JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to $; lint-clean per tools/wiki_latex_lint.py$

2026-05-11T20:06:14Z

Phase N (01b): LaTeX restoration — promote Unicode display-math to <math>; lint-clean per tools/wiki_latex_lint.py

New page

= Neural Field Equations =

{{Audience_Sidebar
| difficulty = Intermediate
| reading_time = 9 minutes
| prerequisites = PDEs; [[Wilson-Cowan_Model|Wilson-Cowan]] and [[Amari_Neural_Field|Amari]]; basic dynamical-systems.
| if_too_basic_see = [[Wilson-Cowan_Coupled_to_Psi]]
| if_you_want_the_math_see = This page
}}

{{Notation
| psi_convention = ψ = scalar field amplitude (NOT the neural-field variable u).
| signature = Mostly-plus (only relevant for the ψ-extended form).
| units = SI for biological observables; cortical position x in metres.
}}

This is the '''index page''' for the family of equations used to model neural-population dynamics in the framework. It collects the canonical models, their relationships, and their roles in the framework.

== The model hierarchy ==

The framework uses a layered hierarchy of models, from single-neuron biophysics up to whole-brain mean-field equations and finally to ψ-coupled extensions:

{| class="wikitable"
! Scale !! Model !! Variables !! Use
|-
| Single neuron (full biophysics) || [[Hodgkin-Huxley_Equations|Hodgkin-Huxley]] || V, m, h, n || Action-potential generation, ion-channel dynamics
|-
| Single neuron (reduced) || [[FitzHugh-Nagumo_Equations|FitzHugh-Nagumo]] || v, w || Pedagogy; phase-plane intuition
|-
| Population (lumped) || [[Wilson-Cowan_Model|Wilson-Cowan]] || E(t), I(t) || Oscillations, multistability; foundation
|-
| Population (spatial) || [[Amari_Neural_Field|Amari]] || u(x, t) || Bumps, waves, working memory
|-
| Cortical column || [[Jansen-Rit_Neural_Mass|Jansen-Rit]] || y0, y1, y2 || EEG/MEG modelling
|-
| Whole-brain network || Jansen-Rit + connectome || y0,i, y1,i, y2,i || TVB, DCM, large-scale dynamics
|-
| ψ-coupled brain || [[Wilson-Cowan_Coupled_to_Psi|WC + ψ]] || u(x,t), ψ(x,t) || Framework target
|}

Each row is well-validated mainstream computational neuroscience; the bottom row is the framework's extension.

== Common structure ==

All these models share a common functional skeleton:

('''Membrane / population time dynamics''') = ('''Self-decay''') + ('''Network input via sigmoid''') + ('''External input''')

The differences lie in:

* '''How many compartments''' (single neuron vs population vs many populations).
* '''Spatial structure''' (point vs field vs network).
* '''Order of ODE''' (1st order for WC/Amari; 2nd order for Jansen-Rit).
* '''Specific nonlinearity''' (sigmoid in all; Hill-function in some variants).

== Cross-walks ==

* '''Hodgkin-Huxley ↔ FitzHugh-Nagumo''' — fast/slow reduction; quasi-steady-state for m.
* '''Many neurons ↔ Wilson-Cowan''' — mean-field limit; sigmoid for firing-rate distribution.
* '''Wilson-Cowan ↔ Amari''' — discrete-to-continuum limit in space.
* '''Wilson-Cowan ↔ Jansen-Rit''' — second-order form of EPSP / IPSP impulse response.
* '''Jansen-Rit columns + connectome ↔ TVB / DCM''' — multi-column network.

== Adding the ψ field ==

The framework adds [[Wilson-Cowan_Coupled_to_Psi|two terms]]:

# '''Source''' coupling: neural firing produces a ψ source. Jψ(x,t) = κJ · f(u(x,t)).
# '''Feedback''' coupling: ψ modulates neural dynamics. + β · ψ(x,t) added to the synaptic-input drive.

