# Bressloff and Rule Modeling Deep Dive Source: https://mesmerprism.com/projects/bressloff-v1-form-constants-deep-dive.html Canonical HTML: https://mesmerprism.com/projects/bressloff-v1-form-constants-deep-dive.html Generated: 2026-05-26 Description: A notebook-style deep dive into Bressloff form constants, Rule flicker modeling, and generated driven neural-field diagnostics after Rule. Markdown: https://mesmerprism.com/projects/bressloff-v1-form-constants-deep-dive.md Plain text: https://mesmerprism.com/projects/bressloff-v1-form-constants-deep-dive.txt BibTeX references: https://mesmerprism.com/projects/bressloff-v1-form-constants-deep-dive.bib CSL JSON references: https://mesmerprism.com/projects/bressloff-v1-form-constants-deep-dive.references.csl.json --- Modeling deep dive # Bressloff and Rule Modeling Deep Dive Geometric hallucination models become useful when their equations stay close to the image. Bressloff's orientation-hypercolumn account asks how cortical symmetry and retino-cortical mapping can generate tunnels, cobwebs, lattices, and honeycombs. Rule, Stoffregen, and Ermentrout ask a different question: how periodic flicker can destabilize a scalar excitatory/inhibitory field into high-frequency period-doubled stripes or lower-frequency hexagonal patterns ([Bressloff et al., 2001](https://doi.org/10.1098/rstb.2000.0769); [Rule et al., 2011](https://doi.org/10.1371/journal.pcbi.1002158)). [Sequential examples](https://mesmerprism.com/projects/bressloff-v1-form-constants-deep-dive.html#examples) [Model split](https://mesmerprism.com/projects/bressloff-v1-form-constants-deep-dive.html#model-split) [Rule flicker model](https://mesmerprism.com/projects/bressloff-v1-form-constants-deep-dive.html#rule) [Rule explorer](https://mesmerprism.com/projects/bressloff-v1-form-constants-deep-dive.html#rule-explorer) [After Rule](https://mesmerprism.com/projects/bressloff-v1-form-constants-deep-dive.html#after-rule) [Driven reports](https://mesmerprism.com/projects/bressloff-v1-form-constants-deep-dive.html#driven-reports) [Source methods](https://mesmerprism.com/projects/bressloff-v1-form-constants-deep-dive.html#source-methods) [Bressloff overview](https://mesmerprism.com/projects/bressloff-v1-form-constants.html) [References](https://mesmerprism.com/projects/bressloff-v1-form-constants-deep-dive.html#references) What this page adds ## Two model families, one calibration record The shorter Bressloff page shows generated visual forms. This deep dive separates the model families behind them: Bressloff's orientation-hypercolumn planforms and retino-cortical mapping, and Rule's periodically forced excitatory/inhibitory field for flicker-induced phosphenes. The two tracks share visual-field questions but not claims. The status tables record which generated targets currently pass, which remain approximate, and which published figures are not reproduced here for rights reasons. ### Figure-use policy Mesmer Prism uses generated model outputs unless an original article figure is clearly open-licensed or reproduced with permission. Generated images cite the source model but are not scans, crops, or reproductions of the paper figures. ### Safety boundary The visuals here are explanatory model outputs, not a photic-stimulation protocol. Frequency parameters are model parameters, not safety-cleared exposure recommendations. Historical problem ## Why these shapes invite a cortical model Reports of geometric visual hallucinations recur around a small vocabulary: tunnels, spirals, rays, cobwebs, lattices, checkerboards, funnels, and honeycombs. The modeling question is not whether those pictures can be drawn. It is whether a plausible cortical system has natural pattern-forming modes that resemble them after the map from visual space to V1 is taken seriously ([Ermentrout and Cowan, 1979](https://doi.org/10.1007/BF00336965); [Bressloff et al., 2002](https://doi.org/10.1162/089976602317250861)). The older scalar cortical-sheet story already explains why stripes, squares, and hexagons matter. Bressloff, Cowan, Golubitsky, Thomas, and Wiener add the retino-cortical transform and V1 orientation preference structure, so some hallucinations become locally oriented contour fields rather than only bright and dark scalar activity. ### Scalar cortical sheet Useful for rings, rays, spirals, and some light-dark lattices because cortical stripes map cleanly through the complex-log retino-cortical transform. ### Orientation hypercolumns