Impact of electrode position on the dynamic range of a human auditory nerve fiber

Objective. Electrodes of a cochlear implant generate spikes in auditory nerve fibers (ANFs). While the insertion depth of each of the electrodes is linked to a frequency section of the acoustic signal, the amplitude of the stimulating pulses controls the loudness of the related frequency band. However, in comparison to acoustic stimulation the dynamic range (DR) of an electrically stimulated ANF is quite small. Approach. The DR of an electrically stimulated ANF is defined as the interval of stimulus amplitudes that causes firing probabilities between 10% and 90%. A compartment model that includes sodium ion current fluctuations as the stochastic key component for spiking was evaluated for different electrode placements and fiber diameters. Main results. The DR is reversely related to ANF diameter. An increased DR is expected to improve the quality of auditory perception for CI users. Electrodes are often placed as close to the center axis of the cochlea as possible. The analysis of the simulated auditory nerve firing showed that this placement is disadvantageous for the DR of a selected ANF. Significance. Five times larger DRs are expected for electrodes close to the terminal of the dendrite or at mid-dendritic placement as opposed to electrodes close to the modiolus.


Introduction
The cochlea operates as a frequency analyzer where, primarily as a consequence of inner ear mechanics, every inner hair cell mimics a band pass filtered microphone by transforming a small frequency band of the acoustical signal into an analog intracellular voltage Lutter 1997, Rattay et al 1998). A single synaptic connection between an inner hair cell and the distal end of an auditory nerve fiber (ANF) converts this analog signal into a digital signal as a train of action potentials (AP) (spikes). According to this principle the spiking rate (spikes/s) of any afferent auditory nerve fiber increases with the loudness of the corresponding frequency.
The physiological task of the main part of the auditory nerve is a data bus that transfers all the APs elicited by inner hair cells to the next neural processing center, the cochlear nucleus. In more anatomical detail, each element of the data bus is a so-called type-1 spiral ganglion cell consisting of a dendrite, a soma, and an axon, where the diameters and the degree of myelination changes along the cell . These details have less impact for synaptic stimulation in the healthy ear in comparison with electrical stimulation where (a) the spike initiation site depends on electrode position, (b) thin degenerated dendrites may not be excited, and (c) polarity sensitivity depends on cochlear status (Shepherd and Javel 1997, Rattay et al 2001a, Undurraga et al 2013, Heshmat et al 2021.
When generated from a cochlear implant (CI), the spiking patterns of the auditory nerve show severe deficits versus natural hearing (Hochmair-Desoyer et al 1984, Sachs et al 1989, Ghitza 1993, Rattay and Lutter 1997. Especially for difficult neural decoding tasks such as speech understanding in noisy environment, every single ANF contributes with its own spiking rate and its temporal changes following the loudness variations in the corresponding frequency region of the acoustic source signal. Spiking probability as a function of stimulus intensity is the key-control element in the inputoutput relation in functional electrical nerve stimulation. The range of intensities where the spiking probability of an electrically stimulated ANF increases Relationship between stimulus intensity, firing efficiency, dynamic range, and relative spread during electrical stimulation with trains of 100 pulses. An integrated Gaussian curve is used to fit the counted spike numbers marked by x. The slope of this firing efficiency curve at threshold can be used to estimate the dynamic range. The spread (magenta) is related to the dynamic range (cyan) by the formula (normalized) dynamic range = 2.56 x RS. (B) During acoustical stimulation, the ratio of stimulus intensities for high (90% of maximum) and low (10% of maximum) firing rates of two typical feline ANFs is 14 (23 dB, red line) and 178 (45 dB, blue line), respectively. For these two examples (redrawn recorded data from figure 4 of Sachs et al 1989), the dynamic ranges of the acoustical stimulation are 2 and 3 orders of magnitude higher than during electrical stimulation of A. SPL: sound pressure level relative to 20 µPa. from 10% to 90% is defined as its dynamic range (DR) and indicates a fiber's individual loudness contribution during CI stimulation (Shepherd and Javel 1997). The firing probability, also called firing efficiency, can be found experimentally as the ratio (number of spikes)/(number of stimulating pulses). In the schematic diagram of figure 1(A) the spiking probability was extracted from five intensitiesá 100 pulses marked by x. The threshold (700 µA) is defined by the 50% firing efficiency and DR was 128 µA, that is 18% if normalized to threshold (128/700). However, the DR during acoustic stimulation (Sachs et al 1989, Rattay andLutter 1997) is about 2-3 orders of magnitude larger in comparison with this example that is typical for electrical stimulation of feline ANFs (Miller et al 1999). In comparison with electrical stimulation recorded acoustic rate-level functions are again of sigmoid shape but a similarity in the slopes requires a logarithmic scaling of the intensity axis ( figure 1(B)). Verveen (1960) introduced relative spread (RS) as another measure for the stochastic nature of an electrically stimulated nerve fiber. He plotted the firing probability in a similar way as shown by the markers (x) in figure 1(A) and noted that the counted spike numbers approximated that of an integrated Gaussian curve which is the cumulative distribution function of N(µ,σ). RS is the standard deviation σ of that Gaussian function divided by threshold µ. Comparison of both measures shows a linear relationship: DR (normalized by threshold) is 2.56 times RS. The factor 2.56 results from evaluating the integrated Gaussian curve (figure 1(A), Tanzer 2021).
