  The human brain contains approximately $10^9$ neurons, each with
approximately $10^3$ connections, synapses, with other neurons. Most sensory,
cognitive and motor functions of our brains depend on the interaction of a
large population of neurons. In recent years, many technologies are developed
for recording large numbers of neurons either sequentially or simultaneously.
An increase in computational power and algorithmic developments have enabled
advanced analyses of neuronal population parallel to the rapid growth of
quantity and complexity of the recorded neuronal activity. Recent studies made
use of dimensionality and model order reduction techniques to extract coherent
features which are not apparent at the level of individual neurons. It has been
observed that the neuronal activity evolves on low-dimensional subspaces. The
aim of model reduction of large-scale neuronal networks is an accurate and fast
prediction of patterns and their propagation in different areas of the brain.
Spatiotemporal features of the brain activity are identified on low dimensional
subspaces with methods such as dynamic mode decomposition (DMD), proper
orthogonal decomposition (POD), discrete empirical interpolation (DEIM) and
combined parameter and state reduction. In this paper, we give an overview of
the currently used dimensionality reduction and model order reduction
techniques in neuroscience.
  This work will be featured as a chapter in the upcoming Handbook on Model
Order Reduction,(P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W. H.
A. Schilders, L. M. Silveira, eds, to appear on DE GRUYTER)

Neurons and Cognition;Numerical Analysis;Numerical Analysis