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Taylor, J., Clarke, C., Schuettler, M. and Donaldson, N. (2012)

The theory of velocity selective neural recording: a study based

on simulation. Medical and Biological Engineering and

Computing, 50 (3). pp. 309-318. ISSN 0140-0118

Link to official URL (if available): http://dx.doi.org/10.1007/s11517-

012-0874-z

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The Theory of Velocity Selective Neural Recording: A

Study Based on Simulation

John Taylor1, Martin Schuettler2, Chris Clarke1 and Nick Donaldson3

1. Department of Electronic and Electrical Engineering, University of Bath, Bath UK

2. Laboratory for Biomedical Microtechnology, Department of Microsystems Engineering –

IMTEK, University of Freiburg, 79110 Freiburg, Germany.

3. Department Medical Physics and Bioengineering, University College London, London WC1 UK.

Corresponding author: John Taylor. Tel +44 (0)1225 38 3910; email j.t.taylor@bath.ac.uk

Abstract- This paper describes improvements to the theory of velocity selective recording and some

simulation results. In this method, activity is different groups of axons is discriminated by their

propagation velocity. A multi-electrode cuff and an array of amplifiers produce multiple neural signals;

if artificial delays are inserted and the signals are added, the activity in axons of the matched velocity are

emphasized. We call this intrinsic velocity selective recording. However, simulation shows that

interpreting the time signals is then not straight-forward and the selectivity Qv is low. New theory shows

that bandpass filters improve the selectivity and explains why this is true in the time domain. A

simulation study investigates the limits on the available velocity selectivity both with and without

additive noise and with reasonable sampling rates and analogue-to-digital conversion (ADC) parameters.

Bandpass filters can improve the selectivity by factors up to 7 but this depends on the speed of the action

potential and the signal-to-noise ratio.

Keywords: Electroneurogram recording, Simulation, Multielectrode cuffs, Velocity selective recording

Total text words: 5848

Words in the abstract: 159

Number of figures: 6

Number of Tables: 2

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1. Introduction

Velocity selective recording (VSR) is a technique which should allow more information to be extracted

from an intact nerve with a recording set-up that does not allow action potentials from single fibres to be

seen at spikes [9, 1]. The method is in essence very simple and relies on taking measurements of a

propagating action potential (AP) at two or more points. The distance between the sample points divided

by the delay between two appearance of the AP provides a measure of the propagation velocity.

Conversely, artificially delaying one or more signals before adding them gives a maximum response at

the matched velocity. This simple idea is not new and various researchers have investigated practical

adaptations of it in the past-e.g. [6, 11, 2, 3].

However, at present the idea has not been demonstrated with naturally-occurring nerve traffic though

experimenters have used multi-electrode cuffs (MECs) to observe appropriate outputs from compound

action potentials [5, 6, 11]. Two papers about the theory of VSR have been published by the same

authors [9, 1]. The first presented a spectral analysis of a single axon in an MEC with a tripolar (double-

differential) amplifier system and the signal processing arrangement shown in Figure 1. The bandpass

filter (BPF) that follows the adder was shown to improve selectivity in the velocity domain, however the

filters were ideal and of infinitesimal bandwidth. We envisage that a useful VSR system might actually

have several signal processing units (Figure 1), each matched to one of the propagation velocities of

functional significance [9].

The second theory paper [1] considered the thermal noise generated by the detection system and

compared its amplitude to that of the signal resulting from the summation of multiple single fibre action

potentials (SFAPs) which were assumed to occur at random times. This allowed the firing rates required

from various sizes of nerve fibre in a given MEC to provide a signal that could be detected above the

background noise to be calculated.

The current paper presents material which supplements and expands our earlier work [9, 1]. In essence it

is a study (by simulation) of improvements in velocity selectivity obtainable by the use of BPFs,

investigating in particular the limitations of the method with and without noise. The intended outcome of

this work is to achieve a significant improvement in the performance of VSR systems so that they can be

employed to discriminate between populations of naturally-occurring APs. As already noted, the

relatively poor velocity selectivity of the VSR systems tested to date has limited their application to

CAPs.

