Firing rate estimation using infinite mixture models and its application to neural decoding

We propose a new neural decoding method using infinite mixture models and nonparametric Bayesian statistics. The proposed method improves decoding performance in terms of accuracy and computation speed. We have successfully applied the proposed method to position decoding from spike trains recorded in a rat hippocampus.

Abstract

Neural decoding is a framework for reconstructing external stimuli from spike trains recorded by various neural recordings. Kloosterman et al. proposed a new decoding method using marked point processes (Kloosterman F, Layton SP, Chen Z, Wilson MA. J Neurophysiol 111: 217–227, 2014). This method does not require spike sorting and thereby improves decoding accuracy dramatically. In this method, they used kernel density estimation to estimate intensity functions of marked point processes. However, the use of kernel density estimation causes problems such as low decoding accuracy and high computational costs. To overcome these problems, we propose a new decoding method using infinite mixture models to estimate intensity. The proposed method improves decoding performance in terms of accuracy and computational speed. We apply the proposed method to simulation and experimental data to verify its performance.

NEW & NOTEWORTHY We propose a new neural decoding method using infinite mixture models and nonparametric Bayesian statistics. The proposed method improves decoding performance in terms of accuracy and computation speed. We have successfully applied the proposed method to position decoding from spike trains recorded in a rat hippocampus.

neural systems encode information into sequences of action potentials called spike trains. Spike trains have been studied widely to reveal how neural systems transmit and process sensory information. One direction of research on spike trains is neural decoding. Neural decoding is a framework for predicting external stimuli from spike trains. Neural decoding not only serves as a clue to the relationship between neurons and external stimuli but is also useful for practical applications such as brain-machine interfaces. For previous research on brain-machine interfaces, see Cunningham et al. (2009).

Some studies have modeled the relationship between spike trains occurring from a specific neural area and specific external stimuli, in a specific experimental setting, and performed neural decoding. For example, Brown et al. (1998) predicted the position of a rat from spike trains recorded from place cells in its hippocampus, and Truccolo et al. (2005) predicted the hand velocity of a monkey from spike trains recorded in its motor cortex. In such studies, neural decoding is performed in two steps. First, we express spike trains in terms of a statistical model that has external stimuli as covariates and we estimate parameters of this model. Since spike trains are sequences of pulse waves, point process models are commonly used to model them. Second, using the estimated model, we reconstruct external stimuli from spike trains. In this step, state-space representation is used for expressing the structures of external stimuli and spike trains.

When we perform neural decoding, multiunit recording is used frequently for recording spike trains. The potential obtained by multiunit recording contains spikes emitted by multiple neurons. Thus, before we model spike trains, we must assign each spike to the corresponding neuron that appears to be source of the spike. Assignment of spikes to neurons is conducted by clustering, and this clustering is called spike sorting. In spike sorting, we sort spikes into clusters based on the similarity of their waveforms.

Therefore, the accuracy of spike sorting has a substantial influence on that of neural decoding. However, spike sorting and neural decoding are treated conventionally as different steps. In other words, neural decoding has been conducted on the assumption that spike sorting assigns spikes to neurons exactly. Hence, in neural decoding, errors and loss of information due to spike sorting have been ignored. Recent research has revealed problems resulting from spike sorting. For example, Ventura (2009) indicated that spike sorting causes bias in the intensity estimation of spike trains.

To address this issue, some new decoding methods without spike sorting have been proposed (Deng et al. 2015; Kloosterman et al. 2014; Ventura 2008). In particular, Kloosterman et al. (2014) proposed a decoding method using marked point processes. Marked point processes describe random occurrences of events to which feature values called marks are attached, and these processes can model not only the spike times but also information about spike waveforms. In their approach, we find mathematical functions for the rate of spikes of certain shapes given the position of the rat. This function is called the intensity function. The intensity function can be estimated from the observed firing patterns and the rat trajectory, which are measured simultaneously. Then, in the decoding step, when we see spikes of certain shapes we can calculate the most likely position for the rat with this function. This method greatly improves decoding accuracy and enables us to reconstruct external stimuli online.

