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==Appendix A: Derivation of a temporal filter for short-term depression==

==Appendix A: Derivation of a temporal filter for short-term depression==

We consider the rate-based dynamics in Eq. （3）for depression-dominated synapses (<math>u^+ \approx U</math>) and for synaptic responses that are much faster than the depression dynamics (<math>\tau_s \ll \tau_d</math>$\tau_s \ll \tau_d$):

We consider the rate-based dynamics in Eq. （3）for depression-dominated synapses (<math>u^+ \approx U</math>) and for synaptic responses that are much faster than the depression dynamics (<math>\tau_s \ll \tau_d</math>):

我们考虑方程式中基于速率的动态。 \ref{poisson} 用于抑郁症主导的突触 <math>u^+ \approx U</math>和比抑郁症动力学快得多的突触反应 ($\tau_s \ll \tau_d$ ):

我们考虑方程式中基于速率的动态。方程式（3）用于抑郁症主导的突触 <math>u^+ \approx U</math>和比抑郁症动力学快得多的突触反应 (<math>\tau_s \ll \tau_d</math>):

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The aim is to derive a filter $\chi$ that relates the output synaptic current $I$ to the input rate $R$.

The aim is to derive a filter<math>\chi</math>that relates the output synaptic current $I$ to the input rate $R$.

Note that because the input rate $R$ enters the equations in a multiplicative fashion the input-output transfer function is non linear. Yet a linear filter can be derived by considering small perturbations $R_1 \rho(t)$ of the firing rate $R(t)$ around a constant rate $R_0$, that is,

Note that because the input rate $R$ enters the equations in a multiplicative fashion the input-output transfer function is non linear. Yet a linear filter can be derived by considering small perturbations $R_1 \rho(t)$ of the firing rate $R(t)$ around a constant rate $R_0$, that is,