studies two sufficiently large samples ofsystematic data on resting

DISTRIBUTION OF RESTING PAIN-REACTIONTHRESHOLDS IN GUINEA-PIGS, WITH ASTATISTICAL CONCEPT OF GRADATION

OF BIOLOGICAL EFFECT*

CLAUDE V. WINDER

Investigation of the minimal contraction of the musculus cuta-neous maximus ("skin twitch") in the guinea-pig, as elicited byradiant thermal stimulation,"' established with reasonable certaintyits nociceptive character,t its susceptibility to the selective action of"central" analgetic drugs, and its probable integration with higher-level neural pain mechanisms. Since the threshold stimulus isamenable to quantitation, the phenomenon 'offered possibilities foracquiring quantitative information on certain objective aspects of thepain mechanism.

In the course of adapting the phenomenon to pharmacologicalstudies two sufficiently large samples of systematic data on restingthresholds were accumulated to enable the present distribution study.

ProcedureThe depilation of the dorsal skin of the animals, the construction and

operation of the radiant thermal stimulator, and the response are describedin detail elsewhere.40 For reasons documented previously, the intensity ofradiant stimulus 'o was selected as preferable to the duration4 15 for the

* From the Research Laboratories, Parke, Davis and Company, Detroit, Michi-gan. This paper is a portion of the material presented at the A. A. A. S. Confer-ence on Medicinal Chemistry, Gibson Island, Maryland, July 22-26, 1946. Theauthor wishes to acknowledge the technical assistance of Lillian Cox and JoanCostantino.

t There is at present no critical evidence as to whether the skin twitch issubserved by an afferent system for "first," "fast," "pricking," "bright" pain, orby one for "second," "slow," "delayed," "burning" pain, as distinguished byvarious authors.

An interesting parallel between the present phenomenon and human pain wasnot pointed out in the preceding report. The skin twitch as elicited is artificiallyfreed from "touch" stimulation, is threshold in grade, and is poorly localized.Analogously in man, "wherever this (pain) is found devoid of specialized touchas in the cornea, glans penis, or viscera," . . . it lacks "any but the most generallocalization,"" particularly at threshold.

YALE JOURNAL OF BIOLOGY AND MEDICINE

dependent parameter of threshold stimulus quantitation. The inferencefrom finding optimal morphine-sensitivity of the threshold when employing acombination of relatively large area of stimulation (730 mm.2 cross-sectionalarea of radiant beam) and relatively short duration (4 seconds), was thatsuch a pattern of emphasis on spatial neural facilitation probably integratesthe phenomenon more closely with higher pain mechanisms,40 hence thesecharacteristics of stimulus were chosen for present observations. Use wasmade of an "interval-splitting" approach to the threshold, with time betweentrial stimuli of two cycles of the particular stimulators used (75 seconds),as previously exculpated.40 Some artificial blackening had failed to decreaseusefully dispersion of data on animals selected for darkness of skin,40such animals without added blackening have been used. The increasedstimulus efficiency, per se, resulting from blackening, was of no inherentusefulness, because the stimulus is quantitated in indirect units (lamp wattage).

The data to be analyzed are pretreatment or resting readings, in studiesto be reported in part elsewhere. Specifically, they are three successivethreshold readings per animal, finished at 20-minute intervals. Such read-ings were made alternately on two animals during the early forenoon and ontwo during the early afternoon, except in those cases where one, or more, ofthe four animals was omitted.

ResultsI. Inter- and intra-animal variability among the three succes-

sive threshold readings. Instability and exclusion from computa-tions of the first reading. All resting readings by two operators ofrespective stimulators, working during periods separated by a yearand a half, are summarized and analyzed in table 1. The absolutevalues of lamp wattage in the two sets (I and II) are not comparablebecause of different efficiencies of the optical systems in the respectivestimulators.

Each set of data shows among animals significant differences,large in comparison with uncontrollable variability in measurement(animal-time discrepance). Also, the second reading tends to beslightly but significantly less than the first, but no significant dif-ference between the second and third was demonstrated.

