Abstract
An increasingly popular approach to the analysis of neural data is to treat activity patterns as being constrained to and sampled from a manifold, which can be characterized by its topology. The persistent homology method identifies the type and number of holes in the manifold, thereby yielding functional information about the coding and dynamic properties of the underlying neural network. In this work, we give examples of highly nonlinear manifolds in which the persistent homology algorithm fails when it uses the Euclidean distance because it does not always yield a good approximation to the true distance distribution of a point cloud sampled from a manifold. To deal with this issue, we instead estimate the geodesic distance which is a better approximation of the true distance distribution and can therefore be used to successfully identify highly nonlinear features with persistent homology. To document the utility of the method, we utilize a toy model comprised of a circular manifold, built from orthogonal sinusoidal coordinate functions and show how the choice of metric determines the performance of the persistent homology algorithm. Furthermore, we explore the robustness of the method across different manifold properties, like the number of samples, curvature and amount of added noise. We point out strategies for interpreting its results as well as some possible pitfalls of its application. Subsequently, we apply this analysis to neural data coming from the Visual Coding-Neuropixels dataset recorded at the Allen Institute in mouse visual cortex in response to stimulation with drifting gratings. We find that different manifolds with a non-trivial topology can be seen across regions and stimulus properties. Finally, we interpret how these changes in manifold topology along with stimulus parameters and cortical region inform how the brain performs visual computation.
Similar content being viewed by others
Data Availability Statement
The data analysis was done on the Visual Coding-Neuropixels dataset from the Allen Institute. This dataset can be found at: https://portal.brain-map.org/explore/circuits/visual-coding-neuropixels
References
(2019) Allen brain observatory—neuropixels visual coding. https://portal.brain-map.org/explore/circuits/visual-coding-neuropixels
Adams H, Emerson T, Kirby M, Neville R, Peterson C, Shipman P, Chepushtanova S, Hanson E, Motta F, Ziegelmeier L (2017) Persistence images: a stable vector representation of persistent homology. J Mach Learn Res 18(1):218–252
Andermann ML, Kerlin AM, Roumis DK, Glickfeld LL, Reid RC (2011) Functional specialization of mouse higher visual cortical areas. Neuron 72(6):1025–1039. https://doi.org/10.1016/j.neuron.2011.11.013
Babichev A, Dabaghian YA (2018) Topological schemas of memory spaces. Front Comput Neurosci 12:27. https://doi.org/10.3389/fncom.2018.00027
Babichev A, Ji D, Mémoli F, Dabaghian YA (2016) A topological model of the hippocampal cell assembly network. Front Comput Neurosci 10:50. https://doi.org/10.3389/fncom.2016.00050
Benjamini Y, Hochberg Y (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing. J Roy Stat Soc: Ser B (Methodol) 57(1):289–300. https://doi.org/10.1111/j.2517-6161.1995.tb02031.x
Billeh YN, Cai B, Gratiy SL, Dai K, Iyer R, Gouwens NW, Abbasi-Asl R, Jia X, Siegle JH, Olsen SR et al (2020) Systematic integration of structural and functional data into multi-scale models of mouse primary visual cortex. Neuron. https://doi.org/10.1016/j.neuron.2020.01.040
Bressloff PC, Cowan JD (2003) A spherical model for orientation and spatial-frequency tuning in a cortical hypercolumn. Philos Trans R Soc Lond B Biol Sci 358(1438):1643–1667. https://doi.org/10.1098/rstb.2002.1109
