HUNTINGTON'S A RANDOM
CHOREA
PROCESS
B. K. Tan*, H. A. K. Mastebroek** a n d IV. H. Zaagman**.
SUMMARY The purpose of the present study was to investigate statistically the irregular nature of the choreatic jerks in Huntington's Chorea. EMG-bursts of certain muscles in a patient with Huntington's Chorea were taken as a measure of the jerks. Statistical analysis of the measurements, i.e. durations of bursts and intervals between bursts, revealed that the irregular nature of the choreatic jerks originate from a random process. Each choreaticjerk appears to be an entirely independent and unpredictable event.
INTRODUCTION T h e c h o r e a t i c m o v e m e n t s in H u n t i n g t o n ' s C h o r e a have been k n o w n as clinically i n v o l u n t a r y a n d irregular j e r k y m o v e m e n t s o f several parts o f the body. W i t h regard to the irregular n a t u r e o f the c h o r e a t i c m o v e m e n t s the question can a p p r o p r i a t e l y be a s k e d : ' A r e the j e r k s r a n d o m events in time?' The present investigation is an a t t e m p t to answer this question and is c o n c e r n e d with the statistical analysis o f the c h o r e a t i c movements.
METHOD This investigation was carried out in only one patient with H u n t i n g t o n ' s C h o r e a , a m a r r i e d w o m a n , aged 51 years, who showed distinct c h o r e a t i c m o v e m e n t s o f several p a r t s o f her body. R e c o r d s o f the c h o r e a t i c m o v e m e n t s , o b t a i n e d with the use o f an accelerometer, were found unsuitable for a q u a n t i t a t i v e statistical analysis. F o r this reason e l e c t r o m y o g r a p h i c bursts o f certain muscles, underlying the choreatic movements were chosen as units o f m e a s u r e m e n t (fig. 1). In this investigation E M G recordings were m a d e o f the c o n t r a c t i o n s o f two muscles, the left gluteus m a x i m u s a n d the left tibialis anterior, representing respectively u p w a r d j e r k s o f the b u t t o c k and dorsiflexion j e r k s o f the left foot. These recordings were m a d e with surface electrodes a n d an E l e m a - S c h 6 n a n d e r writing a p p a r a t u s . * Deltaziekenhuis (Rotterdam Mental Hospital), Poortugaal, The Netherlands. ** Laboratorium voor AIgemene Nat uurkunde der Rij ksuniversiteit Groningen (Dept. of Biophysics), Westersingel 34, Groningen, The Netherlands. Clin. Neurol. Neurosurg., Vol. 79-3
216
_L
k
t
L
~k~
-
,
L
i
-I-
.,a,
ti. 1
L
~',dd_
L
t
.a..aL.
t
k
~l~
-I- t i . 2
Fig. 1. Upper channel: time in seconds, L o w e r channel: sample of E M G record of gluteus maximus ti = interburst-interval, T~ = duration of burst i, -r, = duration of pause between the end of burst i and the beginning of burst i + 1.
Fig. I s h o w s s u c h an E M G
r e c o r d ( l o w e r t r a c i n g ) w i t h the q u a n t i t i e s d e f i n e d as
follows: t~ ---- i n t e r b u r s t - i n t e r v a l i.e. the p e r i o d o f t i m e b e t w e e n the b e g i n n i n g o f b u r s t i a n d the b e g i n n i n g o f burst i + 1 ; T~ = d u r a t i o n o f b u r s t i; Ti
=
d u r a t i o n o f the p a u s e b e t w e e n the e n d o f b u r s t i a n d the b e g i n n i n g o f b u r s t i + 1 ; (burst s t a n d s f o r m u s c l e b u r s t o r m u s c l e c o n t r a c t i o n ; the f o r c e o f a j e r k was n o t m e a s u r e d in the p r e s e n t i n v e s t i g a t i o n a n d t h e r e f o r e n o t i n c l u d e d as a q u a n t i t y ) .
T h e q u a n t i t i e s tj a n d Ti w e r e m e a s u r e d by h a n d w i t h a ruler, a c c u r a t e to the n e a r e s t millimeter. The measurements thus obtained were evaluated with a computer. The following three formulae were used: a. the mean interburst-interval i: 1
(tt + t 2
+ .......
