Coding accuracy on the psychophysical scale

Sensory neurons are often reported to adjust their coding accuracy to the stimulus statistics. The observed match is not always perfect and the maximal accuracy does not align with the most frequent stimuli. As an alternative to a physiological explanation we show that the match critically depends on the chosen stimulus measurement scale. More generally, we argue that if we measure the stimulus intensity on the scale which is proportional to the perception intensity, an improved adjustment in the coding accuracy is revealed. The unique feature of stimulus units based on the psychophysical scale is that the coding accuracy can be meaningfully compared for different stimuli intensities, unlike in the standard case of a metric scale.


∆ =
. I I const (1) As suggested later by Fechner 29 , Weber's law effectively sets the scale for the perceived stimulus intensity since ΔI/I is proportional to Δψ. By integrating Eq. (1) we obtain the well known Fechner's law, stating that the perceived intensity varies as 28 . 0 where k is a proportionality factor and I 0 some reference value. Subsequent investigations found that Eq. (1) holds neither generally nor exactly 24,30 across different sensory modalities. In particular, Weber's factor for human sound intensity discrimination was found to satisfy 26  Here I is the basal sound intensity in W/m 2 , ΔI is the minimum perceptible difference, S ∞ is the value ΔI/I approaches at high intensities, S 0 > S ∞ is the value of ΔI/I at the threshold of hearing and r is a parameter, approximately r = 1/2. Weber's factor in Eq. (3) is no longer constant, but decreases rapidly to a plateau with increasing intensity I. Since the sound intensity and the sound pressure are related by the acoustic impedance =  Z 400 N.s.m −3 as 22 2 the following differential equation follows from Eq.
0 0 provided that the derivative dψ/dp is a good approximation to Δψ/Δp. The solution to Eq. (5) is where = .  c 94 3 so that p = 0 Pa yields ψ = 0 for convenience. Eqution (7) determines Riesz's scale (in arbitrary units) of sound pressure values, correcting the inadequate Fechner's law in Eq. (2) for small sound intensities (pressures). In other words, the value of ψ can be used to measure the sound intensity on the scale which is linearly related to the perception intensity. The standard sound pressure level scale L (given in dB SPL) is essentially equivalent to Fechner's law, since due to Eq. (4) it holds 22 10 0 The Eqs (7) and (8) are approximately proportional to each other for sufficiently high pressure levels ( Fig. 1).

Results
The coding accuracy as a function of the stimulus intensity is significantly affected by the choice of the measurement scale 23 . The question is whether the coding accuracy adaptation to the stimulus distribution (as observed, e.g., in the experiments 5,6,9 ), is preserved under the change of stimulus units. The Fisher information I F (P) as a function of the sound pressure, and the Fisher information I F (ψ ) for the sound intensity measured on Riesz's scale from Eq. (7), are related as Similarly, one may additionally use Eq. (8) to relate, e.g., I F (ψ) and I F (L). The transformation rule in Eq. (9) is well known and can be derived directly from the definition of the Fisher information by using the chain rule for derivatives 15 . Similarly, the stimulus probability density function f(·) satisfies 31 Therefore it follows that any visual alignment between the values of the coding accuracy and the stimulus distribution depends crucially on the choice of units. Even though the square root of the Fisher information transforms analogously to Eq. (10), the potential match between the peaks of I F and f is not preserved under the stimulus scale change because I F and f are often related non-linearly. In fact, it can be shown rigorously that also the global match between the profiles of I F and f is not preserved under the stimulus scale change 32 , unless the stimulus probability density function is exactly proportional to the square root of the Fisher information (known as the Jeffreys prior 33 ).
We illustrate how a specific choice of the stimulus units improves the experimentally observed adaptation of the coding accuracy to the stimulus distribution. We argue that the stimulus scale proportional to the actual perception intensity (the psychophysical scale) is the natural reference frame under which the coding accuracy should be evaluated.
Neurons in the auditory system are reported to adjust their rate-intensity functions in order to improve coding accuracy over high-probability stimulus regions 5,9,10 . The match is not perfect for low sound intensities and a positive bias of maximal coding accuracy towards higher intensities is reported. For example, in the experiment of Watkins and Barbour 9 , the sound level distribution was set to be uniform over -15 dB SPL to 105 dB SPL, with an added 20 dB-wide plateau of high-probability stimulus region ( Fig. 2A, filled area). At every 100 ms during the experiment a new sample was drawn from the distribution to set the amplitude of a pure tone, with its frequency matching the characteristic frequency of the studied neuron (primary auditory cortex of marmoset monkey). The dynamic rate-level function was measured and the coding accuracy (the Fisher information) was calculated ( Fig. 2A, solid line), see Watkins and Barbour 9 for more details. The coding accuracy adaptation was determined for four different positions of the plateau, centered at 5, 25, 45 and 65 dB SPL respectively ( Fig. 2A-D). The peak coding accuracy does not align with frequently occurring low sound intensities ( Fig. 2A).
The same experimental data evaluated on Riesz's scale yield far better alignment of coding accuracy with stimuli statistics, especially for low intensities (Fig. 2E). On the other hand, the existing match for high levels (Fig. 2C,D) is preserved (Fig. 2G,H) due to the similarity of both scales for high intensities (Fig. 1). The match between the stimulus statistics and the coding accuracy can be quantified by the ratio of the maximal Fisher information in the high-probability region to the global maximum of the Fisher information. For the four examined cases of the plateau centered at (5, 25, 45, 65) dB SPL we obtain the following values of this ratio: (0.46, 0.96, 1, 1) on the pressure level scale, and (1, 1, 1, 1) on Riesz's scale. Note that the non-uniform shape of the high-probability regions results from the transformation rule for the probability density function.

