Directed Spectrum Measures Improve Latent Network Models Of Neural Populations
Authors: Neil Gallagher
Kafui Dzirasa
David Carlson
Venue: NeurIPS 2021 Poster (2021)
Keywords: functional brain networks
linear factor models
directed connectivity
directed spectrum

Abstract: Systems neuroscience aims to understand how networks of neurons distributed throughout the brain mediate computational tasks. One popular approach to identify those networks is to first calculate measures of neural activity (e.g. power spectra) from multiple brain regions, and then apply a linear factor model to those measures. Critically, despite the established role of directed communication between brain regions in neural computation, measures of directed communication have been rarely utilized in network estimation because they are incompatible with the implicit assumptions of the linear factor model approach. Here, we develop a novel spectral measure of directed communication called the Directed Spectrum (DS). We prove that it is compatible with the implicit assumptions of linear factor models, and we provide a method to estimate the DS. We demonstrate that latent linear factor models of DS measures better capture underlying brain networks in both simulated and real neural recording data compared to available alternatives. Thus, linear factor models of the Directed Spectrum offer neuroscientists a simple and effective way to explicitly model directed communication in networks of neural populations.

Directed Spectrum Measures Improve Latent Network Models Of Neural Populations
Neil M. Gallagher
Kafui Dzirasa
Department of Neurobiology Duke University
Howard Hughes Medical Institute
Department of Psychiatry and Behavioral Sciences
Durham, NC 27708 neil, wal.aagrrrreereeeeeee..m.o edu
Department of Neurobiology
Duke University
Durham, NC 27710
kafui wwwrasadduue edu
David Carlson
Department of Biostatistics and Bioinformatics
Department of Civil and Environmental Engineering
Duke University
Durham, NC 27708
david. carlsonddukee edu
Abstract
Systems neuroscience aims to understand how networks of neurons distributed throughout the brain mediate computational tasks. One popular approach identify those networks is to first calculate measures of neural activity (e.g. power spectra) from multiple brain regions, and then apply a linear factor model those measures. Critically, despite the established role of directed communication between brain regions in neural computation, measures of directed communication have been rarely utilized in network estimation because they incompatible with the implicit ussumptions of the linear factor model approach. Here, we develop a novel spectral measure of directed communication called the Directed Spectrum (DS). We prove that it is compatible with the implicit assumptions of linear factor models, and provide method to estimate the DS. We demonstrate that latent linear factor models of DS measures better capture underlying brain networks in both simulated and real neural recording data compared available alternatives. Thus, linear factor models of the Directed Spectrum offer neuroscientists a simple and effective way to explicitly model directed communication in networks of neural populations.
1 Introduction
major goal neuroscience to characterize how populations of neurons work together carry out computational tasks []] A well-known example is the biological neural network that identifies low-level visual features such as edges. That network sends signals from the retina to the lateral geniculate nucleus, then from there to the primary visual cortex, then finally to higher cortical visual processing regions 2 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 brain networks) 4 A key aspect of such methods is that they should be interpretable, meaning that a neuroscientist must be able to use the model to draw conclusions about brain function [
help discover latent networks of neural populations (i.e.
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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Unfortunately, many types of common models in the machine learning literature do not offer this type of interpretability (
One class of widely-used models that is considered interpretable for modeling brain networks is Linear Factor Models (LFMs) p 10]. This family includes popular methods like principal component analysis (PCA) and independent component amalysis (ICA), and are used across almost all modalities in neuroscience model brain networks. For example, ICA is regularly used in functional magnetic resonance imaging (IMRI) applications to identify latent brain networks such as the default mode network [T] 12). A common approach in neuroscience is to apply an LFM to some measure or extracted feature from the recorded signal. rather than to the recorded signals themselves, in order to get a model of brain networks that are characterized in the desired measure. For example, power spectra are often calculated from raw electroencephalogram (EEG) recordings before an LFM is applied, yielding brain networks defined by power spectra [3
In multi-site local field potential (LFP) data, LFMs have been applied to identify latent brain networks that are defined by cross-spectral covariance matrix 15
One kind of measure that could be especially informative for defining latent brain networks is directed communication between brain regions. However, LFMs that directly incorporate measures of directed communication are lacking, We believe that this is because existing measures of directed communication, such as Granger causality [16], are incompatible with the implicit assumptions of using LFMs to identify brain networks. This incompatibility between standard measures of directed communication and LFMs discussed further in Sections 2 and 4 In a previous attempt to capture directed communication within an interpretable brain network model, Gallagher et al. [55 modeled networks in terms of phase shifts in spectral content between neural populations, but is unclear such phase shifts are an appropriate proxy for directed transmission of signals, and those models are bottlenecked by significant computational time.
In response to this methodological gap, we introduce a novel measure of directed communication that we will refer as the Directed Spectrum (DS). The Directed Spectrum estimates directed communication between time series in the frequency domain. We prove that these measures are linear function of latent brain networks under reasonable assumptions, making them compatible with LFMs in this application. We then demonstrate that an LFM of DS measures recovers latent networks in simulated dataset where the ground truth networks are known. We compare the performance to several competing directed communication measures, including Spectral Granger Causality [17], and show that using the Directed Spectrum results in significantly higher quality reconstruction of the true brain networks. Finally, we show that latent brain networks identified from real neural data via the Directed Spectrum can decode behaviorally relevant information with much higher fidelity.
