Using Single-Neuron Representations for Hierarchical Concepts as Abstractions of Multi-Neuron Representations
Nancy Lynch
TL;DR
The paper tackles recognition of hierarchically structured concepts in partially information-rich, noise-prone spiking neural networks. It introduces a two-tier abstraction: abstract single-neuron networks $\mathcal{A}_1$ and $\mathcal{A}_2$ and detailed multi-neuron networks $\mathcal{H}$ (high connectivity) and $\mathcal{L}$ (low connectivity), proving that the detailed networks implement the abstract ones through two separate implementation notions. By proving firing guarantees via $\mathcal{A}_1$ and non-firing guarantees via $\mathcal{A}_2$ and establishing precise mappings, the work decomposes complex brain-mechanism analyses into tractable high-level reasoning and lower-level emulation. The results demonstrate how multi-neuron representations can be analyzed as abstractions of single-neuron representations, enabling robust recognition under failures and partial connectivity. This abstraction-based approach provides a framework applicable to broader neural mechanisms and paves the way for analyzing more complex network motifs beyond strictly feed-forward architectures.
Abstract
Brain networks exhibit complications such as noise, neuron failures, and partial synaptic connectivity. These can make it difficult to model and analyze their behavior. This paper describes a way to address this difficulty, namely, breaking down the models and analysis using levels of abstraction. We describe the approach for the problem of recognizing hierarchically-structured concepts. Realistic models for representing hierarchical concepts use multiple neurons to represent each concept [10,1,7,3]. These models are intended to capture some behaviors of actual brains; however, their analysis can be complicated. Mechanisms based on single-neuron representations can be easier to understand and analyze [2,4], but are less realistic. Here we show that these two types of models are compatible, and in fact, networks with single-neuron representations can be regarded as formal abstractions of networks with multi-neuron representations. We do this by relating networks with multi-neuron representations like those in [3] to networks with single-neuron representations like those in [2]. Specifically, we consider two networks, H and L, with multi-neuron representations, one with high connectivity and one with low connectivity. We define two abstract networks, A1 and A2, with single-neuron representations, and prove that they recognize concepts correctly. Then we prove correctness of H and L by relating them to A1 and A2. In this way, we decompose the analysis of each multi-neuron network into two parts: analysis of abstract, single-neuron networks, and proofs of formal relationships between the multi-neuron network and single-neuron networks. These examples illustrate what we consider to be a promising, tractable approach to analyzing other complex brain mechanisms.