Specifics of the Neuron Action Potential Frequency Dependence on the Intensity of the Excitatory Influence

A single formula linking the intensity of the stimulus with the strength of sensation has not yet been found. Neither the Fechner logarithmic dependence nor the Stevens power function can explain the specific dynamics of many psychophysical experimental data. And this is natural, because attempts to replace the multi-stage touch mechanism with a simple “black box” model are initially vulnerable: all stages of stimulus transduction in sensory systems should be separately considered. At the end of the 20th century, neurophysiologists agreed that one of the forms of coding of stimulus intensity is the frequency of action potentials (AР). In line with this paradigm and based on the universal Hodgkin-Huxley model applied to the conditions of a psychophysical experiment, the author developed a mathematical model of the primary receptor neuron. Calculations were made for a neuron with close to real geometric dimensions, physical and physiological properties. Calculations revealed characteristic areas of neuron activity: subthreshold, working, paradoxical and silent zone. A characteristic maximum of the “frequency of APs – stimulus” function was also discovered. Using the proposed model, the “saturation effect” and “cross” behavior of the “stimulus – sensation” curves revealed in psychophysical experiments are easily explained in individuals with different group characteristics of the nervous system. The calculation results showed that individual psychophysical differences are determined by types of combinations of the physical and physiological properties of the receptor. The presented mathematical model can be applied not only to sensory systems, but also to any neurons.

Introduction

In the present work, a primary neuron of a receptor is understood as a place on the sensory path of stimulus transduction, where action potentials (APs) first appear – the universal components of the language of the nervous systems of any living creature. Along with the “primary neuron of a receptor” term, other designations can also be found in the literature: “afferent neurons of receptors”, “afferent nerve fibers of receptors”, simply “sensory receptors”, and others.

The proposed author’s mathematical model of the primary receptor neuron is suitable for a quantitative description of the formation of APs in it both when the free nerve endings are directly acted on (for example, tactile or pain sensitivity), and when it is stimulated indirectly, for example, by the amount of the neurotransmitter released into the synaptic cleft (auditory or visual system). Both with direct and indirect action on the afferent neuron of the receptor, the frequency of AP arising in it is proportional to the stimulus intensity. However, as experimental data show (Willis, 2004), this proportionality is by no means direct, but satisfactorily described by a power function of the form (in (Willis, 2004), the formula is written in words):

where is ν the frequency of the APs, Hz; C is a constant that takes into account the units of measurement of the stimulus S; S, S₀ are the current and threshold values of the stimulus of the given modality; n is the power exponent, which may be less than, equal to, or greater than one for receptors of different modalities (Willis, 2004).

Expression (1) coincides with one of the forms of presenting the so-called power “Stevens law” (1957), which connects the intensity S of the stimulus and the sensory strength R (instead of the frequency ν, there is the sensory strength R in the Stevens formula (1957)). Stevens proposed approximating the psychophysical dependences R(S) by a power function in the middle of the 20th century instead of the approximations previously used everywhere since 1860 according to the so-called logarithmic Fechner’s law (1860):

However, “up to now, the basic psychophysical law has not obtained universal and final recognition either in the form of the logarithmic dependence of Fechner (1860) or in the form of the power dependence of Stevens (1957) between the stimulus magnitude and the sensation intensity. The experimental data obtained by various researchers were not unambiguous: some speak in favor of the logarithmic law, others in favor of the power law” (Ratanova, 2008). It can be added to the quote that it is unlikely that, without knowing the physical mechanisms of the stimulus − sensation transformation, by simply approximating the experimental data by a simple formula, one can “find by chance” the basic psychophysical law connecting the stimulus intensity with the sensation strength. Moreover, a single “psychophysical law” for all sensory modalities without exception is also unlikely, because each sensory system of an organism has its own specifics of transforming stimulus into sensation. Therefore, further search for a simple empirical formula that connects the stimulus intensity with the sensation strength, the “basic psychophysical law”, but does not take into account the physics of the process of transduction of stimulus into sensation, appear to be unpromising. Nevertheless, such search was undertaken after Stevens: for example, Zabrodin proposed combining the “laws” of Fechner (1860) and Stevens (1957) in a single formula in a differential form (Zabrodin, & Lebedev, 1977).

Thus, to identify an adequate formal dependence between the stimulus intensity and the sensation strength, it is necessary to analyze all the stages on the path of transduction of stimulus into sensation. Comparing the dynamics of the “stimulus intensity – sensation strength” curves obtained in psychophysical experiments at the macro level of organisms (Ratanova, 2008; Stevens, 1957) with the dynamics of the “stimulus intensity – frequency of AP impulses” graphs recently obtained at the neural level of receptors (Willis, 2004), the conclusion can be made about their equidistance: both of them are satisfactorily described by a power dependence (see above). Thus, the stage of direct or indirect transformation of stimulus in the primary neurons of the receptor into APs is determinant on the entire path of its transduction into sensation; that is why the present research was started with it.

Method

In the middle of the 20th century, Nobel laureates Hodgkin and Huxley discovered a universal mechanism for the formation of APs in nervous tissue and developed its mathematical model (Hodgkin, & Huxley, 1952). In the author’s works (Fokin, 2017), the general Hodgkin-Huxley model was applied to a particular case of a psychophysical experiment, the conditions of which allow significantly simplifying the original formulas. The thing is that, as a rule, the values of stimulus intensity ​​in a psychophysical experiment are constant and have a short duration (for example, the duration of sounds in acoustic experiments is of the order of one second (see the small print below).

“The research subjects were presented with sound tonal signals with a frequency of 1000 Hz of five intensities (from weak to very strong): 40, 60, 80, 100, 120 dB above a threshold level of 0.0002 bar, with the duration of 1 second, transmitted through the headphones in a random but identical for all research subjects order, 10 times each, with an interval of 12 seconds. In response to a sound of any intensity, the research subject had to press the button with the thumb of his/her right hand as quickly as possible (the reaction time was recorded by an electronic millisecond timer). … Then the subjects made a quantitative (numerical) SE (subjective evaluation – auth.) of the loudness of sounds of the same five intensities. … One sound (60 dB, average in intensity) was a standard sound with the number 10 prescribed to it by the experimentalist, shown to the subject three times before the experiment and subsequently given for evaluation along with other sounds. Sounds of 1s were presented in random order and were evaluated by the subjects quantitatively (by numbers), based on the ratio in volume between the standard and the presented sounds. … He/she was told in advance about the intensity of sounds, how they would be presented, explained what his/her task was and how he/she should act” (Ratanova, 2008).

