Pith. sign in

REVIEW 3 major objections 5 minor 59 references

Investigation of the neural origin of non-Euclidean visual space and analysis of visual phenomena using information geometry

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper argues that visual space is intrinsically hyperbolic because Fisher information in Gaussian neural population codes acts like a stress-energy tensor, forcing constant negative curvature and fixing the Fisher-Rao distance as the…

desk verdict The paper dresses a standard Fisher-Rao geometry result in a neural-origin story, but the central field equation is an algebraic identity and the empirical fit is calibration; the thesis does not survive close reading. read the letter →

arxiv 2505.13917 v1 pith:AVP5UQX4 submitted 2025-05-20 q-bio.NC physics.bio-ph

Debasis Mazumdar , Kuntal Ghosh , Soma Mitra , Late Kamales Bhaumik This is my paper · ORCID
classification q-bio.NCphysics.bio-ph
keywords visualspaceinformationgeometryFisherhyperbolicneuralpopulationcodepsychometricdistanceHelmholtzhoropterMuller-Lyerillusion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the non-Euclidean character of visual space is not an assumption but a consequence of how neurons encode location. Treating each point stimulus as a Gaussian tuning curve, with mean position and width as the encoding parameters, the population code defines a statistical manifold. The Fisher-Rao metric of that manifold is $ds_f^2 = (d\mu^2 + 2d\sigma^2)/\sigma^2$, whose scalar curvature is $R = -1$: visual space is hyperbolic. The paper uses this metric as the psychometric distance and reproduces published data on the error in estimating the distance between two dots, and derives Helmholtz horopters as geodesics of the hyperbolic space. If the account is right, the geometry of visual space is fixed by the Gaussian form of neural tuning curves, giving a neural origin for a classical psychophysical conjecture.

What carries the argument

The engine is the identification of a point stimulus with a Gaussian probability density $p(x;\mu,\sigma)$ and the visual space with the upper half-plane $H_f = \{(\mu,\sigma) \in \mathbb{R}^2 : \sigma > 0\}$. The Fisher-Rao metric computed from this density is $ds_f^2 = (d\mu^2 + 2d\sigma^2)/\sigma^2$, which is the Poincar\'e upper half-plane metric after rescaling $\omega = \mu + i\sqrt{2}\sigma$; the Poincar\'e disk model is obtained by a M\'obius transformation. This metric carries the argument because its constant negative curvature $R=-1$ is the claimed neural origin of hyperbolic visual space, and its geodesics are the claimed loci of the Helmholtz horopter. The metric also supplies the psychometric distance, since for fixed width it reduces to $|\mu_1-\mu_2|/\sigma$, which the paper uses to simulate distance-estimation biases.

What would settle it

Record or simulate population responses to point stimuli at several eccentricities and estimate the local Fisher information from the empirical tuning curves; if the metric does not scale as $1/\sigma^2$ or the scalar curvature is not $-1$, the central claim fails. More directly, if one can find a condition in which tuning curves are clearly non-Gaussian but perceived visual space remains hyperbolic, or vice versa, the proposed neural origin of the curvature is refuted.

Watch

Extended reading notes

Core claim

The central claim is that Fisher information contained in the neural population code serves as an energy-momentum tensor, generating the space-dependent metric tensor and therefore a curved visual space. With Gaussian tuning curves the Fisher-Rao metric becomes $ds_f^2 = (d\mu^2 + 2d\sigma^2)/\sigma^2$, and the scalar curvature computed from the Christoffel symbols and Ricci tensor is constant, $R = -1$, so the visual space is a hyperbolic manifold. The paper recasts the geometric field equation as $R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R - \frac{1}{2}(n-2)g_{\mu\nu} = T_{\mu\nu}$, where $T_{\mu\nu}$ is a stress-energy tensor built from Fisher information, so the presence of information changes curvature. In this space, the Fisher-Rao distance between two Gaussian codes with the same width is $|\mu_1-\mu_2|/\sigma$, and the paper adopts this as the psychometric distance. Simulated distance-estimation errors match the experimental curve up to roughly $4^\circ$ of separation, and beyond that an attention-induced narrowing of the tuning width is added; the resulting correlation with the psychophysical data is $0.9224$. The Helmholtz horopter is derived as a geodesic of this hyperbolic space, matching data for three observers.