+tN)=
1
N X i = 1
(1)
N is the total number of interburst-intervals of the record. b. the standard deviation of the series interburst-intervals:
/ /
N 2 i=l
(h-- 0 2 N -- 1
(2)
This statistic measures the variation of the interburst-intervals around the mean value L c. The coefficient of correlation between succeeding intervals p: (serial correlation):
P
1 (N--I)°2
N--1 v (tl - - i) (tl+l - - t) i= 1
(3)
This statistic gives information about the structure of the interval-series t~. p is defined in such a way that: - - 1 ~ lz~ + 1. (cox and MILLER, 1970; BEVINGTON, 1969). TO illustrate: if in the series of intervals h a short interval is always followed by a long one, most of the terms in (3) will be negative, the series ti then is negatively correlated. A positive correlation
217 on the other hand will be present if short intervals are always followed by short, and long intervals by long ones. If, however, shorter and longer intervals succeed, one another at random (3) will have positive as well as negative terms and the sum will be about zero. in this case, the process, according to which the interburst-intervals t~ are generated, is called a renewal process (cox and MILLER, 1970). It is obvious that for the quantities T~ and "q the same statistics (mean, standard deviation and correlation coefficient) can be defined. T h e p o s s i b i l i t y o f a c o r r e l a t i o n in t h e series T~, t h e d u r a t i o n s o f t h e b u r s t s , will b e i n v e s t i g a t e d in t h i s p a p e r w i t h t h e u s e o f a s o - c a l l e d ' j o i n t - b u r s t l e n g t h h i s t o g r a m ' . W i t h t h i s m e t h o d t h e c o r r e l a t i o n b e t w e e n t~ a n d T ~ , I will a l s o b e i n v e s t i g a t e d . Four separate EMG recordings were made, two of the gluteus maximus muscle and t w o o f t h e left t i b i a l i s a n t e r i o r .
S i n c e all f o u r E M G
recordings gave exactly the same
s t a t i s t i c a l r e s u l t s , o n l y t h e d a t a o f o n e r e c o r d i n g will b e p r e s e n t e d h e r e .
The total
n u m b e r o f b u r s t s in t h i s r e c o r d i n g is 545.
RESULTS I. N o c o r r e l a t i o n s w e r e f o u n d b e t w e e n s u c c e e d i n g i n t e r b u r s t - i n t e r v a l s ti. T h u s t h e p r o c e s s a c c o r d i n g t o w h i c h t h e b u r s t s a r e g e n e r a t e d is, w h a t is c a l l e d , a renewal process.
T h e i n t e r v a l s ti w e r e e x p o n e n t i a l l y d i s t r i b u t e d .
This means that the
r e n e w a l p r o c e s s j u s t m e n t i o n e d , c a n b e classified as a Poisson process, i.e. a process
which describes complete randomness. T h e P o i s s o n d i s t r i b u t i o n o f t h e i n t e r v a l s t~ is i l l u s t r a t e d in fig. 2. A prerequisite for a statistical analysis of the data is that the records of the E M G bursts constitute a stationary series. To ascertain this stationary quality the following investigations were carried out a. with respect to the intensity of the process (number of bursts per unit time) and b. with respect to the structure of the process (as indicated by the correlation coefficient). No significant changes in time should occur, but only variability within certain limits, in order to satisfy the conditions for the process to be stationary (cox and MILLER, 1970; MASTEBROEK, 1974; STORM, 1965). a. The series of N interburst-intervals (N = 544) is divided into 5 consecutive subseries of n = 100 intervals each. For each subseries j (j = 1,2 . . . . 5) the mean interburst-interval ~j (j = 1, 2 . . . . 5) is calculated. ~j should then obey: 2G
where ~ is the mean interburst interval of the total series and cr is the standard deviation of this series. Since ~ = 1.76 sec and a = 0.93 see for the total record, I t j - - [ I = t t J - - 1.76[~ 0.19 for n = 100. b. Calculation of the correlation coefficient for the subseries j (j = 1, 2, .., 5) gives values, which scatter around zero (Table I). This indicates a renewal process. In this case the tzj values may vary between --0.20 and +0.20 (Storm, 1965). From the results presented in Table 1 it is evident that the record forms a stationary series.