Consistency in the amount of (mean) drop in threshold betweenreadings one and two for the two sets of data would be a positivetest of stability of initial experimental conditions. Granted the like-lihoood that the main difference in average measured threshold levelsbetween the two sets is due to the difference in available stimulatingheat per watt of reading as imposed by the respective optical systems,

290

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To test the significance of this difference, it was desirable to usea transformation of the variates which would render the "error"variance of the two sets homogeneous and independent of means.Inspection of table 1 reveals that the means of the two sets and therespective Verror variance differ by the same ratio, approximately1: 2; i. e., error relative to mean watts was essentially the same forthe two samples of data.* The logarithmic transformation was there-fore the obvious one to accomplish the foregoing ends, and at thesame time to yield the desired relative measure of threshold changein the two sets.

Analysis of the resulting "log thresholds" (table 2, A) yields thesame condusions as available before the transformation (table 1).In addition, the now-convenient comparison between the two sets ofdata (table 2, B) shows, at item (1), for the pooled data as wasalready shown for the individual sets, a significant drop betweenreading one and two and no significant change between two andthree. The difference between mean readings with the two opticalsystems hardly needs the proof provided in item (2). The point ofmain in.terest here is item (3), which shows that the logarithmic, orrelative drift between readings is significantly different for the twosets. That is, experimental conditions affecting the small drop inthreshold from reading one to reading two were not identical for thetwo sets of data.

Since these large samples of data, both individually and pooled,failed to suggest a difference between the second and third readings,adequate experimental stability is probably attained at the time ofthe second reading. It would thus seem desirable to calculate themean resting threshold from the second and third readings, leavingthe first reading in all cases for a more general type of experimental

* It should be pointed out that the proportionality of average error to meanvalue for the two samples is the result of artificial imposition of two respectivescales of measurement (stimulators), and does not imply that error by either scalewould be so related to measured value. Such would indicate a "lognormal""6 inter-animal distribution, which is contrary to actual observation (vide ixfra).

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control. But how, quantitatively, is precision in establishing a "rest-ing" threshold affected by the reduction in available data resultingfrom exdusion of the first reading? To supply information on thispoint the two sets of data were subjected to analysis including onlysecond and third readings (table 3). Error per degree of freedomfor set II is seen to be significantly reduced, indeed, in an amountwhich approximately compensates for the loss in degrees of freedomresul.ting from dropping the first readings. Since error was not sig-nificantly reduced in set I, one may infer that the difference betweensets I and II in relative threshold drop after the first reading maypossibly be due to an inconsistency of circumstance affecting the firstreading in set II, not present significantly for set I.

A further point now apparen-t is that error in establishing theresting threshold on the basis of second and third readings is sig-nificantly less in set II than in set I. The difference approximatesthat which might be expected from the fact that though in both casesreadings were made within S watts, this represents greater relativeprecision for the larger wattages involved in set II. There is,reciprocally, a strong suggestion (P = 0.05) of greater variabilityin thresholds among the animals of set II than among those of set I,but such a suggestion of heterogeneity may arise, e. g., from differ-ences in the artificial depilation at the respective periods (vide infra).

The working implication of these analyses is that if under presentconditions comparable data are to be collected over long periods fromanimals of varying sources and incompletely controlled conditions,by different operators of different stimulators, some precision isgained in establishing stable "resting" thresholds of individual ani-mals if the first of three consecutive (20-minute) readings is notincluded in their computation.

II. The distribution of thresholds among amimals. The cis-tributions of resting thresholds for the two sets of animals are shownin Fig. 1. In line with the foregoing analysis, the threshold foreach animal was taken at the average of the second and third read-ings, expressed as watts (A) and log watts (B), respectively. Tofacilitate comparisons between the two samples their respective dis-persions are expressed in terms of standard variates and the distribu-tions reduced to unity. It was noted above that their absolutedispersions were importantly related to the means by the respectivescales of measurement (optical systems).

Application of the X2 test for goodness of fit of the theoretical

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HISTOGRAMS AND THEORETICAL NORMAL DISTRIBUTIONSOF THRESHOLDS AMONG ANIMALS{AVERAGES OF SECOND AND THIRD READINGS)

-4 -3 -. -I 0 I 2 3 4STANDARD MEASURE (STANDARD DEYIATIONS)

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departure from the normal exists after the logarithmictransformation.

