Axonal stimulation affects the linear summation of single-point perception in three Argus II users

Purpose. Retinal implants use electrical stimulation to elicit perceived flashes of light (“phosphenes”). Single-electrode phosphene shape has been shown to vary systematically with stimulus parameters and the retinal location of the stimulating electrode, due to incidental activation of passing nerve fiber bundles. However, this knowledge has yet to be extended to paired-electrode stimulation. Methods. We retrospectively analyzed 3548 phosphene drawings made by three blind participants implanted with an Argus II Retinal Prosthesis. Phosphene shape (characterized by area, perimeter, major and minor axis length) and number of perceived phosphenes were averaged across trials and correlated with the corresponding single-electrode parameters. In addition, the number of phosphenes was correlated with stimulus amplitude and neuroanatomical parameters: electrode-retina and electrode-fovea distance as well as the electrode-electrode distance to (“between-axon”) and along axon bundles (“along-axon”). Statistical analyses were conducted using linear regression and partial correlation analysis. Results. Simple regression revealed that each paired-electrode shape descriptor could be predicted by the sum of the two corresponding single-electrode shape descriptors (p < .001). Multiple regression revealed that paired-electrode phosphene shape was primarily predicted by stimulus amplitude and electrode-fovea distance (p < .05). Interestingly, the number of elicited phosphenes tended to increase with between-axon distance (p < .05), but not with along-axon distance, in two out of three participants. Conclusions. The shape of phosphenes elicited by paired-electrode stimulation was well predicted by the shape of their corresponding single-electrode phosphenes, suggesting that two-point perception can be expressed as the linear summation of single-point perception. The notable impact of the between-axon distance on the perceived number of phosphenes provides further evidence in support of the axon map model for epiretinal stimulation. These findings contribute to the growing literature on phosphene perception and have important implications for the design of future retinal prostheses.


Introduction
Retinitis pigmentosa (RP) is an inherited degenerative disease of the eye that is estimated to affect one in 4,000 individuals worldwide (Hamel, 2006).Although recent advances in gene and stem cell therapies (e.g., Russell et al., 2017, da Cruz et al., 2018; for a recent review see McGregor, 2019) as well as retinal sheet transplants (e.g., Foik et al., 2018, Gasparini et al., 2019; for a recent commentary see Beyeler, 2019) are showing great promise as near-future treatments for early-stage RP, electronic retinal prostheses continue to be a pertinent option for later stages of the disease (Beyeler et al., 2017b).
Retinal prostheses typically acquire visual input via an external camera, which is then translated into electrical pulses sent to a microstimulator implanted in the eye (Weiland et al., 2016).The stimulator receives the information, decodes it, and stimulates the surviving retinal neurons with electrical current, thus evoking the perception of flashes of light ("phosphenes").The most widely adopted retinal implant thus far is the Argus II Retinal Prosthesis System (Vivani Medical, Inc; formerly Second Sight Medical Products, Inc.), which was the first retinal implant to obtain regulatory approval in the US and Europe, and has been implanted in roughly 500 individuals worldwide (Luo and da Cruz, 2016).
A series of papers demonstrated that phosphenes elicited by stimulating a single Argus II electrode have a distinctive shape that is relatively consistent over time (Nanduri et al., 2008;Luo et al., 2016;Beyeler et al., 2019b).Phosphene shape has been shown to depend strongly on the retinal location of the stimulating electrode, predominantly elongated along the trajectory of the underlying nerve fiber bundle (Rizzo et al., 2003;Beyeler et al., 2019b).In addition, phosphene appearance varies systematically with stimulus amplitude and frequency (Horsager et al., 2009;Nanduri et al., 2012;Sinclair et al., 2016) to the extent that a simple computational model can predict phosphene shape across a wide range of stimulus parameters (Granley and Beyeler, 2021).However, less is known about how phosphenes combine when multiple electrodes are stimulated.Early research suggested that repeated paired stimulation resulted in reproducible phosphenes as participants perceived "similar" phosphenes on 66% of trials (Rizzo et al., 2003).But more recent studies indicated that phosphenes tend to merge in nontrivial ways.For instance, Wilke et al. (2011b) highlighted the importance of electric crosstalk between electrodes in determining the response to simultaneous stimulation of multiple electrodes.Horsager et al. (2011) found that elicited percepts were affected by other stimulating electrodes (even after temporally staggering pulses to remove electric field interactions) and demonstrated a linear combination of threshold currents for simultaneous stimulation.Using a suprachoroidal prosthesis, Sinclair et al. (2016) found that bipolar electrode configurations produced percepts that were similar in appearance to the summation of the phosphenes that were elicited from the two individual electrodes using a monopolar configuration.Most recently, Yücel et al.
(2022) identified several factors that might limit the spatial resolution of prosthetic vision, which included retinal damage, electrode-retina distance, and the inadvertent stimulation of nerve fiber bundles.To avoid electric crosstalk and aid the perceptual merging of multi-electrode phosphenes, some researchers (Beauchamp et al., 2020;Oswalt et al., 2021;Christie et al., 2022) considered sequential stimulation paradigms.
However, sequential stimulation does not always lead to perceptually intelligible forms or objects; often participants are only able to trace an outline of the perceived shape, and their interpretation of the shape relies heavily on this basic outline (Christie et al., 2022).Therefore, understanding how multi-electrode stimulation can be leveraged to produce form vision (that is, a fundamental aspect of visual perception that enables humans to recognize spatial patterns and objects) remains an open challenge for the field of visual prosthetics.
Here we aim to study the consistency and predictability of the (presumably fundamental) building blocks of form vision: the percepts elicited by single-and pairedelectrode stimulation.While single-electrode stimulation is relatively well understood (Nanduri et al., 2008;Sinclair et al., 2016;Luo et al., 2016;Beyeler et al., 2019b;Granley and Beyeler, 2021), it remains to be demonstrated whether this knowledge can be extended to predict phosphene appearance elicited by paired-electrode stimulation.
Specifically, the axon map model (Beyeler et al., 2019b;Granley and Beyeler, 2021) predicts that the probability of seeing two phosphenes should increase with increasing distance between their axon bundles (as opposed to distance on the retinal surface), but no empirical studies have validated this hypothesis.Moreover, recent computational models of prosthetic vision assume linear summation of phosphenes (Spencer et al., 2019;de Ruyter van Steveninck et al., 2022), but this has yet to be demonstrated empirically.Therefore, to assess whether phosphenes sum linearly and to determine which neuroanatomical and stimulus parameters may be predictive of paired-phosphene appearance, we retrospectively analyzed an extensive psychophysical dataset collected with the help of three Argus II users.