Therefore, when constructing a mathematical model of the primary neuron of a receptor for the conditions of a psychophysical experiment, one can take into account neither adaptation processes nor the change in the stimulus intensity over time. Such conditions of the psychophysical experiment made it possible to significantly simplify the differential equations of the Hodgkin-Huxley model and obtain their analytical solution for the frequency of AP impulses (Fokin, 2017).

According to the electrical engineering rules (Detlaf A.A., et al., 2007), the following equations can be written for the electrical circuit of an excitable Hodgkin-Huxley cell (see Fig. 1):

or:

Figure 1. The electrical circuit of the membrane of the primary neuron of a receptor. As contrasted with the Hodgkin-Huxley scheme (1952), there is no “leakage” branch here; the leakage currents through potassium and sodium ion channels at rest are taken into account in the corresponding branches in the scheme. The directions of currents, electromotive forces (EMFs) and voltages are set according to the rules of electrical engineering (Detlaf, 2007). See the details in the text.

After some simple transformations we get:

The general solution of this type of differential equations with constant coefficients is presented in (Kamke, 2003). The coefficients in the above equation can be considered constant, because the capacity of the neuron membrane, sodium and potassium EMF are individually stable for each organism, at least for a long time, whereas the resistance of the membrane ion channels does not change during certain time intervals that make up the duration of AP. It is for these time intervals ∆t_d, ∆t_p and ∆t_r (see Fig. 2, the description is given below) that the solutions below will be written. Substituting our initial data and simplifying the solution of the differential equation from (Kamke, 2003), we get:

From this equation we can express the duration of any time interval during which its coefficients remain constant:

According to the last formula and in view of the fact that the duration of all action potentials, except for the first and the last ones, is the same for each stimulus in the psychophysical experiment (see Fig. 2), we can write for the AP frequency:

X_s = R_res.K/R_(res.+s)Na(S), is constant during

X_(s+p) = X_s+ X_p= X_s+ R_res.K/R_pNa, is constant during

X_r = X_s∙R_(res+rep)K/R_res.K, is constant during

where C_m is the electrical capacitance of the neuron membrane of the receptor, F;

R_res.K is the total electrical resistance of potassium ion leakage channels at rest; it remains constant throughout the duration of the AP Δt_ap = , Ohm;

is the total electrical resistance of potassium ion leakage channels and voltage-dependent potassium channels which open upon repolarization; it remains so for a period of time  Ohm;

R_(res.+s)Na(S) is the total electrical resistance of sodium ion leakage channels and stimulus-dependent sodium channels which open upon the depolarization of the neuron membrane under the influence of the stimulus S; it remains so throughout the duration of the AP Δt_ap = , Ohm;

R_pNa is the total electrical resistance of potential-dependent sodium ion channels, opening upon the depolarization of the neuron membrane of the receptor after the membrane potential reaching a threshold value, , and closing upon the neuron membrane potential reaching the peak value, ; it remains so for a period of time  (see Figure 2), Ohm;

E_Na, E_K are the sodium and potassium EMFs corresponding to the equilibrium sodium and potassium potentials, mV; for test calculations, it is set: E_Na = 60 mV; E_K = −90 mV;

X with the indices – potassium-sodium membrane parameter or translated stimulus (new terms and quantities) equal to the ratio of the total electrical resistance of single potassium ion channels of the neuron membrane to the total electrical resistance of individual sodium channels regardless of the mechanisms of their opening/closing at the current time: X(t) = R_K(t)/R_Na(t); X(t) depends on the stimulus intensity S, the transfer function of the neuron, its physiological properties and time. The term “translated stimulus” or “translated action” for X was chosen due to the fact that during the transduction process, the physical intensity of any stimulus is “translated” into the universal language of the nervous system – into the APs of the primary neurons of the receptor, the frequency of which is specifically determined by the value of X_s. The quantity X_s itself is formed under the action of a real physical stimulus; and it is, so to speak, the first part of its “translation” into the language of the organism. Thus, the “potassium-sodium membrane parameter” is another name for X_s, it is a kind of universal stimulus, on which the frequency of the AP of the primary neuron of the  receptor depends.

It is clear that, in the psychophysical experiment, for each S_i(t) = const, X_i(t) will remain constant periodically during three characteristic time intervals which constitute the total time of the AP Δt_ap (see Figure 2):

1. Δt_d – when the membrane potential changes from U_cK (closure potential of the potential-dependent potassium channels) to U_t (threshold potential of the neuron); in this case, X_s is determined by the constant resistances of the sodium and potassium leakage channels (at rest) plus the total resistance of the stimulus-dependent sodium channels that have opened under the action of the stimulus S = const, and is denoted by X_s;
2. Δt_p – when the membrane potential changes from U_t to U_p (the maximum value of the membrane potential of the neuron); in this case, X_(s+p) is determined by the resistances listed in item 1 plus the total resistance of the potential-controlled sodium channels that have opened at U_t ;
3. Δt_r – when the membrane potential changes from U_p to U_cK; in this case, X_r is determined by the resistances listed in item 1 plus the total resistance of the potential-controlled potassium channels that have opened at U_p (the potential-controlled sodium channels were inactivated in this case).

Here, the AP frequency is calculated without taking into account the first and last impulses and therefore does not depend on the resting potential, U_res, but depends only on the threshold value of the membrane potential, U_t, and the closure potential of the potassium channels, U_cK (see Figure 2). The thing is that, under the stimulus preserving its action, the membrane potential of a stimulus-dependent cell, as a rule (Kamkin, & Kiseleva, 2004), does not have time to decrease to the resting potential (U_res) during repolarization for each AP, each new depolarization begins slightly higher than U_res (see Figure 2). It is this point of transition from repolarization to depolarization, i.e. from the decrease to the growth in the membrane potential in the process of AP generation under the action of an adequate stimulus constant in time, that was conventionally called the “potential of closing the potential-dependent potassium channels” – U_cK . It should be noted here that for the proposed mathematical model of a stimulus-dependent cell, the specific value of U_cK , selected between the resting potential, U_res, and the threshold potential, U_t, is not important in the sense that it does not influence the qualitative dynamics of the dependence ν(X(S)), but determines only the numerical value of the frequency of APs: the higher the potential for closing the potassium channels, the higher the frequency of AP impulses ν.