III. The distribution of change in threshold in indiudualanimals. A conclusion analogous to that above would not necessarilyhold for the moment-to-moment fluctuations in threshold in indi-

296


vidual animals. To provide sample information on this problem astudy was made of the distribution of uncontrolled change in thresh-

HISTOGRAMS AND THEORETICAL NORMAL DISTRIBUTIONSOF CHANGES IN THRESHOLDS IN ANIMALS(FIRST READING MINUS AVERAGE OF SECOND AND THIRD)

A.-AS WATTS. SET I SETX

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old in the animals, for a uniform experimental circumstance; viz.,the change from first reading to the mean of the better stabilizedsecond and third readings. In Fig. 2 these threshold changes are

distributed, as were thresholds themselves above, both before andafter logarithmic transformation of the original readings.

N. NO. ANIAALS * 133X- MEAN a 5.25. ST. DEVN. . 10.6

297

298 YALE JOURNAL OF BIOLOGY AND MEDICINE

TABLE 4

TESTS OF GOODNESS OF FIT OF THEORETICAL NORMAL FREQUENCY C.URVE TO ACTUALDISTRIBUTIONS OF THRESHOLDS (AVERAGES OF SECOND AND THIRD READINGS).

Readings Data Degreesexpressed set of P

as: number X2 freedom (Approximate)Watts 1 0.54 2 0.77

II 0.90 2 0.65Additive* 1.44 4 0.83Pooledt 1.03 2 0.62

Log watts I 2.72 2 0.26II 9.90 2 0.007

Additive* 12.62 4 0.012Pooledt 11.36 2 0.004

* Additive character of X' and its degrees of freedom for cumulation of evidencefrom a series of samples.

t Enabled by use of standard measures of respective data sets (see figures) asclass marks for frequency distributions.

TABLE 5

TESTS OF GOODNESS OF FIT OF THEORETICAL NORMAL FREQUENCY CURVE TODISTRIBUTIONS OF THRESHOLD CHANGES IN ANIMALS

(FIRST READING MINUS AVERAGE OF SECOND AND THIRD).

Readings Data Degreesexpressed set of P

as: number X' freedom (Approximate)Watts 1 2.25 2 0.33

II 8.21 2 0.017Additive 10.46 4 0.032Pooled 6.02 2 0.050

Log watts 1 3.18 2 0.211I 5.46 2 0.066

Additive 8.64 4 0.070Pooled 1.92 2 0.38

Notation as for preceding table.

Application of the X2 test for goodness of fit of the theoreticalnormal distribution (table 5) revealed no significant departure fromnormal for either scale of expression in the smaller sample (set I;this was also the case for resting thresholds, as seen in table 4).However, the larger sample, the additive X2 for the two samples,


and particularly the pooled relative distribution, all exhibited sig-nificant departure from normal before but not after logarithmictransformation (5 per cent level). That is, in contrast to the dis-tribution of "resting" thresholds among animals, changes in theirindividual thresholds were more nearly lognormal18 than normal indistribution.

IV. Tests for possible influences on threshold of time lapse indays, time of day, shade of (dark) skin color, body weight, and sex.The data of set I were subjected to analysis of covariance- "35 of

TABLE 6

DISTRIBUTION OF ANIMALS OF SET I ACCORDING TO COLOR AND ORDER OF READING.

Time of uAnial Fur colorday number Gray Brindle Brown Black Total

A.M. ~~~6 12 2 19 39A.M. ll I 3 16 7 12 38

P.M. ~ lilt 1 8 3 17 29P.M. IVt 4 11 3 9 27

Total [ 14 47 15 57 1 3 3* Early forenoon. Readings made alternately, animal II following animal I by

10 minutes.t Early afternoon. Alternate readings, with IV following III by 10 minutes.NoTE: Readings distributed over 74 days. No statistically significant color-

time of day influence on threshold or regression on time lapse in days could bedemonstrated by detailed analysis of covariance.

threshold (average of second and third readings in watts) with cal-endar day succession. Days were numbered successively as theindependent variable. Data were stratified according to shade ofskin color and time of day, as explained in table 6, for purposes oftesting for possible influences of these two factors while concurrentlyadmitting greater potential precision to the test for regression ofthreshold on time-lapse in days.

The results were entirely negative. Heterogeneity among the("adjusted") mean thresholds for the 16 subclasses of color andtime of day was indeed somewhat less than would be expected frominter-animal variability within subclasses (F = < 1, with 15 and 1 16degrees of freedom). The (average) correlation coefficient for atime lapse-threshold relationship was not significant (r = 0.139,with 116 degrees of freedom) over the period of 74 days studied.

