Quantum and Non-Quantum Formulation of Eye’s Adaptation to Light’s Intensity Increments

Context and background: A quantum formulation of vision in vertebrates was proposed in the early 1940s. The number of quanta useful for enabling vision was found. The time interval required for their absorption, however, was never specified. In the early 1950s, experimental data on the effects of light’s intensity increment on vision indicated that the quantum formulation is true only at low light’s intensities. In this case, a vaguely described signaling adaptation mechanism was invoked to explain the separation between vision at low and high intensities, accompanied by the switch from rod to cones as photoreceptors. Motivation: In this article, we want to prove the validity of the non-totally-quantum formulation and unveil the nature of the signaling adaptation mechanism. Hypothesis: To accomplish our proof, we hypothesize that the amount of energy transferred and conserved in light’s interaction with the eyes is given by the product of light’s intensity (or power) times its period. Method: We construct and use the plots of the trends of light’s intensity increments and the corresponding changes in the axon’s membrane capacitance versus adapting intensity. Results: We find that 1) the average solar light’s intensity is the critical value that separates low from high light’s intensity regimes in vision, and 2) changes in the capacitance of the axon’s membrane enable the signaling adaptation of vision when light’s intensity changes. Conclusions: We prove the validity of the non-totally-quantum formulation and unveil the nature of the signaling adaptation mechanism. Our proof is supported by the model based on light’s intensity times period as being the energy conserved in light-matter interaction This model suggests that 1) all the waves in the electromagnetic spectrum, at the correct intensity for each frequency, could be used to produce the effects of optogenetics in diagnostics and therapy, and 2) it takes seconds to minutes to see details in the dark when light is switched off.


