The Fundamental Psycho-Physical Law and Its Modern Formulation

(Report at the annual meeting of the Institute of Biological Physics,
December 17, 1920)

P. P. Lazarev

Every process of sensation is composed of the following separate phenomena: first, the conduction of an external stimulus to the apparatus connected with the nerve endings of the sense organs (this part of the whole process is the physics of the sense organs, and for vision and hearing its study was completed in the brilliant works of Helmholtz); the first stage of the process is followed by a second, when the stimulus, having reached the cells of the nerve endings, by means of chemical reactions causes their irritation and then carries the irritation through the nervous system all the way to the cortical centers of the brain, whose excitation translates a definite physiological process into consciousness. The material phenomena in the terminal apparatus of the sense organs, in the nerves, and in the centers constitute the object of an extensive part of biological physics, which we have called the ionic theory of excitation and which has already reduced, in part, the process in the nerve endings to well-studied physicochemical reactions. Finally, the third link in the phenomena of sensation is the purely psychic process, the study of which is possible only by the methods of introspection developed in psychology.

The physical and physicochemical processes in the sensing organs, as has already been indicated, are at present studied in general outline, and, in any case, the method of study here has been clearly determined; likewise in psychology the subjective phenomena following the irritation of the sense organs have been investigated in sufficient detail. What remains is to establish the connection between the possible material processes in the excited organs and the sensation produced by them; yet comparatively recently it was still possible to assert, as E. Hering noted, that “the doctrine of the relations between matter and spirit, between body and soul, is a domain of philosophical dispute without a firm foundation, without solid principles and method, which are necessary for the success of investigations.”

Thanks to the works of Fechner, the field of psychophysics acquired certain scientific foundations—and the subsequent works of physicists, physiologists, and psychologists made it possible to establish more firmly the general principles of the connection between the science of matter and the science of spirit. Already in 1870, in his [[unclear: continuation cut off at bottom of page]]

the remarkable speech “Über das Gedächtniss als eine allgemeine Funktion der organ. Materie,” Hering says: “If the physiologist who studies the nervous system stands between the physicist and the psychologist, and if the former takes as the basis of his investigation the firm causal continuity of all material processes, while, on the other hand, the psychologist seeks the laws of conscious life by the inductive method and in doing so makes the assumption of an unshakable regularity the starting point of his arguments, and, finally, if self-observation teaches the physiologist that his conscious life depends on the experiences of his body and that his body is, within certain limits, subject to his will—then he can only admit that this mutual dependence between spiritual and material life is also lawful, and that a connection has been found which unites the science of matter with the science of consciousness.

Considered from this point of view, the phenomena of consciousness are represented as a function of the material changes of organized substance; and in order to eliminate any misunderstanding, it is necessary to emphasize (although this in itself is inherent in the concept of a function) that the material processes in the cerebral cortex are functions of consciousness, because if two variables in their transformations depend on one another in a definite way, so that with a change in one there simultaneously occurs a change in the other, then one, as is known, is called a function of the other. This expresses only the fact that both variables—matter and consciousness—stand in the relation of cause and effect, of ground and consequence.”

Hence it is clear that between the simplest psychical act, for example, the sensation of the weight of a load, and the external cause that has acted upon it—in the given case, the weight of the load—there must be a relation expressible by the equality

\[ E=\varphi(J), \]

where \(E\) is the magnitude of the sensation, \(J\) is the weight of the load. At the same time, as is clear, the magnitude \(\varphi(J)\), changing with the change in the magnitude \(J\), must not depend on the choice of units for \(J\).

In establishing the relation between \(E\) and \(J\), Fechner was able to use Weber’s excellent experiments on the muscular and cutaneous sense, in which the increment of the magnitude of the stimulus (for example, the increment of a load \(\Delta J\)), divided by the magnitude of the stimulus \(J\) at a barely noticeable increment of sensation, remained constant, so that

\[ \frac{\Delta J}{J}=\mathrm{Const.}=k \ldots \ldots (1)^{1)} \]

\(^{1)}\) It is not without interest to note here that, for the evaluation of musical intervals, the ratio of the numbers of vibrations of sounds is of importance, as was first discovered by the famous mathematician-academician Euler, who in this field was Fechner’s predecessor. Euler’s work on this subject was printed in Petersburg in 1739.