Figure 2. Dynamics of temporal change in the membrane potential of the receptor depending on the value of the adequate stimulus S > S_res: for S < S_crit., the impulsing frequency of APs increases with the growth of the stimulus due to the predominant decrease in the duration of the depolarization component of AP – Δt_d; for S > S_crit., it decreases due to the predominant increase in the duration of the repolarization component of AP – Δt_r.

As already discussed in (Fokin, 2017), due to the fact that, in reality, the stimulus-dependent sodium and voltage-dependent sodium and potassium channels do not open/close at the same time (Ikeda et al., 2018), U_t and U_cK are some conventional values of the membrane potential at which it is observed: for U_t, an inflection on the curve U_m(t) in the depolarization region; for U_cK, the decrease in membrane potential (repolarization) is changed to its growth (depolarization). In reality, the “lengthiness” in time and the heterogeneity over the membrane surface of the opening/closing of the stimulus-dependent and voltage-dependent ion channels will lead to some dependence of U_t and U_cK on the stimulus intensity. These nuances were not taken into account in the present study, but they can be taken into account in the development of the model. Now it will be assumed that U_t and U_cK in the above formulas are constant for a given primary neuron of a receptor.

Thus, instead of simple empirical formulas of Fechner (2) or Stevens (1), proposed in the past to describe the relationship between stimulus intensity and sensory strength for the whole organism, the authors have obtained a more complex expression of the “basic psychophysical law” even at the receptor level (3-6). In addition, it turned out that during the transduction of any analog external stimulus into, so to speak, “digital” AP pulse frequency, non-linearity inevitably occurs and there is even a maximum on the calculated curve ν_ap(X(S)), see Figures 3, 4. This is caused by the fact that, with an increase in the stimulus intensity, the characteristic time intervals  from which the period Δt_ap of the AP is composed, behave differently:  decreases;  increases;  changes slightly (see Figures 2-4 and formulas 3-6, 4*-6*).

Results

THEORY AND CALCULATION

For clarity, let us carry out a test calculation of the dependence ν_ap(X(S)) with real physiological parameters. Based on the characteristic sizes of neurons (20 µm) and the cable properties of their membranes (Katz, 1966), for test calculations, the resistance of sodium and potassium ion channels were calculated (Fokin, 2017): R_res.Na = 1.196∙10⁹ Ω; R_res.K = 8.547∙10⁷ Ω; R_(res.+rep.)K = 6∙10⁶ Ω, R_pNa = 4.274∙10⁶ Ω. In this case, the resistance ratio, of course, will be inverse to the ratio of conductivities of the corresponding channels: R_resK/R_resNa = 1/14. Resistances turned out to be so “uneven”, because they are interconnected with each other and with the surface area of ​​the cell, i.e. its membrane. The membrane electric capacity was also taken to be characteristic for the nerve tissue (Katz, 1966): C_m ≈ 62.68 pF (for details see Fokin, 2017a).

For the test calculations, the following values ​​of the characteristic membrane potentials and EMFs were taken: U_res = −80 mV; U_cK = −65 mV; U_t = −45 mV; U_p = 35 mV; E_K = −90 mV; E_Na = 60 mV. The signs of potentials and EMFs are set here according to the generally accepted notion that, at rest, the intracellular environment is electronegative with respect to the extracellular one. However, formulas (2-4) were obtained in accordance with the sign rules of electrical engineering (Detlaf, & Yavorskii, 2007, p. 252) based on an electrical circuit similar to that from (Hodgkin, & and Huxley, 1952, p. 501) provided that the “leakage” branch is taken into account in the corresponding branches of sodium and potassium currents (Figure 1). Therefore, in formulas (3-6), the values ​​of the characteristic potentials and EMFs should be inserted with the following signs (Fokin, 2017): U_res= 80 mV; U_cK = 65 mV; U_t= 45 mV; U_p= −35 mV; E_K = 90 mV; E_Na = 60 mV. After substituting test values to formulas (3-6) and simple simplifications, the following expressions are obtained:

The time intervals Δt_d, Δt_p, Δt_r in formulas (4*-6*) are expressed already in milliseconds.

Figure 3 shows the dynamics of change of the factors in formulas (4*-6*) depending on the translated stimulus, X_s. The factors in front of the logarithms in all three formulas decrease monotonically with increasing X_s – dash-dotted curves in Figures 3a, 3b and 3c, whereas the logarithm values (continuous lines) behave in different directions. The logarithm in the first time interval ( , formula 4*) decreases hyperbolically with increasing X_s; in the second ( , formula 5*), decreases very slowly and almost linearly; in the third ( , formula 6*), increases steeply closer to the maximum value of X_s. Besides, the logarithm in formulas (4, 4*) determines the lower threshold value of the membrane potential: for the values ​​of the translated stimulus X_s ˂ 0.4286 ≈ X_t , the AP will not be formed, because the membrane potential never reaches its threshold value; in this test example, it is U_t = −45 mV, whereas the value of the expression under the logarithm sign will go beyond its domain of definition: it will become negative, which makes no sense.

In formulas (6, 6*), the logarithm determines the upper limit of the translated stimulus (in the test example, it is X_slim ≈ 2.85, whereas X_rlim ≈ 0.2, respectively), above which the formation of AP pulses also ceases, but for another reason: the membrane potential never reaches the potential for closing potential-dependent potassium channels during repolarization; the value of the expression under the logarithm sign will also go beyond its domain of definition – it will become

Figure 3. Dynamics of change of factors in formulas (4*- 6*) depending on the translated stimulus, X_s: 3a – formula (4*); 3b – formula (5*); 3c – formula (6*). Continuous lines denote the values of the corresponding logarithms, dash-dotted lines represent the factors in front of the logarithms. For other parameter values in formulas (4-6), the qualitative run of the curves will be the same as in Figures (3a-3c).negative, which makes no sense. The dynamics of the dependence of all three time intervals , constituting the time of the AP Δt_ap, on the translated stimulus X_s is presented in Figure 4.