Participants
This study involved three blind participants (one female and two male) with severe RP, ranging from 41 to 70 years in age at implantation (Table 1).Participants were chronically implanted with the Argus II Retinal Prosthesis System as part of an interventional feasibility trial (clinicaltrials.govNCT00407602; completed).
All psychophysical experiments were carried out at least six months after device implantation.The study was approved by the Institutional Review Board (IRB) at each participant's clinical site and was conducted under the tenets of the Declaration of Helsinki.Informed consent was obtained from the participants after explanation of the nature and possible consequences of the study.
Due to their geographic location, the participants were not directly examined by the authors of this study.Instead, initial experimental procedures were sent to the clinical site, and trained field clinical engineers performed the experiments as specified.
Raw collected data was then sent to the authors for subsequent analysis.Table 1: Participant details: sex (M: male, F: female), preoperative visual acuity (VA) categorized as either bare light perception (BLP) or no light perception (NLP), the age range at implantation, and the number of years that participants had been blind prior to implant surgery (self-reported).Years blind for Participant 1 was unknown due to gradual loss of vision.

Stimuli
Argus II consists of a 6 × 10 grid of platinum disc electrodes, each 200 µm in diameter, subtending 0.7 • of visual angle (Luo and da Cruz, 2016).Electrodes were spaced 575 µm apart.In day-to-day use, an external component is worn by the user, consisting of a small camera and transmitter mounted on a pair of glasses.The camera captures video and sends the information to the visual processing unit (VPU), which converts it into pulse trains using pre-specific image processing techniques (camera mode).
All stimuli described in this study were presented in direct stimulation mode, where stimuli were sent from the VPU directly to each electrode, without involving the external camera.Stimuli were charge-balanced, cathodic-first, square-wave pulse trains with 0.45 ms phase duration and 250 ms total stimulus duration.Stimulus amplitudes, frequencies, and the number of stimulated electrodes varied based on the design of each experiment.Stimuli were programmed in Matlab 7 (Mathworks, Inc.) using custom software, and pulse train parameters (i.e., the electrode(s) to be stimulated, current amplitude, pulse width, inter-pulse interval, and overall stimulus duration) were sent directly to the VPU, which then sent the stimulus commands to the internal portion of the implant using an inductive coil link.The implanted receiver wirelessly received these data and sent the signals to the electrode array via a small cable.

Perceptual thresholds
Perceptual thresholds for individual electrodes were measured using an adaptive yes/no procedure.Custom software was utilized to measure perceptual thresholds on each electrode through a hybrid method combining an adaptive staircase and constant stimuli approach, using charge-balanced, biphasic 20 Hz pulse trains (de Balthasar et al., 2008).The experiment involved five sessions, where each electrode was tested 12 times, interspersed with 32 catch trials across sessions to assess the false alarm rate, with stimulus amplitudes adjusted based on a Weibull function fit to current data.Data from sessions where the false alarm rate exceeded 20% were deemed unreliable and excluded from the analysis.See Appendix A for a more detailed description of the procedure.

Phosphene drawings
Participants were asked to perform a drawing task upon electrical stimulation of the retina.Participants were comfortably seated in front of a touchscreen monitor whose center was horizontally aligned with the participant's head.The distance between the participant's eyes and the monitor was 83.8 cm for Participant 1, 76.2 cm for Participant 2, and 77.5 cm for Participant 3.
Each stimulus was presented in 5-10 trials randomly amongst other stimuli with different frequency and/or amplitude levels.The stimulus frequency ranged from 6 Hz to 120 Hz, and the amplitude was between 1.25 times threshold to 7.5 times threshold.
Within each trial, either one or two electrodes were randomly selected and stimulated; if two electrodes were selected, they were stimulated simultaneously.After delivering each stimulus and before moving to the subsequent trial, participants were asked to trace the perceived shape on the touchscreen monitor.The drawing data was recorded and converted into a binary shape data file using Matlab, and stored for future analysis.
All psychophysical experiments were carried out by local field clinical engineers at each participating site, and the results were forwarded to the authors.
This yielded 3587 phosphene drawings across three participants.To make the collected phosphene drawings amenable to automated image analysis, we manually inspected all drawings (see Appendix B for details) to make sure that: • all drawn contour lines were closed (e.g., when drawing a circle, the starting point of the drawing must touch the endpoint); • small specs (i.e., phosphene with size smaller than 10 pixels) that appeared in less than 50% of trials for a particular electrode were not counted as additional phosphenes.
As part of this procedure, 13 drawings were removed.The remaining 3574 drawings (2717 single-electrode drawings and 857 paired-electrode drawings; see Since the validity and reliability of the experiment relied on the ability of our participants to accurately draw the perceived phosphenes, a control task was conducted where participants were asked to feel six different tactile shapes made of felt with a cardboard background, and then draw them on a touchscreen (Beyeler et al., 2019b).
As the shape of these tactile targets was known and we asked participants to repeat each drawing five times, we were able to determine each participant's drawing error and bias.A detailed description of this task can be found in the Appendix S2 of Beyeler et al. (2019b).In short, this control established baseline drawing variability for each participant, against which we could compare electrically elicited phosphene drawing variability to determine the stability of phosphene appearance.

