# Equivalence of neural field dynamics with different embedding dimensionality

**Abstract**

Early recordings of cats’ somatosensory and visual cortices suggested that the cortical sheet is organized in vertical columns of functionally similar neurons. The columnar organization of the cortex motivated the design of spatially structured models of neural population dynamics, namely neural field models, where the spatial dimensions corresponded to the two dimensions of the cortical sheet. Later it has been shown that not only the physical position of the neurons can be used to parametrize their function, but also their location in more abstract “embedding spaces”, for example a space of “neuron celltypes”.

Decoupling the embedding space from the physical position of neurons on the cortical sheet begs the question of uniqueness of the embedding space. Given an embedding space and neural dynamics thereon, are there other embedding spaces which can express the same behavior? Is there a “natural” choice for the embedding space?

In this work, we show that the answer to the second question is highly nontrivial because given some neural dynamics, we can find multiple equivalent embedding spaces in different dimensions. We illustrate this using a toy example of a 2 (or 3)-dimensional embedding that is mapped to an equivalent 1-dimensional embedding. This is done by applying measurable bijective mappings \(S : [0,1]^2 \mapsto [0,1]\), which lead, as we show, to well-behaved numerical simulation schemes. In this, we exploit the key concept of “locality”, which expresses the property of (possibly fractal) neural fields, in which neighbouring neurons in the embedding space *tend* to have similar functions.