This document summarizes a study that aimed to replicate the results of a previous paper on selectively stimulating neuronal fibers or cell bodies using different asymmetric biphasic current waveforms. The study developed a multi-compartment Hodgkin-Huxley neuronal model in MATLAB and simulated the response of populations of neurons to different stimulus waveforms. The results showed that an anodic-leading asymmetric biphasic waveform selectively activated fibers, while a cathodic-leading waveform preferentially activated cell bodies, consistent with the previous study.

1. Cell body versus fiber neuronal activation in response to
asymmetric biphasic injected current waveforms
Esha John, Wenze Li, Tasha Nagamine, Steven Yoon
Abstract
The goal of this study was to replicate the results found by McIntyre and Grill in their paper Selective Microstimulation of
Central Nervous System Neurons [3]. This paper explored how electrical stimulation parameters affected neuronal excitation.
In particular, we sought to verify that a biphasic, anodic-leading asymmetrical waveform can be used to selectively stimulate
the fibers of a neuron, while a biphasic, cathodic-leading asymmetrical waveform preferentially stimulates cell bodies.
Key words: Microstimulation, Neural model, BCI
1 Introduction
McIntyre and Grill conducted a comprehensive study
regarding the effect that electrical stimulus waveform
has on neuronal activation. The results they obtained
through computational models demonstrate that alter-
ing the waveform of an injected current pulse into the
brain can selectively stimulate either fibers of passage or
cell bodies. This has important implications for improv-
ing the treatment of brain disorders utilizing microstim-
ulation and the performance of BCIs.
Nonlinear conductance properties of the cell membrane
make it possible to design a stimulus that can selec-
tively initiate action potentials in targeted neuron pop-
ulations. Because electrical activation of fibers may re-
sults in unwanted downstream activation of neural net-
works, it is important to design a stimulus that can se-
lectively stimulate cell bodies. This selectivity of action
potential initiation is a function of electrode-to-neuron
distance, stimulus duration, and stimulus polarity. Us-
ing these relationships, the parameters for a novel stim-
ulation waveform can be designed.
1.1 Current-Distance Relationship
Threshold stimulus for action potential initiation de-
pends on the electrode-to-neuron distance. The thresh-
old stimuli for both initial segment and axon node were
experimentally determined by applying a range of am-
plitudes for a given electrode-to-neuron distance.
Two types of motoneurons were considered in the study,
type S (innervating slow twitch muscle fibers) and type
Fig. 1. Current-Distance Relationship. Corresponds to figure 2 in [3]
FR (innervating fast twitch fatigue resistant muscle
fibers). Because both types of CNS neurons were ob-
served to have identical current-distance relationships,
the study lumps the initial segment of FR and S as the
cell body, and the axon nodes of FR and S as the cell
fiber.
Preprint submitted to Automatica 9 December 2013

2. 1.2 Population-Based Approach
To properly assess selective activation of targeted pop-
ulations, a population-based simulation approach was
needed. In NEURON, fifty fibers of passage (25 10 µm,
15 µm diameter) and fifty cells (25 FR-type, 25 S-type)
were distributed randomly in the center of a 300 µm di-
mension cube. In MATLAB, a model was set at a ran-
dom relative distance from the electrode and simulated
repeatedly to approximate a randomly distributed pop-
ulation.
Fig. 2. Randomly Distributed Population-Based Approach
Corresponds to figure 8 in [[3]]
This study tested selectivity for six different stimulation
waveforms, and the results are summarized in Fig.(fig3).
If V m from a compartment was observed to be over the
threshold voltage Vthres = 0, then it was considered ac-
tivated. For a particular stimulus design, the percentage
of activated neurons was displayed as a function of stim-
ulus amplitude.
Fig. 3. I/O relations for neuron populations stimulated by a
monopolar electrode. Corresponds to figure 7 in [3].
1.3 Stimulation
Two important observations can be made from Fig.(fig3)
with direct implications towards optimal stimulation
design. First, as observed in A and B, rectangular
monophasic pulse stimuli has high selectivity between
fibers and cells. Second, as observed in C and D, charge-
balanced, symmetric biphasic pulse stimuli from the
same electrode reduces selectivity between fibers and
cells.
To retain high selectivity while being charge-balanced,
the duration of the primary phase must be long while
the secondary phase must be short. This is to make
sure the proper activation site is excited during the pri-
mary phase of the stimulus. As observed in E and F, the
charge-balanced biphasic stimuli waveforms were able to
retain selectivity at desired levels.