2 Linear factor models (LFMs) for identifying brain networks
The term linear factor model
18 describes any model that seeks to approximate a data vector æ, as,
xr = Z!1 x + t
The vectors x (1)), (1),,.... ("),,..,,"") (2)
are the different latent factor loadings, E, is the additive noise term, and
2 is the activation score of the jth factor in the 1hth sample. æ, represents single sample from larger dataset, [æ1,.... ... N. Many well-known models, such as PCA 19. ICA 20), and nonnegative matrix factorizaton (11, are LFMs. IFMs are frequently used for modeling and discovering brain
networks [10] In that context, it is straightforward to interpret the scores, z as the activation levels for set of brain networks. Likewise, we can interpret the factor loading ())) as the observable signature of the jth brain network. LFMs are quite flexible; a wide variety of assumptions can be made regarding the structure of the factors, the scores, and the noise distribution 6 9 (2). The fact that LFMs are interpretable and adaptable to most data types has made them a standard model class for studying latent brain networks.
Note: For simplicity, the subscript n will be dropped from all variables for the remainder of this document. In cases where a variable is constant over all samples, such as the factor loadings U0), will be explicitly stated.

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2.1 Compatibility between linear factor models and measures of neural activity
is common to model measures of neural activity recordings rather than modeling the raw recording time-series directly. In this work, we assume that the vector a contains measures calculated from multi-channel time-series, V ; ukl", where K is the number of channels and v. E IR"" is the row of V corresponding to recording from single channel c. The frequency domain representations of these quantities will be marked by a tilde (e.g. (e(w) the frequency domain (See) for each channel 13
representation of t22. A typical choice for x is to convert the observed data into a set of power spectra
Seo(w) = (
Modeling these measures in LFM leads to the latent factor loadings xú) being described in terms of power spectra associated with each channel. The chosen measures dictate the representation the discovered networks, so we want to relevant and easy-to-ime toitter-itt measures.
When an LFM used model latent brain networks, there is an implicit assumption that the measures contained in æ are linear functions of the latent brain networks. This is seen in ① where the observed measure is linear function of the latent factor score ZO) within some noise tolerance. We let (-) represent a function, such as 2 that produces the set of measures from our observed neural data V.
Definition 1.
We call X(-) linear function of latent brain networks if
X(V) = 200060) = Eÿ ((:
where Q6) represents the component of the neural data (V) that is due to the jth network, normalized to represent a score of In this way, (12()) represents the expected value of the measure if the j"h network was active with a score of in isolation.
can shown that power spectra are theoretically consistent with this requirement (see Supplemental Section (A). but there many measures that cannot obey this assumption. For example, the Pearson correlation coefficient and Granger causality are two measures that cannot obey this assumption (see Section 4 Including such measures in LFMs will lead to suboptimal representations of latent brain networks.
3 Modeling directed communication in brain networks with LFMs
Measures capturing directed communication between neural populations are rarely included in LFMs despite their frequent use in neuroscience (24 We believe this is because standard measures of directed communication, including Granger causality [16] 17). are not linear functions of brain networks and therefore are poorly modeled by LFMs (see Section 4). Nonetheless, measures of directed communication such Granger causality are vital for the study of brain networks 22 28
Below, we introduce novel measure of directed communication, which we refer to as the Directed Spectrum (DS). In order to derive the Directed Spectrum, first build general time-series model of latent brain networks in Section 3.1 and then define how latent networks in the model combine to produce the observed signal in Section 3.2 The Directed Spectrum is formally defined in Section 3.3 captures communication in a similar manner to Granger causality while also being compatible with linear model of brain networks.
3.1 Modeling brain networks as independent VAR processes
We outline a model where the output for each latent network is defined by a vector autoregressive (VAR) process. For additional background on VAR models, see Supplemental Section B The model outlined below represents a general framework for understanding what the DS measures capture, though inferring it directly would be quite challenging. We begin by assuming that we have J latent brain networks that each generate vector timeseries outputs. Rather than slicing by channel as in Section 2.
we define the output series as VO) [o] ...., (2) where (.()) RR is the column of Vt) representing signal for all channels time t. The output series is associated with VAR process,
( =
(1.0) () ((-T) + t
of ~ ~ Z(150))
(4)
3

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()) We refer to of as the innovations in the jth network time 4, and assume that they are drawn iid from a zero mean Gaussian distribution. Each network has a single covariance structure over all samples (E()) that is scaled by an activation score (Z(3) define the innovation distribution of the current sample. The set of autocovariance matrices for the jth network "A.......A A.p are also assumed be constant over all samples. The innovation terms represent new signal introduced
into the network that cannot be explained by the past network outputs (v (()-))..... (t-p,) We can represent this network in the frequency domain given standard assumptions regarding the stability of the VAR model (29).
(5)
(w) = H (w)) ())(w) (1- ET- A SSw SSSS 2n, (5)
where (w) RK and (w) E RK are the frequency domain representations of the series 0(") ....., Bi and ol TT respectively. The network transfer matrix (H()))) (w:) is assumed to
be constant over all samples.