218 TABLE 1
Subseries
~j
Eij - - il
1.83 1.56 1.76 1.76 1.89
0.07 0.20 0.00 0.00 0.13
pj
no.
1 2 3 4 5
~.05 --0.18 0.13 -4).03 0.09
Since the correlation coefficients vary about zero the burst generation process represents a renewal process. Such a process can be described completely by the interburst-interval distribution p(t) (cox and MILLER, 1970) which is defined as follows: a
b
0.IC p(t)
p(t)
0.05
O.10
L
L
O.O1 Q05
0.005
1M
O.O1 10
2'0 30 CHANNEL NUMBER
O.OC
10
20 ~, CHANNEL NUMBER
Fig. 2. Histogram of the interburst-intervals h. In fig. 2a the vertical scale is linear. In fig. 2b the vertical scale is logarithmic. Channelwidth = 0.2 sec. p(t) dt is the probability for the generation of the onset of a burst in a time dt around t, given the preceding burst to occur at a time t = 0. The histogram, as calculated from the values of h is given in fig. 2. The vertical scale is linear in fig. 2a and logarithmic in fig. 2b. Fig. 2b shows that, apart from a 'silent period' during which no bursts were being generated the distribution is linear on a logarithmic scale. Hence the interburst-interval distribution can be fitted by a negative exponential curve. Such a distribution of intervals belongs to a Poisson process through which events are determined in a random manner. 2. T h e r e w e r e n o c o r r e l a t i o n s b e t w e e n s u c c e e d i n g b u r s t l e n g t h s T~. T h e r e f o r e , t h e p r o c e s s a c c o r d i n g t o w h i c h t h e b u r s t l e n g t h s a r e g e n e r a t e d , is a r e n e w a l p r o c e s s . T h e b u r s t l e n g t h s T, w e r e d i s t r i b u t e d a c c o r d i n g t o a G a m m a - t y p e f u n c t i o n as s h o w n in fig. 4b.
distribution
219 The series T~, durations of the bursts, was analysed by means of a so-called 'joint-burstlength histogram'. Such a histogram (fig. 3)
TII
Ti÷l
]
Ti
ETCETERA
Ti ÷1
Fig. 3. was obtained by plotting successively T~ (on the horizontal scale) against T~+I (on the vertical scale), T~+~ against T~+2, and so on until all 545 bursts were included (fig. 4a). This histogram shows that
30 26 A 22 I.-
v
18
~- 14 n.." Z)
rn 10
.,
m . =
•
. . . . . . . . .
m.m.m
_
~.
• .
m
,.n
r ~. - m l
.
~. m . . .
.
.
L
- .
.
.
.
.
~ = = r . - . •
,
L
.
.
.
.
.
22 0.20 p(T)
26
BURST i(T i )
0.15 010 005
2
,.,_I-L ~ , - , 8t ~ [ ~ l ~CHANNEL 14 20 26 NUMBER
, 32
Fig. 4a. Joint-burstlength histogram (for explanation see text). Fig. 4b. Burstlength histogram (Gamma-type distribution). Channelwidth 0.067 sec.
there is no correlation between consecutive burst lengths. A burst of any length on the horizontal scale will not be succeeded by a burst of a definite duration of time but - on the average - will always be followed by a burst with a mean length of about 0.56 sec. From the finding that there are no correlations between T~ values the conclusion can be drawn that the generation process for the burst lengths is a renewal process. The burstlength histogram p(T) as shown in fig. 4b represents a distribution of the Gamma-type. The mean burst length is 0.56 sec and the standard deviation 0.32 sec.
220 3. There were no c o r r e l a t i o n s between interburst-intervals t~ a n d the directly succeeding burst lengths Ti+l. As regards the relationship between the measurements t~ and T~+~, the question can be asked whether for instance a long interburst-interval h will always be succeeded by a long (or a short) burst length TI+I.
4c
Tp.1
1~
•
....