1.4 Other Considerations
McIntyre and Grill determined that action potential ini-
tiation is not affected by intracellular resistivity value of
the axon, cell body, and dendrites. Because the geome-
try and active conductance of dendrite trees also had no
effect, the dendrites were modeled as a passive circuit.
Among conductance values, only gNa of the axon node
and initial segment had noticeable effects on selectivity.
It may be of interest to see if changing gNa values for the
fiber or cell body would significantly boost the selectivity
of a stimulus design.
2 Objectives
First, we aim to recreate the described NEURON model
in MATLAB code. The model will demonstrate action
potential initiation and propagation behavior in terms
of compartment number and time.
Second, we aim to demonstrate population selectiv-
ity of the novel stimulus waveforms by repearing the
population-based simulation approach as described in
the paper.
Stimulus 1: Anodic first (duration = 0.2 ms), cathodic
second (duration = 0.02 ms)
Stimulus 2: Cathodic first (duration = 1 ms), anodic
second (duration = 0.1 ms)
3 Methods
To replicate the findings described by McIntyre and
Grill, a Hodgkin-Huxley multi-compartment cable
2

3. Fig. 4. Novel Stimuli
model was implemented using MATLAB. The neu-
ral model included a 15-node myelinated axon, an
initial segment, a 6-compartment soma, and a three-
dimensional branching dendritic tree. For a more de-
tailed description of the model geometry, refer to the
appendix. Each compartment was represented by an
equivalent electrical circuit. The dendritic tree was
represented by linear membrance capacitances and con-
ductances in parallel. Due to contributions from ion
channels, the soma, initial segment, and axon were all
modeled using nonlinear membrane dynamics.
The results of this study were obtained by applying
an extracellular current pulse and determining whether
or not neural activation occurred. To find the mem-
brane potential of the active compartments, three cur-
rent sources had to be taken into account: intracellular
current due to the applied extracellular potential, in-
tercompartmental membrane current, and the Hodgkin-
Huxley ion currents. The dendrites were modeled as pas-
sive currents and ionic currents were neglected.
3.1 Extracellular Electric Field
The stimulating electrode was modeled as an extracel-
lular point source in an infinite homogeneous medium.
The extracellular potential at each segment is thus given
by
V (n) =
Iextρext
4π([X(n) − Xe]2 + [Y (n) − Ye]2 + [Z(n) − Ze]2)1/2
(1)
Here, Xe, Ye, and Ze are the coordinates of the electrode
relative to compartment n, Iext is the amplitude of the
stimulating current pulse, and ρext is the extracellular
resistivity.
3.2 Intracellular and Intercompartmental Currents
The equations describing the intracellular current due to
applied extracellular voltage and the intercompartmen-
tal currents took the same general form and are given
by
I(n) =
V (n − 1) − V (n)
Ri(−)
+
V (n + 1) − V (n)
Ri(+)
(2)
For intracellular current due to the external voltage,
I(n) = Iint(n) and V (n) is given by (1). For intercom-
partmental current, I(n) = Im(Vm, n, t) and V (n) =
Vm(n) (transmembrane potential). Ri(−) is the inter-
segmental resistance between the n and n - 1 compart-
ment and Ri(+) is the corresponding resistance between
the n and n + 1 compartment.
This cable equation had to be modified in three special
cases where branching geometries occurred in the model.
For branching dendrites, current was given by
I(n) =
V (n − 1) − V (n)
Ri(−)
+
V (n + 1)branch1 − V (n)
Ri(+)
+
V (n + 1)branch2 − V (n)
Ri(+)
(3)
For soma compartment S[0]
I(S[0]) =
[V (IS) − V (S[0])]
RIS
+
RS
RS + 5RS[0]
V (S[0])
RS[0]
+
5
j=1
V (S[j])
Rs
−
V (S[0])
RS[0]
(4)
For soma compartments S[i], i [1, 5]
I(S[i]) =
[V (D[i]) − V (S[i])]
RD
+
RS[0]
RS + 5RS[0]
V (S[0])
RS[0]
+
5
j=1
V (S[j])
Rs
−
V (S[i])
RS
(5)
Finally, at each compartment the transmembrane volt-
age Vm(n, t) was calculated using the nonlinear differen-
tial equation
Cm(n)
dVm(n, t)
dt
+ Gm(Vm, n, t)Vm(n, t)
− Im(V + m, n, t) = Iint(n) (6)
Here, Gm(Vm, n, t) represents the membrane conduc-
tance.
