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 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.