..~:~_'-_--.._~_~- : : • ....
~ m . . . . . .
..
~
.
.
.
"['l
.
.
.~
.
~ ~ - . ~'.'L--LI~.
.
.
.
.
" ¢
.
.
.
l
....
S --
+ m I l
I l m
t 1
Fig. 5. Joint histogram of interburst-intervals t~ and burstlength T~+l (for explanation see text). The joint histogram (fig. 5), obtained by plotting tj (on the horizontal scale) against T~÷~ (on the vertical scale), h÷l against T~÷z, and so on, shows that there is no correlation between the quantities t~ and T~÷~, again indicating a renewal process.
DISCUSSION
On the basis o f the a b o v e m e n t i o n e d results the conclusion can be d r a w n , that the irregular n a t u r e o f the c h o r e a t i c j e r k s - at least in this patient with H u n t i n g t o n ' s C h o r e a - is truly r a n d o m . The r a n d o m n a t u r e applies to b o t h the sequence o f the choreatic j e r k s as well as to the d u r a t i o n o f these jerks. Each c h o r e a t i c j e r k is thus a c o m p l e t e l y i n d e p e n d e n t and u n p r e d i c t a b l e event. C o n f i r m a t i o n o f these results is needed by similar studies o f m o r e patients with this disease. A l t h o u g h these results were o b t a i n e d from only one patient with H u n t i n g t o n ' s C h o r e a , it would seem that the r a n d o m n a t u r e - whatever significance it m a y have, a p o i n t not c o n s i d e r e d in this study - o f the c h o r e a t i c j e r k s is quite likely an inherent feature o f the disease and not confined to this patient alone. F o r it would a p p e a r t h a t r a n d o m events d o n o t constitute so rare a p h e n o m e n o n in clinical medicine. F o r instance, DE JONG a n d BURNS (1967) f o u n d evidence for a r a n d o m process in
221 P a r k i n s o n ' s disease as regards the d i s t r i b u t i o n o f lesions within some area o f the central n er v o u s system. Ventricular r h y t h m in patients with atrial fibrillation also shows a r a n d o m nature
(BOOTSMAet al., 1970). F u r t h e r m o r e , statistical m e a s u r e m e n t s
o f the d u r a t i o n s o f s o m e specific E E G - p a t t e r n s such as: 1. p a r o x y s m s o f bilateral s y n c h r o n o u s spikes and waves and the intervals between them. 2. bursts o f a l p h a waves a n d the intervals between them. 3. the intervals between the typical discharges o f subacute leuco-encephalitis reveals a r a n d o m d i s t r i b u t i o n o f these patterns
(.STRACKEE-KUIJERet al., 1959).
ACKNOWLEDGEMENT
T h e a u t h o r s wish to t h a n k Mrs. J e a n n e t van A b s - K n i g g e for m easu r i n g the records, Mr. B. K a m p s for d r a w i n g the figures and Mrs. A. Keulemans-Visser for her assistence with the E M G recordings.
REFERENCES
BEVINGTON,PH. R. (1969) Data reduction and error analysis for the Physical Sciences. McGraw-Hill, New York-London. BOOTSMA,B. K., HOELEN,A. J., STRACKEE, J. and MEIJLER, F. L. (1970) Analysis of R-R intervals in patients with atrial fibrillation at rest and during exercise. Circulation, 41,783. cox, D. R. and MILLER,H. D. (1970) The theory of stochastic processes. Methuen & Co., London. DE JONG, J. D. and DELISLEBURNS,B. (1967) Parkinson's Disease - A random process. The Can. Med. Ass. J., 97, 49. MASTEBROEK,H. A. K. (1974) Stochastic structure of neural activity in the visual system of the blowfly. Thesis, University of Groningen, Groningen. STORM, R. (1965) Wahrscheinlichkeitsrechnung, Mathematische Statistik und Statistische Qualit/itskontrolle. V.E.B. Fachbuchverlag, Leipzig. STRACKEE-KUIJER,A., DENIERVAN DER GUN, J. J. and STRACKEE,J. (1959) Statistical relations in the duration of some patterns in the EEG. EEG Clin. Neurophysiol. 11,620.