3.2 Modeling observed signal as a superposition of latent transmitted signals
In order to complete our model of latent brain networks, must relate it to the observed signal. We model each observed sample by a VAR in the frequency domain, analogous to the way we modeled latent network outputs,
( = H (w))(W).
We assume that the K channels in our data are partitioned into a set of non-overlapping groups, G = (b, ....... For example, each group can be a single channel to model all inter-channel relationships, or could be all channels within given brain region. We partition the observed data and the transfer matrix H (w) and innovations o (w) into these groups.
(b(w)
Hbb(w) Hoclw) Heb
Hba(w) (w)
(e(w)
Heelw) H(w) = Hab(w) Had(w)
Hea (w)
((w)- = (d(w)
Had(w)
(7)
[b((w)
Ebb
Ebe. Exe
End
Ta(w)
Eab
End
(ww) = @a(w)
E
Eab
Edc
Ead
We assume that the networks outputs and corresponding VAR parameters are partitioned in the same way. The ordering of these groups is arbitrary. Without loss of generality, we will focus on modeling communication between groups b and
We examine how our VAR models represent communication between groups by noting that
Te(w) = Egeg
This shows that (e(w) is sum of contributions from each group g, including the contribotiiition from c. If the innovations in b and are independent, then Hes (w))@w)))))) would unambiguously represent the signal in that can be attributed to innovations b. If the innovations in b and e are not independent, then there is ambiguity regarding how to assign 'responsibility' for the observed signal in c. Specifically, the component of the innovations in that is correlated with innovations in e not necessary to explain the observed signal in e in expectation, since the innovations in are sufficient explain this part of the observed signal. We will assume that only the uncorrelated component of the innovations in b contribute to the observed signal in c.
Definition 2. The transmitted signal represents the contribution of the innovations in b to the ob-
served signal
TSp-e(ww) = Heb(w)) (ōb(w)- Eoc
b b
(9)
Egec ( (o)
c

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By restricting TSb-c(w) to only convey uncorrelated innovations, we construct a more conservative estimate of the effect of b on This is not the only choice that could made; for example, we could use the full innovations, but we believe that using the uncorrelated innovations provides the cleanest interpretation.
We complete our model of the observed signal ((w) with the following two assumptions: Assumption 1. The innovations associated with any network are independent of the innovations in all other networks.
The transmitted signals of the observed data are the sum of the corresponding transmitted signals for the latent networks,
Assumption 2.
TS = = TS
(10)
where the transmitted signal is defined for the latent networks as
TS
H (w) (w) Ege Heg (W) 20c (())
be Lee
( بلي
(w).
b b
(11)
b=c
Assumption 1 can be thought of enforcing the independence of inputs each network. Assumption 2 is equivalent to the following two statements; 1) the propagation of signal within each network is independent of propagation in each other network, and 2) the observed outputs occur in some medium where simultaneously occurring phenomena obey the laws of superposition, which is property of many neural activity measurement modalities (e.g., electrical potentials. fluorescence). By combining these two assumptions with (8) network outputs,
we get the result that the observed data are superposition of the
(e(w)) = Eges Heg (w) (w)
= Egeg TS (ww)
(12)
= -gee Eラ TS = Lges ES-1 Hiy (w)
(
( (w) (w) Ej 50 (w).
(w))
(13)
Note in (12) and 13 summing the transmitted signals to over all source groups, including causes the conditioning terms our definitions of the transmitted signal to cancel out. Because the relationship above does not contain noise term, con-yossiologiaal sources of signal would simply be represented as additional networks in the model. This completes our model of the relationship between latent brain networks and observed signals.
3.3
The Directed Spectrum
The model defined above provides framework for determining whether a measure of neural recordings can be considered linear function of latent brain networks.
Definition We define our measure, the Directed Spectrum (DS), as the second moment of the transmitted signal,
DSb-re(w) = (
b) =
(14)
Σbjc Ebb De
where Ebje represents the innovation variance for conditioned on the innovations in c.
The Directed Spectrum corresponds to the portion of the power spectrum for that is explained by signal that originated in b (see Supplemental Section for more details). It a linear function the latent networks when data is appropriately modeled described above (for proof, see Supplemental Section D).
The Directed Spectrum can be estimated efficiently by fitting VAR model to the observed data or via factorization of the cross-spectral matrix associated with the observed data 224 3Q BT. Additional algorithmic details can be found in Supplemental Section F.1 These DS measures can be calculated according to (14) for data partitioned into any number of groups greater than one. In some situations, it is desirable to calculate the DS values separately for each pair of groups, by estimating (w) and Σ using reduced models that contains only two groups per model. We refer to this method as the Pairwise Directed Spectrum (PDS). One benefit of the PDS is that
5

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simplifies the handling of missing channels from the data, since the PDS for the non non missing groups of channels can still be calculated without any adjustments. Code for calculating the Directed Spectrum and Pairwise Directed Spectrum is provided in the supplemental material (MATLABB) and at hitttss:: http://gshh.....cc.eca.///aaaaar spectrum (Python).
