  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.
