,
the four axon models (Table 2) were joined to multiple nerve bundles that lie in different depths underneath
the stimulating electrode. The nerve bundles had a length of 15cm and were centered under the cathode
at depths (from skin) between 0.6cm and 1.5cm with 0.1cm spacing (see Fig. 1). Each of the ten nerve
bundles consisted of 100 axons with diameters distributed according to the bimodal distribution in human
nerve bundles with peaks at 6µm and 13µm [46, 41]. The minimal axon diameter was 4µm and the maximal
diameter was 16µm. Recruitment Rec was defined as the percentage of axons that were activated in all
nerve bundles that consisted in total of 1000 axons (10 nerve bundles each 100 axons). Axons with different
diameters had different internodal distances ranging from 0.4mm to 1.6mm. The first nodes of all axons in
each nerve bundle were aligned with each other. Initial fiber activation was verified in order to make sure
that the AP was not initiated at the nerve model boundary. The threshold of the TP to detect activation was
set to 0mV. Additionally, only axons with propagating APs were counted as activated by ensuring that after
detection of the initial AP also at all other nodes the TP was above 0mV.
Nerve models A, B and C (Table 2) were implemented in MATLAB (The Mathworks Inc., Natick, MA)
and Model D in NEURON . Parameters in all nerve models were used as published (references are given
in Table 2). The underlying equation of nerve models A to C is given in (2). Vn(t) is the TP at node n (nodes
of Ranvier) and time t, Ve,n(t) is the extracellular potential, Ii,n(t) is the ionic current, Cmis the membrane
capacitance, and Gais the conductance of the axoplasm (equivalent circuit given on the bottom of Fig. 1).
Nerve model D (MRG) has a more complex structure and also takes into account the extracellular potentials
at non-nodal compartments between the nodes of RanvierVe,n−n(t).
The link between the FE model and the nerve models was established by assigning the time dependent,
spatially interpolated potentials from the FE modelVFE,n(t) to the corresponding extracellular potentials of the
nerve modelVe,n(t) =VFE,n(t). When using nerve model D additionally the non-nodal extracellular potentials
were interpolated in the FE model and assigned to the axon models Ve,n−n(t) =VFE,n−n(t).
2.2 Experimental measurements
Experimental measurements were performed on three human volunteers (age: 25-28, one female, two male) in
order to verify the TES model. Two main aspects of the TES model were verified with two sets of experiments:
motor thresholds were measured in order to compare strength-duration curves (section 3.4), and isometric
twitch forces were measured in order to compare recruitment-duration curves (section 3.5) with results of the
In all experiments rectangular, monophasic current regulated pulses were applied with a Compex Motion
Stimulator . The motor point of the Flexor Digitorum Superficialis that articulates the middle finger was
identified with a stainless steel probe with 0.5cm tip diameter. The probe was moved over the muscles until
the point that required the least current to generate minimal movement of the middle finger was identified.
Because surface motor points move depending on the configuration of the arm, the arm was set up in the iso-
metric condition that was used during the force measurements. Following the identification of the motor point,
the active electrode (cathode) (5cm by 5cm, hydrogel) was placed centered over the identified motor point
and the indifferent electrode (anode) was placed at the wrist. In order to avert potentiation 300 stimulation
pulses were applied prior to data capture.
The motor thresholds were determined by palpation of the region over the muscle. We stimulated with
single pulses of 0.05, 0.1, 0.3, 0.5, 0.7, 1, and 2ms duration. The amplitude of the stimulation pulse was
increased in 0.3mA steps for each PD until motor activation was felt by the examiner. The resting periods
between applying the different pulse durations were 20s.
After a resting period of 1min the isometric twitch forces of the middle finger were measured with the
dynamic grasp assessment system (DGAS) . Single stimulation pulses with an amplitude of 20mA and
with PDs of 0.05, 0.1, 0.3, 0.5, 0.7, 1, and 2ms were randomly applied every 2.5s to 5s. Each PD was applied
a total of six times (randomized). An extract of the raw twitch force measurements is shown in Figure 2. Each
data series was normalized to its maximal value in order to obtain recruitment-duration curves.
