1 Multitaper Analysis of Evolutionary Spectra from Multivariate Spiking Observations Anuththara Rupasinghe and Behtash Babadi Abstract—Extracting the spectral representations of the neural proposed to quantify the energy-frequency-time distributions processes that underlie spiking activity is key to understanding [17]–[26]. One notable example is the evolutionary power how the brain rhythms mediate cognitive functions. While spec- spectral characterization [22], [23], which defines a non- tral estimation of continuous time-series is well studied, inferring the spectral representation of latent non-stationary processes stationary spectral density matrix in order to quantify the based on spiking observations is a challenging problem. In this local spectral energy distributions at each instant of time for paper, we address this issue by developing a multitaper spectral a multivariate process. This characterization has a physical estimation methodology that can be directly applied to multi- interpretation similar to that of stationary Fourier spectral variate spiking observations in order to extract the evolutionary analysis, and reduces to the classical power spectra when spectral density of the latent non-stationary processes that drive spiking activity, based on point process theory. We establish the processes are stationary [22]. A unified approach that theoretical bounds on the bias-variance trade-off of the proposed considers multivariate spiking observations driven by non- estimator. Finally, we compare the performance of our proposed stationary latent processes is lacking, but highly desired due technique with existing methods using simulation studies and to the emerging demands of modern neuronal data analysis. application to real data, which reveal significant gains in terms In this work, we close this gap by developing a framework of the bias-variance trade-off. to estimate the evolutionary spectral density (ESD) matrix Index Terms—Point process model, multivariate non stationary of a multivariate non-stationary latent process, given spiking latent process, evolutionary spectral density matrix, multitaper observations. We model the spiking observations as multiple analysis, binary spiking observations realizations of point processes with logistic links to the latent continuous processes. We then pose the problem of spectral I. INTRODUCTION estimation within a multitapering framework. Multitapering is Eural oscillations are known to play a significant role in a widely-used PSD estimation technique with desirable bias- N mediating the cognitive and motor functions of the brain variance trade-off performance [27], due to the usage of the [2]–[4]. The advent of high-density electrophysiology record- discrete prolate spheroidal sequences (dpss) [28] as data tapers, ings [5]–[7] from multiple locations in the brain has opened which are known for their minimal spectral leakage [29]–[31]. a unique window of opportunity to probe these oscillations We employ a state-space model to characterize the dynamics at the neuronal scale. In order to exploit such experimental of the evolutionary spectra, with the underlying states pertain- data for inferring the mechanisms of brain function, spectral ing to the eigen-spectra of the multivariate latent processes analysis techniques tailored for such neuronal spiking data are [25]. We derive an Expectation-Maximization (EM) algorithm required [8]. for efficiently computing the maximum a posteriori (MAP) Existing techniques for spectral analysis of neuronal data estimate of the latent variables and smoothed states given use point process theory [8]–[10] to capture the spiking the spiking observations, which we then use to construct the statistics. Due to the time-domain smoothing procedures used ESD matrix. We establish theoretical bounds on the bias- arXiv:1906.09359v1 [cs.IT] 22 Jun 2019 by existing techniques [11]–[13] for recovering the latent pro- variance performance of our proposed estimator, and recover cesses that drive spiking activity, the resulting power spectral the favorable asymptotic properties of the classical multitaper density (PSD) estimates undergo distortion in the spectral framework. domain. Alternative methods to directly estimate the PSD from We compare the performance of our proposed method to spiking data have recently been proposed in [14], [15]. existing techniques through simulation studies and application These existing methods consider univariate spiking obser- to experimentally-recorded neuronal data. We present two vations and assume the latent process to be second-order simulated case studies using non-stationary multivariate au- stationary during the observation period. However, it is known toregressive processes, whose dynamics are inspired by neural that the brain