Robust Estimation of Sparse Narrowband Spectra from Binary Neuronal Spiking Data

Abstract

Objective

Characterizing the spectral properties of neuronal responses is an important problem in computational neuroscience, as it provides insight into the spectral organization of the underlying functional neural processes. Although spectral analysis techniques are widely used in the analysis of noninvasive neural recordings such as EEG, their application to spiking data is limited due to the binary and non-linear nature of neuronal spiking. In this paper, we address the problem of estimating the power spectral density of the neural covariate driving the spiking statistics of a neuronal population from binary observations.

Methods

We consider a neuronal ensemble spiking according to Bernoulli statistics, for which the conditional intensity function is given by the logistic map of a harmonic second-order stationary process with sparse narrowband spectra. By employing sparsity-promoting priors, we compute the maximum a posteriori estimate of the power spectral density of the process from the binary spiking observations. Furthermore, we construct confidence intervals for these estimates by an efficient posterior sampling procedure.

Results

We provide simulation studies which reveal that our method outperforms the existing methods for extracting the frequency content of spiking data. Application of our method to clinically recorded spiking data from a patient under general anesthesia reveals a striking resemblance between our estimated power spectral density and that of the local field potential signal. This result corroborates existing findings regarding the salient role of the local field potential as a major neural covariate of rhythmic cortical spiking activity under anesthesia.

Conclusion

Our technique allows to analyze the harmonic structure of spiking activity in a robust fashion, independently of the local field potentials, and without any prior assumption of the spectral spread and content of the underlying neural processes.

Significance

Other than its usage in the spectral analysis of neuronal spiking data, our technique can be applied to a wide variety of binary data, such as heart beat data, in order to obtain a robust spectral representation.

I. Introduction

Spectral analysis of time-series recorded from the brain, such as electroencephalography (EEG), has long been used for monitoring and characterizing brain activity in both clinical and research settings. Presence of specific oscillations in the EEG has been identified as the neural correlate of a variety of cognitive functions. Examples include the occipital alpha rhythms [1] and the somatomotor mu-rhythms [2]. Benefiting from the well-developed theory of spectral analysis of time-series, the spectral EEG signal processing techniques have been proven successful for diagnosis purposes such as the identification of epilepsy seizures [3], [4] and sleep disorders [5], [6].

Analysis of data from noninvasive recordings is limited by the low spatial resolution of the measurement mechanism, as the sensors record the integrated electrical activity of a large population of neurons in the brain. With the development of invasive recording procedures, acquisition of Local Field Potential (LFP) and single- and multi-unit recordings have also been made possible [7], [8]. The LFP captures the electrical activity of a more localized population of neurons compared to EEG, and single- and multi-unit recordings capture the neural activity at the neuronal level. Although EEG signal processing techniques can be readily applied to LFP recordings, analysis of spike recordings has set forth various signal processing challenges due to their binary nature [9].

In recent years, the theory of point processes has been successfully employed to model and analyze binary spiking data [10], [11], [12]. These models provide a mathematically principled framework to relate the observed neuronal responses to the underlying covariates such as the sensory stimuli. In most of these applications, the point processes are used to model the neuronal responses in the time domain by enforcing temporal smoothness. The few exceptions which aim at calculating a frequency domain representation of the spiking data often proceed by computing an estimate of the spiking rate (as a continuous function) and then analyzing the spectral properties of the estimated rate. The spiking rate estimation techniques range from simple smoothing of the spiking histogram [13], [14], [15] to more sophisticated models which use generalized linear Gaussian state-space models to estimate the conditional intensity function (CIF) of the point process using Kalman filtering and smoothing techniques [16], [17]. The objective of these techniques is to provide a smoothed estimate of the spiking rate as a surrogate function whose power spectral density (PSD) is interpreted as the spectral representation of the spiking data. However, this interpretation has three immediate shortcomings. First, it is known that smoothing in the time domain results in blurring in the frequency domain [18], and hence these techniques are limited in terms of their spectral resolution. Second, spectral estimation requires estimating the second-order statistics of the underlying time-series, and even if the spiking rate is estimated accurately, the second-order statistics may not be. Third, these techniques are blind to the low-dimensional structure of neural data in conditions such as sleep [19], anesthesia [20], and epileptic seizures [21]. This low-dimensional structure is often manifested as sparsity in the spectral domain.

In this paper, we address these shortcomings by casting the problem of spectral estimation from binary spiking data in the traditional discrete-parameter harmonic spectral estimation framework, where the objective is to estimate the second moments of a harmonic process driving the spiking activity. To this end, we model the spiking statistics of the underlying neurons by a conditional Bernoulli point process model, where the CIF is formed by mapping a stationary harmonic process through a logistic link function. Given the spiking data and considering sparsity-promoting priors, we compute the maximum a posteriori (MAP) estimate of the PSD of the harmonic process using the Expectation-Maximization (EM) algorithm. In addition, we construct confidence intervals for these estimates via sampling from the posterior distribution. Simulation studies concerning spiking data driven by sparse harmonic and autoregressive (AR) processes as well as application to real spiking data from anesthesia illustrate the superior performance of our proposed technique as compared to several existing techniques. Although our motivation stemmed from neuronal spiking data, it is worth noting that our modeling and estimation framework can be applied to any binary data modeled by point processes, such as the heart beat [22], [23], in order to extract a sparse spectral representation of the data.

The rest of the paper is organized as follows: In Section II, we introduce our model for the spiking activity of a population of neurons driven by a harmonic process. In Section III, we derive the sparse MAP estimator of the PSD associated with the harmonic process. Section III-C discusses the construction of confidence intervals for the PSD estimate based on the Metropolis-Hastings sampling. Section IV provides simulation results comparing our sparse PSD estimates with those obtained by existing methods for extracting the PSD of spiking data. Furthermore, we apply our estimator to real multi-unit recordings of spiking activity under general anesthesia. This is followed by our discussion and concluding remarks in Sections V and VI, respectively.

II. Preliminaries and Problem Formulation

Let (0, T] be an observation interval during which the spiking activity of a neuron is recorded. For t ∈ (0, T] let N(t) be a point process representing the number of spikes in (0, t] and H_t denote the spiking history in the interval (0, t). We define the Conditional Intensity Function (CIF) of a point process N(t) as [24]:

$$\lambda (t\mid H_t):=\underset{\mathrm{\Delta }\to \infty }{lim}\frac{P(N(t+\mathrm{\Delta })-N(t)=1\mid H_t)}{\mathrm{\Delta }}$$ (1)

In order to discretize the continuous-time point process, we consider bins of length Δ such that T = KΔ, for some integer K. Assuming that Δ is small enough, the probability of having two or more spikes in an interval of Δ becomes negligible and the point process can be approximated in the k^th bin by a Bernoulli random variable n_k with success probability of λ_k:= λ(kΔ|H_kΔ)Δ, for 0 ≤ k ≤ K. This assumption is biophysically plausible due to the absolute refractory period of neurons, and a choice of Δ ~ 1 ms is typically sufficient to ensure that at most one spike occurs in any bin [10].

In general, oscillatory behavior of the neuronal spiking can be directly attributed to the oscillatory nature of the CIF. Our objective is to develop a method to estimate the PSD of the CIF from the observed binary spiking data. We consider an ensemble of L neurons driven by the same CIF, and denote the observed spike trains by ${\{n_k^{(\ell )}\}}_{\ell =1,k=1}^{L,K}$. The CIF, in turn, is modeled by a second-order stationary random process. We consider a simplified model where ${\{x_k\}}_{k=1}^K$ be a realization of the second-order stationary process with mean μ. In our model, we consider a logistic link for the CIF, such that ${\lambda }_k=\frac{1}{1+exp(-x_k)}$. In summary, the model can be expressed as:

$$\{\begin{array}{l}{\lambda }_k=\frac{1}{1+e^{-x_k}},1\le k\le K\hfill \\ n_k^{(\ell )}~\text{Bernoulli}({\lambda }_k),1\le k\le K,1\le \ell \le L\hfill \end{array}$$ (2)

and the objective is to estimate the PSD of x_k given the observations ${\{n_k^{(\ell )}\}}_{\ell =1,k=1}^{L,K}$.