Discussion
The described adaptation of neural coding precision to the local stimulus distribution results in a more efficient representation of the environment 5,9 . However, the investigation of coding strategy should also take the actual perception intensity into the account 11 . In all likelihood, coding precision expressed by employing the psychophysical scale (such as Riesz's scale) is more useful and natural than when evaluated in the standard metric system (such as dB SPL). The reasoning is that Riesz's scale is linear in the true perception intensity as described in the Methods section. Consequently, the smallest noticeable increment in perception Δψ is proportional to a fixed value on Riesz's scale, and this value is constant for all stimulus intensities. Hence the unique feature of a stimulus unit based on the psychophysical scale is that the coding accuracy evaluated in such units can be meaningfully compared for different stimuli intensities -unlike the metric scale case. Even if coding precision varies with the stimulus intensity on the metric scale substantially, these variations might be immaterial provided that the actual difference in sensation falls within the smallest noticeable increment.
Note that if Weber's law was valid for the sound intensity perception, the dB SPL scale would correspond to the exact psychophysical scale. From this point of view the shifted-logarithm in Eq. (7) represents a seemingly  negligible correction. We have shown, however, that the difference between Riesz's and sound pressure level scales affects the coding accuracy adjustment substantially. Ries'z correction, ψ ∝ log(const. + I), to the purely logarithmic Fechner's law in Eq. (2) has a long history and is more fundamental and general, going beyond the case of the sound intensity perception. See [34][35][36] for a detailed account. For example, the equation for Riesz's scale as a function of the sound pressure in Eq. (7) is formally identical to the psychophysical mel scale 37 , which describes the perception intensity for sound frequency. Both the mel and Riesz's psychophysical scales thus follow Weber's law for large stimuli values only.
Our message, however, reaches beyond the topic of psychophysical scales and auditory neuroscience. We argue that the coding accuracy is generally a relative quantity, with respect to chosen units, a fact whose consequences seem to have been neglected in the experimental research. The expected matching of stimulus statistics with the coding accuracy is thus not absolute and does not hold in different unit systems. The coding accuracy reflects the spread of estimated stimulus values, which is affected not only by the stochastic nature of neural responses but also by the arbitrarily chosen unit system for stimulus quantification. In addition, we believe that coding accuracy should generally be evaluated on the scale which is linearly proportional to the internal representation of the stimulus, i.e., proportional to the actual perception intensity.
Finally, it is worth noting that different ways to asses the neural coding efficiency were developed over the decades. A substantial part of the literature employs Shannon's measure of information 38 to determine the absolute scale on neuronal performance 39 . By treating the neuronal system as an information channel, and by maximizing the mutual information between stimuli and responses, one obtains the optimal stimulus distribution, as for example in [40][41][42][43][44] . Under the assumption of vanishing response variability, the optimal stimulus distribution is proportional to I F [45][46][47][48][49][50][51] , which is known to be invariant under coordinate transformations 33 . Heuristically, one may view this result as providing support for the idea of high coding precision matching high probability stimulus regions 40 . Unlike the local method of Fisher information described in this paper, however, the information theory determines the complete (global) form of the stimulus distribution.