Needed for cobwebs, honeycombs, and lattice tunnels that are better described as locally oriented contours than as only bright and dark regions. ### Flicker E/I field Rule's model belongs beside Bressloff's track, not inside it: the core object is a periodically forced excitatory/inhibitory field with scalar spatial kernels. Bressloff model family ## Orientation-hypercolumn planforms The implementation keeps Bressloff's family as model_family = bressloff_orientation_hypercolumn . It reuses retino-cortical rendering where the mathematics calls for it, but does not treat later flicker models as Bressloff paper presets. ### Position map \[ \begin{aligned} x &\simeq \log r_R \\ y &\simeq \theta_R \end{aligned} \] Circles, rays, and logarithmic spirals in visual space become vertical, horizontal, and oblique stripes in cortical coordinates. ### Orientation map \[ \phi = \phi_R - \theta_R, \qquad \phi_R = \phi + \theta_R \] Local retinal contour orientation is measured relative to angular visual-field position. The renderer uses the equivalent drawing convention when it places contour glyphs. ### Hypercolumn field \[ \frac{\partial a(\mathbf r,\phi,t)}{\partial t} = -a(\mathbf r,\phi,t) + \mu \int w(\mathbf r,\phi;\mathbf r',\phi')\, \sigma(a(\mathbf r',\phi',t))\, d\mathbf r'\,d\phi' \] The weight separates local isotropic coupling inside a hypercolumn from anisotropic long-range coupling between similarly tuned orientation patches. ### Planform modes \[ a(\mathbf r,\phi) \sim \sum_j c_j\,u(\phi-\phi_j)\, \exp(i\,\mathbf k_j\cdot\mathbf r) \] Rolls, squares, rhombs, and hexagonal branches are selected from finite wavevector sets and then mapped into retinal space. Sequential examples ## Generated cells from the current implementation These loops are static exports from the Rust/browser implementation. They are generated images, not scans of the source papers, and they should be read as qualitative model illustrations unless a row explicitly says it has passed a calibration check. Cell 01 ### 2002 roll mode: cortical stripes before remapping The simplest Bressloff example is a cortical roll: a striped activity pattern in cortical coordinates. In the visual field, the same family can become rings, rays, or spirals depending on stripe orientation. Preset fig5_roll_cortical Source Bressloff et al. 2002 Figure 5 Model family Bressloff orientation hypercolumn Cell 02 ### Lattice tunnel through the retino-cortical map The lattice tunnel shows why the mapping step matters. A structured V1 planform becomes a tunnel-like field when projected into visual-field coordinates, making the radial expansion of the retino-cortical map visible. Preset fig7_lattice_tunnel Source Bressloff et al. 2002 Figure 7 Status qualitative rendered target pass Cell 03 ### Single inverse map: non-contoured square cells Figures 29 and 30 in the 2001 paper show scalar activity before local contour orientation is drawn. The implementation thresholds bright and dark regions, maps them back to visual-field coordinates, and leaves the orientation-glyph layer out. Preset fig29_square_noncontoured Source Bressloff et al. 2001 Figure 29, Table 2 Status rendered target pass; threshold geometry approximate Cell 04 ### Double map: square even contours The square even case adds the local orientation field, so the output is closer to a cobweb of contour fragments than to a scalar checkerboard. The current renderer draws the target family, while the local branch selector still prefers a roll branch under the same diagnostics. Preset fig31_square_even Source Bressloff et al. 2001 Figure 31 Status rendered target review; branch diagnostic mismatch Cell 05 ### Rhombic even branch as a cleaner calibration case The rhombic example is currently the cleanest agreement case: the rendered family and branch selector both identify the rhombic target. That makes it a useful baseline for later figure-level image metrics. Preset fig33_rhombic_even Source Bressloff et al. 2001 Figure 33 Status rendered target pass; branch diagnostic pass Cell 06 ### Zero-phase hexagonal even branch The zero-phase hexagonal even example gives a honeycomb-like retinal image. It tests whether the preset registry can distinguish hexagonal phase partners instead of treating every three-wave branch as the same picture. Preset fig35_hex_zero_even Source Bressloff et al. 2001 Figure 35 Status rendered target pass Cell 07 ### Triangular odd branch and the current fidelity limit The triangular