(\(k\) is a constant).

Fechner further assumes that, for a barely perceptible increment of sensation, the magnitude of the increment of sensation \(\Delta E\) is constant and does not depend on the magnitude of the sensation \(E\) corresponding to the stimulus \(J\), so that, with a certain choice of units for \(E\), one may write

\[ \Delta E = \frac{\Delta J}{J}\ldots\ldots\text{(II)}. \]

Assuming that relation (II) is valid up to the limit—which can be true only approximately—and allowing the summation of separate increments of sensations \(dE\), Fechner finds that

\[ dE = \frac{dJ}{J} \]

or

\[ E = \lg J + B\ldots\ldots\text{(III)}, \]

where \(B\) is a constant.

If \(J_o\) denotes the threshold magnitude of the stimulus, barely perceived by human consciousness (when \(E=0\)), then the preceding expression (III) takes the following form

\[ E = \lg \frac{J}{J_o}\ldots\ldots\text{(IV)}. \]

In this form the law fully satisfies the requirement that \(E\) be independent of the choice of units for \(J\), and is the general expression of the famous Fechner law. We shall call law (I) Weber’s law.

Relation (I), or (IV), has enormous application in various fields of science. Fechner succeeded in extending the domain of application of the law by showing that it is approximately fulfilled for vision, hearing, and the other sense organs; further, Pfeffer found that the motion of bacteria obeys Fechner’s law; Loeb succeeded in applying this law to the threshold stimulation of unicellular organisms by salt solutions; finally, repeated attempts were made to apply this law to economic phenomena. All this indicates that at the basis of the law there lies an extremely general relation, connecting all phenomena of stimulation into one common whole, and it was natural to try to verify the consequences of this law by precise quantitative experiments.

This verification could most easily and most accurately be carried out on vision, and it is to this field that the first precise experiments belong.

Fechner himself already clearly saw that his law could be only approximate, and that with small stimuli a deviation of the phenomena from the law must occur. Indeed, if the law were exact, then in the absence of an external stimulus (when \(J=0\)) the barely perceptible increment of the stimulus \(\Delta J\) would have to represent a threshold

the value of the light intensity. Meanwhile, if formula (I) is correct, then for \(J=0\) also \(\Delta J=0\), and consequently the threshold of light intensity is equal to zero. This is not in fact observed and, consequently, Weber’s formula, from which Fechner derived his law, has only an approximate significance. Fechner attempted to correct this defect by writing the formula of the psycho-physical law for vision as follows:

\[ \frac{\Delta J}{J+\alpha}=K=\mathrm{Const.}\ . . .\ (V). \]

(here \(\alpha\) is a small positive constant quantity). For large values of the stimulus \(J\), when \(\alpha\) disappears in comparison with \(J\), this law coincides with law (I), and only when \(\alpha\) is small is its influence significant. For \(J=0\), \(\Delta J\) is equal to the threshold value of the stimulus (equal to \(K\alpha\)).

The difference between formulas (I) and (V) becomes very clear if we represent them graphically, laying off \(J\) on the \(X\)-axis, and \(\Delta J\) on the \(Y\)-axis (Fig. 1).

Fig. 1.

\(OJ\) graphically represents the relation \(\dfrac{\Delta J}{J}=K\), \(AJ_2\) the relation \(\dfrac{\Delta J}{J+\alpha}=k\);

\(\alpha\), according to the formula, must represent a certain brightness of light and be equal to the brightness of the subjective light which arises in the eye as a result of irritation of the retina from internal causes. \(\alpha\) received the name of the intrinsic light of the retina; by measuring \(\Delta J\) at \(J=0\) and knowing from experiments with bright light the magnitude \(K\) (when the quantity \(\alpha\) may be neglected), Volkmann determined the magnitude of the intrinsic light of the retina. This magnitude proved to be equal to the brightness of light sent into the eye by a surface of black velvet illuminated from a considerable distance by a candle.

