Directed Spectrum Measures Improve Latent Network Models Of Neural Populations

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 . 1 Introduction . A major goal in neuroscience is to characterize how populations of neurons work together to carry out computational tasks [ 1 ] . 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 , 3 ] . 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 . brain networks ) [ 4–6 ] . 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 [ 7 ] . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . Unfortunately , many types of common models in the machine learning literature do not offer this type of interpretability [ 8 ] . One class of widely-used models that is considered interpretable for modeling brain networks is Linear Factor Models ( LFMs ) [ 9 , 10 ] . This family includes popular methods like principal component analysis ( PCA ) and independent component analysis ( ICA ) , and are used across almost all modalities in neuroscience to model brain networks . For example , ICA is regularly used in functional magnetic resonance imaging ( fMRI ) applications to identify latent brain networks such as the default mode network [ 11 , 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 [ 13 , 14 ] . In multi-site local field potential ( LFP ) data , LFMs have been applied to identify latent brain networks that are defined by a 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 is discussed further in Sections 2 and 4 . In a previous attempt to capture directed communication within an interpretable brain network model , Gallagher et al . [ 15 ] modeled networks in terms of phase shifts in spectral content between neural populations , but it is unclear if 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 to as the Directed Spectrum ( DS ) . The Directed Spectrum estimates directed communication between time series in the frequency domain . We prove that these measures are a 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 a 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 xn as , xn = ∑J j=1 Z ( j ) n x ( j ) + n. ( 1 ) The vectors x ( 1 ) , . . . , x ( J ) are the J different latent factor loadings , n is the additive noise term , and Z ( j ) n is the activation score of the jth factor in the nth sample . xn represents a single sample from a larger dataset , [ x1 , . . . , xN ] . Many well-known models , such as PCA [ 19 ] , ICA [ 20 ] , and nonnegative matrix factorizaton [ 21 ] , are LFMs . LFMs are frequently used for modeling and discovering brain networks [ 10 ] . In that context , it is straightforward to interpret the scores , Z ( j ) n , as the activation levels for a set of brain networks . Likewise , we can interpret the factor loading x ( j ) 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 , 22 ] . 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 x ( j ) , it will be explicitly stated . 2.1 Compatibility between linear factor models and measures of neural activity . It 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 x contains measures calculated from multi-channel time-series , V = [ v1 , . . . , vK ] ᵀ , where K is the number of channels and vc ∈ RT is the row of V corresponding to a recording from a single channel c. The frequency domain representations of these quantities will be marked by a tilde ( e.g . ṽc ( ω ) is the frequency domain representation of vc ) . A typical choice for x is to convert the observed data into a set of power spectra ( Scc ) for each channel [ 13 , 23 ] , Scc ( ω ) ≡ E [ |ṽc ( ω ) |2 ] . ( 2 ) Modeling these measures in an LFM leads to the latent factor loadings x ( j ) being described in terms of power spectra associated with each channel . The chosen measures dictate the representation of the discovered networks , so we want to use relevant and easy-to-interpret measures . When an LFM is used to model latent brain networks , there is an implicit assumption that the measures contained in x are linear functions of the latent brain networks . This is seen in ( 1 ) where the observed measure is a linear function of the latent factor score Z ( j ) within some noise tolerance . We let X ( · ) represent a function , such as ( 2 ) , that produces the set of measures from our observed neural data V . Definition 1 . We call X ( · ) a linear function of latent brain networks if X ( V ) = X ( ∑J j=1 Z ( j ) Ω ( j ) ) = ∑J j=1 Z ( j ) X ( Ω ( j ) ) , ( 3 ) where Ω ( j ) represents the component of the neural data ( V ) that is due to the jth network , normalized to represent a score of 1 . In this way , X ( Ω ( j ) ) represents the expected value of the measure if the jth network was active with a score of 1 in isolation . It can be shown that power spectra are theoretically consistent with this requirement ( see Supplemental Section A ) , but there are many measures that can not obey this assumption . For example , the Pearson correlation coefficient and Granger causality are two measures that can not 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 [ 24–26 ] . 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 as Granger causality are vital for the study of brain networks [ 27 , 28 ] . Below , we introduce a novel measure of directed communication , which we refer to as the Directed Spectrum ( DS ) . In order to derive the Directed Spectrum , we first build a 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 . It captures communication in a similar manner to Granger causality while also being compatible with a 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 , even 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 V ( j ) = [ v ( j ) 1 , . . . , v ( j ) T ] , where v ( j ) t ∈ RK is the column of V ( j ) representing signal for all channels at time t. The output series is associated with a VAR process , v ( j ) t = ∑pj τ=1A ( j ) τ v ( j ) ( t−τ ) + σ ( j ) t , σ ( j ) t ∼ N ( 0 , Z ( j ) Σ ( j ) ) . ( 4 ) We refer to σ ( j ) t as the innovations in the j th network at time t , and assume that they are drawn iid from a zero mean Gaussian distribution . Each network has a single covariance structure over all samples ( Σ ( j ) ) that is scaled by an activation score ( Z ( j ) ) to define the innovation distribution of the current sample . The set of autocovariance matrices for the jth network { A ( j ) 1 , . . . , A ( j ) pj } are also assumed to be constant over all samples . The innovation terms represent new signal introduced into the network that can not be explained by the past network outputs { v ( j ) ( t−1 ) , . . . , v ( j ) ( t−pj ) } . We can represent this network in the frequency domain given standard assumptions regarding the stability of the VAR model [ 29 ] , ṽ ( j ) ( ω ) = H ( j ) ( ω ) σ̃ ( j ) ( ω ) , H ( j ) ( ω ) ≡ ( I − ∑pj τ=1A ( j ) τ e−iτω ) −1 , 0 ≤ ω ≤ 2π , ( 5 ) where ṽ ( j ) ( ω ) ∈ RK and σ̃ ( j ) ( ω ) ∈ RK are the frequency domain representations of the series v ( j ) 1 , . . . , v ( j ) T and σ ( j ) 1 , . . . , σ ( j ) T , respectively . The network transfer matrix ( H ( j ) ( ω ) ) is assumed to be constant over all samples .

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.