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Deterministic models of nerve signals (i.e. SFAPs) are used exclusively in the study, the effects of

random signal generation being reserved for presentation in a subsequent paper.

2. Methods

2.1 Basic principles

The response of an MEC to a propagating SFAP has been discussed in detail elsewhere and is only very

briefly summarised here [9, 1]. The input to the MEC is a trans-membrane action potential function

(TMAP), Vm(t), with the corresponding spectrum Vm(f). The resulting SFAP is a propagating wave with

the time dependence of the underlying TMAP function, the relationship between the two being explained

in [9]. For the purpose of simulation, we represent the TMAP function and its spectrum by the Fourier

transform pair [1]:

Vm(t) = Atne-Bt

1

2

!

)(

n

m

fjB

A

n

fV

… (1)

where A, B and n are constants and f is frequency (the symbology has been preserved from [9]). This

function has been proposed as a suitable basis for the simulation of mammalian ENG [8, 7, 10]. The

output Y(f, v), which is a function of both frequency and velocity, is obtained by treating the MEC as a

linear time-invariant system with transfer function H(f, v):

f

1

2

0

2

!

sin4

sin

sin

,,,

d

n

a

e

m

fjB

An

v

fd

R

R

v

f

v

d

fN

VvfHvfGvfY

…. (2)

This equation describes the output of a cuff with N tripoles, electrode pitch d and propagation velocity v.

Ra, the intra-axonal resistance per unit length, has been assumed to be large compared to Re, the extra-

axonal resistances per unit length inside the cuff. The artificial time delays are multiples of (see [1] for

a full explanation). Equation 2 is the product of the spectrum of the TMAP (Vm(f)), the transfer function

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of one tripole (Ho(f,v)), and the transfer function of the delay-and-add block (G(f,v)). At matched

velocities (i.e. where τ = d/v and v = vo in eqn (2)), eqn (2) reduces to:

1

0

2

0

2

!

sin4,

n

a

e

fjB

An

v

fd

R

R

NvfY

… (3)

The output of the system Y(f,v) is a function of two variables and it was pointed out in [9] that if f is

fixed by passing the output through a bandpass filter (so that f = f0), Y becomes a function of propagation

velocity v only, enabling the velocity selectivity profile (see the tuning curves in [9]) to be calculated.

We define a velocity quality factor, Qv, by analogy with linear systems in the frequency domain [1]:

33

0

vv

v

Qv

… (4)

where v0 is the matched (i.e. peak) velocity and v3+ and v3- are the velocities at which the output has

fallen to 1/√2 (-3 dB) of the peak value. Close to the matched velocities, the velocity selectivity is

dominated by the function G(f,v) and an approximate formula for Qv was derived in [1]:

0

0

v

64.2

dfN

Qv

… (5)

Qv and its approximation (eqn (5)) will be used to characterise velocity profiles in this paper.

2.2 The intrinsic velocity spectrum (IVS)

If bandpass filtering is not applied, as noted above, the output will depend on v and on frequency

dependent elements in the system including the spectral properties of the input signal (i.e. the TMAP

function) and the characteristics of the channel. For these reasons, unlike the bandpass filtered version,

the resulting time domain function is difficult to interpret. Simulating in the time domain, Fig 2 is the

time-domain output of the adder in Fig 1 when the system is stimulated with a TMAP resulting in an

SFAP propagating at a velocity of 30 m/s. Two TMAP functions are considered, based on equation (1),

with the parameters given in Figs 2(b) and 2(c) (the scaling parameter A has been adjusted in both cases

so that the peak amplitudes of the functions are normalised to unity). The matched velocity vo is treated

as a parameter leading to the family of curves shown in Fig 2(a). The peak value is reached when the

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artificial delays exactly match (cancel) the naturally-occurring delays at which point the output signal

has the same form as a single SFAP, with amplitude multiplied by N [1].

Fig 2(b) shows the IVS of the system, stimulated by an SFAP generated by TMAP#1 (and repeated in

Fig 2(c) for TMAP#2). This is a plot of the peak values of the output time record (Fig 2(a)) as a function

of propagation velocity after the tripole signals have been subjected to delay and add operations only.