Phosphene shape descriptors
We used the measure module of scikit-image (version 0.18.3, https:// scikit-image.org)to automatically extract phosphenes (connected regions) and their corresponding centroids from each drawing.Phosphene shape was quantified using four parameter-free shape descriptors commonly used in image processing: area, perimeter, major axis length, and minor axis length (Nanduri et al., 2008).An example is shown in Fig. 1.These descriptors are based on a set of statistical quantities known as image Figure 1: An example of a phosphene drawing and five shape properties of the phosphene.A) Phosphene described by major axis length (red) and minor axis length (blue).B) Phosphene described by area (white), perimeter (red), and the number of distinct regions (green).
moments (Hu, 1962).For an M × N pixel grayscale image, I(x, y), where x ∈ [1, M ] and y ∈ [1, N ], the raw image moments M ij were calculated as: x i y j I(x, y). (1) Raw image moments were used to compute area (A = M 00 ) and the center of mass (x, ȳ) = (M 10 /M 00 , M 01 /M 00 ) of each phosphene.
Phosphene perimeter was calculated using an algorithm described in Benkrid et al. (2000), which approximates the length of each phosphene's contour as a line running through the centers of connected border pixels.
The distribution of raw shape descriptors for all participants is given in Appendix C. Phosphene orientation was previously shown to depend mostly on the retinal location of the stimulating electrode (Beyeler et al., 2019b) and was thus excluded from the main analysis.However, the interested reader is referred to Appendix D for the supplemental analysis.

Estimation of electrode-fovea distance and inter-electrode distance
Electrode-fovea distances and inter-electrode distances were estimated using the pulse2percept software (Beyeler et al., 2017a).Following Beyeler et al. (2019b), each participant's implant location was estimated based on the fundus images taken before and after surgery by extracting and analyzing retinal landmarks (e.g., foveal region and optic disc).Image pixels were converted into retinal distances using Argus II interelectrode spacing information.The implant image was then rotated and transformed such that the raphe fell on the horizontal axis and the fovea was the origin of the new coordination system.The stimulated implant was placed on a simulated map of axonal nerve fiber bundles (Fig. 2), which was modeled based on ophthalmic fundus photographs of 55 sighted participants (Jansonius et al., 2009).Since the fovea is the origin in the stimulated implant's coordinates, the electrode-fovea distance was measured as the distance between an electrode and the origin.
Inter-electrode distance measurements were adapted from Yücel et al. ( 2022) to investigate the effect of axonal stimulation on perceived phosphene shapes, in which the distance between two electrodes was divided into two, nearly orthogonal components: • between-axon distance (green lines in Fig. 3): the shortest distance between the center of the more nasal electrode to the closest axon of the more temporal electrode; • along-axon distance (blue curves in Fig. 3): the distance from the center of the temporal electrode, along the nasal electrode's closest axon, up to the point where the nasal electrode's between-axon line reached the temporal electrode's axon.

Estimation of electrode-retina distance
Electrode-retina distances were estimated from post-surgical optical coherence tomography (OCT) images collected with either Cirrus HD-OCT (Carl Zeiss Inc) or Topcon 3D-OCT 1000 (Topcon Inc).The SD-OCT scans were obtained 6 months after implantation of Participants 1 and 2, and 13 months after implantation of Participant 3.
When performing OCT scanning, the opaque metal electrodes prevent image  acquisition directly underneath the corresponding electrode.However, based on the length of the shadow between the electrode and the retinal surface, it is possible to estimate the electrode-retina distance of that electrode (Ahuja et al., 2013).A single grader manually measured the electrode-retina distance by counting the number of pixels from the center of the shadow on the retinal pigment epithelium to the implant (Fig. 4).
These pixel counts were then converted to microns, using the known electrode diameter as a reference to calibrate the pixel-to-micron ratio based on the width of each electrode shadow's gap in the OCT images.Distances of poorly imaged electrodes were excluded from the dataset.
Details about each participant's estimated electrode-fovea distances and electroderetina distances are given in Table 3. Welch's t-test was used to compare differences in stimulus and neuroanatomical parameters across participants.There was no statistical difference between the averaged electrode-fovea distance across different participants (for
To control for individual drawing bias and variance (Beyeler et al., 2019b) as well as facilitate statistical analysis, we transformed the data as follows: • All independent variables (i.e., amplitude, frequency, electrode-retina distance, electrode-fovea distance, between-axon distance, and along-axon distance) were standardized across all participants.
• The dependent variables, which describe phosphene shape (i.e., area, perimeter, major/minor axis lengths), were expressed as multiples of the shape descriptors elicited by a "standard" pulse train (amplitude: 2× threshold, frequency: 20 Hz).This procedure was performed separately for each participant, but considered drawings from all recorded electrodes of that participant, in order to account for drawing bias and variance.For instance, the area of an individual phosphene was normalized by the phosphene area averaged across all drawings of a particular participant when one of their electrodes was stimulated with the standard pulse.
• Shape descriptors were first extracted from each individual phosphene in each drawing, before they were averaged across trials of the same electrode and stimulus combination, in order to eliminate repeated measures of the same data point.
Averaging in this fashion across trials reduced the 3574 drawings to 379 data points (278 single-electrode percepts, 101 paired-electrode percepts).
• Data points that fell more than 2.5 standard deviations away from the mean were considered outliers and were removed from all further analyses.In total, 26 data points were removed from single-electrode analyses, and no data points were removed from paired-electrode analyses.The remaining 353 data points (252 singleelectrode percepts, 101 paired-electrode percepts) were included in all analyses.
• Feature descriptors were transformed using a power of 1/n to keep the residuals normally distributed.Specifically, we used n = 3 for area and n = 2 for perimeter, major axis length, and minor axis lengths.All residuals were verified for normality using Quantile-Quantile plots (see Appendix E).
Partial correlation plots for the shape descriptors are given in Appendix E, along with their linear fits.
A series of multiple linear regression and partial correlation analyses were conducted within participants (Hou et al., 2023), while linear mixed-effects analyses (with stimulus and neuroanatomical parameters as fixed effects and participants as a random effect) were performed across participants.