oscillations that drive neuronal spiking are non- oscillations. These studies reveal that the proposed method stationary and may exhibit rapid changes corresponding to the outperforms two of the widely used methods for deriving brain state or behavioral dynamics [12], [16]. spectral representations from spiking data. Finally, we apply Non-stationary time series analysis has been well studied for our proposed estimator to multi-unit spike and local field multivariate continuous signals and various methods have been potential (LFP) recordings from a human subject undergoing The authors are with the Department of Electrical & Computer Engi- general anesthesia [12], [14]. Our proposed method corrob- neering, University of Maryland, College Park, MD 20742. E-mails: {raar, orates existing hypotheses on the relation between the LFP behtash}@umd.edu. This work has been supported in part by the National signals and spiking dynamics, by providing a high resolution Science Foundation Award No. 1807216 and the National Institutes of Health award No. 1U19NS107464-01. This work was presented in part at the IEEE characterization of the underlying spectrotemporal couplings. Data Science Workshop, Minneapolis, MN, USA, June 2–5, 2019 [1]. The rest of the paper is organized as follows: We present our 2 problem formulation in Section II, followed by the proposed where ck,j,n = Ak,j (ωn) aj,n and dk,j,n = Ak,j (ωn) bj,n estimation framework in Section III. We provide our theoret- are real-valued random variables. Further, the evolutionary ical results on the bias-variance performance of the proposed spectrum [22] at time k is defined as estimator in Section IV. Our simulation studies are presented ψ (ω ) dω = A (ω ) 2 E dZ (ω ) 2. in Section V, followed by application to experimentally- k,j n n | k,j n | | j n | recorded data in Section VI. Finally, we close the paper by Hence, according to our model, the ESD function can be our concluding remarks in Section VII. approximated by, π II. PROBLEM FORMULATION ψ (ω )= E[(c + id )(c id )]. k,j n N k,j,n k,j,n k,j,n − k,j,n Let N(t) and H(t) denote the point process representing the number of spikes and spiking history of a neuron in [0,t), The Spectral Density Matrix of a multivariate stationary 1 ∞ −iℓω respectively, where t [0,T ] and T denotes the observation random process is defined as, Ψ(ω)= 2π ℓ=−∞ e Γ(ℓ), duration. The Conditional∈ Intensity Function (CIF) [32] of a where Γ( ) is the covariance matrix of the process [33]. Fur- · P point process N(t) is defined as: ther, considering a vector valued orthogonal increment process ⊤ Z(ω) = [Z1(ω),Z2(ω), ,ZJ (ω)] , the spectral density P [N(t + ∆) N(t)=1 Ht] ··· E H λ(t Ht) := lim − | . (1) matrix is characterized by Ψ(ω)dω = [dZ(ω)dZ(ω) ]. We | ∆→0 ∆ extend this to the evolutionary spectra, and formulate the ESD To discretize the continuous process, we consider time bins matrix at time k and frequency ωn for our model as, of length ∆, small enough that the probability of having two π Ψ (ω )= E[(c + id )(c id )⊤], or more spikes in an interval of length ∆ is negligible. Thus, k n N k,n k,n k,n − k,n the discretized point process can be modeled by a Bernoulli ⊤ where ck,n = [ck,1,n, ck,2,n, ,ck,J,n] and dk,n = process with success probability λk := λ(k∆ Hk)∆, for 1 | ≤ [d , d , , d ]⊤. ··· k K, where K := T/∆ is an integer (with no loss of k,1,n k,2,n ··· k,J,n ≤ Further, we assume the processes to be quasi-stationary generality). We refer to λk as CIF hereafter for brevity. In a similar fashion, we consider spiking observations [25], the J-variate random process to be jointly stationary K in windows of length W . The total data duration, K is from an ensemble of J neurons, with CIFs λk,j k=1, for j = 1, 2, ,J. Suppose that for each neuron,{ L}indepen- divided into M non overlapping segments of length W , with ··· dent realizations of the spiking activity are observed. The K = MW . Thus, the vector process [xk,1, xk,2, , xk,J ] is ··· collection of the binary spiking observations are represented assumed to be jointly stationary for (m 1)W +1 k mW , − ≤ ≤ as (l) L,K,J . We model the th CIF by a logistic link 1 m M. Simplifying the previous model under this nk,j l,k,j=1 j ≤ ≤ { } ⊤ quasi-stationarity assumption, for (m 1)W +1 k mW , to a latent random process, xj = [x1,j , x2,j , , xK,j ] , ··· 1 m M we get, − ≤ ≤ which needs not be stationary in general. Thus, the matrix ≤ ≤ X x x x represents a -variate random N−1 = [ 1, 2, , J ]K×J J 2π process. ··· x = µ + (p cos(ω k) q sin(ω k)), k,j m,j N m,j,n n − m,j,n n Accordingly, for 1 j J, 1 k K and 1 l L, n=1 ≤ ≤ ≤ ≤ ≤ ≤ X we have where p and q are real-valued random variables. (l) m,j,n m,j,n n Bernoulli(λk,j ), (2) k,j ∼ Defining Xm,j = [x(m−1)W +1,j , x(m−1)W +2,j , , ⊤ N ··· where λk,j = logistic(xk,j )=1/(1+exp( xk,j )).