Table 1 Dynamics of the different model variants, where ht is the cortical RNN state, zt the readout and ct cerebellar feedback

From: Cerebellar-driven cortical dynamics can enable task acquisition, switching and consolidation

 

No feedback

Readout feedback

Cerebellar feedback

No feedback (cerebellar readout)

ht

αht−1 + Whhf(ht−1) + Wihxt

αht−1 + Whhf(ht−1) + Wihxt + Wzhzt

\(\alpha {{{{\bf{h}}}}}_{t-1}+{{{{\bf{W}}}}}_{hh}f({{{{\bf{h}}}}}_{t-1})+{{{{\bf{W}}}}}_{ih}{{{{\bf{x}}}}}_{t}+{{{{\bf{W}}}}}_{{{{\mathcal{C}}}}h}{{{{\bf{c}}}}}_{t}\)

αht−1 + Whhf(ht−1) + Wihxt

zt

Wrdtf(ht)

Wrdtf(ht)

Wrdtf(ht)

\({{{\mathcal{C}}}}(f({{{{\bf{h}}}}}_{t}))\)

ct

NA

NA

\({{{\mathcal{C}}}}(f({{{{\bf{h}}}}}_{t-1}))\)

NA

  1. For the experiments presented here we set \(f=\tanh\) and \({{{\mathcal{C}}}}\) is the cerebellar feedforward network with one hidden layer, \({{{\mathcal{C}}}}\left(f\left({{{\bf{h}}}}\right)\right)={{{{\bf{W}}}}}_{{{{\rm{PF}}}}}{f}^{{{{\mathcal{C}}}}}\left({{{{\bf{W}}}}}_{{{{\rm{MF}}}}}f({{{\bf{h}}}})\right)\). Whh RNN recurrent weights; Wih stimulus-to-RNN weights, Wrdt (cortical) readout weights; \({{{{\bf{W}}}}}_{{{{\mathcal{C}}}}h}\), cerebellar-to-RNN weights, WMF cerebellar mossy fibre weights, WPF cerebellar parallel fibre weights; \({f}^{{{{\mathcal{C}}}}}\) set as ReLU.
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