E8 Graded → spike-rate re-quantisation — the hand-off (spine complete)
The retina re-encodes the graded signal as a ganglion spike RATE: a thresholded (rheobase), bounded (depolarisation-block ceiling), monotone clock that tracks the slow envelope and never re-introduces the carrier — the collapse is preserved end-to-end.
What this rung establishes (the spine closes)
The eye's last step is a re-quantiser, not a mixer. The frozen R19 neuron whose slow recovery set the band now fires — each spike a membrane up-crossing, an all-or-none fold-crossing event — and re-encodes the graded ~Hz signal as a ganglion spike RATE. The code is thresholded, bounded, and monotone, and it tracks the slow envelope while never re-introducing the carrier. With E8 the down-conversion spine E0→E8 is complete. [F][V]
A thresholded, bounded, monotone clock
- Rheobase floor
- silent at g≤0.20, fires at g≥0.30 (the switch must clear its fold)
- Monotone re-encoding
- rate rises across the band, 0.00783 → 0.01133
- Near-periodic clock
- CV(ISI) ≈ 0.00017–0.00024 (≪ 1)
- Depolarisation-block ceiling
- at drive +1.00 the rate falls to 0.000167 (silent)
- Carrier-invariance
- rate spread 1.53% across a 20× carrier span (rate ⟂ carrier)
The full-range f–I is a bandpass in drive: a rheobase floor and a depolarisation-block ceiling, monotone only inside a finite band. The central result closes the loop with E5/E7 — fix the envelope and sweep the carrier and the rate is invariant (1.53%); raise the envelope and the rate rises; the spike train sits in the neural low band. The ~13-order down-conversion is preserved end-to-end. [F]
Where γ sits (read-only) and the hand-off
γ shifts the f–I position (rheobase) via spinodal(γ), not the band (that is τ, E7) and not the signal (the drive): at fixed drive a higher-γ gene fires slower (CNGB3 0.009167 down to RHO 0.007833). Because γ moves the rate about as much as the drive here, E8 makes no ‘γ negligible’ claim. The felt percept of sight is out of scope — it hands off to the mind volume behind the firewall. [F][V]
Grades (VP-SPEC C3 — honest)
| [F] | The rate code is thresholded (rheobase), bounded (depolarisation-block ceiling — a bandpass in drive), and monotone; the carrier stays gone (rate ⟂ carrier, rate ∝ envelope); the train sits in the neural low band ⇒ the collapse is preserved end-to-end. |
| [V] | Silent below the rheobase, firing above; strictly monotone across the band; CV(ISI) ≪ 1; depolarisation block at high drive; rate invariant to the carrier, rising with the envelope. |
| [L] | The frozen Neuron β=0.5/τ_f=1.0/τ_s=40.0; γ read byte-equal from the atlas (read-only offset). |
| [O] | The absolute Hz/spike-count, the f–I shape, the single-stage collapse of the retinal network, and the felt percept (→ mind volume). Each obstacle named. |
Reproducibility
Every number on this page is the code’s own output. The transcript below is the verbatim, hash-pinned stdout of the listed module(s); tools/gate_volume.py re-runs them and asserts HTML↔code drift 0.
research/E8-graded-to-spike-rate/run.pysha256 9f5ea6bd12f7f7489bd81978…
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E8 — GRADED → SPIKE-RATE RE-QUANTISATION (the hand-off; the ladder's last rung)
==========================================================================
(0) THE RE-QUANTISER IS THE FROZEN R19 NEURON (spike = membrane up-crossing = fold-crossing event):
inherited readers: Neuron.spikes / Neuron.rate_hz / dominant_freq ; β=0.5, τ_f=1.0, τ_s=40.0; SEED=19
graded photoreceptor signal → the slowly-varying DRIVE ; output → the spike RATE (a discrete count). [F]
(1) THE RHEOBASE FLOOR — from a SILENT hyperpolarised rest (drive=B=-1.0), a graded increment g:
g=0.00 drive=-1.00 rate=0.000000 (SILENT)
g=0.10 drive=-0.90 rate=0.000000 (SILENT)
g=0.20 drive=-0.80 rate=0.000000 (SILENT)
g=0.25 drive=-0.75 rate=0.007833 (firing)
g=0.30 drive=-0.70 rate=0.008667 (firing)
→ SILENT at g≤0.20, FIRES at g≥0.30: the all-or-none switch must be driven across its fold to
spike (the E2/E4 condition read forward). The rate code has a FLOOR (rheobase). [V]
(2) THE MONOTONE CODE — above the floor, the RATE rises strictly with the graded amplitude g:
g=0.25 drive=-0.75 rate=0.007833
g=0.30 drive=-0.70 rate=0.008667
g=0.40 drive=-0.60 rate=0.009500
g=0.50 drive=-0.50 rate=0.010167
g=0.60 drive=-0.40 rate=0.010500
g=0.70 drive=-0.30 rate=0.010833
g=0.80 drive=-0.20 rate=0.011167
g=0.90 drive=-0.10 rate=0.011333
→ strictly monotone increasing across the operating band (0.25→0.9): the graded
amplitude is RE-ENCODED as a firing rate (0.00783→0.01133). [F]/[V]
(3) A GENUINE QUANTISER — the output is a near-periodic point process (regular inter-spike clock):
g=0.40 nspikes= 57 mean ISI= 104.76 CV(ISI)=0.00017
g=0.60 nspikes= 63 mean ISI= 94.59 CV(ISI)=0.00024
g=0.80 nspikes= 67 mean ISI= 90.00 CV(ISI)=0.00012
→ CV(ISI) ≪ 1 (a clean clock): the continuous graded input is re-quantised into a discrete
spike train whose RATE carries the signal — not a graded continuation. [F]/[V]
(4) THE HONEST CEILING — the FULL-range f–I is NON-monotone (block on BOTH sides of the band):
drive=-1.00 rate=0.000000 (hyperpol. floor → SILENT)
drive=-0.30 rate=0.010833 (operating band → firing)
drive=+0.00 rate=0.011333 (near peak → firing)
drive=+1.00 rate=0.000167 (depol. block → SILENT)
→ the rate code is a BANDPASS in drive: a rheobase FLOOR and a depolarisation-block CEILING
(the switch fails to flip below, sticks ON above). Monotone only INSIDE a finite band.