In view of the great fundamental significance of the question of the applicability of Fechner’s law, König and Brodhun undertook a systematic study of Fechner’s law for vision both in white light and in monochromatic rays.

Ingeniously designed and carefully executed investigations proved that the experimental data give a more complex result than is given by Fechner’s law, and that the observed points lie on the curve \(CBDJ_2\), which at small light intensities \(J\) differs sharply from the straight line \(AJ_2\), expressing the law

\[ \frac{\Delta J}{J+\alpha}=K, \]

and only for considerable \(J\) does observation coincide with the straight line.

Helmholtz’s brilliant works made it possible to explain these deviations and made possible a further verification of the psychophysical law for vision. Helmholtz’s idea is as follows: first of all, as is easily established by self-observation, in complete darkness the field of vision is never entirely dark. If, by passing a constant electric current of a definite direction (so that the anode of the current is applied to the eye), the excitability of the optic nerve is weakened, then, upon closing the circuit, it is easy to notice that the field of vision becomes darker than without passing the current, and that upon repeated closing and opening of the circuit the phenomenon of darkening of the field appears especially vividly.

A careful study of the field of vision in absolute darkness showed Helmholtz that the brightness of the field is not constant and that the intrinsic light of the retina \(a\) depends on the position of the point on the retina. From this Helmholtz arrives in the following way at a generalized Fechner law. “Let,” writes Helmholtz, “\(a\) be the objective brightness of light which would be necessary in order to produce at a given place of the retina the very same stimulation as can be found in the intrinsic light of the retina. Since this latter is patchy, \(a\) must have different values in different parts of the retina. Let the area of those places of the retina in which the intrinsic light lies in the interval from \(a\) to \(a+da\) be equal to \(\varphi\,da\), where \(\varphi\) in general is a function of \(a\). We consider the magnitude of the increment of sensation when the brightness \(\Delta J\) of the objective intensity of light \(J\) changes, as the sum of all the separate actions which correspond to the separate degrees of brightness \(da\), and, according to Fechner’s law, we put

\[ \Delta E=\Delta J\int_{0}^{a}\frac{\varphi\,da}{J+\alpha}\ . . . . . . . . .\ (VI)^{1)}. \]

(here \(0\) and \(a\) are the limits of the intrinsic light of the retina).

Expression VI, as Helmholtz shows, after transformations will take the form

\[ \Delta E=\Delta J\left[\frac{A_0}{J+J_0}+\frac{A_1}{(J+J_0)^2\,(J+J_2)}+\ . . . . . . .\right], \]

where \(A_0, A_1\ . . .\ J_0, J_2\ . . .\) are constants.

\(^{1)}\) We write everywhere \(\Delta E\) instead of \(dE\) in Helmholtz, and, moreover, replace the letter \(r\) by \(J\).

If we restrict ourselves to two terms of the series, we obtain a hyperbola, very closely approximating the curve observed by König and Brodhun and shown in Fig. 1.

A further verification of the Helmholtz–Fechner law was made by us in a series of works, in which it was possible, first of all, to prove the validity of the summation, proposed by Helmholtz, of the separate expressions

\[ \frac{\varphi\, da}{J+a} \]

for equal parts of the field, if the field does not exceed the yellow spot; further, it was possible quantitatively to establish the difference in the distribution of intrinsic light in the retina and to show that the laws of threshold stimulation are also quantitatively derived from the Fechner–Helmholtz equation; thus, for example, the relation between the magnitude of the illuminated area \(S\) and the threshold of stimulation \(\Delta J\) is expressed as \(\Delta J \cdot S = K\).