odd target is useful because it exposes a limit of the first-pass implementation. The renderer can draw the requested triangular family, but the present stability and branch diagnostics do not yet reproduce the full higher-order odd-hexagonal selection described in the paper. Preset fig36_triangle_odd Source Bressloff et al. 2001 Figure 36 Status rendered target review; higher-order branch work deferred Calibration reports ## What the current registry can say The implementation currently separates source-figure identity, rendered target, contour mode, and branch-selection diagnostics. That is deliberately conservative: the page can say that a generated figure resembles a source target without pretending that all stability curves, phase conventions, or higher-order amplitude terms have already been recovered. 24 Bressloff paper and convenience presets 17 rendered target passes 7 review targets for branch or stability fidelity Bressloff geometry calibration ## Generated stills against private numeric targets This surface shows generated implementation stills and public-safe metric summaries. Original paper scans and crops are not displayed here; when a private source-derived profile exists, only derived numeric comparison fields are loaded. Format available after report load Generated stills available after report load Source comparisons available after report load Mean edge density available after report load Generated-still metric summary appears here when the report asset is available. Rule flicker model family ## A separate scalar E/I simulation track Rule, Stoffregen, and Ermentrout model flicker-induced phosphenes using a forced excitatory/inhibitory neural field and then study spatial instabilities over flicker frequency and stimulus parameters. The public implementation names this as model_family = rule_flicker_ei and keeps it separate from Bressloff's orientation-hypercolumn registry. ### E/I field \[ \begin{aligned} \tau_e \frac{\partial U_e}{\partial t} &= -U_e + F(a_{ee}K_e * U_e - a_{ei}K_i * U_i - \theta_e + g_e S(t)) \\ \tau_i \frac{\partial U_i}{\partial t} &= -U_i + F(a_{ie}K_e * U_e - a_{ii}K_i * U_i - \theta_i + g_i S(t)) \end{aligned} \] The implementation uses two scalar fields over a periodic two-dimensional domain with normalized Gaussian kernels. ### Flicker stimulus \[ S(t) = A\,H\!\left(\sin\!\left(\frac{2\pi t}{T}\right)-\vartheta\right) \] Period \(T\), amplitude \(A\), threshold, smoothing, and inhibitory feed-forward fraction are exposed as explicit parameters. ### Temporal check \[ C(T)\approx -1,\quad C(2T)\approx 1 \qquad\text{versus}\qquad C(T)>0 \] The qualitative report compares frames one and two forcing periods apart, which is enough for a first period-doubling sanity check. ### Floquet boundary check \[ \dot v = D_u f(u_0(t),t;k)\,v, \qquad v(T)=M(k)v(0), \qquad \lambda\in\sigma(M(k)) \] The current report computes first-pass monodromy boundary markers and Figure 8 curve comparisons. Published-axis reproduction remains calibration work, not a settled claim. ### Rule model status - Implemented: qualitative high-frequency and low-frequency presets, one-period/two-period correlation checks, and Fourier-family readouts. - Partial: sweep maps, monodromy hints, and domain-normalized Figure 8 beta-axis comparison. - Deferred: exact digitized reproduction of published phase-boundary figures. - Not claimed: prediction of a viewer's hallucinations from stimulus frequency. ### Explorer status The public explorer loads static JSON reports when they are present. If a report asset is absent, the qualitative Rule presets and claim ledger remain the public surface rather than showing repeated empty map states. Rule Dynamics Explorer v1 ### Synchronized flicker, E/I response, and spatial modes This explorer shows the two qualitative Rule regimes as linked views: stimulus pulse, reduced excitatory/inhibitory response, scalar cortical field, dominant Fourier family, the one-period versus two-period temporal check, and a simulator-backed sweep report where available. It is an explanatory model view, not a stimulus protocol. Model family rule_flicker_ei Active preset High-frequency stripes Response period doubled Pattern depth 0.82 High-frequency stripes Low-frequency hexagons Pause View Cortical field Retinal projection Time Amplitude 0.80 I drive 0.00 #### Stimulus and E/I traces stimulus excitatory inhibitory #### Fourier mode readout Stripe axis 0.92 Hexagonal triplet 0.18 #### Period check C(T) -0.99 C(2T) 0.98 Stimulus period 55 ms #### Frequency sweep strip Sweep report summary appears here when the JSON asset is available. 