2.3 Strength-duration curves, rheobase, and chronaxie
Rheobase Irhand chronaxie Tchfrom strength-duration curves calculated with the TES model were compared
with own experimentally obtained and previously published experimental rheobase and chronaxie values. In
the TES model strength-duration curves were obtained by calculating threshold currents Ithfor PDs of 0.05,
0.1, 0.3, 0.5, 0.7, 1, and 2ms. The threshold amplitudes were determined using bisection search with an
accuracy of 0.01mA. At threshold only the thickest axon model closest to the electrode was activated (axon
diameter: 16µm, depth: 0.6cm). In the experiments motor thresholds were measured as described in section
2.2. Lapicque’s equation Ith= Irh/(1−exp(−PD/Tch))  was fit to the measured strength-duration data in
order to obtain Irhand Tch. R-square (R2) values between the fitted curves and the actual strength-duration
data were calculated to check the accuracy of the fit (all values were below 0.8%).
2.4 Influence of tissue and stimulation parameters on chronaxie
The influence of tissue properties on the chronaxie was investigated in order to find out how the chronaxie
changes for different tissue thicknesses, tissue properties, electrode sizes, and nerve depths. The aim was to
investigate by computer modeling which parameters cause the large range of chronaxie values (200-700µs)
observed in strength-duration measurements with surface electrodes (far-field situation).
Tissue thicknesses of the forearm model were changed in the range of values that cover most human
forearms [50, 45, 53]. The range of thicknesses was: for skin from 1 to 3mm, fat from 2 to 30mm, muscle
from 20 to 60mm, cortical bone from 4 to 8mm, and bone marrow from 4 to 8mm. The range of resistivity
values that were tested are summarized in Table 1 (columns Min and Max) and cover the range of values that
can be expected in practical applications of TES [12, 45, 10, 40]. Electrode size was kept at 5cmx5cm when
changing tissue thicknesses and tissue properties. Afterwards, chronaxie values for electrode sizes between
0.1cm x 0.1cm and 7cm x 7cm and two nerve depths of 0.6cm and 1.5cm were calculated. Electrode sizes
below 0.5cm x 0.5cm are usually not used in TES and were included to allow a comparison of our simulated
chronaxie values with publications that use point sources as electrodes. The changes in chronaxie values were
calculated for all four nerve models (A to D).
2.5 Recruitment-duration curves and time constants
Simulated recruitment-duration curves were compared with experimentally obtained recruitment curves by
comparing the corresponding time constants τ . The time constants τsimof the recruitment-duration curves
from the TES model were compared with the time constants τexpof the experimentally measured recruitment-
duration curves (twitch forces). Both time constants were calculated by fitting equation (3) to the recruitment
data as suggested in .
Rec = Recsat(1−e−(PD−PD0)/τ)
for PD ? PD0
Rec = 0 for PD < PD0
Rec is the recruitment, Recsatis the value where the recruitment saturates, PD represents the stimulation PD,
PD0is the threshold PD above which an AP is generated, and τ is the time constant of the rising recruitment.
R-square (R2) values between the fitted curves and the actual recruitment data were calculated to check the
accuracy of the fit (all values were below 1.5%).
3.1 Chronaxie of TES model
in Fig. 3 (section 2.3). The calculated chronaxie values for the different nerve models are summarized in Table
2. The previously published chronaxie values that were obtained experimentally are shown in the same Table
2 and were determined for implantable systems where small electrodes were close to the nerve. For all nerve
models the chronaxie values of the TES model were higher compared to the published chronaxie values. The
chronaxie values using nerve models A, B, and D were in the range of experimentally obtained chronaxie
values for TES, which are between 200µs and 700µs [48, 15, 23]. The largest increase was found in nerve
model D, where the chronaxie increased by 205% from 150µs to 457µs. The values with nerve model C
(33µs) were too short compared with the experimental range of 200µs to 700µs.
3.2 Influence of permittivities (capacitance)
The influence of the capacitive effects on Rec was investigated with the TES model using nerve model D.