299


In the second set of data, sex and body weight were also recorded.In this set, lack of significant difference in mean threshold betweenthe first and last chronological halves of the data again failed toevidence a drift over the lapse of 60 days involved. Covariance ofthreshold on body weight, with the data stratified according to sexand whether forenoon or afternoon (table 7), yielded no significantpositive information.

TABLE 7

DISTRIBUTION OF ANIMALS OF SET 11* ACCORDING TO COLOR, TIME OF DAY READ,AND SEX.

Time of | Fur colorday Sex J Gray Brindle and Brown Black Total

1 42 18 61A.M 354 21 | 60

P.M. d 1 6 29 22 57_________ 9 3 31 17 51

Total 14 137 78 | 229** 3 animals excluded by failure to record sex.NOTE: Body weights ranged between 222 and 435 gm.; mean, 317.2 gm.

No statistically significant color-time of day-sex influence on threshold, or regressionof threshold on body weight could be demonstrated by detailed analysis of covariance.

Thus no influence on the data could be demonstrated for timeof day, lapse of 2 or 3 months' time, shade of skin color (all dark),sex, or body weight of animal within the range studied.

V. An erratic influene according to day or group of days.Finally, classification of the data by day and time of day (table 8)

TABLE 8

ANALYSIS OF VARIANCE OF 200 THRESHOLDS (AVE. OF 2ND. AND 3RD. READINGS)OF SET II, CLASSIFIED ACCORDING TO DAY (50 DAYS) AND SEQUENCE

OF READING (4- ANIMALS READ DURING DAY).

Source of Degrees of Mean square Variance ratiovariation freedom (Variance) (F)

Total 199Days 49 3081.56** 2.77Sequence within day 3 261.34 <1Discrepance 147 1114.31

Mean threshold, 2 3 3.2 watts# Significant at 1 %o level (P = 0.01 or less).

300


revealed the only clue concerning a specific, outstanding source ofvariability among animals. There is an important influence asso-ciated with day, or group of days, of observation. Since there wasno regular regression on lapse of days (vide supra), this must havebeen some erratic influence more or less randomly distributed.

DiscussionThe small drift downward in threshold on successive readings,

which may continue very gradually after the second reading for aslong as four hours40 is speculative as to its nature. That a diurnalchange in the animal is importantly involved is discounted by failureof a significant difference in results among animals according to thetime of day read. In this respect the present results with an objectivecomponent of the pain reaction in animals agree interestingly withthose of Hardy et al.,20 and Schumacher, Goodell, Hardy, andWolff83 with pain perception in man. The drift is thus apparentlyassociated with the experimental conditions.40 In any event, evenincluding the first and particularly unstable reading, it is so small asnot seriously to complicate experimentation, and furthermore, thefirst reading can be excluded from computations.

The finding that fluctuations in threshold in individual animals,as sampled by variability in the above drift, are more nearly log-normally than normally distributed is of both practical and funda-mental interest. From the former point of view, log thresholdchanges would yield better estimates of the average and would bemore amenable to standard statistical procedures than changesexpressed as watts or other stimulus-energy units. Specific justifica-tion thus arises for the logarithmic treatment of threshold changeswhich was originally adopted40 for more general reasons.

Of more fundamental interest is the inference from their approxi-mately lognormal distribution that random factors influencingthreshold fluctuations operate to an important extent in quasi (videinfra) proportion to, or quasi relative to, the general peripherallymeasured threshold level. This seems very probably in close rela-tion to a well-known property of sensory perception, viz., that withinlimits, for linear progression of central neural effect an approximately(quasi) logarithmic (geometric, exponential, relative) progressionof stimulus is required (Weber-Fechner law). That is, the approxi-mately lognormal fluctuations observed in peripheral threshold prob-

301


ably reflect random fluctuations predominantly central in origin; or,correlatively, random fluctuations in central limen require, withinlimits, quasi exponential adjustments in peripheral stimulus energyfor liminal effects. The a priori likelihood of random lability in thecomplex central mechanism dominating that in the periphery wouldindeed be granted.

Such inferences are, furthermore, in line with Bishop's7 conten-tion that a potential subliminal margin of peripheral pain-receptoractivity exists, particularly in relation to reaction as distinguishedfrom perception. Visualize an approximately exponential fluctuationof the upper border of the peripheral energizing background for thissubliminal activity as central limen varies arithmetically.