3.3 Hodgkin-Huxley Ionic Currents
The Hodgkin-Huxley model was employed to find
Gm(Vm, n, t) and the corresponding ionic currents. For
each ion, the ionic current can be described by the
following equations.
Iion = gion(Vm − Eion) (7)
INa = gNam3
h(Vm − ENa)
INa,p = gNa,pm3
h(Vm − ENa)
IK = gK n4
(Vm − EK ) (8)
IK,s
1 = gK,ss(Vm − EK ) (axon node)
IK,s = gK,sq2
(Vm − EK ) (soma)
3

4. Fig. 5. Cable model of a spinal motoneuron Corresponds to
figure 1 in [[3]]
gion is the maximum conductance for an ion channel
multiplied by a gating variable. In general, each gating
parameter ω is governed by the following equations
dw
dt
= αw(1 − w) − βw(w∞ − w) =
w∞ − w
τw
(9)
τw =
1
αw + βw
(10)
w∞ =
αw
αw + βw
(11)
3.3.1 Axon Node
The Q10 scaling factor in this section serve to convert
parameters initially defined at 20 ◦
C to the correspond-
ing values at the model temperature 36 ◦
C.
1
From eq. 4 in [4]
Fast Na+
current:
gNa,f = 3.0
S
cm2
ENa,f = 67.0mV
Q10,m = 2.2
Q10,h = 2.9
αm =
1.86(Vm + 8.4)
1.0 − exp(
−(Vm+8.4)
10.3
)
(12)
βm =
−0.086(Vm + 12.7)
1.0 − exp( Vm+12.7
9.16
)
(13)
αh
2 =
0.0336(Vm + 101)
1.0 − exp( Vm+101
11.0
)
(14)
βh =
2.3
exp(
−(Vm+18.8)
13.4
) + 1.0
(15)
Persistent Na+
current:
gNa,p = 0.01
S
cm2
ENa,p = 67.0mV
Q10,m = 2.2
αm =
1.86(Vm + 31.4)
1 − exp(
−(Vm+31.4)
10.3
)
(16)
βm
3 =
0.086[−(Vm + 25.7)]
1 − exp( Vm+25.7
9.16
)
(17)
Slow K+
current:
gK,s = 0.08
S
cm2
EK,s = −67.0mV
Q10,s = 3.0
αs =
0.00122(Vm + 2.5)
1.0 − exp(
−(Vm+2.5)
23.6
)
(18)
βs =
−0.00074(Vm + 70.1)
1.0 − exp( Vm+70.1
21.8
)
(19)
Leakage current:
EL = −67.0mV
gL = 0.08
S
cm2
2
αh taken from table 1 in [4]
3
βm taken from table 1 in [4]
4

5. 3.3.2 Initial Segment
Na+
current:
gNa = 0.5
S
cm2
ENa = 50.0mV
αm =
4.0 − 0.4Vm
exp(Vm−10.0
−5.0
) − 1.0
(20)
βm =
−14.0 + 0.4Vm
exp(Vm−35.0
5.0
) − 1.0
(21)
αh =
0.16
exp(Vm−37.78
−18.14
)
(22)
βh =
4.0
exp(Vm−30.0
−10.0
) + 1.0
(23)
Fast K+
current:
gK,f = 0.10
S
cm2
EK,f = −75.0mV
αn =
0.2 − 0.02Vm
exp( Vm−10.0
−10.0
) − 1.0
(24)
βn =
0.15
exp( Vm−33.79
71.86
) − 0.01
(25)
Leakage current:
EL = −65.0mV
gL = 0.0066
S
cm2
3.3.3 Soma
Na+
current:
gNa = 0.1
S
cm2
ENa = 50.0mV
αm =
7.0 − 0.4Vm
exp(Vm−17.5
−5.0
) − 1.0
(26)
βm =
−18.0 + 0.4Vm
exp(Vm−45.0
5.0
) − 1.0
(27)
αh =
0.15
exp(Vm−34.26
18.19
)
(28)
βh =
4.0
exp(Vm−40.0
−10.0
) + 1.0
(29)
Fast K+
current:
gK,f = 0.035
S
cm2
EK,f = −75.0mV
αn =
0.4 − 0.02Vm
exp( Vm−20.0
−10.0
) − 1.0
(30)
βn =
0.16
exp( Vm−33.79
66.56
) − 0.032
(31)
Slow K+
current:
EK,s = −75.0mV
αq =
3.5
exp(Vm−55.0
−4.0
) + 1.0
(32)
S motoneuron:
gK,s = 0.012
S
cm2
βq =
0.015
exp(Vm+50.0
−0.001
) + 1.0
(33)
FR motoneuron:
gK,s = 0.035
S
cm2
βq =
0.0035
exp(Vm+50.0
−0.001
) + 1.0
(34)
Leakage current:
EL = −65.0mV
S motoneuron:
gL = 0.0022
S
cm2
FR motoneuron:
gL = 0.0066
S
cm2
3.3.4 Dendrite
Leakage current:
EL = −65.0mV
S motoneuron:
gL = 0.000022
S
cm2
FR motoneuron:
gL = 0.000083
S
cm2
4 Results
To generate figures 7E and 7F presented in [3], the same
population-based approach was taken with 40 neurons
(20 cells and 20 fibers of passage) generated at random
positions within a 300 µm cube. Current amplitudes
ranging from 1 to 1000 µA were considered utilizing a
5

6. Fig. 6. Plot of membrane voltage of axon compartments in
response a monophasic current stimulus of amplitude 100µA.