4 Related measures of directed communication
The Directed Spectrum is a necessary development because similar measures typically used in neuroscience are not linear functions of latent brain networks. We first examine Granger causality, which measures the degree to which one signal or group of signals helps predict future values of another signal or group [16]. Spectral Granger causality is modification that separates Granger causality into components associated with individual frequencies [77] It a leading spectral measure of directed communication in neuroscience [24] 255, and we view it as conceptually the closest
measure to the Directed Spectrum. The (unconditional) spectral Granger causality from b to e can be defined in terms of the Directed Spectrum,
[Sca(w)] fo-e(w)- In [Se DDDS
(15)
It is also possible to calculate conditional spectral Granger causality that accounts for all other groups in the recording before assigning influence from b to e [32. Neither form of spectral Granger causality is linear function of the network models described in Section To see this, consider the case where only the jdh network is present without any other signals being added to it. Note that changing Z would then have no impact on fb->((w) because both the numerator and denominator in (15) would scale equally with z Standard (non-spectral) Granger causality is equivalent to a scaled integral over all frequencies of the spectral Granger causality [17. meaning that it also would remain constant as Z changes.
There are several other spectral measures of directed communication used in neuroscience, including phase slope index (33, partial directed coherence R4. and the directed transfer function [35.. Each of these can not be a linear function of latent brain networks under reasonable assumptions. We offer the following brief explanations here, with more details in Supplemental Section
E The phase slope
index cannot capture bidirectional transmission of signal (all signal is considered unidirectional) [36]. Partial directed coherence and the directed transfer function are both based in VAR models like the Directed Spectrum, but they rely solely on the transfer matrix properties and do not use the innovations. As such, they are inveeiivvrriant and are not linear functions of the brain networks.
There are number of other non-spectral measures of directed communication used in neuro- science 25 26]. The cross-correlatinn function between activity in different neural populations has been used to identify directed communication between neuronal populations 37 In these applications it has been used to identify the lag associated with unidirectional transmission between populations, and is considered unreliable in situations involving bidirectional communication [25]. Transfer entropy provides information theoretic measure of directed communication. In our application transfer entropy suffers from the same problem as Granger causality, where scaling the activation of a single network results in the same value 38 39). We discuss methods for measuring directed communication between neural populations further in Supplemental Section
One final method worth mentioning here is dynamic causal modeling (DCM) [0] DCM produces a generative model to explain observed neural activity based on latent connectivity. Directed connectiv- ity is represented by parameters of the model, which contrasts with our application where measures of directed connectivity are calculated first before being used as features in an LFM. DCM has even been extended in order to represent spectral content in neural signals 41 42
One substantial
difference between DCM and the approach we suggest here is that DCM not designed to segregate
the directed connectivity out into subnetworks in an unsupervised manner. Instead, DCM is typically used to identify differences connectivity between two known conditions
5 Directed Spectrum improves identification of latent networks
To test the effectiveness of the Directed Spectrum and LFMs, we generated dataset of simulated latent networks. The dataset contains 10,000 independent recordings, each containing 5 channels

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A
B
C
D
E
Network 1-5 Network 30
N 5
Figure 1: Graphical representation of the simulated latent networks. The five channels represented by the letters A, B, C, D, and E. A box around a channel indicates the corresponding network induces oscillations in that channel. Arrows indicate propagation of signal some delay. Networks and have a predominant frequency of 5 Hz (both in purple), and Network 2 has a predominant frequency of 30 Hz (in blue).
Simulation
Network 1
Network 2
Network 3
+
+
w-
E
0.0 02 0.4
06
0.8
1.0
0.0 0.2 0.4
0.6 0.8 1.0
0.0 0.2
0.4
0.8
1.0
0.0
0.2
0.4
0.8
1.0
Time (s)
Figure 2: Simulated recordings are sum latent network contributions. The "observed" recording (left) is sum of signals generated by each of the three networks. Each panel contains signals associated with the regions over the same one second period. Black scale bars indicate the period for oscillations associated with each network, and are placed to show that the distance between peaks in the signal is approximately one period.
sampled for five seconds at 500 Hz. The "observed" recordings were generated as a sum of contribu- tions from three independent networks. Each network was associated with vector autoregressive (VAR) model that generated network contributions as a different random draw for each recording with parameters that were fixed over all recordings, Each VAR model was designed to generate a particular pattern of oscillations and delayed directional transmission of signals. See Figure for the network properties and Figure 2 for visualization of the superposition. See Supplemental Section F.2 for additional details on the design of those VAR models.