Each curve in the time record shown in Fig 2(a) has three peaks, labelled , and , two positive and

one negative, corresponding to the phases of the tripolar SFAP. Whilst it is possible to calculate the IVS

using the height of any of these peaks as the amplitude, this paper considers only the two larger-

amplitude peaks, and . The resulting spectra peak at the same matched velocity (40 m/s), but have

different selectivities as shown in Fig 2(b), where the two IVS plots are shown together with the values

of Qv calculated from the figure using eqn (4). These values of intrinsic velocity selectivity are used as

baseline references for the enhancements described in the next section.

From the above observations, it is clear that if this method is to be used to separate neural signals in

terms of velocity, certain problems of interpretation arise:

1. The measured IVS depends on the point in the time response of the delayed and summed signal used

to make the measurement;

2. The IVS profile depends on the properties of the TMAP function, in particular, shorter responses

result in larger values of Qv;

3. The actual selectivity obtainable is quite low and declines with increasing velocity. This accords

with the theory presented below (see also [1]).

These issues are considered in the next section.

2.3 Improved Velocity Selective Recording Using Bandpass Filters (BPFVS)

The theory presented in our earlier papers and summarized above described the operation of a delay-

matched VSR system in the velocity domain. However in order to understand better the generation of the

IVS using this method and especially the enhancements possible using bandpass filters (BPFVS), a time

domain interpretation is very useful. Consider the representation of the tripolar amplifier outputs in Fig 1

by N, equally-spaced rectangular pulses of arbitrary height h and duration δt. The delay between adjacent

tripolar outputs is τ and that this is related to the SFAP propagation velocity v and the inter-electrode

spacing d as follows:

v

d

… (6)

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Clearly if the N pulses are summed and δt < τ, the result will be a train of pulses of height h and spacing

τ – δt. If , on the other hand, artificial delays are introduced to cancel the naturally-occurring delays as

shown in the signal processing unit in Fig 1, the summed output will be a single pulse of duration δt and

height N.h. Between these extreme cases lie intermediate states where the externally applied delay is not

exactly equal to τ and hence the delay is not exactly matched. The rapidity of the transition through these

intermediate states determines the velocity selectivity and Qv. As the inserted delay is increased from

zero, the pulses begin to overlap and it is easy to see that the value of inserted delay required for this to

happen depends on both τ and on δt, the width of the pulses themselves. So, in order to produce a

summed output of particular amplitude, the value of the inserted delay is required to be closer to τ than

would be the case for wider pulses (adding more pulses-increasing N-has a similar effect). This is

observed as an increase in intrinsic velocity selectivity for narrower pulses. In summary, we can draw

the following initial conclusions regarding intrinsic delay matched VSR:

1. The intrinsic velocity selectivity, Qv is inversely proportional to the width of the tripolar SFAP

pulses and therefore strongly dependent on the spectral content of the underlying TMAP

function;

2. Qv is proportional to N, the order of the system (number of tripoles).

The proposed approach to reducing the sensitivity to the TMAP is to place a BPF at the output of each

tripolar amplifier of the VSR system. The impulse response of a bandpass filter is a burst of damped

sinewaves whose frequency is the centre frequency of the BPF while the duration of the burst depends

on the filter order and relative bandwidth. For example, Fig 3(a) shows the response of an 8th order

Butterworth BPF of centre frequency 1 kHz and relative bandwidth 20% to a narrow pulse of unit

amplitude. Since the same type of response is elicited by stimulating a BPF with a tripolar action

potential, the ‘delay matching’ process is transformed into matching delayed sinewaves rather than

complex SFAP waveforms as in the intrinsic case. Unlike the SFAP waveform itself, the BPF output has

no dependence on the characteristics of the TMAP except for its amplitude and its exact position in the

time record. The addition of BPFs in this way therefore allows the measurement of velocity selectivity to

be decoupled from the spectral properties of the TMAP and to be controlled by means of the centre

frequency of the filters which is to some extent a free parameter. Note that although the proposed

arrangement requires a BPF at the output of each tripolar channel, in practice, due to the linearity of the

processes involved, the summation and filtering operations can be reversed. This leads to the much

simpler and more practical arrangement of Fig 1, where only a single BPF is required.