The bandpass STRUCTURE is forced; the absolute band edges/rates are a named [O]. [V]+[O]
(5) THE CARRIER STAYS GONE (no mixing) — re-quantiser tracks the ENVELOPE, not the carrier:
(5a) FIX the slow envelope (drive=−0.3), sweep the carrier frequency f_c only:
carrier f_c= 0.0 rate=0.010833
carrier f_c= 0.5 rate=0.011000
carrier f_c= 1.0 rate=0.010833
carrier f_c= 2.0 rate=0.010833
carrier f_c= 5.0 rate=0.010833
carrier f_c= 10.0 rate=0.010833
→ rate spread across a 20× carrier span = 1.53% ⇒ rate ⟂ carrier (a mixer would track it). [F]
(5b) FIX the carrier (f_c=2), raise the depolarising envelope across the rising band:
envelope drive=-0.70 rate=0.008667
envelope drive=-0.60 rate=0.009500
envelope drive=-0.50 rate=0.010167
envelope drive=-0.40 rate=0.010500
envelope drive=-0.30 rate=0.010833
envelope drive=-0.20 rate=0.011167
→ rate rises with the envelope (0.00867→0.01117) ⇒ rate ∝ envelope. [F]/[V]
⇒ the hand-off carries the slow ENVELOPE and NEVER re-introduces the carrier: the E5/E7
~13-order down-conversion is PRESERVED across the re-quantisation. [F]
(6) THE RATE LIVES IN THE LOW BAND — the spike train's dominant frequency IS the rate, ≪ carrier:
operating point drive=−0.3: rate_hz=0.010833 dominant_freq(train)=0.010833
vs carrier f_c= 2.0: rate/carrier = 0.005417 (orders below)
vs carrier f_c=10.0: rate/carrier = 0.001083 (orders below)
→ the train's frequency content sits at the RATE, in the neural low band — the re-quantisation
does not undo the collapse. What crosses to the brain is the low-band rate. [F]
(Absolute Hz is the ladder's standing [O]; only the band PLACEMENT is forced.)
(7) WHERE γ SITS (READ-ONLY) — γ shifts the f–I POSITION (rheobase), not the band, not the signal:
raising a gene's γ raises its spinodal (the switch is harder to flip) ⇒ at fixed drive the rate
drops; γ is a STRUCTURAL excitability OFFSET. (γ moves the rate ~as much as the drive here, so
E8 makes NO 'γ negligible' claim — γ is read byte-equal from the atlas only to show the offset):
CNGB3 γ=1.2425 spinodal=0.5331 dwell∝γ^1.5=1.2591 rate(drive=−0.3)=0.009167
RPE65 γ=1.2842 spinodal=0.5601 dwell∝γ^1.5=1.3230 rate(drive=−0.3)=0.008833
GUCY2D γ=1.3650 spinodal=0.6138 dwell∝γ^1.5=1.4498 rate(drive=−0.3)=0.008333
RHO γ=1.4719 spinodal=0.6873 dwell∝γ^1.5=1.6234 rate(drive=−0.3)=0.007833
(band channel = τ [E7]; signal channel = graded drive [E8]; γ = structural offset. [F]/[V])
LADDER PLACEMENT (E8 closes the down-conversion spine E0→E8):
ν_light ~10¹⁴ Hz ──(E=hν → ONE flip; R19 event-detector, §E2)──▶ discrete flip-events
discrete flips ──(cascade low-pass, cutoff f_c=β/(2π·τ), §E7)──▶ graded ~Hz band
graded band ──(THIS rate re-quantisation; floor+monotone+ceiling, carrier gone)──▶ ~spike RATE
spike RATE ──(what seeing FEELS like)─────────────────────────────────▶ mind volume (firewall)
the rate carries the slow envelope, never the carrier; the ~13-order collapse is preserved end-to-end.
LEARNED: the eye's last step is a RE-QUANTISER, not a mixer. The graded transduction signal is
turned into an all-or-none spike train whose RATE encodes the graded amplitude — a
THRESHOLDED (rheobase), BOUNDED (depolarisation-block ceiling), MONOTONE code that tracks
the slow envelope and leaves the carrier discarded (E2) and filtered (E7). The down-
conversion survives end-to-end because colour/place are geometry and the rate is the
envelope, never the carrier. The absolute Hz, the f–I shape, the single-stage collapse,
and the felt percept (→ mind volume) are the named [O].
sha256: bf47f1d0ea6c85f471b516461afb7feb0537996511b20b89f7e2ab13a7df594d