051 — Bayesian cue combination in a PING network

Status: proposal — not yet run. This entry pre-registers the hypotheses, design, and pass/fail criteria before any data is collected, so the result cannot be reverse-engineered into a confirmation.

Abstract

A leaky-integrate-and-fire PING network is given two noisy sensory cues about the same hidden variable and trained, by plain gradient descent on a squared-error loss, to report a single estimate. The question is whether the trained network spontaneously performs Bayes-optimal precision-weighted averaging — weighting each cue by its reliability — without that arithmetic ever being written into the architecture. If it does, the entry then asks a sharper, riskier question owed to the sampling-school reading (see ar007): does the network’s gamma rhythm carry the posterior uncertainty, with tighter E-cell bands when both cues are reliable and looser bands when either is noisy? This is the project’s first contact between the PING machinery and the uncertainty-representation literature (Ernst & Banks 2002; Ma, Beck, Latham & Pouget 2006).

Background: cue combination as Bayes

Two cues $s_A, s_B$ about a latent $s$, each a noisy observation $s_A = s + \epsilon_A$, $\epsilon_A \sim \mathcal{N}(0, \sigma_A^2)$, and likewise for $B$. With a flat prior and independent Gaussian noise, the posterior over $s$ is Gaussian, and its mean is the precision-weighted average of the two cues:

Precisions ($1/\sigma^2$) add; the more reliable cue pulls the estimate harder; the combined estimate is strictly tighter than either cue alone. This is the calculation human observers were shown to perform near-optimally in visual–haptic size judgement (Ernst & Banks 2002), and the operation that probabilistic population codes make linear in neural activity (Ma et al. 2006). The full derivation and the worked Gaussian case are in ar007 — Uncertainty & Bayesian inference in the cortex.

The point of this notebook is that nothing in the network is told $\sigma_A$ or $\sigma_B$. The weights $1/\sigma^2$ that optimal combination requires would have to be inferred, per trial, from the statistics of the input itself — a noisier cue produces a broader, lower population bump — and applied by the recurrent dynamics. Whether gradient descent finds that solution is an empirical question.

Hypotheses

• H1 (primary) — the computation emerges. A PING network trained with BPTT and an $L_2$ loss on the two-cue task produces readout estimates $\hat{s}$ consistent with Bayes-optimal precision-weighted averaging, without precision-weighting being built in.
• H2 (secondary, the conjecture) — the rhythm is legible. Gamma-band dispersion in the E-cell raster tracks the analytical posterior variance $\sigma_*^2$: tighter bands when both cues are reliable, looser bands when either cue is noisy.
• H3 (tertiary) — the two confidences agree. Two independent uncertainty read-outs — the readout-implicit posterior width (spread of population activity) and the raster band dispersion — co-vary trial by trial. Where they diverge localises where in the network uncertainty is actually represented.
• H4 (control) — the rhythm is the substrate. Against a non-rhythmic conductance control (same COBANet, driven to an asynchronous-irregular operating point with no gamma cycle), the temporal uncertainty channel of H2/H3 is absent — the control has no band to be tight or loose around. If the control nonetheless matches PING on H1, precision-weighting is generic to the conductance network; if PING additionally carries legible posterior width that the control cannot, the rhythm is what provides it.

Setup

Inputs. Two input populations $A$ and $B$, ≈ 50 neurons each, with Gaussian tuning curves tiling $s \in [-1, 1]$. On each trial the latent $s$ is drawn uniformly; each population is driven by a bump centred on its own corrupted cue value $s_A = s + \epsilon_A$ (resp. $s_B$), with $\epsilon_A \sim \mathcal{N}(0, \sigma_A^2)$ independent of $\epsilon_B$. The reliabilities $\sigma_A, \sigma_B$ are test-time knobs — a less reliable cue is delivered as a broader, lower-gain bump, so the network must read reliability off the input statistics, never from a label.

Network. Both input populations project to the PING E-cells through learned weights $W_\text{in}$; the recurrent E↔I loop is the standard COBANet PING substrate used throughout this collection. The gamma rhythm is the network’s own, not imposed.

Non-rhythmic control. The same task is trained on a second copy of the same COBANet driven to a non-rhythmic, asynchronous-irregular operating point — the V&S-style regime of nb050 (fixed fan-in, $I\to I$ coupling, per-cell independent drive), which produces a broadband spectrum and no gamma cycle. This is the same PING-vs-non-rhythmic contrast nb050 used for the balanced state, here repurposed as the control for uncertainty representation: the two networks share architecture, readout, loss, and training schedule, and differ only in whether a rhythm exists. The control has no gamma band, so the temporal channel that H2/H3 measure is structurally unavailable to it — any uncertainty it represents must live in the rate-amplitude channel (population bump width/gain), the PPC mechanism, which needs no oscillation.

Readout. Population-vector decode of $\hat{s}$ over an integer number of gamma cycles (so the estimate is phase-consistent), with a plain linear readout run in parallel as a sanity check. The population-activity spread around $\hat{s}$ gives the readout-implicit posterior width used in H3.