In general, the PSD of second-order stationary processes can be characterized using the Spectral Representation Theorem [18]. This theorem implies that for the zero-mean, second-order stationary time series x_k −μ with spectral density function S(ω), there exists a continuous, orthogonal increment, and complex process Z(ω) such that

$$x_k-\mu ={\int }_{-\pi }^{\pi }{e}^{j\omega k}dZ(\omega )$$ (3)

where the integral is in the Riemann-Stieltjes sense and E{|dZ(ω)|²} = S(ω)dω. The function S(ω) is referred to as the PSD. Several nonparametric estimation techniques, such as the Welch’s method and multitaper estimate [18], exist to estimate S(ω) given a finite sequence of observations ${\{x_k\}}_{k=1}^K$. In our setting, due to the non-linearity of the model, these techniques cannot be directly applied. We therefore consider a discrete approximation to the PSD by assuming that the process Z(ω) defines a discrete-parameter harmonic process, i.e., it is constant over intervals of length $\frac{\pi }{N}$ for large enough N [18]. With this assumption, we can replace Z(ω) by a jump process in [0, π) with jumps of $\frac{\pi }{N}(a_i+j{b}_i)$ at ${\omega }_i=\frac{i\pi }{N}$, where a_i and b_i are some random variables, for i = 1, 2, …, N − 1, and $\frac{\pi }{N}$ is a normalization factor. Given that the process x_k is real, and invoking the symmetry Z(ω) = Z^*(−ω), we can express the integral in Eq. (3) as:

$$x_k-\mu =\frac{2\pi }{N}\sum _{i=1}^{N-1}\left(a_{i}cos({\omega }_{i}k)-b_{i}sin({\omega }_{i}k)\right).$$ (4)

where ${\omega }_i:=\frac{i\pi }{N}$. Using the property E{|dZ(ω)|²} = S(ω)dω, the PSD at ω_i for i = 1, 2, …, N − 1 can be expressed as:

$$S({\omega }_i)=\frac{{\pi }^2}{N^2}\mathbb{E}\{a_i^2+b_i^2\}.$$ (5)

Letting x = [x₁, x₂, …, x_K]^T ∈ ℝ^K, $\mathbf{v}={[\frac{N}{2\pi }\mu ,a_1,b_1,a_2,b_2,\dots ,a_{N-1},b_{N-1}]}^T\in {ℝ}^{2N-1}$, and defining A ∈ ℝ^K×(2N−1) as

$$\mathbf{A}:=\frac{2\pi }{N}\left[\begin{array}{cccccc}1& cos(\frac{\pi }{N})& -sin(\frac{\pi }{N})& \dots & cos\left(\frac{(N-1)\pi }{N}\right)& -sin\left(\frac{(N-1)\pi }{N}\right)\\ 1& cos(\frac{2\pi }{N})& -sin(\frac{2\pi }{N})& \dots & cos\left(\frac{2(N-1)\pi }{N}\right)& -sin\left(\frac{2(N-1)\pi }{N}\right)\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 1& cos(\frac{K\pi }{N})& -sin(\frac{K\pi }{N})& \dots & cos\left(\frac{K(N-1)\pi }{N}\right)& -sin\left(\frac{K(N-1)\pi }{N}\right)\end{array}\right]$$ (6)

we can express Eq. (4) as follows:

$$\mathbf{x}=\mathbf{Av}.$$ (7)

It is worth noting that the matrix A resembles the DFT/DCT synthesis matrices; however, it is not in general full rank. A would have full column rank (resp. full row rank) if K ≥ 2N−1 (resp. K ≤ 2N−1). We further assume that the process Z(ω) is Gaussian, and hence the variables $v_i~\mathcal{N}(0,{\sigma }_i^2)$, for i = 2, 3, ⋯, 2N−1. Note that v_i’s are independent due to the orthogonality of the increments of Z(ω). According to Eq. (5), we have $S({\omega }_i)=\frac{{\pi }^2}{N^2}({\sigma }_{2i}^2+{\sigma }_{2i+1}^2)$, which corresponds to the discrete PSD approximation at ${\omega }_i=\frac{i\pi }{N}$, for i = 1, ⋯,N −1, with N controlling the degree of approximation. Since we are interested in the oscillatory behavior of the CIF, estimation of μ (i.e., the DC component) is not of particular importance. Nevertheless, in order to have a consistent prior on all the elements of v, we assume an independent Gaussian prior on μ such that $v_1~\mathcal{N}(0,{\sigma }_1^2)$.

III. Bayesian estimation of the PSD

As a result of our formulation in Section II, estimating the PSD of x_k is reduced to estimating the parameters $\mathit{\theta }:={[{\sigma }_1^2,{\sigma }_2^2,\dots ,{\sigma }_{2N-1}^2]}^T$. Using a Bayesian formulation, we will perform the parameter estimation in a computationally efficient way. In addition, we can enforce the sparsity of the PSD by incorporating sparsity-promoting priors on θ. To this end, we use an exponential prior with parameter γ for the elements of θ resulting in a log-prior $log{f}_{\mathit{\theta }}(\mathit{\theta })=(2N-1)log\gamma -\gamma {\sum }_{i=1}^{2N-1}{\sigma }_i^2$. Note that the log-prior is akin to the ℓ₁-norm of θ modulo constants, which is known to promote sparsity. Using the shorthand notation

$$\mathcal{D}:={\{n_k^{(\ell )}\}}_{\ell =1,k=1}^{L,K},$$ (8)

the maximum a posteriori (MAP) estimate of θ is defined as:

$$\begin{gathered} {\widehat{\mathit{\theta }}}_{\text{MAP}}=\underset{\mathit{\theta }}{argmax}(log{f}_{\mathit{\theta }\mid \mathcal{D}}(\mathit{\theta }\mid \mathcal{D})) \\ =\underset{\mathit{\theta }}{argmax}\left(logP(\mathcal{D}\mid \mathit{\theta })+log{f}_{\mathit{\theta }}(\mathit{\theta })\right) \end{gathered}$$ (9)

A. MAP estimation via the Expectation-Maximization algorithm

Expressing P(𝒟|θ) solely in terms of the data 𝒟 results in an intractable function of θ. However, if the vector v was known, the log-likelihood of the complete data could be expressed as:

$$\begin{gathered} logf(\mathcal{D},\mathbf{v}\mid \mathit{\theta })=logP(\mathcal{D}\mid \mathbf{v},\mathit{\theta })+log{f}_{\mathbf{v}\mid \mathit{\theta }}(\mathbf{v}\mid \mathit{\theta }) \\ =\sum _{k=1}^K\sum _{\ell =1}^L{n}_k^{(\ell )}{(\mathbf{Av})}_k-log(1+exp({(\mathbf{Av})}_k))-\sum _{i=1}^{2N-1}\left(\frac{v_i^2}{2{\sigma }_i^2}+\frac{1}{2}log{\sigma }_i^2\right)+\text{cnst}. \end{gathered}$$ (10)

where cnst. stands for terms which are not functions of v or θ. We can thus use the Expectation-Maximization (EM) algorithm to calculate the MAP estimate in (9) [25].