In their method, Kloosterman et al. used kernel density estimation to estimate intensity functions of marked point processes. However, the use of kernel density estimation for intensity estimation causes problems. First, there is arbitrariness in the choice of band matrices because there is no efficient way to decide on these values. Second, accuracy is reduced. If the dimensions of the marks are high, then the accuracy of kernel density estimation becomes poor, resulting also in poor decoding accuracy. In addition, they separately estimate density by kernel density estimation and then take the ratio of these densities, so that this two-step procedure results in an amplification of estimation error. Third, since the computational costs for likelihood calculation are large, decoding consumes too much time.

We propose a new decoding method to overcome these problems. Instead of using a nonparametric approach, we take a parametric approach for intensity estimation. When we model spike trains as point processes, we generally assume that each neuron responds to covariate values in a specific range and generates action potentials with characteristic features. For example, in decoding rat movement from place cell activity, we assume that each place cell fires when the animal passes through a specific location and that the waveform of the action potential has characteristic features due to the positional relationship between the cell and the measuring device. Under this assumption, the intensity function defined on the product space of marks and covariates appears to be multimodal, with peaks corresponding to each neuron. Hence, if we model the intensity functions using parametric models, mixture models appear to be suitable. As a simple plan, we can model intensity function as a scalar multiple of a finite Gaussian mixture model and estimate parameters with maximum likelihood estimation. However, likelihood maximization for such models is difficult. Predicting covariate values requires intensity as a function of covariates, not as a function of time. For this reason, we cannot use the commonly used expectation-maximization algorithm for likelihood maximization of mixture models. Consequently, parameter estimation and model choice are difficult when we use finite mixture models.

Therefore, instead of finite mixture models, we consider infinite mixture models. Intensity estimation using infinite mixture models has been proposed in nonparametric Bayes contexts (Ishwaran and James 2004; James 2003; Lo and Weng 1989). This approach uses weighted gamma processes as priors and enables flexible modeling for intensity. In another direction, Taddy and Kottas (2012) proposed intensity estimation using a Dirichlet process prior. This method resembles density estimation using Dirichlet process infinite mixture models. In particular, Kottas et al. (2012) took the latter approach to estimate spike train intensity. However, the latter approach cannot estimate intensity as a function of covariates. Hence, infinite mixture models have not been used for neural decoding.

In this report, we select infinite mixture models using weighted gamma processes. We replace the part of intensity estimation based on kernel density estimation in the existing method with this approach to improve decoding performance. In the proposed method, we estimate the parameters with a blocked Gibbs sampler, as described in Ishwaran and James (2004). Using an infinite mixture model does not require determination of model complexity. Moreover, since estimated intensity is expressed parametrically, the computational cost of decoding is improved. By applying two methods to simulated and experimental data, we show that the proposed method performs better than the existing method in terms of decoding accuracy and computational speed.

This report is organized as follows. In methods, we 1) review existing techniques of neural decoding, explain our problem settings, and review point process models and 2) propose an intensity estimation procedure using infinite mixture models. In results, we 1) apply the proposed and existing methods to simulated data and compare performances and 2) apply the proposed method to experimental data recorded in the hippocampus of a rat. We provide additional comments in discussion.

METHODS

Neural Decoding

Problem settings.

Kloosterman et al. (2014) proposed a decoding method using marked point processes. This method does not require spike sorting in advance, resulting in a substantial improvement in decoding accuracy. Later, Deng et al. (2015) extended this method to a state-space model. In this report, we denote this approach as “the existing method.” Both Kloosterman et al. (2014) and Deng et al. (2015) applied their decoding method to decode a rat’s movement from multiunit activity of its place cells. The place cell is a neuron in the hippocampus that generates spikes when an animal passes a specific location (O’Keefe and Dostrovsky 1971). This is the problem we consider in this report.

