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Schottdorf M, Keil W, Coppola D, White LE, Wolf F. Random Wiring, Ganglion Cell Mosaics, and the Functional Architecture of the Visual Cortex. PLoS Comput Biol 2015; 11:e1004602. [PMID: 26575467 PMCID: PMC4648540 DOI: 10.1371/journal.pcbi.1004602] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/19/2015] [Accepted: 10/14/2015] [Indexed: 12/11/2022] Open
Abstract
The architecture of iso-orientation domains in the primary visual cortex (V1) of placental carnivores and primates apparently follows species invariant quantitative laws. Dynamical optimization models assuming that neurons coordinate their stimulus preferences throughout cortical circuits linking millions of cells specifically predict these invariants. This might indicate that V1's intrinsic connectome and its functional architecture adhere to a single optimization principle with high precision and robustness. To validate this hypothesis, it is critical to closely examine the quantitative predictions of alternative candidate theories. Random feedforward wiring within the retino-cortical pathway represents a conceptually appealing alternative to dynamical circuit optimization because random dimension-expanding projections are believed to generically exhibit computationally favorable properties for stimulus representations. Here, we ask whether the quantitative invariants of V1 architecture can be explained as a generic emergent property of random wiring. We generalize and examine the stochastic wiring model proposed by Ringach and coworkers, in which iso-orientation domains in the visual cortex arise through random feedforward connections between semi-regular mosaics of retinal ganglion cells (RGCs) and visual cortical neurons. We derive closed-form expressions for cortical receptive fields and domain layouts predicted by the model for perfectly hexagonal RGC mosaics. Including spatial disorder in the RGC positions considerably changes the domain layout properties as a function of disorder parameters such as position scatter and its correlations across the retina. However, independent of parameter choice, we find that the model predictions substantially deviate from the layout laws of iso-orientation domains observed experimentally. Considering random wiring with the currently most realistic model of RGC mosaic layouts, a pairwise interacting point process, the predicted layouts remain distinct from experimental observations and resemble Gaussian random fields. We conclude that V1 layout invariants are specific quantitative signatures of visual cortical optimization, which cannot be explained by generic random feedforward-wiring models.
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Affiliation(s)
- Manuel Schottdorf
- Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
- Bernstein Center for Computational Neuroscience, Göttingen, Germany
- Bernstein Focus for Neurotechnology, Göttingen, Germany
- Faculty of Physics, University of Göttingen, Göttingen, Germany
- Institute for Theoretical Physics, University of Würzburg, Würzburg, Germany
| | - Wolfgang Keil
- Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
- Bernstein Center for Computational Neuroscience, Göttingen, Germany
- Bernstein Focus for Neurotechnology, Göttingen, Germany
- Faculty of Physics, University of Göttingen, Göttingen, Germany
- Center for Studies in Physics and Biology, The Rockefeller University, New York, New York, United States of America
| | - David Coppola
- Department of Biology, Randolph-Macon College, Ashland, Virginia, United States of America
| | - Leonard E. White
- Department of Orthopaedic Surgery, Duke Institute for Brain Sciences, Duke University, Durham, North Carolina, United States of America
| | - Fred Wolf
- Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
- Bernstein Center for Computational Neuroscience, Göttingen, Germany
- Bernstein Focus for Neurotechnology, Göttingen, Germany
- Faculty of Physics, University of Göttingen, Göttingen, Germany
- Kavli Institute for Theoretical Physics, Santa Barbara, California, United States of America
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Reichl L, Heide D, Löwel S, Crowley JC, Kaschube M, Wolf F. Coordinated optimization of visual cortical maps (I) symmetry-based analysis. PLoS Comput Biol 2012; 8:e1002466. [PMID: 23144599 PMCID: PMC3493482 DOI: 10.1371/journal.pcbi.1002466] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/21/2011] [Accepted: 02/24/2012] [Indexed: 11/18/2022] Open
Abstract
In the primary visual cortex of primates and carnivores, functional architecture can be characterized by maps of various stimulus features such as orientation preference (OP), ocular dominance (OD), and spatial frequency. It is a long-standing question in theoretical neuroscience whether the observed maps should be interpreted as optima of a specific energy functional that summarizes the design principles of cortical functional architecture. A rigorous evaluation of this optimization hypothesis is particularly demanded by recent evidence that the functional architecture of orientation columns precisely follows species invariant quantitative laws. Because it would be desirable to infer the form of such an optimization principle from the biological data, the optimization approach to explain cortical functional architecture raises the following questions: i) What are the genuine ground states of candidate energy functionals and how can they be calculated with precision and rigor? ii) How do differences in candidate optimization principles impact on the predicted map structure and conversely what can be learned about a hypothetical underlying optimization principle from observations on map structure? iii) Is there a way to analyze the coordinated organization of cortical maps predicted by optimization principles in general? To answer these questions we developed a general dynamical systems approach to the combined optimization of visual cortical maps of OP and another scalar feature such as OD or spatial frequency preference. From basic symmetry assumptions we obtain a comprehensive phenomenological classification of possible inter-map coupling energies and examine representative examples. We show that each individual coupling energy leads to a different class of OP solutions with different correlations among the maps such that inferences about the optimization principle from map layout appear viable. We systematically assess whether quantitative laws resembling experimental observations can result from the coordinated optimization of orientation columns with other feature maps.