Load-bearing premise

The whole construction rests on treating a point stimulus as encoded by a bell-shaped Gaussian tuning curve whose peak marks position and whose width marks uncertainty, and on assuming that this bell shape survives unchanged from the retina through the dLGN to the parietal cortex.

Editorial extensions

If this is right

  • Visual space in frameless or dark-room conditions is hyperbolic with constant curvature $-1$, so Euclidean geometry is not the right model for perceived spatial relations in such conditions.
  • The Fisher-Rao distance is the appropriate psychometric distance, meaning that perceived separation between two points should scale inversely with the width of the underlying neural tuning curves.
  • The Helmholtz horopter is a geodesic of this hyperbolic space, and the same construction can account for parallel and distance alleys and the M\"uller-Lyer illusion in one framework.
  • Any factor that changes the tuning width $\sigma$, such as attention, eccentricity, or receptive-field size, changes the metric and hence the perceived distance, giving a concrete neural handle on geometric visual illusions.
  • If the tuning curves deviate from Gaussian under rich background cues, the visual space loses constant curvature, which the paper connects to measured curvature varying from elliptic in near space to hyperbolic in far space.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable consequence of the paper's logic is that manipulating tuning width without changing stimulus position, for example by adaptation or pharmacological intervention, should measurably distort perceived distance in the direction predicted by the $1/\sigma$ Fisher-Rao distance.
  • If the argument is read as a general mechanism rather than a vision-specific one, the same Fisher-Rao construction would predict non-Euclidean perceptual geometry in any modality whose population codes are approximately Gaussian, such as auditory or haptic space.
  • The derivation depends on the Gaussian form of tuning curves, so a natural extension is to compute the Fisher-Rao metric and curvature for empirically measured non-Gaussian population codes; the sign and magnitude of the curvature would then be a physiological observable rather than an assumption.
  • The field-equation analogy suggests that the perceived curvature of space could be read as a proxy for the local information content of the neural code, connecting information-theoretic measures of coding efficiency to psychophysical geometry.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes that the non-Euclidean, hyperbolic structure of visual space arises from the Fisher information contained in Gaussian neural population codes. The authors model the population response to a point stimulus as a Gaussian tuning curve, compute the Fisher-Rao metric on the half-plane of parameters (mean, standard deviation), obtain constant scalar curvature R = -1, and introduce an information-geometric field equation (Eq. 23) in which Fisher information plays the role of a stress-energy tensor. They then fit the model to Oleksiak et al.'s distance-estimation data by iteratively adjusting the tuning-curve width, and they simulate Helmholtz horopters as geodesics of the hyperbolic visual space using Luneburg's transformation. The paper concludes that Fisher information creates the curved visual space and that the Fisher-Rao distance is the correct psychometric distance.

Significance. If the central claim were correct, the paper would provide a neural-origin mechanism for the long-studied hyperbolic geometry of visual space, connecting population coding, information geometry, and Luneburg's classical model. The mathematical computation of the Gaussian Fisher metric and its constant negative curvature is correct, and the formal connection between the Fisher-Rao upper half-plane and the Poincaré disk is a useful pedagogical observation. The paper also demonstrates the possibility of reproducing psychophysical distance-error data and horopter shapes within a hyperbolic-geometry framework. However, the advertised causal mechanism—that Fisher information acts as a stress-energy tensor and thereby generates curvature—is an algebraic identity rather than a physical or neural mechanism, and the empirical agreement is obtained by fitting a free parameter. These issues undermine the paper's central claim as stated.