Since subjective pain has not been a convenient field of sensationfor demonstration of the Weber-Fechner approximation in terms ofgraded sensation, the present distribution data may thus be consideredas evidence of the approximation's operation in an c!bjective reactionintegrated with pain.

The largeness of variations in threshold among animals as com-pared with variability of change within animals40 tempts one tosuppose that the normal (non-lognormal) distribution of thresholdsamong animals represents superimposition of an additional and pre-dominant set of random factors operating independently of theperipherally measured threshold, peripheral to the neural centers,and by assuming functional similarity at the various synaptic levels(vide infra), as far peripherally as the receptor or surroundingdermal tissue. Such general possibilities come to mind, and in rela-tion to the erratic day-to-day influence observed, as varying sourcesof animals, nutritional states, detailed manners of daily depilation byrotated attendants, etc.

One is led to predict that with adequate attention to uniformityin such details, central, hence presumably approximately lognormaldispersing influences among animals might be unmasked into pre-dominance as in the case of intra-animal fluctuations. This thoughtis supported by reported observation of a lognormal distribution ofhuman dermal weight perception thresholds (referred to in ref. 18).In comparing thresholds among depilated animals under presentconditions, however, the wattage (energy) scale (of a given stimu-lator) without transformation is the appropriate one.

Lack of demonstrable influences on threshold of time of day,body weight (and,/or age), or sex is in interesting agreement with

302


analogous results of Wolff et al.20 33 on the human pain-perceptionthreshold.

A statistical concept of quantitative biological gradation. Thefundamental question arises as to the site or sites, in passing fromstimulus to central effect, where excitation 'becomes an approximatelyexponential function of stimulus energy. Receptor impulse-dischargeinto sensory nerve fibers is more nearly proportional to the logarithmof -the stimulus energy than to a linear scale thereof, not only forthe single sensory unit-sensory nerve fiber with its endings-(e. g.,th'e frog muscle proprioceptor,27 the arthropod photoreceptor,24 themammalian muscle proprioceptor,28 and the audi'tory receptor,19 butalso for multiple-fiber preparations, e. g., of frog musdle proprio-ceptors,3' ' mammalian "pressure" receptors,4 and frog cutaneousnociceptors (ref. 26, intercepts of Fig. 6). The carotid sinus pres-soreceptors"10' and pulmonary stretch receptors' may be exceptions,but complications of tangential forces in a hollow viscus enter intotheir consideration.

Quantitative data on the eye are instructive in passing on throughsynaptic relays. It has already been noted above that in the arthro-pod eye, where the photosensitive cell-nerve fiber junction is a simpleone without opportunity for spatial summation, nerve fiber dischargeapproximates an exponential function of stimulus; the same can besaid for the local electric response of this eye, presumably an expres-sion 'of local excitation.22 Now it is of interest to note that theelectrical manifestation of ganglionic excitation in the mammalianretina (the retinal potential) where spatial summation mechanismsare highly operative, 'bears a similar relationship to stimulus.12' 13, 21Furthermore, the same approximation holds across the two-stage,spatially summating synaptic relay into single fiber23 and multifiber2optic nerve discharge.

One might at this point be tempted to attribute the approximatelylogarithmic transformation, so to speak, of stimulus energy to thelevel of the peripheral receptor and look upon successive junctionalsites as linear relays, but there is evidence to the contrary. Forexample, from experiments in which the receptor level was by-passedby controlled, direct stimulation of afferent nerve fibers, Bishop,Heinbecker, and O'Leary8 recorded data which show a similarapproximate exponential relationship between number of afferentimpulses and peripheral effect, thus carrying the present considera-tion from beyond the receptor, through the centers, and to the

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effector. Perhaps analogous to this circumstance of the Weber-Fechner approximation in physiology is the well-known approxi-mately exponential form of dose-effect curves in pharmacology,fairly irrespective of site or level of initial action.

The first temptation thus gives way to entertainment of the seem-ingly greater likelihood that at various (e. g., junctional) sites afairly uniform law operates in the summation and relaying ofexcitation, but in such a manner that the form of relationship betweeninitial stimulus and progressing excitation does not necessarily sufferincremental qualitative change from one relay to the next.