The permittivities (εr) of electrode, skin, fat, and muscle were changed in the range of published experimental
values (Table 1). The results were produced for a stimulation pulse amplitude of 15mA and a PD of 0.3ms
(values that are commonly applied on forearms using TES). Permittivity changes at the electrode-skin inter-
face had no influence on Rec (<0.1%). Skin and fat permittivities changed Rec by 2%. The muscle permittivity
had the largest influence with 5%. Strength-duration and recruitment-duration curves were therefore calcu-
lated for the published range of muscle permittivities (Table 1) in order to identify an upper limit for the
influence of the capacitive effects.
Increasing the muscle permittivity shifted the strength-duration to slightly higher values (Fig. 3). The
chronaxie was increased from 457µs (static volume conductor) by 2% to 466µs for εr= 1.2e5 and by 3.6%
to 474µs for εr= 25e5. These changes are small compared to the large variations of chronaxie values in
experimental measurements of 200µs to 700µs [48, 15, 23].
The recruitment-duration curves for the smallest and largest muscle permittivity are depicted in Fig. 4.
The curves show the percentage of axons that are activated in the nerve model for different pulse amplitudes
and durations. The influence on the time constants τsimwas largest for small pulse amplitudes. At 5mA
the time constant changed from 408µs to 433µs (6%) when increasing the permittivity from 1.2e5 to 25e5.
However, the changes in recruitment due to the capacitive effects of the muscle are small compared to changes
in recruitment caused by PD or pulse amplitude changes.
Fig. 4 also shows the recruitment-duration curve described by a model that uses a static approximation (ca-
pacitive effects of the electrode-skin interface and the bulk tissues neglected) of equation (1) for the FE model.
It can be seen that the curves are nearly congruent with the curves obtained using the model considering the
3.3 Influence of tissue and stimulation parameters on chronaxie
The influence of different tissue and stimulation parameters on the chronaxie was investigated. Changing the
forearms (see section 2.4) resulted in chronaxie changes below 1.1% for all four nerve models (percentage
was calculated relative to the chronaxie values in Table 2). Only the thickest fat layer (30mm) had a larger
influence on the chronaxie (<6.3%). This is due to the spread of the current in the thicker fat layer that
influences more nodes of the nerve models simultaneously (see Discussion).
Changes of electrode size and nerve depth have a larger influence on the chronaxie. The results using nerve
model D are shown in Table 3 where it can be seen that the chronaxie values are in the range from 220µs to
574µs. In general, smaller electrodes and more superficial nerves result in smaller chronaxie values and vice
versa. For nerve models A to C smaller electrodes and superficial nerves also generated smaller chronaxie
values, however, the effect was less pronounced compared to nerve model D. The range of chronaxie values
was 124µs to 171µs using nerve model A, 112µs to 149µs using nerve model B, and 27µs to 36µs using
nerve model C.
3.4 Comparison of simulated with experimental strength-duration curves
The strength-duration curves that were calculated with the TES model containing the four tested nerve models
(A-D) were compared with experimentally obtained motor threshold amplitudes (mean and standard devia-
tion) for different PDs (see Fig. 3). The TES model with nerve model D matched best the experimental
measurements for all measured PDs. The thresholds obtained with nerve models A, B, and C were all at least
a factor of four higher. This was also indicated by the rheobasic currents (Table 2) that were too high for nerve
models A, B, and C compared to the experiments.
3.5 Comparison of simulated with experimental recruitment-duration curves
The time constants τsim(section 2.5) of the recruitment curves from the TES model were compared with
the time constants τexpof the experimentally obtained recruitment curves. The TES models including nerve
model A, B, or C used a current amplitude of 90mA and the TES model with nerve model D used a current
amplitude of 20mA. The current amplitude for models A, B, and C had to be increased due to the higher
rheobasic currents of these nerve models (Table 2).