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In order to determine the velocity spectrum obtained by delay matching with BPFs, it is sufficient to

calculate the peak value of the output as a function of inserted delay. As already noted, the transient

response of a BPF is a burst of damped sinewaves (see Fig 3(a)) and so the overall response of the

system, f(t), close to the point of exact match can be represented as the sum of N sinewaves, a simplified

example of which is shown in Fig 3(b). Each of the three sinewaves shown in the figure represents the

sinewave at the centre of the burst at the output of each BPF. For simplicity, the origin of the time axis

has been placed at the centre of the set of sinewaves (so that when the signals are matched they peak at t

= 0) and the amplitudes of the sinewaves have been normalised to unity. The period of the sinewaves is

T, which is the inverse of f0, where f0 is the centre frequency of the BPF and the offset due to the passage

of the AP along the MEC is τ. The result of adding the N sinewaves where N is odd (similar principles

apply for N even) is:

2

2

1

cos ...cos ...

1

cos)(

000

N

tt

N

ttf

… (7)

where τ = d/v0 and ω0 = 2πf0. This function reaches a maximum at t = 0 for any values of N and τ and so,

to consider the selectivity, eqn (7) is rewritten:

2

2

1

cos...1...

1

cos)(

000

NN

tf

t

… (8)

So, for example, for N = 3:

003

cos21)(

t

tf

for N = 5:

0

2

0

05

cos2 cos21)(

t

tf

and so on.

After some manipulation, it is possible to show for the general case that:

...

2

cos2

2

cos2

2

cos2)(

5

0

2

3

3

0

1

2

1

0

0

N

N

N

N

N

tN

CCtf

where the series terminates when the exponent is zero. The sum of this standard finite series is given in

eqn (9).

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) 2/ sin(

) 2/( sin

0

0

N

… (9)

Eqn (9) is just the function G from eqn (2) except that, as already noted, the delay (τ) shown in eqn (2)

has been normalised so that it actually represents offsets from the matched value. This is the formula

derived in [1] by spectral (velocity domain) analysis, showing that the two views of the method are

consistent.

2.4 The influence of analogue bandwidth and analogue-to-digital conversion

Figure 4 shows the frequency spectrum of a 9-tripole IVS system with an inter-electrode spacing (d) of 3

mm, SFAP derived from both the TMAPs considered in this paper and matched at a propagation velocity

of 30 m/s. The sample rate is 100 ks/s and the resolution of the analogue to digital converter (ADC) is 10

bits, both requirements being realisable in a practical system. In the matched case (i.e. v = v0), as already

noted, the transfer function of the system is just that for a single tripole multiplied by N (see eqns (2) and

(3)):

0

22

0

2

0

/ sin/ sin4,

v fdvv fd

R

R

NfvH

a

e

m

The transmission zeros of this function occur at the following frequencies:

... 3, 2, 1, 0, ;

0

mm

d

v

f

… (10)

where the sequence extends in theory to the Nyquist limit (fs/2, where fs is the sampling frequency) but in

practice is likely to be limited by the finite bandwidth of the analogue channels, the constraints of ADC

and noise. Examination of these spectra indicates that the amplitude of TMAP #2 declines much more

slowly with frequency than that of TMAP #1. As a result, due to the wider bandwidth of the TMAP #2

spectrum, the dynamic range is reduced compared to TMAP #1, easing the demands on ADC. Operation

at higher frequencies, approaching the Nyquist limit is therefore possible for TMAP #2 if the analogue

channel bandwidth permits it. This in turn permits much higher levels of selectivity, since this is

proportional to frequency (see eqn (5)). In summary, for the two TMAPs considered, the limiting factor

for TMAP #1 is ADC resolution, whereas in the case of TMAP #2 it is determined by analogue

bandwidth and the Nyquist limit. It should be noted, of course that any selectivity enhancements are

subject to the degrading effects of noise. This aspect is dealt with in Section 3 below.