major comments (3)
  1. [Section 3, Eqs. (20)–(23)] The central mechanism is an algebraic identity. Defining ρμν = Rμν + gμν (Eq. 20) and Tμν = ρμν − (1/2)gμνρ (Eq. 22), one obtains by substitution Tμν = Gμν − (n−2)/2 gμν, where Gμν is the Einstein tensor. Hence Eq. (23) is identically satisfied for every Riemannian metric and imposes no constraint on the geometry. In the paper's n = 2 visual space the Einstein tensor vanishes identically, so Eq. (23) reduces to 0 = 0. Consequently the claim that Fisher information 'creates' the curved visual space is not supported: the constant negative curvature R = −1 follows directly from the Gaussian ansatz in Eq. (12), not from any dynamical equation involving information. The statement in the text that 'the curvature of the space changes when information appears in it' is also unsupported because no information-dependent variation of the metric is ever computed.
  2. [Section 3.1, Steps 4–5 and Figure 6] The claimed empirical verification is a curve fit, not a prediction. The authors first compute distance errors using σe from Eq. (24), find disagreement beyond 4°, and then 'keep on modifying the value of σe iteratively till the difference between the simulated and experimental values of the error reaches a minimum.' The reported correlation 0.9224 between simulated and experimental curves is therefore obtained by tuning a free parameter per data set. The subsequent attentional-gain analysis (Eq. 25) does not provide an independent test; it solves for the attentional width σA using the already-fitted σR. This procedure cannot distinguish the Fisher-Rao metric from any other distance function with a suitably adjusted width parameter.
  3. [Section 2 and Step 4, Section 3.1] The load-bearing assumption that neural tuning curves are exactly Gaussian, with the same width parameter retained from retinal ganglion cells through dLGN to LIP, is asserted rather than derived. The DOG-based response equations (3)–(5) contain factors of (1 − exp(...)) and incomplete gamma functions; the step to a normalized Gaussian population code is an additional idealization. Because the Fisher metric (Eq. 12) and the scalar curvature R = −1 depend entirely on this Gaussian form, the paper's central geometric results rest on an unvalidated phenomenological assumption. No neural-recording evidence or formal derivation is supplied to justify the claim that the Gaussian shape is preserved unchanged along the dorsal pathway.
minor comments (5)
  1. [Section 2, references] There is a typographical error: 'Enviol et al.' should be 'Einevoll et al.' (reference [24]).
  2. [Section 3.1] The sentence 'Further to note that the interaction of attentional influence and the neural response has two contrastive natures' is duplicated verbatim in consecutive paragraphs; one occurrence should be deleted.
  3. [Figure 6 caption] The caption refers to 'σe as given by Equation (20)', but the relevant formula for σe is Equation (24); the cross-reference should be corrected.
  4. [Eq. (9)] The expansion of the Kullback-Leibler divergence contains unclear notation ('1/P2(x; θ)') and the signs in the intermediate expression are not clearly justified; a cleaner derivation with explicit Taylor expansion would improve readability.
  5. [Reference [57]] Reference [57] appears to have the same title as reference [56] (Zajaczkowska), which is likely a citation error; the actual title of [57] should be verified.

Circularity Check

3 steps flagged · score 8.0 of 10

The central claim that Fisher information acts as a stress-energy tensor and generates curvature is an algebraic identity: Eq. (23) follows from the definition of Tμν, and in the 2D visual space it reduces to 0 = 0.

  1. self definitional [Section 3, Eqs. (20)-(23), around 'We can further rewrite Equation (20) as...']
    "Rµν + gµν = ρµν (20) ... We can further rewrite Equation (20) as, Rµν + gµν− 1 2gµνρ = ρµν− 1 2gµνρ = Tµν (22) ... finally, we can write Equation (22) as, Rµν− 1 2gµνR− 1 2(n−2)gµν = Tµν (23) ... Therefore the curvature of the space changes when information appears in it."

    Tµν is not an independent gravitational source. Eq. (22) defines Tµν via ρµν, and Eq. (20) defines ρµν = Rµν + gµν. Substituting that definition into Eq. (22) yields exactly Eq. (23); the field equation is therefore an algebraic identity, not a mechanism by which Fisher information creates curvature. In the n = 2 visual space the Einstein tensor vanishes and n − 2 = 0, so Eq. (23) reduces to 0 = 0 and imposes no constraint on the metric. The constant negative curvature R = −1 is a property of the Gaussian Fisher–Rao metric, Eq. (12), not a dynamical consequence of a stress-energy tensor.

  2. fitted input called prediction [Section 3.1, Step 5 and simulation discussion after Figure 6]
    "Then we keep on modifying the value of σe iteratively till the difference between the simulated and experimental values of the error reaches a minimum. The simulated curve is shown by the curve presented by the (–o) line ... The correlation between the experimental (-*) and simulated curve (-o) is computed and found to be 0.9224."

    The 'simulated' error curve is obtained by adjusting σe (and hence σR via Eq. (25)) until the simulated error matches the experimental error from Oleksiak et al. That makes the reported 0.9224 correlation a measure of the quality of a parameter fit, not an independent prediction of the Fisher–Rao metric. Because the fitted width parameter is used to produce the very data being compared, this step does not provide independent empirical confirmation of the hyperbolic visual space.