Figure 4. The duration of the AP impulse, Δt_ap, and its components: depolarization – Δt_d, peak – Δt_p and repolarization – Δt_r, depending on the magnitude of the stimulus expressed through the dimensionless potassium-sodium parameter X_S of the membrane (translated stimulus).

The dependence Δt_ap(X_S) has a minimum due to the fact that, for large values of the ongoing stimulus, the remaining open stimulus-dependent sodium channels decelerate the repolarization process. The contribution of each of the factors in the formulas (4* -6*) is shown in Figure 3.

Since the frequency ν_ap of AP impulses is inversely proportional to their periods Δt_ap (see formula 3), the dependence ν_ap(X(S)) will have a corresponding maximum. The characteristic curve ν_ap(X_S), constructed according to the data of test calculation, is presented in Figure 5.

By changing the above physiological or rather physical parameters of the primary neuron of the receptor, one can obtain at the output not only different thresholds of susceptibility to an adequate stimulus, but also different slopes of the curve ν(X(S)), different coordinates of the maximum on it, and other individual features of sensory sensitivity discovered in psychophysical experiments at the macro level of the organism. For example, a change in only one parameter of the model – the

Figure 5. The calculated graph of the dependence of the frequency of APs ν on the intensity X_S of the translated stimulus, in the conditions of the psychophysical experiment: each X_S = const during a short period of time (~ 1s). The curve ν(X_S) is not smooth (analog), but “stepwise” (“digital”) – see view 1, because the membrane resistance changes discretely when opening/closing individual sodium or potassium channels. In the subthreshold region and the pessimum zone, the neuron does not respond to the APs stimuli, although for different reasons (see the text and Fokin, 2017a for details).

Results and discussion

The theoretical maximum of the dependence ν_ap(X(S)) under natural conditions, when the intensity of external stimuli, as a rule, is in the working range of values (see Figure 5), is difficult to detect unless one specifically sets such a goal.

However, psychophysical experiments in the area of high stimulus intensities (Ratanova, 2008) indicate its presence, at least, for the individuals with a “weak nervous system” (according to the terminology of Pavlov (1952) and the motional technique of Nebylitsyn (1966)). For example, it is known that the difference in sensations stably correlates with differences in such a physiological indicator of the organism as the amplitude of the evoked potentials of the cerebral cortex. So, experiments showed that “for some people, with an increase in the stimulation intensity, the amplitude of the evoked potentials increases all the time, while for others in the same range of intensities, at some point, when the stimulation intensity increases, the amplitude growth stops or even the amplitude decreases” (Ratanova, 2008, p. 96). However, psychophysicists have searched for the explanation of this phenomenon not at the level of receptors, but at the level of the central nervous system. An assumption has been put forward concerning the existence of a central mechanism for controlling intensity or a sensory “filter”, the purpose of which is to protect the nervous system from the effects of superstrong stimulation. Depending on the sensitivity of the nervous system, this mechanism is activated in different people at different levels of stimulation intensity, which explains the observed differences (Petrie et al., 1960).

Not denying the possibility of the existence of a hypothetical “central control mechanism”, through which the signals from receptors are modified in order to achieve the best adaptive result for each particular organism with its inherent psychophysiological parameters, it should be noted, however, that the above differences between individuals can be formed already at the receptor level, because it is at the receptor level that the nonlinearity of the conversion of the stimulus intensity to the sensation strength is physiologically formed (see Figure 5). For a final clarification of this issue, additional experiments are necessary at the neurophysiological level, which make it possible to reveal the presence or absence of a maximum on the curve ν_ap(S).

Another indirect confirmation of the correctness of the proposed mathematical model of the primary neuron of a receptor is the experimental data of psychophysicists, indicating significant individual differences in the slope of the “sensation strength – stimulus intensity” curves R(S) in the individuals with different sensitivity thresholds (Ratanova, 2008). Contrary to the theoretical hypothesis of Nebylitsyn-Ilyin (Nebylitsyn, 1966; Ilyin, 2004), the experimental curves R(S) do not shift equidistantly to the right along the abscissa axis with increasing sensitivity threshold, but intersect already in the first half of the range of the stimulus values due to different slopes (Ratanova, 2008; Figure 6_RT, 6_SE).

Figure 6_RT. “Cross” dynamics of experimental dependences of reaction time (RT) on the loudness of the presented sounds in subjects with a “strong” and “weak” nervous system according to Pavlov I.P. Source: (Ratanova T.A., 2008).

Figure 6_SE. “Cross” dynamics of experimental dependences of the subjective evaluation (SE) of the loudness of the presented sounds in subjects with a “strong” and “weak” nervous system according to Pavlov I.P. Source: (Ratanova T.A., 2008)

The model proposed in the present paper can explain the above phenomenon already at the receptor level. For example, with an increase in the threshold value of the membrane potential U_t of the primary neuron of the receptor from -50 mV to -40 mV, which corresponds to a decrease in its sensitivity, the curves ν_ap(X(S)) calculated using formulas (3-6) will have different slopes and intersect already in the first half of the range of values of the translated stimulus X_s(S) (see Fig. 6).

Figure 6. Calculated dependences of the frequency of impulses of the AP receptor (V) on the magnitude of the translated effect (X) and characteristic differences in their dynamics for organisms with weak (V50) and strong (V40) nervous systems (according to Pavlov’s terminology (1952)). V40, V45 and V50 are AP impulse frequencies at the threshold receptor membrane potentials (U_t) of -40, -45, and -50 mV, respectively. Moreover, it is conventionally accepted that a threshold potential (receptor response threshold) of -50 mV characterizes a weak nervous system, while -40 mV, a strong one (a difference of about 20%).

Since the frequency of impulses of the AP receptor V and the magnitude of the translated action X are the “sources” for the formation of sensation strength (R) depending on the stimulus intensity (S), one can observe a qualitative agreement between the calculated V(X) and experimental R(S) dynamics, obtained by psychophysicists for various receptors taking into account the strength-weakness of the nervous systems (NS) of the research subjects (for example, Ratanova, 2008, pp. 189-191 – for acoustic receptors). The calculated dependences are extended due to the paradoxical region characteristic of individuals with a weak nervous system, although not shown on the above experimental graphs, but described in detail in the monograph (Ratanova, 2008, pp. 92-106, 140-147).

