Granley et al. (2021): Pulse parameters shape phosphene appearance

The axon map model of [Beyeler2019] predicts one phosphene shape per electrode, regardless of the pulse train. [Granley2021] adds three scaling factors, fit to psychophysical and electrophysiological data: amplitude, frequency, and phase duration modulate brightness (\(F_\mathrm{bright}\)), spatial extent (\(F_\mathrm{size}\), scaling \(\rho\)), and streak length (\(F_\mathrm{streak}\), scaling \(\lambda\)).

This example qualitatively recreates Fig. 3 of [Granley2021]. Each row sweeps one pulse parameter on one Argus II electrode:

• amplitude makes the phosphene brighter and larger,

• frequency makes it brighter but not larger,

• phase duration shortens the axonal streak.

import matplotlib.pyplot as plt

from pulse2percept.implants.retina import ArgusII
from pulse2percept.models.retina import BiphasicAxonMapModel
from pulse2percept.stimuli import BiphasicPulseTrain
from pulse2percept.units import xTh

One electrode, one model

rho and lam (um) are the baseline spread across and along axons. lam = 800 um (longer than in the paper) elongates the baseline phosphene to about 2.5:1, so the streak effect in the bottom row is visible. The grid covers the region electrode A4 of a right-eye Argus II projects to.

ELECTRODE = 'A4'
BASE_FREQ = 5     # Hz
BASE_AMP = 1      # xTh
BASE_PDUR = 0.45  # ms

model = BiphasicAxonMapModel(implant=ArgusII(), rho=200, lam=800,
                             xrange=(-10.5, 1.5), yrange=(-2, 10), step=0.1)
model.build()


def predict(freq, amp, pdur):
    """Brightest frame of the percept for one biphasic pulse train"""
    train = BiphasicPulseTrain(freq, amp * xTh, pdur)
    return model.predict_percept({ELECTRODE: train}).data[..., 0]

The three sweeps

Amplitude is in multiples of perceptual threshold (xTh), defined at 0.45 ms phase duration.

In this model, threshold falls as phase duration grows (a0 * pdur + a1, Eq. 3): a fixed 1 xTh at 100 ms would be ~200x the threshold-scaled amplitude at 0.45 ms and saturate. The phase-duration row therefore divides amplitude by the same factor, so phase duration acts only through \(F_\mathrm{streak}\).

AMPS = [1, 2, 3, 4, 5, 6]              # xTh, at 5 Hz / 0.45 ms
FREQS = [5, 10, 20, 40, 80, 120]       # Hz, at 1 xTh / 0.45 ms
PDURS = [0.1, 1, 5, 25, 50, 100]       # ms, at 5 Hz, amplitude compensated

scale = model.spatial.bright_model.scale_threshold
pdur_amps = [BASE_AMP * scale(BASE_PDUR) / scale(pdur) for pdur in PDURS]

rows = [
    ('Increasing amplitude',
     [f'{a:g}' + r'$\times$Th' for a in AMPS],
     [predict(BASE_FREQ, a, BASE_PDUR) for a in AMPS]),
    ('Increasing frequency',
     [f'{f:g} Hz' for f in FREQS],
     [predict(f, BASE_AMP, BASE_PDUR) for f in FREQS]),
    ('Increasing phase duration',
     [f'{t:g} ms' for t in PDURS],
     [predict(BASE_FREQ, a, t) for a, t in zip(pdur_amps, PDURS)]),
]

All 18 panels share one gray scale, [0, vmax], with vmax the brightest pixel in the figure (the 120 Hz panel, ~9x the 5 Hz reference). Per-panel autoscaling would make every panel equally bright.

vmax = max(frame.max() for _, _, frames in rows for frame in frames)

fig, axes = plt.subplots(3, 6, figsize=(11, 6.5), facecolor='k')
for row_axes, (label, titles, frames) in zip(axes, rows):
    for ax, title, frame in zip(row_axes, titles, frames):
        ax.imshow(frame, cmap='gray', vmin=0, vmax=vmax)
        ax.set_title(title, color='w', fontsize=10, pad=4)
        ax.set_xticks([])
        ax.set_yticks([])
    row_axes[0].set_ylabel(label, color='w', fontsize=11)
fig.tight_layout()

Amplitude (top) increases size and brightness. Frequency (middle) increases brightness only; the outline is identical across the row. Phase duration (bottom), with brightness and width held fixed, shortens the streak by about half between 0.1 and 100 ms.

What this does not establish

• The three factors are phenomenological fits to a few Argus I/II subjects, not a biophysical model. They are linear or single power laws and extrapolate poorly beyond the ranges swept here.

• v0.11 uses an Argus II refit of the [Horsager2009] phase-duration threshold relation, so the bottom row does not exactly reproduce the published panel. The compensating amplitudes come from the model’s own scale_threshold.

• Brightness is in arbitrary units; compare only within this figure.

• With several electrodes active, the model sums linearly; real multi-electrode percepts do not.