CHOREA
PROCESS
B. K. Tan*, H. A. K. Mastebroek** a n d IV. H. Zaagman**.
SUMMARY The purpose of the present study was to investigate statistically the irregular nature of the choreatic jerks in Huntington's Chorea. EMG-bursts of certain muscles in a patient with Huntington's Chorea were taken as a measure of the jerks. Statistical analysis of the measurements, i.e. durations of bursts and intervals between bursts, revealed that the irregular nature of the choreatic jerks originate from a random process. Each choreaticjerk appears to be an entirely independent and unpredictable event.
INTRODUCTION T h e c h o r e a t i c m o v e m e n t s in H u n t i n g t o n ' s C h o r e a have been k n o w n as clinically i n v o l u n t a r y a n d irregular j e r k y m o v e m e n t s o f several parts o f the body. W i t h regard to the irregular n a t u r e o f the c h o r e a t i c m o v e m e n t s the question can a p p r o p r i a t e l y be a s k e d : ' A r e the j e r k s r a n d o m events in time?' The present investigation is an a t t e m p t to answer this question and is c o n c e r n e d with the statistical analysis o f the c h o r e a t i c movements.
METHOD This investigation was carried out in only one patient with H u n t i n g t o n ' s C h o r e a , a m a r r i e d w o m a n , aged 51 years, who showed distinct c h o r e a t i c m o v e m e n t s o f several p a r t s o f her body. R e c o r d s o f the c h o r e a t i c m o v e m e n t s , o b t a i n e d with the use o f an accelerometer, were found unsuitable for a q u a n t i t a t i v e statistical analysis. F o r this reason e l e c t r o m y o g r a p h i c bursts o f certain muscles, underlying the choreatic movements were chosen as units o f m e a s u r e m e n t (fig. 1). In this investigation E M G recordings were m a d e o f the c o n t r a c t i o n s o f two muscles, the left gluteus m a x i m u s a n d the left tibialis anterior, representing respectively u p w a r d j e r k s o f the b u t t o c k and dorsiflexion j e r k s o f the left foot. These recordings were m a d e with surface electrodes a n d an E l e m a - S c h 6 n a n d e r writing a p p a r a t u s . * Deltaziekenhuis (Rotterdam Mental Hospital), Poortugaal, The Netherlands. ** Laboratorium voor AIgemene Nat uurkunde der Rij ksuniversiteit Groningen (Dept. of Biophysics), Westersingel 34, Groningen, The Netherlands. Clin. Neurol. Neurosurg., Vol. 79-3
216
_L
k
t
L
~k~
-
,
L
i
-I-
.,a,
ti. 1
L
~',dd_
L
t
.a..aL.
t
k
~l~
-I- t i . 2
Fig. 1. Upper channel: time in seconds, L o w e r channel: sample of E M G record of gluteus maximus ti = interburst-interval, T~ = duration of burst i, -r, = duration of pause between the end of burst i and the beginning of burst i + 1.
Fig. I s h o w s s u c h an E M G
r e c o r d ( l o w e r t r a c i n g ) w i t h the q u a n t i t i e s d e f i n e d as
follows: t~ ---- i n t e r b u r s t - i n t e r v a l i.e. the p e r i o d o f t i m e b e t w e e n the b e g i n n i n g o f b u r s t i a n d the b e g i n n i n g o f burst i + 1 ; T~ = d u r a t i o n o f b u r s t i; Ti
=
d u r a t i o n o f the p a u s e b e t w e e n the e n d o f b u r s t i a n d the b e g i n n i n g o f b u r s t i + 1 ; (burst s t a n d s f o r m u s c l e b u r s t o r m u s c l e c o n t r a c t i o n ; the f o r c e o f a j e r k was n o t m e a s u r e d in the p r e s e n t i n v e s t i g a t i o n a n d t h e r e f o r e n o t i n c l u d e d as a q u a n t i t y ) .
T h e q u a n t i t i e s tj a n d Ti w e r e m e a s u r e d by h a n d w i t h a ruler, a c c u r a t e to the n e a r e s t millimeter. The measurements thus obtained were evaluated with a computer. The following three formulae were used: a. the mean interburst-interval i: 1
(tt + t 2
+ .......
