ANALYSIS OF INTERVAL FLUCTUATION OF THE SENSORY NERVE IMPULSE

Transcription

1 ANALYSIS OF INTERVAL FLUCTUATION OF THE SENSORY NERVE IMPULSE Department SUSUMU HAGIWARA* of Physiology, Tokyo Medical and Dental University, Tokyo A considerable amount of work has been done on the rhythmic impulse of the sensory nerve fiber. But most of them are concerned only with the regular property, and the irregularity has not yet been fully treated from the statistical standpoint. The present work was done to analyse the rhythmic sensory impulse as a stationary time series. METHOD Excised sciatic-sartorius or sciatic-semitendineus preparation of the Japanese toad or of the bullfrog was used. In these preparations the operation for isolationg single nerve fibers can be done at a point on the nerve mm. away from the muscle. In a few cases the sciatic-gastrocnemius preparation was also employed. Operation was made under a binocular microscope, during which the preparation was often brought under a high power microscope and single fibers of 8-10 u were chosen. About half of these fibers were afferent. Most of the afferent fibers were the so-called tonic fibers (2) originating from the muscle spindles, and some of them were the phasic fibers arising apparently from the fascia. The present observation was performed exclusively on the response of the tonic fibers. The muscle was laid in a Petri-dish filled with Ringer solution. One end of the muscle was fixed and the load of various weights was applied to the muscle through a thread tied to the other end of the muscle. Action currents from single fibers were led directly from the operated region of the nerve by suspeneding this region in the air with one of the leading electrodes. They were continuously recorded for sec. with the aid of an electro-magnetic oscillograph. The intervals between successive nerve impulses were measured by the time mark of 100 per second. Standard deviation of the error of measurement never exceeded 1 msec., being much smaller than that of the interval variation. RESULT Even under the constant tension the average interval of the successive impulses increased gradually. This phenomenon is usually known as the sensory. adaptation. To illustrate the temporal course of the interval variation, the interval between the two successive impulses was plotted against the time when the Received for publication April 3, The preliminary report of this investigation was published in 1950 (1). 234

2 SENSORY NERVE IMPULSE 235 latter of these two impulses took place. In fig. 1, A is one of these illustrations taken from the experiment in which the sartorius was stimulated by a constant tension. The average interval also varied with the change of stimulus intensity. This is shown in B of the same figure. In this figure each trace shows the time course of the interval variation when the discharge became stationary (10 seconds after the application of tension). (1) Stationary State: The successive intervals named as T1, T2,..., Tn become a stochastic time series. For applying the statistical method to this time series it is necessary that it should be a stationary one, but owing to the sensory adaptation the series did not generally become stationary as a whole. Therefore analysis was made by dividing it into several stationary subseries (3). Most of these subseries were composed of more than 50 intervals and a few of them less than 30 as in the case illustrated in A of fig. 1. FIG. 1 (left). Time course of the interval fluctuation (see text). FIG. 2 (right). Histograms of intervals. A and B obtained from the same muscle stimulated by 100 gram and 300 gram respectively. Abscissae: interval in msec. Ordinates: percentage of total number of intervals. Series of histograms in A and B were taken at various stadium of adaptation. (2) Auto-Correlation: In the first step of the analysis, correlations among intervals in a series were examined, and correlation coefficients between Ti and Ti itself, Ti+1, T+2,...,Ti+n. were computed. They are usually called the auto-correlation coefficients and denoted with p0, p1, p2,...and pn. In all the cases examined, none of them showed significantly any positive or negative value except that po is identically equal to 1. The relation is illustrated in B of fig. 4. From this result T1, T2,...is considered to be a random series. Thus our investigation has come to be concentrated on the analysis of the law of interval distribution. (3) Mean Interval and Standard Deviation: The histograms obtained from the various series are shown in fig. 2 and 3. To know their character the relation between mean interval (Ė) and standard deviation (s) was at first ex-