1 more flagged steps
  1. self citation load bearing [Section 3.2, derivation of horopter geodesics after Eqs. (26a)-(26b)]
    "In a past publication [36] we derived the locus of the geodesics of the hyperbolic visual space by solving the geodesic equations ... The general equation of the geodesics in the visual space represented by the upper half plane model is obtained as [36], 2σ 2 + µ2−Cµ = 1 ... Therefore, we obtain the equation of the Helmholtz horopters in the Poincare disc model using the same Mobius transformation[36]."

    The load-bearing identification of Helmholtz horopters with geodesics of the visual space is imported from the authors' own arXiv preprint [36] rather than derived or verified here. Reference [36] is not an external, machine-checked, or independently reproduced result, so the paper's application section leans on a self-citation chain for the central geometric identification. The earlier preprint invokes the same upper half-plane Fisher–Rao model, so it cannot independently ground the present claim of a neural origin for hyperbolic visual space.

full rationale

The Gaussian Fisher–Rao metric calculation (Eq. 12), the Christoffel symbols, and the scalar curvature R = −1 are self-contained mathematical steps and are not circular: given a Gaussian population code with mean μ and width σ, the standard information-geometric computation does yield a hyperbolic metric. What is circular is the advertised mechanism connecting Fisher information to curvature: Eqs. (20)–(23) define the 'stress-energy tensor' Tμν in terms of the Ricci tensor and metric, so Eq. (23) is an identity rather than a field equation that lets Fisher information generate curvature; in the 2D visual space it reduces to 0 = 0. The empirical comparison in Section 3.1 also fits σe to the experimental error before reporting the correlation, so that match is a fitted-input result rather than an independent prediction. Finally, the horopter-as-geodesic result is taken from the authors' own earlier preprint [36], making that application load-bearing on self-citation. These are specific, quotable reductions; the paper nevertheless has independent mathematical content in the Fisher metric computation itself, so the overall circularity is substantial (8/10) rather than total (10/10).

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The ledger shows that the paper's contribution is an interpretive application: it postulates Gaussian population codes, identifies the visual space with their (mu, sigma) parameter space, and then imports a known hyperbolic metric and a definitional field equation. The distance-error validation requires fitting sigma_e to the target data. No new physical entities such as particles, forces, or dimensions are introduced; the stress-energy tensor T_mu_nu is a formal object defined by the equations, not an independent entity with a falsifiable handle.

free parameters (2)
  • Retinal / tuning-curve scale sigma_e (iteratively adjusted) = varies with eccentricity; reconstructed in Fig. 6c
    In Section 3.1, sigma_e is initialized by Eq. (24) and then 'keep on modifying the value of sigma_e iteratively till the difference between the simulated and experimental values of the error reaches a minimum.' The reported correlation r = 0.9224 is therefore a fit, not a prediction. The attention scale sigma_A is derived from this sigma_e through Eq. (25), not measured.
  • DOG parameters (sigma_e0, sigma_s, w, I(L)) = sigma_e = 0.6 deg, sigma_s = 1.2 deg, w = 0.5, I = 200 spikes/s; sigma_s = 1.6 sigma_e in Step 1
    These are chosen constants from prior models ([24], [46]) and not fitted to the target data, but they determine the width of the tuning curve and therefore the metric. They are inputs, not measured values.
assumptions (4)
  • domain assumption A point stimulus is represented by a univariate Gaussian population code p(r; mu, sigma) whose mean mu encodes location and whose standard deviation sigma encodes uncertainty.
    Stated in Sections 2 and 3 around Eq. (7). The paper references prior literature [27] on tuning curves but does not measure or derive this for mid-level vision; the Gaussian form is the premise on which the Fisher metric is built.
  • domain assumption Each Gaussian population code is a point in the upper half-plane (mu, sigma) with sigma > 0, and the visual space is this statistical parameter space.
    Introduced in Section 3 around Eq. (6). This is a modeling choice that makes the hyperbolic upper half-plane the natural geometry.
  • domain assumption The Gaussian shape of the population code is preserved without distortion from retina through dLGN to LIP.
    Step 4 in Section 3.1 says the two tuning curves are identical and 'progress keeping the shape unchanged'; no direct neural evidence is given for dLGN or LIP in this paper.
  • ad hoc to paper The information-geometric field equation (Eq. 23), with Fisher information as stress-energy tensor T_mu_nu, is a physically meaningful analogue of Einstein's equation for visual space.
    Eqs. (18)-(23) define T_mu_nu via rho_mu_nu, and Eq. (20) identifies rho_mu_nu with R_mu_nu + g_mu_nu. The resulting equation is an algebraic identity; the paper supplies no independent physical or neurophysiological justification for treating it as a dynamical field equation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Investigation of the neural origin of non-Euclidean visual space and analysis of visual phenomena using information geometry." pith.science (2026). https://pith.science/paper/AVP5UQX4