It should be noted here that the result shown in Figure 6 is obtained only under a simultaneous proportional change in both the threshold values of the membrane potential U_t and the closure potential of the potential-dependent potassium channels U_cK such that, for example, their ratio always equals 1.4 = U_cK/U_t. The numerical value of the ratio is not important here, the main thing is that, when the value of U_t changes, the value of U_cK changes proportionally. If one changes only the response threshold of the primary neuron of the receptor U_t, leaving the closure potential U_cK of the potassium channels constant, then the curves ν(X_s(S)) will not intersect. The dependencies will simply equidistantly shift to the right along the abscissa axis as in the theoretical Nebylitsyn-Ilyin model (1966-2004), which contradicts the experimental data of psychophysicists (Ratanova, 2008). The above calculation options indirectly predict that the sensitivity threshold value of the primary neuron of the receptor is somehow connected with the value of the closure potential of its voltage-dependent potassium channels. To confirm or refute this circumstance, additional neurophysiological experimental studies of primary neurons of receptors with different response thresholds are necessary

Transition from the dependence ν(X_s) to the dependence ν(S)

Now let us consider the relationship between the translated stimulus X_s and the real physical stimulus S, because without this link the desired dependence ν_ap(S) remains undetermined. Obviously, with the linear dependence X_s(S), although the curves “AP frequency – intensity of the translated stimulus”, ν(X_s), and “sensation strength – intensity of the physical stimulus”, R(S), are shifted with respect to each other, they are equidistant in form.

However, there are data supporting more complex X_s(S) relationships. Thus, experimental psychophysical curves R(S) were obtained not only convex in the direction of growth of R, similar to the calculated curves ν(X_s), but also concave or almost linear (Stevens, 1957; Ratanova, 2008). Even for the simplest case of the direct impact of the stimulus on the primary neuron of the receptor, for example, during tactile or painful sensations, it is unlikely that one can write down the dependence X_s(S) in one equation, because, most probably, the curve X_s(S) has a characteristic S-shape. For more complex receptors, for example, the auditory one, it is necessary to take into account the specifics of the stimulus transformation at each stage of its path to the primary neuron. That is, it will be necessary to write down several equations, each of which may have its own coefficients that differ among different individuals.

The dependence X_s(S) in the conditions of a psychophysical experiment is actually the dependence of the total electrical resistance of the stimulus-dependent sodium channels of the neuron membrane on the stimulus magnitude: X_s = R_res.K/R_(res.+s)Na(S), which can be successfully identified using modern microbiological methods. For example, in (Kamkin, & Kiseleva, 2004), an experimental dependence was obtained of the incoming sodium current strength on the elongation of a cardiomyocyte when it is stretched by various forces. The value of the potential difference between the extra- and intracellular media was the same for all variants of stretching. Since, according to Hooke’s law, the elongation of the tissue in the elastic region is directly proportional to the tensile force, whereas, according to Ohm’s law, the current strength at constant voltage is inversely proportional to the resistance, for the cardiomyocyte, the dependence R_(res.+s)Na(S) was actually determined, where the tensile force acts as a stimulus. The same can be reproduced not with stretching, but with pressure on the pain or tactile receptors. The same manipulations can also be carried out with thermoreceptors, light-sensitive receptors, etc. Further, after determining the physical model of the influence of stimulus intensity on the electrical resistance of the membrane of the primary neuron of receptors and its mathematical expression, the obtained experimental data are substituted into it and the limits of individual coefficients can be calculated. Thus, one can differentiate between:

a) individual features that are manifested during transduction of the stimulus at all stages of its path to the primary neuron of the receptor;

b) differences determined by individual variations of the electrochemical properties of the neuron itself.

The authors have already decided on the mathematical model of the primary neuron of receptors; some features of the stages of stimulus transduction are considered below.

As a rule, the electrical resistance of the membrane of a stimulus-sensitive cell decreases with increasing intensity of action due to opening of stimulus-controlled sodium ion channels (Fokin, 2017):

where R_sNa(S,Ψ) is the electrical resistance of sodium channels opened under the stimulus action, Ohm; Ψ is the matrix of individual properties of the neuron, also affecting the dynamics of changes in electrical resistance. For S = 0, no stimulus-dependent sodium channel is opened, and R_sNa is equal to the resistance of the lipid layer of the membrane, which is several orders of magnitude greater than R_resK and, accordingly – matrix of individual properties of the neuron, also affecting the dynamics of changes in electrical resistance. At S = 0, not a single stimulus-dependent sodium channel was opened and R_sNa is equal to the resistance of the lipid layer of the membrane, which is several orders of magnitude greater than R_resK and, accordingly, R_resK/R_sNa(S=0) → 0, whereas X_s(S=0) = R_resK/R_resNa= X_res.

R_1Na is the electrical resistance of a single stimulus-dependent sodium channel of the neuron membrane of the receptor, Ohm. To date, the electrical conductivity of many types of cation channels has been experimentally measured and amounts to about 10-35 pS (Kamkin, & Kiseleva, 2004), which corresponds to electrical resistance of ~ 0.1- 0.3 TΩ;

N_sNa(S,Ψ) is the number of opened stimulus-dependent sodium channels of the neuron membrane of the receptor when a constant stimulus of the magnitude of S is applied to it, pcs.; Ψ is the matrix of individual properties of a neuron that affect the dynamics of the opening of sodium channels during stimulation.

The ratio R_resK/R_resNa of “|leakage resistances” can be designated as the potassium-sodium rest parameter – X_res; for the test neuron, it will be equal to: X_res= R_resK/R_resNa = 1/14 ≈ 0.07143 (see above). The other notation in (7) is the same as in the previous formulas.

The expression for the potassium-sodium parameter X_s of the membrane in formula (7) in terms of the stimulus-dependent total resistance R_sNa can be conveniently used to obtain the experimental dependences X_s(R_sNa(S)), because in such experiments, the total electrical characteristics of the membrane are operated with (Kamkin, & Kiseleva, 2004). However, the integrated characteristic R_sNa(S,Ψ), obtained experimentally, will not tell anything about the mechanisms of the process, nor about the criteria for individual differences, although it will be useful for testing theoretical models.