We generated model of the latent networks in the simulated dataset by applying LFM to DS measures. We calculated the Directed Spectrum for frequencies between I Hz and 50 Hz at Hz intervals, between all directed pairs of channels. A non-negative matrix factorization model (i.e. LFM) was trained using the Itakura-Saito (IS) divergence loss, with an LI penalty on the factor loadings and scores, and multiplicative update steps for optimization 22 The IS divergence loss was chosen because it is related to the gamma distribution and is more appropriate loss for power than mean squared error loss. After training, the non-negative matrix factorization (NMF) factors were re-ordered to correspond to the matching network in the simulation, based on maximizing the average Spearman's correlation between the estimated and network activation scores. For comparison, we repeated this process with the Pairwise Directed Spectrum (PDS), unconditional spectral Granger causality (GC) [17), conditional spectral Granger causality (cGC) [32), phase-slope index (PSI) [33], directed transfer function (DTF) [35. and partial directed coherence (PDC) [4 features substituted for DS features. We also use the difference between (unconditional) Granger causality values in either direction as suggested by Roebroeck et al. 44 as an additional comparison method. Because both this difference and PSI represent the direction of shared information with sign (i.c. positive/negativve,, set negative values to zero and leave the opposite direction positive
7

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Table 1: Spearman's 's correlation between latent network activation estimates and true activation scores. GC: unconditional Granger causality; cGC: conditional Granger Causality; GCdiff: difference between Granger causality directions; PSI: phase slope index; DTF: directed transfer function; PDC: partial directed coherence; DS: Directed Spectrum: PDS: Pairwise Directed Spectrum. Values in
[brackets] represent the 95% confidence interval [45
Measure
Network
Network
Network 3
GC
0,485 [0.468, 0.502]
0.442 [0.424, 0.459]
0,281 [0.261, 0.300]
eGC
0.475 [0.458, 0.49/]
0.293 /0.274, 0.312]
0.127 [0. 108, 0.147]
GCdiff
0.554 [0.539, 0.569]
0.501 [0.484, 0.517]
0.444 [0.427, 0.462]
PSI
0.387 [0.368, 0.405]
0.135 [0.115, 0,155]
0.248 [0.229, 0.268]
DTF
0.426 [0.408, 0.443)
0.131 [0.111, 0.151]
0.542 [0.526, 0.557]
PDC
0.560 [0.545, 0.575]
0.154 /0. 134, 0.174]
0,445 [0.427, 0.462]
DS
0.920 10,916, 0,923]
0.905 [0.901, 0.909]
0.927 [0.924, 0.930]
PDS
0.908 [0.904, 0.912]
0.917 [0.913, 0.920]
0.916 [0.913, 0.920]
Network 1
Network
Network 3
Network
Network 2
Network
2
DS Estimate
GC Estimate
1.5
1.5
0.5
0.5
0.5
0.5
1
0.5
0.5
0.5
0.5
True Score
True Score
Figure 3: True vs. estimated network scores. The estimated scores for the model trained using DS features plotted in blue, the estimated scores for model trained unconditional Granger causality features are plotted in red. In all plots, scaling factor was applied to the estimated scores minimize the mean squared error. The line in black demonstrates the expected trend for a model that perfectly recovers the latent network scores.
for both of those comparison methods; this allowed to continue NMF identify latent networks.
We evaluated how well each model recovered the underlying network activation scores using Spear- man's correlation. The model DS features performed significantly better in this regard than the "non-linear" features (see Table [ The PDS model also performed significantly better than the "non-linear" models, and gave comparable results to the DS model. We have visualized the estimated scores along with the true scores for each window for the DS features and unconditional Granger causality features in Figure 3 We see that the score estimates of the DS model are much more tightly spread around the true score values, with the relationship between true scores and estimated scores being very weak in the model of GC features.
We also tested how robust these results were to violations of the model assumptions described in Section and to variations in recording window length. Violations of the model assumptions did reduce model performance somewhat, but the DS features still performed better than the comparison methods. Shorter window lengths reduced performance in all models, but again the DS models performed better at all window lengths. Full details and results for these claims are in Supplemental Sections G and H
Finally, we tested whether the models produced accurate and interpretable representations of the true latent networks from dataset. The model of DS features successfully recovers all directional influences of each network in the corresponding factors (see Figures 4 ST Spurious detection of network components was limited to connections between true nodes in the network at the frequency associated with the network, and these spurious influences were small relative to the true detected

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A
B
C
D
D
E
A
DS
GC
E
E
50
25
50
25 50
25
Freq. (Hz)
Freq. (Hz)
Figure 4: Estimated directional influences Network 2 for a model DS features (blue) and unconditional Granger causality features (red). Within each grid, plot corresponds to signal that is being transmitted from the channel listed on the left to the channel listed above.
influences. While the model of GC features did also recover all directional influences of the networks underlying the simulated data, incorrectly spread the directional influences associated with each network out over multiple estimated networks. This resulted in the GC model identifying more spurious influences with greater relative amplitudes.