Carlsson G (2009) Topology and data. Bull Am Math Soc 46(2):255–308. https://doi.org/10.1090/S0273-0979-09-01249-X
Chaudhuri R, Gercek B, Pandey B, Peyrache A, Fiete I (2019) The intrinsic attractor manifold and population dynamics of a canonical cognitive circuit across waking and sleep. Nat Neurosci 22(9):1512–1520. https://doi.org/10.1038/s41593-019-0460-x
Choi H, Choi S (2007) Robust kernel isomap. Pattern Recogn 40(3):853–862. https://doi.org/10.1016/j.patcog.2006.04.025
Cohen J (2013) Statistical power analysis for the behavioral sciences. Academic press, New York
Curto C (2017) What can topology tell us about the neural code? Bull Am Math Soc 54(1):63–78. https://doi.org/10.1090/bull/1554
Curto C, Itskov V (2008) Cell groups reveal structure of stimulus space. PLoS Comput Biol 4(10):e1000205. https://doi.org/10.1371/journal.pcbi.1000205
Dabaghian Y, Mémoli F, Frank L, Carlsson G (2012) A topological paradigm for hippocampal spatial map formation using persistent homology. PLoS Comput Biol 8(8):e1002581. https://doi.org/10.1371/journal.pcbi.1002581
de Vries SE, Lecoq JA, Buice MA, Groblewski PA, Ocker GK, Oliver M, Feng D, Cain N, Ledochowitsch P, Millman D et al (2020) A large-scale standardized physiological survey reveals functional organization of the mouse visual cortex. Nat Neurosci 23(1):138–151. https://doi.org/10.1038/s41593-019-0550-9
Edelsbrunner H, Harer J (2010) Computational topology: an introduction. American Mathematical Soc, New York
Gallego JA, Perich MG, Miller LE, Solla SA (2017) Neural manifolds for the control of movement. Neuron 94(5):978–984. https://doi.org/10.1016/j.neuron.2017.05.025
Gao P, Ganguli S (2015) On simplicity and complexity in the brave new world of large-scale neuroscience. Curr Opin Neurobiol 32:148–155
Ghrist R (2008) Barcodes: the persistent topology of data. Bull Am Math Soc 45(1):61–75. https://doi.org/10.1090/S0273-0979-07-01191-3
Giusti C, Pastalkova E, Curto C, Itskov V (2015) Clique topology reveals intrinsic geometric structure in neural correlations. Proc Natl Acad Sci 112(44):13455–13460. https://doi.org/10.1073/pnas.1506407112
Hatcher A (2002) Algebraic topology. Cambridge University Press, Cambridge
Jones JP, Palmer LA (1987) An evaluation of the two-dimensional gabor filter model of simple receptive fields in cat striate cortex. J Neurophysiol 58(6):1233–1258. https://doi.org/10.1152/jn.1987.58.6.1233
Jun JJ, Steinmetz NA, Siegle JH, Denman DJ, Bauza M, Barbarits B, Lee AK, Anastassiou CA, Andrei A, Aydın Ç et al (2017) Fully integrated silicon probes for high-density recording of neural activity. Nature 551(7679):232–236
Ko H, Cossell L, Baragli C, Antolik J, Clopath C, Hofer SB, Mrsic-Flogel TD (2013) The emergence of functional microcircuits in visual cortex. Nature 496(7443):96–100
Ko H, Mrsic-Flogel TD, Hofer SB (2014) Emergence of feature-specific connectivity in cortical microcircuits in the absence of visual experience. J Neurosci 34(29):9812–9816
Kriegeskorte N, Mur M, Bandettini PA (2008) Representational similarity analysis-connecting the branches of systems neuroscience. Front Syst Neurosci 2:4. https://doi.org/10.3389/neuro.06.004.2008
Leinweber M, Ward DR, Sobczak JM, Attinger A, Keller GB (2017) A sensorimotor circuit in mouse cortex for visual flow predictions. Neuron 95(6):1420–1432
Mante V, Carandini M (2005) Mapping of stimulus energy in primary visual cortex. J Neurophysiol 94(1):788–798. https://doi.org/10.1152/jn.01094.2004
Marshel JH, Garrett ME, Nauhaus I, Callaway EM (2011) Functional specialization of seven mouse visual cortical areas. Neuron 72(6):1040–1054. https://doi.org/10.1016/j.neuron.2011.12.004