If, without confining ourselves to the fovea centralis, we wished to give laws for stimuli also in the peripheral part of the field, we would find that equation VI must be generalized further, and the summation must be applied separately both to the region occupied in the illuminated portion by rods and to the region occupied by cones; and the general expression of the law assumes the form proposed by us,

\[ \Delta E = \Delta J K_1 \int_{0}^{a} \frac{\varphi\, d\alpha}{J+\alpha} + \Delta J K_2 \int_{0}^{b} \frac{\psi\, d\beta}{J+\beta} \ .\ .\ .\ (\mathrm{VIa}); \]

\(\alpha\) and \(\beta\) denote the intrinsic light in cones and rods, \(\varphi\) and \(\psi\) the corresponding areas occupied by intrinsic light with brightness lying between \(\alpha\) and \(\alpha+d\alpha\), or \(\beta\) and \(\beta+d\beta\); \(K_1\) and \(K_2\) are constants. Law (VIa) can provide an explanation for all phenomena of the increase of sensations also at the periphery of the retina.

If for light we thus obtain expression (VIa), which quite well conveys the law of stimulation for weak and medium brightnesses, then at strong brightnesses a further correction is required, attempts to introduce which were first made by Helmholtz; moreover, law (VI), derived for vision, may prove incorrect for hearing, taste, etc. Finally, from the formulation of Fechner’s law it is not clear from the outset what we should regard as the stimulus for the temperature sense, the sense of pressure, for electric current, etc. In fact, as the stimulus of the body in the last case one may take either the current strength \(i\) or the energy proportional to \(i^2\). All this, therefore, eliminates the generality of the connection for all stimuli, the generality of the law expressed by Fechner’s formula. Therefore we made attempts to approach more closely the analysis of the fundamental psychophysical law and to give it such a formulation as would be applicable to all sense organs and to all kinds of stimuli. The initial considerations were as follows. In order to give a general law connecting the magnitudes of sensation and stimulus, it is necessary for the stimu-

one should take not the external stimuli, but those processes, identical for all irritated organs, in the first endings, which lead to the primary excitation of the nerves and to the subsequent transmission of excitation along the fiber. At the present time we must regard ions as such primary excitants, and their concentration may be the magnitude of the primary stimulus for the nerve. External stimuli lying outside the organism, such as light, heat, sound, and electricity, can create irritation only insofar as they influence the content of ions in the organ. And since organs sensitive to stimuli always contain ions within themselves, it is sufficient to take the concentration of ions as the magnitude of irritation in Fechner’s law and to assume that the irritation of the various irritated elements is summed, in order to obtain a general expression for the psychophysical law.

We shall write it, by analogy with Fechner’s law, in the form:

\[ \sum_{0}^{n}\frac{\Delta C'_1}{C'_1}=K=\Delta E \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (\mathrm{VII}) \]

where \(C'_1\) is the concentration of ions in the organ, \(\Delta C'_1\) is the increment of this concentration that is barely noticeable to sensation. The summation from \(0\) to \(n\) is performed over all excited elements, and each term \(\frac{\Delta C'_1}{C'_1}\) is an elementary excitation or sensation \(\Delta e\). This law, expressed in the most general form and not involving specific stimuli, must be applicable to all kinds of irritation, and we shall show below how it is in fact applied in these cases.

First of all we shall dwell on the application of the law to vision. The photochemical process in vision has been elucidated by our works, and the concentration of irritating ions obtained in light is equal to

\[ C''_1=\frac{\alpha_1}{\alpha_2}kJC \]

where \(\alpha_1\) and \(\alpha_2\) are the coefficients of the direct light and the reverse dark reactions, \(k\) is the absorption coefficient, and \(C\) is the concentration of the photosensitive pigment.