Directed Spectrum improves brain state decoding of neural recordings
We next tested how effectively networks identified by the Directed Spectrum decode behaviorally relevant variables from a dataset of neural recordings, a proxy for determining whether they reflect "real" networks. The dataset consists of local field potentials simultaneously recorded from 11 brain regions 26 mice (originally published in (77) ) In each recording session local field potentials were recorded from mouse while it was exposed to three different behavioral contexts of successively increasing stress levels: resting the home cage, exploring an open field, and tail suspension test. The tail suspension test is commonly used to investigate learned helplessness [46]
In order train networks that could be used to decode behavioral context, we first divided the dataset into time windows with duration of second. For each time window, we calculated the Directed Spectrum for all directed pairs of brain regions, for frequencies from I 56 Hz at Hz intervals. A nonnegative matrix factorization model with LI regularization was trained using the IS divergence objective [47 in order identify latent brain networks from the DS measures. multinomial logistic regression classifier with L1 regularization was then applied to the latent factor scores for decoding the behavioral context. In order to obtain estimates of the spread of the decoding performance and choose optimal hyperparameter values, 5-fold nested cross-validatio procedure was used, where each mouse was associated with only one split. The average one-vs-all area under the reciever operating characteristic curve (AUC) across all three behavioral contexts was used as the evaluation metric. The hyperparameters tuned during cross- cross-validation were the number of NMF factors (20, 40, 80). NMF regularization strength (1000, 100, 10, O), and logistic regression regularization strength (10, 1). Similar to the experiment described in Section 5 the procedure was repeated using the Pairwise Directed Spectrum (PDS) and the comparison measures listed in that section. The one comparison method that was not used here is conditional Granger causality; in our tests, obtaining stable conditional spectral Granger causality estimates methods outlined by Barnett and Seth 24 requires too much computation time to be practical for this application.
rodent LFP data via the
Our training procedure resulted in a one-vs-all AUC for each of the behavioral contexts, for each of the 5 splits, for each of the measure types (see Table 2 We performed 2-factor repeated measures

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ANOVA and determined that there was significant difference associated with the measure types (F 92.9, p .001). A Tukey's HSD post-hoe test revealed that there was not a significant difference between models of the pairwise and non-pairwise versions of DS (p -90), but that both performed significantly better than models of the other measures (p S .001 for all).
Table 2:
Behavioral Context Decoding Performance. The columns 'HC AUC', 'OF AUC", and 'TS AUC' report the mean and standard error of the one-vs-all AUC across splits for the homecage, open field, and tail suspension behavioral contexts, respectively. The 'Mean AUC" column reports the average across the mean AUCs reported for each behavioral context.
Measure
Mean AUC
HC AUC
OF AUC
TS AUC
GC 0.828
0.825 + 0.019
0.824 0.022
0.835 + 0.010
GCdiff
0.795
0.795 0.013
0.798 0.023
0.791 + 0.007
PSI
0.674
0.676 : 0.014
0.684 = 0.015
0.661 = 0.016
DTF
0.755
0.774 0.017
0.774 0.015
0.717 = 0.022
PDC
0.717
0.733 + 0.021
0.778 = 0.014
0.639 = 0.008
DS
0.908
0.894
0.916 0.012
0.915 = 0.007
PDS
0.919
0.909 + 0.014
0.915 0.014
0.932 + 0.005
7 Discussion
We have shown that our novel measure of directed communication, the Directed Spectrum (DS), is a linear function of latent brain networks under reasonable assumptions, and so is compatible with L.FMs for characterizing latent brain networks. We saw drastically improved recovery of networks in simulated data with known ground truth. In real neural recordings, the Directed Spectrum improved decoding of neurally relevant environmental variables. These results demonstrate that the Directed Spectrum allows for more accurate and interpretable models of latent networks defined by directed communication.
A limitation that could be explored in future work is that the Directed Spectrum assumes that the network states are stationary over a single time window (see Supplemental Section B This limitation is common to almost all methods that assess spectral content of neural activity. Because of this, the Directed Spectrum is only theoretically grounded when applied time windows that are shorter than the time expected for substantial change to be occur in the latent brain state. In practice, time window lengths of one to five seconds have been considered a relatively stable period studies of emotional processing 48 49
We have only explored LFMs here, but variety of nonlinear latent factor models exist for modeling latent brain networks 5 50). Nonlinear models are especially useful in applications where predictive performance is the only priority, such as brain-computer interfaces. When the primary goal is to drive scientific understanding of the brain, nonlinear models are less desirable because it is challenging to relate the parameters of such models relevant conclusions about brain function [8]. We believe that LFMs and the Directed Spectrum are an appropriate, efficient, and reproducible baseline approach in both scientific and prediction based applications, while noting that other more expressive models may lead to even better predictive performance at the cost of interpretability.
The development of the Directed Spectrum provides straightforward way to generate linear models of latent brain networks defined by directed communication. We view this as an important advancement because directed communication between neural populations is critical component of the way neuroscientists understand brain networks. Furthermore, the Directed Spectrum is likely useful for studying latent networks many other fields as well (e.g. latent networks of directed internet traffic). We view the major strength of the Directed Spectrum is that it enhances the expressiveness of LFMs by allowing them to accurately model directed communication within networks. Thus, we have expanded the capabilities of LFMs while retaining the level of interpretability that make them attractive models practice.
10

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Acknowledgments and Disclosure of Funding
We thank the anonymous reviewers for their suggestions to evaluate how robust the methods are to assumption violations, which led to the additional experiments of Supplemental Sections Gand H
Research reported in this manuscript was supported by the National Institute of Biomedical Imaging and Bioengineering and the National Institute of Mental Health through the National Institutes of Health BRAIN Initiative under Award Number R01 EB026633, the National Institute Mental Health of the National Institutes of Health under Award Numbers R01 MH125430 and RO1 MH120158 58, and the Hope for Depression Research Foundation.
