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= Jansen-Rit Neural Mass =

{{Audience_Sidebar
| difficulty = Intermediate
| reading_time = 7 minutes
| prerequisites = ODEs; [[Wilson-Cowan_Model|Wilson-Cowan]]; some EEG/MEG familiarity.
| if_too_advanced_see = [[Wilson-Cowan_Model]]
| if_you_want_the_math_see = This page
}}

{{Notation
| signature = Mostly-plus (irrelevant here).
| units = Voltages in mV; time in ms; gain factors A, B in mV; rates a, b in 1/s.
}}

The '''Jansen-Rit neural-mass model''' (Jansen and Rit, 1995) is a lumped model of a cortical column. It treats three interacting populations — '''pyramidal cells''', '''excitatory interneurons''', '''inhibitory interneurons''' — as point masses, each described by a second-order ODE.

Jansen-Rit is the most widely-used neural-mass model in EEG/MEG source modelling. Multi-column networks built from Jansen-Rit units underlie tools like '''The Virtual Brain''' (TVB) and many dynamic-causal-modelling (DCM) workflows.

== Statement ==

Three coupled second-order ODEs for the membrane potentials y0, y1, y2 of pyramidal, excitatory-interneuron, and inhibitory-interneuron populations:

d2y0/dt2 = A · a · S[ y1 − y2 ] − 2a · dy0/dt − a2 · y0

d2y1/dt2 = A · a · [ p(t) + C2 · S[C1 y0] ] − 2a · dy1/dt − a2 · y1

d2y2/dt2 = B · b · C4 · S[C3 y0] − 2b · dy2/dt − b2 · y2

== Components ==

{| class="wikitable"
! Symbol !! Meaning !! Typical value
|-
| y0 || Pyramidal-cell membrane potential || mV
|-
| y1 || Excitatory-interneuron potential || mV
|-
| y2 || Inhibitory-interneuron potential || mV
|-
| A || Excitatory PSP amplitude || 3.25 mV
|-
| B || Inhibitory PSP amplitude || 22.0 mV
|-
| a || Inverse excitatory time constant || 100 /s (10 ms)
|-
| b || Inverse inhibitory time constant || 50 /s (20 ms)
|-
| C1–C4 || Intra-column connection strengths || C ≈ 135 (with C1=C, C2=0.8C, C3=C4=0.25C)
|-
| S(v) || Sigmoidal firing function: S(v) = 2 e0 / (1 + exp(r(v0 − v))) || e0 ≈ 2.5 /s, r ≈ 0.56 /mV, v0 ≈ 6 mV
|-
| p(t) || External input (e.g. thalamic) || 120–320 /s (white-noise band)
|}

The '''EEG signal''' is identified with y1 − y2 — the net excitatory minus inhibitory drive onto the pyramidal cell — because pyramidal-cell post-synaptic potentials in apical dendrites are the dominant source of the scalp EEG.

== Derivation sketch ==

# A second-order ODE of the form (d/dt + a)2 · y = A · a · u(t) is the impulse response of an '''excitatory post-synaptic potential''' (EPSP): a single delta-function input u = δ(t) produces y(t) = A · a · t · exp(−a · t), the canonical α-function PSP shape.
# Each population's net membrane potential is the convolution of its PSP impulse response with the firing rate of its inputs.
# The pre-synaptic firing rate is the sigmoidal function S of the pre-synaptic population's membrane potential.
# Three populations (pyramidal, excitatory interneuron, inhibitory interneuron) connected by C1–C4 close the loop.

The compact second-order-ODE form is just the differential equivalent of "PSP impulse response convolved with input rate".

== Dynamics ==

For appropriate parameters, Jansen-Rit produces:

* '''α rhythm''' (~ 10 Hz) — the default state at biologically-realistic parameters.
* '''Slow oscillations''' at higher inhibitory gain B.
* '''Spike-wave discharges''' (epilepsy-like) at extreme parameter values.
* '''Bifurcation transitions''' between regimes as p(t) is varied — used to model state transitions (waking ↔ sleep, normal ↔ epileptic).

Jansen-Rit was originally designed to match the visual-evoked-potential waveform; its parameters were tuned to fit empirical visual ERPs. Its later widespread use for spontaneous-EEG modelling was a fortunate generalisation.

== Sanity-check limits ==

* '''C = 0''': decoupled populations; each relaxes to zero. ✓
* '''Linear regime''' (small y, sigmoid linearised): three decoupled damped oscillators with frequencies ~ a, ~ b. ✓
* '''B = 0''' (no inhibition): runaway excitation; no oscillation. ✓
* '''A = 0''' (no excitation): no activity. ✓

== Multi-column extension ==

Networks of Jansen-Rit units connected by long-range projections (the '''connectome''') form the basis of large-scale brain models:

:<math>\frac{d^2 y_{0,i}}{dt^2} = A\,a\,S[\,y_{1,i} - y_{2,i}\,] - 2a\,\frac{dy_{0,i}}{dt} - a^2\,y_{0,i} + \sum_j K_{ij}\,S[\,y_{0,j}(t - \tau_{ij})\,]</math>

with delay τij and coupling Kij from diffusion-MRI tractography.

This is the architecture of '''The Virtual Brain''' (Sanz Leon et al. 2013) and similar large-scale tools.

== Connection to ψ ==

In the framework, Jansen-Rit units source ψ via their firing-rate output: Jψ(x,t) ∝ S[y1(x,t) − y2(x,t)]. ψ propagates non-locally and feeds back via an additive β · ψ term in the input to each column. Coherent collective oscillations across many Jansen-Rit columns produce the strongest ψ source — connecting the framework to standard whole-brain neuroimaging.

== Experimental status ==

Jansen-Rit is '''mainstream computational neuroscience''':

* Fit to visual-evoked potentials (the original 1995 work).
* Used in DCM for spontaneous EEG / MEG (Friston-group software).
* Underpins TVB and similar whole-brain simulators.
* Validates against frequency-band statistics, ERP waveforms, and parametric pharmacological-modulation studies.

== See Also ==

* [[Wilson-Cowan_Model]]
* [[Amari_Neural_Field]]
* [[Hodgkin-Huxley_Equations]]
* [[Neural_Field_Equations]]
* [[Wilson-Cowan_Coupled_to_Psi]]

== References ==

* Jansen, B. H., Rit, V. G. (1995). "Electroencephalogram and visual evoked potential generation in a mathematical model of coupled cortical columns." ''Biological Cybernetics'' 73: 357–366.
* David, O., Friston, K. J. (2003). "A neural mass model for MEG/EEG: Coupling and neuronal dynamics." ''NeuroImage'' 20: 1743–1755.
* Sanz Leon, P., et al. (2013). "The Virtual Brain: A simulator of primate brain network dynamics." ''Frontiers in Neuroinformatics'' 7: 10.

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

Jansen-Rit Neural Mass - 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 $; lint-clean per tools/wiki_latex_lint.py$