Model Reference¶
Wilson-Cowan¶
The Wilson-Cowan model describes coupled excitatory (E) and inhibitory (I) neural populations:
$$ \begin{aligned} \tau_E \dot{E} &= -E + S_e(a_{EE} E - a_{EI} I + P_E) \ \tau_I \dot{I} &= -I + S_i(a_{IE} E - a_{II} I + P_I) \end{aligned} $$
where $S_e$, $S_i$ are sigmoid activation functions:
$$ S(x) = \frac{1}{1 + e^{-k(x - \theta)}} $$
Parameters¶
| Parameter | Default | Range | Description |
|---|---|---|---|
aee |
10.0 | 0–20 | E→E coupling strength |
aei |
10.0 | 0–20 | I→E coupling strength |
aie |
10.0 | 0–20 | E→I coupling strength |
aii |
2.0 | 0–20 | I→I coupling strength |
Pe |
-2.0 | -10–10 | External input to E |
Pi |
-8.0 | -10–10 | External input to I |
ke |
1.0 | 0.1–5 | E sigmoid gain |
ki |
1.0 | 0.1–5 | I sigmoid gain |
thetae |
4.0 | 0–10 | E sigmoid threshold |
thetai |
4.0 | 0–10 | I sigmoid threshold |
Typical Dynamics¶
- Low external drive → stable fixed point (low activity)
- Increased drive → limit cycle (oscillations)
- Strong E→E, weak I→E → epileptic-like runaway activity
FitzHugh-Nagumo¶
A simplified model of neuronal excitability, capturing the essence of Hodgkin-Huxley dynamics with two variables:
$$ \begin{aligned} \dot{v} &= v - \frac{v^3}{3} - w + I \ \dot{w} &= \varepsilon (v + a - b w) \end{aligned} $$
Parameters¶
| Parameter | Default | Range | Description |
|---|---|---|---|
a |
0.7 | -1–2 | Recovery offset |
b |
0.8 | 0–2 | Recovery gain |
epsilon |
0.08 | 0.001–1 | Time scale separation |
I |
0.5 | -2–2 | External current |
Typical Dynamics¶
- $I < I_{crit}$: stable fixed point (resting state)
- $I > I_{crit}$: limit cycle (repetitive firing)
- The cubic nullcline ($\dot{v}=0$) and linear nullcline ($\dot{w}=0$) intersect to produce these transitions
MPR (Quadratic Integrate-and-Fire)¶
The Montbrió-Pazó-Roxin model derives a firing-rate description for a population of quadratic integrate-and-fire neurons with Lorentzian-distributed excitability:
$$ \begin{aligned} \dot{r} &= \frac{\Delta}{\pi} + 2 r v \ \dot{v} &= v^2 + \bar{\eta} + J r + I(t) - (\pi r)^2 \end{aligned} $$
where $r$ is the population firing rate and $v$ is the mean membrane potential.
Parameters¶
| Parameter | Default | Range | Description |
|---|---|---|---|
delta |
1.0 | 0.01–5 | Lorentzian half-width Δ |
eta_bar |
-5.0 | -20–10 | Mean excitability η̄ |
J |
15.0 | -20–30 | Synaptic coupling J |
I |
0.0 | -10–10 | External input I(t) |
Key Phenomena¶
- Saddle-node bifurcation: Increasing η̄ or J creates/annihilates pairs of fixed points
- Bistability: For Δ=1, η̄≈-5, J≈15 there are three coexisting fixed points: a stable low-activity node, a saddle, and a stable high-activity focus
- Limit cycles: Beyond the bistable region, oscillatory firing-rate dynamics emerge
Paper Reference¶
Montbrió, E., Pazó, D., & Roxin, A. (2015). Macroscopic description for networks of spiking neurons. Physical Review X, 5(2), 021028. DOI: 10.1103/PhysRevX.5.021028
Reproducing the Paper¶
See the MPR Demo for interactive reproductions of Figures 1 and 2 from the paper.