Introduction
Recent experimental results in infrared (IR) spectroscopy [1] and in the description of the interaction between IR light and capacitors [2] show that light's intensity (power/area) plays a critical role in establishing the amount of energy transferred to capacitors and conserved in the interaction with them. Specifically, through the law of conservation of energy it was found that the magnitude of the product of light's power P times light's period τ , i.e. Pτ (where 1 τ ν = , and ν is frequency), equals the magnitude of the electrical, thermal, and mechanical energy, or combination thereof, transferred from the electromagnetic (EM) wave to the capacitors. In this article, we show that Pτ is the energy conserved also in the mechanism of vision in vertebrates, which involves the nervous system and its complex network of components acting as capacitors. In proving this conclusion we solve the dispute between the quantum and non-quantum formulation of eye's adaptation to light's intensity increments.
In order to explore the consequences of the hypothesis that Pτ is the energy conserved in the mechanism of vision in vertebrates, we briefly review the process of vision [3]. The nerve cells, or axons, can be depicted as capacitors in which the axon's membrane acts as the dielectric layer, whereas the intra-and extra-cellular fluids, inside and outside the axons, respectively, act as electrodes. These fluids consist of Na + , K + , Ca 2+ , and Cl − ions in solution. When light hits the eye, a decrease of the cyclic guanosine monophosphate (cGMP) concentration takes place, which closes the Na + and K + ion channels [3]. This phenomenon triggers the hyperpolarization from −40 mV to −70 mV of the axon's membrane. Afterwards, the closure of the Ca 2+ gate causes the Ca 2+ concentration to drop, generate a cascade of biochemical processes [3] [4], and depolarize the axon from its resting potential of −70 mV. If sufficient energy is supplied to depolarize the axon to the threshold potential of ~−55 mV (a voltage jump of ~15 mV), an action potential ( ap V ) is fired. In turn, the action potential polarizes the axon up to +40 mV at the synaptic terminal. The total voltage jump from −70 mV to +40 mV has a magnitude of ~110 mV. A wave of action potentials thus generated can travel along the axons to the axon terminal, and then to the optic nerve [3]. The production rate of the action potentials is related to light's intensity.
We hypothesize that, during the formation of the action potentials, light's energy is transformed into electrical energy 2 1 2 CV , where C is the capacitance of the axon's membrane [5], and V is the depolarization voltage from the cell's resting potential of −70 mV to the threshold potential of ~−55 mV. The depolarization voltage V has a magnitude of ~15 mV, and is the voltage needed to fire  contribute in the dispute  arisen about a quantum or a non-quantum formulation of vision in vertebrates  involving signaling through the axon's membrane embedded between the intraand extracellular fluids. The dispute evolved between the early 1940s and 1950s with Hecht et al. [6] [7], and Müller [8] [9] as major protagonists. Arguing with photons, or quanta of light, in a low light's intensity regime, Hecht and coworkers found that a minimum threshold of about 8 photons is required to trigger vision [6] [7]. On the other hand, Müller found, through experimental data, that different levels of light's intensity require different photochemical adaptation [3] processes. This finding challenges Hecht's quantum formulation because, in Müller's words, "For different adapting intensities different proportionality terms would relate the number of quanta absorbed and the stimulus intensity" [9]. The stimulus intensity thus changes the number of quanta absorbed by the eye in a defined time interval. Specifically, at low light's intensity, for which transduction is carried out by rod cells, the process of adaption is different than at large light's intensity, where the transduction process is carried out by cone cells. At low light's intensity, the predictions of Hecht and coworkers are confirmed in Figure 3 in [9]. At large light's intensity, however, the absorption constant changes, and the response of the eye departs from the trends at low light's intensity, and thus from the prediction by Hecht et al. [6] [7]. However, neither does Müller provide an explanation of his findings, nor does he address other questions generated by his results, such as: 1) What is the adaptation process that adjusts the behavior of the eye to increasing light's intensity? 2) What factor decides the magnitude of the light's intensity that separates its low and high regimes, shifts the transduction process from rod to cone cells, and establishes the regime in which the formulation by Hecht et al. [6] [7] is not valid?
In this article, we reproduce the results in Figure 3 in [9], and attempt to respond to the questions listed above that were neglected by Müller. Our strategy consists of using the law of conservation of energy, along with the assumption that Pτ is the energy conserved in the mechanism of vision in vertebrates. We propose that the adaptation process employed by the eye to adjust its behavior to increasing light's intensity consists of changing the magnitude of the axon's membrane capacitance C: small values of C are pursued at small light's intensities, whereas larger values are pursued as light's intensity increases and approaches a critical value 0 I . Thus, larger C values, associated to larger light's intensities, are also associated to larger rates of production of action potentials. The ability of the axon's membrane to change its capacitance underlines that, unlike the capacitors used in electronics, which have a fixed capacitance, the "capacitors" in the nervous system have a variable capacitance. We identify the critical intensity 0 I with the average intensity of solar light: 136 mW/cm 2  Furthermore, we find that the above-mentioned singularity at 0 I I = separates the low and high regimes of the adapting intensity I. Finally, we picture the adaptation mechanism to light's intensity increments as consisting of changes in the axon's membrane capacitance C. In low light's intensities, these changes in C are slow, as they are related to the slow synthesis of rhodopsin in rods, while in intense light the changes in C are fast, as they are related to the fast synthesis of the opsins in cones. This picture explains, for instance, why it takes seconds to minutes to the eye to define the details of objects in the dark, when light is suddenly switched off. The need of a certain time interval to implement the adaptation mechanisms is evident also in other adaptation processes, e.g. chemotaxis [3], i.e. the directional motion of cells towards a source of a chemical gradient.
Other adaptation processes requiring time are described in [10].
Our findings related to Pτ as being the energy conserved in the mechanism of vision in vertebrates, suggest a possible application to optogenetics [10] [11].
This technique exploits the ability of light to activate viruses and generate a potential across cellular membranes [12] [13]. Assuming Pτ to be the amount of energy conserved in light-matter interaction, we can estimate the energy effective in activating the viruses. For example, with typical optogenetics parameters, such as blue light at wavelength 450 nm λ = , and power 3.5 mW P = [11] [14], it is possible to activate specific axon terminals in the parabrachial nucleus [11]. The energy Pτ in this case is 5.25 aJ, and the depolarization voltage of 15 mV, needed to fire the action potential, is achieved if the capacitance in the axon's membrane is ~46 fF. On the other hand, with green light at 561 nm λ = and 12 mW P = , it is possible to silence, or deactivate, the axon terminals in the parabrachial nucleus [11]. The energy supplied by this green light to the virus is 22.44 aJ Pτ = . Achieving a voltage smaller than the depolarization voltage 15 mV V = , necessary to avoid firing the action potential, requires a capacitance of the axon's membrane as large as ~200 fF, in agreement with the arguments on adaptation and related time intervals discussed in the previous paragraph [10] [12]. From these two examples we infer that, if energy is the major issue explaining the effectiveness of optogenetic tools, than activation and silencing of the axon terminals in the parabrachial nucleus, as described in the examples above, could be successfully achieved by radio waves at, e.g. 20 ns τ = , activation 0.26 nW P = , and silencing 1.12 nW P = , respectively. This hypothesis would be the basis for "radio-genetics" using radio waves (not just low-frequency magnetic fields as in [15]).

