Title: Predicting how and when hidden neurons skew measured synaptic interactions,
Abstract: A major obstacle to understanding neural coding and computation is the fact
that experimental recordings typically sample only a small fraction of the
neurons in a circuit. Measured neural properties are skewed by interactions
between recorded neurons and the "hidden" portion of the network. To properly
interpret neural data and determine how biological structure gives rise to
neural circuit function, we thus need a better understanding of the
relationships between measured effective neural properties and the true
underlying physiological properties. Here, we focus on how the effective
spatiotemporal dynamics of the synaptic interactions between neurons are
reshaped by coupling to unobserved neurons. We find that the effective
interactions from a pre-synaptic neuron $r'$ to a post-synaptic neuron $r$ can
be decomposed into a sum of the true interaction from $r'$ to $r$ plus
corrections from every directed path from $r'$ to $r$ through unobserved
neurons. Importantly, the resulting formula reveals when the hidden units
have---or do not have---major effects on reshaping the interactions among
observed neurons. As a particular example of interest, we derive a formula for
the impact of hidden units in random networks with "strong"
coupling---connection weights that scale with $1/\sqrt{N}$, where $N$ is the
network size, precisely the scaling observed in recent experiments. With this
quantitative relationship between measured and true interactions, we can study
how network properties shape effective interactions, which properties are
relevant for neural computations, and how to manipulate effective interactions.