Mastrogiuseppe F, Ostojic S (2018) Linking connectivity, dynamics, and computations in low-rank recurrent neural networks. Neuron 99(3):609–623. https://doi.org/10.1016/j.neuron.2018.07.003
Munkres JR (2018) Elements of algebraic topology. CRC Press, London
Naitzat G, Zhitnikov A, Lim LH (2020) Topology of deep neural networks. J Mach Learn Res 21(184):1–40
Pachitariu M, Steinmetz N, Kadir S, Carandini M et al (2016) Kilosort: realtime spike-sorting for extracellular electrophysiology with hundreds of channels. BioRxiv
Saxena S, Cunningham JP (2019) Towards the neural population doctrine. Curr Opin Neurobiol 55:103–111. https://doi.org/10.1016/j.conb.2019.02.002
Seabold S, Perktold J (2010) statsmodels: Econometric and statistical modeling with python. In: 9th Python in science conference
Seabrook TA, Burbridge TJ, Crair MC, Huberman AD (2017) Architecture, function, and assembly of the mouse visual system. Annu Rev Neurosci 40:499–538. https://doi.org/10.1146/annurev-neuro-071714-033842
Shepard RN, Chipman S (1970) Second-order isomorphism of internal representations: Shapes of states. Cogn Psychol 1(1):1–17. https://doi.org/10.1016/0010-0285(70)90002-2
Siegle JH, Jia X, Durand S, Gale S, Bennett C, Graddis N, Heller G, Ramirez TK, Choi H, Luviano JA et al (2021) Survey of spiking in the mouse visual system reveals functional hierarchy. Nature 592(7852):86–92. https://doi.org/10.1038/s41586-020-03171-x
Singh G, Memoli F, Ishkhanov T, Sapiro G, Carlsson G, Ringach DL (2008) Topological analysis of population activity in visual cortex. J Vis 8(8):11–11. https://doi.org/10.1167/8.8.11
Sizemore AE, Phillips-Cremins JE, Ghrist R, Bassett DS (2019) The importance of the whole: topological data analysis for the network neuroscientist. Netw Neurosci 3(3):656–673. https://doi.org/10.1162/netn_a_00073
Stringer C, Pachitariu M, Steinmetz N, Carandini M, Harris KD (2019) High-dimensional geometry of population responses in visual cortex. Nature 571(7765):361–365. https://doi.org/10.1038/s41586-019-1346-5
Tenenbaum JB, De Silva V, Langford JC (2000) A global geometric framework for nonlinear dimensionality reduction. Science 290(5500):2319–2323. https://doi.org/10.1126/science.290.5500.2319
Tipping ME, Bishop CM (1999) Probabilistic principal component analysis. J R Stat Soc Ser B (Stat Methodol) 61(3):611–622. https://doi.org/10.1111/1467-9868.00196
Tralie C, Saul N, Bar-On R (2018) Ripser.py: a lean persistent homology library for python. J Open Sour Softw 3(29):925. https://doi.org/10.21105/joss.00925
Van Hooser SD, Heimel JAF, Chung S, Nelson SB, Toth LJ (2005) Orientation selectivity without orientation maps in visual cortex of a highly visual mammal. J Neurosci 25(1):19–28
Wasserman L (2018) Topological data analysis. Ann Rev Stat Appl 5:501–532. https://doi.org/10.1146/annurev-statistics-031017-100045
Zomorodian A (2010) Fast construction of the vietoris-rips complex. Comput Graph 34(3):263–271. https://doi.org/10.1016/j.cag.2010.03.007
Author information
Authors and Affiliations
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Code availability
The code for all our analyses is available at the following GitHub link: https://github.com/KBeshkov/-geod_top
Additional information
Communicated by Benjamin Lindner.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Beshkov, K., Tiesinga, P. Geodesic-based distance reveals nonlinear topological features in neural activity from mouse visual cortex. Biol Cybern 116, 53–68 (2022). https://doi.org/10.1007/s00422-021-00906-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00422-021-00906-5