The E Step

Suppose that at iteration r, we have an estimate of θ, denoted by ${\widehat{\mathit{\theta }}}^{(r)}={[{{\sigma }_1^2}^{(r)},{{\sigma }_2^2}^{(r)},\cdots ,{{\sigma }_{2N-1}^2}^{(r)}]}^T$. Given that in the complete data (𝒟, v), the vector v is unobserved, in the E step we calculate the function

$$\begin{gathered} Q\left(\mathit{\theta }\mid {\widehat{\mathit{\theta }}}^{(r)}\right):={\mathbb{E}}_{\mathbf{v}\mid \mathcal{D},{\widehat{\mathit{\theta }}}^{(r)}}\{logf(\mathcal{D},\mathbf{v}\mid \mathit{\theta })\}+log{f}_{\mathit{\theta }}(\mathit{\theta }) \\ ={\mathbb{E}}_{\mathbf{v}\mid \mathcal{D},{\widehat{\mathit{\theta }}}^{(r)}}\{log{f}_{\mathbf{v}\mid \mathit{\theta }}(\mathbf{v}\mid \mathit{\theta })\}+log{f}_{\mathit{\theta }}(\mathit{\theta })+\text{cnst}. \\ =\sum _{i=1}^{2N-1}\left(-\frac{1}{2}log{\sigma }_i^2-\frac{1}{2{\sigma }_i^2}{\mathbb{E}}_{\mathbf{v}\mid \mathcal{D},{\widehat{\mathit{\theta }}}^{(r)}}\{v_i^2\}-\gamma {\sigma }_i^2\right)+\text{cnst}. \end{gathered}$$ (11)

where, similar to (10), we have used the conditional independence P(𝒟|v, θ) = P(𝒟|v). In (11), the term cnst. represents all terms which are not functions of θ.

In order to compute the expectation in (11), the distribution of v|𝒟, θ̂^(r) or its samples are required. However, calculating the distribution of v|𝒟, θ̂^(r) involves computing intractable integrals, and sampling from v|𝒟, θ̂^(r) by numerical methods such as the Metropolis-Hastings is not computationally efficient considering that it has to be carried out in every iteration. As a result, Monte Carlo methods are not computationally efficient when N is large.

As shown in [12], [10], the density of v|𝒟, θ̂^(r), which is proportional to the product of the Gaussian density v|θ̂^(r) and a Binomial 𝒟|v, can be well approximated by a multivariate Gaussian density $\mathcal{N}({\mathit{\mu }}_{\mathbf{v}}^{(r)},{\mathbf{\sum }}_{\mathbf{v}}^{(r)})$. Noting that the mean and mode of a multivariate Gaussian coincide, and that the Hessian of its natural logarithm is equal to $-{({\mathbf{\sum }}_{\mathbf{v}}^{(r)})}^{-1}$, we calculate the mode of f_{v|𝒟,θ̂^(r)} (v|𝒟, θ̂^(r)) as the mean of the Gaussian approximation and the Hessian of log f_{v|𝒟,θ̂^(r)} (v|𝒟, θ̂^(r)) evaulated at the mode as $-{({\mathbf{\sum }}_{\mathbf{v}}^{(r)})}^{-1}$. For ${\mathit{\mu }}_{\mathbf{v}}^{(r)}$, we have:

$$\begin{gathered} {\mathit{\mu }}_{\mathbf{v}}^{(r)}=\underset{\mathbf{v}}{argmax}log{f}_{\mathbf{v}\mid \mathcal{D},{\widehat{\mathit{\theta }}}^{(r)}}(v\mid \mathcal{D},{\widehat{\mathit{\theta }}}^{(r)}) \\ =\underset{\mathbf{v}}{argmax}(logP(\mathcal{D}\mid \mathbf{v})+log{f}_{\mathbf{v}\mid {\widehat{\mathit{\theta }}}^{(r)}}(\mathbf{v}\mid {\widehat{\mathit{\theta }}}^{(r)})) \\ =\underset{\mathbf{v}}{argmax}\left(\sum _{k=1}^K\sum _{\ell =1}^L{n}_k^{(\ell )}{(\mathbf{Av})}_k-log(1+exp({(\mathbf{Av})}_k))-\sum _{i=1}^{2N-1}\frac{v_i^2}{2{{\sigma }_i^2}^{(r)}}\right). \end{gathered}$$ (12)

The maximization problem in (12) is concave, and the Hessian is negative definite. Hence, we can use the Newton’s method to efficiently compute ${\mathit{\mu }}_{\mathbf{v}}^{(r)}$. The method is summarized in Algorithm 1. The stopping condition 𝒮_N can be either a convergence constraint or a limit on the number of iterations.

Letting ${\mathbf{a}}_i^T$ denote the ith row of the matrix A, the inverse covariance can be computed as:

$${\left({\mathbf{\sum }}_{\mathbf{v}}^{(r)}\right)}^{-1}=L{\mathbf{A}}^T\mathbf{FA}+\text{diag}\left\{\frac{1}{{{\sigma }_1^2}^{(r)}},\cdots ,\frac{1}{{{\sigma }_{2N-1}^2}^{(r)}}\right\}$$ (13)

where

$$\mathbf{F}=\text{diag}\left\{\frac{e^{-{\mathbf{a}}_1^T{\mathit{\mu }}_{\mathbf{v}}^{(r)}}}{{(1+e^{-{\mathbf{a}}_1^T{\mathit{\mu }}_{\mathbf{v}}^{(r)}})}^2},\cdots ,\frac{e^{-{\mathbf{a}}_K^T{\mathit{\mu }}_{\mathbf{v}}^{(r)}}}{{(1+e^{-{\mathbf{a}}_K^T{\mathit{\mu }}_{\mathbf{v}}^{(r)}})}^2}\right\}$$ (14)

Algorithm 1.

Newton’s method for finding ${\mathit{\mu }}_{\mathbf{v}}^{(r)}$

Input: ensemble average spiking dataa ${\overline{n}}_k:=\frac{1}{L}{\sum }_{\ell =1}^L{n}_k^{(\ell )}$, for k = 1, 2, ⋯, K, current parameter estimate θ̂(r), Newton’s stopping condition 𝒮N.
Output: ${\mathit{\mu }}_{\mathbf{v}}^{(r)}$
1:	m(0) = 0.
2:	iteration number i = 0.
3:	while ¬𝒮N do
4:	i ← i + 1
5:	x = Am(i−1).
6:	${\lambda }_k=\frac{1}{1+e^{-x_k}}$, for 1 ≤ k ≤ K.
7:	$\mathbf{q}={\left[\frac{m_1^{(i-1)}}{{{\sigma }_1^2}^{(r)}},\dots ,\frac{m_{2N-1}^{(i-1)}}{{{\sigma }_{2N-1}^2}^{(r)}}\right]}^T$.
8:	calculate the gradient as g = LAT (n̄ − λ) − q
9:	$\mathbf{U}=\text{diag}\left\{\frac{1}{{{\sigma }_1^2}^{(r)}},\dots ,\frac{1}{{{\sigma }_{2N-1}^2}^{(r)}}\right\}$.
10:	$\mathbf{G}=\text{diag}\left\{\frac{e^{-x_1}}{{(1+e^{-x_1})}^2},\dots ,\frac{e^{-x_K}}{{(1+e^{-x_K})}^2}\right\}$.
11:	calculate the Hessian as H = −LATGA − U.
12:	m(i) = m(i−1) −H−1g.
13:	end while
14:	${\mathit{\mu }}_{\mathbf{v}}^{(r)}={\mathbf{m}}^{(i)}$.

^an̄_k is often referred to as the Peristimulus Time Histogram (PSTH).

Going back to (11), using the solutions of (12) and (13), we have $E_{\mathbf{v}\mid \mathcal{D},{\widehat{\mathit{\theta }}}^{(r)}}\{v_i^2\}={({({\mathit{\mu }}_{\mathbf{v}}^{(r)})}_i)}^2+{({\mathbf{\sum }}_{\mathbf{v}}^{(r)})}_{i,i}$, which we denote by $E_i^{(r)}$ for notational simplicity.

