Lattice representations suggest novel organizing principles for odorant and neural spaces

Understanding how small biological nervous systems organize stimulus-response relationships requires analytical tools capable of extracting interpretable structure from graded, noisy, and heterogeneous data. In this work we present an interpretable neuro-symbolic modelling and methodological framework based on K-Formal Concept Analysis, an extension of Formal Concept Analysis to real-valued contexts, to uncover organizational principles in the chemosensory system of Caenorhabditis elegans. Using K-FCA we show how to extract interpretable lattice-structured distributed representations of odorant profiles and neural activities from neural activity data, suggesting that C. elegans' sensorium uses lattice-structured spaces for encoding sensorial information. Our methodological approach combines three key contributions: (i) a cross-validated robustness analysis demonstrating that the conceptual hierarchies extracted by K-FCA remain stable under data perturbations; (ii) an automated, data-driven procedure for selecting the parameter φ, enabling reproducible exploration of graded relations without reliance on expert tuning; and (iii) an integrated analysis of excitatory and inhibitory responses using the algebras R‾_min,+ and R‾_max,+, which reveals functional patterns not captured by standard linear or nonlinear embeddings. Applied to calcium activity recordings from 23 odorants and 11 sensory neuron pairs, our framework uncovers coherent multi-level conceptual structures and identifies neurons-most notably ASJ-that consistently switch between activation and inhibition in a context-dependent manner. These results show that algebraic and order-theoretic representations can expose organizing principles of neural and stimulus spaces that remain inaccessible to geometric embeddings, suggesting new avenues for interpretable modelling and biologically informed neural architectures.