The expression of X_s through the number of newly opened sodium channels when the cell is effected will help to understand the mechanisms of the corresponding processes and the possible causes of individual differences at the level of perception of the stimulus. For example, a mechanical effect on a cell is perceived by its cytoskeleton (Kamkin, & Kiseleva, 2004). Moreover, if the magnitude of the mechanical stimulus does not go beyond the elastic region of operation of the cytoskeleton of the stimulus-sensitive cell, for example, cardiomyocyte (Kamkin, & Kiseleva, 2004), then, upon the removal of the impact, the electrical resistance of its membrane returns to its original state, i.e. is completely restored. Thus, even if the full form of the dependence N_sNa(S, Ψ) is unknown, one can still definitely say that for different elastic properties of the cytoskeleton of cells, the number of opened stimulus-dependent sodium channels in their membranes will be different for the same stimulus value. This means that the threshold value S_t of the stimulus will also be different, of course, provided that X_st is the same for them. Here, the threshold value S_t of the stimulus is understood to be such its value, under the action of which such a number of  sodium channels opens that is necessary to achieve the threshold potential (U_t) of the primary neuron of the receptor. Thus, one can already explicitly select from the array of internal individual characteristics of the neuron Ψ one of its components, which is responsible for the elastic properties of the cytoskeleton. By analogy with the coefficient responsible for the elastic properties of a spring in physics, it can be denoted by the Latin letter “k” and called “stiffness of the cytoskeleton”. A cell can deform under the action of tensile, compressive, or shear forces. If the values of the impacts are in the elastic region, then, according to Hooke’s law, the strain value Δd will be directly proportional to the applied force – in this case, the stimulus S:

For any of the above types of deformations, mechanosensitive sodium channels will open in an amount proportional to the displacement of its cytoskeleton (and membrane, respectively) from its resting position, i.e. proportional to Δd. Thus, it can be written:

where k is the stiffness of the cytoskeleton of the membrane of the primary neuron of the receptor, [unit of the impact S/unit of the deformation Δd]; as will be shown below, the notation form indicated in (9) will be valid for many sensory modalities, therefore a generalized name for k is proposed – “stimulus perception coefficient by the neuron surface”:

F(S,Ψ₁) is a function taking into account the influence of all other individual parameters of the neuron on the dynamics of opening of sodium channels, except for k; Ψ₁ is a matrix of the individual properties of a neuron that affect the dynamics of opening of sodium channels during stimulation even without taking into account the stiffness coefficient k of the cytoskeleton.

That is, the larger the stiffness of the cytoskeleton, the less the cell membrane will be deformed under the same magnitude of the stimulus, and, accordingly, a smaller number of stimulus-controlled sodium channels will open.

Formula (9) is written for the simplest case, when the impact is performed directly on the primary neuron of the receptor, for example, as in tactile or pain receptors. If there are “mediators” between the stimulus and the primary neuron, such as hair cells in the auditory receptor, then k will reflect the stiffness of the cytoskeleton of the hair cells, the electrical resistance of which also varies depending on the level of sound pressure deforming the hairs. In such complex cases, each stage of stimulus transduction should be considered separately in their sequence up to the primary neuron of the receptor. However, in response to deformations, only a continuous generator potential arises in the hair cells, which stimulates the release of the mediator into the synaptic cleft with the primary auditory neuron, and already in the latter, AP arises in an amount proportional to the concentration of the mediator in the synaptic cleft and the reaction rate constant. That is, in addition to the same stimulus for all individuals – the magnitude of sound pressure repeatedly converted along the way to the auditory neuron, there are also individual coefficients of the auditory neurons themselves – the rate constants of the reaction of the mediator with the corresponding receptors of the membrane. The greater the reaction rate constant, the faster the threshold will be reached for the onset of generation of APs by the auditory neuron at the same concentration of the mediator in the synaptic cleft, i.e. at the same sound pressure. Thus, the reaction constant will be one of the parameters that determine the threshold value of the sound volume. It is interesting that the formal dependence of the number of opened stimulus-dependent sodium channels on the stimulus intensity (the sound pressure value) and the constant of the mediator reaction rate in the synaptic cleft of the auditory neuron will be the same as in formula (9), only k will denote the reaction rate constant and be in the numerator, where S is the sound pressure level. Thus, the volume of perceived sound will depend not only on a single objective parameter of the stimulus – the sound pressure level, but also on a number of subjective parameters that are individual for each organism, which can be used to explain individual differences obtained by psychophysicists (Ratanova, 2008).

Formula (9) will also be valid for thermoreceptors and photosensitive receptors, but it will be necessary to consider the coefficients in each case separately. It should also be noted that many neurons of the nervous system are excited through chemical synapses, and as mentioned above, for this case, formula (9) is also valid.

Substituting (9) into (7) and taking into account previously accepted notation, one can obtain:

By combining the constant individual parameters R_resK/(R_1Na·k) of the receptor into one constant K_R , one can obtain:

The identification of the explicit form of the function F(S,Ψ₁) requires additional research that is beyond the scope of the present paper. A priori, it can be only said that F(S,Ψ₁) will depend on the cell geometry, the number of stimulus-dependent sodium channels per unit area of ​​its membrane, the maximum value of the stimulus, and other factors. While the form of the function F(S,Ψ₁) remains unclear, it is excluded from formulas (10, 10*) and equated to unity. Further, the consequences of variations in the electrochemical parameters of the neuron (see the characteristics of the test neuron above) and the coefficient k responsible for the dynamics of the stimulus perception by the neuron surface are considered.

As mentioned above, formulas (10, 10*) are valid for many sensory modalities, therefore a generalized stimulus value will be used, expressed as a percentage of the maximum value – 100%. Obviously, for F(S,Ψ₁) = 1, the dependence X_s(S,k) will be linear with respect to S with the constant parameter k for each calculation option. The function X_s(S,k) will intersect the ordinate axis at the point X_res= R_resK/R_resNa (for the test data, X_res = 1/14 ≈ 0.07143) for any values of the stimulus perception coefficient k. Further, two calculation options are possible:

1. If the value of the coefficient k_e of stimulus perception by the neuron surface is known from the experiment, then substituting it and the threshold value X_st of the potassium-sodium parameter of the membrane into formula (10), one can express the threshold value of the stimulus S_t (X_st is calculated from formula (4) when equating the numerator of the expression under the logarithm sign in this formula to 0):

Next, for each value of the suprathreshold stimulus S_i > S_t , one can find the corresponding value X_si according to the formula (10*) and, substituting the obtained result into the formulas (3-6), one can find the array of the sought-for frequencies of the impulses of the APs ν_api (X_si(S_i)), generated by the primary neuron of the receptor in response to the stimuli S_i.

