[ 22O ]
THE RESPONSE OF A SENSE ORGAN TO A
HARMONIC STIMULUS
BY J. W. S. PRINGLE AND V. J. WILSON* The Zoological Laboratory, University of Cambridge
(Received 29 August 1951)
(With Plate 9 and Eight Text-figures)
INTRODUCTION
Since the classical researches of Adrian (1926), in which electrical amplification techniques were first used to detect the pattern of impulses which pass up a sensory nerve when its end-organ is stimulated, a large number of different types of receptor have been studied by this method. Its attraction is that it gives direct information about the quality of stimulus to which the ending responds and also, when the activity of a single nerve fibre can be observed, provides quantitative measurements of the relationship between the magnitude of the stimulus and the frequency of the nervous response. From such experiments have come the most important generaliza-tions about the properties of receptors (Adrian, 1928). The electrical recording technique in nerves and nervous tracts is now the most useful method available to the physiologist not only for the investigation of sense organs but also in the analysis of the working of the rest of the nervous system.
From another point of view, however, this powerful technique has so far proved disappointing. Detailed analysis of the properties of sense organs, nerves and muscles has not led to an increase in the accuracy of our knowledge of the physiological composition of the patterns of movement which make up the behaviour of the intact animal. The work done by Sherrington and his co-workers on the reflex activity of the spinal vertebrate remains the basis for subsequent investigations of animal move-ment at this level of analysis. The concepts introduced by him have provided the framework for analytical researches such as those of Gray & Lissmann (1946) on the toad, and of Pringle (1940) on Arthropods. Workers in this field have been concerned to demonstrate qualitatively the existence of reflexes or inherent movement patterns and to show how these are related to the normal behaviour of the intact animal. In few cases has the reflex or basic movement pattern been given quantitative definition, and in none has the attempt been made to analyse further this quantitative description in terms of the properties of the sense organs, nerves, muscles and portions of the central nervous system which combine to produce the observed activities of the preparation.
The difficulties of such a further analysis are considerable. Not the least is the lack of detailed knowledge of the anatomy of the nervous system of most of the
Animals which have been studied, and the great numerical complexity of the cells whose co-ordinated functioning produces the response observed. The Arthropods have the advantage over the Vertebrates in this respect; the work on Crustacea (Wiersma, 1941) and insects (Pringle, 1939) has shown that, at any rate on the motor side of the reflex arc, the number of nerve cells which have to be considered may be conveniently small. But lacking also has been the theoretical basis for the synthesis of an understanding of the functioning of the whole of a complicated dynamic system from a knowledge of the functioning of each of its parts. Help has recently come to biology in this matter from an unexpected source. The efforts of engineers to design satisfactory and stable automatic control machinery have led to the develop-ment of a comprehensive theory of such self-regulating devices or servomechanisms, and it has come to be recognized that the long-established self-regulatory properties of living organisms (Cannon, 1939; Holmes, 1948 and earlier papers) imply a similarity in mode of functioning between organisms and some of these man-made machines. The similarity resides ultimately in the mathematical expressions adequate to describe the functioning of the two types of system. While there is not yet agree-ment among biologists about the value of some of the analogies which may be drawn between the functioning of the living and the non-living, there can be no doubt that much may be learned from the methods used by control-system engineers in the analysis and synthesis of self-regulating machinery. The neurophysiologist, who is still at the stage of preliminary analysis, can be grateful for any hints that may enable him to plan his experiments economically, and to acquire his results in a form that will make easier the ultimate task of synthesis of an understanding of the working of the whole of the nervous system and, ideally, of the whole of the living body.