140 ms Regime low frequency Modes hexagonal Peak 0.00 120 ms Regime one to one Modes hexagonal Peak 0.00 85 ms Regime transition Modes mixed Peak 0.00 65 ms Regime stripe onset Modes stripe Peak 0.00 55 ms Regime period doubled Modes stripe Peak 0.00 Low-frequency mode available after report load Transition mode available after report load High-frequency mode available after report load #### Frequency-amplitude map Dense sweep-map summary appears here when the JSON asset is available. Floquet boundary summary appears here when the JSON asset is available. Point available after report load Regime available after report load Spatial score available after report load Temporal score available after report load Floquet available after report load Note available after report load #### Rule Figure 8 refined boundary curves Figure 8 curve comparison appears here when the report assets are available. Parameter set available after report load Branches available after report load Points available after report load Beta RMS available after report load Continuity available after report load Beta map available after report load #### A. Flicker input The forcing term \(S(t)\) supplies periodic drive with exposed amplitude and period. It is a temporal input before any spatial pattern is visible. #### B. Homogeneous E/I response The reduced traces show the driven excitatory and inhibitory populations. Their relative timing sets the gain available to spatial modes. #### C. Spatial perturbation growth The scalar field view shows pattern amplitude riding on the E/I response. The amplitude and I-drive sliders expose weak, patterned, and suppressed cases. #### D. Low-frequency branch Longer forcing periods are represented as one-to-one hexagonal responses, matching the qualitative low-frequency island. #### E. High-frequency branch Shorter periods favor stripe-like modes with a two-cycle temporal signature: \(C(T)\) is negative while \(C(2T)\) returns positive. #### F. Floquet boundary map The sweep strip and frequency-amplitude map now read simulator reports. Sign-change markers show where the monodromy threshold conditions cross on the current grid, and the report now refines beta-axis crossings into first-pass boundary curve points. Exact published-axis calibration remains the next rigorous layer. The live field and trace controls are an explanatory browser projection of the current qualitative presets. The sweep strip, map, and monodromy hints read Rust simulator reports when the JSON assets are available. Cell 08 ### High-frequency flicker: period-doubled stripes The high-frequency qualitative preset represents the stripe island in Rule's Figure 4. The period-doubling signature is temporal: after one forcing cycle, foreground and background approximately exchange; after two cycles, the pattern matches again. Preset rule_fig4_high_freq_stripes Stimulus 55 ms representative period Check rT = -0.99 , r2T = 0.98 Cell 09 ### Lower-frequency flicker: one-to-one hexagons The lower-frequency qualitative preset represents the hexagonal island. The current version uses a 120 ms representative rather than claiming exact reproduction of the 110 ms panel in the paper. Preset rule_fig4_low_freq_hexagons Stimulus 120 ms qualitative representative Check rT = 0.30 , r2T = 0.90 After Rule ## Driven neural fields, not one straight line The lineage after Rule splits into driven-input problems and architecture extensions. Rule models diffuse periodic flicker. Nicks, Cocks, Avitabile, Johnston, and Coombes model spatial forcing and orthogonal response in Billock-Tsou-style effects. Tamekue, Prandi, and Chitour treat MacKay stimuli as localized inputs that break symmetry and act as controls in an Amari neural field. Bolelli and Prandi then combine localization with time-periodic input, giving a current framework for flickering visual stimuli whose model contours depend on frequency and inhibition ([Nicks et al., 2021](https://doi.org/10.1137/20M1366885); [Tamekue et al., 2024](https://doi.org/10.1137/23M1616686); [Bolelli and Prandi, 2025](https://doi.org/10.1007/s10851-025-01257-7)). A second branch complicates the cortical architecture itself: pinwheel lattices, long-range connections, color dimensions, contrast polarity, and perceptual grouping. These do not replace the Bressloff simulator; they show