Initial segment(n=16) is the compartment closest to stimu-
lating electrode.
logarithmic scale. Activation threshold was determined
for each neuron in the simulation and used to construct
a curve of current amplitude versus percent activation
of neuron type.
Figure 6 shows the response of the axon compartments to
a 100 µA monophasic current stimulus. The electrode is
closest to the initial segment (n = 16). Thus it is observed
that the initial segment fires first and all the segments
follow in succession as the impulse is propagated along
the axon. It was observed that the membrane potential
was restored to resting in about 1.5 ms.
Figure 7 shows the percentage of fibers and cell bodies
activated versus the stimulating current amplitude us-
ing the two different biphasic waveforms. Figure 7A was
generated using a charge balanced biphasic waveform
with an anodic stimulus followed by a cathodic stimu-
lus. This corresponds to Figure 7E in the McIntyre and
Grill paper. The trend observed is similar in both fig-
ures. The activation percentage of both cells and fibers
increase as function of stimulus amplitude. However, we
see that a greater percentage of cells were activated for
each stimulus amplitude unlike what was seen in their
paper. In Figure 7B, we see the responses to a charge
balanced biphasic stimulus with the cathodic stimulus
leading the anodic stimulus. This figure matches Figure
7F in the other paper rather accurately. The cell bodies
are selectively activated by this waveform as compared
to the fibers. The paper demonstrates that the two dif-
ferent waveforms selectively stimulate either cell body or
fiber; however, it did not offer any explanation for why
this might be expected beyond a vague and implication
that this was due to nonlinear conductance properties.
Fig. 7. Final results. Input-output relations for populations
of neurons stimulated by a monopolar electrode. Each pop-
ulation of cells consisted of 10 FR and 10 S motoneurons,
and each population of fibers consisted of 10 10µm and 10
5µm diameter fibers of passage. Percentages of activated neu-
rons from random distributions are displayed as a function
of stimulus amplitude. (A)Anode first(pd=0.20ms), cathode
second(pd=0.02ms) charge balanced biphasic stimulus; (B)
cathode first(pd=1.00ms), anode second(pd=0.10ms) charge
balanced biphasic stimulus; (C)activation of cells for two
stimulus; (D)activation of fibers for two stimulus
This makes it difficult to determine what aspect of our
model resulted in this selectivity not being seen. How-
ever, we do see a shift in the percentage activation in
both the cell bodies and the fibers from one waveform to
the other. Figure 7C shows the response of the cell bod-
ies to the different waveforms. The red trace shows the
response to the cathodic stimulus followed by the an-
odic stimulus. As we can see a higher percentage of cells
are activated by this stimulus. Similarly in Figure 7D,
we see that the percentage of fibers activated by the an-
odic first, cathodic second stimulus is higher than with
the other stimulus which is exactly as predicted by the
McIntyre and Grill paper. The type of waveform seems
to make a larger difference to activation of cell bodies
than for the fibers.
We were able to replicate the trends observed regarding
the percentage activation of both cell bodies and fibers
to the stimulating amplitude. We were even able to show
that moving from an anodic first, cathodic second stim-
ulus to an cathodic first, anodic second stimulus resulted
in an increase in the activation of cell bodies and a de-
crease in the activation of fibers. However, we were un-
able to show that the anodic first stimulus selectively
activated the fibers over the cell bodies.