From peripheral nerve fiber stimulation experiments Verveen estimated log RS = −1.5 −0.8 * log d (diameter d in µm), which specifies how RS decreases with diameter (Verveen 1962). The dependence of RS on fiber diameter is an important feature for electrical hearing with CI as the dendrite diameter is about half of the axon diameter in the normal human ear , but the dendrite diameter is often reduced in cases of severe hearing deficits (Heshmat et al 2020). However, the RS dependence on fiber diameter, as reported for long homogenous axons, is disturbed by irregularities, such as a short dendrite, an unmyelinated soma region or the curvature of the ANF pathway (Nadol 1990, Rattay et al 2001a, 2013, Potrusil et al 2012, 2020, Rask-Andersen et al 2012. Moreover, the spike initiation site depends on the position of the stimulating electrodeas well as the polarity and amplitude of the stimulus (Bai et al 2019, Shepherd and Javel 1997, Rattay et al 2001a, Undurraga et al 2013. Thick fibers are easy to stimulate (Blair andErlanger 1933, Rattay 1990), which favors the axon versus the dendrite, but on the other hand the threshold decreases with electrode-fiber distance and the stimulating electrode can be placed closer to the thin dendrite than to the axon.
These contradictory circumstances were the driving forces to analyze the impact of electrode placement on the DR of healthy and degenerated human ANFs in a pilot study with a rather simple cable model. A noise current was added in every ANF compartment with an active membrane (nodes of Ranvier, non-myelinated terminal, somatic region). Each segment's simulated noise current is proportional to the square root of the number of its sodium channels (Sigworth 1980, Rattay 2000. As proof of concept a straight ANF with feline morphometry was stimulated with a monopolar electrode above the center of the dendrite and the calculated RS values were compared with the reported data of Verveen as well as with feline ANF recordings (Miller et al 1999). In the next step the same ion channel dynamics were applied to the thicker human ANF which, in comparison with cat, has a longer dendrite and a non-myelinated somatic region. In addition, the ANF pathway was more realistic in curvature.
RS as a proportional replacement of DR was evaluated for possible electrode positions in a human cochlea for normal and degenerated dendrites; mono-and biphasic pulses of both polarities were applied. Finally, we show that DR increases with the electrode distance to the central axis of the cochlea.

Methods
A type 1 spiral ganglion cell, which connects an inner hair cell with a neuron in the cochlear nucleus, consists of several subunits as schematically shown in figure 2(A) with parameters listed in table 1. Each of these subunits (terminal, internode, node of Ranvier, etc) is simulated as a single compartment with exception of four compartments for the non-myelinated presomatic segment. As seen in figure 2(A), the cell consists of segments with active membranes with high (marked red) and low ion channel density (pink). Additionally, there are myelinated internodes (gray) where the membrane is assumed to be passive, which means no voltage sensitive ion channel gating mechanisms are considered. The high ion channel density, here modeled as ten-fold Hodgkin Huxley membrane conductance, is needed for signal amplification along the neural pathway (Rattay 2000, Rattay andDanner 2014). This sodium channel density is comparable to nodes of Ranvier in axons of the mammalian peripheral nerve system Aberham 1993, Rattay et al 2002). Beside the peripheral terminal and the nodes of Ranvier, the high sodium concentration is applied in the pre-and post-somatic compartment.
