hyperbolic-neural-population-geometry-computation

name	hyperbolic-neural-population-geometry-computation
description	Hyperbolic geometry framework for hippocampal neural population activity. Provides theoretical construction of hyperbolic tuning curves, connects neural decoding to associative memory via Modern Hopfield Network, and introduces hyperbolic-space associative memory with significantly larger capacity. Use when studying hippocampal encoding, hyperbolic cognitive maps, memory capacity optimization, or neural population geometry.
category	neuroscience
tags	["hyperbolic-geometry","hippocampus","neural-decoding","associative-memory","hopfield-network","cognitive-map","memory-capacity","spatial-encoding"]
arxiv_id	2606.10238
paper_title	Hyperbolic Neural Population Geometry Benefits Computation
authors	["Dennis Wu","Yi-Chun Hung","Braden Yuille","James E. Fitzgerald","Han Liu"]
published_date	2026-06-08T00:00:00.000Z
conference	ICML 2026

Hyperbolic Neural Population Geometry Benefits Computation

Summary

This paper provides a theoretical framework explaining why hippocampal neural population activity exhibits hyperbolic geometry. It:

1. Proposes plausible construction of hippocampal tuning curves inducing hyperbolic geometry
2. Connects neural decoding to associative memory (Modern Hopfield Network → MMSE estimator)
3. Introduces hyperbolic-space associative memory with larger capacity

Key insight: Animals encode spatial information as latent hyperbolic cognitive maps, improving both memory capacity and decoding accuracy.

Core Contributions

1. Hyperbolic Tuning Curve Construction

Hippocampal place cells encode spatial location through tuning curves. The paper shows:

• Place cell firing rates follow a distribution that statistically induces hyperbolic geometry
• Construction based on exponential decay of firing rate with distance from place field center
• Mathematical proof: this tuning structure naturally maps neural activity to hyperbolic space

# Conceptual tuning curve model
def place_cell_tuning(x, center, sigma):
    """Hippocampal place cell tuning curve.
    Exponential decay induces hyperbolic geometry.
    """
    distance = np.abs(x - center)
    return np.exp(-distance / sigma)

# Hyperbolic distance encoding
def hyperbolic_embedding(positions, curvature=-1):
    """Embed positions in hyperbolic space.
    Curvature < 0 enables exponential expansion of space.
    """
    # Map 2D positions to Poincaré disk
    r = np.linalg.norm(positions, axis=1)
    return positions / (1 + r**2)  # Poincaré disk model

2. Neural Decoding ↔ Associative Memory Connection

Key theorem: Modern Hopfield Network update rule computes Minimum Mean-Squared Error (MMSE) estimator