The principles of the Theory of Control are now well described in a number of text-books, such as that of Macmillan (1951). The methods used in the analysis of a complicated piece of automatic machinery repay a short examination in order to see how they are related to the methods of the physiologist in his investigations of the living organism. The engineer has available to him several points in the control circuit at which, with the aid of instruments, he can measure the state of the machinery. He can also apply controlled ' disturbances' which alter the state of the machinery in a manner which can be measured. He may be obliged for various reasons to make his measurements without disturbing the natural running of the machine. On the other hand, it may be possible for him to stop its normal operation, or the operation of portions of it, in order to study the properties of the components in partial or com-plete isolation. If the machinery is fully automatic it will involve, in some part of its mechanism, a ' closed-loop' connexion between its constituent parts; that is, a part A whose operation is controlled by another part B will also, through a greater or smaller number of intermediate stages, control part B. This is a universal feature of all fully automatic or self-regulatory systems. More than one such closed-loop may be present in the machinery, and the engineer may be able to 'open' one or more of these by disconnecting one part from another in order to make his measurements. These analytical procedures have their exact counterparts in the methods of the physiologist—the preliminary observation using techniques which do not affect the
222 J. W. S. PRINGLE AND V. J. W I L S O N
animal as a whole; the application of 'stimuli' producing 'responses' of variou? kinds; the interruption of normal working by anaesthetics: and the disconnexion of parts by dissection. Were it not for the confusion caused by the growth of two different terminologies to describe the same procedures, the similarity would no doubt be more widely appreciated.
One of the most obvious ways in which the technique of the biologist is inferior to that of the engineer is in the nature of the disturbances or stimuli used. The engineer, with the goal of mathematization always before him, uses disturbances whose time course is planned to facilitate subsequent mathematical manipulation of the results. To the biologist a stimulus often means merely a quantitatively undefined change in the intensity of some quality in the environmental situation. When it is more exactly specified, as in physiological experiments, the definition usually covers only the magnitude of the change and does not concern itself with the rate of change. The change may be made as rapidly as experimental conditions allow, and where the rate of response is slow this approximation to a sudden 'transient' is adequate for the purpose. But it is a type of disturbance which leads to results in a form which is not ideal for all purposes, and the form is particularly unsuitable when the objective is a synthesis of an understanding of the dynamic operation of the whole of the mechanism or the behaviour of the animal as a whole.
For such a synthesis it is necessary that the properties of each of the component parts of a control circuit be expressed in a form which makes possible the combination of the results into a statement or representation of the properties of the whole circuit. With this in mind, engineers have developed the concept of the 'transfer function', which may be either a mathematical expression or a graphical representation of such an expression, and with the aid of which it is possible to describe accurately and fully the response which is given by the component when a disturbance of any known time course is applied to it. A mathematical expression naturally carries with it an assumption of accuracy which may be unjustified in biological work, and graphical forms may be more suitable in this context; but with either the procedure is well worked out for the synthesis of an understanding of the properties of the whole from a knowledge of the properties of the parts once these are expressed in suitable form. A note of the different forms of transfer function is included as an appendix to this paper.
As will be clear from a perusal of the procedures summarized in the Appendix, the derivation of the transfer function of a component from results obtained by the use of transient disturbances is laborious and carries the danger of serious errors. For linear systems (a linear system is defined as one in which the magnitude of the response is directly related to the magnitude of the stimulus) the use of harmonic (sinusoidal) disturbances greatly simplifies the subsequent calculations. With this type of stimulus the response is always also sinusoidal; the properties of the com-ponent can therefore be described in terms of three parameters, a simple propor-tionality of magnitude and a phase relationship between input and output for each frequency of harmonic stimulation. A single vector diagram thus contains a full statement of the results in a form which may be used directly for subsequent synthesis procedures. Although the assumption of linearity is often unjustified in biological systems when a wide range of stimulus intensities has to be considered, it can serve to initiate an understanding of the behaviour of the system and to indicate the nature of the effects produced by departures from linearity. Transfer functions for systems containing non-linear elements apply adequately to a particular amplitude of stimulus or input disturbance, and provided this limitation is borne in mind they may be used to build up a picture of the functioning of the whole.
An investigation of the properties of a sense organ from this point of view is described in this paper, the object being not to reveal any essentially new feature of sensory neurophysiology, but to re-describe the well-known phenomenon of sensory adaptation in a manner which may be of more direct value in an understanding of the functioning of the whole response mechanism of which the sense organ is the initial part. The work was undertaken more in order to test the possibilities of the method than with the intention of pursuing the investigation to a high degree of accuracy. For this reason a sense organ was chosen which is easy to study in isolation and from which single nerve-fibre oscillograph records can be obtained with a minimum of dissection.