Applying Kirchhoff 's law (the sum of all currents is zero) to the nth compartment of the electric network of figure 2 where the first two terms are capacitance current and ionic current across the membrane and the next two terms describe the intracellular current flow to the left and right compartment, respectively. Introducing and V rest are the intracellular, extracellular and resting potential, respectively) leads to the following system of differential equations (Rattay 1999, 2000, Rattay et al 2002. For the first (last) compartment equations (1) and (2) have a reduced form because of less neighbors. The membrane surface A n of every compartment has to be calculated to find C m,n = A n .c m,n (c m,n is the specific membrane capacitance; note that the capacitance of N layers of membranes is proportional to 1/N). In order to calculate the extracellular potentials V e , needed in equation (2), we assumed an infinite homogeneous extracellular medium. Ignoring capacitance effects of tissue, the extracellular potential was calculated as where r is the center to center distance between a compartment of interest and a spherical electrode. ρ e = 300 Ohm cm was assumed to be the mean resistivity of the extracellular medium (Rattay et al 2001a).
The ion membrane current is governed by the gating mechanisms of specific voltage sensitive ion channels. It consists of two components where i ion is the ionic membrane current density and I noise,n represents ion channel current fluctuations in active compartments. The effective noise current measured in µA is assumed to be proportional to the square root of the number of sodium channels within a compartment where GAUSS is a Gaussian noise current term (mean = 0, σ = 1) that changes its value every 2.5 µs, knoise = 0.00125 µA mS −1/2 is a factor common to all compartments, A n denotes membrane area in cm 2 , and g Na is the maximum sodium conductance (g Na = 1200 mS cm −2 for the regions simulated with 10-fold channel density, g Na = 120 mS cm −2 for the human soma, g Na = 0 for all myelinated internodes and cat's soma (Rattay 2000).
In compartments with passive membranes (internodes) the term I ion of equations (1) and (2) is a current with constant conductance I ion = g m A n V n /N, where membrane conductance g m is 1 mS cm −2 and the number of insulating layers of cell membranes N is 40 and 80 for dendrite and axon, respectively (Rattay et al 2001a).
The ionic currents of active membranes were simulated with original Hodgkin Huxley kinetics (Hodgkin and Huxley 1952) at a temperature of 28.9 • C (Motz and Rattay 1986) that caused a spike duration as observed in feline ANFs at 37 • C. The system of ordinary differential equations was computed in C++ using the backward Euler method with time steps of 2.5 µs.

Results
The first stimulations are calculated with monophasic pulses in order to demonstrate which regions of an ANF respond with depolarization or hyperpolarization depending on the polarity of the stimulus.

Dynamic range of a feline ANF
In our simple model approach a single parameter knoise for Gaussian N(0,1) controls all stochastic effects. In a proof of concept knoise was set to 0.001 25 (with unit µA mS −1/2 ). The ANF consisted of a dendrite (diameter d1 = 1 µm), three node-internode combinations with an internode length of 150 µm (Liberman and Oliver 1984), a myelinated spherical soma (d = 15 µm, covered with13 sheets of membrane; Rattay et al 2013) and a 4 mm long axon (d2 = 2 µm, internode length = 300 µm). The electrode center was 300 µm above the center of the dendrite and the axis of the cell was a straight line (figure 3, left). For a period of 0.5 ms before stimulus onset the enlarged plot of membrane voltage accurately demonstrated the decrease of membrane voltage fluctuations with increase of diameter as predicted by theory (figure 3, bottom). In this example where d2 = 2d1, the surface of the axonal node is twice the size of the dendrite node but its noise current increases only by a factor of √ 2. Indeed, the Vm range decreased from 2.4 to 1.7 which is exactly the theoretical ratio √ 2/1. Cathodic pulses of 100 µs duration resulted in a threshold current of 99.7 µA, RS = 5.05% and DR = 12.86 µA (12.9% normalized to threshold). This RS is between Verveens 3% and data of Miller et al (1999) where RS values between 2% and 20% (mean 6.6%) based on single ANF recordings were reported. Consequently, we used the same knoise = 0.001 25 for the human case where no single fiber recordings are possible because of ethical reasons.
