Table 1.

LTP and LTD asymptotic behaviors

	LTP	LTD
$\langle \delta w\rangle \ll \overline{\delta w^{th}}$	0	0
$\langle \delta w\rangle ~\overline{\delta w^{th}}$	$\lambda {\rho }_{\text{post}}\left(\frac{1}{2}{\langle \delta w_{\text{LTP}}\rangle }_{t_{\text{post}}}-\frac{1}{2}\overline{\delta w_{\text{LTP}}^{\text{th}}}+\frac{\sigma {\left(\delta w_{\text{LTP}}\right)}_{t_{\text{post}}}}{\sqrt{2\pi }}\right)$	$-\lambda {\rho }_{\text{pre}}\left(\frac{1}{2}{\langle \delta w_{\text{LTD}}\rangle }_{t_{\text{pre}}}-\frac{1}{2}\overline{\delta w_{\text{LTD}}^{\text{th}}}+\frac{\sigma {\left(\delta w_{\text{LTD}}\right)}_{t_{\text{pre}}}}{\sqrt{2\pi }}\right)$
$\langle \delta w\rangle \gg \overline{\delta w^{\text{th}}}$	$\lambda {\rho }_{\text{post}}\left({\langle \delta w_{\text{LTP}}\rangle }_{t_{\text{post}}}-\overline{\delta w_{\text{LTP}}^{\text{th}}}\right)$	$-\lambda {\rho }_{\text{pre}}\left({\langle \delta w_{\text{LTD}}\rangle }_{t_{\text{pre}}}-\overline{\delta w_{\text{LTD}}^{\text{th}}}\right)$

The middle and last columns contain the asymptotic expression of LTP and LTD terms for the three possible regimes corresponding to the cases in which the fast variables are well below $\langle \delta w\rangle \ll \overline{\delta w^{\text{th}}}$, close to $\langle \delta w\rangle ~\overline{\delta w^{\text{th}}}$, or well above $\langle \delta w\rangle \gg \overline{\delta w^{\text{th}}}$ the slow variables. Depending on the past activity of the synapse and on the parameter set, any combination of these LTP/LTD terms can occur and describe the current plasticity pattern.