multi-scale-information-geometry-neural

name

multi-scale-information-geometry-neural

description

Multi-scale information geometry framework for analyzing neural population codes. Extends Fisher information metric across stimulus coarse-graining scales to reveal mutual information structure. Use when analyzing: (1) neural population coding geometry, (2) Fisher information limitations in neural data, (3) representational geometry from first principles, (4) mutual information estimation from neural responses, (5) diffusion model-based neural encoding analysis. Trigger: multi-scale Fisher, information geometry neural, representational geometry, neural population code, Fisher information metric, coarse-graining, mutual information neural.

Multi-Scale Information Geometry for Neural Populations

Based on Azeglio et al. (2026), arXiv:2605.06304.

Core Insight

The Fisher information metric is purely local — two neural codes can have identical Fisher information everywhere yet differ markedly in global discriminability. A unique multi-scale Riemannian geometry emerges from axioms about how distances contract under stimulus coarse-graining (isotropic Gaussian diffusion).

Key Mathematical Framework

Multi-Scale Fisher Information Matrix

G(x) = ∫₀^∞ [φ_t * J_t](x) dt

Where:

• φ_t = isotropic Gaussian kernel of variance t
• J_t(x) = Fisher information of the diffused conditional p_t(r|x)
• * = spatial convolution

Four Axioms Deriving the Geometry

1. Contraction: Coarse-graining reduces discriminability → distances contract
2. Locality: Contraction depends only on p(r|x) and first derivatives at x
3. Sufficiency: Invariant to sufficiency-preserving transformations of r
4. Zero baseline: If r ⊥ x, then G(x) = 0

Fundamental Identity: Mutual Information = Expected Magnification

I(R; X) = (1/2) · E_x[Tr(G(x))]

Mutual information equals the expected rate at which squared distances increase under infinitesimal stimulus perturbations. Well-encoded directions are expanded; poorly encoded directions are contracted.

Estimation via Diffusion Models

The metric tensor can be estimated using conditional diffusion models:

dxᵀ J_t(x_t) dx = (1/2t²) · E_{r,r'}[(x̂(x_t,r) - x̂(x_t,r'))ᵀ dx]²

Where x̂(x_t, r) = E[x | x_t, r] is the posterior mean estimated by a diffusion model.

Comparison: Fisher vs Multi-Scale Fisher

Property	Fisher J(x)	Multi-Scale G(x)
Scope	Local (infinitesimal)	All scales
Mutual info link	None	Exact: I = ½E[Tr(G)]
Model artifacts	Sensitive to unconstrained directions	Robust (data-constrained)
Tuning curve discrimination	Identical for bell/monotonic	Correctly distinguishes

Practical Application Pipeline

Step 1: Fit Encoding Model

Train a model to predict neural responses r from stimulus x (e.g., CNN for visual stimuli).

Step 2: Compute Diffused Fisher at Multiple Scales

For each scale t_k (discretized):

• Compute J_t(x) from the diffused distribution p_t(r|x)
• Use Tweedie's identity to relate score to posterior mean

Step 3: Integrate Across Scales

G(x) ≈ Σ_k [φ_{t_k} * J_{t_k}](x) · Δt_k

Approximate the expectation over z via Monte Carlo sampling.

Step 4: Analyze Eigenvectors

Leading eigenvectors of G(x) identify stimulus directions contributing most to information transmission. These are:

• V1: Spatially localized, edge-like (fine structure)
• V4: Broader, spatially distributed (global structure)

Key Findings from Visual Cortex Analysis

• Applied to macaque V1/V4 recordings (Papale et al. dataset)
• 50 most reliable neurons per area, InceptionV3 encoding model
• V1 correlation: 0.75, V4 correlation: 0.71
• Multi-scale geometry shows clear V1/V4 differentiation
• Fisher eigenvectors show no clear differentiation (high-frequency noise artifacts)

When to Use This Framework

• Instead of Fisher: When global discriminability matters, not just local sensitivity
• For high-dimensional stimuli: When encoding models are weakly constrained in some directions
• For comparing cortical areas: When seeking robust, model-independent geometric features
• For mutual information: When needing exact MI-geometry correspondence

Limitations

• Requires fitting an encoding model first
• Computationally intensive for very high-dimensional stimuli
• Diffusion model estimation adds approximation error at small t
• Assumes isotropic Gaussian diffusion (may not match all experimental designs)

Related Concepts

• Čencov's theorem (uniqueness of Fisher metric)
• Tweedie's identity (score-posterior relationship)
• Heat equation (diffusion process)
• Riemannian geometry on stimulus space
• Neural population coding