+tN)=
1
N X i = 1
(1)
N is the total number of interburst-intervals of the record. b. the standard deviation of the series interburst-intervals:
/ /
N 2 i=l
(h-- 0 2 N -- 1
(2)
This statistic measures the variation of the interburst-intervals around the mean value L c. The coefficient of correlation between succeeding intervals p: (serial correlation):
P
1 (N--I)°2
N--1 v (tl - - i) (tl+l - - t) i= 1
(3)
This statistic gives information about the structure of the interval-series t~. p is defined in such a way that: - - 1 ~ lz~ + 1. (cox and MILLER, 1970; BEVINGTON, 1969). TO illustrate: if in the series of intervals h a short interval is always followed by a long one, most of the terms in (3) will be negative, the series ti then is negatively correlated. A positive correlation
217 on the other hand will be present if short intervals are always followed by short, and long intervals by long ones. If, however, shorter and longer intervals succeed, one another at random (3) will have positive as well as negative terms and the sum will be about zero. in this case, the process, according to which the interburst-intervals t~ are generated, is called a renewal process (cox and MILLER, 1970). It is obvious that for the quantities T~ and "q the same statistics (mean, standard deviation and correlation coefficient) can be defined. T h e p o s s i b i l i t y o f a c o r r e l a t i o n in t h e series T~, t h e d u r a t i o n s o f t h e b u r s t s , will b e i n v e s t i g a t e d in t h i s p a p e r w i t h t h e u s e o f a s o - c a l l e d ' j o i n t - b u r s t l e n g t h h i s t o g r a m ' . W i t h t h i s m e t h o d t h e c o r r e l a t i o n b e t w e e n t~ a n d T ~ , I will a l s o b e i n v e s t i g a t e d . Four separate EMG recordings were made, two of the gluteus maximus muscle and t w o o f t h e left t i b i a l i s a n t e r i o r .
S i n c e all f o u r E M G
recordings gave exactly the same
s t a t i s t i c a l r e s u l t s , o n l y t h e d a t a o f o n e r e c o r d i n g will b e p r e s e n t e d h e r e .
The total
n u m b e r o f b u r s t s in t h i s r e c o r d i n g is 545.
RESULTS I. N o c o r r e l a t i o n s w e r e f o u n d b e t w e e n s u c c e e d i n g i n t e r b u r s t - i n t e r v a l s ti. T h u s t h e p r o c e s s a c c o r d i n g t o w h i c h t h e b u r s t s a r e g e n e r a t e d is, w h a t is c a l l e d , a renewal process.
T h e i n t e r v a l s ti w e r e e x p o n e n t i a l l y d i s t r i b u t e d .
This means that the
r e n e w a l p r o c e s s j u s t m e n t i o n e d , c a n b e classified as a Poisson process, i.e. a process
which describes complete randomness. T h e P o i s s o n d i s t r i b u t i o n o f t h e i n t e r v a l s t~ is i l l u s t r a t e d in fig. 2. A prerequisite for a statistical analysis of the data is that the records of the E M G bursts constitute a stationary series. To ascertain this stationary quality the following investigations were carried out a. with respect to the intensity of the process (number of bursts per unit time) and b. with respect to the structure of the process (as indicated by the correlation coefficient). No significant changes in time should occur, but only variability within certain limits, in order to satisfy the conditions for the process to be stationary (cox and MILLER, 1970; MASTEBROEK, 1974; STORM, 1965). a. The series of N interburst-intervals (N = 544) is divided into 5 consecutive subseries of n = 100 intervals each. For each subseries j (j = 1,2 . . . . 5) the mean interburst-interval ~j (j = 1, 2 . . . . 5) is calculated. ~j should then obey: 2G
where ~ is the mean interburst interval of the total series and cr is the standard deviation of this series. Since ~ = 1.76 sec and a = 0.93 see for the total record, I t j - - [ I = t t J - - 1.76[~ 0.19 for n = 100. b. Calculation of the correlation coefficient for the subseries j (j = 1, 2, .., 5) gives values, which scatter around zero (Table I). This indicates a renewal process. In this case the tzj values may vary between --0.20 and +0.20 (Storm, 1965). From the results presented in Table 1 it is evident that the record forms a stationary series.