ψ itself satisfies the relativistic field equation:

:<math>\Box\psi - m^2\psi - \lambda\psi^3 = \alpha\,F_{\mu\nu}F^{\mu\nu} + J_\psi</math>

So the full ψ-coupled Amari (or Wilson-Cowan or Jansen-Rit) system is:

:<math>\tau\,\frac{\partial u(\mathbf{x},t)}{\partial t} = -u + \!\int\! w(\mathbf{x}-\mathbf{x}')\,f\bigl(u(\mathbf{x}',t)\bigr)\,d^n x' + h(\mathbf{x},t) + \beta\,\psi(\mathbf{x},t)</math>

:<math>J_\psi(\mathbf{x},t) = \kappa_J\,f\bigl(u(\mathbf{x},t)\bigr)</math>

:<math>\Box\psi - m^2\psi - \lambda\psi^3 = \alpha\,F_{\mu\nu}F^{\mu\nu} + J_\psi</math>

Three coupling parameters:

{| class="wikitable"
! Parameter !! Meaning !! Estimated magnitude
|-
| κJ || Firing → ψ-source strength || ~ 10−6 (natural units)
|-
| β || ψ → firing-rate strength || ~ 10−3 (small but detectable)
|-
| α || EM → ψ vertex (universal) || See [[Effective_Field_Theory_of_Consciousness]]
|}

== Properties of the loop ==

# '''β > 0''' (positive feedback) → trance, kundalini, mystical-state regime. Self-sustaining oscillation: firing → ψ → more firing.
# '''β < 0''' (negative feedback) → meditative quietude regime. ψ suppresses firing locally.
# '''ψ-resonance at ω* = √(m2 + k2)''' for plane-wave perturbations — practitioners frequency-lock to specific values.

These regime distinctions are the framework's mapping from neural-field dynamics to phenomenology of altered states. See [[Practice_to_Theory_Translation_Table]].

== Sanity checks ==

{| class="wikitable"
! Limit !! Should recover !! Status
|-
| β = 0 || Pure Wilson-Cowan / Amari; no ψ → brain coupling || ✓
|-
| κJ = 0 || ψ decoupled from brain; just classical scalar field || ✓
|-
| α = 0 || ψ doesn't couple to EM either; decoupled scalar || ✓
|-
| β = 0 AND κJ = 0 AND α = 0 || Mainstream neuroscience + standard QFT; no psionics || ✓
|-
| Single neuron, no coupling || HH or LIF biophysics || ✓
|-
| Slow-firing limit || f(u) ≈ linear; system linearises || ✓
|-
| Sigmoid → step function || Heaviside threshold; classical threshold dynamics || ✓
|}

== Experimental status ==

* The baseline neural-field equations (HH, FN, WC, Amari, Jansen-Rit) are all '''mainstream, validated, undisputed'''.
* The ψ-coupling terms (κJ, β) are '''framework extensions'''; their values are estimated from anomalous-cognition experiments and meditation neuroscience. They are testable via precision EEG/MEG under controlled ψ-source conditions.
* The most direct test: does meditative coherence produce small but systematic correlations with anomalous-cognition signals beyond what classical neural-field dynamics predict?

== See Also ==

* [[Wilson-Cowan_Model]]
* [[Amari_Neural_Field]]
* [[Hodgkin-Huxley_Equations]]
* [[FitzHugh-Nagumo_Equations]]
* [[Jansen-Rit_Neural_Mass]]
* [[Wilson-Cowan_Coupled_to_Psi]]
* [[Sanity_Check_Limits]] (next page in the Undergrad Physics reading path)
* [[Effective_Field_Theory_of_Consciousness]]
* [[Practice_to_Theory_Translation_Table]]

== References ==

* Coombes, S., Beim Graben, P., Potthast, R., Wright, J. (eds.) (2014). ''Neural Fields: Theory and Applications.'' Springer.
* Bressloff, P. C. (2012). "Spatiotemporal dynamics of continuum neural fields." ''Journal of Physics A: Mathematical and Theoretical'' 45: 033001.
* Deco, G., Jirsa, V. K., Robinson, P. A., Breakspear, M., Friston, K. (2008). "The dynamic brain: From spiking neurons to neural masses and cortical fields." ''PLoS Computational Biology'' 4: e1000092.

[[Category:Psionics]]
[[Category:Equations]]
[[Category:Neuroscience]]

Neural Field Equations - Revision history

JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to ; lint-clean per tools/wiki_latex_lint.py

JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to $; lint-clean per tools/wiki_latex_lint.py$