3 236 S. HAGIWARA amined and Ė and s were compute from the following formulas (4) As is noticed from A of fig. 3, s is nearly zero when Ė is below 50 msec. Above this value of Ė, s increases with the increase of Ė. This increase is not linear but the rate itself becomes larger (as shown in table 1). TABLE 1 Fm. 3. A: Relation between mean interval in msec. (abscissae) and standard deviation in msec. (ordinates). B: Relation between mean interval in msec. (abscissae) and skewness (ordinates). (4) Skewness: As is seen in A of fig. 4, some of the histograms showed an obvious asymmetry, especially in the range of larger mean interval. To inves tigate this the skewness k was computed by the following formula (4): When the mean frequency of impulse is low, k generally shows a significantly positive value. But the higher the former becomes, the smaller becomes the latter, and finally k takes even a negative value though it is not enough to be significant. In B of fig. 3 the relation between k and is illustrated as in the case of s. This Ė shows that the skewness is highly correlated with the mean interval, and leads to a conclusion that the third moment is also determined by the mean interval. FIG. 4. A: Distribution curves of intervals. B: Correlograms for two groups of interval. Abscissae: auto-correlation coefficient. Ordinates: Order of interval. I NTERVAL (MSEC.)

4 SENSORY NERVE IMPULSE 237 (5) Higher Moments: The number of the sample was usually not large enough to give stable values of the higher moments. Therefore a direct comparison was made among the histograms whose mean intervals were not significantly different from each other. Tested by the x2-method, deviations among them did not become generally significant, which sampling was made from records of the same preparation, and even from those of different preparations deviations were not very often significant. The above fact means that each of the distributions of intervals should be characterized by a single parameter. DISCUSSION The initiation of sensory impulse is considered to be due to a depolarization of the terminal nerve fiber caused by a potential such as the generator potential of the muscle spindle. Therefore the consideration is limmited to the occasion where spikes of the terminal nerve fiber are elicited by a constant current. When an impulse appears in the nerve by a constant current, the nerve is thrown into the stadium of depressed threshold. After a while it gradually recovers and becomes responsive to the applied current, then the next impulse occurs. And in exactly the similar manner successive impulses can be obtained. Thus the intevals of these successive impulses should become smaller with the increase of stimulus intensity (here the adaptation and other transient phenomena are not taken into account). The interval must be irregular if any amount of fluctuation exist in the process of initiation of impulse. At present we have not any certain knowledge about the origin of such a fluctuation except for some suppositions to be mentioned later. In the following paragraphs an equation of the interval distribution will be derived one of such suppositions, which are described as follows: (1) The recovery process of the threshold of the nerve fiber is expressed as Aeclt when the preceding impulse has appeared at t=0, where A is the rheobase of the fiber and c is a constant in reference to this fiber. (2) An impulse should occur when the following condition is satisfied: Aecit=S+X(t), S being the stimulus intensity, X(t) an additional random factor. Here we assume that X(t) is a random normal process, the mean being 0 and the standard deviation a being a constant which is not dependent on S, but only on the nerve fiber.

5 238 S. HAGIWARA where Then, 1-ƒ³nƒ t becomes the probability in which no impulse occurs between t and tn+i, if any response has not occurred in the preceding intervals. When P (S, t) ƒ t expresses the probability that the interval of impulse is between t and t+ƒ t, it is given by the product of the probabilities that no impulse occurs in the preceding intervals and the probability that impulse does occur in the interval from t to t+ƒ t, or At the early stage of this investigation the calculation was made by the equation [2] using the step-by-step method, in which 5 msec. was taken as a values of time unit. The most probable values of c and A: ƒð were chosen by the trial and error method. These values show a little variation among the different preparations. In the case given in fig.5, the curves were calculated by taking the different values as the stimulus intensity and by assuming c=100 msec. and A: ƒð: 1: 1. Comparing these curves with those shown in fig.4, it will be seen that the equations are sufficient to express the distribution of interval, especially in the following points: (1) The relative value of the standard deviation to the mean becomes larger with the increase of the latter. FIG.5. Distribution curves calculated from equation [2] (see text). (2) The skewness shows large positive values for the large mean, but becomes smaller with the decrease of the mean. A mathematically more complete form of equation [2] can be written (10) as; Since ƒ t can be taken as small as desired, we can expand each logarithmic term and use only the first term of expansion in each case, or i.e. it becomes:

6 SENSORY NERVE IMPULSE 239 then we obtain: A similar group of curves can be obtained from equation [3] In spite of this good fitness of the equations the based assumptions have some difficulties. The first difficulty is that the estimated values of i are rather large, relative to the rheobase of the nerve fiber. These values can not be explained exclusively from the thermal noise of the membrane resistance. Fatt and Katz (5) demonstrated the spontaneous excitation of the terminal nerve fiber at the motor ending. They concluded that even in a very fine terminal nerve fiber the standard deviation of fluctuation of the resting potential due to the thermal noise can not become such a large value. The second difficulty is that: According to Katz (6) the tonic afferent fiber of M. ext. long. dig. IV arises from more than one muscle spindle and some of impulses coming from each spindle occasionally fails to transmit at the branching region. If this is also the case with the sartorius the fluctuation of interval is not originated from the single pacemaker but from the several, among which some interaction may exist. In the present stage a further speculation is thought to be of little use, though the above assumption is rather simple to explain such a mechanism. However, the relation between the spontaneous oscillation of the threshold in the medullated nerve fiber (7, 8, 9, 10) and this phenomenon will be an interesting problem to be considered. Lastly, an additional description will be made on the relation between the fluctuation of impulse interval and Weber's law. It is demanded by Weber's law that the central effect of the sensory impulse have a linear relation with its frequency because the latter has a similar relation with the logarithm of stimulus intensity in the case of no fluctuation. If there exists some fluctuation, the relation between the mean frequency and the logarithm of stimulus intensity should not be linear any longer. Our results however, have indicated that its deviation becomes larger especially in the range of low frequency i.e. of low stimulus intensity. This may be one of the possible explanations of the fact that Weber's law deviats specially during a low stimulus intensity. During the preparation of this paper a similar kind of work has been published by Buller, Nicholls and Strom. Their experimental results were quite similar to those of ours. They also interpreted the fluctuation from the standpoint of thermal agitation in the terminal nerve fibers. The equation upon which their consideration based was E=E0+(a/t). If we take early two terms in a expansion of our equation (E=Aec/t), we can obtain the above equation. SUMMARY (1) The interval fluctuation of impulses of the tonic nerve fiber from the muscle spindle of a Japanese toad is investigated by method of the time series analysis. (2) Dividing the interval series into several stationary subseries and computing the serial correlation coefficients, we find that these subseries can be

7 240 S. HAGIWARA considered as a random process. (3) In these stationary subseries, not only the absolute value of the standard deviation but also its relative value to the mean interval becomes larger with the increase of the mean interval. (4) High correlation among the moments of the interval distribution makes us conclude that each of the distributions obtained from the same preparation can be characterized by a single parameter. (5) Assuming a random fluctuation such as that due to the thermal agitation the equation of distribution is calculated. And comparison is made between the observed and the calculated histograms. ACKNOWLEDGEMENT The author wishes to acknowledge to Prof. Y. Katsuki and Prof. T. Wakabayashi for their important advices, to Dr. I. Tasaki for his kind advice on the technique of the single fiber method, to Dr. M. Ogawara for his precious suggestion on the mathematical treatment and to Mr. H. Uchiyama for his kindly correctioning English of this paper. REFERENCES 1. HAGIWARA, S. Report. Physiograph. Sci. Inst. Tokyo Univ. 4: 28, KOBAYASHI, Y., OSHIMA, K. AND TASAKI, I. J. Physiol. 117: 152, WOLD, H. A study in the analysis of stationary time series. Uppsala, SNEDECOR, G. W. Statistical method FATT, P. AND KATZ, B. J. Physiol. 117: 109, KATZ, B.J. Physiol. 111: 248, PECHER, C. C. R. Soc. Biol. 122: 87, PECHER, C. Arch. Internat. Physiol. 49: 129, ERLANGFR, J., BLAIR, E. A. AND SCHOEPFLE, G. M. Amer. J. Physiol. 135: 705, LANDHL, H. D. Mathematical Biophysics, 3: 141, BULLER, A. J., NICHOLLS, J.G. AND STROM, G. J. Physiol. 122: 409, 1953.