Table 2.

The adjoint spiking network to Table 1 that computes the adjoint variable ${\lambda }_I$ needed for the gradient [Eq. (2)]. The adjoint variables are computed in reverse time (i.e., from $t=T$ to $t=0$) with ${}^{\prime }=-\frac{\mathrm{d}}{\mathrm{d}t}$ denoting the reverse time derivative. ${({\lambda }_V^{-})}_{n(k)}$ experiences jumps at the spikes times $t_k^{\text{post}}$, where n(k) is the index of the neuron that caused the kth spike. Computing this system amounts to the backpropagation of errors in time. The initial conditions are ${\lambda }_V(T)={\lambda }_I(T)=0$ and we provide ${\lambda }_V^{-}$ in terms of ${\lambda }_V^{+}$ because the computation happens in reverse time

Free dynamics	Transition condition	Jump at transition
$\begin{array}{cc}\hfill {\tau }_{\text{mem}}{\lambda }_V^{\prime }& =-{\lambda }_V-\frac{\partial l_V}{\partial V}\hfill \\ \hfill {\tau }_{\text{syn}}{\lambda }_I^{\prime }& =-{\lambda }_I+{\lambda }_V\hfill \end{array}$	$\begin{array}{cc}\hfill t-t_k^{\text{post}}& =0\hfill \\ \hfill \text{for any}& k\hfill \end{array}$	$\begin{array}{cc}\hfill {({\lambda }_V^{-})}_{n(k)}& ={({\lambda }_V^{+})}_{n(k)}+\frac{1}{{\tau }_{\text{mem}}{({\dot{V}}^{-})}_{n(k)}}[\vartheta {({\lambda }_V^{+})}_{n(k)}\hfill \\ \hfill & +{\left(W^{\top },({\lambda }_V^{+}-{\lambda }_I)\right)}_{n(k)}+\frac{\partial l_{\text{p}}}{\partial t_k^{\text{post}}}+l_V^{-}-l_V^{+}]\hfill \end{array}$