Amplitude and frequency modulation affect single-point perception differently
Consistent with the literature on single-electrode phosphene drawings (Nanduri et al., 2008;Luo et al., 2016;Beyeler et al., 2019b), phosphene shape greatly varied across participants and electrodes, but was relatively consistent across trials of a single electrode.Single-electrode stimulation reliably elicited phosphenes in all three participants, who reported seeing a single phosphene on 86.8% of trials, two phosphenes on 13.0% of trials, and three or more phosphenes on the remaining trials.
Fig. 5 shows the mean images for each electrode, obtained by averaging the drawings for each electrode across trials obtained with a particular current amplitude (Fig. 5 rows; expressed as a multiple of the threshold current).Mean images were then centered over the corresponding electrode in a schematic of the participant's implant to reveal the rich repertoire of elicited percepts across electrodes (see Appendix F).
Whereas Participant 1 mostly drew blobs and wedges, which grew larger as the stimulus amplitude was increased, Participant 2 reported seeing exclusively lines and arcs, which got longer with increasing amplitude.The effect of amplitude on phosphene shape was most apparent for Participant 3, where phosphenes that appeared as lines and arcs near threshold turned into blobs and wedges as amplitude was increased.
First reported by Nanduri et al. (2012), pulse frequency seemed to affect phosphene shape differently than amplitude (Fig. 6).Whereas phosphenes that were located close to the center of vision (denoted by □ in Fig. 6) did not noticeably change in shape, more eccentric phosphenes turned from blobs at 6 Hz to rectangles at 60 Hz (Participant 1), or from short streaks at 6 Hz to orders-of-magnitude longer arcs at 60 Hz (Participant 3).

Factors affecting phosphene shape during single-electrode stimulation
To more systematically investigate how different stimulus and anatomical parameters affect phosphene shape in single-electrode stimulation, we considered how the four shape descriptors (area, perimeter, major axis length, and minor axis length; see Methods, Section 2.5) could be predicted by different stimulus parameters (i.e., amplitude and frequency) and neuroanatomical parameters (i.e., electrode-retina distance and electrode-fovea distance).To address this, shape descriptor values were first averaged across trials and normalized per participant (see Methods, Section 2.8).
We first performed a multiple linear regression and partial correlation analysis for each participant (top three sections in Table 4), corrected for multiple comparisons with the Bonferroni method.Consistent with Nanduri et al. (2012), we found that stimulus amplitude strongly affected phosphene area in two out of three participants (p < .001) and minor axis length (p < .001),suggesting that phosphenes tended to get larger with increasing amplitude.However, amplitude did not significantly modulate phosphenes drawn by Participant 2 (also visually evident in Fig. 5).Stimulus frequency had no significant effect on phosphene shape in Participants 1 and 2, but strongly (β > .3,r > .6)and significantly (p < .001)modulated phosphene perimeter, major axis length, and minor axis length in Participant 3.
In terms of neuroanatomical parameters, we considered an electrode's distance to the fovea (i.e., retinal eccentricity) and distance to the retina (i.e., height).Electroderetina distances (labeled "ERD" in Table 4) were non-zero only in Participant 1, where larger ERDs led to smaller phosphenes (p < .05).Interestingly, we found that electrode-fovea distance (labeled "EFD" in Table 4) significantly modulated shape in all three participants.For Participant 1, more eccentric phosphenes tended to be more elongated (p < .001)but not necessarily larger.For Participants 2 and 3, more eccentric phosphenes tended to be larger overall (affecting all shape descriptors with p < .05 or smaller).
To determine which of these correlations were general trends that reached significance across all three participants, we also fitted a linear mixed-effects to all data (bottom section in Table 4), corrected for multiple comparisons (Bonferroni), with "Participant" as a random effect.This analysis revealed that larger stimulus amplitudes tended to elicit larger (p < .001)and "blobbier" phosphenes (by means of increased minor axis length; p < .001).Perhaps driven by Participant 3's data, increased stimulus frequencies tended to elicit slightly larger and more extended/less compact phosphenes, by means of the overly increased perimeter (β = .156,r = .617)and major/minor axis lengths (β > .12,r > .4;p < .001).Increasing retinal eccentricity (EFD) had a similar effect, leading to slightly larger and more elongated phosphenes, by means of the overly increased perimeter (β = .107,r = .469;p < .001)and major axis length (β = .128,r = .479;p < .001).Partial correlation plots can be found in Appendix E.2.Table 4: Single-electrode phosphene shape predicted by amplitude (Amp), frequency (Freq), electrode-fovea distance (EFD), and electrode-retina distance (ERD; only non-zero for Participant 1).Participant-specific analyses (top three sections) were conducted using a multiple linear regression (β: standardized regression coefficient) and partial correlation analysis (r: partial correlation coefficient).All-participant analysis (bottom section) was conducted using a linear mixed-effects model fit on all data, with "Participant" as a random factor.Intercepts (not shown) were included in the analysis.The variance inflation factor of all predictors was smaller than 2, suggesting minimal multicollinearity.N denotes the number of data points included in each analysis, where each data point represents the mean shape descriptor of the phosphenes elicited with a particular stimulus amplitude and frequency on a particular electrode, averaged across trials.*: p < .05,**: p < .01,***: p < .001.Significant effects are marked in bold (corrected for multiple comparisons using the Bonferroni method).