The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
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15


Reviews:
[{"review_id": "pX6rNl7lCdh", "reviewer": "Reviewer_3jsZ", "summary": "The authors define the Directed Spectrum (DS) of multivariate timeseries data. It is shown that under certain conditions the DS is a linear function of latent brain networks. In synthetic data, the DS (and NMF) was better able to recover ground truth networks compared to a set of other directed measures. In neural data from mice, the network activations derived from DS are better able to predict behavioral state.", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The motivation for the manuscript, careful consideration of the interplay between LLFMs and measures of neural data, and results were all clear. The DS is well motivated and is put into the context of other directed measures. I have one broad suggestion in terms of the presentation of Section 3 and some minor questions and suggestions.\n\nIn my opinion, the main weakness  is that Section 3 is very dense and relies on a number of definitions and results that are only in the appendix, which made it difficult to read through as a self-contained section. I think it would benefit the clarity of Section 3 to walk the reader through some definitions (e.g., how to interpret $H^{(j)}(\\omega)$) in Eq 5 and the interpretations of the terms in certain definitions (e.g., Eq 9). Some of these are in the appendix, but I would suggest using some of the additional page to make this section more self-contained.\n\n- I found that using $x$ for both the data vector and latent factors in Eq 1 to be confusing. I would suggest changing the latent factor variable.\n- In lines 131-134, I think it would be helpful to have more exposition about when it could be useful to define groups versus single channels.\n- For Section 6, is it possible to better differentiate whether the DS is better for decoding as an engineered features versus whether some small subset of the inferred networks are better for decoding. Since there are only 3 behavioral contexts, it would be interesting if a small set of network activations were responsible for decoding versus if many networks are being used.\n- Should equations 12 and 13 be part of one multiline equation?\n- Line 278: L1 factorization -> L1 regularization", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors address limitations of the DS based on its assumptions of stationarity and its use with LLFMs. It would be good to know how sensitive the method is to violations of Assumptions 1 or 2 as they likely don't hold exactly in neural data. If there are weak correlations in the innovations or if the transmitted signals combine sub-additively, how quickly do the estimates break down?", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}, {"review_id": "oUxgbU3YZn3", "reviewer": "Reviewer_mATF", "summary": "The authors present a method for estimating the directional influences between channels in order estimate the existence and structure of distinct, latent networks. They call their metric the \"directed spectrum\" and demonstrate its superiority over Granger causality to estimate directed influences between recording channels. The motivation for their approach is the paucity of directed influence metrics that operate in conjunction with latent variable models of neural population activity.", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The paper is well written and flows nicely. The metric itself is fairly simple and it would be surprising if it didn't already exist under some other name. Although the authors appear to know and cite the relevant literature for this topic. One point of concern may be that the metric could have unfavorable sampling properties. For example, it is well known that common estimators for the cross-spectral matrix may be either inconsistent and/or high-variance. Does their estimator inherit these properties? Also, could the authors comment on how the metric might be influenced by model mismatch (eg.  generative models with non-linear dynamics, Poisson measurements).\n\nWhile I don't mind some of their basic assumptions I find the arguments about innovations and independence (in particular in lines 148-153) somewhat weak. It is not at all obvious that brain networks wouldn't want to bounce existing information back-and-forth. The authors would have to substantiate this claim more throughly. That said, I would prefer to accept that this is a basic assumption of the model and being clear-eyed about how that may impact results rather than swallow the claim that somehow the brain prefers to operate this way.\n\nI must admit that I am dubious of directed measures of neural population activity in general as I find it difficult to imagine a brain with recurrent connections showing anything other than directed activity in both directions. For example, the LFP data the authors present in the Supplement seems to indicate that none of the other brain areas direct their activity to the nucleus accumbens. Would we not expect VTA to act upon NAc? That does not appear to be the case based on the DS measure. How would the authors have us understand this inconsistency?\n\nLastly, I'm glad that the authors have explored this method and it's great that it performs so well on the classification task but I'm curious about the authors' thoughts on what new science can be done with this approach. Is this merely a tool for functional network discovery or did they have something else in mind?\n\nAs an aside, the authors may be interested in some recent work by Yousefi et al. (2019) describing the \"global coherence\" in which the authors develop an explicit latent variable model for network activity that is also designed to learn the identity of latent networks.\n\nA. Yousefi, R. S. Fard, U. T. Eden and E. N. Brown, \"State-Space Global Coherence to Estimate the Spatio-Temporal Dynamics of the Coordinated Brain Activity,\" 2019 41st Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), 2019, pp. 5794-5798, doi: 10.1109/EMBC.2019.8856634.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "yes.