jansen_rit.md

Jansen-Rit model

$$\begin{equation*} \begin{aligned} d X_t &= \dot X_t d t \\\ d Y_t &= \dot Y_t d t \\\ d Z_t &= \dot Z_t d t \\\ d \dot X_t &= \left[A a \left(\mu_x(t) + \mbox{Sigm}(Y_t - Z_t)\right) - 2a \dot X_t - a^2 X_t\right] d t + \sigma_x d W^{(1)}_t\\\ d \dot Y_t &= \left[A a \left(\mu_y(t) + C_2\mbox{Sigm}(C_1 X_t)\right) - 2a \dot Y_t - a^2 Y_t\right] d t + \sigma_y d W^{(2)}_t\\\ d \dot Z_t &= \left[B b \left(\mu_z(t) + C_4\mbox{Sigm}(C_3 X_t)\right) - 2b \dot Z_t - b^2 Z_t\right] d t + \sigma_z d W^{(3)}_t, \end{aligned} \end{equation*}$$

with initial condition

$$(X_0,Y_0,Z_0, \dot X_0, \dot Y_0, \dot Z_0)=(x_0,y_0,z_0, \dot x_0, \dot y_0, \dot z_0) \in R^6$$

where

$$\mbox{Sigm}(x) := \frac{\nu_{max}}{1 + e^{r(v_0 - x)}},$$

and

$$C_1 = C, \quad C_2 = 0.8C, \quad C_4 = C_3 = 0.25C.$$