Predicting two-point perception from single-point perception
When two electrodes were stimulated simultaneously, participants reported seeing a single phosphene on 53.1% of trials, two phosphenes on 43.4% of trials, and three or more phosphenes on the remaining trials.Three or more phosphenes were generally encountered when single-electrode stimulation itself produced more than one phosphene.
Representative examples of phosphene drawings for different electrode pairs are shown in Fig. 7.
When paired-electrode stimulation produced two distinct phosphenes (Fig. 7, left), their shape resembled the linear summation of the phosphenes reported during singleelectrode stimulation.For instance, as shown in Row 1 of the left panel in Fig. 7, Participant 1 perceived a long arc when electrode E1 was stimulated and an oval when electrode A10 was stimulated.When both E1 and A10 were stimulated concurrently, the resulting phosphene appeared as an arc alongside an oval.Similarly, in Row 9 of the left panel, Participant 3 perceived a tilted line for electrode E6 and a small triangle for electrode D7.Then during the simultaneous stimulation of electrodes E6 and D7, the resulting shape preserved the original form of the individual phosphene shapes.
A colored version of this figure that superimposes the outline of the single-electrode phosphenes on the paired-electrode phosphenes is given in Appendix G.
When paired-electrode stimulation produced a single phosphene (Fig. 7, right), the phosphenes reported during single-electrode stimulation appeared to merge into a unified shape.For instance, as shown in Row 2 of the right panel in Fig. 7, Participant 1 perceived a blob for electrode C7 and a right-leaning straight line for electrode D7.When both C7 and D7 were stimulated simultaneously, the participant saw a larger blob tilted rightward.Similarly, in Row 6, electrode B4 elicited a small dot, and electrode F4 elicited a long arc; and simultaneous stimulation yielded an arc-shaped phosphene, appearing as a cohesive shape formed by connecting the two individual shapes.Table 5: Paired-electrode phosphene shape predicted by amplitude (Amp), electrodefovea distance (EFD), and electrode-retina distance (ERD; only non-zero for Participant 1).Participant-specific analyses (top three sections) were conducted using a multiple linear regression (β: standardized regression coefficient) and partial correlation analysis (r: partial correlation coefficient).All-participant analysis (bottom section) was conducted using a linear mixed-effects model fit on all data, with "Participant" as a random factor.Intercepts (not shown) were included in the analysis.The variance inflation factor of all predictors was smaller than 3, suggesting minimal multicollinearity.N denotes the number of data points included in each analysis, where each data point represents the mean shape descriptor of the phosphenes elicited with a particular stimulus amplitude on a particular electrode pair, averaged across trials.*: p < .05,**: p < .01,***: p < .001.Significant effects are marked in bold (corrected for multiple comparisons using the Bonferroni method).