The normalized recruitment-duration curves obtained from simulations are shown in Fig. 5. The time
constants τsimwere 189µs with nerve model A, 164µs with nerve model B, 19µs with nerve model C, and
476µs with nerve model D. The experimentally measured recruitment curves from the upper extremities are
shown in Fig. 6. The time constants τexpof these curves were 489µs in volunteer one, 240µs in volunteer
two, and 380µs in volunteer three. The time constants τsimobtained with the TES model containing nerve
measurements. The value of τsimderived from using nerve model C was more than a factor of ten shorter than
the shortest τexp.
We developed a two step model that describes the total dynamics from the applied stimulation current pulse
to nerve recruitment for TES. The model enabled us to find out if published nerve models that were used
in many studies for near-field stimulation with implantable electrodes are also suitable to describe far-field
stimulation (TES). This was unclear because of the large discrepancy between the chronaxie values of pub-
lished nerve models (30µs and 150µs) and chronaxie values obtained with surface electrodes (200µs and
700µs [48, 15, 23]). The simulation results in Table 2 show that the chronaxie values were increased when
using the tested nerve models in the TES model. The chronaxie values were increased up to 205% compared
with the chronaxie from publications, which were obtained with electrodes close to the axon. The capacitive
changes are not the reason for the increase as they had only a small influence (<6%). The results show that
the electrode sizes and the electrode/nerve distance have the largest influence on chronaxie values amongst
the tested parameters (Table 3). Smaller electrodes and smaller electrode/nerve distances resulted in smaller
chronaxie values. This was also found in studies that investigated implantable electrodes close to the axon
[44, 45]. Using different combinations of these two parameters in the TES model resulted in a range of 220µs
to 574µs when using nerve model D that described best the published experimental range (200µs and 700µs)
compared with the other tested nerve models (A to C). Possible reasons why nerve model D compared best
with experiments are discussed in section 4.1.
duration curves (Figs. 5 and 6) were compared with experimental measurements. The cylindrical geometry
(tissue thicknesses), which we used (see Methods) was specified such that it compared well to intermediate
values of the three human volunteers (this was achieved using MRI scans in the same three human volun-
teers in an earlier study ). The results in Fig. 3 show that only the strength-duration curve using nerve
model D (MRG-model) compared well with the experiments. The rheobasic currents using nerve models A,
B, and C were too high. The recruitment-duration curves using nerve models A, B, and D compared well with
experiments. The reason why not all nerve models compared well with experimental data can be partially
explained due to different parameter values that were used in some nerve models (see section 4.1). Until now
the MRG-model was exclusively used for implanted ES systems . With our investigations we could show
that the MRG-model can also be used to model transcutaneous ES where the electrode/nerve distances are
much larger than in implanted systems.
4.1 Parameter changes in used nerve models
It was investigated if changes in the parameters of nerve models A, B, and C can increase the chronaxie to
values found with nerve model D. The parameters from nerve model D that mainly influence the chronaxie
(membrane capacity, membrane resistivity, and axoplasm resistivity ) were applied to nerve models A,
B, and C. The chronaxie values of nerve models A, B, and C were at the most increased by 20%. Therefore,
most probably, the double cable structure, with explicit representation of the nodes of Ranvier, paranodal, and
internodal sections  was responsible for the longer chronaxie and not the different parameters of nerve
We showed that the TES models using nerve models A, B, or C had too high a rheobase (section 3.4 and
Fig. 3). The reason might be the different parameters of model D compared with models A, B, and C. The
nodal leakage conductance gLwas 7mS/cm2in D, but 30.3mS/cm2in A and B, and 128mS/cm2in C. The
axoplasmatic resistivity ρiwas 70Ωcm in D instead of 110Ωcm in A and B, and 54.7Ωcm in C. Applying
the parameters from model D in model A and B lowered the thresholds at 0.2ms PD from 36mA to 10mA in
A and from 34mA to 9mA in B. These thresholds are closer to the experimentally obtained thresholds (see
Fig. 3) of 6.3mA. Applying the parameters from model D in model C did not significantly change the motor
threshold found in the TES model containing nerve model C.
4.2 In which cases can the capacitive effects of the volume conductor be neglected?
The high variability of the electrode-skin interface and the published bulk tissue capacitances were found to
have a minor influence on recruitment in TES (Fig. 4). Therefore, the capacitive effects can be neglected,
which is equivalent to setting the time derivative term in equation (1) to zero yielding the Laplace equation.