The M Step

In the M step, we maximize Q(θ|θ̂^(r)) with respect to θ. The function Q(θ|θ̂^(r)) is quasi-concave over the positive orthant with a unique maximizer. We have:

$$\frac{\partial }{\partial {\sigma }_i^2}Q(\mathit{\theta }\mid {\widehat{\mathit{\theta }}}^{(r)})=-\frac{1}{2{\sigma }_i^2}+\frac{E_i^{(r)}}{2{\sigma }_i^4}-\gamma =0$$ (15)

Noting that γ and $E_i^{(r)}$ are both positive, solving the quadratic equation $2\gamma {\sigma }_i^4+{\sigma }_i^2-E_i^{(r)}=0$ for 1 ≤ i ≤ 2N−1 in terms of ${\sigma }_i^2$’s and picking the positive root gives the updated parameter vector as:

$${({\widehat{\mathit{\theta }}}^{(r+1)})}_i={{\sigma }_i^2}^{(r+1)}=\frac{-1+\sqrt{1+8\gamma E_i^{(r)}}}{4\gamma },$$ (16)

for i = 1, 2, ⋯, 2N −1. It is worth noting that if no prior on ${\sigma }_i^2$’s is used, the EM algorithm can be used similarly to calculate the Maximum Likelihood (ML) estimate of θ given the data 𝒟. In that case, the update rule of the EM algorithm is simply given by:

$${({\widehat{\mathit{\theta }}}^{(r+1)})}_i={{\sigma }_i^2}^{(r+1)}=E_i^{(r)},1\le i\le 2N-1$$ (17)

Algorithm 2 summarizes the MAP estimation of the PSD. The EM stopping condition 𝒮_EM again can be either a convergence condition or a limit on the number of iterations. A random vector with small positive elements can be considered as the initialization point θ⁽⁰⁾. It is worth noting that the maximization problem of Eq. (9) is not concave or quasiconcave in general. Hence, the EM algorithm may converge to a local maximum, depending on its initialization. However, our numerical analysis suggests that initializing the EM algorithm with θ⁽⁰⁾ taking small positive values, results in meaningful and interpretable estimates as shown in our simulations and real data analysis in Section IV.

Algorithm 2.

MAP estimate of the PSD

Input: Ensemble spike observations $\mathcal{D}={\{n_k^{(\ell )}\}}_{\ell =1,k=1}^{L,K}$, exponential prior hyperparameter γ, EM stopping condition 𝒮EM, EM initialization θ(0), frequency spacing of the PSD estimate as the number of bins N in [0, π].
Output: N−1 uniform samples in (0, π) of the PSD associated with the ensemble CIF.
1:	Construct the matrix A as in Eq. (6).
2:	Iteration number r = 0.
3:	while ¬𝒮EM do
4:	Using Algorithm 1 with θ(r), solve the optimization problem of (12) to calculate the mean of the Gaussian approximation, i.e. ${\mathit{\mu }}_{\mathbf{v}}^{(r+1)}$.
5:	Using θ(r) and ${\mathit{\mu }}_{\mathbf{v}}^{(r+1)}$, calculate (13) as the covariance inverse of the Gaussian approximation, i.e. ${({\mathbf{\sum }}_{\mathbf{v}}^{(r+1)})}^{-1}$.
6:	Update θ based on (16) to get θ̂(r+1)
7:	r ← r + 1
8:	end while
9:	Using the last updated parameter vector θ̂(r), calculate the PSD estimates ${\widehat{S}}_i=\frac{{\pi }^2}{N^2}({{\sigma }^2}_{2i}^{(r)}+{{\sigma }^2}_{2i+1}^{(r)})$ for 1 ≤ i ≤ N − 1.

Remark 1

It is worth noting that if there exists any prior information on the maximum frequency content of the data, this information can be incorporated into the model in order to reduce the computational cost. For instance, neural data is often sampled at rates much higher than the significant frequency content. Suppose f_spc is the frequency spacing we require, and we know the maximum frequency content would not be larger than f_max. Thus, The number of bins in [0, $\frac{f_s}{2}$) corresponding to this spacing is $N_{\text{spc}}=\lceil \frac{f_s}{2f_{\text{spc}}}\rceil$, and we want to focus on the first $N_{\text{max}}=\lceil \frac{f_{max}}{f_{\text{spc}}}\rceil$ bins. In this case, the matrix A in Eq. (6) can be reduced to an M × (2N_max−1) matrix rather than a M×(2N_spc−1) matrix. This modification can greatly reduce the computational cost of the problem.

Remark 2

In general, the spectral resolution of a possibly infinite stationary signal depends on the number of acquired samples, i.e. the main lobe width of the sampling window. Similarly, the number of spiking samples K externally limits the frequency resolution in the spectrum estimate of the neural covariate. In order not to confuse this resolution with $\frac{\pi }{N}$, we keep referring to $\frac{\pi }{N}$ as the frequency spacing in the estimated spectrum rather than the spectrum resolution. As mentioned before, N represents the desired number of spectrum estimate samples in [0, π) and controls the degree of approximation.

Remark 3

Note that the as long as K ≥ 2N_max −1, the full column rank virtue of the matrix A guarantees stable estimates of the spectra from Eqs. (12) and (13). When K ≤ 2N_max−1, the matrix will only have full row rank, which may result in instability of the Newton’s algorithm. Although the ℓ₁- regularization in this case may mitigate the latter shortcoming (i.e., γ in Eq. (15)), we assume in what follows that the number of observations K satisfies K ≥ 2N_max − 1.

B. Hyper-parameter selection

We choose the optimal value of the hyper-parameter γ using cross-validation. We will use a two-fold cross-validation algorithm [26] to this end. We divide the ensemble into two groups, thereby partitioning the data into 𝒟₁ and 𝒟₂. In this case, the cross-validation criterion for each value of γ is the likelihood of 𝒟₁ (res. 𝒟₂) given the estimated parameter vector θ̂ using 𝒟₂ (res. 𝒟₁). Considering the generic data set 𝒟, we have:

$$\begin{gathered} P\left(\mathcal{D}\mid \widehat{\mathit{\theta }}\right)=\underset{\mathbf{v}\in {ℝ}^{(2N-1)}}{\int \cdots \int }f(\mathcal{D},\mathbf{v}\mid \widehat{\mathit{\theta }})d{v}_{1}d{v}_2\cdots d{v}_{2N-1} \\ =\underset{\mathbf{v}\in {ℝ}^{(2N-1)}}{\int \cdots \int }P(\mathcal{D}\mid \mathbf{v})f_{\mathbf{v}\mid \widehat{\mathit{\theta }}}(\mathbf{v}\mid \widehat{\mathit{\theta }})d{v}_{1}d{v}_2\cdots d{v}_{2N-1} \\ ={\mathbb{E}}_{\mathbf{v}\mid \widehat{\mathit{\theta }}}\left\{P(\mathcal{D}\mid \mathbf{v})\right\}. \end{gathered}$$ (18)

We also have:

$$P(\mathcal{D}\mid \mathbf{v})=\prod _{k=1}^K{\left(\frac{1}{1+e^{-{\mathbf{a}}_k^T\mathbf{v}}}\right)}^{\sum _{\ell =1}^L{n}_k^{(\ell )}}{\left(1-\frac{1}{1+e^{-{\mathbf{a}}_k^T\mathbf{v}}}\right)}^{\sum _{\ell =1}^L(1-n_k^{(\ell )})}.$$ (19)

Due to the independent Gaussian priors on the elements of v in our model, we have v|θ̂ ~ 𝒩(0, diag{θ̂}).

Algorithm 3.