If we allow that the visual pigment is decomposed not only by light but also by heat, and that this latter decomposition gives a concentration of ions \(\gamma\), then the total concentration is \(C'_1=C''_1+\gamma=\frac{\alpha_1 kJC}{\alpha_2}+\gamma\); hence, by the fundamental psycho-physical law (VII), we find

\[ \Delta E=\sum_{0}^{n}\Delta e=\Delta J\sum_{0}^{n}\frac{1}{J+\frac{\gamma\alpha_2}{\alpha_1 kC}}\ldots \ldots (\mathrm{VIII}). \]

i.e. we obtain an equation which, as is easy to show, represents the same relation as Helmholtz’s equation VI, and in which the role of \(a\) is played by the quantity \(\dfrac{\gamma a_2}{c_1 k C}\). The difference between the relations lies only in the fact that the Helmholtz relation assumes the individual light-receiving elements of the retina to be infinitely small, whereas in our formula they are allowed finite dimensions; this explains why, instead of the integral sign, we have the summation sign.

All the consequences that were confirmed by the experiments of König and Brodhun, and by our own, are fulfilled in our formula (VIII), deduced from a generalized psychophysical law. Starting from the formula

\[ \sum_{0}^{n} \frac{\triangle C'}{C'} = K, \]

it is easy to show that the magnitude \(\triangle J\), as \(J\) increases, must increase, departing from linearity; and it is precisely in this that the deviations due to dazzling consist. Indeed, if the light is so bright that practically almost all the visual substance has decomposed, then a further increase in the intensity of the light, having practically no effect on the change in the concentration of the irritating substances, gives no new sensations, since \(\triangle J\) is not sufficient to evoke an increment of decomposition and the increment of sensation associated with it.

In order to pass from the generalized Fechner law to the form which it must have for auditory sensations, we must remember that, in auditory sensations, in contrast to visual ones, stimulation of the ear by a tone of a definite pitch always causes a noticeable vibration of only one cortical fiber in the inner ear, a fiber tuned in resonance with the incident tone; together with the fiber the cortical cells lying on the fiber also enter into vibration, whereas the remaining fibers situated nearby remain practically at rest. According to our theory, the motion of the fiber causes the decomposition of sound-sensitive substances in the cells of the cortical organ and causes the appearance of irritating ions acting upon the endings of the auditory nerve. Thus each simple tone (without overtones) gives irritation to only one nerve fiber, which for vision should correspond to the irritation of one of the cones or rods.

In formula VII, in this case, only one term remains, and, calling the intensity of the sound \(U\), and denoting by the constant \(\beta\) the quantity \(\dfrac{\gamma a_2}{c_1 k C}\), we find for the sound-sensitive reactions in the ear, completely analogous to the light-sensitive reactions, relations identical in form with the irritation of a single visual element:

\[ \triangle E = \triangle c = \frac{\triangle U}{U + \beta} = \mathrm{Const} = K \ . \ . \ . \ (IX) \]

the relation between \(\triangle U\) and \(U\) for hearing is, according to the theory, linear (see Fig. 1).

whereas \(J\) and \(\triangle J\) for vision have a hyperbolic relation. Experiment has confirmed this difference between hearing and vision. The reason for this difference in Fechner’s law for vision and hearing lies in the fact that in hearing it is possible, and almost constantly occurs, that only a single element, a single cell and a single nerve fiber are stimulated; we shall call this stimulation point-like, whereas in visual perceptions we have to do with areal stimulations, with stimulations encompassing the entire surface of the retina. As is evident from formula IX, in the ear there must be observed, when the organ is at complete rest, a constant irritation depending on the decomposition of sound-sensitive substances by heat and perceived as a very weak tone. We shall call this tone the cochlea’s own tone. Work in our laboratory has succeeded in proving the complete validity of formula IX.

First of all, it proved possible to discover that when a very strong sound acted upon the ear and caused a strong decomposition of substances in the auditory cells, producing a consecutive subjective sound in the ear that lasted for several weeks, then at the same time the magnitude \(\beta\) also increased, representing the cochlea’s own sound, dependent on the chemical processes that arise in the ear and that are superimposed upon the processes produced at the given moment by sound. Thus there was obtained an excellent experimental confirmation of the theory, and the existence of the cochlea’s own sound was established. In certain individuals the relation between \(U\) and \(\triangle U\) was sometimes expressed not by a linear relation, but approached that relation which was also observed in light sensations. This could be explained only by the fact that the fibers of the organ of Corti are so closely connected with one another that the vibration of one of them produces a joint vibration of the neighboring fibers. In this case the sound stimulation, generally speaking belonging to the type of point-like stimulations, passes into an areal one, and accordingly the line representing the relation between \(\triangle U\) and \(U\) changes from a straight line into a hyperbola (see Fig. 1).