Here, we summarize the problem and define some variables. Consider the situation in which the movement of the rat and the spike train of its place cells are observed simultaneously. Let (0, T] be an observation interval. Let t be the time and x_t the position of the rat at t. Although the rat movement {x_t, 0 < t ≤ T} is a continuous process, we cannot generally observe this process continuously. Hence, assume that x_t is observed only at t = r Δ, r = 1, …, R, where Δ is a sampling interval and R = T/Δ. Suppose that the spike train is observed by an extracellular multiunit recording at a high sampling rate compared with Δ. Thus consider that the spike train is obtained continuously. Let κ denote a feature of each spike, for example, peak amplitude, spike width, or principal components, calculated from the waveform of the spike. Suppose that we observe n spikes (t₁, κ₁), …, (t_n, κ_n) in (0, T] and let $x_i\hspace{0.17em}:=\hspace{0.17em}{x}_{t_i}$. In fact, since x_t is observed only at t = rΔ, r = 1, …, R, we cannot know exact values of $x_{t_i}$. Hence, we define x_i as a linear interpolant calculated from x_(r−1)Δ and x_rΔ if t_i ∈ ((r − 1)Δ,rΔ].

State-space representation.

We use state-space representation to decode a rat movement from spike trains of place cells. We consider that the position of the rat is a hidden variable and the spike train is an observable variable. Under this state-space representation, our aim is to calculate the posterior of the rat position at each time step given its spike train.

We assume a state-space structure as

$$\begin{array}{lll}{x}_{r\Delta }|x_{(r-1)\Delta }\hfill & ~\hfill & p\left(x_{r\Delta }|x_{(r-1)\Delta }\right),\hfill \\ \text{spikes\hspace{0.17em}in\hspace{0.17em}}((r-1)\Delta ,r\Delta ]|x_{r\Delta }\hfill & ~\hfill & p\left(\text{spikes\hspace{0.17em}in\hspace{0.17em}}((r-1)\Delta ,r\Delta ]|x_{r\Delta }\right).\hfill \end{array}$$ (1)

Here, spikes in ((r − 1)Δ,rΔ] indicate the set {(t_i,κ_i)|t_i ∈ ((r − 1)Δ,rΔ]}. The system model p(x_rΔ|x_(r−1)Δ) describes how the rat moves over time. The observation model p(spikes in ((r − 1)Δ,rΔ]|x_rΔ) describes how a spike occurs given the present position of the rat. Since these models are generally unknown, we must estimate these models from the observational data. For the system model, Deng et al. (2015) use an AR(1) model such that

$$x_{r\Delta }=a{x}_{(r-1)\Delta }+b{ϵ}_r,{ϵ}_r~\text{Normal}(0,1).$$ (2)

The parameters in Eq. 2 are decided from the true trajectory of the rat. For the observation model, we suppose that the spikes are generated from some point process. In the proposed method, we estimate the intensity function that characterizes the point process using infinite mixture models. We explain details of the latter estimation procedure in Proposed Method.

Under this state-space representation, the one-step predictive distribution $p(x_{r\Delta }|{\mathcal{H}}_{(r-1)\Delta })$ and posterior distribution $p(x_{r\Delta }|{\mathcal{H}}_{r\Delta })$, where ${\mathcal{H}}_{r\Delta }$ is the history of this process up to rΔ, are calculated as follows:

One-step predictive distribution:

$$p(x_{r\Delta }|{\mathcal{H}}_{(r-1)\Delta })={\displaystyle \int p}(x_{r\Delta }|x_{(r-1)\Delta })p(x_{(r-1)\Delta }|{\mathcal{H}}_{(r-1)\Delta })\text{d}{x}_{(r-1)\Delta },$$ (3)

Posterior distribution:

$$p(x_{r\Delta }|{\mathcal{H}}_{r\Delta })=\frac{p(\text{spikes\hspace{0.17em}in\hspace{0.17em}}((r-1)\Delta ,r\Delta ]|x_{r\Delta })p(x_{r\Delta }|{\mathcal{H}}_{(r-1)\Delta })}{{\displaystyle \int p}(\text{spikes\hspace{0.17em}in\hspace{0.17em}}((r-1)\Delta ,r\Delta ]|x_{r\Delta })p(x_{r\Delta }|{\mathcal{H}}_{(r-1)\Delta })\text{d}{x}_{r\Delta }}\mathrm{.}$$ (4)

Given an initial distribution of x₀, we can calculate the posterior distribution of x_rΔ, r = 1, …, R by updating distributions in accordance with Eqs. 3 and 4.

