Eye Model: Optical–Phototransduction–Synaptic–Ganglion Mathematical Formulation

The following system modularly describes the physical and mathematical components of the eye model for contrast-enhanced pattern stimulation under photopic PERG conditions: optical transfer, phototransduction chemistry, membrane currents, synaptic transmission, and ganglion firing.

Optical input and retinal illumination

• Pattern stimulus:

𝐼(𝐱, 𝑡) = 𝐼₀[1 + 𝐶 ⋅ 𝑠(𝐱, 𝑓_s) ⋅ sin , ⁣(2𝜋𝑓_t𝑡)]

Here, 𝐼_o is the photopic mean light intensity, 𝐶 is the contrast, 𝑠(𝐱, 𝑓_s) is the spatial frequency filter (e.g., 1.6 cpd), and 𝑓_t is the temporal frequency (e.g., 4 Hz).

• Pupil-lens optical transmission:

𝐸_ret(𝐱, 𝑡) = 𝜂_opt ⋅ 𝐴_pupil ⋅ 𝐼(𝐱, 𝑡) ⋅ 𝑇_λ

Here, 𝜂_opt is optical efficiency, 𝐴_pupil is pupil area, and 𝑇_λ is wavelength transmission coefficient.

Phototransduction chemistry and current gates

• Opsin–G protein–PDE cascade (activity modification):

𝑑𝑅^∗ / 𝑑𝑡 = 𝛼_R𝐸_ret− 𝛽_R𝑅^∗, 𝑑𝐺^∗ / 𝑑𝑡 = 𝛼_G𝑅^∗ − 𝛽_G𝐺^∗, 𝑑PDE / 𝑑𝑡 = 𝛼_P𝐺^∗ − 𝛽_PPDE

• cGMP pool and Ca²⁺ feedback:

𝑑[cGMP] / 𝑑𝑡 = 𝑘_syn (1 + 𝛾_Ca(Ca‾ − [Ca²⁺]) − 𝑘_hydPDE [cGMP]

• CNG door opening probability and current:

𝑝_CNG = [cGMP]ⁿ / [𝐾ⁿ_CNG + [cGMP]ⁿ] , 𝐼_CNG = 𝑔_CNG 𝑝_CNG(𝑉_m − 𝐸_CNG)

• Photoreceptor membrane equation (rod/cone):

𝐶_m𝑑𝑉_m / 𝑑𝑡 = −(𝐼_CNG + 𝐼_leak + 𝐼_HC + 𝐼_Ca + 𝐼_K)

Here, 𝐼_leak = 𝑔_leak(𝑉_m − 𝐸_leak), horizontal cell feedback can be modeled with the 𝐼_HC inclusion term dependent on the environmental mean.

• Internal Ca²⁺ dynamics:

𝑑[Ca²⁺] / 𝑑𝑡 = 𝐼_Ca,in / 𝑧𝐹𝑉_cell − 𝑘_upt[Ca²⁺] + 𝑘_rel𝑆_store

and for SR depots

𝑑𝑆_store / 𝑑𝑡 = = 𝑘_fill[Ca²⁺] − 𝑘_rel𝑆_store

Bipolar synaptic transmission and postsynaptic potentials

• ON pathway (metabotropic, reversed sign):

𝐼_syn,ON = 𝑔_ON 𝜎 , ⁣([Glu]₀ − Δ[Glu](𝑉_m)) (𝑉_BP− 𝐸_ON)

Here, 𝜎(⋅) represents sigmoidal transfer, and Δ[Glu] represents glutamate decrease due to photoreceptor output.

• OFF path (ionotropic, same sign):

𝐼_syn,OFF = 𝑔_OFF 𝜎 , ⁣(Δ[Glu](𝑉_m)) (𝑉_BP− 𝐸_OFF)

• Bipolar membrane equation:

𝐶_BP𝑑𝑉_BP / 𝑑𝑡 = −(𝐼_syn,ON + 𝐼_syn,OFF + 𝐼_leak,BP), 𝐼_leak,BP = 𝑔_leak,BP(𝑉_BP− 𝐸_leak,BP)

Ganglion cell: firing pattern and output

• Simple LIF (discharge condition):

𝐶_RGC𝑑𝑉_RGC / 𝑑𝑡 = −𝑔_L(𝑉_RGC− 𝐸_L) + 𝐼_syn,BP(𝑡) , if 𝑉_RGC ≥ 𝑉_th ⇒ spike, 𝑉_RGC → 𝑉_reset, 𝑡 → 𝑡 + 𝑡_ref

• HH-derivative (optional channel detail):

𝐶_RGC𝑑𝑉_RGC / 𝑑𝑡 = −(𝑔_Na𝑚³ℎ(𝑉_RGC − 𝐸_Na) + 𝑔_K𝑛⁴(𝑉_RGC− 𝐸_K) + 𝑔_L(𝑉_RGC− 𝐸_L)) + 𝐼_syn,BP

gate variables:

𝑑𝑥 / 𝑑𝑡 = 𝛼_x(𝑉_RGC) (1 − 𝑥) − 𝛽_x(𝑉_RGC) 𝑥, 𝑥 ∈ {𝑚, ℎ, 𝑛}

• Ignition rate and PERG components:

𝑓_RGC(𝑡) = Σ_k𝛿(𝑡 − 𝑡_k) ⇒ PERG(𝑡) = ℱ(𝑓_RGC(𝑡))

≈ 𝑎 𝑁35(𝑡) + 𝑏 𝑃50(𝑡) + 𝑐 𝑁95(𝑡)

Here, ℱ(⋅) represents the linear–weak linear functional representing the recording/optical nerve–electrode response; 𝑎, 𝑏, 𝑐 are the component weights.

Phase and time calibration under PERG conditions.

• Temporal period alignment (4 Hz):

𝑇 = 1 / 𝑓_t = 250 ms, 𝑡_k ∈ {35,50,95} (ms, in-loop reference)

• Phase lock error function (for component timings):

Parameter calibration, 𝜃 = {𝛾_Ca, 𝑘_syn, 𝑘_hyd, 𝑔_ON, 𝑉_th, 𝑡_ref}

𝜃^∗ = arg min_𝜃 𝒥

• Amplitude normalization and adaptation metric:

𝐴_p^norm = 𝐴_p / max_d 𝐴_p(𝑑) , 𝜅_adapt = 𝐴_p(𝑑₁) − 𝐴_p(𝑑_N) / 𝐴_p(𝑑₁)

Here, 𝑝 ∈ {𝑃50, 𝑁95}, 𝑑₁ is the first loop, and 𝑑_N is the last loop.

Parameter-output sensitivity summary

• Optical-chemical bond:

∂𝑁35 / ∂𝑘_hyd > 0, ∂𝑃50 / ∂𝑔_ON < 0 (Phase positive peak earlier)

• Synaptic-neural connection:

This formulation establishes a complete chain, starting from optical energy transfer and extending to second messenger dynamics, membrane currents, and synaptic-neural output. The phase-locked error function and sensitivity derivatives provide a practical optimization framework for calibrating time-phase coherence and amplitude-adaptation relationships under photopic PERG conditions.