Methods
We use the law of conservation of energy to describe the transfer of energy from light to axons as follows: World Journal of Condensed Matter Physics where P and τ are light's power and period, respectively, C is the capacitance of the axon's membrane, and V is the depolarization voltage needed to fire the action potential. For most axons, 15 mV V = . The transfer of energy from light to the axon occurs indirectly through a complex biochemical process known as the visual cascade [3] [4]. In order to start the visual cascade, the energy Pτ in Equation (1) needs to be at least as large as the activation energy of the biochemical reactions involved in the visual cascade. To maximize the probability of such a match, the vertebrate's eye evolved such as to adapt the capacitance in the axon's membrane to the critical intensity ), requires 2.14 pF C = . This value is in good agreement with the baseline capacitance of few pF found in retinal axon membranes [5]. This baseline capacitance changes when light induces an axon's depolarization process, thus an action potential [5].
Typical magnitudes of capacitance changes are of the order of With this information we begin the investigation of the effects of low light's intensity on the eyes, searching in particular the conditions needed to supply the depolarization voltage V of 15 mV required to drive the axon from the resting potential of −70 mV to the threshold potential of −55 mV, and so fire action potentials. As previously mentioned, to investigate these effects we follow a procedure similar to that adopted in [9], which consists of plotting  Figure 1(a).
This plot is similar to the one in Figure 3 of [9], reproduced in Figure 1(b), World Journal of Condensed Matter Physics which Müller obtained from experimental data of intensity discrimination. In addition, in Table 1 we report the values of the capacitance C obtained from Equation (1)  Thus, we compute C as . In [9] there is no graph corresponding to Figure 1(c).
from the 9 experimental data points reported in , which is the same as the fitting line in Figure 1 pacitance change [5]. The values in the abscissa are the same as in Figure 1 Table 1. . We note that 0 I is the critical intensity, identified with the average intensity of solar light: 136 mW/cm 2 , and 0 I I I ∆ = − . The computations start from low light's intensity at 1 mW/cm 2 , and move to the critical intensity 0 I , and beyond. In column 5, the numbers that are the real part of complex numbers are highlighted with an asterisk (*). Complex numbers arise because of the singularity at 0 I reported in Figure 1(a). The presence of the singularity at 0 I signifies that for

Results
We compare Figure 1(a) with Figure 3 in [9], reproduced in Figure 1(b). Figure 3 of [9], and in Figure 1(b), consist of 9 data points obtained after averaging over many trials in order to minimize the errors.

Müller's results in both
The points in Figure 3 of [9], (reproduced in our Figure 1(b)) do not exhibit any singularity as the one appearing at 0 I I = in Figure 1(a). Instead, the points in Figure 3 of [9], follow the continuous fitting line shown in both Figure 1(a) and Figure 1(b). With an offset of +0.9, this fitting line overlaps the data points we provide from our computations in Figure 1 is required to justify the trends of the experimental data points reported by Müller [9]. Indeed, with logharitmic functions but without the singularity, we would expect the points in Figure 1(a) and Figure 1 Figure 1(a). The two functions are the same, except that an offset of +0.9 is applied to the fitting line in Figure 1(a). The presence of the singularity at 0 I signifies that for is a complex number, whose real part is reported in Figure 1(a) and  Figure 3 of [9], and in our Figure 1(b), we observe a continuous exponential line without singularity for . The absence of a G. Scarel singularity can be explained by assuming that in Müller's experiment the critical intensity 0-Mueller I was much less than 136 mW/cm 2 so that for the 7 points on the right of the graph in Figure 3 of [9], and in our Figure 1(b), the condition 0 I I ∆ < is verified. We thus conclude that 2 0-Mueller 136 mW cm I I <  . The fact that the data points in Müller's experiment require an exponential fitting function and those in our numerical experiment a logharitmic function, defined in an interval containing a singular point at 0 I I = , might be adopted to explain the offset of +0.9 existing between the data in Figure 1(b) and those in Figure  1(a). Figure 1(c) illustrates some of the consequences of the change in the capacitance C of the axon's membrane occurring when the vertebrate's eye interacts with light of intensity I. Specifically, in Figure 1(c) we report [5] is the critical capacitance change, and the abscissa is the same as in Figure 1(a). Thus, we are describing the effects of the changes in C in the light intensity conditions examined in Figure 1(a). To offer a quantitative description of these changes, in Table 1 we report the values of C, for the same intensities I considered in Figure 1(a). No singularity is observed with the capacitance changes because 0 C ∆ has a fixed non-zero value. As a consequence, the trend in 0 10 is linear, as illustrated in Figure 1(c). Table 1 also shows that, in the light intensity range between 1 mW/cm 2 to 136 mW/cm 2 , the values of the capacitance vary between 15 fF to ~2 pF. The observed trends suggest that the change of the photoreceptors from rods, at low light's intensity, to cones, in bright light, does not introduce singularities in the changes of magnitude of the capacitance.