Two-fold cross-validation for optimizing the hyper-parameter γ

Input: Two subsets of data 𝒟1 and 𝒟2, and a set of candidate values of γ given by Γ.
Output: Optimal value of the hyperparameter γopt.
1:	for each test value of γ do
2:	Estimate θ̂1 using 𝒟1 from Algorithm 2.
3:	Draw R samples v1, …, vR from 𝒩(0, diag{θ̂1}).
4:	Estimate ℒ2|1:= P(𝒟2|θ̂1) using Monte Carlo sampling from (19).
5:	Repeat steps 2 to 4 interchanging the roles of 𝒟1 and 𝒟2 to calculate ℒ1|2.
6:	$ℒ(\gamma )=\frac{1}{2}({ℒ}_{2\mid 1}+{ℒ}_{1\mid 2})$.
7:	end for
8:	γopt = arg maxγ∈Γ ℒ(γ).

Thus, the expectation in (18) can be estimated in arbitrary precision using the Monte Carlo method [26] by drawing R samples v₁, …, v_R from 𝒩(0, diag{θ̂}) and calculating the sample average of P(𝒟|v) in (19), i.e., $P(\mathcal{D}\mid \widehat{\mathit{\theta }})\simeq \frac{1}{R}{\sum }_{i=1}^{R}P(\mathcal{D}\mid {\mathbf{v}}_r)$. Note that in order to ensure numerical stability, we compute log P(𝒟|v_r), which from Eq. (19) takes an additive form over k. Algorithm 3 summarizes the steps of the cross-validation algorithm to determine the optimal value of the hyper-parameter γ_opt among a set of test values.

C. Constructing confidence intervals

It is possible to construct confidence intervals for the estimated PSD values {Ŝ₁, ⋯, Ŝ_N−1} obtained from Algorithm 2 by sampling from the density f_θ|𝒟(θ̂|𝒟). We have:

$$f_{\widehat{\mathit{\theta }}\mid \mathcal{D}}(\widehat{\mathit{\theta }}\mid \mathcal{D})\propto P(\mathcal{D}\mid \widehat{\mathit{\theta }})f_{\mathit{\theta }}(\widehat{\mathit{\theta }})\propto \underset{g(\widehat{\mathit{\theta }},\mathcal{D},\gamma )}{\underbrace{{\mathbb{E}}_{\mathbf{v}\mid \widehat{\mathit{\theta }}}\{P(\mathcal{D}\mid \mathbf{v})\}{e}^{-\gamma \sum _{i=1}^{2N-1}{\sigma }_i^2}}}$$ (20)

We can therefore use the Metropolis-Hastings algorithm [26] to sample from f_θ|𝒟(θ̂|𝒟). The expectation term in g(θ̂,𝒟, γ) can be estimated using the Monte Carlo sampling procedure explained in Section III-B. Algorithm 4 summarizes the Metropolis-Hasting algorithm for our sampling purpose. For simplicity, we have considered a Gaussian proposal density q(u|w), i.e. u|w ~ 𝒩(u,Σ_q), where Σ_q is a diagonal matrix in ℝ^{(2N−1)×(2N−1)} with diagonal elements proportional to the estimated parameter vector θ̂. It is worth noting that g(θ̂,𝒟, γ) is zero for any vector θ̂ which is not element-wise non-negative. Thus, if the normally distributed candidate z in line 3 of Algorithm 4 has negative components, it would be discarded.

Algorithm 4.

Constructing confidence intervals for ${\{{\widehat{S}}_i\}}_{i=1}^{N-1}$

Input: Neural spiking data 𝒟, cross-validated hyperparameter γopt, MAP estimate of θ (θ̂MAP), number of samples M, symmetric proposal density function q(.|.).
Output: confidence intervals for ${\{{\widehat{S}}_i\}}_{i=1}^{N-1}$.
1:	Initialize ϑ(0) = θ̂MAP.
2:	while m ≤ M do
3:	Generate a candidate z for the next sample by sampling from the density q(z|ϑ(m−1)).
4:	Calculate the acceptance ratio $\alpha :=\frac{g(\mathbf{z},\mathcal{D},{\gamma }_{\mathit{opt}})}{g({\mathit{\vartheta }}^{(m-1)},\mathcal{D},{\gamma }_{\mathit{opt}})}$.
5:	If α ≥ 1, accept z as the next sample; otherwise, accept z as the next sample with probability α.
6:	If z is accepted as the next sample set ϑ(m) = z; otherwise set ϑ(m) = ϑ(m−1).
7:	m ← m + 1
8:	end while
9:	Transform each sample ϑ(m), 1 ≤ m ≤ M, into a sample for Ŝi as ${\widehat{S}}_i^{(m)}=\frac{{\pi }^2}{N^2}({{\sigma }^2}_{2i}^{(m)}+{{\sigma }^2}_{2i+1}^{(m)})$ for 1 ≤ i ≤ N − 1.
10:	Construct confidence intervals at a level 1−ν for ${\{{\widehat{S}}_i\}}_{i=1}^{N-1}$ using the samples ${\{{\widehat{S}}_i^{(m)}\}}_{m=1,i=1}^{M,N-1}$.

IV. Application to Simulated and Real data

In this section, we first demonstrate the performance of our method in two simulated settings. The two settings correspond to CIFs from noisy two-tone line spectra and an autoregressive process, respectively. We compare the performance of our method with three existing techniques: 1) calculating the periodogram of the spiking data of each neuron and averaging over the periodograms across neurons, which we refer to as PER-PSD [27]; 2) smoothing the ensemble average spiking (PSTH) and computing the PSD of the resulting smoothed PSTH, which we refer to as the PSTH-PSD; 3) Using a state-space model to estimate the CIF, followed by computing the PSD of the estimated CIF [12], which we will refer to as SS-PSD. Finally, we apply our method to ECoG data from a human subject under general anesthesia and compare the extracted PSD from the spiking data to that of the LFP.

It is worth noting that the class of spectra which is identifiable from neuronal spiking data, is generally limited by the spiking rate in the PSTH. In a way, the spiking rate represents the amount of information available for inference procedures such as spectral estimation. The higher the PSTH spiking rate is, the larger and more complex the class of identifiable spectra would be. In order to conduct simulation studies and perform comparisons with existing methods in a setting akin to real neuronal data, we have limited the simulation setting to PSTH spiking rates of %5–%10 and sparse spectra which can potentially be identified under these low spiking rates.

A. Spike trains driven by a noisy dual-tone signal

Consider the dual-tone signal

$$x(t)=1.48cos(2\pi f_{0}t)+0.685cos(2\pi f_{1}t)+0.17n(t)-5.7$$ (21)

with f₀ = 1 Hz, f₁ = 10 Hz, and n(t) representing a zero-mean white Gaussian noise with unit variance. When sampled at f_s = 300 Hz, the discretized data forms x_k. The bias term of −5.7 is chosen to make sure that the resulting spiking rate is low enough and consistent with real-world neuronal spiking rates. The tones at f₀ and f₁ are chosen as a model of neuronal spiking modulated by slow and alpha oscillations, respectively. We consider K = 1000 samples of x_k and simulated the spiking data for L = 10 neurons based on our model in (2). The signal x_k and the raster plot of the ensemble are shown in Figure 1–(a) and 1–(b), respectively. The average spiking rate of the ensemble from the PSTH is 0.056.

Fig. 1.

(a) Dual-tone signal x_1:K (b) Raster plot of the ensemble.

Figure 2 shows the results obtained by our proposed method as well as the PER-PSD, PSTH-PSD and SS-PSD methods. Figure 2–(a) shows the PSTH smoothed via two Gaussian kernels: a narrow kernel (green trace) and a wide kernel (dotted red trace). Figure 2–(b) shows the normalized multitaper estimate [28], [18], [29] of the PSD corresponding to the smoothed PSTH values shown in panel 2–(a) as well as the PER-PSD estimate. The multitaper method is arguably the most reliable nonparametric spectral estimation technique, as it addresses the estimation bias and variance trade-off in an optimal fashion [28]. The spectral resolution of the multitaper method is chosen as 0.125Hz. The means of all spiking signals in PER-PSD method, smoothed n̄_ks in PSTH-PSD method, and x̂_k|Ks in SS-PSD method are subtracted prior to calculating the periodogram or multitaper estimate to make sure the significant oscillatory components would not get dominated by the DC component. Also, all of the PSD estimates are normalized for the purpose of comparison.