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2.5 The significance of bandpass filter order

Fig 5 demonstrates the effect and importance of filter order on the behaviour of a VSR system. As

described above, the bandpass filter added at the output of an IVSR system is required to produce a burst

of damped sinusoidal oscillation at the centre frequency of the BPF when stimulated with a wide band

signal such as an SFAP. For this to happen, the filter must reject out-of-band signals that would

otherwise corrupt its response. In particular, the ‘baseband’ (borrowing some terminology from

communication systems) SFAP signal generated from TMAP#1 peaks at about 1.5 kHz and has a much

larger amplitude than the ‘harmonics’ further up the frequency scale. The figure shows that this is still

true for a p = 2 Butterworth bandpass filter since the filtered spectral amplitude at frequencies in the

range 0-5 kHz is actually larger than in the passband. This suggests the need for a BPF with p = 4, at

least. The small insets in the corners of the figure show the effect of filter order in the time domain. For

p = 2, for example, the filter output is dominated by the SFAP input signal, which is consistent with the

frequency domain situation. By contrast, for p = 8 the input signal has been suppressed and the output is

a symmetrical burst of damped sinusoids. Note however that p cannot be increased without limit

because, as the filter order is increased, the time domain response becomes longer. As discussed in

Section 2.6, this tends to increase the prevalence of false ‘image’ outputs. In addition, the requirement

for higher filter order tends to increase the ratios of the digital filter coefficients, making realisation more

difficult.

2.6 Generation of spurious responses

One of the effects of using BPFs to enhance the performance of IVSR is the potential for the generation

of spurious velocity spectral responses (‘images’). These occur because single tripolar SFAP waveforms

are replaced by bursts of damped sinewaves of frequency fo, the centre frequency of a BPF. Therefore,

although a global maximum occurs when all the signals are matched, with or without BPFs, the potential

exists for subsidiary, ‘image’ responses to be generated when the signals are mismatched by integer

multiples of the period of the sinewaves, 1/fo. Images will be generated if and only if:

W ≥ N

… (11)

where W is the number of cycles of sinewave in each burst and N is order of the system (i.e., the number

of tripolar channels being delay-matched) and that the velocity at which the images are formed is given

by:

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o

image

fq

d

v

/

… (12)

where d is the inter-electrode spacing, τ is the naturally-occurring delay and q is an integer. Note that the

velocities at which image responses occur depend on fo, the centre frequency of the BPF, in contrast to

an output resulting from an excited population which will appear at the same velocity irrespective of fo.

In addition, for practical values of the other parameters in eqn (10), the images tend to occur at velocities

that lie outside the band of interest (say 20 – 120 m/s). For these reasons, images are easily distinguished

from genuine outputs, as illustrated in Fig 6.

3. Results

3.1 Simulated results without noise

In order to demonstrate the effects of adding BPFs to a delay-matched IVS system, the MATLAB

simulations shown in Fig 2 were repeated with a single bandpass filter of centre frequency f0 placed at

the output of the system, as shown in Fig 1. The SFAPs were generated using TMAP#1 and TMAP#2

and the system was noiseless. The filter was an 8th-order digital Butterworth BPF and centre frequencies

of 1 kHz, 2 kHz, 4 kHz, 8 kHz or 16 kHz and relative bandwidth 20% were used. The velocity spectra

are plotted in Fig 6 (the amplitudes of the plots corresponding to the two TMAPs have been normalized)

and show good responses at the matched velocities. It can be shown that it is possible to obtain

satisfactory responses for BPFs with centre frequencies up to the Nyquist frequency (50 kHz in this

case). The responses to SFAPs generated from both the TMAPs are identical in the sense that the values

of Qv measured at the matched velocities are the same in both cases. These results support the assertion

that the bandpass filtered velocity selectivity depends only on N, f0 and v and some physical constants,

not on the characteristics of the TMAP function, as is the case for IVS. The values of Qv are listed in

Table 1 together with values calculated from eqn (4). The calculated values fit the simulated ones very

well.