@misc{pith2026250513917,
  author       = {Pith},
  title        = {Pith review of: Investigation of the neural origin of non-Euclidean visual space and analysis of visual phenomena using information geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AVP5UQX4}},
  note         = {Machine review of arXiv:2505.13917}
}
read the original abstract

The present paper aims to develop a mathematical model concerning the visual perception of spatial information. It is a challenging problem in theoretical neuroscience to investigate how the spatial information of the objects in the physical space is encoded and decoded in the neural processes in the brain. In the past, researchers conjectured the existence of an abstract visual space where spatial information processing takes place. Based on several experimental data it was conjectured that the said psychological manifold is non-Euclidean. However, the consideration of the neural origin of the non-Euclidean character of the visual space was not explicit in the models. In the present paper, we showed that the neural mechanism and specifically the Fisher information contained in the neural population code plays the role of energy-momentum tensor to create the space-dependent metric tensor resulting in a curved space described by a curvature tensor. The theoretical prediction of information geometry regarding the emergence of curved manifolds in the presence of the Fisher information is verified in the present work in the domain of neural processing of spatial information at mid-level vision. Several well-known phenomena of visual optics are analyzed using the notion of non-Euclidean visual space, the geodesics of the space, and the Fisher-Rao metric as the suitable psychometric distance.

Figures

Figures reproduced from arXiv: 2505.13917 by the authors.

Figure 1
Figure 1. (a) Response functions of retinal ganglion cells [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. (a) Stimulus used in the fixate trial by Oleksiak et.al.[31]. (b) Experimental data of [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Encoding of two point stimulus by the Gaussian-shaped tuning curve. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Ray diagram of projection of a point stimulus at point ‘a’ in the physical space onto [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: (a) computer-generated retinal projection of two dots. (b) The center surrounds the RF [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Simulation of error in distance estimation. (a) Experimental values obtained in [7] [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: (a)Variation of the scale factor of the bell-shaped spatial attentional spotlight [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]
Figure 8
Figure 8. Figure 8: Helmholtz’s horopter [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]
Figure 9
Figure 9. Figure 9: Coordinates of the physical and the visual space. (a) The physical space is represented [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]
Figure 10
Figure 10. Figure 10: Simulated frontal plane horopters. Experimental data for three subjects are collected [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

59 extracted references · 56 canonical work pages

  1. [1]

    Vision: A computational investigation into the human represen- tation and processing of visual information

    David Marr. Vision: A computational investigation into the human represen- tation and processing of visual information . MIT press, 2010. 26

    work page 2010

  2. [2]

    Space and geometry in the light of physiological, psychological and physical inquiry

    Ernst Mach. Space and geometry in the light of physiological, psychological and physical inquiry. Open Court Classics, 1906

    work page 1906

  3. [3]

    Mathematical analysis of binocular vision

    Rudolf Karl Luneburg. Mathematical analysis of binocular vision. Princeton University Press, 1947

    work page 1947

  4. [4]

    The luneburg theory of binocular visual space

    Albert A Blank. The luneburg theory of binocular visual space. JOSA, 43(9):717–727, 1953

    work page 1953

  5. [5]

    The global structure of visual space , volume 1

    Tarow Indow. The global structure of visual space , volume 1. World Scien- tific, 2004

    work page 2004

  6. [6]

    Methods of information geometry , volume 191

    Shun-ichi Amari and Hiroshi Nagaoka. Methods of information geometry , volume 191. American Mathematical Soc., 2000

    2000

  7. [7]

    Gibbs vs boltzmann entropies

    Edwin T Jaynes et al. Gibbs vs boltzmann entropies. American Journal of Physics, 33(5):391–398, 1965

    work page 1965

  8. [8]

    Dissi- pation: The phase-space perspective

    Ryoichi Kawai, Juan MR Parrondo, and Christian Van den Broeck. Dissi- pation: The phase-space perspective. Physical review letters, 98(8):080602, 2007

    work page 2007

Show all 59 references
  1. [9]