MATERIAL AND METHODS
The thoracic legs of the American cockroach, Periplaneta americanaL.., bear on the tibia and femur a series of large spines each of which is innervated by a single sensory nerve fibre of large diameter. The response of this ending to mechanical stimulation was described by Pumphrey (1936), who drew attention to the fact that the rate of adaptation to a constant deflexion of the spine is slow compared with that of the tactile receptors described by Adrian & Zotterman (1926). After some preliminary experiments with various tibial spines, a choice was made of the single large spine on the dorsal surface of the end of the femur of the metathoracic leg, which gave consistent results in many individuals.
Impulses in the sensory fibre supplying this spine may be recorded with platinum wire electrodes inserted through the cuticle at opposite ends of the femur without any dissection, and the isolated femur preparation survives for sufficiently long for experiments to be performed. The simplicity with which single nerve-fibre records may be obtained with this preparation is a great advantage.
224 J- W. S. PRINGLE AND V. J. W I L S O N
The electrical amplifying and recording system is standard. A Grass P4 pre-amplifier was used together with the built-in amplifiers of the Cossor 339A oscilloscope.
The generation of a sinusoidally varying mechanical stimulus of variable frequency was achieved by means of a steel strip, clamped at one end and at the other carrying a heavy weight, which thus oscillated in a horizontal plane (Text-fig. 1). A copper
Wax block
'_/_ I
Text-fig. I. Diagram of experimental apparatus.
To transmit the stimulus to the femoral spine, connexion was made between its tip and the steel strip by a short length of fine elastic, and the preparation on its wax block was then adjusted by means of a micromanipulator so that the elastic was just taut at the end of the swing of the oscillating strip. The leg spines of the cockroach have a small freedom of movement in their sockets and this movement produces slight excitation of the sensory ending, but the more effective and natural method of excita-tion when it is pressing against an object on the ground is with the spine against its limiting stop in the socket. In this position (which was used in the experiments) its movement is negligible and the adequate stimulus is a change of tension. The sinusoidal changes of stimulus intensity were thus transmitted to the spine without distortion. The peak tensions applied in this way were about 5 g.
Stimulus,
Text-fig. 2. Response of the femoral spine ending to a low-frequency harmonic stimulus superimposed on a steady tension.
METHOD OF ANALYSIS Typical oscillograph records are shown in PI. 9.
The analysis of these records is carried out as follows. A zero point in the cycle of stimulation is first chosen, conveniently at the narrowest part of the stimulus trace (minimum tension on the spine). By measurement from this point a graph is plotted showing the relationship between instantaneous impulse frequency (the reciprocal of the interval between one impulse and the next) and the instant of time to which this refers. In Text-figs. 2-4 each frequency measurement is referred to the instant of time half-way between the two impulses.
2 2 6 J. W. S. PRINGLE AND V. J. W I L S O N
3
Stimulus
f=&85
50
-30
JT 20
10
Sec
10 1-5
Text-fig. 3. Response of the femoral spine ending to harmonic stimuli of different amplitudes.
/•=O27
RESULTS
The records of PI. 9 show, as may be expected, that the sensory ending is excited only when the tension on the spine reaches a threshold value. To avoid this effect, which makes the frequency graphs discontinuous, the harmonic stimulus was some-times superimposed on a steady tension sufficiently great to prevent the frequency of impulses falling to zero. The record from which Text-fig. 2 is plotted was obtained with this arrangement; the frequency graph then becomes sinusoidal with super-imposed random fluctuations.
It will be noted that the maximum response regularly precedes the maximum of the stimulus. Experiments were carried out to determine how the amplitude and phase-advance angle varied: (1) with change in the amplitude of the stimulus; (2) with change in the frequency of the stimulus.
Text-fig. 3 shows a plot of an experiment in which the tension was allowed to fall just below threshold at the troughs of the stimulus cycle and in which the frequency plot is therefore discontinuous. The stimulus amplitude was varied at constant frequency. The peak response occurred at a point in the cycle corresponding to a phase advance of approximately 45 ° and showed a tendency to occur earlier as the amplitude of the stimulus was increased. The amplitude relationship also shows some non-linearity.