As is easy to understand, in persons whose Corti fibers are closely connected with one another, the discrimination of tones lying close to one another must be diminished.

This is in fact observed, as our investigations have shown; moreover, with practice there is observed a transition of the hyperbola relating \(\triangle U\) and \(U\) into a straight line, corresponding to an improvement in the discrimination of the pitches of neighboring sounds.

In this latter case we are, as it were, tuning the ear, just as we tune the wooden soundboards of stringed instruments.

Passing on to taste, we must indicate that both earlier work and our own investigations have established first of all that there are four kinds of receptive apparatuses, giving the sensation of sour, salty, sweet, and bitter taste. These apparatuses are independent of one another and can be stimulated separately. When the corresponding stimuli act at once upon the entire surface

of the tongue, we obtain a plane stimulation, and, consequently, the relation between the concentration of the stimulating substances and its increment, as required by ionic theory, must be hyperbolic. This has been found in experiments in our laboratory.

Finally, using the formula of the generalized psycho-physical law (VII), it is easy to show that, for sufficient strength of stimulation by an electric current on the surface of the skin, the ratio \(\frac{\Delta i}{i}\) must be constant, as was also found earlier by A. V. Leontovich.

We have considered only one aspect of the phenomena in stimulation, namely, either only the succession of two stimuli of different strength one after the other (hearing, taste), or their simultaneous action upon a sense organ (vision), and in doing so we selected stimuli that produced sensations barely distinguishable in strength. But it is possible gradually to intensify one and the same stimulus (to increase, for example, the intensity of light) and to establish such a rate of its change that we shall barely sense the process of change; as we have been able to show in this case for vision, we have

\[ \frac{dJ}{dt}\cdot \frac{1}{J}=\mathrm{Const}=K. \]

More complex problems may also be posed, where \(\Delta E\) will depend on a number of factors, for example, on the plane distribution, on color, and so forth; in this case \(E\) will be a function of all these separate factors, and the task of a general study of Fechner’s generalized law is the investigation of these factors.

Assuming that the initial stimulation is equal to zero, we shall find the threshold stimulation, which, in the case of the action of ions, leads directly to Loeb’s law

\[ \frac{C_1'}{C_2' + a}=\mathrm{Const}. \]

As a special case of this latter, there appears Nernst’s law of stimulation

\[ C_1'=\mathrm{Const}. \]

Thus all the laws of stimulation proposed up to now are united by the new generalized psycho-physical law.

Generalizing the law further, it may be extended to the action of some sense organs upon others, as was done by us for the influence of the organ of vision and of hearing; moreover, in complete agreement with experiment, we find that

\[ \Delta E=k_1\frac{\Delta J}{J}+k_2\frac{\Delta U}{U} \]

where \(k_1\) and \(k_2\) are constants.

The laws of the muscular sense, the sense of temperature, and the complex phenomena of the sense of equilibrium and the general sense remain unstudied from the point of view of the generalized psycho-physical law. This constitutes the nearest and most important task of psychophysics.

In conclusion, we would like to note one circumstance that brings the results obtained in the study of the fundamental psychophysical law closer to the conclusions of modern atomistics.

We have seen that the magnitude of the increment of sensation, $\Delta E$, is always a finite, strictly determined quantity, depending in a definite way on the magnitude of the stimuli. The complete sensation is thus composed of elements of finite dimensions, analogous to quanta of energy, and, thanks to Fechner’s law, we may with full justification speak of atoms or quanta of sensation, as we already noted back in 1913.