However, the capacitive effects of the volume conductor have to be considered in the model for the following
- Investigations of time dependent voltage drops in the skin layer (which have a slow rise time): Such inves-
tigations are relevant if new pulse stimulation technologies, as for example presented in , are being
developed. For such cases a model considering the capacitive effects allows one to optimize the power
consumption because both the time dependent currents and the time dependent voltages can be simulated
- Investigation of voltage regulated stimulation: It is important to note that only a simulation that incorporates
the tissues’ capacitance is able to produce reasonable values for the extracellular potential at the axon for
voltage regulated stimulation. The voltage drop in the skin layer increases over time because of the high
skin capacitance  and thus the extracellular potential at the nerve significantly drops during the applied
course of the pulse.
4.3 Spatial position of the nodes of ranvier
Axons with different diameters do not have the same internodal distance. As a consequence the nodes do
not lie at the same position underneath the electrode and could lead to different activation thresholds. To
investigate this we shifted the node of an axon by 0.1mm steps within the internodal distance and could only
observe very small changes of the motor thresholds (<0.01mA). The reason that the shifts did not have an
influence was that the activation peaks were wider than the internodal distance.
4.4 Model limitations
The presented TES model has limitations that should be noted:
- In the presented TES model, nerve activation is calculated in two consecutive steps (FE model and nerve
model). The coupling between the two steps is established by interpolating the potential field calculated
using the FE model along the axons at Ve,n(t) (extracellular potential). This interpolation is discrete in
space and time which could introduce inaccuracies. The spatial interpolation is conducted at the axon
models’ nodes of Ranvier for axon models A-C. In nerve model D additionally an interpolation at the
paranodal and the internodal sections was performed. To ensure a good accuracy of this interpolation
the FE mesh size was refined until no significant change (<1%) was found in the resulting potential
distribution. The temporal interpolation was performed in 1µsec steps, which is much shorter than the PD
(> 50µsec in our model), helping to ensure numerical accuracy.
- The two calculation steps of the TES model (FE model and nerve model) are performed in one direction.
This means that the extracellular potential Ve,n(t) affects the axons’ TP, but the influence of the TP on
the extracellular potential is neglected. Both directions were taken into account for the first time in 
using a bidomain model. It was shown that the TP can influence the extracellular potential in direct muscle
stimulation, however, it was not shown if the generation of APs in adjacent muscle fibers is significantly
influenced. The method is computationally expensive and was therefore used on a simplified volume con-
ductor which was coupled with two muscle fibers . Since, the presented TES model contains a more
detailed volume conductor and 1000 axons, the solution could currently not be computed in reasonable
- The non-linear dependence of the bulk tissue properties to current density was neglected. This is not a major
concern as it was shown that with current-regulated pulses the non-linear properties of the bulk tissues can
be neglected .
- Dispersion of the bulk tissues was neglected. Sensitivity studies  showed that a wide range of tissue
properties did not influence neural activation. This indicates that dispersion can be neglected, too.
- The exact location where the APs are initiated in TES cannot be generalized due to geometrical and physi-
ological diversity. We accounted for that by using multiple nerve bundles at different depths.
A FE model was combined with previously published active nerve models to a TES model. The TES model
allows to describe the total dynamics from the applied stimulation current pulse to nerve recruitment and
serves as a tool to investigate the influences from the geometry, the tissue properties, and new stimulation
distance influence the chronaxie. Our results show that the chronaxie is also in the far-field situation mainly
influenced by the electrode size and the electrode/nerve distance. With electrode sizes between (0.1cm and
7cm) and electrode/axon distances between 0.6cm and 1.5cm chronaxie values between 220µs and 574µs
were obtained. The capacitive effects, variations of the tissue resistivities, and variations of the tissue thick-
nesses have a minor influence.
Simulated strength-duration and recruitment-duration curves using the MRG-nerve-model (model D)
compared well with experimental measurements. We conclude from these results that the MRG-model can be