Jansen-Rit Neural Mass
Jansen-Rit Neural Mass
Notation on this page
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
| 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:
- $ {\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})\,] $
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.
