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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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