Text-fig. 4 shows a plot of the records of an experiment in which the frequency of stimulation was varied at constant amplitude. The phase advance angle of the maximum response for the three frequencies of stimulation (0-85, o-6, and 0-27 cycles/sec.) is, respectively, 45, 51 and 490, and the amplitudes of the peaks of the response are in the ratio 54 : 47 : 29. These results are plotted on the graphs of Text-figs. 7 and 8.
DISCUSSION
The results obtained do not cover a sufficiently wide range of stimulus frequencies and amplitudes to permit an accurate evaluation of the transfer function for this sense organ, nor to show quantitatively how this changes with stimulus amplitude owing to the non-linearity of the ending. They do, however, show that with a slow harmonic stimulus the ending produces a phase advance in the plot of nerve impulse frequency against that of the stimulus. Owing to the presence of a random scatter in the timing of individual impulses the estimates of phase advance can only be made approximately, and for the same reason frequencies can only be used which are sufficiently slow to produce a considerable number of impulses at each cycle of stimulation. At higher frequencies, which ultimately produce only one or two impulses per cycle, it would be necessary to adopt a different measure of nerve activity to indicate quantitatively the excitation of the ending.
228 J. W. S. P R I N G L E AND V. J. W I L S O N
to transient stimulation and the phase advance using harmonic stimulation is dis-cussed in the Appendix.
The non-linearity of the response of the sense organ is shown in Text-fig. 3, the phase advance increasing and the proportionality of magnitude decreasing as the stimulus is increased. Evidence from other types of mechanical sensory ending (Matthews, 1933; Fessard & Sand, 1937), in which the frequency of nerve impulses has been shown to be approximately proportional to the logarithm of the stimulus intensity, also suggest that this element in the reflex arc cannot be regarded as a linear component.
The existence of the phase-advancing property in sense organs has a number of interesting implications when their role in the intact animal is considered. Reflex arcs whose function is to regulate the level of some aspect of vital activity are known to occur in many animals. The best known of them is perhaps the myotatic reflex of the vertebrate, in which the stretch produced in muscle sense organs by an increase in tension on the muscle as a whole leads to an increase in the tone of the muscle (Creed, Denny-Brown, Eccles, Liddell & Sherrington, 1932). The control sequence in this reflex includes the delay involved in nervous conduction to and from the central nervous system, and the central and neuromuscular delays. The theoretical analysis of automatic control systems shows clearly that a regulating system of this sort with finite delays in the control sequence is unstable and gives rise to oscillations or ' hunting' if some form of compensation is not included. The performance is made worse if there is any inertia in the load against which the muscle is working. The introduction of a phase advance in one of the components in the control sequence provides the required compensation, provided that the ratio of derivative to direct response of the component is greater than the total delay time and that the overall gain of the system (e.g. in this case the muscle tension produced reflexly by a given stimulus tension) is not too great (Macmillan, 1951). This function of sensory adaptation has been recognized by Merton (1951), who has demonstrated its quantita-tive significance in the intact reflex in man by studies of the timing of the silent period in the motor discharge which occurs when a stimulus in the form of an increase in tension is applied to the muscle.
particular control mechanism, but a reduction of the derivative response in the sense organs certainly favours the generation of rhythmic oscillations by such a mechanism. Harmonic stimulation of mechanical receptors is most nearly approached under natural conditions in the swimming movements of fish. If reflex arcs similar in function to the myotatic reflex of mammalian limbs are present in the trunk neuro-muscular system of fishes (see Gray, 1936), the relation of these reflexes to the inherent rhythm of activity of the spinal cord (i.e. whether they will reinforce or antagonize the inherent rhythm) will be determined by the phase relationships round the reflex arc, and to these the properties of the sense organs will make a significant contribution.
