Exploratory preprint · Open academic deposit
Rate-level robustness under quantisation, on connectome-derived graphs
Do firing rates survive quantisation because the graph is special, or because the arithmetic makes it so? Four datasets, a null model, and an answer associated with density — with all the reservations such a result imposes.
What is observed
I instantiate simplified LIF models from four graphs derived from published connectomes, and run each one twice, in lockstep and under identical conditions: once in integer arithmetic (int64 storage, 16 fractional-bit grid, ArithSpec substrate), once in float64. On all four graphs, at 5,000 steps, the two levels of description come apart:
| Graph | Neurons | Edges | Spikes identical | Rate correlation (5,000 steps) |
|---|---|---|---|---|
| FlyWire, adult Drosophila (subsample) | 1,000 | 24,788 | 0% | 0.999739 |
| C. elegans, hermaphrodite | 419 | 4,681 | 0% | 0.999891 |
| Drosophila larva | 2,952 | 110,140 | 0% | 0.999666 |
| Mouse retina (inner plexiform layer) | 1,104 | 181,724 | 0% | 0.999173 |
Exact spikes are preserved on no graph, while the per-neuron rate vector stays correlated above 0.999. On the FlyWire subsample, this separation holds over one million steps (200 checkpoints; mean rate stable at 0.05638, no silent neuron), and the same run gives an identical SHA-256 fingerprint on x86-64 and on a Raspberry Pi 5 (native ARM64), for the reference seed and five further seeds, after a double execution on each platform.
Two complementary tests on synthetic networks indicate which way the mechanisms push. Spatial averaging strengthens the separation: with several dendritic compartments, the rate correlation rises from 0.997976 to 0.999998. Temporal accumulation degrades it: with multi-timescale synaptic dynamics (facilitation and depression), the rate correlation collapses to −0.48 at Q8, then recovers to 0.9925 at Q12 and 0.9948 at Q16, still below the simple case. That is why no robustness is claimed at Q8 on the connectomes.
The question and its trap
Spiking models instantiated from connectome-derived graphs preserve their firing rates under quantisation. Two explanations compete:
The trap is a classic one: without a point of comparison, what belongs to the arithmetic gets attributed to the graph. The control therefore consists in generating random graphs matched on out-degree — the same number of outgoing edges per neuron, the same weights, but randomised targets — and then comparing.
What the null model controls for
The null model preserves only the out-degree and the global weight distribution. It preserves neither in-degree, nor reciprocity, nor cell types, nor spatial geometry, nor community structure. It therefore does not isolate “topology” in general — only a difference relative to this precise null. Every conclusion must remain at that scale.
The result, connectome by connectome
source: paper E, Table 2 (permutation test)
See the data
| Connectome | n | Edges | Syn/neuron | Observed | Permuted null | Delta | N | p |
|---|---|---|---|---|---|---|---|---|
| C. elegans | 419 | 4,681 | 11.2 | 0.9991 | 0.9996 ± 0.0003 | −0.0004 | 1,000 | 0.915 |
| FlyWire | 1,000 | 24,788 | 24.8 | 0.9999 | 0.9997 ± 0.0001 | +0.0002 | 1,000 | 0.005 |
| Drosophila larva | 2,952 | 110,140 | 37.3 | 0.9984 | 0.9836 ± 0.0005 | +0.0148 | 100 | 0.010 |
| Mouse retina | 1,104 | 181,724 | 164.6 | 0.9975 | 0.8225 ± 0.0657 | +0.1750 | 100 | 0.010 |
In this sample of four graphs, the sparser the network, the more generic the observed robustness — chance does just as well. The denser it is, the more the real structure appears to play a part. As if, at low density, temporal averaging alone were enough to smooth the errors, independently of the wiring. This is a reading compatible with the data, not an established mechanism: density is not isolated from the other factors.
How to read this result
Exact wording matters more here than elsewhere. These observations are compatible with differences relative to the nulls tested. They confirm neither a causal dependence on topology, nor a density transition, nor an intrinsic property of the wiring.
Density, topology, weights, species, developmental stage and measurement modality all vary jointly across this sample of four datasets. None of them is isolated. The mouse retina result is a strong association within this dataset, not an identified causal mechanism.
The connectome data are compatible with a transition somewhere between roughly 11 and 25 synapses per neuron. A complementary synthetic sweep (500 neurons, 10 densities from 5 to 200 synapses per neuron, 5 topologies, 5 seeds, 250 configurations) points the same way: no exception from 30 synapses per neuron onward. But this estimate rests on four connectomes and does not separate density from the factors that vary with it: no established transition is claimed.
Three statistical reservations
- For the larva and the mouse retina, the p = 0.010 values sit at the resolution floor of a 100-permutation test — the true value may be smaller, but the test cannot say so. These effects are provisional.
- The table entries use 1,000 time steps, whereas the paper’s main measurements use 5,000. The observed/permuted comparison is at equal duration within the table, but not from one section to another. Since the proposed mechanism is precisely temporal averaging, these deltas remain provisional until the control is re-run at 5,000 steps.
- On the mouse retina, the per-neuron rate distribution is close to uniform: this is the regime in which the Pearson correlation becomes ill-conditioned, and the concordance re-analysis covered the real graphs only, not the permuted ones. Part of the +0.175 delta may therefore be a correlation artefact rather than a topology effect: this delta is provisional on two counts.
Scope and limitations
- The scope is RG0 — structural graph only. No biological function, no cognition, no evolutionary conservation, no generalisation to living systems.
- These are structural graphs derived from connectomes, not biological connectomes. The weights are edge counts from reconstruction, not conductances.
- The mouse retina connectome uses a contact-area matrix made bidirectional — not chemical synapses with annotated polarity. Bidirectionalisation adds a reciprocity that the null also destroys.
- The branch described as Q16.16 is stored in int64 on a 16 fractional-bit grid: it is not a strict INT32 data path.
- Q8 was tested only on synthetic networks, where one multi-timescale configuration collapses (−0.48): no robustness is claimed at Q8 on the connectomes, and nothing licenses extrapolation below Q6 (the fragility threshold of the sweeps lies between Q4 and Q6).
- The 1,000-neuron FlyWire subsample is selected by degree: it over-represents highly connected neurons, which plausibly acts in the direction of the reported effect. The FlyWire release used to build it is not pinned in the paper.
- The margin-to-error criterion ranks the spike-exact contract on synthetic networks (AUC 0.961) and the rate contract less well (0.808); it has no discriminant power on the connectome-derived graphs and certifies neither contract.
- The determinism check (double execution, same fingerprint) verifies the implementation; it is not evidence of the observed robustness.
- Inference only, fixed weights: learning under quantised arithmetic is not tested here.
- No hardware sizing rule. No commercial neuromorphic processor was run; replay on RISC-V was done in emulation only (QEMU).
- The full permutation distributions are not characterised beyond mean and standard deviation. Nested null models remain to be done.
- The measurements come from experiments already run in this programme (except the mouse retina): their re-analysis establishes internal consistency, not independent confirmation.