    Stochastic time evolution, information ge- ometry, and the cram´er-rao bound

    Sosuke Ito and Andreas Dechant. Stochastic time evolution, information ge- ometry, and the cram´er-rao bound. Physical Review X, 10(2):021056, 2020

    2020

  2. [10]

    Measuring fisher information accurately in correlated neural pop- ulations

    Ingmar Kanitscheider, Ruben Coen-Cagli, Adam Kohn, and Alexandre Pouget. Measuring fisher information accurately in correlated neural pop- ulations. PLoS computational biology, 11(6):e1004218, 2015

    2015

  3. [11]

    Information thermodynamics on causal networks

    Sosuke Ito and Takahiro Sagawa. Information thermodynamics on causal networks. Physical review letters, 111(18):180603, 2013

    2013

  4. [12]

    Information is physical

    Rolf Landauer. Information is physical. Physics Today, 44(5):23–29, 1991

    1991

  5. [13]

    The physical nature of information

    Rolf Landauer. The physical nature of information. Physics letters A, 217(4- 5):188–193, 1996

    1996

  6. [14]

    Information, physics, quantum: The search for links

    John Archibald Wheeler. Information, physics, quantum: The search for links. Feynman and computation, pages 309–336, 2018

    2018

  7. [15]

    Operationally invariant information in quantum measurements

    ˇCaslav Brukner and Anton Zeilinger. Operationally invariant information in quantum measurements. Physical review letters, 83(17):3354, 1999. 27

    1999

  8. [16]

    Information flow in entangled quantum systems

    David Deutsch and Patrick Hayden. Information flow in entangled quantum systems. Proceedings of the Royal Society of London. Series A: Mathemati- cal, Physical and Engineering Sciences , 456(1999):1759–1774, 2000

    1999

  9. [17]

    Emergent general relativity from fisher information met- ric

    Hiroaki Matsueda. Emergent general relativity from fisher information met- ric. arXiv preprint arXiv:1310.1831, 2013

    2013 arXiv

  10. [18]

    Riemannian geometry of fluctuation theory: An in- troduction

    Luisberis Velazquez. Riemannian geometry of fluctuation theory: An in- troduction. In Journal of Physics: Conference Series , volume 720, page 012005. IOP Publishing, 2016

    2016

  11. [19]

    The visual repre- sentation of space in the primate brain

    Stefan Dowiasch, Andre Kaminiarz, and Frank Bremmer. The visual repre- sentation of space in the primate brain. Neuroforum, 28(4):199–209, 2022

    2022

  12. [20]

    Cortical high-density counterstream architectures

    Nikola T Markov, M ´aria Ercsey-Ravasz, David C Van Essen, Kenneth Knoblauch, Zolt ´an Toroczkai, and Henry Kennedy. Cortical high-density counterstream architectures. Science, 342(6158):1238406, 2013

    2013

  13. [21]

    Receptive fields of single neurones in the cat’s striate cortex

    David H Hubel and Torsten N Wiesel. Receptive fields of single neurones in the cat’s striate cortex. The Journal of Physiology, 148(3):574, 1959

    1959

  14. [22]

    Two cortical visual systems, volume 549

    Leslie G Ungerleider. Two cortical visual systems, volume 549. MIT press, 1982

    1982

  15. [23]

    Separate visual pathways for per- ception and action

    Melvyn A Goodale and A David Milner. Separate visual pathways for per- ception and action. Trends in neurosciences, 15(1):20–25, 1992

    1992

  16. [24]

    Mathematical models for the spatial receptive-field organization of nonlagged x-cells in dorsal lateral geniculate nucleus of cat

    Gaute T Einevoll and Paul Heggelund. Mathematical models for the spatial receptive-field organization of nonlagged x-cells in dorsal lateral geniculate nucleus of cat. Visual neuroscience, 17(6):871–885, 2000

    2000

  17. [25]

    Non-linear model of neural responses in cat visual cortex in computational models of visual processing

    DJ Heeger. Non-linear model of neural responses in cat visual cortex in computational models of visual processing. Landy MS, Movshon JA, 1991

    1991

  18. [26]

    Quantitative analysis of cat retinal ganglion cell response to visual stimuli

    Robert W Rodieck. Quantitative analysis of cat retinal ganglion cell response to visual stimuli. Vision research, 5(12):583–601, 1965