It has often been pointed out that the Weber-Fechner law andallied relationships are but approximations applying only to inter-mediate segments of total effect ranges, and the same is certainly ingeneral true of the logarithmic dose-effect approximation in drugaction. In cases where the experimental range has been wide enough,the nature of failure of the law has fairly uniformly been a gradualdecrease in slope of the log stimulus- (or dose-) effect curve at theupper end, and commonly at the lower, thus resulting in a basicallysigmoid curve reminding one of a cumulative probability or fre-quency curve. This is apparent at one or both ends of studiedranges in three of the four works referred to above on single sensoryfiiber discharge, in two of the four on multifiber discharge, in thework of Chaffee et al.,12' 13 and of Hartline 21, 22 on the retinal poten-tial, in that of Hartline23 and of Adrian and Mathews2 on single andmultifiber optic nerve discharge, and in the data of Bishop et al.8That it is a common experience in pharmacology also is evident fromlooking through publications in this field.

One is thus led naturally to the proposition that cumulativestatistical probability is a very basic law in so-called quantitative gra-dation of physiological and pharmacological phenomena. This willbe recognized as closely related, e. g., to Hecht's25 concept concerningvisual acuity and brightness difference threshold, and Shackell's34 andGaddum's" ideas concerning the actions of certain drugs. Thepresent thought, however, is more accommodating in scope: perhapsthe main basis for criticism of the concepts of Hecht, Shackell, andGaddum has -been their limitation to a single level of frequency orprobability integration, thus excluding concurrent gradation withindiscrete units; it is proposed that a: simple extension of integratedprobability to successive lower levels of discrete units accommodatesunitary gradation at a given level and removes any necessity for

304


choice between all-or-none and graded action of units such as is feltby Rosenblueth.29To illustrate, a scale of stimulus energy will progress with a

characteristic cumulative probability or relative frequency of randomassociation with some specific physico-chemical activation on oneterminal of a sensory unit (afferent fiber). A second order of fre-quency distribution of such first order probability ranges will existfor the several terminals of the unit, along the energy scale, accord-ing to their random diversity of average susceptibility. The sum-mated terminal activation for the sensory unit will thus progress alongthe energy scale as the second order cumulation of first ordercumulations, which is analogous to summation of individual sampledistributions into an eventual population distribution in experimentalprocedure, and does not inherently require deviation in form of thesecond order cumulation from that of the first. This meets a require-ment pointed out by citations above; viz., that a common law beapplicable at successive stages of excitation without inherentlyrequisite incremental qualitative change in the overall relation.Similarly, a third order cumulation will be of the distribution ofsensory units converging to the second neurone, according to theaverages of their respective average terminal distributions, and so onto the central or end effect.

Concurrence of divergence with convergence of pattern does notalter the basic concept. Neglect or ignorance of intermediate pointsof cumulation does not inherently affect the end result. The conceptis noncommittal as to the nature of the successive changes. Concur-rence of graded efficiency of temporal summation, as with increasingrates of quantal actions, would seemingly only alter the scale.

Now it is more unlikely than likely that the first-order distribu-tions will be "normal" with respect to the energy scale, or any orderof distribution normal with respect to the preceding order of cumula-tion. For example, physico-chemical transmission of one cumulationto next order units may impose hyperbolic or exponential distortions.The well-known nearly lognormal distribution of all-or-none effectson dose of drug is instructive. Factors underlying accommodationand adaptation would dispose toward a roughly lognormal distribu-tion along a stimulus scale. The well-known existence of subliminalexcitation would cut off more or less of the lower tail of distributions,and perhaps to a lesser extent limitation of impulse discharge rate bynerve fiber refractory period would lower the upper tail. Qualita-

305


tively dissimilar systems brought into intermediary action sucessivelyalong the scale would result in distortion. Perhaps of great impor-tance is convergence onto the chain of cumulations of an externalbackground of support such that the measured contribution startshigh on the cumulation curve for the particular relay level. Aninteresting presumptively supportive illustration of this corollary ofthe general concept is provided by comparing mildly skewed carotidsinus pressoreceptive stimulus-response curves, where importantknown background support is eliminated,88 39 with nearly exponen-tial depressor nerve curves where heavy carotid sinus support is leftintact to decrease compensatorily.8

From such influences more or less skew and kurtosis would beexpected, and indeed in directions such that a considerable lower-intermediate portion of the final cumulation curve would approachan exponential form; that is, such that the expedient of logarithmictransformation of the energy scale (and sometimes both energy andresponse scales) would render approximately linear a considerable,lower-intermediate portion of the curve. But the logarithmic scaleis relative to an origin. Thus, the degree and region of the linearapproximation is dependent on the origin of measurement, henceperipheral background against which the measured energy is applied.

