Figure S1: Estimated Networks 1 & 3 from linear factor models of DS (Top) and Granger causality (Bottom) for simulated data experiment. Each panel shows a grid of DS or Granger causality (GC) features associated with the indicated network estimate. Within each grid, a plot corresponds to signal that is being transmitted from the channel listed on the left to the channel listed at the top. See Figure 1 for a description of the true networks. Each subplot represents the DS from the region listed on the left to the region listed on top. Power spectra are reasonable to model using a linear factor model because they satisfy Definition 1 under reasonable assumptions. We will use Scc(ω) to refer to the spectral power of the signal vc(t) at frequency ω, and vc(ω) to refer to the frequency domain representation of vc(t) at ω.

A very common assumption when dealing with neural timeseries recordings is that the recorded signal within each windownisapproximately stationary,and isappropriately modeled asaVAR process [24,29,34,35]. The autoregressivematrices,A1,A2,...,Ap, define the how the previous signal valuesinfluence vt. The cross-spectral matrix can be factorized into aunique set of VAR parameters H(ω) and Σ [53]. The causal component of power (Hcb(ω)Σb|cH cb) is exactly the Directed Spectrum as defined in Section3.3. C bc(ω)Cbc(ω+δω), (42) where I() represent the imaginary component of the expression in parentheses,F is a group of sequential frequencies forwhich thePSIisbeing calculated, andδω isthefrequencyresolution of therecording. Wenote thatthepowerspectrum andother elements ofthecross-spectral matrix should both scale linearly with the network activationZ(j) (for more details see Supplemental Section A).

While some biological neural networks are well known, we expect that the vast majority remain undiscovered due to the enormous variety of tasks the brain performs. Many methods have been developed to help discover latent networks of neural populations (i.e.