Table 1.

Dynamics of the different model variants, where h_t is the cortical RNN state, z_t the readout and c_t cerebellar feedback

	No feedback	Readout feedback	Cerebellar feedback	No feedback (cerebellar readout)
ht	αht−1 + Whhf(ht−1) + Wihxt	αht−1 + Whhf(ht−1) + Wihxt + Wzhzt	$\alpha {\mathbf{h}}_{t-1}+{\mathbf{W}}_{hh}f\left({\mathbf{h}}_{t-1}\right)+{\mathbf{W}}_{ih}{\mathbf{x}}_t+{\mathbf{W}}_{\mathcal{C}h}{\mathbf{c}}_t$	αht−1 + Whhf(ht−1) + Wihxt
zt	Wrdtf(ht)	Wrdtf(ht)	Wrdtf(ht)	$\mathcal{C}\left(f\left({\mathbf{h}}_t\right)\right)$
ct	NA	NA	$\mathcal{C}\left(f\left({\mathbf{h}}_{t-1}\right)\right)$	NA

For the experiments presented here we set $f=\tanh$ and $\mathcal{C}$ is the cerebellar feedforward network with one hidden layer, $\mathcal{C}\left(f\left(\mathbf{h}\right)\right)={\mathbf{W}}_{\mathrm{PF}}{f}^{\mathcal{C}}\left({\mathbf{W}}_{\mathrm{MF}}f\left(\mathbf{h}\right)\right)$. W_hh RNN recurrent weights; W_ih stimulus-to-RNN weights, W_rdt (cortical) readout weights; ${\mathbf{W}}_{\mathcal{C}h}$, cerebellar-to-RNN weights, W_MF cerebellar mossy fibre weights, W_PF cerebellar parallel fibre weights; $f^{\mathcal{C}}$ set as ReLU.