It would be difficult to choose between cumulative probabilitycurves on a logarithmic dose or stimulus scale and hyperbolic effectcurves on an arithmetic dose or stimulus scale for empirically sum-marizing many sets of data.29' 30, 31, 32, 37 The present concept choosesthe former as fundamentally more explanatory. It is interestingthat the hyperbolic concept led to the necessity of assuming a linearrelationship (number of secretory impulses vs. epinephrine secretion)in an otherwise non-linear family of relationships, some presumablyinvolving the same chemical transmitter.30 As pointed out above, onthe other hand, the probability-cumulation concept accommodates asuccession of relays without inherently requisite change in overalllaw.

The present concept obviously implies a fundamental similaritybetween gradation of a quantitative effect and the frequency distribu-tion of a qualitative effect, such as mortality percentage. Each isbased on the overall cumulative frequency of suscepti;bility of discreteunits of successive statistical population levels, along the scale of anindependent variable. From the practical point of view of measure-ment, 'however, a distri'bution percentage commonly cannot be

306


determined in the former case because of lack of a measurablemaximum, hence the probability scale so useful in the latter cannotbe applied in the former. Such partially satisfactory devices as thelogarithmic scale seem inevitable. Thus, it would be most imprac-tical in the present type of experiment to determine wattage at whichmaximal pain-reaction occurs, for the purpose of transforming thewattage scale to an inverse probit scale in terms of which randomfluctuations in central limen would be more precisely normallydistributed.

Summary

1. A statistical analysis was made of resting, threshold, radiantthermal stimulus energies (lamp wattage) required for eliciting thenociceptive contraction of the musculus cutaneous maximus, whichprevious work40 had placed in close integration with central painmechanisms. The data consisted of three successive "readings" at20-minute intervals on 365 animals, 133 by one operator and stimu-lator, and 232 by another operator and stimulator during a sub-sequent period. Due to a variable small drop between the first andsecond readings, the second and third were used in computing theresting threshold without significant loss in precision due to reductionin degrees of freedom.

2. Changes in thresholds in individual animals as measured bythe difference between the first and the average of the second andthird readings in watts were not normally distributed, but were(approximately?) so after logarithmic transformation of the originalreadings in watts. This approximately "lognormal" distribution ofthreshold changes points the way to analysis of experimental designsbased on threshold changes. It was inferentially related to theWeber-Fechner approximation through a presumed predominantlycentral origin of random fluctuations entailing correspondingapproximately exponential fluctuations in peripheral liminal stimuluson a subliminal margin of receptor activation.

3. The distribution of thresholds among animals was approxi-mately normal without transformation. This finding was inferen-tially related to the greater absolute varialbility among than withinanimals-to peripheral random factors superimposed upon and dom-inating central factors, such as a demonstrated erratic day-to-day

307


influence of animal source, condition, and/or manner of depilation,etc.

4. Without significant influence on the thresholds were: elapseof as many as 74 days of observation, time of day, shade of (dark)skin color, body weight, and/or age within the range studied, and sex.

5. A statistical concept of gradation of so-called quantitativebiological effects is proposed, based on successive levels of cumulativefrequency distributions of preceding frequency distributions, biologi-cally skewed and kurtotic in specific natural and experimental man-ners, and so fortuitously providing various degrees of success to suchdevices as the Weber-Fechner approximation.

REFERENCES

1 Adrian, E. D.: Afferent impulses in the vagus and their effect on respiration.J. Physiol., 1933, 79, 332-58.

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ADDENDUM: Referring to tables 4 and 5 the "interaction" or "discrepance" X2's(additive minus pooled) are non-significant and significant, respectively, for distribu-tions of thresholds (averages of second and third) and for changes (first minus averageof second and third), in the two samples. This reflects lack of demnstrated hetero-geneity in nature of distribution between the two samples after the first reading, butpresence of heterogeneity in the change after first reading, as already inferred.

studies two sufficiently large samples ofsystematic data on resting

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