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==== Topology of population dynamics ====
[[Dimensionality reduction]] and [[topological data analysis]], have revealed that the population code is constrained to low-dimensional manifolds<ref>{{cite journal| title=Neural population dynamics during reaching|first1=MM|last1=Churchland|first2=JP|last2=Cunningham |first3=MT|last3=Kaufmann|first4=JD|last4=Foster|first5=P|last5=Nuyujukian|first6=SI|last6=Ryu|first7=KV|last7=Shenoy|journal=Nature|issue=487|pages=51-56|date=2012|doi=10.1038/nature11129}}</ref>, sometimes also referred to as [[attractors]]. The position along the neural manifold correlates to certain behavioral conditions like head direction neurons in the anterodorsal thalamic nucleus forming a ring structure<ref>{{cite journal |last1=Chaudhuri |first1=R |last2=Gercek |first2=B |last3=Pandey |first3=B |last4=Peyrache |first4=A |last5=Fiete |first5=I |title=The intrinsic attractor manifold and population dynamics of a canonical cognitive circuit across waking and sleep |journal=Nature Neuroscience |date=2019 |issue=22 |pages=1512-150 |doi=10.1038/s41593-019-0460-x}}</ref>, [[grid cells]] encoding spatial position in [[entorhinal cortex]] along the surface of a [[torus]]<ref>{{cite journal |last1=Gardner |first1=RJ |last2=Hermansen |first2=E |last3=Pachitariu |first3=M |last4=Burak |first4=Y |last5=Baas |first5=NA |last6=Dunn |first6=BA |last7=Moser |first7=MB |last8=Moser |first8=EI |title=Toroidal topology of population activity in grid cells |journal=Nature |date=2022 |issue=602 |pages=123-128 |doi=10.1038/s41586-021-04268-7}}</ref>, or [[motor cortex]] neurons encoding hand movements<ref>{{cite journal |last1=Gallego |first1=JA |last2=Perich |first2=MG |last3=Miller |first3=LE |last4=Solla |first4=SA |title=Neural Manifolds for the Control of Movement |journal=Neuron |date=2017 |volume=94 |issue=5 |pages=978-984 |doi=10.1016/j.neuron.2017.05.025}}</ref> and preparatory activity<ref>{{cite journal |last1=Churchland |first1=MM |last2=KV |first2=Shenoy |title=Preparatory activity and the expansive null-space |journal=Nature Reviews Neuroscience |date=2024 |issue=25 |pages=213-236 |doi=10.1038/s41583-024-00796-z}}</ref>. The low-dimensional manifolds are known to change in a state dependent manner, such as eye closure in the [[visual cortex]]<ref>{{cite journal |last1=Morales-Gregorio |first1=A |last2=Kurth |first2=AC |last3=Ito |first3=J |last4=Kleinjohann |first4=A |last5=Barthelemy |first5=FV |last6=Brochier |first6=T |last7=Gruen |first7=S |last8=van Albada |first8=S |title=Neural manifolds in V1 change with top-down signals from V4 targeting the foveal region |journal=Cell Reports |date=2024 |volume=43 |issue=7 |page=114371 |doi=10.1016/j.celrep.2024.114371}}</ref>, or breathing behavior in the [[ventral respiratory column]]<ref>{{cite journal |last1=Bush |first1=NE |last2=Ramirez |first2=JM |title=ventral respiratory column |journal=Nature Neuroscience |date=2024 |issue=27 |pages=259-271 |doi=10.1038/s41593-023-01520-3}}</ref>.
=== Sparse coding ===
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