Modelling of Compound Nerve Action
Potentials in H e a l t h and Disease
BUDAGAVI MADHUKAR,
I . S. N . MURTHY
AND
A. G.
RAMAKRISHNAN
Department of Elec, Engg., Indian Institute of Science, Bangalore - 560 012, India
nents. Thus,
Abstract-This paper reports the results of application of Hurthy-Prasad technique in extracting the
principal component fror cwpound nerve action
potentials (CNAP). The DCT of the CNAP is pole Zero
modelled using the Steiglitz-UcBride method. This
model is expanded into a set of partial fractions
eachof order ( 2 . 2 ) . The sum of the monophasic functions recovered frcm the relevant fractions in the
inverse process gives the major potential in the
response. In the case of patients’ responses, this
improvesthe main component by rejecting the sraller
peaks due to dispersion by demyelination. The technique works well in waveforms deformed moderately.
The improved waveshape is extremely useful in cases
where the original waveshape prevents the exact
location of arrival of nerve impulse.
where np is t h e number of monophasic
components in t h e main response (not
necessarily t h e first np components of x( n ) )
and np t nr = m. The delineation of t h e
principal response is achieved using the
following principle, proved in [2]: The disc r e t e cosine transform (DCT) of a bell
shaped monophasic w a v e can be approximated as t h e impulse response of a system
function with 2 poles and 2 zeros, ie., of
order (2,2).
Let X(k), k = O , l , ...( N- 1) be t h e DCT of
t h e signal x(n). X(k) is approximated as the
impulse response of a system, X(z), of
o r d e r (2mJm). X(z) is given by t h e rational
t r a n s f e r function,
I. I NTRODUCTION
The s t u d y of compound nerve action
potentials (CNAP) is extremely useful in t h e
diagnosis and assessment of neurological
disorders. The typical response consists of
the integrated potential due to all the fast
conducting A fibres. In healthy individuals,
it is easy to identify t h e onset of t h e main
potential and measure the nerve conduction
time accurately. I n t h e case of patients
with demyelinating neuropathies, different
nerve fibres conduct at different velocities
and there is dispersion of t h e CNAP. Hence,
there is uncertainty in locating the time of
arrival of t h e nerve impulse. Based on the
model proposed in (11 for ECGs, we present
a technique in this paper, which delineates
the different components of the waveform
and t h u s helps to separate the main potential from other peaks, artifacts or otherwise. The latencies of all the peaks are also
obtained from the model.
X(z) =
X(Z)
m
I: ICi+
i=l
(ai
(1)
t
bi z
-1
)
............................
1
-
2ri cos
ei 2 - l
t
q ~i , =
1,2
3
(4)
r2z-2
~ ~ / AThe
~ ~ constants
.
ci = - x i ( 0 )
where each xi(n) i s a monophasic wave. A
s u b s e t of these components add upto p ( n ) ,
the principal p a r t of the CNAP and r ( n ) is
the ensemble of all the remaining compo-
0-7803-0785-2/92$03.00OEEE
=
where c ci
B
determined as
The CNAP is approximated as a sum of m
number of monophasic components :
... .m
(3)
where t h e coefficients Ai’s and Bi’s a r e
Steiglitz-McBride
estimated using
the
method (31. This model is expanded into a
unique set of partial fractions (PF) each of
o r d e r (2,2) and a monophasic function is
recovered from each one of these fractions
in t h e inverse process. The PF expansion
of ( 3 ) is given by
11. THEORY
x(n) = 2 x i ( ” ) , i = 1 , 2 ,
- 2m
Bo t B 1 z - l t ... + BZmz
............................
- 2m
1 t A 1 z - l t ... t AZmz
,...,m
ci a r e
(5)
The sum of all t h e above m component
waves gives the complete signal x(n). The
selection of t h e relevant n fractions making u p p ( n ) is based on tRe fact t h a t the,
pole nearest to the unit circle corresponds
to the peak due to t h e principal component. Thus the components corresponding to
2600
t h e cluster formed by the pole angle 6i (i
= 1,2, np) a r e selected such t h a t they lie
within t h e angular region (6-6,/2)
< 6i <
(ePt6,/Z) where 6p is the angle corresponding to the pole nearest to the unit circle
and 0, is t h e maximum angular width. A
value for e, is chosen depending upon t h e
duration of the principal potential, t h e
sampling frequency and the total number
of samples in t h e input signal.
Median nerves of 5 normal subjects and
5 leprosy patients were electrically stimulated and CNAPs w e r e recorded j u s t above
...,
the
to the
artery*
These potentials were represented using
the above model and t h e main responses of
the A fibres were extracted.
111. R ESULTS
A system of o r d e r (6,6) or (8,8) w a s
found to be adequate to model all t h e data.
Fig. l a shows the CNAP from a normal subject and the corresponding reconstructed
model output. Figs. l b and l c show the
separated monophasic components and t h e
extracted principal potential obtained by
regrouping the pertinent fractions. Fig. 2a
shows the CNAP from a leprosy patient and
the
Output*
2b
gives the reconstructed principal response
for the patient. The main potential usually
requires 2 o r at most 3 monophasic functions. A one to one relatimship is found t o
exist between the pole pattern in t h e
z-plane and component wave pattern in t h e
time signal. The latency of t h e peaks can
be easily obtained from the model since a
pole angle 6 is directly related to t h e peak
sample number of t h a t component.
IV. CONCLUSION
The method is general and provides a
way to develop efficient models to waveforms with well defined peaks. The technique can be successfully used to separate
H-reflex, F response, etc. from t h e direct
M-potentials in t h e case of stimulation of
motor nerve trunks. Similarly t h e different
waves in the auditory brainstem evoked
response (ABER) attributed to specific
s t r u c t u r e s in the ascending auditory pathway can be singled out. This could have
potential applications in neurology.
R EFERENCES
[I] I. S. N. Murthyand G. S. S. D. Prasad, “Analysis
of ECG from pole-zero models,“ IEEE Trans. Biomed
Eng. , in press.
121 1,s. N. Murthy, G. S. S. D. Prasad and M. R. S.
Reddy, “Pole-zero models for ECG,” unpublished.
[3] K. Steiglitz and L. E. McBride, “A technique for
the identification of linear systems,” IEEB Trans
Automat. Contr, Vol. 10, pp. 461-464, Oct. 1965.
1000
Pig. 1c
Pig. la
-250
-200
-200
-1 500
-600
0
165
-600
0
330
165
330
-95
-800
-400
L
0
165
-
0
165
1
-21
330
0
...
165
330
-
Pig. la. Original (
) and the reconstructed (
) CNAP from a normal. lb. The
) and
separated fractionalcorponents. lc. The extracted principal potential. 2a. Original (
the reconstructed (
) CNAP from a patient. 2b. The extracted principal potential.
For all the figures, x-axis is sample no. and y-axis is arbitrary amplitude.
...
260I
330