    1965

  19. [27]

    Information processing with population codes

    Alexandre Pouget, Peter Dayan, and Richard Zemel. Information processing with population codes. Nature Reviews Neuroscience, 1(2):125–132, 2000

    2000

  20. [28]

    Organization of the dorsal lat- eral geniculate nucleus in the mouse

    Daniel Kerschensteiner and William Guido. Organization of the dorsal lat- eral geniculate nucleus in the mouse. Visual neuroscience, 34:E008, 2017. 28

    2017

  21. [29]

    Wiring and rewiring of the retinogeniculate synapse

    Y Kate Hong and Chinfei Chen. Wiring and rewiring of the retinogeniculate synapse. Current opinion in neurobiology, 21(2):228–237, 2011

    2011

  22. [30]

    Population coding of visual space: comparison of spatial representations in dorsal and ventral pathways

    Anne B Sereno and Sidney R Lehky. Population coding of visual space: comparison of spatial representations in dorsal and ventral pathways. Fron- tiers in computational neuroscience, 4:159, 2011

    2011

  23. [31]

    Distance estimation is influenced by encoding conditions

    Anna Oleksiak, Mirosława Ma ´nko, Albert Postma, Ineke JM van der Ham, Albert V van den Berg, and Richard JA van Wezel. Distance estimation is influenced by encoding conditions. Plos one, 5(3):e9918, 2010

    2010

  24. [32]

    Peripheral vision and perceived visual direction

    S Mateeff and A Gourevich. Peripheral vision and perceived visual direction. Biological Cybernetics, 49(2):111–118, 1983

    1983

  25. [33]

    Effects of blur and eccentric- ity on differential spatial displacement discrimination

    A Toet, HP Snippe, and JJ Koenderink. Effects of blur and eccentric- ity on differential spatial displacement discrimination. Vision Research, 28(4):535–553, 1988

    1988

  26. [34]

    Retinal metric: a stimulus distance measure derived from population neural responses

    Ga ˇsper Tka ˇcik, Einat Granot-Atedgi, Ronen Segev, and Elad Schneidman. Retinal metric: a stimulus distance measure derived from population neural responses. Physical review letters, 110(5):058104, 2013

    2013

  27. [35]

    Rao’s distance measure

    Colin Atkinson and Ann FS Mitchell. Rao’s distance measure. Sankhy¯a: The Indian Journal of Statistics, Series A , pages 345–365, 1981

    1981

  28. [36]

    Representation of 2d frame less visual space as a neu- ral manifold and its information geometric interpretation

    Debasis Mazumdar. Representation of 2d frame less visual space as a neu- ral manifold and its information geometric interpretation. arXiv preprint arXiv:2011.13585, 2020

    2011 arXiv

  29. [37]

    Curvature of binocular visual space

    Albert A Blank. Curvature of binocular visual space. an experiment. JOSA, 51(3):335–339, 1961

    1961

  30. [38]

    A new method for determining the per- sonal constants in the luneburg theory of binocular visual space

    Genichi Hagino and Ichiro Yoshioka. A new method for determining the per- sonal constants in the luneburg theory of binocular visual space. Perception & Psychophysics, 19(6):499–509, 1976

    1976

  31. [39]

    Direct mea- surement of the curvature of visual space

    Jan J Koenderink, Andrea J Van Doorn, and Joseph S Lappin. Direct mea- surement of the curvature of visual space. Perception, 29(1):69–79, 2000

    2000

  32. [40]

    Convergence of probability measures

    Patrick Billingsley. Convergence of probability measures . John Wiley & Sons, 2013. 29

    2013

  33. [41]

    Hyperbolic geometry

    James W Anderson. Hyperbolic geometry . Springer Science & Business Media, 2006

    2006

  34. [42]

    Axioms for geodesics and their implications

    Herbert Busemann. Axioms for geodesics and their implications. In Studies in Logic and the F oundations of Mathematics , volume 27, pages 146–159. Elsevier, 1959

    1959

  35. [43]

    Uniformity of monkey striate cortex: a parallel relationship between field size, scatter, and magnification factor

    David H Hubel and Torsten N Wiesel. Uniformity of monkey striate cortex: a parallel relationship between field size, scatter, and magnification factor. Journal of Comparative Neurology, 158(3):295–305, 1974