Fig. 2.

Noisy dual-tone CIF model for a neuronal ensemblef: (a) Normalized smoothed PSTH using Gaussian kernels with small and large variances (b) Normalized PER-PSD estimate and normalized multitaper estimate of the PSD corresponding to the smoothed PSTHs (c) Estimate of x_k using state-space smoothing (d) Normalized multitaper estimate of the PSD of x̂_k|K (e) Raw PSTH of the data n̄_k with 0.056 spiking rate (f) Normalized PSD estimate using the proposed method after 130 EM iterations together with %95 confidence intervals.

The PER-PSD method considers each spiking signal as samples of a stationary signal, and does not make use of the ensemble average (PSTH). Since periodogram is not a consistent estimator of the PSD and needs further smoothing, the average of the resulting periodograms across the realizations is calculated [27]. Hence, the PER-PSD method can be viewed as the average of the periodogram PSD estimates obtained from individual neurons. Figure 2–(b) (purple trace) shows the PER-PSD estimate. Although a significant peak is retrieved at 1Hz, the 10Hz component is not recovered due to the high variability of the estimate.

The estimate corresponding to the narrow smoothing kernel in PSTH-PSD (Figure 2–(b), green trace) detects the two peaks at 1 Hz and 10 Hz, but has a high variability in higher frequencies. In addition, two spurious peaks at 2 Hz and 9 Hz are detected in the PSD. On the other hand, the estimate corresponding to the wide smoothing kernel (Figure 2–(b), dotted red trace) has a smaller variability but misses the 10 Hz component of the data. These results show the high sensitivity of the PSTH-PSD approach to the choice of the smoothing kernel. Comparing to PER-PSD, the PSTH-PSD method results in estimates with lower variability as it forms PSD estimates by employing the more informative ensemble average signal (PSTH) rather than the spiking data of the individual neurons.

Figure 2–(c), shows the estimate of x_k using state-space smoothing [12]. The state-space framework corresponding to our model is given by:

$$\{\begin{array}{l}{x}_k=x_{k-1}+{\epsilon }_k,1\le k\le K\hfill \\ {\epsilon }_k\stackrel{\text{iid}}{\sim }𝒩(0,{\nu }^2),1\le k\le K\hfill \\ {\lambda }_k=\frac{1}{1+e^{-x_k}},\hfill \\ n_k^{(\ell )}~\text{Bernoulli}({\lambda }_k),1\le k\le K,1\le \ell \le L\hfill \end{array}$$ (22)

Using a forward/backward filtering, this method computes the MAP estimate of x_k given all the data, denoted by x̂_k|K, which is plotted in Figure 2–(c). The parameter ν² is estimated via the EM algorithm. Similar Gaussian density approximations to (12) and (13) have been used in [12] specially in the filtering and smoothing parts. As observed from Figure 2–(c), the estimate x̂_k|K correlates with the smoothed PSTH estimate via a wide kernel (Figure 2–(a), dotted red trace). The normalized multitaper estimate of the PSD of x̂_k|K, namely the SS-PSD estimate, is shown in Figure 2–(d). Similar to the PSTH-PSD estimate using a wide Gaussian kernel, the 10 Hz cannot be recovered using this method.

The raw PSTH of the data ${\overline{n}}_k=\frac{1}{L}{\sum }_{\ell =1}^L{n}_k^{(\ell )}$, 1 ≤ k ≤ K is shown in Figure 2–(e), followed by our estimate of the PSD in Figure 2–(f). We have chosen N = 1200 corresponding to a frequency binning of 0.125 Hz for f_s = 300 Hz, which is comparable to the design resolution of the multitaper method used for the PSTH-PSD and SS-PSD methods. We have used the first 140 bins (0.125 Hz to 17.375 Hz) in constructing the matrix A, in order to reduce the computational complexity (N_max = 140). Furthermore, the 95% confidence intervals for the identified oscillatory components are calculated using Algorithm 4 with M = 1000 samples and shown in Figure 2–(f) (gray hulls). The upper confidence bound at f = 10 Hz is at ≈ 1.4 and is truncated in the graph for graphical convenience. The cross-validated value for γ using Algorithm 3 is 10⁻⁴. Clearly, both of the tones at 1 Hz and 10 Hz are recovered (unlike the PSTH-PSD with a wide kernel and the SS-PSD), while the irrelevant frequencies are significantly suppressed (unlike the PER-PSD method and the PSTH-PSD with a narrow kernel). Note that we have used 130 EM iterations to obtain the estimate. Figure 3 shows the estimated PSD vs. EM iterations. The dominant frequency of 1 Hz is detected at around iteration 30, and by continuing the EM iterations the component at 10 Hz is eventually discovered at around iteration 100. A MATLAB implementation of our algorithm, as well as the existing ones, producing Figure 2 is archived on the open source repository GitHub and made publicly available [30].

Fig. 3.

PSD vs. EM iterations corresponding to Figure 2–(f).

The foregoing simulation demonstrated the superior performance of our method in application to ensemble neuronal activity. An intriguing comparison setting is to assess the performance of our method, as well as the existing ones, when applied to spiking data from a single neuron (L = 1). As mentioned earlier, we consider spiking data from a single neuron with sufficiently high spiking rate in order for the PSD estimation methods to have meaningful results. To this end, for the single neuron simulation, we change the bias term of −5.7 in Eq. (21) for the noisy dual-tone neural covariate to −3.7. This results in the average spiking rate of 0.05 for the single neuron spike train shown in Figure 4–(e). All the other parameters, including those of the dual-tone neural covariates, are the same as those in the foregoing simulation setting.

Fig. 4.

Noisy dual-tone CIF model for a single neuron: (a) Normalized smoothed spiking signal using Gaussian kernels with small and large variances (b) Normalized PSD estimates corresponding to the smoothed spiking signal (c) Estimate of x_k using state-space smoothing (d) Normalized multitaper estimate of the PSD of x̂_k|K (e) Single neuron spiking signal n_k with 0.05 spiking rate (f) Normalized PSD estimate using the proposed method after 300 EM iterations together with %95 confidence intervals.

Figure 4 shows the estimated PSDs using the different methods applied to the spike train from a single neuron. Since the output of the narrow kernel PSTH-PSD method is nearly identical to that of the PER-PSD method in the case of a single neuron, we have only shown the PSD estimate of the PSTH-PSD method in Figure 4–(b). Similar to the foregoing simulation results using an ensemble of L = 10 neurons, we observe that the narrow kernel PSTH-PSD method identifies the two frequency peaks at 1Hz and 10Hz. However, the estimate also contains significant redundant peaks at 2Hz, 4Hz, and around 9Hz, while having a high variability at higher frequencies. Furthermore, both the wide kernel PSTH-PSD method and the SS-PSD method smooth the spiking data to the degree that results in missing the 10Hz peak in the PSD. In contrast, as shown in Figure 4–(f) and Figure 5, our proposed method recovers the two peaks and only contains a minor redundant low-frequency component in the PSD estimate. Note that since only one spiking realization is employed by our algorithm, it takes more EM iterations to decode the smaller peak at 10 Hz in this case, as compared to the foregoing simulation (~ 200 EM iterations in Figure 5 vs. ~ 100 EM iterations in Figure 3).

Fig. 5.

PSD vs. EM iterations corresponding to Figure 4–(f).