In any analysis of the phase relationships round a complete reflex arc the integrative properties of junctions have also to be considered. The liberation of a chemical mediator of synaptic or neuromuscular transmission whose concentration is related to the frequency of arriving impulses implies a phase lag in this part of the control sequence, and the rates of production and destruction of such a mediator provide another variable which will affect the overall performance of the reflex arc. The slow potential changes observed by Barron & Matthews (1938) to accompany the initiation of impulses at the spinal origin of motor fibres also provide an intermediate point at which the level of excitation may be measured, and the evidence that the frequency of impulses here as in a sense organ is related both to the magnitude of the preceding event and to its rate of change suggests that the derivative response or phase-advancing property is present in this type of excitable nerve cell as well as in the sensory fibres.
230 J. W. S. PRINGLE AND V. J. W I L S O N
impulse is supposed to preserve its identity in its passage through the synaptic maze. It postulates that the smooth potential wave-forms which can be recorded from the cortex are the sum, not of a large number of impulse spikes, but of a number of synchronized potential changes in the neurons each of which has an approximately sinusoidal time-course, and each similar in nature to the slow potential waves which were observed by Barron & Matthews (1938) in the dorsal and ventral horns of the spinal cord.
The analysis clearly becomes inapplicable in the case of junctions where there is appreciable synchronization between presynaptic and postsynaptic impulses, just as it fails in the case of sense organs at stimulus frequencies which produce only a few impulses per cycle. But it is not clear that such synchronization across synapses is as common under normal conditions as it is when the presynaptic volley is pro-duced artificially by maximal stimulation of the sensory nerve trunk. Detailed neurophysiological examination of reflex connexions between single nerve fibres in the central nervous system has as yet been reported only in the case of giant fibres (Wiersma & Turner, 1950), whose synapses are not concerned with maintained activity in the muscles.
APPENDIX
The form and derivation of the transfer function
The transfer function for an element in a control system is ideally defined in the form of a mathematical expression with the aid of which differential equations may be derived for the behaviour of the element for any type of input disturbance. Procedures are available for the derivation of transfer functions from the results of experiments using either transient or harmonic (sinusoidal) inputs, but the com-putation involved in the use of results of the latter type is much less cumbersome. Transfer functions in mathematical form are commonly expressed in one of three ways:
(1) With the Heaviside differential operator D ( = d/dt).
(2) In terms of the parameter p (sometimes s) defined by the Laplace
transfor-mation ra>
- W ) } = / ( / > ) = e-»>F(t)dt. J o
-(3) In terms of complex numbers in the form M{jw).
Graphical representations of transfer functions are based on the Argand diagram,
using the identity M(Jw) ^ Ag_^
When the proportionality of magnitude (^4) and the phase angle (tfi) between input and output are known from observations over a range of input frequencies, a vector diagram in the complex plane may be plotted direct in polar co-ordinates and used in synthesis procedures without any attempt to remove experimental error by the fitting of mathematical formulae to the observed measurements.
a transient stimulus produced by suddenly increasing the tension on the spine by means of a rapidly acting electric relay. To the points which represent the instan-taneous frequency of impulses in the sensory nerve (plotted as previously described) has been fitted a smooth curve of the equation
frequency = 24 + i65e~12'+ i2oe~11' + 42£~0'176'. (1) The fit is good within the limits of natural scatter of the observations.
350
-300
250
200
-150
100
50
Stimulus
- J 1 I L I 1 ' I I 1 ' 0-5
Sec 1-5
Text-fig. 5. Response of the femoral spine ending to a transient stimulus. The points are plotted from measurement of the oscillograph record and the curve is that of Equation (i).
150
-100
50
10 15 20
Sec.
Text-fig. 6. Continuation on a longer scale of the record plotted in Text-fig. 5.
232 J. W. S. P R I N G L E AND V. J. W I L S O N
with initial conditions of zero activity in the nerve. Under such conditions the Heaviside differential operator D can replace/" to give the transfer function in form (1) above. The response of this sense organ thus appears to involve the second and third derivative of the stimulus as well as the first.