218 TABLE 1
Subseries
~j
Eij - - il
1.83 1.56 1.76 1.76 1.89
0.07 0.20 0.00 0.00 0.13
pj
no.
1 2 3 4 5
~.05 --0.18 0.13 -4).03 0.09
Since the correlation coefficients vary about zero the burst generation process represents a renewal process. Such a process can be described completely by the interburst-interval distribution p(t) (cox and MILLER, 1970) which is defined as follows: a
b
0.IC p(t)
p(t)
0.05
O.10
L
L
O.O1 Q05
0.005
1M
O.O1 10
2'0 30 CHANNEL NUMBER
O.OC
10
20 ~, CHANNEL NUMBER
Fig. 2. Histogram of the interburst-intervals h. In fig. 2a the vertical scale is linear. In fig. 2b the vertical scale is logarithmic. Channelwidth = 0.2 sec. p(t) dt is the probability for the generation of the onset of a burst in a time dt around t, given the preceding burst to occur at a time t = 0. The histogram, as calculated from the values of h is given in fig. 2. The vertical scale is linear in fig. 2a and logarithmic in fig. 2b. Fig. 2b shows that, apart from a 'silent period' during which no bursts were being generated the distribution is linear on a logarithmic scale. Hence the interburst-interval distribution can be fitted by a negative exponential curve. Such a distribution of intervals belongs to a Poisson process through which events are determined in a random manner. 2. T h e r e w e r e n o c o r r e l a t i o n s b e t w e e n s u c c e e d i n g b u r s t l e n g t h s T~. T h e r e f o r e , t h e p r o c e s s a c c o r d i n g t o w h i c h t h e b u r s t l e n g t h s a r e g e n e r a t e d , is a r e n e w a l p r o c e s s . T h e b u r s t l e n g t h s T, w e r e d i s t r i b u t e d a c c o r d i n g t o a G a m m a - t y p e f u n c t i o n as s h o w n in fig. 4b.
distribution
219 The series T~, durations of the bursts, was analysed by means of a so-called 'joint-burstlength histogram'. Such a histogram (fig. 3)
TII
Ti÷l
]
Ti
ETCETERA
Ti ÷1
Fig. 3. was obtained by plotting successively T~ (on the horizontal scale) against T~+I (on the vertical scale), T~+~ against T~+2, and so on until all 545 bursts were included (fig. 4a). This histogram shows that
30 26 A 22 I.-
v
18
~- 14 n.." Z)
rn 10
.,
m . =
•
. . . . . . . . .
m.m.m
_
~.
• .
m
,.n
r ~. - m l
.
~. m . . .
.
.
L
- .
.
.
.
.
~ = = r . - . •
,
L
.
.
.
.
.
22 0.20 p(T)
26
BURST i(T i )
0.15 010 005
2
,.,_I-L ~ , - , 8t ~ [ ~ l ~CHANNEL 14 20 26 NUMBER
, 32
Fig. 4a. Joint-burstlength histogram (for explanation see text). Fig. 4b. Burstlength histogram (Gamma-type distribution). Channelwidth 0.067 sec.
there is no correlation between consecutive burst lengths. A burst of any length on the horizontal scale will not be succeeded by a burst of a definite duration of time but - on the average - will always be followed by a burst with a mean length of about 0.56 sec. From the finding that there are no correlations between T~ values the conclusion can be drawn that the generation process for the burst lengths is a renewal process. The burstlength histogram p(T) as shown in fig. 4b represents a distribution of the Gamma-type. The mean burst length is 0.56 sec and the standard deviation 0.32 sec.
220 3. There were no c o r r e l a t i o n s between interburst-intervals t~ a n d the directly succeeding burst lengths Ti+l. As regards the relationship between the measurements t~ and T~+~, the question can be asked whether for instance a long interburst-interval h will always be succeeded by a long (or a short) burst length TI+I.
4c
Tp.1
1~
•
....
..~:~_'-_--.._~_~- : : • ....