6

7. 5 Discussion
Replicating the selectivity results in [3] for charge-
balanced, biphasic stimuli proved challenging. One of
the major hurdles we encountered was the presence of
errors in several of the equations for the gating variables
in the axon node model (see appendix for a list of in-
correct equations). The Hodgkin-Huxley model is very
sensitive to input parameters; small mistakes in any of
the equations resulted in unstable results in the axon
node. We were able to eventually remedy these mistakes
by cross-referencing all the gating variable equations
with other publications ([1],[4]) and making corrections
as needed. However, the results obtained from our simu-
lation do not mirror those results obtained by McIntyre
and Grill, and we cannot rule out the possibility that
some error persists in the equations for one or more of
the gating variables.
We also found it necessary to choose a sufficiently
small integration time step when solving the nonlinear
differential equations used in the model. Time steps
larger than 0.0001 ms resulted in truncation error that
produced unstable models, with gating variables and
transmembrane voltages quickly approaching infinity.
This was an order of magnitude higher than the time
step used by McIntyre and Grill, which was 0.001 ms
while implementing the Crank-Nicholson implicit inte-
gration method (a second-order Runge-Kutta method)
in the NEURON simulation package. Our results were
obtained using the Euler method, which is subject to
higher truncation errors, as well as the need for smaller
time steps and thus much longer computational run-
times. We predict that our simulation would be much
more computationally efficient and less prone to unsta-
ble behavior if we had implemented more sophisticated
integration method.
One of the limitations we discovered in the simulation
that was not discussed in [3] was sensitivity to electrode
position relative to the model neuron. From equation (1)
it is clear that at small distances the intracellular po-
tential becomes very large; above a certain threshold of
injected current the model becomes unstable and trans-
membrane voltage grows without bound. This presented
an issue when attempting to replicate figure 7F, in which
stimulus amplitude is varied in order to obtain a percent
activation curve for the population of neurons. Because
electrode position is generated randomly between trials,
stimuli with larger amplitudes were more likely to result
in instability.
We believe that small errors in the equations we used
for the gating variables are the most probably explana-
tion for the inability of our simulation to replicate the
findings from McIntyre and Grill’s figure 7E. After find-
ing that our model did not yield the expected result, we
began a more extensive literature search for deviations
between the equations given to us and those in other
papers. In [2], we found that equations (22), (28), (33),
and (34), did not match the corresponding equations
(A8), (A16), (A23), and (A26) given on page 419. Pro-
hibitively long runtimes (over 8 hours) for the simula-
tion prevented us from attempting to use the equations
from [2].
A Incorrect Equations
Because several of the equations presented in [3] for the
axon node currents were incorrect, the corresponding
equations were taken from another publication and the
rest of the model equations cross-referenced to ensure
accuracy [4],[1]. The incorrect equations are presented
below.
αh =
−0.0336(Vm + 101)
1.0 − exp(−(Vm+101)
11.0 )
(A8)
βm =
−0.086(Vm + 12.7)
1.0 − exp(Vm+12.7
9.16 )
(A12)
gK,s = gK,s(Vm − EK) (A13)
As mentioned in the discussion, [2] had a few discrepan-
cies for the initial segment and soma equations.
αh,IS =
0.16
e
−(Vm−37.78)
18.14
(A8)
αh,soma =
0.15
e
−(Vm−34.26)
18.19
(A16)
βq,soma,S =
0.015
e
−(Vm+50)
0.001 − 1
(A23)
βq,soma,F R =
0.0035
e
−(Vm+50)
0.001 − 1
(A26)
B Model Parameters
The following tables describing the model parameters
were taken from [3].
References
[1] Kristian Hennings, Lars Arendt-Nielsen, and Ole Andersen.
Breakdown of accommodation in nerve: a possible role for
persistent sodium current. Theoretical Biology and Medical
Modelling, 2(1):16, 2005.
[2] Kelvin E Jones and Parveen Bawa. Computer simulation of
the responses of human motoneurons to composite 1a epsps:
effects of background firing rate. Journal of neurophysiology,
77(1):405–420, 1997.
7

8. [3] Cameron C McIntyre and Warren M Grill. Selective
microstimulation of central nervous system neurons. Annals
of biomedical engineering, 28(3):219–233, 2000.
[4] J¨urgen R Schwarz, Gordon Reid, and Hugh Bostock. Action
potentials and membrane currents in the human node of
ranvier. Pfl¨ugers Archiv, 430(2):283–292, 1995.
8