Changing just the polarity and applying a positive pulse needs about double the electrode current to elicit an AP which is initiated in the axon as both the complete dendrite and the soma are hyperpolarized during pulse application (figure 3). In comparison to cathodic stimulation the RS is reduced from about 5% to 3%, which is a consequence of spike initiation in the thicker axon.

Dynamic range for electrode positions close to an ANF in a human cochlea
CIs are commonly inserted at the round window into scala tympani, the lower cavity of the cochlea. The electrodes are positioned either close to the outer wall or central close to the modiolus but can also be placed in between (figure 4). In comparison with the feline ANF, in human the dendrite is longer and a bit thicker, there is a non-myelinated pre-somatic region, the soma is larger and non-myelinated, and the axon is thicker and longer (table 1, Rattay et al 2013).
In contrast to the straight cell axis of the feline example a curved two-dimensional ANF as introduced by earlier work (Rattay et al 2001a) was evaluated to study more realistic electrode distances Dendrite diameters are often reduced in persons with hearing deficits. A recent study reported a peak in the dendritic (inner) diameter histogram at d1 = 0.4 µm and diameters decreased to 0.3 µm were recorded in a profound deaf ear (Heshmat et al 2020). Table 2 shows the threshold and RS values for monophasic 100 µs pulses for a normal ANF with parameters of table 1 and the corresponding data when just the dendrite diameter is reduced to half of the standard value. As expected, RS increases for all positions and both polarities if the diameter is reduced (table 2, degenerated ANF).
For cathodic pulses RS is largest for the terminal electrode and the values decrease with the electrode distance from the terminal. For anodic stimulation the excitation process is more complicated and the spike initiation site is commonly further away from the electrode as predicted by the activating function concept (Rattay 1986(Rattay , 2013. The mid-dendritic electrode has the largest RS for an anodic pulse, followed by a much smaller RS for the terminal position. The large anodic RS for the mid-dendritic position is related to the more complicated excitation process which results in spikes that initiate both in the dendrite and in the axon, see figure 5. The characteristic dependence of the DR (DR = 2.56 times RS) on diameter d1, electrode position and polarity is also shown in figure 6. According to the reversed recruitment order thicker fibers have lower thresholds (Blair andErlanger 1933, Rattay 1990) which makes it more likely for spikes to be elicited in the axon. On the other hand, an electrode can be placed closer to the dendrite resulting in a competition between dendrite and axon for the site of spike initiation. Accordingly, the low threshold value for the last electrode position in table 2 is essentially lower than at the dendrite. As mentioned before, the shape of the scala tympani may hinder in many cases to set an electrode very close to an ANF axon (compare figure 4).
The ratio of axon diameter d2 and dendrite diameter d1 has a large impact on the decision whether the spike is generated in the dendrite or in the axon. While for d1 > 0.6 µm the terminal electrode position preferentially initiates dendritic spikes (figures 6(A) and (B)), a change to the axon is shown in figure 6(C).
In the next example, we studied the spike initiation site for stimulation with biphasic pulses, 50 µs per phase, for the mid-dendritic electrode position. Two cases are compared concerning the polarity of the first phase. Cathodic first stimulation needs less current versus anodic first and for both polarities spikes are elicited in the terminal P0 by the anodic phase ( figure 7). Surprisingly, in both shown cases, the anodic phase is more effective as it is the phase which elicits the spike. However, it is a combination of the ANF pathway relative to the electrode position and the two phases which lets P0 win the threshold  In case E two spikes are generated, one in dendrite one in the axon; however, when the dendritic spike arrives at the axon it is stopped by collision and generates a weak (subthreshold) second excitation. The large jitter at threshold (A), (C) is tremendously reduced already at 1.2 times threshold (B), (E). The green rectangle indicates a 240 µs longer delay for spikes generated in the dendrite. 10 runs per case, electrode in mid-dendritic position, d1 = 1.35 µm.
competition. Several other compartments, marked by dashed lines in figure 7, are candidates for the spike initiation site as they are close to threshold voltage during the stimulation period. A small increase of the electrode current would make one of them the winner during the first phase. For cathodic first stimulation, P1 reached the highest membrane voltage Vm in the first phase, followed by P2 and P4. The inset on the top of figure 7 shows Vm of P0 and P1 at the same baseline (without the vertical shift for each compartment) and demonstrates that Vm at P1 (dashed green line) essentially exceeds Vm at P0 during the stimulation time. A small variation of the noise signal can initiate the spike in P1. Note that the second phase was able to stop spike excitation in P1, but for comparison a quite similar condition enables to generate the P0 spike in the anodic first phase (figure 7(B)) in spite of the repolarizing second phase. For the anodic first pulse the axon compartment C2 is the second candidate for the spike initiation site.