2. If the value of the threshold stimulus S_t is known from the experiment, then substituting it and the threshold value X_st of the potassium-sodium parameter of the membrane into formula (10), one can express the coefficient k of stimulus perception by the neuron surface:

Further steps are similar to those described in item (1).

Test calculations were carried out using the second algorithm, i.e. several variants of the lower threshold values of the stimulus were set and each of them was combined with the variants of the threshold potential of the primary receptor neuron shown in Figure 5. Figures (6-8) present the most interesting results from the point of view of coincidence of the calculated dynamics ν_ap(S) with the dynamics R(S), obtained in psychophysical experiments (Ratanova, 2008).

Figure 7. The calculated dependences of the translated stimuli X on the real physical stimulus S, expressed as a percentage of its maximum value S_max. The horizontal lines X₀40, X₀45, X₀50 are the threshold values of the translated stimulus in neurons with threshold potentials of −40, −45, and −50 mV, respectively. The inclined lines characterize the dynamics of changes of X(S) for different coefficients k of stimulus perception by the neuron surface for neurons with threshold potentials of −40, −45 and −50 mV (X40s15, X45s15, X50s15 and X50s25, respectively). For the three upper curves, the coefficients k are such that the threshold value of the physical stimulus S_t is the same (S_t = 0.15·S_max) for all three neurons with different threshold potentials (−40, −45 and −50 mV). It is also shown that an increase in k leads to a decrease in the slope of the graph X(S) and an increase in the threshold value of the physical stimulus: the line X50s15(S): k ≈ 0.01316 → S_t = 0.15·S_max; the line X50s25(S): k ≈ 0.02083 → S_t= 0.25·S_max. Moreover, the threshold potential of the neuron remains unchanged: U_t = −50 mV.

Regardless of the coefficient k of stimulus perception by the neuron surface, all lines X(S) start with the same value – the free term in the equations (10, 10*): X_res = R_resK/R_resNa, which for the test receptor is equal to X_res = 1/14 ≈ 0.07143. The question concerning possible individual differences in X_res remains open. The greater k for mechanosensitive receptors, the smaller the inclination angle between the X(S) line and the abscissa axis, because k is in the denominator of formula 10 (it should be noted that for chemo-, light-, and heat-sensitive receptors, the dependence on the corresponding coefficients will be direct, see above). From a physical point of view, this means that a cell with a stiffer cytoskeleton has a greater threshold value of the physical stimulus S_t, of course, under the same threshold value of the membrane potential U_t (see Figure 7, lines X50s15 (S) and X50s25 (S)). A change in the threshold potential leads to a change in the lower threshold value of the translated stimulus X_St. The larger the value of |U_t|, i.e. the closer the threshold potential to the resting potential of the cell, the smaller the lower threshold value of the translated stimulus X_St, which is determined by equating to zero the numerator of the fraction under the logarithm sign in formulas 4 and 4*. This can be seen in Figure 6: the threshold value of the translated stimulus X₀40 (|U_t | = 40 mV) is higher than X₀50 (|U_t| = 50 mV). At the intersection of the line X(S) with the level of the threshold value of the translated stimulus X₀, the value of the threshold physical stimulus S_t is obtained (see Figure 7).

Besides the lower threshold value X_St , the translated stimulus also has an upper threshold value X_Slim (see Figure 5), which is determined by equating the numerator of the fraction under the logarithm sign in formulas 6 and 6* to zero. In Figure 7, the levels of upper threshold values  X_Slim are not shown; their values depend on the value of the threshold potentials in the same way as the lower ones: the larger the value of |U_t|, the lower the upper threshold value of the translated stimulus X_Slim. Obviously, the greater the angle of inclination of the line X(S), determined by the value of the corresponding coefficients of the stimulus perception (not necessarily mechanical, see above) by the cell surface, the smaller the critical value S_crit and the upper threshold S_lim of the physical stimulus corresponding to X_Scrit and X_Slim in Figure 5.

Recall that the sensation reaching its upper limit, after which its strength begins to decrease even with increasing stimulus intensity, is called the “satiation effect” by psychophysicists and is explained by the presence of hypothetical “central filters” that prevent too strong stimulation from being transmitted to the upper parts of the brain (Petrie et al., 1960). The results obtained using the proposed mathematical model make it possible to go without the use of “central filters” by detecting the saturation phenomenon (the pair of X_Scrit and S_crit) for primary neurons with certain properties even at the receptor level (see Figures 5, 8 and 9). Note that the saturation effect is characteristic only for individuals with a weak nervous system (in the terminology of Pavlov (1952) and in the motor technique of Nebylitsyn (1966)), i.e. for those whose primary neuron of the receptor has a lower value of the lower threshold of sensitivity (a greater value in the absolute value, respectively): U_t = −50 mV (Figures 8, 9).

Even if not all of the research subjects experimentally reveal the saturation effect at the macro level of the organism, then it is hardly possible to observe the limit level of the stimulus (X_Slim and S_lim) in the natural conditions, because it has no adaptive significance, and, therefore, had to be eliminated by natural selection. However, one can try to detect the limit level of stimulation on individual neurons outside the organism, because it is not known whether it is excluded in principle, for example, by small slopes of the lines X(S) (see Figure 7), or the magnitudes of the stimuli in the natural conditions are too small to achieve it, but the primary neuron of the receptor will still retain its working capacity even for S > S_lim. This issue requires its solution at the experimental neurophysiological level.

It should be noted that when the stimulus reaches its lower threshold value S_t, the primary neuron of the receptor immediately begins to generate AP pulses with a certain, minimum possible frequency of the order of several tens of Hertz (Figure 8), which cannot be lowered due to the specific features of the analog-to-digital conversion (Fokin, 2017). However, the minimum possible frequency may differ for neurons with different values of characteristic potentials.

