Neural circuits construct distributed representations of key variables—external stimuli or internal constructs of quantities relevant for survival, such as an estimate of one’s location in the world—as vectors of population activity. Although population activity vectors may have thousands of entries (dimensions), we consider that they trace out a low-dimensional manifold whose dimension and topology match the represented variable. This manifold perspective enables blind discovery and decoding of the represented variable using only neural population activity (without knowledge of the input, output, behavior or topography). We characterize and directly visualize manifold structure in the mammalian head direction circuit, revealing that the states form a topologically nontrivial one-dimensional ring. The ring exhibits isometry and is invariant across waking and rapid eye movement sleep. This result directly demonstrates that there are continuous attractor dynamics and enables powerful inference about mechanism. Finally, external rather than internal noise limits memory fidelity, and the manifold approach reveals new dynamical trajectories during sleep.
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The code is available at https://fietelab.mit.edu/code/.
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The authors thank D. Tank, S. Lewallen and R. Low for insightful discussions, and F. Caccuci, T. Wills, S. Deneve, Y. Burak, J. Murray, J. Pillow, L. Paninski and M. Sahani for comments on the work or manuscript. I.F. is grateful to G. Prasad for pointing her to the field of topological data analysis several years ago, and to W. Bialek for raising the possibility of unsupervised discovery of encoded variables from neural data, also several years ago. This work was supported in part by grants from the NIH (U01-NS094330-03), the Simons Foundation (SCGB and the International Brain Laboratory) and the Howard Hughes Medical Institute through the Faculty Scholars Program to I.F., and by the Canadian Research Chair in Systems Neuroscience (245716), a CIHR Project Grant (155957), a NSERC Discovery Grant (RGPIN-2018-04600) and the IRDC (108877-001) to A.P. Part of this work was performed by R.C. and I.F. in residence at the Simons Institute for the Theory of Computing at UC Berkeley, where R.C. was a Google Research Fellow.
The authors declare no competing interests.
Peer review information: Nature Neuroscience thanks Vivek Jayaraman, Kate Jeffery, Sung Soo Kim, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
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Integrated supplementary information
Low-dimensional structure in the head direction circuit.
Distortion of manifold by linear methods and by inhomogeneous sampling.
Toroidal manifold in simulated grid cell activity.
Decoding performance does not strongly depend on embedding dimension or number of spline knots and performance is good with ~20 neurons and several minutes of data.
Wake dynamics for other animals.
Spiking variance and tuning curves explained by decoding.
Ring manifold preserved during REM across animals.
REM occupancy and flows.
REM decoding across animals.
Waking and REM states and dynamics replicated in a continuous attractor model.
Loss of ring and higher-dimensional nREM manifold across animals.
nREM decoding and dynamics controls.
Decoding, dynamics, and trajectory statistics during nREM for Mouse 25 are similar to those observed for Mouse 28.
Qualitative reproduction of nREM trajectory, dynamics, and statistics in continuous attractor ring network model.
Supplementary Figs. 1–14 and Supplementary Notes 1–4.
Wake manifold. Three-dimensional Isomap embedding of the waking manifold from Mouse 28, session 140313.
REM manifold. Three-dimensional Isomap embedding of the REM manifold from mouse 28, session 140313.
Waking dynamics. Dynamics during 200 s of waking (projection generated using Isomap). The moving trace shows 1 s (most recent point is darkest). The blue points in the background show the waking manifold.
REM dynamics. Dynamics during 200 s of REM sleep (projection generated using Isomap). The moving trace shows 1 s (most recent point is darkest). The green scatter plot in the background shows the REM manifold. Note that REM sleep intervals are typically quite short, so the 200 s comprises multiple concatenated intervals.
nREM manifold. Three-dimensional Isomap embedding of the nREM manifold from mouse 28, session 140313, shown in mustard yellow, along with the waking manifold for comparison, shown in blue.
nREM dynamics. Dynamics during 200 s of nREM sleep (projection generated using Isomap). The moving trace shows 1 s (most recent point is darkest). The mustard yellow scatter plot in the background shows the nREM manifold, and the blue points show the waking manifold.