Thresholds and RS are listed in table 3 for both polarities of biphasic pulses. All electrode positions with exception of the soma have lower thresholds for cathodic first pulses. In comparison with table 2, the thresholds (again with exception of the soma) are essentially higher for the short 50 µs phase. In contrast to monophasic pulses the RS differences are similar for terminal and mid-dendrite position. For the biphasic case, terminal and mid-dendritic position have more than five times higher RS values as at the axon.
As seen in table 3 electrodes in mid dendritic position have a rather low threshold and a high DR value which favors this electrode placing, both for normal and reduced dendrite diameter. Note that lower thresholds are important because there will be less spread of excitation to other frequency regions. Moving the electrode towards the soma let increase the threshold before it is reduced again at the axon. However, the DR for the soma and axon position is quite low. Moreover, the electrode can be placed close to the axon only in the basal turn of the cochlea. On the other hand, the mid dendritic position is not working for ANFs which lost their dendrites. A new electrode system with two distinctly activated contacts per frequency region could be a solution for a placement optimized for the individual neural status of a CI user.

Loudness discrimination
During evolution, the mammalian auditory system refined three key methods to reach its remarkable sensitivity. Aside from the well-known mechanical frequency separation along the cochlea (tonotopical principle) and the temporal coding (phase locking of ANFs spiking times), there is a mechanism related to stochastic resonance in order to detect weak signals (subthreshold signal + noise make the signal detectable, similar to the dithering effect in electrical engineering). At the threshold of hearing the contribution of sound pressure vibrations to the motion of inner hair cell stereocilia is one order of magnitude below that of stochastic thermal forces of water molecules causing Brownian motion of about 2 nm at the stereocilia tip (Svrcek-Seiler et al 1998). ANFs with high spontaneous rates, which receive their spontaneity from a high number of calcium channels in the presynaptic membrane of low threshold ribbon synapses (Wichmann and Moser 2015), enable the Figure 6. Spiking efficiency with color coded spike initiation site for an electrode at the terminal (A)-(C), at mid-dendrite (D), (E), and soma position (F). Cathodic stimulation: Even for a thin dendrite of diameter d1 = 0.7 µm almost all spikes are initiated in the dendrite (A) but reduction to d1 = 0.6 µm favors spike generation in the axon, especially for stronger pulses. Further reduction of d1 to 0.5 µm initiate most spikes in the axon (C). A shift of the electrode from terminal to mid-dendrite position increases axonal spike initiation ((A) vs (D)). Note the decreased dynamic range (marked as green lines) for mid-dendritic (D) versus terminal (A) electrode position and the increase of dynamic range for reducing diameter (A)-(C). For anodic stimulation the dynamic range of mid-dendritic electrode position is larger than the cathodic ((E) vs (D)) as well as for anodic soma position ((E) vs (F)). Stimulation with 100 µs pulses, 500 pulses for each bin, intensities relative to threshold. perception of quite weak sound. The perception of such a weak sound and the estimation of its intensity are not coded by the spiking rate. Instead, the signals at the threshold of hearing are perceived by detecting temporal regularities hidden in the stochastic spiking patterns of ANFs with high spontaneous rates which can be understood by analyzing the interspikehistograms of various nerve fibers that are sensitive to specific frequencies (Rattay et al 1998a). All three key mechanisms work in concert to discriminate loudness and frequency of sound in a huge range of 120 dB, although the spiking rate variability of a single ANFs is restricted to a much smaller interval of about 50 dB which is even smaller if the increase is reduced according the 10%-90% interval applied for DR in figure 1(B).
