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Pre-Filtering of Stimuli for Improved Energy
Efficiency in Electrical Neural Stimulation
Francesc Varkevisser∗, Amin Rashidi∗, Tiago L. Costa∗, Vasiliki Giagka∗† and Wouter A. Serdijn∗‡
Email: {f.varkevisser,amin.rashidi,t.m.l.costa,v.giagka,w.a.serdijn}@tudelft.nl
∗Section Bioelectronics, dept. Microelectronics, Delft University of Technology, Delft, The Netherlands
†Technologies for Bioelectronics Group, dept. of System Integration and Interconnection Technologies
Fraunhofer IZM,Berlin, Germany
‡Neuroscience Dept., Erasmus Medical Center, Rotterdam, The Netherlands
Abstract—This work proposes a guideline for designing more
energy-efficient electrical stimulators by analyzing the frequency
spectrum of the stimuli. It is shown that the natural low-pass
characteristic of the neuron’s membrane limits the energy trans-
fer efficiency from the stimulator to the cell. Thus, to improve the
transfer efficiency, it is proposed to pre-filter the high-frequency
components of the stimulus. The method is validated for a
Hodgkin-Huxley (HH) axon cable model using NEURON v8.0
software. To this end, the required activation energy is simulated
for rectangular pulses with durations between 10 µs and 5 ms,
which are low-pass filtered with cut-off frequencies of 0.5-50 kHz.
Simulations show a 51.5% reduction in the required activation
energy for the shortest pulse width (i.e., 10 µs) after filtering at
5 kHz. It is also shown that the minimum required activation
energy can be decreased by 11.04% when an appropriate pre-
filter is applied. Finally, we draw a perspective for future use of
this method to improve the selectivity of electrical stimulation.
Index Terms—neuron modeling, electrical stimulation, neuro-
modulation, energy transfer efficiency, frequency spectrum
I. INTRODUCTION
Implantable electrical stimulators have proven to be life-
changing devices for many patients. Commercially available
pacemakers, cochlear implants, and deep brain stimulators
are some successful examples. However, there are safety and
practical constraints on the energy that can be transferred to
and/or stored on such devices [1]. As most of this energy is
used by the stimulator circuitry, its power efficiency can play
a role in prolonging the functional lifetime of a stimulator
device (e.g., battery-powered implants [2]), improving the
quality of prosthetic function (e.g., more stimulation sites in
visual prostheses [3]), or miniaturizing wireless implants (e.g.,
smaller receiving aperture in mm-sized implants [4], [5]).
Improving the stimulation energy efficiency requires a good
understanding of neural reactions to the stimulation pulses.
Activation behavior of neuronal cells has been researched
extensively, using a wide variety of analytical and bio-realistic
models with different levels of accuracy and complexity [6]–
[9]. Hodgkin and Huxley (HH) derived a mathematical de-
scription of the non-linear conductance behavior of voltage-
gated ion channels in the membrane of a squid giant axon
[6]. McNeal’s model [7] and the MRG model [8] use single
and double cable models, respectively, with HH-channels in
This work is part of the INTENSE project funded by the Dutch Research
Council (NWO) under grant number 17619, and the Moore4Medical project
funded by the ECSEL Joint Undertaking under grant number H2020-ECSEL-
2019-IA-876190
Fig. 1: The linearized membrane model for sub-threshold conditions in the
leaky integrate-and-fire model, with membrane capacitance Cm, leakage
conductance gL, rest potential Vrest, and the input stimulus x(t).
the active sections to describe extracellular activation of a
myelinated axon fiber. In these models, the HH-like channel
mechanisms are tuned to reproduce activation behavior mea-
sured in experimental data.
Even though the aforementioned models can give accu-
rate descriptions of neuron activity, the non-linearity in the
ion gate descriptions requires differential equation solvers
to predict the response to a given stimulus. Therefore, the
computational cost is high, especially in the case of morpho-
logically accurate models or in multi-cell simulations [10].
Simplifying the cell membrane to a linear model makes the
prediction of the responses much easier, and a large portion
of response phenomena can be accurately explained using the
linearized model [11]. A widely used model that assumes
linear membrane properties is the Leaky Integrate-and-Fire
(LIF) model [9]. In the model, sub-threshold depolarization
of the membrane is described by a passive parallel network
of a membrane capacitance and a membrane resistance, see
Fig. 1. Due to the RC network, the depolarization in sub-
threshold conditions as a response to an extracellular stimulus
is described by an equivalent membrane time constant, τe.
When the membrane voltage reaches a predefined threshold
value, the model generates an action potential — a spike —
and resets the membrane voltage, Vm, to its rest value. This
simplified model neglects more complex known properties of
cell activation. Nevertheless, it is still widely used to describe
cell dynamics in electrical stimulation [12]. For instance, [13]
summarizes how τehas been used in the literature to estimate
strength-, charge-, and energy-duration curves in time-domain
analysis.
