 modified as follows: $$d_{x}\left( {t + \Delta t} \right) = max\left\{ 0,\ {\ d}_{x}(t) + g\frac{b + v_{x}\left( t \right)}{s + {\sum_{y}{v_{y}\left( t \right)}}} + {N\left( 0,\sigma^{0} \right)}_{x}(t) \right\}$$  For two alternatives, this 