3.2. Simulated results with additive noise

Zero-mean white Gaussian noise was added to the system in a manner consistent with the approach

adopted in [1] (i.e., 11 sources of uncorrelated voltage noise were introduced, one at the input to each

monopolar channel). These noise sources represent the total noise present in each channel referred to the

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input. As noted in [1] the total input-referred noise is the sum of several individual sources that are

assumed to be independent and uncorrelated. The signal-to-noise ratio (SNR) was defined as the ratio of

the r.m.s. value of signal (i.e. TMAP) to noise over a sequence of standard length (10.23 ms,

corresponding to 1024 samples), referred to the input.

In order to test the effect of the noise on the system and in particular on the ability of the BPFs to

increase the velocity selectivity compared to IVS, the simulations described in Section A. were repeated

with varying levels of additive white Gaussian noise. The results are presented in Table 2 for three

values of SNR (1, 10, 100) for each of the two TMAPs. The frequency in column A for each value of

SNR gives the maximum frequency (fomax) at which an intelligible output is obtainable from a BPF

centred at that frequency. Once fomax has been determined, the maximum available velocity selectivity

(Qv) can be calculated from eqn (10) (column B) and the enhancement factor found (i.e. compared to

IVS-column C). In general TMAP#2 performs better than TMAP#1, due to the wider bandwidth of the

signal. There is thus more energy at higher frequencies in SFAPs generated from TMAP#2, whilst the

additive noise has the same spectral density at all frequencies and for both TMAPs. In the worst case

considered, with SNR set to unity, there is no enhancement in Qv for TMAP#1, whilst for TMAP#2 a

modest enhancement of about 3.5 is possible. For SNR = 10 the values increase to 2 and 7 respectively

and 4 and 7 respectively for SNR = 100, although in this case, higher values could be obtained for

TMAP#2 if more bandwidth were available.

4. Discussion

4.1 Limitations of the theory

The use of bandpass filters to improve the selectivity of VSR systems was first proposed in [9, 1] and the

current paper extends and develops these methods. After briefly summarising the basic theory of the

method, the effects of key filter parameters, in particular centre frequency and filter order are examined.

The proposed system uses digital bandpass filters and for the purposes of this paper these were

implemented in software, using MATLAB. Since it is likely that a real VSR system would use similar

BPFs, whether implemented in software or hardware, the results of the simulations presented in this

paper can be considered to be quite reliable and representative.

4.2 Limits of available selectivity with and without noise

In Section 2.4 it was suggested that the factors limiting available velocity selectivity are (i) the spectral

bandwidth of the TMAP function, (ii) the analogue bandwidth of the signal capture/acquisition system,

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(iii) the resolution of the digitisation process (ADC) and, (iv), noise. Furthermore, if digitisation with 10-

bits resolution is assumed, in the absence of noise, the level of available velocity selectivity is limited by

(i) and (ii). It was then demonstrated that in the case of TMAP #1, ADC resolution set the limit whilst

for TMAP #2, with its wider spectrum, analogue bandwidth (which in turn is limited by the sampling

rate/Nyquist frequency) was the limiting factor. However, these considerations are purely hypothetical

and the introduction of noise tends to alter the situation completely. Uncorrelated white Gaussian noise

was added to each monopolar input as suggested in [1]. Since the input-referred noise spectrum is flat in

contrast to those of the TMAPs, the presence of noise has most influence at higher frequencies where the

signal spectral density is lowest. The resulting bandwidth restriction in turn limits the available velocity

selectivity, both IVS and FVS. So, although the effect of bandpass filtering removes the influence of the

TMAP spectrum on the value of Qv at a particular frequency, the maximum available value of Qv will

depend on the TMAP function, because of the effects of noise. In spite of the restrictions imposed by

noise, very useful enhancements in selectivity are nonetheless predicted for both TMAPs, especially

TMAP #2, as detailed in Table 2.

4.3 The effect of image responses

The formation of spurious responses (‘images’) was discussed in Section 2.6. For the example shown in

Fig 6, d = 3mm, τ = 100 μs (i.e. for vo = 30 m/s) and so for the 16 kHz BPF, velocity spectral images

would be expected at 13.3 m/s, 18.5 m/s and 80 m/s. The lower velocity images are clearly visible in

both figures, the higher velocity signal being outside the range shown. As already noted, the amplitude

and range of occurrence of these images depend on N and W, while W depends on the BPF parameters,

especially the filter order and relative bandwidth and can often be quite small in comparison with

genuine outputs from excited populations. Note that there is no possibility of confusing a genuine output

with an image since the former will occur at the same velocity in the spectra calculated at all filter

outputs whilst images will appear at velocities dependent on fo.