From a different perspective, the LIF model suggests a first-
order low-pass transfer function for the neuron’s membrane,
which forms the basis of this work. In this paper, we show that
the low-pass characteristic of the neuron affects the stimulation
Fig. 2: Diagram of the signal transfer from the stimulator to the neuron.
efficiency dependent on the stimulation waveform. Thus, it is
proposed to employ this knowledge to improve the perfor-
mance of electrical stimulators. In the following sections, the
basic concept of the proposed method is explained, and some
circuit design guidelines are provided. Then, the potential
energy savings are validated using a computational model
of an axon fiber. Finally, the conclusion is provided with a
discussion on present limitations and future improvements.
II. BAS IC CO NC EP T
The basic concept of this work originates from the fact
that the neural membrane has a low-pass characteristic in sub-
threshold conditions. As a consequence, high-frequency com-
ponents of a stimulus do not contribute to the depolarization
of the cell. Therefore, we propose to pre-filter the stimuli to
save energy and reduce the energy delivered to the tissue. As
explained in the next section, the losses in the pre-filtering
process can be negligible with careful circuit design. Here,
the stimuli and the filtering effects are analyzed in the Laplace
domain. The total signal transfer function is depicted in Fig. 2,
where X(s) is the unfiltered stimulus in the frequency domain,
G(s) is the introduced pre-filter for energy saving, X′(s) is
the pre-filtered input signal, H(s) represents the low-pass filter
(LPF) transfer function of the neuron, and Y(s) is the effective
signal delivered to the cell. In this work, both G(s) and H(s)
are first-order low-pass filters with corner frequencies fc,G and
fc,H , respectively. One can derive fc,H from Fig. 1 as fc,H
= 1/(2πτe), with τe=Cm/gL. The energy delivered to the
cell and the energy of the input signal are calculated using (1)
and (2), respectively. For the case without filtering, G(s) = 1.
Consequently, the energy transfer efficiency is defined by (3).
Eout =Z∞
−∞
|Y(s)|2df (1)
Ein =Z∞
−∞
|X′(s)|2df (2)
ηE=Eout
Ein
·100% (3)
If fc,G is chosen such that the total transfer function is
dominated by H(s), the input energy (i.e., the energy of X′(s))
is reduced with a small effect on the energy of Y(s). In
that case, when the stimulation pulse is short, non-effective
high-frequency components can be pre-filtered, which will
improve energy efficiency. This is illustrated in Fig. 3 for the
example of a pre-filter corner frequency at fc,G = 3·fc,H and
a rectangular stimulation pulse with a duration of 1.25·τe. In
this particular case, ηEis 42.9% and 52.8% for the unfiltered
and pre-filtered cases, respectively, indicating an efficiency
increase of (52.8 −42.9)/42.9 ·100% = 23.1%. It can be
10-1 10010 1102
Normalized Frequency
0
0.2
0.4
0.6
0.8
1
1.2
Normalized Power Density
|X(s)|2
G(s)
H(s)
(a)
10-1 10010 1102
Normalized Frequency
0
0.2
0.4
0.6
0.8
1
1.2
Normalized Power Density
|X (s)|2
|Y(s)|2 with prefiltering
|Y(s)|2 without prefiltering
(b)
Fig. 3: (a) Spectrum X(s) of rectangular input signal x(t), neuron’s bandwidth
H(s) and pre-filter bandwidth G(s). (b) Spectrum X’(s) of the pre-filtered input
signal x’(t) and the spectra of the resulting output signals with and without
pre-filtering. In this example, the frequency axis is normalized to fc,H,fc,G
= 3·fc,H , and the signal is a rectangular pulse with a duration of 1.25·τe.
imagined that for long pulses, almost all frequency compo-
nents fall within the natural bandwidth of the membrane, and
thus filtering will have a negligible effect on energy saving.
Finally, if the corner frequency of the pre-filter is chosen too
low, it will limit the bandwidth of the delivered energy to the
tissue and thereby increase the required activation energy.
There are multiple approaches to estimate the neurons’
equivalent time constant [14]. The first method uses curve
fitting to the Strength-Duration (SD) curve, either to the
’hyperbolic’ SD relationship (4) [15] or to the ’logarithmic’
SD relationship (5) [9], where I0— the rheobase current — is
the current threshold for infinitely long pulses, PW the pulse
width, and τethe equivalent time constant of the cell [14].
Another method uses (6), where Q0is the charge threshold
for short pulses and I0is the rheobase current [14]. Due to the
linearization of the model, the time constant will only be an
approximation. Therefore, all three time-constant estimation
methods are sufficiently accurate.