Factors affecting phosphene shape during paired-electrode stimulation
We wondered whether these stimulus and neuroanatomical parameters could also explain the shape of phosphenes elicited by paired-electrode stimulation.As participants would frequently draw multiple phosphenes during paired-electrode stimulation (Fig. 7), we extracted each shape descriptor for each individual phosphene.Then, we summed all phosphenes' corresponding shape descriptor within each drawing in order to account for the variable number of perceived phosphenes.Finally, we averaged each shape descriptor of each drawing across trials (see Methods, Section 2.8).
The results are shown in Table 5, and partial correlation plots can be found in Appendix E.2.Similar to the single-point results (Table 4), electrode-fovea distance affected phosphene shape in two out of three participants (p < .001),generally increasing the perimeter and major axis length of more eccentric phosphenes.However, in contrast to the single-point results, amplitude (i.e., the average amplitude across the two stimulated electrodes) had a less definitive effect on phosphene shape, only increasing the minor axis length (p < .05)for Participant 1.Unfortunately, all paired-electrode drawings were collected at 20 Hz, thus stimulus frequency could not be included in the analysis.
Naturally, we asked to what extent the phosphene shape elicited by paired-electrode Table 6: Paired-electrode phosphene shape descriptors predicted by the sum of the corresponding single-electrode shape descriptors.Participant-specific analyses (top three rows) were conducted using a simple linear regression (β: standardized regression coefficient) and partial correlation analysis (r: partial correlation coefficient).All-participant analysis (bottom row) was conducted using a linear mixed-effects model fit on all data, with "Participant" as a random factor.Intercepts were not included in the analysis, because if the value of a predictor (sum of the single-electrode phosphene shapes) was zero, the corresponding value of the dependent variable (the paired-electrode phosphene shape) should also be zero.N denotes the number of data points included in each analysis, where each data point represents the mean shape descriptor of the phosphenes elicited on a particular electrode pair, averaged across trials.*: p < .05,**: p < .01,***: p < .001.Significant effects are marked in bold (Bonferroni-corrected for multiple comparisons).
stimulation could be predicted by the phosphene shape elicited during single-electrode stimulation.To answer this question, we conducted a simple linear regression (Table 6) where each shape descriptor from a paired-electrode stimulation trial (e.g., the sum of phosphene areas when Electrodes A and B were simultaneously stimulated) was regressed on the same shape descriptor from a single-electrode stimulation trial (e.g., phosphene area elicited by Electrode A plus phosphene area elicited by Electrode B).
In short, we found that each paired-electrode shape descriptor could be predicted by the sum of the two corresponding single-electrode shape descriptors (Table 6; p < .001).
Across all participants, shape descriptors tended to sum linearly, with the β values suggesting that phosphenes elicited by paired-electrode stimulation appeared larger than the average of their single-electrode counterparts, but smaller than their sum.Table 7: Number of perceived phosphenes predicted by amplitude (Amp), electrode-fovea distance (EFD), electrode-retina distance (ERD), between-axon distance, and along-axon distance.Participant-specific analyses were conducted using multiple linear regression (β: standardized regression coefficient) and partial correlation analysis (r: partial correlation coefficient); all-participant analysis was conducted using a linear mixed-effects model fit on all data, with "Participant" as a random factor.Intercepts (not shown) were included in the analysis.The variance inflation factor of all predictors was smaller than 3.1, suggesting minimal multicollinearity.N denotes the number of data points included in each analysis, where each data point represents the mean number of perceived phosphenes elicited with a particular stimulus amplitude on a particular electrode pair, averaged across trials.*: p < .05,**: p < .01,***: p < .001.Significant effects are marked in bold (Bonferroni-corrected for multiple comparisons).
single phosphene, even though the two electrodes may be far apart on the retina.
To test this hypothesis, we split retinal distance into two, almost orthogonal components: "between-axon" distance, which spreads the current radially from the more nasal electrode until it reaches the more temporal electrode's closest axon, and "along-axon" distance, which walks along the axon from that point until it reaches the more temporal electrode (Fig. 3; this works even for pairs on opposite sides of the raphe).
During the preliminary stage of this study, we experimented with a number of similar formulations of splitting these two components, and all of them gave similar results.
Consistent with the axon map model (Beyeler et al., 2019b), we found a significant correlation between the number of perceived phosphenes and the between-axon distance (p < .05,Table 7).Along-axon distance, on the other hand, was not significantly correlated with the number of perceived phosphenes (p > .05).
To further demonstrate the predictive power of the between-axon distance, we constructed two sets of models and compared their Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) scores: • Model A: phosphene number = f (along-axon distance, additional factors) • Model B: phosphene number = f (between-axon distance, additional factors) where "additional factors" consisted of the stimulus parameters (e.g., amplitude) and neuroanatomical parameters (e.g., electrode-fovea distances).As is evident in Table 8, we found strong evidence that models relying on between-axon distance (Model B) significantly outperformed models relying on along-axon distance (Model A) in two out of three participants, as well as in the all-participant analysis.8: Predicting the number of perceived phosphenes during paired-electrode stimulation based on the along-axon distance (Model A) and between-axon distance (Model B).Participant-specific analyses (top three sections) were conducted using a simple linear regression; all-participant analysis (bottom section) was conducted using a linear mixed-effects model fit on all data, with "Participant" as a random factor.Performance is measured using the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) scores, where lower scores indicate better performance.A difference in AIC scores (∆AIC) or BIC scores (∆BIC) of less than 2 suggests that there is substantial support for both models (i.e., there is no clear preference for one over the other).2 ≤ ∆AIC < 7 or 2 ≤ ∆BIC < 6 indicates some evidence against the model with the higher AIC.∆AIC ≥ 7 or ∆BIC ≥ 6: strong evidence against the model with the higher AIC/BIC.In practical terms, the model with the lower AIC/BIC is significantly better in terms of the balance between goodness of fit and model simplicity (highlighted in bold).Amp: stimulus amplitude; EDR: electrode-fovea distance; ERD: electrode-retina distance.

Discussion
In this study, we set out to investigate the relationship between single-point and two-point perception of Argus II users.Our results suggest that two-point perception can be predicted by the linear summation of single-point perception, supporting the notion of independent stimulation channels.We also found that the number of perceived phosphenes increased with the between-axon distance of two stimulating electrodes, but not the along-axon distance, thus providing further evidence in support of the axon map model for epiretinal stimulation (Rizzo et al., 2003;Nanduri, 2011;Beyeler et al., 2019b).
These findings contribute to the growing literature on phosphene perception and have important implications for the design of future retinal prostheses, as they may inform the optimal surgical placement of an epiretinal implant (Beyeler et al., 2019a;Bruce and Beyeler, 2022) and constrain AI-based stimulus optimization algorithms (Granley et al., 2022;de Ruyter van Steveninck et al., 2022;Relic et al., 2022).