~ m . . . . . .
..
~
.
.
.
"['l
.
.
.~
.
~ ~ - . ~'.'L--LI~.
.
.
.
.
" ¢
.
.
.
l
....
S --
+ m I l
I l m
t 1
Fig. 5. Joint histogram of interburst-intervals t~ and burstlength T~+l (for explanation see text). The joint histogram (fig. 5), obtained by plotting tj (on the horizontal scale) against T~÷~ (on the vertical scale), h÷l against T~÷z, and so on, shows that there is no correlation between the quantities t~ and T~÷~, again indicating a renewal process.
DISCUSSION
On the basis o f the a b o v e m e n t i o n e d results the conclusion can be d r a w n , that the irregular n a t u r e o f the c h o r e a t i c j e r k s - at least in this patient with H u n t i n g t o n ' s C h o r e a - is truly r a n d o m . The r a n d o m n a t u r e applies to b o t h the sequence o f the choreatic j e r k s as well as to the d u r a t i o n o f these jerks. Each c h o r e a t i c j e r k is thus a c o m p l e t e l y i n d e p e n d e n t and u n p r e d i c t a b l e event. C o n f i r m a t i o n o f these results is needed by similar studies o f m o r e patients with this disease. A l t h o u g h these results were o b t a i n e d from only one patient with H u n t i n g t o n ' s C h o r e a , it would seem that the r a n d o m n a t u r e - whatever significance it m a y have, a p o i n t not c o n s i d e r e d in this study - o f the c h o r e a t i c j e r k s is quite likely an inherent feature o f the disease and not confined to this patient alone. F o r it would a p p e a r t h a t r a n d o m events d o n o t constitute so rare a p h e n o m e n o n in clinical medicine. F o r instance, DE JONG a n d BURNS (1967) f o u n d evidence for a r a n d o m process in
221 P a r k i n s o n ' s disease as regards the d i s t r i b u t i o n o f lesions within some area o f the central n er v o u s system. Ventricular r h y t h m in patients with atrial fibrillation also shows a r a n d o m nature
(BOOTSMAet al., 1970). F u r t h e r m o r e , statistical m e a s u r e m e n t s
o f the d u r a t i o n s o f s o m e specific E E G - p a t t e r n s such as: 1. p a r o x y s m s o f bilateral s y n c h r o n o u s spikes and waves and the intervals between them. 2. bursts o f a l p h a waves a n d the intervals between them. 3. the intervals between the typical discharges o f subacute leuco-encephalitis reveals a r a n d o m d i s t r i b u t i o n o f these patterns
(.STRACKEE-KUIJERet al., 1959).
ACKNOWLEDGEMENT
T h e a u t h o r s wish to t h a n k Mrs. J e a n n e t van A b s - K n i g g e for m easu r i n g the records, Mr. B. K a m p s for d r a w i n g the figures and Mrs. A. Keulemans-Visser for her assistence with the E M G recordings.
REFERENCES
BEVINGTON,PH. R. (1969) Data reduction and error analysis for the Physical Sciences. McGraw-Hill, New York-London. BOOTSMA,B. K., HOELEN,A. J., STRACKEE, J. and MEIJLER, F. L. (1970) Analysis of R-R intervals in patients with atrial fibrillation at rest and during exercise. Circulation, 41,783. cox, D. R. and MILLER,H. D. (1970) The theory of stochastic processes. Methuen & Co., London. DE JONG, J. D. and DELISLEBURNS,B. (1967) Parkinson's Disease - A random process. The Can. Med. Ass. J., 97, 49. MASTEBROEK,H. A. K. (1974) Stochastic structure of neural activity in the visual system of the blowfly. Thesis, University of Groningen, Groningen. STORM, R. (1965) Wahrscheinlichkeitsrechnung, Mathematische Statistik und Statistische Qualit/itskontrolle. V.E.B. Fachbuchverlag, Leipzig. STRACKEE-KUIJER,A., DENIERVAN DER GUN, J. J. and STRACKEE,J. (1959) Statistical relations in the duration of some patterns in the EEG. EEG Clin. Neurophysiol. 11,620.