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3"}, "extra_text": {}}, {"review_id": "35xq0rD3xT", "reviewer": "Reviewer_pzTq", "summary": "This study introduces a new metric (Directed Spectrum or DS) that is compatible with linear latent factor methods (LLF) and, therefore, can be used for latent network discovery with such methods. First, the authors show that there exists a linear relationship between the DS between channels and the DS in latent networks. Using simulated data, the authors estimate latent networks by applying NMF directly to the DS estimates. They show stronger correlations with latent activations recovered by their method, as compared to other traditional metrics like spectral GC, PDC, DTF, and the like. A specific comparison reveals that GC estimates far more spurious connections as compared to the DS metric. Finally, the authors also show that, in a neuroscience dataset, the latents estimated by DS features better predict the behavioral states as compared to latents estimated by traditional metrics. ", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Preamble: I am quite familiar with measures of directed communication applied in neuroscience, and routinely employ these with our research data. But I am not an expert on the theoretical underpinnings of these measures. I will, therefore, append each critique with a confidence score (10/highest to 1/lowest), to enable the authors to optimally address my criticisms in the rebuttal, considering space constraints.\n\nOriginality: 7/10 \n\nQuality: 7/10\n\nClarity: 6/10\n\nSignificance: 6/10\n\na) Regarding the comparison with spectral GC: \n\ni) There is a particular challenge with GC when inferring causality because the GC often estimates weights for both forward and (spurious) backward connections. In this case, differencing the estimates has been found to provide some control over the spurious backward connections (Roebroeck, Formisano, Goebel, 2005). I would suggest the authors use this metric (Fy->x(w) - Fx->y(w)) for their comparison rather than GC per se. \n\nii) Another aspect that was unclear from the main paper was whether the authors used pairwise GC or conditional GC, although both have been defined in Supplemental section C.3. If they had used pairwise GC, could they also provide results with conditional GC (or both)?\n\niii) Is it possible that GC performs better with data from longer time windows? Fig 3 could be plotted for different lengths of windows (e.g. with mean and errorbors, rather than showing the estimates for every window/sample).\n\n*Confidence: 10 \n\n\nb) On a related note. Please show the directed influence spectra (analog of Fig.4) with the other approaches also, at least in summary form. Did these other approaches perform comparably with DS? The proportion of spurious connections estimated with all approaches should be quantified for comparison.\n\n*Confidence: 10\n\n\nc) (line 160) The authors indicate that \"Assumption 1 can be thought of as enforcing the independence of inputs to each network\". How valid is this assumption in a neurobiological setting? Of course, some components of the innovations may reflect \"independent noise\", but visual or other sensory inputs are likely to be highly correlated among networks (e.g. the LGN projects to multiple high level visual areas, in parallel). Can the authors clarify, and perhaps explore to what level the model is tolerant to a failure of this assumption?\n\n*Confidence: 10 \n\n\nd) The behavioral context decoding section needs to be updated significantly. One would need to see what accuracies can be obtained simply from the \"raw\" timeseries data or from simple spectral features extracted from these data, for each context. Having established this baseline, it would then make sense to show that DS features outperform these simpler features in terms of decoding behavioral state. Otherwise, the results could be simply explained by DS+NMF producing the least \"distortion\" in the raw data (given that the other methods are non-linear).\n\n*Confidence: 10 \n\n\ne) The authors seek to provide a thorough and detailed description, but some equations and derivations could be made clearer, at least in the Supporting Information. For example, the \"transmitted signal\" (based on the conditional density) introduced in equation 9 -- which is at the heart of the model -- could be derived in the supporting information with a clear explanation of how each term maps on to the stated assumptions in the immediately preceding paragraph.\n\n*Confidence: 8\n\n\nf) Given that the main contribution of the paper seems to be theoretical, the precise relationship between DS and traditional metrics should be clearly described, at least in the Supplemental information. Such a derivation would ideally follow the Supplemental Section E on the relationship with the cross-spectral matrix and its factorization. While Supplemental section C presents the equations associated with other metrics, such a description would allow the reader to also understand the relative advantages and limitations of the DS metrics vis-a-vis traditional metrics. \n\n*Confidence: 7\n\n\ng) On a related note, a few more empirical simulations could be presented that explore the limits of the proposed metric, and its tolerance to violations of key assumptions. For example, what happens when latents mix non-linearly. Or network interactions characterized by operations other than scaling and delay? In the brain, it is known that the phase of slow oscillations modulates the amplitude of faster oscillations. Could these types of interactions also be captured by DS, and better than DTF or GC? \n\n*Confidence: 7\n\n\nh) As an empirical control it would be nice to see a baseline comparison with latent linear factor models directly applied to the time series data, followed by directed influence estimation with DS or other traditional methods. In the case of DS I guess the order will not matter significantly, given linearity assumptions. But how do the other methods fare when NMF is applied to the data followed by, say, GC estimation on the latents compared to NMF on the channel-wise GC estimates? Are there certain interactions that can be recovered only by the latter method vs the former?\n\n*Confidence: 6\n\n\ni) I am not sure the sample subscript 'n' adds anything that cannot be understood without it. It shows up in every term and clutters up the equations, and gets dropped by mistake occasionally (Supplemental Section S1). I'd suggest removing it.\n\n*Confidence: 6\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes.", "review_text": "", "extra_scores": {"ethical_concerns": "No.", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}]

Author rebuttal:

Meta-review: Through a productive discussion with the authors, the reviewers came to a borderline/accept consensus. The proposed "directed spectrum" measure is simple (in fact, the authors say it was an introduced as an intermediate quantity in a 1982 paper by Geweke), but appears to have some nice properties. While some reviewers praised the clarity of this paper, Reviewer 3jsZ and I found the paper rather dense (and the unnecessary subscripts didn't help).  I hope the authors will work to improve this aspect of the paper before publication. 
Decision: Accept (Poster)