In summary, these two simulation studies demonstrate the superior performance of our algorithm as compared to several existing techniques. In addition, they highlight the difference of considering the ensemble spiking data vs. single-neuron spiking data for PSD estimation. If the PSTH corresponding to an ensemble of low spiking neurons is rich enough to identify a specific spectral structure, in order to get comparable results using data from a single neuron, the spiking rate must be chosen high enough to account for the lack of multiple realizations. In other words, an ensemble of low spiking neurons can provide much more information than considering each of them in isolation. This observation explains the performance gap between the PER-PSD and PSTH-PSD estimates shown in Figure 2–(b); the PER-PSD method forms periodogram estimates using single neuron spiking data rather than the PSTH, and hence exhibits inferior performance in comparison to the PSTH-PSD methods.

B. Spike trains driven by an AR(6) process

In the second set of simulations, we examine a more complex scenario where the driving signal x_k is generated from a 6^th order autoregressive (AR) process. Figure 6–(a) shows the PSD corresponding to the AR process with third-order poles at ${\omega }_1=\frac{\pi }{20}=0.1571$ rad and ${\omega }_2=\frac{\pi }{5}=0.6283$ rad with magnitudes of 0.997 and 0.999, respectively. The sample realization of this process of length K = 500 as well as the raster plot of the spike trains for L = 10 neurons are depicted in Figures 6–(b) and –(c), respectively. Similar to the previous simulation, a significant negative mean is added to the AR signal to make the PSTH spiking rate close to real neuronal spiking rates. The average spiking rate of the PSTH corresponding to the ensemble in Figure 6–(c) is 0.058.

Fig. 6.

(a) PSD of the dual peak AR process (b) Sample realization of the AR process (c) Raster plot of the ensemble.

Figure 7 shows the estimated PSDs using the different methods. The spacing of our estimate is 0.105 rad corresponding to N = 300 bins in our model, out of which 100 bins have been considered to cover 0.0105 rad to 1.0395 rad frequency range (N_max = 100). Similar to the previous example, the PER-PSD estimate in Figure 7–(b) (purple trace) is so noisy that the two peaks at 0.1571 rad and 0.6283 rad are comparable to noisy retrieved components. Also in the PSTH-PSD method, we observe that the multitaper estimate corresponding to the narrow kernel in 7–(b) tends to have a high variability and detects undesired peaks around 0.4 rad and 0.8 rad, comparable in magnitude to the correctly identified peaks at 0.1571 rad and 0.6283 rad. While reducing the variability at higher frequencies, the wide smoothing Gaussian kernel has resulted in dismissing the peak at 0.6283 rad. Thus, the inevitable effect of tuning the width of the smoothing kernel persists in this example as well. Figures 7–(c) and 7–(d) show the results of the SS-PSD method in time and frequency domains, respectively. Again, we observe a correlation between the estimated x̂_k|K in Figure 7–(c) and the smoothed n̄_k using a wide Gaussian kernel in Figure 7–(a). However, the SS-PSD method similarly dismisses the peak at 0.6283 rad. In addition, the recovered peak at 0.1571 rad is dominated by other falsely recovered low frequency components due to the temporal smoothing nature of the estimated x̂_k|K. Figures 7–(e) and 7–(f) respectively show the PSTH and the output of our method for 100 EM iterations, with the cross-validated value of 0.045 for γ. The convergence of the EM algorithm follows a similar pattern to that of the preceding section and is depicted in Figure 8. Similar to the previous simulation setting, M = 1000 samples are used for constructing 95% confidence intervals (grey hulls). The upper confidence bound at ω = 0.1571 rad is at ≈ 1.2 and is truncated in the graph for graphical convenience. As observed in Figure 7–(f), the two peaks are perfectly recovered while the undesired frequency components are nearly estimated zero.

Fig. 7.

AR(6) generated CIF model for a neuronal ensemble: (a) Normalized smoothed PSTH using Gaussian kernels with small and large variances (b) Normalized PER-PSD estimate and normalized multitaper estimate of the PSD corresponding to the smoothed PSTHs (c) Estimate of x_k using state-space smoothing (d) Normalized multitaper estimate of the PSD of x̂_k|K (e) Raw PSTH of the data n̄_k with 0.058 spiking rate (f) Normalized PSD estimate using the proposed method after 100 EM iterations together with %95 confidence intervals.

Fig. 8.

PSD vs. EM iterations corresponding to Figure 7–(f).

C. Application to neuronal spiking data from anesthesia

Finally, we apply our proposed algorithm on multi-unit recordings from a human subject under Propofol-induced general anesthesia (data from [31]). The data set includes the spiking activity of 41 neurons as well as the LFP recorded from a patient undergoing intra-cranial monitoring for surgical treatment of epilepsy using a multichannel micro-electrode array implanted in temporal cortex [31]. Recordings were conducted during the administration of Propofol for induction of anesthesia. The experimental protocol under which the data was collected is extensively explained in [31]. Given that the the original multi unit recordings are oversampled at a rate of 1 KHz, to reduce computational complexity, the spike recordings were downsampled by the factor of 40, and the sampling rate of the LFP signal was reduced from the original 250 Hz to 25 Hz. A time frame of 50 s is considered containing K = 1250 samples of the downsampled multi-unit recordings and the LFP signal. In our analysis, we have considered L = 27 neurons with at least two spikes in the 50 s time frame. Figure 9 shows the raster plot of the neuronal ensemble. The average spiking rate of the population from the PSTH is given by 0.1064.

Fig. 9.

Raster plot of the neuronal ensemble corresponding to multi-unit recordings from a human subject under Propofol-induced general anesthesia.

Figure 10 shows the results of the different PSD estimation techniques. For our method, we have chosen a spacing of 0.02 Hz which corresponds to N = 625 frequency bins in our model, considering the reduced sampling rate of 25 Hz. Given that the relevant frequencies modulating neuronal spiking under anesthesia pertain to slow oscillations [31], we have considered the first 100 frequency bins in our method covering 0.02 Hz to 2 Hz (N_max = 100). Figures 10–(a) and 10–(b) respectively show the smoothed PSTH with the two Gaussian kernels and their corresponding PSD estimates as well as the PER-PSD estimate. As was the case in our simulation studies, the PER-PSD estimate is noisy everywhere. Also, the PSTH-PSD estimate corresponding to the narrow kernel contains considerable variability in high frequencies. In contrast, the PSTH-PSD estimate using the wide smoothing kernel significantly suppresses the PSD components beyond 0.4 Hz. Figures 10–(c) and –(d) show the estimates of x_k|K and the PSD using the SS-PSD method. Similar to the preceding simulation studies, low frequency components dominate the PSD estimate due to the heavy time-domain smoothing in estimating x_k|K.

Fig. 10.

Neuronal spiking data from anesthesia: (a) Normalized smoothed PSTH using Gaussian kernels with small and large variances (b) Normalized PER-PSD estimate and normalized multitaper estimate of the PSD corresponding to the smoothed PSTHs (c) Estimate of x_k using state-space smoothing (d) Normalized multitaper estimate of the PSD of x̂_k|K (e) Raw PSTH of the data n̄_k with 0.1064 spiking rate (f) Normalized PSD estimate using the proposed method after 100 EM iterations together with %95 confidence intervals (g) Recorded LFP signal (h) Normalized multitaper estimate of the PSD corresponding to the recorded LFP signal.

The PSTH and the output of our method after 100 EM iterations ensuring convergence are shown in Figures 10–(e) and 10–(f), respectively. The EM convergence vs. iteration is shown in Figure 11. The cross-validated value for γ is 0.075, and M = 1000 samples are used in Algorithm 4 to construct 95% confidence intervals (grey hulls). The upper confidence bound at f = 0.42 Hz is at ≈ 1.35 and is truncated in the graph for graphical convenience. Figure 10–(g) and 10–(h) show the LFP and its multitaper PSD estimate respectively. The PSD of the LFP signal shows a dominant peak around 0.42 Hz, with a few others extending to 0.8 Hz. Strikingly, the PSD obtained by our method is the most similar to the PSD of the LFP: the PSTH-PSD fails to suppress the high frequency variability (narrow kernel) or dismisses the PSD peaks beyond 0.42 Hz (wide kernel). The SS-PSD method recovers dominant low frequency component which do not exist in the PSD of the LFP signal. This result corroborates the findings of [31] that the neuronal spiking under general anesthesia is highly phase-locked to the LFP signal, and hence the LFP can be considered as a salient neural covariate driving the spiking of the nearby cortical neuronal ensemble.