what it abstracts away ([Veltz et al., 2015](https://link.springer.com/article/10.1186/s13408-015-0023-8); [Faugeras et al., 2022](https://doi.org/10.5802/crmath.289); [Carroll and Bressloff, 2018](https://doi.org/10.1137/16M1076125); [Sarti and Citti, 2015](https://doi.org/10.1007/s10827-014-0540-6)). Branch Source Input type Model output Use in Mesmer Prism Spontaneous form constants Ermentrout-Cowan; Bressloff et al. No visual input or parameter shift Planforms mapped to tunnels, spirals, lattices, and cobwebs Current simulator core Diffuse flicker Rule et al. 2011 Spatially homogeneous time-periodic forcing Frequency-dependent phosphene regimes Rule panel and explorer Spatial forcing Nicks et al. 2021 Periodic spatial input Orthogonal response and 2:1 resonance Nicks generated diagnostic report Localized static input Tamekue et al. 2024 MacKay rays or target plus localized control Input-biased orthogonal contours MacKay generated diagnostic report Localized time-periodic input Bolelli and Prandi 2025 Flickering localized input Periodic state and contour width versus frequency/inhibition Bolelli generated diagnostic report Pinwheel architecture Veltz et al. 2015 More realistic V1 lattice and long-range connections Symmetry-restricted planforms and hexagonal robustness Architecture caveat Color V1 Faugeras et al. 2022 Hue and saturation dimensions Spatio-chromatic planforms and localized snaking states Extension layer Contrast gradients Carroll and Bressloff 2018 Contrast-polarity/orientation field Gradient-direction encoding "Not hallucination only" sidebar Perceptual units Sarti and Citti 2015 Visual input plus neurogeometry Eigenmodes as perceptual units Grouping note ### Spatial forcing and orthogonal response Nicks et al. provide the bridge from diffuse flicker toward Billock-Tsou-style sensory-induced hallucinations. Under the retino-cortical map, rings and arms become orthogonal cortical stripe families, and a spatially forced neural field can support an orthogonal response when the resonance conditions are right. ### Localized MacKay input Tamekue, Prandi, and Chitour recast the MacKay branch as an input-output problem. The stimulus is represented as a cortical input through the retino-cortical map, and localized information breaks the global symmetry of a regular funnel or tunnel pattern. ### Time-periodic localized input Bolelli and Prandi are the current endpoint for this page: localized geometric input and periodic forcing are studied in one neural-field framework. Frequency remains a model parameter, not a safe exposure recommendation. ### Architecture and extensions Pinwheel lattices, color dimensions, contrast-polarity fields, and perceptual-unit models show that V1 neural-field mathematics is broader than hallucination geometry. The page should use them to clarify the simulator's abstractions, not to broaden its claims. ### Generated visual policy No full PDFs, paper figure scans, page renders, or source-figure crops are hosted for this branch. New visuals should be generated in Mesmer Prism style unless a paper figure is clearly reusable and captioned with its license, source, and change notes. Driven report surface ## The implemented continuation is source-target diagnostic, not calibrated The current public assets expose three generated driven-field report families. The MacKay report covers localized stationary input. The Bolelli report covers localized time-periodic input, period-lock diagnostics, and an accepted source-side principal-pole width convention with generated decay-width fit diagnostics and Figure 5 source-equation curve samples. This is not source-panel calibration. The Nicks report covers 2:1 orthogonal-response amplitude diagnostics, Appendix-B kernel-derived coefficient tables, source-equation Figure 8 boundary residual checks, and a source-derived acceptance policy. None of these are paper-figure reproductions or calibration claims. ... registered driven examples in the public-safe model registry. ... implemented entries with generated report targets. ... partial entries awaiting stronger source-target metrics. ... JSON report families mirrored into the website assets. Claim level diagnostic/source-target; not calibrated Formats loading report formats... Source loading generated report JSON... Computational provenance ## What the original papers say about software The source record is mixed. Some papers name concrete software stacks; others mainly expose equations, symmetry