Ith(P W ) = I0·(1 + τe
P W )(4)
Ith(P W ) = I0
1−exp (−P W/τe)(5)
τe=Q0
I0
(6)
III. CIRCUIT IMPLEMENTATION
Based on the previous section, it is possible to achieve
system-level energy savings by adequately implementing a
programmable
pre-filter
Waveform
generator
Amp Current
Driver
Scaled Supply
Rail(s)
Stimulation controller
Shape
Amp.
PW
fC
H-Bridge
Stimulator Circuit
Fig. 4: Block diagram of a current-controlled stimulation output stage with a
programmable low-pass filter
pre-filter block before delivering the stimuli to the tissue.
Figure 4 illustrates one example of the stimulator circuit
implementation in which a stimulation waveform is applied to
a programmable low-pass filter before delivering the stimuli
through a current driver. Here, it is assumed that the current
driver linearly converts the pre-filtered signal to the stimulation
current. In case of close-to-ideal loading of the filter (i.e., with
either a large input impedance in case of a voltage filter or a
small input impedance in case of a current filter), the power
consumption of the LFP can be optimized to be negligible.
Since the bandwidth of the target neuron is dependent on
many factors (e.g., type of nerve, distance from the nerve,
orientation with respect to the nerve, the type of tissue in
between the electrodes and the nerve, etc.), it is proposed to
program the LPF corner frequency (fc) by the stimulation
controller. In other words, it is proposed to apply a more
efficient and selective stimulation waveform by introducing
a new single stimulation parameter, fc, in addition to the
already known pulse parameters (e.g., duration and amplitudes
of the stimuli in anodic and cathodic phases). Notably, the
applied stimulation waveform won’t be rectangular anymore,
and scaling of the supply rail(s) is recommended. The authors
have discussed this in [16].
IV. VALIDATION
A. Validation Method
To validate the proposed concept, the current thresholds
for different pulse configurations are analyzed in a Hodgkin-
Huxley (HH) axon cable model. The axon is modeled in the
NEURON v8.0 software [17]. It consists of 101 active nodal
sections connected by 100 passive inter-nodal sections, rep-
resenting a myelinated axon. The active sections contain five
HH-like ion channels — transient and persistent sodium and
potassium, and A-type potassium (Kv3.1) — which were tuned
to the response of a human cortical L5 pyramidal cell in [18].
The model parameters are listed in the Appendix. Extracellular
electrical stimulation is modeled for a point-source electrode
at a distance of 100µm above the center node of the axon.
The extracellular potential along the axon is calculated using
(7), where σis the extracellular conductivity, and ris the
distance to the electrode. The potential is applied to each active
segment using NEURON’s ’extracellular’ mechanism. During
each simulation step, the potential is scaled according to the
amplitude of the applied stimulus.
Ve(r) = 1
4πσr (7)
All simulations were executed using implicit Euler integra-
tion using a time step of 0.1 µs. A binary search algorithm
is employed to determine the threshold amplitude for a pulse
configuration to an accuracy of 10−2µA. If the membrane
voltage of the outermost node crosses 0 mV, the stimulus
is considered supra-threshold. Rectangular, monophasic, ca-
thodic pulses are used as stimuli, with stepped pulse widths
(PWs) in steps of 10 µs for the range 10 µs to 500 µs and
steps of 100 µs for the range 500 µs to 5 ms.
To estimate the equivalent membrane time constant (τe),
the strength duration data of the unfiltered pulses (0.1-5 ms in
steps of 0.1 ms) is fitted to (4) and (5) using non-linear least
squares.
Next, the rectangular pulses were pre-filtered with a 1st-
order low-pass filter with corner frequencies ranging from 0.5
to 50 kHz, and the activation thresholds for the resulting pulses
were determined for a range of PWs between 10 µs and 5 ms.
The required energy for activation for each pulse configura-
tion is calculated using (8) [19], where I(t)is the applied
stimulus at threshold intensity. This calculation assumes a
purely resistive load of the stimulator system.
Eth =Z∞
0
I(t)2dt (8)
Finally, we assess the relative optimal energy thresholds
between filtered and unfiltered pulses by comparing the mini-
mum required activation energy for both cases using (9), where
Efc,xis the energy curve for the pulses filtered at a frequency
of x kHz, and Eunf iltered is the energy curve for the unfiltered
pulses. The energy curves are compared using their minimal
values because the pre-filter alters the effective PW in the time
domain.
∆Ex=min(Efc,x)
min(Eunfiltered)−1·100% (9)
B. Validation Results
The strength-duration curve for the unfiltered pulses in
Fig. 5 depicts the minimum required amplitude for a certain
PW to excite the neuron. The fitted curves of (4) and (5) are
also plotted in Fig. 5. For the exponential fit, τeis 0.22 ms,
and for the hyperbolic fit, it is 0.18 ms. These correspond to an
equivalent corner frequency of the membrane of 723 Hz and
884 Hz, respectively. The estimation of the membrane’s cut-
off frequency has been used to choose the range of pre-filter
corner frequencies in this validation.
