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q^{\uparrow}(t) =  f \, q^{\downarrow}(t)  - \tau \cdot \frac{ d q^{\uparrow}}{ dt }  - J \cdot \tau\beta \cdot \frac{d p}{dt}

\tau \geq  0 , \quad \beta \geq 0,  \quad  0 \leq f \leq 1

q^{\uparrow}_j (t) = \sum_i^{n_i} f_{ij} q^{\downarrow}_i(t)  - \tau_j \cdot  \frac{ d q^{\uparrow}_j}{ dt }  - J\beta_j \cdot  \tau_j \cdot  \frac{d p_j}{dt}

\tau_j \geq  0 ,  \quad \beta_j \geq 0,  \quad f_{ij} \geq  0 , \quad \sum_i^{N^{\uparrow}} f_{ij} \leq 1

Q^{\uparrow}_j (t) = \sum_i^{n_i} f_{ij} Q^{\downarrow}_i(t)  - \tau_j \cdot \big[ q^{\uparrow}_j(t) - q^{\uparrow}_j(0) \big]  - J_j \cdot  \tau\beta_j \cdot  \big[ p_j(t) - p_j(0) \big]

\tau_j \geq  0 ,  \quad \beta_j \geq 0,  \quad f_{ij} \geq  0 , \quad \sum_i^{N^{\uparrow}} f_{ij} \leq 1