arguments, numerical methods, or figure descriptions. Mesmer Prism uses that provenance to decide what kind of implementation is credible here, while keeping the public outputs as generated diagnostics and source-target comparisons rather than original-code reproductions. Source track Original-author workflow named in the paper record How Mesmer Prism treats it Bressloff form constants Analytic retinocortical map, Euclidean symmetry, orientation-hypercolumn planforms, and bifurcation analysis; no named figure-code stack identified. Independent Rust implementation with generated planforms and explicit comparison reports, not original-code reuse. Rule flicker E/I field AUTO through XPPAUT, Floquet/monodromy stability calculations, and custom two-dimensional periodic-grid simulations. Rust sweep and Floquet diagnostics stay first-pass until report-backed source-axis fits support stronger language. Nicks spatial forcing MATLAB simulations, FFT-based pseudo-spectral convolution, ode45 , and XPPAUT checks for reduced amplitude-equation stability. Current reports implement reduced amplitude-equation and boundary diagnostics; full half-space field simulations remain deferred. Tamekue-Prandi-Chitour MacKay model Julia visualization/solver workflow and fixed-point iteration for stationary Amari neural fields. The Rust MacKay report keeps a compact fixed-point diagnostic and records generated metrics without claiming source-panel reproduction. Bolelli-Prandi periodic input Mathematica for principal-pole calculations and Julia for nonlinear mean-field simulations. Pole-width formulas are source-target diagnostics; Figure 5 source-equation curves are generated from equations, and generated decay-width fits share the convention only when the fit gate passes. True source-panel digitization is reserved for later plot-image comparison. Veltz pinwheel architecture Trilinos, FFTW, PETSc, petsc4py , Newton-Krylov/GMRES, Arnoldi eigensolvers, BDF integration, and large meshes. Pinwheel dynamics are treated as a high-cost architecture extension, not near-term Rust report work. Faugeras-Song-Veltz color field Julia, BifurcationKit.jl, KrylovKit.jl, pseudo-arclength continuation, CUDA.jl, and GPU FFTs. Color hallucinations and localized snaking remain deferred until the project has a continuation/color architecture layer. Carroll-Bressloff and Sarti-Citti Mathematica algebraic checks for Carroll-Bressloff; mean-field discretization, affinity matrices, eigenvectors, and MCMC-style estimation for Sarti-Citti. Both stay as adjacent perceptual-function tracks unless this project expands beyond driven hallucination-style diagnostics. ### Why this matters The original papers do not share one software lineage. Some are analytic, some are continuation-heavy, and some use MATLAB, Julia, or Mathematica for specific parts of the workflow. A single Rust simulator should therefore expose model-family reports and validation status rather than pretending all figures come from one executable model. ### Current claim level The generated assets here are diagnostics, calibration targets, and source-target comparisons. They are not produced by the authors' original scripts, and they are not presented as calibrated reproductions of private or permission-bound source figures. ### Implementation consequence Near-term work should keep tightening Bolelli source-equation diagnostics and Nicks region-boundary residual policies. Veltz/Faugeras continuation, GPU, color, and pinwheel stacks should wait until the report layer can support that complexity. Claim ledger ## What is solid, partial, or deferred Claim Current level Next work Bressloff and Rule are separate model families. Implemented in public metadata and page structure. Design a cross-model registry only when more families need shared indexing. Bressloff examples reproduce named visual families. Qualitative generated target checks for the current preset catalog. Digitize source figures and add image metrics. Rule high-frequency and low-frequency examples show different regimes. Qualitative seeded E/I simulations with temporal-correlation checks, Fourier-family scores, and first sweep-map diagnostics. Calibrate exact Rule Figure 5 and Figure 6 parameter locations. The public article explains Rule's temporal mechanism visually. Client-side explorer links flicker, E/I traces, mode family, period checks, sweep strip, dense map, and monodromy hints. Calibrate dense Rule Figure 6 and Figure 8 