-1160
- 1-40
- 120
r
60
50
•40
30
20
10
/
y
i i i i i
-- 100
- 80
- 60
- 20
UJ-lTlf
Text-fig. 7. Observed (dotted) and calculated (continuous) curves for magnitude proportionality A and phase angle <p for the response of the femoral spine ending to a harmonic stimulus, t
To derive the harmonic response locus from the transfer function in the form of expression (2) we may substitute jco for/) and express the result in the form This gives
V [ (5 7- 7 8 9 0 ^ +a,* (486-35 l a ^
UUi) ~
x expr . fa) (486 - 3 5 1 w
2
)] .
L 57-789^2 J-3-i3'3ws
233
results from the experiment of Text-fig. 4 are also shown on these graphs, to indicate the correlation between the results obtained with the two types of input stimulus. It should be borne in mind that the experiments were done with preparations from two different individuals.
200 Scale of A
Text-fig. 8. Vector plot of the observed (dotted) and calculated (continuous) response of the femoral spine ending to a harmonic stimulus.
SUMMARY
1. The response is described of the tactile ending in a femoral spine of the leg of Periplaneta to a harmonic (sinusoidal) mechanical stimulus of low frequency. The peak frequency of impulses in the sensory nerve precedes the maximum tension of the stimulus.
2. This result is shown to be a corollary of the adaptation shown by the sensory response to a transient stimulus.
3. The concept of the 'transfer function' is discussed in relation to neuro-physiology. Its value is explained as a means of describing the dynamic properties of component parts of the nervous system when the objective is an understanding of the functioning of the complete reflex arc.
234 J- W. S. PRINGLE AND V. J. W I L S O N
REFERENCES
ADMAN, E. D. (1926). The impulses produced by sensory nerve endings. J. Physiol. 61, 49-72. ADRIAN, E. D. (1928). The Basis of Sensation. London.
ADRIAN, E. D. (1933). Afferent impulses in the vagus and their effect on respiration. J. Physiol. 79, 332-58.
ADRIAN, E. D. & ZOTTBRMAN, Y. (1926). The impulses produced by sensory nerve endings. Part 3. Touch and pressure. J. Physiol. 61, 465.
BARRON, D. H. & MATTHEWS, B. H. C. (1938). The interpretation of potential changes in the spinal cord. J. Physiol. 93, 276-321.
CANNON, W. B. (1939). The Wisdom of the Body. New York.
CREED, R. S., DENNY-BROWN, D., ECCLES, J. C , LIDDELL, E. G. T. & SHERRTNGTON, C. S. (1932). Reflex Activity of the Spinal Cord. Oxford.
FESSARD, A. & SAND, A. (1937). Stretch receptors in the muscles of fishes. J. Exp. Biol. 14, 383-404, GRAY, J. (1936). Resistance reflexes in the eel. J. Exp. Biol. 13, 180.
GRAY, J. & LISSMANN, H. W. (1946). The co-ordination of limb movements in the Amphibia. J. Exp. Biol. 33,
133-42-HOLMES, S. J. (1948). Organic Form and Related Biological problems. University of California Press. LORENTE DE No (1938). Analysis of the activity of chains of internuncial neurons. J. Neurophysiol. I,
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MACMILLAN, R. H. (195 I). An Introduction to the Theory of Control. Cambridge University Press. MATTHEWS, B. H. C. (1933). Nerve endings in mammalian muscle. J. Physiol. 78, 1-53.
MERTON, P. A. (1951). The silent period of a muscle in die human hand. J. Physiol. 114, 183-98. PITTS, R. F. (1942). The function of components of the respiratory centre. J. Neurophysiol. 5, 403-13. PRINGLE, J. W. S. (1939). The motor mechanism of die insect leg. J. Exp. Biol. 16, 220-31. PRINGLE, J. W. S. (1940). The reflex mechanism of the insect leg. J. Exp. Biol. 17, 8.
PUMPHREY, R. J. (1936). Slow adaptation of a tactile receptor in the leg of the common cockroach. J. Physiol. 87, 6 P.
WIERSMA, C. A. G. (1941). The efferent innervation of muscle. Biol. Symposia, 3, 259.
WIERSMA, C. A. G. & TURNER, R. S. (1950). The interaction between the synapses of a single motor fiber. J. Gen. Physiol. 34, 137.
EXPLANATION OF PLATE 9
PLATE 9
a