The psychoacoustic loudness estimations during acoustical and electrical stimulation have to work quite differently. Note that electrical activation of voltage sensitive calcium channels in the pre-synaptic membrane causes a late 'D' spike, recorded ∼2.2 ms after the stimulating pulse in healthy cats (figure 3 in Javel and Shepherd 2000), which evidently includes synaptic transmission time, but it only happens in functioning hair cells. Moreover, the pulses of most CI strategies are not synchronized (phase locked) with the acoustic signal but appear at a constant frequency, defined by the cycle time of the CI, that hinders phase locking with the acoustic source signal. Missing two of the three coding principles, phase locking and support by spontaneous firing, hampers to decode the neural pattern generated by CIs. In Figure 7. Spike initiation with biphasic pulses about 10% above threshold, 50 µs per phase, from an electrode in mid-dendritic position. Spike initiation is here always in the terminal P0 during the anodic phase. Several other compartments, marked by dashed lines, are candidates for spike initiation sites if stimulus intensity is increased. Insert at top of (A) contrasts size of membrane voltage Vm at P0 against P1 during pulse application, demonstrating a high Vm of P1 at the end of the first pulse which fails to generate a spike in the way as it was possible by the magenta Vm curve of P0 in (B). addition, the DR of a single ANF is 2-3 orders of magnitude smaller for electrical vs acoustical stimulation.
For a single active electrode, the psychometric intensity discrimination is in the order of 5% of the DR corresponding to 20 steps for intensity discrimination. This, however, significantly varies between CI users (6-45 steps) (Nelson et al 1996) and depends on the number and status of all ANFs that are excitable by the electrode of interest. A typical DR of 10 dB expressed as the difference between threshold and maximum acceptable loudness level is a result of the number of spiking ANFs rather than the DR contribution of individual ANFs.
In our study, we simplified the analysis of loudness perception by studying the impact of the electrode position along the first part of a target ANF on its loudness contribution via spiking rate. According to the presented modelling study electrode placement has a large impact on the DR of a target ANF. A preferred electrode location for a larger DR is close to the dendrite but without being close to the soma or axon.
The model predicts a surprisingly large polarity dependence of the RS ratio for the electrode positions terminal and mid-dendrite. According to table 2 cathodic pulses cause 2.3 times larger RS for terminal versus mid-dendrite, whereas for anodic pulses middendrite has the largest RS with a ratio of 3.5 for these positions. Halving the dendrite diameter, a consequence of degeneration related with hearing deficits (Heshmat et al 2020), showed an RS increase close to the theoretical value of √ 2 and caused quite similar RS ratios for terminal and mid-dendrite electrode positions.
This extreme polarity dependence of monophasic pulses for terminal as well as mid-dendritic electrode positions is changed in direction of average values for biphasic stimulations (tables 2 and 3) with rather small differences concerning the leading pulse polarity. Biphasic pulses are commonly used in CIs. For such pulses, model evaluations predict for both terminal and mid-dendritic positions a five times larger DR compared to electrodes close to the soma or to the axon (table 3). This advantage of a large DR depends, however, on the degeneration status of the dendrite. Degeneration of an ANF means either the loss of its dendrite or a reduction in diameter (Nadol 1990, Wu et al 2019. In cochlear regions where the greater part of ANFs lost their whole dendrites it is not advantageous to place the electrode close to the outer wall (Heshmat et al 2020).

Impact of dendrite diameter
In a healthy ear ANFs with low spontaneous firing rates have smaller diameters (Gleich and Wilson 1993). This type has an essentially larger acoustical DR than the high spontaneous rate fibers (Winter et al 1990). Especially the low spontaneous ANFs lose the synaptic connection to the inner hair cells during extreme sound exposure and aging. Moreover, a change in their postsynaptic membranes is mentioned (Liberman and Liberman 2015) which may affect threshold and DR during electrical stimulation, especially for spikes elicited at the terminal. If thereby the number of sodium channels is reduced in the terminal, our model approach predicts an additional increase of electrical DR that is not explicitly investigated in the current study. In contrast to the mentioned reports, our previous work has shown a general decrease of dendrite diameters as a consequence of hearing loss in human ears (Heshmat et al 2020). The histograms of dendrite diameters of the control group developed a Gaussian like distribution with a minimum diameter d of 1 µm and a peak at d = 2 µm, whereas for profound and severe hearing loss, this peak is shifted to smaller d and the distributions became multimodal showing a second prominent peak at d = 0.4 µm (Heshmat et al 2020). While the thinnest ANFs (0.3 < d < 0.5 µm) need the smallest synaptic stimulus to initiate an AP , Heshmat et al 2020 their reversed recruitment order for electrical stimulation demonstrates high thresholds (Blair andErlanger 1933, Rattay 1990 ,  table 3). Consequently, even electrodes placed close to a thin dendrite often elicit the AP in the axon which is demonstrated by the red regions appearing in figure 6(C) for terminal electrode placement. Thus the trend of spike initiation in the axon hinders a continuous increase of DR at decreasing d for d < 0.7 µm (figures 6 (A)-(C)).