Discussion
Our work contributes in solving the dispute between the quantum [6] [7] and the non-totally quantum formulation [8] [9] of the phenomenon of vision.
Originally, the dispute evolved between the early 1940s and 1950s with Hecht et al. [ [9], and in our Figure 1(b). Müller, however, was unable to explain the trend of his experimental data, which appear to follow an exponential rather than a logarithmic function. Probably due to this mathematical mismatch, Müller's conclusions were not granted further consideration in the scientific community. Our results in Figure 1(a), on the other hand, reproduce Müller's experimental data and, in addition, are obtained using logarithmic functions. To construct the data reported in Figure 1 Equation (1), where C is the capacitance of the membrane in the axon, and V is the depolarization voltage of ~15 mV needed to trigger the action potential. Equation (1) sheds light to the adaptation mechanism implied in Müller's experimental data and in our Figure 1(a), suggesting that the capacitance C of the axon's membrane is the only parameter that can change within the axon to enable the eyes to adjust to the energy Pτ transferred from light. The adjustment needs to occur such as to generate the depolarization voltage ~15 mV V necessary to trigger the action potentials: this is the criterion that enabled us to derive the values of C in Table 1. Changing the magnitude of C in the axon requires sufficient time (few seconds to minutes) to enable the appropriate type and amount of ions to actually penetrate the axon's membrane. This fact elucidates why our eyes require some time to adjust while passing from vision in high to low light's intensity. More specifically, low light's intensity needs very small capacitances, large 0 C C ∆ ratios (see Table 1), and possibly large time intervals to allow the membrane to reach such small capacitances. Other phenomena tes-G. Scarel tify that light intensity plays a significant role in adaptation mechanisms. For example, the unicellular green alga Chlamydomonas can swim toward or away from light. In this process, called phototaxis, this alga produces a photoreceptor current when illuminated by light. This photoreceptor current was found to increase with the light's intensity and with intensity's rate of increase [16].
The validity of Equation (1) in explaining vision's adaptation to different light's intensities has further implications, including the fact that it is a proof that the magnitude of the energy Pτ is the amount of energy transferred and conserved during the interaction between light and matter. The quantity Pτ should therefore be considered in optogenetics. This research-and therapy-technique is usually performed using blue or other visible light [11]. Adopting Pτ in optogenetics seems to suggest that the same amount of energy transferred by, e.g. , which is within the frequency range of amateur radio transmitters). It is noticeable that the power with radio waves is 6 orders of magnitude lower than that with blue light. Thus, it is reasonable to ask whether the same results of optogenetics with blue or other visible light could be achieved with radio waves with the proper power. In other words, we might ask 1) whether optogenetics is related to the specific frequency of the EM waves used, or to the energy that these EM waves transfer, and 2) what is the impact of the wavelength of the EM waves on the size of the optogenetics samples, or on the ability of focusing the EM wave on the target. Pursuing the answer to these questions is the objective of research in progress.

Conclusion
The hypothesis that Pτ is the energy conserved in the mechanism of vision in vertebrates enables us to support the non-totally quantum formulation of the mechanism of vision, and thus, that the existence of a minimum threshold of about 8 photons is not sufficient to justify the start of the vision process in vertebrate's eyes. In addition, our results unveil the adaptation mechanism required by the vertebrate's eyes to adjust to different levels of light's intensity. This me-