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Affiliation(s)
- Lars Reichl
- Max-Planck-Institute for Dynamics and Self-Organization, Göttingen, Germany
- Bernstein Center for Computational Neuroscience, Göttingen, Germany
- Bernstein Focus Neurotechnology, Göttingen, Germany
- Faculty of Physics, Georg-August University, Göttingen, Germany
- * E-mail: (LR); (FW)
| | - Dominik Heide
- Max-Planck-Institute for Dynamics and Self-Organization, Göttingen, Germany
- Frankfurt Institute of Advanced Studies, Frankfurt, Germany
| | - Siegrid Löwel
- Bernstein Focus Neurotechnology, Göttingen, Germany
- School of Biology, Georg-August University, Göttingen, Germany
| | - Justin C. Crowley
- Carnegie Mellon University, Department of Biological Sciences, Pittsburgh, Pennsylvania, United States of America
| | - Matthias Kaschube
- Frankfurt Institute of Advanced Studies, Frankfurt, Germany
- Physics Department and Lewis-Sigler Institute, Princeton University, Princeton, New Jersey, United States of America
| | - Fred Wolf
- Max-Planck-Institute for Dynamics and Self-Organization, Göttingen, Germany
- Bernstein Center for Computational Neuroscience, Göttingen, Germany
- Bernstein Focus Neurotechnology, Göttingen, Germany
- Faculty of Physics, Georg-August University, Göttingen, Germany
- Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California, United States of America
- * E-mail: (LR); (FW)
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Keil W, Wolf F. Coverage, continuity, and visual cortical architecture. NEURAL SYSTEMS & CIRCUITS 2011; 1:17. [PMID: 22329968 PMCID: PMC3283456 DOI: 10.1186/2042-1001-1-17] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/02/2011] [Accepted: 12/29/2011] [Indexed: 12/01/2022]
Abstract
BACKGROUND The primary visual cortex of many mammals contains a continuous representation of visual space, with a roughly repetitive aperiodic map of orientation preferences superimposed. It was recently found that orientation preference maps (OPMs) obey statistical laws which are apparently invariant among species widely separated in eutherian evolution. Here, we examine whether one of the most prominent models for the optimization of cortical maps, the elastic net (EN) model, can reproduce this common design. The EN model generates representations which optimally trade of stimulus space coverage and map continuity. While this model has been used in numerous studies, no analytical results about the precise layout of the predicted OPMs have been obtained so far. RESULTS We present a mathematical approach to analytically calculate the cortical representations predicted by the EN model for the joint mapping of stimulus position and orientation. We find that in all the previously studied regimes, predicted OPM layouts are perfectly periodic. An unbiased search through the EN parameter space identifies a novel regime of aperiodic OPMs with pinwheel densities lower than found in experiments. In an extreme limit, aperiodic OPMs quantitatively resembling experimental observations emerge. Stabilization of these layouts results from strong nonlocal interactions rather than from a coverage-continuity-compromise. CONCLUSIONS Our results demonstrate that optimization models for stimulus representations dominated by nonlocal suppressive interactions are in principle capable of correctly predicting the common OPM design. They question that visual cortical feature representations can be explained by a coverage-continuity-compromise.
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Affiliation(s)
- Wolfgang Keil
- Max-Planck-Institute for Dynamics and Self-organization, Am Fassberg 17, D-37077 Göttingen, Germany
- Bernstein Center for Computational Neuroscience, Am Fassberg 17, D-37077 Göttingen, Germany
- Georg-August-University, Faculty of Physics, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany
- Kavli Institute for Theoretical Physics, Santa Barbara, CA 93106-4030, USA
| | - Fred Wolf
- Max-Planck-Institute for Dynamics and Self-organization, Am Fassberg 17, D-37077 Göttingen, Germany
- Bernstein Center for Computational Neuroscience, Am Fassberg 17, D-37077 Göttingen, Germany
- Georg-August-University, Faculty of Physics, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany
- Kavli Institute for Theoretical Physics, Santa Barbara, CA 93106-4030, USA
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