Fig. 11.

PSD vs. EM iterations corresponding to Figure 10–(f).

V. Discussion

The preceding section demonstrated the superior performance of our algorithm on simulated data as well as real data recordings. In order to explain this performance gap as well as to characterize the computational cost of our algorithm, two discussion points are in order.

A. Pursuit domain comparisons of the PSD estimators

The significant performance gain of our proposed PSD estimation framework over methods such as the PSTH-PSD or PER-PSD mainly stems from the difference in the underlying pursuit domains. The PSTH-PSD and PER-PSD methods generate the spectral estimates by forming a second-order combination of the data from a finite collection of binary sets, i.e., Fourier transforms of the PSTH ${\overline{n}}_k=\frac{1}{L}{\mathrm{\sum }}_{l=1}^L{n}_k^{(l)}$ or the spiking data of each neuron $n_k^{(l)}$, in which $n_k^{(l)}\in \{0,1\}$. Given that the PSTH signal takes values in the set {0, $\frac{1}{L}$, ⋯, $\frac{L-1}{L}$, 1} and the spiking of each neuron is a binary variable, the pursuit domains of the PSTH-PSD and PER-PSD algorithms are limited to small and finite subsets of ℝ. Perhaps the poor performance of the PER-PSD compared to the PSTH-PSD method is due to the fact that the former generates estimates using a second-order function of variables in {0, 1}, while the latter does so using the richer set {0, $\frac{1}{L}$, ⋯, $\frac{L-1}{L}$, 1}.

In contrast, our MAP estimator employs the same observations from the small subset {0, $\frac{1}{L}$, ⋯, $\frac{L-1}{L}$, 1}, but performs inference of the latent variables x, v, and θ directly over the richer set ℝ. To this end, the MAP estimates are obtained by solving an optimization problem seeking a spectral estimate with elements in ℝ that is consistent with the observed data and sparse priors in the Bayesian sense. Therefore, by searching over a much richer set, the MAP-based PSD estimator outperforms methods such as PSTH-PSD or PER-PSD. The SS-PSD method also searches for the latent variable x in ℝ, but due to enforcing smoothness of x_k in the time domain, generates spectral estimates which undergo distortion in the spectral domain.

B. Computational comparisons of the PSD estimators

The memory requirement of our approach is 𝒪(KN_max), as we need to store the matrix A ∈ ℝ^{K×(2N_max−1)}. For each EM iteration, we have a concave optimization problem in Eq. (12) which includes a logistic log-likelihood as its objective function plus a quadratic term. As Newton’s method is widely used for logistic as well as approximately quadratic regression problems, we have chosen to use it as our main optimization algorithm. To achieve quadratic convergence, the Newton’s method requires the calculation and inversion of the Hessian of the objective function. Calculation of the Hessian in line 11 of Algorithm 1 requires $\mathcal{O}(K{N}_{max}^2)$ operations as G is diagonal and its inversion in line 12 has a computational cost of the order $\mathcal{O}(N_{max}^3)$. To reduce the computational cost, we can also use quasi-Newton methods such as BFGS. While enjoying a super-linear convergence, these methods require 𝒪(KN_max) operations to calculate the gradient and $\mathcal{O}(N_{max}^2)$ operations for the Hessian approximation [32]. For each EM iteration, the calculation of $E_i^{(r)}$ in Eq. (16) for i = 1, ⋯, 2N_max−1, requires a matrix inversion with a cost of $\mathcal{O}(N_{max}^3)$ operations. However, since we only need the diagonal elements of the inverse, this complexity can be reduced by methods in [33], in case N_max is large. In summary, the computational complexity of Algorithm 2 is $\mathcal{O}(K{N}_{max}^2)$, if Newton’s method is used, under the assumption of K ≥ 2N_max −1. The PSTH-PSD, SS-PSD, and PER-PSD algorithms, however, have a complexity of 𝒪(K log K), thanks to the underlying FFT procedure, in computing K samples of the PSD. There are other steps in our method, such as the cross-validation for the selection of the sparsity hyperparameter γ (Algorithm 3), and constructing confidence intervals (Algorithm 4). However, these steps involve multiple runs of Algorithm 2, and are common in MAP-based inference algorithms such as the SS-PSD algorithm. In light of the preceding comparison, the performance gain of our proposed algorithm comes with an increase of $\mathcal{O}\left(\frac{N_{max}^2}{logK}\right)$ in computational complexity.

VI. Conclusion

In this paper, we considered the problem of computing the power spectral density of the neural covariates underlying spiking data. Existing methods first compute an estimate of the ensemble PSTH or CIF through a temporal smoothing procedure. Then, the PSD estimates of the smoothed PSTH or CIF are computed as the spectral representation of the data. This two-step procedure, although results in a smoothed estimate of the spiking rate, distorts the frequency content of the data. In addition, existing technique do not exploit the underlying sparsity of the frequency content of the data in favor of estimation accuracy.

In order to address these issues, we considered a model where the neuronal ensemble is driven by a harmonic second-order stationary process through a logistic link and according to Bernoulli statistics. We integrated techniques from point process modeling and spectral estimation of second-order stationary process in order to perform the PSD estimation in a Bayesian framework. Our proposed technique enjoys from several features which improve over existing techniques for obtaining spectral representations of neuronal spiking data. First, we directly estimate the PSD from spiking data by regressing the second-order statistics of the underlying process to the observed data, without any time-domain smoothing. Second, motivated by the spectral sparsity of biological signals such as EEG and LFP, we incorporated sparsity-enforcing priors in our PSD estimation. Third, we provided an algorithm for constructing confidence intervals for the PSD estimates.

We compared our proposed method with existing techniques for computing spectral representations of point processes using simulated as well as real data. As for the simulated data, we considered neuronal ensembles driven by oscillatory covariates in the form of dual-tone signals as well as autoregressive processes. Our results showed that the proposed method significantly outperforms the aforementioned existing techniques. Application of our method to multi-unit recordings from a patient undergoing anesthesia showed that the estimated PSD of the neuronal ensemble using our method exhibits a striking resemblance to the PSD of the corresponding LFP signal. This results confirms the findings of [31], which show that the spiking dynamics under general anesthesia is governed by the LFP as a salient neural covariate.

The complexity of the spectra which are identifiable from ensemble neuronal observations is limited by the average spiking rate in PSTH, i.e., the amount of available information for inference purposes. A potential limitation of spectral estimation from binary data is thus in extracting complex (and not necessarily sparse) spectral structures under low neuronal spiking rates. Nevertheless, we have demonstrated the significant performance gain of our proposed method over the existing techniques in estimating sparse narrowband spectral structures detectable under low neuronal spiking rates. This performance gain, however, comes at the cost of a higher computational complexity.

Our technique is particularly useful in analyzing the harmonic structure of spiking activity independently of the local field potentials, without any prior assumption of the spectral spread and content of the underlying neural processes. Our method can be modified in a straightforward fashion to handle other spiking models such as Poisson statistics. In addition, although we have posed the problem in the neuronal spiking data application, our algorithm can be applied to a wide variety of binary data, such as heart beat data, in order to obtain a robust spectral representation. In the spirit of easing reproducibility, we have archived a MATLAB implementation of our method on the open source repository GitHub and made it publicly available [30].