    1974

  36. [44]

    Multi-area visuotopic map complexes in macaque striate and extra-striate cortex

    Jonathan R Polimeni, Mukund Balasubramanian, and Eric L Schwartz. Multi-area visuotopic map complexes in macaque striate and extra-striate cortex. Vision research, 46(20):3336–3359, 2006

    2006

  37. [45]

    A new psychophysical estimation of the receptive field size

    Arash Yazdanbakhsh and Simone Gori. A new psychophysical estimation of the receptive field size. Neuroscience Letters, 438(2):246–251, 2008

    2008

  38. [46]

    Theory of edge detection

    David Marr and Ellen Hildreth. Theory of edge detection. Proceedings of the Royal Society of London. Series B. Biological Sciences, 207(1167):187–217, 1980

    1980

  39. [47]

    Feature-based attention influences motion processing gain in macaque visual cortex

    JC Treue, S Martınez Trujillo. Feature-based attention influences motion processing gain in macaque visual cortex. Nature, 399:575–579, 1999

    1999

  40. [48]

    Feature-based attention increases the se- lectivity of population responses in primate visual cortex.Curr Biol, 14:744– 751, 2004

    JC Martınez Trujillo and S Treue. Feature-based attention increases the se- lectivity of population responses in primate visual cortex.Curr Biol, 14:744– 751, 2004

    2004

  41. [49]

    Parallel and serial neural mecha- nisms for visual search in macaque area v4

    NP Bichot, AF Rossi, and R Desimone. Parallel and serial neural mecha- nisms for visual search in macaque area v4. Science, 308:529–34, 05 2005

    2005

  42. [50]

    Attention to both space and feature mod- ulates neuronal responses in macaque area v4

    CJ McAdams and J.H Maunsell. Attention to both space and feature mod- ulates neuronal responses in macaque area v4. J. Neurophysiol, 83:1751– 1755, 2000

    2000

  43. [51]

    Spatial attention effects in macaque area v4

    C.E Connor, D.C Preddie, J.L Gallant, and D.C Van Essen. Spatial attention effects in macaque area v4. J Neurosci, 17:3201–3214, 1997

    1997

  44. [52]

    Attention to both space and feature modulates neuronal responses in macaque area v4

    J Moran and R Desimone. Attention to both space and feature modulates neuronal responses in macaque area v4. Science, 229:782–784, 1985. 30

    1985

  45. [53]

    Dynamic shifts of visual receptive fields in cortical area mt by spatial attention

    T Womelsdorf, K Anton-Erxleben, F Pieper, and S Treue. Dynamic shifts of visual receptive fields in cortical area mt by spatial attention. Nat Neurosci, 9:1156 – 1160, 2006

    2006

  46. [54]

    Dreceptive field shift and shrinkage in macaque middle temporal area through attentional gain modu- lation

    T Womelsdorf, K Anton-Erxleben, and S Treue. Dreceptive field shift and shrinkage in macaque middle temporal area through attentional gain modu- lation. J.Neurosci, 28(36):8934–8944, 2008

    2008

  47. [55]

    Helmholtz

    H.V . Helmholtz. Treatise on Psychological Optics, volume 3. Optical Soci- ety of America, 1925

    1925

  48. [56]

    Experimental test of luneburg’s theory

    A Zajaczkowska. Experimental test of luneburg’s theory. horopter and alley experiments. J.Opt.Soc.of America, 46.No.7:514–527, 1956

    1956

  49. [57]

    Experimental test of luneburg’s theory

    Ojha Rahul Kumar, Mandal Mainak, Mazumdar Debasis, and Ghosh Kuntal. Experimental test of luneburg’s theory. horopter and alley experiments.11th International Conference, Spatial Cognition, Tubingen, Germany , pages 514–527, 2018

    2018

  50. [58]

    Modelling muller-lyer illusion using information geometry

    Mazumdar Debasis, Mitra Soma, Ghosh Kuntal, and Bhaumik Kamales. Modelling muller-lyer illusion using information geometry. 3rd In- ternational Conference on Data Intelligence and Cognitive Informatics, Tirunelveli, India, pages 514–527, 2022

    2022

  51. [59]

    Place units in the hippocampus of the freely moving rat

    John O’keefe. Place units in the hippocampus of the freely moving rat. Experimental Neurology, 51 issue (1):78–109, 1976. 31

    1976

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.