phase boundaries. Later work extends the lineage toward driven neural fields. Public taxonomy separates spatial forcing, localized MacKay input, localized time-periodic input, pinwheel architecture, color extensions, contrast-gradient encoding, and perceptual grouping. MacKay, Bolelli, and Nicks now have generated first-pass diagnostic reports. Add source-derived numeric targets before calling any driven output calibrated. The original papers used one shared software stack. Not claimed here. The source record names different workflows across papers, including XPPAUT/AUTO, MATLAB, Julia, Mathematica, PETSc/Trilinos, and BifurcationKit. Keep implementation provenance separate from generated-report validation. Spatial forcing, localized input, and time-periodic input can explain every viewer's percept. Not claimed here. Requires participant-level data, stimulus calibration, and safety review outside this public model note. Flicker frequency predicts individual visual experience. Not claimed here. Requires participant data, safety constraints, and calibrated stimulus hardware. Figure rights ledger ## Which figures can be hosted here Figure source Public-page treatment Rights basis Generated Mesmer Prism images Hosted here. Created by the Mesmer Prism renderer from the described model families. Rule et al. 2011 PLOS figures Can be reproduced or adapted with attribution and change notes. PLOS Computational Biology articles are covered by Creative Commons Attribution reuse terms. Bressloff et al. 2001 Royal Society figures Reference only unless permission or a specific open license is verified. Royal Society journal reuse is routed through its permissions process unless the article license already permits the use. Bressloff et al. 2002 Neural Computation figures Reference only unless MIT Press permission or a specific open license is verified. MIT Press provides a rights and permissions process for journal reuse. Ermentrout and Cowan 1979 Springer figures Reference only unless Springer permission or a specific open license is verified. Springer Nature distinguishes openly licensed material from content requiring reuse permission. Bressloff 2012 IOP figures Check the article first page; treat as permission-required unless clearly open. IOP permissions guidance treats adaptations as permission-bound unless the source license allows reuse. Nicks et al. 2021 SIAM figures Reference only; redraw concepts for public diagrams. Version of record is SIAM-published and the user-provided PDF was a review-purpose manuscript. Tamekue et al. 2024 SIAM figures Reference only; redraw MacKay examples unless permission is obtained. Version of record is SIAM-published; the arXiv manuscript is not treated as figure-reuse permission. Bolelli and Prandi 2025 figures Reusable only with CC BY attribution and caption-level third-party checks. The Springer article is open access under CC BY 4.0 with a third-party material caveat. Veltz et al. 2015 figures Reusable only with CC BY attribution and caption-level third-party checks. The Journal of Mathematical Neuroscience article is distributed under CC BY 4.0. Faugeras et al. 2022 figures Reusable only with CC BY attribution and caption-level third-party checks. The Comptes Rendus Mathematique article page states CC BY 4.0. Carroll and Bressloff 2018 SIAM figures Reference only unless permission is obtained. SIAM copyright applies to the version of record. Sarti and Citti 2015 figures Reference only; redraw concepts. No clear reusable figure license was identified in the local copy; treat Springer version as permission-required unless verified otherwise. References ## Sources and implementation lineage ### Model papers - Ermentrout, G. B., and J. D. Cowan. "[A Mathematical Theory of Visual Hallucination Patterns](https://doi.org/10.1007/BF00336965)." Biological Cybernetics 34 (1979). - Bressloff, P. C., J. D. Cowan, M. Golubitsky, P. J. Thomas, and M. C. Wiener. "[Geometric Visual Hallucinations, Euclidean Symmetry and the Functional Architecture of Striate Cortex](https://doi.org/10.1098/rstb.2000.0769)." Philosophical Transactions of the Royal Society B 356 (2001). - Bressloff, P. C., J. D. Cowan, M. Golubitsky, P. J. Thomas, and M. C. Wiener. "[What Geometric Visual Hallucinations Tell Us About the Visual Cortex](https://doi.org/10.1162/089976602317250861)." Neural Computation 14, no. 3 (2002). - Bressloff, P. C. "[Spatiotemporal Dynamics of Continuum Neural