Neural Field Equations

Neural Field Equations

Audience

Difficulty	Intermediate

Notation on this page

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:

Scale	Model	Variables	Use
Single neuron (full biophysics)	Hodgkin-Huxley	V, m, h, n	Action-potential generation, ion-channel dynamics
Single neuron (reduced)	FitzHugh-Nagumo	v, w	Pedagogy; phase-plane intuition
Population (lumped)	Wilson-Cowan	E(t), I(t)	Oscillations, multistability; foundation
Population (spatial)	Amari	u(x, t)	Bumps, waves, working memory
Cortical column	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	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 two terms:

1. Source coupling: neural firing produces a ψ source. J_ψ(x,t) = κ_J · f(u(x,t)).
2. Feedback coupling: ψ modulates neural dynamics. + β · ψ(x,t) added to the synaptic-input drive.

ψ itself satisfies the relativistic field equation:

$ \Box \psi -m^{2}\psi -\lambda \psi ^{3}=\alpha \,F_{\mu \nu }F^{\mu \nu }+J_{\psi } $

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

$ \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) $

$ J_{\psi }(\mathbf {x} ,t)=\kappa _{J}\,f{\bigl (}u(\mathbf {x} ,t){\bigr )} $

$ \Box \psi -m^{2}\psi -\lambda \psi ^{3}=\alpha \,F_{\mu \nu }F^{\mu \nu }+J_{\psi } $

Three coupling parameters:

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

1. β > 0 (positive feedback) → trance, kundalini, mystical-state regime. Self-sustaining oscillation: firing → ψ → more firing.
2. β < 0 (negative feedback) → meditative quietude regime. ψ suppresses firing locally.
3. 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

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 (κ_J, 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.