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The Bayesian sampling in a canonical recurrent circuit with a diversity of inhibitory interneurons
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Builds a nonlinear ring-attractor canonical circuit with E, PV and SOM neurons and shows analytically that E+PV implements Langevin sampling of stimulus posteriors while adding tuning-dependent SOM inhibition upgrades it to Hamiltonian sampling.
Problem
How diverse inhibitory interneurons (PV and SOM) in canonical recurrent cortical circuits contribute to sampling-based Bayesian inference of latent stimulus posteriors remains poorly understood, especially for nonlinear circuit dynamics.
Approach
A nonlinear ring-attractor circuit with divisive global PV inhibition and additive, stimulus-tuned SOM inhibition; perturbation and eigen-analysis project the dynamics onto a stimulus-feature manifold, where E+PV yields Langevin sampling and SOM feedback yields a mixed Langevin-Hamiltonian sampler, requiring SOM to receive no direct feedforward input.
Key results
- The reduced E+PV circuit samples the likelihood via Langevin dynamics, with cross-correlation matching the predicted exponential decay exp(-g_EF t/tau_E).
- Adding SOM neurons with no direct feedforward input to SOM realizes Hamiltonian-like sampling that yields faster decay of sample autocorrelation and KL divergence than the no-SOM circuit.
- The optimal feedforward weight for Langevin sampling is derived as w_EF = 2*sqrt(3)/3 * F_E, and the full circuit with fixed weights samples posteriors of varying uncertainty when in the linear regime.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add tuning-dependent SOM inhibition to E neurons (ablation) | E + PV circuit · Stimulus posterior | sampling speed | Helped |
| Remove SOM neurons (ablation) | E + PV + SOM circuit · Stimulus posterior | sampling algorithm | Mixed |
| Add SOM neurons (ablation) | E + PV circuit · Bimodal stimulus posterior | sampling capability | Helped |
| Increase PV-to-E inhibition (ablation) | E + PV circuit · Stimulus posterior | sampling speed | Helped |
| Vary SOM-to-E inhibition (ablation) | E + PV + SOM circuit · Stimulus posterior | sampling speed | Mixed |
| Remove direct feedforward input to SOM neurons (ablation) | E + PV + SOM circuit · Stimulus posterior | Hamiltonian sampling | Helped |
| Tune feedforward and SOM-to-E weights to the sampling condition | E + PV + SOM circuit · Stimulus posterior | posterior matching | Helped |
| Operate the circuit in the linear input-output regime | E + PV + SOM circuit · Stimulus posteriors with varying uncertainty | posterior matching | Helped |
| Increase inter-circuit coupling | Coupled circuits · Bivariate stimulus posterior | prior precision | Helped |
| Add SOM neurons to the coupled circuits (ablation) | Coupled circuits without SOM neurons · Bivariate stimulus posterior | sampling speed | Helped |
Surprising findings
- SOM inhibition produces temporal oscillations between samples (imaginary part of eigenvalues), and sampling speed is a non-monotonic function of the SOM inhibitory weight.
- Hamiltonian sampling is only realized if SOM neurons receive no direct feedforward input, which is consistent with neuroanatomy.
Limitations
- VIP neurons are not included in the model.
- The circuit stores only a uniform stimulus prior due to neuron homogeneity; non-uniform priors need heterogeneity not analyzed.
- Bimodal posterior sampling is not comprehensively linked to posteriors and the model infers only static stimuli.
Experimental setup
Metrics: cross-correlation of samples KL divergence to posterior JS divergence eigenvalues of sampling dynamics
Baselines: linear sampling circuit (Echeveste et al.) circuit without SOM neurons
Builds on: Langevin sampling Hamiltonian Monte Carlo canonical ring attractor network divisive normalization neural sampling hypothesis
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