The calculated activation energy thresholds for the different
pre-filters and PWs are depicted in the energy-duration curves
in Fig. 6, where the y-axis is normalized to the minimal value
of the unfiltered pulse. The PW depicted in Fig. 6 is the PW of
signal x(t)before applying the pre-filter. The corresponding
values of ∆Eare listed in Table I. In the small pulse width
range, it can be seen that filtering with low corner frequencies
μ
Fig. 5: Strength-Duration curve for the unfiltered pulses, together with the
fitted curves for the hyperbolic and exponential relationships [14].
f
c
Fig. 6: Energy-Duration relationships for the filtered and unfiltered rectangular
pulses, normalized to the optimal value for the unfiltered pulse.
(≤1 kHz) increases the activation energy threshold compared
to the unfiltered pulses. The required activation energy is
minimized when using an fcof 5 kHz. Further increasing
the fcreduces the energy saving and the energy-duration
relationship approaches the unfiltered case. As expected and
discussed above, the pre-filtering is more effective for shorter
pulses and shows a lower improvement for longer pulses. For
example, at a PW of 10 µs, pre-filtering at 5 kHz reduces the
required energy by 51.1%, while at 5 ms, the improvement is
<1%. It should be noted that the filtering changes the effective
PW of the pulses in the time domain. After the end of the
original pulse, the amplitude decays with e−2πfct.
TABLE I: Relative minimum required energy of filtered pulses
fc[kHz] 0.5 1.0 5.0 10.0 20.0 50.0
∆E[%] 22.76 1.79 -11.04 -8.7 -5.63 -2.63
V. DISCUSSION AND CONCLUSION
In this work, we introduced the concept of pre-filtering the
stimuli to reduce the required activation energy in electrical
stimulation. The foundation of the principle is based on the
frequency-domain analysis of the stimulation signals and the
linearized cell membrane. The potential energy reduction was
validated in a single axon fiber model that included non-linear
membrane dynamics.
The presented analysis is focused on energy reduction and
is limited to monophasic rectangular stimulation pulses. How-
ever, the frequency-domain analysis and pre-filtering could be
used for a broader range of analyses and applications, which
will be shortly discussed here.
First, the current analysis assumes a purely resistive load
to the system. However, it is known that the electrode-tissue
interface introduces capacitive components in series with the
tissue resistance [20]. In the frequency domain, one could
model this as a high-pass filter to enhance the analysis. The lo-
cation of the corner frequency of this filter depends on multiple
factors, including electrode material and geometry. In electrode
design for electrical stimulation, capacitive charging should be
minimized to prevent harmful electrochemical reactions [20],
which would lead to a low corner frequency of the high-pass
characteristic.
Second, pre-filtering might be used to increase the selectiv-
ity of the stimulation. We focused on a single isolated cell
at a fixed distance for the analysis. However, it is known
that the equivalent time constant depends on many factors,
including axon fiber diameter, electrode size, and the distance
from the cell [14]. Careful design and tuning of the pre-filter
could reduce activation thresholds for the target cells while
increasing the thresholds of non-target cells. In this way, a
smaller group of non-target cells will be excited, and therefore
the associated adverse effects of stimulation might be reduced.
Third, the effect of pre-filtering should be analyzed for more
complex stimuli. For example, frequency-domain filtering of a
rectangular pulse does change the effective pulse width in the
time domain due to the after-pulse decay. This might affect
the usability of the filtering for biphasic pulses and repetitive
pulsing. Additionally, the analysis should be extended to non-
rectangular or arbitrary waveforms.
Finally, the relation between time- and frequency-domain
properties should be explored in more detail. One example is
the relation between the total energy and maximum amplitude.
From Fig. 3, one can see that even though the effect of pre-
filtering on signal Y(s) is small, there is always some degree of
modulation. Translating this change to the time domain would
be helpful for further understanding the pre-filtering effects.
Next to that, it would be useful to identify the optimal ratio
between the pre-filter corner frequency and the membrane’s
time constant.
APPENDIX
The model parameters are listed in Table II, the membrane
dynamics are based on the models presented in [18].
TABLE II: Parameters used in the axon model
Symbol Description Value
ρaAxial resistivity 100 Ω·cm
σoExtracellular conductivity 0.276 S·m−1
doDiameter myelin sections 1.25 µm
dnDiameter nodal sections 0.93 µm
LnLength nodal sections 1 µm
LmLength myelin sections 59 µm
Cm,n Membrane capacitance nodal sections 1 µF·cm−2
Cm,m Membrane capacitance myelin sections 0.02 µF·cm−2
Rm,n Membrane resistance nodal sections 33.3 kΩ·cm2
Rm,m Membrane resistance myelin sections 1.125 MΩ·cm2
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