  Recent developments in experimental neuroscience make it possible to
simultaneously record the activity of thousands of neurons. However, the
development of analysis approaches for such large-scale neural recordings have
been slower than those applicable to single-cell experiments. One approach that
has gained recent popularity is neural manifold learning. This approach takes
advantage of the fact that often, even though neural datasets may be very high
dimensional, the dynamics of neural activity tends to traverse a much
lower-dimensional space. The topological structures formed by these
low-dimensional neural subspaces are referred to as neural manifolds, and may
potentially provide insight linking neural circuit dynamics with cognitive
function and behavioural performance. In this paper we review a number of
linear and non-linear approaches to neural manifold learning, by setting them
within a common mathematical framework, and comparing their advantages and
disadvantages with respect to their use for neural data analysis. We apply them
to a number of datasets from published literature, comparing the manifolds that
result from their application to hippocampal place cells, motor cortical
neurons during a reaching task, and prefrontal cortical neurons during a
multi-behaviour task. We find that in many circumstances linear algorithms
produce similar results to non-linear methods, although in particular in cases
where the behavioural complexity is greater, nonlinear methods tend to find
lower dimensional manifolds, at the possible expense of interpretability. We
demonstrate that these methods are applicable to the study of neurological
disorders through simulation of a mouse model of Alzheimers Disease, and
speculate that neural manifold analysis may help us to understand the
circuit-level consequences of molecular and cellular neuropathology.