Intensity discrimination increases with electrode distance from modiolus
A study on intensity discrimination of 14 CI users compared two types of electrode placements (a) close to modiolus, using the Clarion HiFocus electrode array with an electrode positioning system in order to place the electrodes as close as possible to the axons of ANFs and (b) lateral, using a standard Clarion spiral electrode array where the electrodes are close to the terminals of ANFs (Kreft et al 2004, Wardrop et al 2005. Surprisingly large was the effect of the electrode placement on DR and intensity discrimination: subjects with the electrodes close to modiolus had an average of only nine discriminable intensity steps as compared to 23 steps in the other group. For both electrode systems comparable thresholds were reported but the DR of the lateral CI was essentially larger because of much higher maximum acceptable loudness levels in all subjects and for all tested stimulus pulse rates (comp. Figure 1 in Kreft et al 2004). These findings underline our hypothesis that an increased DR of single ANFs essentially supports the loudness discrimination of CI users.

Limitations of the model
The purpose of the study is a first approach of the DR of electrically stimulated human ANFs and its dependence on electrode positions in the scala tympani. A more accurate model should be based on the geometry and the tissue depending electrical conductances of the human cochlea as well as on morphometric details of ANFs including their threedimensional pathways (Rattay et al 2001b, Potrusil et al 2012, 2020, Rask-Andersen et al 2012, Kalkman et al 2014, Bai et al 2019, Heshmat et al 2020. Moreover, the impact of stimulation strategies, e.g. duration and shape of pulses, mono-, bi-and tripolar electrode configurations and the stimulation frequency, was not included but should be studied (Heshmat et al 2021, Recugnat et al 2021. Considering more details enables better predictions, e.g. (a) for possible electrode positions close to the axon and the soma which vary with the cochlear turns, (b) on the effect of reduced number of myelin layers or (c) on the impact of spatial fine structure of ANFs.
A fundamental problem is the modeling concept for the noisy currents to simulate post-stimulus and interval histograms as observed in electrically stimulated ANFs (Hartmann et al 1984, Hochmair-Desoyer et al 1984, Van den Honert and Stypulkowski 1984, Shepherd and Javel 1997. In our study, we used a computationally efficient approach (Rattay 2000) which assumes a noise current with constant rms (root mean square) amplitude that reflects the prominent impact of each single sodium channel in the cell membrane (Sigworth 1980). A single parameter, knoise, defines the stochasticity of every compartment via its maximum sodium ion conductance. A more realistic fit incorporates how current fluctuations depend on ion channel gating (Rubinstein 1995) or introduces correction terms for fiber diameters and membrane voltage (Badenhorst et al 2016). In comparison with the last two methods, our approach overestimates the current fluctuations in the resting state.
However, the chosen knoise parameter fit the RS of feline ANFs which are a bit smaller in diameter than the human ones. Reported RS mean values for cat monophasic, monopolar stimulation were 6.6% (Miller et al 1999), 9% (Shepherd and Javel 1997) and 5%-10% (Dynes 1996). Our model test for a straight feline ANF (d1 = 1 µm, d2 = 2 µm, electrode at mid-dendrite, figure 3) resulted in RS of 5% and 3% for cathodic and anodic pulses, respectively. The cathodic 3% RS value is in exact accordance with Verveen's formula for peripheral axons. Verveen did not observe polarity dependent differences in his data. According to table 2 (RS = 1.05%), the human axon diameter of 2.67 µm underestimates the RS = 1.44% of Verveen's formula. Comparing all these RS data, a judgement of our results point to a bit larger knoise value as our values are comparable with the lower border of the reported RS range. Tests showed a rather linear relationship between knoise and RS for small changes (Tanzer et al 2021) which makes it easy to calculate an adjustment factor to be in accordance with reported RS of a specific article. Moreover, RS was reported to be quite independent of pulse duration (Rubinstein 1995).

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.