The equation
$$ v_{i+1}=h_i(x_i,v_i,F_i). $$can be used to train a feed-forward neural network with inputs \( v_i \) and outputs \( v_{i+1} \) at a time \( t_i \). But we can think of this also as a recurrent neural network with inputs \( v_i \), \( x_i \) and \( F_i \) at each time step \( t_i \), and producing an output \( v_{i+1} \).
Noting that
$$ v_{i}=v_{i-1}+\Delta t\left(F_{i-1}-\eta v_{i-1}-x_{i-1}\right)=h_{i-1}. $$we have
$$ v_{i}=h_{i-1}(x_{i-1},v_{i-1},F_{i-1}), $$and we can rewrite
$$ v_{i+1}=h_i(x_i,h_{i-1},F_i). $$