Training. BPTT with surrogate gradients (the ar006 recipe), $L_2$ loss $\lVert \hat{s} - s \rVert^2$. Crucially, $\sigma_A, \sigma_B$ are sampled per trial across a range during training, so the network sees varied and mixed reliability and cannot collapse to a fixed weighting. Optimal behaviour, if it appears, is the cheapest way to minimise loss over that distribution — not something the loss names.

A caveat that shapes how H2/H3 should be read. The $L_2$ loss rewards only the point estimate $\hat{s}$; the posterior width $\sigma_*^2$ is needed for nothing the loss measures. Computing the weighted mean (H1) forces the network to represent the input reliabilities $\sigma_A, \sigma_B$ instrumentally — that much is load-bearing — but there is no gradient pressure to represent the output width at all. So any $\sigma_*^2$ that shows up in the raster or the readout (H2, H3) is emergent and free, a structural by-product of the dynamics rather than a trained quantity. That is precisely what makes the PING-mechanism conjecture interesting, but it also means H2/H3 may have no teeth under this loss. If the emergent signal is weak or absent, a follow-up should add a task that requires uncertainty — a confidence read-out scored on calibration, a cost-asymmetric loss, or temporal integration where propagating $\sigma^2$ pays — to give uncertainty representation something to be selected for.

Tests and pass conditions

T1 and T4 run on both networks; T2 (which needs a gamma band) runs on PING only; T3 runs on both, using each network’s available channels.

A clean result also reproduces the two qualitative Ernst–Banks signatures inside T1: as one cue is degraded, $\hat{s}$ shifts towards the more reliable cue by the precision-predicted amount, and the combined estimate is tighter than either single-cue estimate.

Planned figures. (1) $\hat{s}$ vs $\hat{s}^*$ scatter across the $\sigma_A \times \sigma_B$ grid with the unit line, PING and control overlaid. (2) Cue-shift curves: estimate vs cue conflict at several reliability ratios, against the Bayesian prediction. (3) Band dispersion vs $\sigma_*^2$ (PING). (4) Readout width vs $\sigma_*^2$ for PING and control side by side — the panel that shows whether the control carries uncertainty in the rate channel at all. (5) Example rasters at a reliable and an unreliable operating point, PING (banded) above control (scattered).

Falsification map

The three hypotheses are nested, so the failure points are diagnostic rather than fatal-or-nothing:

• T1 fails. The network is not doing cue combination at all. Abandon the Bayesian framing for this architecture — H2 and H3 are then moot.
• T1 passes, T2 fails. The inference happens but is not written into the raster. The professor’s conjecture (H2) is wrong as stated: uncertainty is computed but encoded somewhere other than gamma-band structure.
• T2 and T3 diverge. The two confidence channels disagree, which localises the representation — uncertainty lives in the readout population’s spread but not in the rhythm’s dispersion (or vice versa). This is the most informative outcome: it says where the network keeps its uncertainty.

The non-rhythmic control (T4) then resolves what the rhythm specifically contributes. Note that passing T1 already requires representing the input reliabilities, so a control that passes T1 is never literally without an interior uncertainty metric — the question is only whether it represents the posterior width in any legible form:

• Control matches PING on T1. Precision-weighting the mean is generic to the conductance network — the rhythm is not needed to compute the combined estimate. Expected, and consistent with the PPC account (Ma et al. 2006).
• Control fails T1 under the time budget. The rhythm acts as an inference accelerant: within a few-cycle window PING reaches the weighted estimate while the asynchronous control mixes too slowly. This is the Aitchison–Lengyel (ar007) angle — gamma as the momentum of a fast sampler — and would be a stronger claim than the representational one.
• PING tracks $\sigma_*^2$, control does not (in any channel). The rhythm is the uncertainty substrate — the strong form of the conjecture. Posterior width is legible only where there is a cycle to disperse around.
• Both track $\sigma_*^2$ in the rate channel, only PING also in band timing. The likeliest real outcome: the rhythm supplies a redundant, more legible second read-out rather than a unique one. Uncertainty is represented either way; PING just shows its work.

What’s at stake

If H1 holds, this is a concrete instance of the sampling-school claim that probabilistic computation can fall out of trained recurrent E/I dynamics (Echeveste et al. 2020, via ar007) — reached here by ordinary gradient descent rather than by a sampling objective. If H2 also holds, it ties the project’s PING work directly to uncertainty representation: the gamma rhythm would be doing double duty as both the network’s clock and its confidence gauge. The most likely real outcome — T1 passes, H2 partly holds — is itself the interesting one, because the divergence in T3 is what would tell us where the uncertainty is.

Next steps

1. Implement the task and runner. A two-cue input generator (Gaussian-tuned populations with per-trial reliability), the population-vector readout over whole gamma cycles, and the runner at src/notebooks/nb051.py with hardcoded recipe (tier + modal-gpu only). The runner trains two networks on the identical task — the PING substrate and the nb050 non-rhythmic control — sharing readout, loss, and schedule.
2. Pilot at the tiny tier to confirm the network trains to non-trivial accuracy on a single reliability level before opening the mixed-reliability regime.
3. Run T1 first — it gates everything. Only if the precision-weighting is real do T2, T3, and the T4 contrast carry meaning.
4. If H2/H3 come out weak, add a confidence-requiring task variant (calibration-scored confidence output, cost-asymmetric loss, or temporal integration) so that posterior width is something the loss selects for rather than an incidental by-product.