Phosphene shape is well predicted by stimulus and neuroanatomical parameters
Although a link between neuroanatomical parameters such as electrode-retina distance and perceptual thresholds has been well established in the literature (Mahadevappa et al., 2005;de Balthasar et al., 2008;Ahuja et al., 2013;Pogoncheff et al., 2023), research examining the effect of these parameters on the shape of elicited phosphenes has been limited.
We found that phosphenes tended to appear larger and rounder as stimulus amplitude increased (Table 4), which is consistent with previous considerations about the current spread in the retina (de Balthasar et al., 2008;Granley and Beyeler, 2021;Yücel et al., 2022).However, in contrast to Nanduri et al. (2012), we found that stimulus frequency also affected phosphene size (Table 4 and Fig. 6).Perhaps driven by Participant 3's data, increased stimulus frequencies tended to elicit slightly larger and more extended/less compact phosphenes, by means of overly increased perimeter and major axis length.This relationship between stimulus frequency and phosphene size partially agrees with data from suprachoroidal prostheses, where phosphenes tend to appear thicker or rounder as the stimulation rate increases (Sinclair et al., 2016).
In addition, we found that increased electrode-fovea distance (i.e., retinal eccentricity) led to slightly larger and more elongated phosphenes (Table 4).While more eccentric phosphenes may be elongated along the trajectory of the underlying nerve fiber bundles (Beyeler et al., 2019b), the increased size may be a consequence of ganglion cell receptive fields increasing with eccentricity (Curcio and Allen, 1990).This would agree with psychophysical (Freeman and Simoncelli, 2011;Stingl et al., 2013) and computational considerations (Song et al., 2022), but is an as-of-yet unpublished finding about the appearance of phosphenes elicited by epiretinal implants.Indeed, most phosphene models assume a constant scaling factor between retinal and visual field coordinates (Horsager et al., 2009;Nanduri, 2011;Beyeler et al., 2019b).

Two-point perception is the linear sum of single-point perception
This study demonstrates that the shape of phosphenes elicited by paired-electrode stimulation is well predicted by the linear summation of the shape of their corresponding single-electrode phosphenes (Table 6), supporting the notion of independent channels for phosphene perception.Specifically, β values in Table 6 suggest that phosphenes elicited by paired-electrode stimulation were smaller than the sum of their single-electrode counterparts.This finding is partially consistent with Christie et al. (2022), who showed that the phosphene elicited by electrode "quads" was similar to phosphenes elicited by individual electrodes that belonged to the quad, with Wilke et al. (2011a), who showed that single-electrode phosphenes consisting of round dots and lines added up to more complicated patterns when stimulated simultaneously, and with Barry et al. (2020), who reported that multi-electrode percepts in the Orion cortical implant were perceived to be smaller and simpler than the predicted combination of single-electrode phosphene shapes.
The observed linear summation of single-electrode phosphenes provides valuable empirical evidence for future computational model development.Many computational models of prosthetic vision (Chen et al., 2009;Perez-Yus et al., 2017;Sanchez-Garcia et al., 2019;Granley and Beyeler, 2021) assume a linear relationship between stimulus parameters (e.g., amplitude) and phosphene appearance (e.g., brightness).The same is true for the stimulus generation procedure that underlies "video mode" in Argus II.
Here we were able to provide empirical evidence for this assumption and detail the factors that affect phosphene appearance during paired-electrode stimulation.These results may thus inform recent AI-based stimulus optimization algorithms (Spencer et al., 2019;de Ruyter van Steveninck et al., 2022;Relic et al., 2022;Granley et al., 2022), which aim to select the optimal stimulation parameters on each electrode based on their predicted effect on phosphene appearance.These insights may also benefit the prediction of phosphene shape in multi-electrode stimulation scenarios (Zrenner et al., 2010;Shivdasani et al., 2017), which aim to arrange individual phosphenes into more complex patterns, with the ultimate goal of producing form vision to support activities such as reading and recognizing objects.
However, it should be noted that multiple phosphene patterns may not automatically group into perceptually intelligible objects (Stingl et al., 2015;Shivdasani et al., 2017;Christie et al., 2022).This "binding problem" (Roelfsema, 2023) also extends to cortical implants.Although a recent study with intracortical electrodes (Chen et al., 2020) showed that macaques could successfully identify the intended shape of a patterned electrical stimulus, human participants implanted with the same technology could not always do that (Fernández et al., 2021).Indeed, human participants implanted with cortical surface electrodes required a dynamic stimulation strategy to allow for perceptual grouping (Beauchamp et al., 2020).

The number of perceived phosphenes depends on the axonal distance in paired-electrode stimulation
While it is not surprising that two electrodes separated by a large retinal distance might produce two distinct phosphenes (Yücel et al., 2022), here we were able to split retinal distance into two (nearly orthogonal) components: between-axon distance, which measures how far the electric current must spread away from an axon bundle until it reaches another electrode, and along-axon distance, which measures how far the electric current must spread along an axon bundle until it reaches another electrode.We found that models relying on between-axon distance consistently outperformed models relying on along-axon distance when predicting the number of perceived phosphenes (Table 8).
This result provides the first computational evidence that paired-electrode epiretinal stimulation is more likely to elicit two distinct phosphenes as the distance between their underlying axon bundles increases (as opposed to retinal distance alone), and provides further evidence in support of the axon map model for epiretinal stimulation (Rizzo et al., 2003;Nanduri, 2011;Beyeler et al., 2019b).This result has important clinical implications.First, it suggests that a user's axon map should be considered when deciding on an intraocular surgical placement of the array (Beyeler et al., 2019a), as phosphenes tend to appear elongated in the direction of the nerve fiber bundle that underlies the stimulating electrode (Beyeler et al., 2019b).As the probability of seeing two phosphenes increases with between-axon distance, the largest number of phosphenes should be produced by an implant whose placement maximizes the sum of between-axon distances between all pairs of electrodes in the array.In other words, electrodes that stimulate the same axon bundle (i.e., with zero between-axon distance) are redundant and should therefore be avoided (Beyeler et al., 2017b).Second, rather than arranging their electrodes on a rectangular grid in an attempt to efficiently tile the retinal surface, future epiretinal implants should strive to place every electrode on a different nerve fiber bundle in an attempt to efficiently tile the axon map, which in turn efficiently tiles the visual field (Bruce and Beyeler, 2022).The same principle may be applied to cortical implants, where future devices could arrange electrodes such that they efficiently tile the visual field rather than the cortical surface.