If one considers the frequency of AP of the primary neuron of the receptor as the basis for sensation formation, then the curves V40 (S15p40) and V50 (S25p50) in Figure 8 reflect atypical cases of their dynamics, because it was revealed in a number of psychophysical experiments (Ratanova, 2008) that highly sensitive individuals (low threshold S_t , curve V40 (S15p40)), as a rule, have a weak nervous system (U_t = −50 mV), whereas low sensitive (high threshold S_t, curve V50 ( S25p50)), a strong one (U_t = −40 mV). That is, the above curves are exceptions to the general rule; and in the theory of Nebylitsyn (1966) and the early works of Ilyin (2004), there were no such exceptions at all. However, “atypical” individuals were sometimes still found in psychophysical scaling (Ratanova, 2008).

Figure 8. The calculated dependences of the AP frequency of the primary neuron of the receptor on the value ν_ap(S) of the physical stimulus for combinations of threshold values S_t of physical stimuli and threshold potentials U_t of the neuron, shown in Figure 6. It can be seen that for the same value S_t = 0.15·S_max, but different U_t= −40, −45, −50 mV, the curves ν_ap(S) do not behave identically. See details in the text.

In order for the curves V40 (S15p40) and V50 (S25p50) to become “typical”, they need to be interchanged, which was done in other variants of calculation shown in Figure 8. Here, increased sensory sensitivity (S_t = 15, S15p50 in Figure 9) is manifested in the persons with a weak nervous system (U_t = −50, in Figure 9 it is denoted by V50), whereas the decreased one (S_t =19, in Figure 9 by S19p40), with a strong one (U_t = −40, in Figure 9 it is denoted by V40). The above curves ν_ap(S) intersect in the first half of the stimulus value range, which is typical for experimental psychophysical curves R(S), “sensation strength in its dependence on the stimulus intensity.

Figure 9. The calculated dependences of the AP frequency of the primary neuron of the receptor on the value ν_ap(S) of the physical stimulus for the following combinations of parameters: a) V50 (S15p50) → S_t = 0.15·S_max, U_t = −50 mV; b) V45 (S17p45) → S_t = 0.17·S_max, U_t = −45 mV; c) V40 (S19p40) → S_t = 0.19·S_max, U_t = −40 mV. The presented calculated curves ν_ap(S) correspond to the typical dynamics of changes in sensation strength due to the intensity of the physical stimulus, R(S), for individuals with a strong (U_t = −40 mV) and weak (U_t = −50 mV) nervous system (according to classification of Pavlov (1952) and Nebylitsyn (1966)). See details in the text

Such “cross” dynamics could not be explained in any way either from the point of view of the “laws” of Fechner (1860) or Stevens (1957) or from the point of view of the Nebylitsyn-Ilyin theory (1966-2004). This happens because in the indicated theories, the type uniformity of transformation of the stimulus into the sensation within the “black box” model was postulated, whereas for different research subjects R(S) differ only in the threshold S_t.

Conclusions

In fact, as the proposed mathematical model shows, there are several independent steps of regulating the dynamics of ν_ap(S) and R(S), respectively, two of which are considered in the present paper: the first is at the level of stimulus perception by the surface of the corresponding stimulus-sensitive cells, the second, at the level of electrochemical properties of the primary neuron of the receptor. As a result of test calculations performed using the author’s mathematical model with the real physiological properties of the neuron, the characteristic dynamics of the ν_ap(S) dependence agreeing with the experimental ones were obtained (Ratanova, 2008; Figures 6_ВР, 6_СО), which indirectly confirms the adequacy of the proposed model.

In addition, the dependence ν_ap(S) already at the level of the primary neuron of the receptor has a characteristic maximum (see Figure 5), which allows explaining another experimental psychophysical phenomenon, the “saturation effect”. It consists in the fact that after reaching a certain level, a further increase in the stimulus no longer leads to an increase in the sensation strength or even leads to its weakening (Ratanova, 2008). This cannot be explained with the help of traditional monotonous psychophysical dependencies. In fact, as the proposed mathematical model shows, even the dependence ν_ap(S) is not simple and monotonous, but consists of several “competing” equations (see formulas 3-6). Moreover, two of them have asymptotic limits that determine the lower and upper thresholds of sensory sensitivity (see Figures 3, 4).

The variant with the same stiffness coefficient k of the membrane cytoskeleton for all neurons with different threshold values ​​of the membrane potential: U_t = −40; −45 and −50 mV is of interest. It is quite possible, although unlikely from a physiological point of view, that the membranes of all primary neurons of receptors of a certain modality are “manufactured” according to the same genetic program, not distorted by natural selection. Even if this is so and k_i = const, then, due to different threshold values ​​of the membrane potential of the primary neuron, the lower thresholds of the sensitivity of the receptor to a physical stimulus will also be different. For the above test values ​​of U_t and k_i = const, as shown by elementary calculations by formulas (4*, 11), the lower thresholds of sensitivity to a physical stimulus will differ from each other as follows: S_t(U_t = −40 mV)/S_t(U_t = −45 mV) ≈ 1.2, i.e by 20%; S_t(U_t = −50 mV)/S_t(U_t = −45 mV) ≈ 0.82, i.e. about 18%. The real limits of individual variability of cytoskeleton stiffness and threshold values ​​of the membrane potential of primary neurons of the receptor should be revealed by the future neurophysiological experimental studies, but whatever their results, the author’s mathematical model is capable of working with any initial data.

PERSPECTIVE

Despite indirect confirmation of the adequacy of the proposed mathematical model of the primary neuron of a receptor by the results of psychophysical experiments at the macro level of the entire organism, experimental confirmation of it at the micro level of individual neurons and receptors is also required.

The proposed model requires further development, because many questions remain open: what is the form of the dependence R_sNa(S,Ψ); at what stage of the dependence ν_ap(X_s(S)) do stimulatory-controlled sodium channels end; what is the contribution of each stage of stimulus transduction on its way to the primary neuron for sensory systems with a complex structure and how can the stimulus transformations be formalized on each of them; is there a relationship between the threshold value of the potential of the primary neuron and the closure potential of the potential-dependent potassium channels and which factors determine it; what are the real dynamics of opening stimulus-dependent sodium and closing potential-dependent potassium ion channels of the primary neuron of a receptor and what is the error introduced into the model by the accepted assumption of the instantaneous opening/closing of these channels; and so on.

COMPETING INTERESTS:

Declarations of interest: none.

AUTHOR CONTRIBUTIONS:

The author approved the final version of the manuscript and agrees to bear responsibility for all aspects of the work. All persons appointed by the authors are entitled to authorship, and all those who are entitled to authorship are listed.

FUNDING

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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