Fig 2. Expansion of effective interactions into contributions from hidden paths.

A. Graphical representation of Eq (2). The linear response of the hidden network, ${\widehat{\Gamma }}_{h,h^{\prime }}\left(\omega \right)$, has been expanded as a series (corresponding to the grey hidden nodes and links between them), such that each term in the overall series can be interpreted as a contribution from a path through which the pre-synaptic neuron r′ is able to send a signal to post-synaptic neuron r via 1, 2, etc. hidden neurons. This expression holds for any pair of neurons in the recorded subset. B. Quantitative expressions for each diagram in the series can be read off by assigning the shown factors for each hidden neuron node and each link between neurons, recorded or hidden, and multiplying them together. (No factor is assigned to the recorded neuron nodes). γ_h is the gain of neuron h and ${\widehat{J}}_{i,j}\left(\omega \right)$ is the true interaction from j to i in the frequency domain.