In conclusion, this paper presents a practical method to improve the performance of VSR systems

compared to currently available methods. The use of BPFs provides a significant increase in velocity

selectivity which will form the basis of new systems to widen the application of the method beyond the

current state-of-the-art which is restricted to the recording and analysis of CAPs.

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References

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List of Figure Captions

Figure 1. This shows a multielectrode cuff (MEC) connected to a tripolar (double differential) amplifier

array. The N tripolar outputs (where N is typically about 10) are digitised and processed in the signal

processing unit on the right of the figure. For an action potential (AP) propagating symbolically

downwards, voltages are induced sequentially at the electrodes which result in the appearance of tripolar

output signals. The delay between the appearances of successive outputs depends on the propagation

velocity of the AP and the inter-electrode spacing of the MEC and allows a simple calculation of the AP

velocity. The calculation process is carried out in the digital domain using the programmable delays

indicated. Finally, the bandpass filter (BPF) is used to improve the velocity selectivity as described in the

text. We refer to the velocity spectrum calculated at the output of the adder as the intrinsic velocity

spectrum (IVS) and that at the output of the BPF, the filtered velocity spectrum (FVS).

Figure 2. Time and intrinsic velocity domain responses of the system shown in Fig 1 (output of the

adder), with a propagation velocity of 30 m/s. Subplot (a) shows the time domain response of the system,

each curve corresponding to a different matched velocity, the SFAP being generated from TMAP#1. The

three peaks of the waveforms are labelled , & . Subplot (b) shows the intrinsic velocity spectrum

(IVS) for TMAP#1 derived from Fig 2(a) measured at points α (top curve) and β (bottom curve). Subplot

(c) shows the same calculation using TMAP #2. The parameters of the TMAP function (x(t) = Atne-Bt)

are included.

Figure 3. Plot (a) shows the time record of the output of an 8th order Butterworth bandpass filter of 20%

relative bandwidth stimulated with a narrow pulse of unit amplitude. Plot (b) shows three sinusoids, each

representing the central half-cycle of the output of a channel BPF. The amplitudes have been normalised

to unity and the delay offset τ is the naturally-occurring inter-channel delay. N sinusoids are summed to

calculate the velocity selectivity of the system.

Figure 4. Frequency domain spectrum (FFT) of a 9-tripole system driven by SFAPs generated by

TMAPs #1 and #2. The propagation velocity is 30 m/s and the system is matched at that velocity. Note

that the spectral amplitude generated by TMAP #1 declines with frequency much more rapidly than that

generated by TMAP #2. The sampling rate is 100 ks/s.

Figure 5. Time domain responses (insets) and corresponding bandpass filtered spectra at the output of a

VSR system (derived from TMAP #1 and matched at 30 m/s-see Fig 4) compared to the unfiltered (IVS)

spectrum. The unfiltered output is passed through Butterworth BPFs with centre frequency (fo) 8 kHz

and order (p) 2, 4 and 8 and with 20% relative bandwidth. For the p = 2 filter the spectral amplitude at

frequencies in the ‘baseband’ range 0-5 kHz is larger than at fo, suggesting significant breakthrough in

the passband and the need for a BPF with p = 4, at least.

Figure 6. Bandpass filtered version of the time domain IVS plot shown in Fig 2 and comparing the

differences between SFAPs generated using TMAP functions #1 (dashed line) and #2 (dotted line) with

a propagation velocity of 30 m/s. The filters are 8th order Butterworth digital units with centre

frequencies f0 are (a) 1 kHz, (b) 2 kHz, (c) 4 kHz, (d) 8 kHz, (e) 16 kHz and (f) 32 kHz. There is no

significant difference between the two responses and the corresponding values of Qv are 1.4, 2.9, 5.7.

The velocity step is 1 m/s and there is no additive noise. Note the formation of spurious ‘image’