Limitations and future work
Despite the ability of our work to highlight important factors that guide the appearance of phosphenes elicited by retinal implants, it is important to note that our linear analyses cannot identify nonlinear predictors of phosphene shape.Future studies could thus focus on nonlinear (but still explainable) machine learning models (Pogoncheff et al., 2023).In addition, due to data availability, our analyses are currently limited to single-and paired-electrode stimulation in three participants.However, to achieve form vision, it will be important to stimulate more than two electrodes at a time for each participant.Therefore, future studies should investigate whether this linear summation can be extended to more than two electrodes across the Argus II and the broader retinal implant population.
Subretinal electronic chips allow blind patients to read letters and combine them to words.Proceedings of the Royal Society B: Biological Sciences, 278(1711):1489-1497.
Publisher: Royal Society.potentially be judged as two independent, connected regions by the image processing software, thereby accidentally doubling the number of reported phosphenes and halving their reported size.
Fortunately, we identified only twelve drawings with this issue.To fix them, we manually identified four endpoints of the broken contour line (Fig. B3

Appendix B.2. Phosphene drawings with other artifacts
Fourteen phosphene drawings had other artifacts, such as tiny specs (less than 10 pixels in size) that were not part of any other discernible shape, and were subsequently removed (Fig. B4).

Figure 2 :
Figure 2: A) Participant 2's fundus image with Argus II implant placed over the retinal surface.B) Participant 2's simulated implant placed on the simulated axonal map.

Figure 3 :
Figure 3: Axonal distances (adapted from Yücel et al., 2022).A) The between-axon distance (green line) and the along-axon distance (blue curve) when two electrodes are on the same side of the raphe, B) and when two electrodes are on different sides.

Figure 4 :
Figure 4: A) Participant 1's retinal implant fundus image.The cyan arrow marked the current scanning area, and the green electrode array was superimposed onto the original image for better electrode visualization.B) Participant 1's OCT b-scan.Each electroderetina distance (vertical blue line) was represented by the length between the center of the shadow on the retinal surface (horizontal blue line) and the implant (white line).

Figure 5 :
Figure 5: Single-electrode phosphene drawings as a function of stimulus amplitude (rows; expressed as multiples of the threshold current).Mean images were obtained by averaging drawings from individual trials aligned at their center of mass (Appendix F).Averaged drawings were then overlaid over the corresponding electrode in a schematic of each participant's implant (pulse2percept 0.8.0.dev0,Beyeler et al., 2017a).Pulse train frequency was 20 Hz for all participants.Squares (□) indicate the estimated location of the fovea.

Figure 6 :
Figure 6: Single-electrode phosphene drawings as a function of pulse train frequency.Mean images were obtained by averaging drawings from individual trials aligned at their center of mass (Appendix F).These averaged drawings were then overlaid over the corresponding electrode in a schematic of each participant's implant (pulse2percept 0.8.0dev;Beyeler et al., 2017a).Shown are only those electrodes for which drawings at all stimulus frequencies were available.Stimulus amplitude was 1.5 times threshold for Subjects 1 and 2, and 1.25 times threshold for Participant 3. Squares (□) indicate the estimated location of the fovea.

Figure 7 :
Figure 7: Left: Representative examples of single phosphenes combining linearly without overlap during paired-electrode stimulation.Right: Representative examples of phosphenes merging and overlapping during paired-electrode stimulation.Mean images were obtained by averaging drawings from individual trials, each phosphene aligned at its trial-averaged center of mass (Appendix F).

3. 5 .
Factors affecting the number of perceived phosphenes during paired-electrode stimulation Yücel et al. (2022) previously demonstrated that the probability of perceiving two distinct phosphenes increases with inter-electrode distance.However, the axon map model(Beyeler et al., 2019b) makes a more nuanced prediction: participants should be more likely to see two distinct phosphenes as the distance between two nerve fiber bundles increases ("between-axon" distance; as opposed to distance on the retinal surface; see Methods).Under this model, paired-electrode stimulation with a short between-axon distance should activate the same nerve fiber bundles and thus lead to , Panel A) and connected them (Panel B), then used scipy.ndimage.binaryfill holes() to fill the area (Panel C).

Figure B3 :
Figure B3: Procedure for fixing phosphene drawings with broken contour lines.

Figure B4 :
Figure B4: Example phosphene drawings with small specs (artifacts) that were removed from the dataset.

Figure E2 :
Figure E2: The Q-Q plots of residuals for paired-electrode stimulation.

Figure E3 :
Figure E3: Partial correlation plots of area, perimeter, major axis length, or minor axis length correlated with amplitude, frequency, electrode-fovea distance, and electrode-retina distance across all participants in single-electrode stimulation.

Figure E4 :
Figure E4: Partial correlation plots of normalized phosphene shape elicited by pairedelectrode stimulation, correlated with standardized stimulus amplitude and neuroanatomical parameters (arbitrary units).

Figure E5 :
Figure E5: Partial correlation plots of the number of distinct phosphene regions correlated with stimulus parameters and electrode-retina properties across all participants in pairedelectrode stimulation.

Figure F1 :
FigureF1: Procedure for stacking drawings when the number of phosphenes differed across trials.Note that this procedure only applies to the visualizations in Figures5-7, as our statistical analysis did not require phosphene stacking.