Fields](https://doi.org/10.1088/1751-8113/45/3/033001)." Journal of Physics A: Mathematical and Theoretical 45, no. 3 (2012). - Rule, M., M. Stoffregen, and B. Ermentrout. "[A Model for the Origin and Properties of Flicker-Induced Geometric Phosphenes](https://doi.org/10.1371/journal.pcbi.1002158)." PLOS Computational Biology 7, no. 9 (2011). ### Driven-field continuation - Billock, V. A., and B. H. Tsou. "[Neural Interactions Between Flicker-Induced Self-Organized Visual Hallucinations and Physical Stimuli](https://doi.org/10.1073/pnas.0610813104)." PNAS 104, no. 20 (2007). - Nicks, R., A. Cocks, D. Avitabile, A. Johnston, and S. Coombes. "[Understanding Sensory Induced Hallucinations: From Neural Fields to Amplitude Equations](https://doi.org/10.1137/20M1366885)." SIAM Journal on Applied Dynamical Systems 20, no. 4 (2021). - Tamekue, C., D. Prandi, and Y. Chitour. "[A Mathematical Model of the Visual MacKay Effect](https://doi.org/10.1137/23M1616686)." SIAM Journal on Applied Dynamical Systems 23, no. 3 (2024). - Bolelli, M. V., and D. Prandi. "[Neural Field Equations with Time-Periodic External Inputs and Some Applications to Visual Processing](https://doi.org/10.1007/s10851-025-01257-7)." Journal of Mathematical Imaging and Vision 67 (2025). ### Architecture and perception extensions - Veltz, R., P. Chossat, and O. Faugeras. "[On the Effects on Cortical Spontaneous Activity of the Symmetries of the Network of Pinwheels in Visual Area V1](https://link.springer.com/article/10.1186/s13408-015-0023-8)." Journal of Mathematical Neuroscience 5 (2015). - Faugeras, O. D., A. Song, and R. Veltz. "[Spatial and Color Hallucinations in a Mathematical Model of Primary Visual Cortex](https://doi.org/10.5802/crmath.289)." Comptes Rendus Mathematique 360 (2022). - Carroll, S. R., and P. C. Bressloff. "[Symmetric Bifurcations in a Neural Field Model for Encoding the Direction of Spatial Contrast Gradients](https://doi.org/10.1137/16M1076125)." SIAM Journal on Applied Dynamical Systems 17, no. 1 (2018). - Sarti, A., and G. Citti. "[The Constitution of Visual Perceptual Units in the Functional Architecture of V1](https://doi.org/10.1007/s10827-014-0540-6)." Journal of Computational Neuroscience 38, no. 2 (2015). See also the [arXiv record](https://arxiv.org/abs/1406.0289). ### Public implementation and context - Mesmer Prism. "[Bressloff V1 Form Constants Lab](https://github.com/MesmerPrism/bressloff-v1-form-constants)." MIT-licensed Rust and browser implementation used for these generated assets. - Mesmer Prism. "[Driven Neural Fields Implementation Plan](https://github.com/MesmerPrism/bressloff-v1-form-constants/blob/main/docs/DRIVEN_NEURAL_FIELDS_IMPLEMENTATION_PLAN.md)." Public-safe implementation plan for generated MacKay, Bolelli, and Nicks diagnostics. - Mesmer Prism. "[Original Author Software Methods](https://github.com/MesmerPrism/bressloff-v1-form-constants/blob/main/docs/ORIGINAL_AUTHOR_SOFTWARE_METHODS.md)." Public-safe provenance note for software and numerical methods named in the source papers. - BifurcationKit contributors. "[BifurcationKit.jl](https://github.com/bifurcationkit/BifurcationKit.jl)." Public Julia package named in the continuation lineage for the color-hallucination work. - Mesmer Prism. "[Bressloff V1 Form Constants](https://mesmerprism.com/projects/bressloff-v1-form-constants.html)." Public overview page for the same implementation. - karacsm. "[V1-sim](https://github.com/karacsm/V1-sim)." Public notebook lineage for V1 activity and retino-cortical visualization. - CountYourCulture. "[Form Constants and the Visual Cortex](https://isomerdesign.com/countyourculture/2011/03/13/form-constants-visual-cortex/)." Public visual explainer for the form-constant comparison lineage. - Hewitt et al. "[Stroboscopically Induced Visual Hallucinations: Historical, Phenomenological, and Neurobiological Perspectives](https://doi.org/10.1093/nc/niaf020)." Neuroscience of Consciousness (2025). - Figure-rights sources. [PLOS licenses and copyright](https://journals.plos.org/ploscompbiol/s/licenses-and-copyright); [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/); [Royal Society permissions](https://royalsociety.org/journals/permissions/); [MIT Press rights and permissions](https://mitpress.mit.edu/rights-permissions/); [Springer Nature rights and permissions](https://www.springernature.com/gp/partners/rights-permissions-third-party-distribution); [IOP permissions FAQ](https://publishingsupport.iopscience.iop.org/permissionsfaq/).