In summary, decoding is conducted in two steps as follows: 1) Estimate the intensity function: model spike trains of place cells as marked point processes, including the position of the rat as a covariate, and estimate intensity. 2) Calculate the posterior of the rat movement: reconstruct the rat trajectory using state-space representation whose observable variable is the spike train and whose state variable is the position.

Point process.

A point process is a stochastic process that expresses the random occurrence of events. Modeling a time series as a point process requires the use of a proper representation of point processes depending on the type of the data. When each event in the data has some characteristic values, marked point process representation is suitable for modeling the series of events. Marked point processes describe random occurrences of events to which feature values called marks are attached. For example, such processes are used for expressing earthquakes or financial transactions. In an earthquake analysis, the magnitude of each quake can be considered as a mark.

Any orderly point process can be characterized by its conditional intensity function (see, e.g., Daley and Vere-Jones 2003). A conditional intensity function, defined as the instantaneous conditional risk of the occurrence of an event given its previous history, determines the probability structure of a point process uniquely. Hence, instead of estimating the probability structure of point processes directly, we generally estimate the conditional intensity function from data.

The conditional intensity function of a marked point process is defined as follows:

$$\begin{array}{l}\lambda (t,\kappa |x_t,{\mathcal{H}}_t)\\ \hspace{0.17em}:=\hspace{0.17em}\underset{\Delta t,\Delta \kappa \to 0}{\lim }\frac{\text{E}\left[\begin{array}{c}\text{number\hspace{0.17em}of\hspace{0.17em}events\hspace{0.17em}whose\hspace{0.17em}mark}\\ \text{\hspace{0.17em} \hspace{0.17em}within\hspace{0.17em}}(\kappa ,\kappa +\mathrm{\Delta }\kappa ]\text{occurred\hspace{0.17em}at\hspace{0.17em}}(t,t+\Delta t]\end{array}|x_t,{\mathcal{H}}_t\right]}{\Delta t\Delta \kappa },\end{array}$$ (5)

where $\kappa \hspace{0.17em}\in \hspace{0.17em}\mathcal{K}$ is a mark attached to each event, x_t is a covariate, and ${\mathcal{H}}_t$ is the history of this process up to t. This ${\mathcal{H}}_t$ contains all information regarding previous occurrences of events and covariate values. The integral of $\lambda (t,\kappa |x_t,{\mathcal{H}}_t)$ with respect to κ corresponds to the intensity function of the point process ignoring marks:

$$\Lambda (t|x_t,{\mathcal{H}}_t):={\displaystyle {\int }_{\mathcal{K}}\lambda }(t,\kappa |x_t,{\mathcal{H}}_t)\text{d}\kappa \mathrm{.}$$ (6)

This $\Lambda (t|x_t,{\mathcal{H}}_t)$ is called the intensity function of the ground process of this point process. For example, the intensity function of an inhomogeneous marked Poisson process modulated by a covariate x_t can be expressed by some function λ(κ|x) as

$$\lambda \left(t,\kappa |x_t,{\mathcal{H}}_t\right)=\lambda (\kappa |x_t),$$

$$\Lambda \left(t|x_t,{\mathcal{H}}_t\right)={\displaystyle {\int }_{\mathcal{K}}\lambda }(\kappa |x_t)\text{d}\kappa \mathrm{.}$$ (7)

The likelihood of the point process is obtained by using the conditional intensity function. Let (0, T] be an observation interval. Suppose that we observe (t₁, κ₁), …, (t_n, κ_n) from a marked point process with the intensity function $\lambda (t,\kappa |x_t,{\mathcal{H}}_t)$. Then, the likelihood is given by

$$\mathcal{L}=\exp \left(-{\displaystyle {\int }_0^T{\displaystyle {\int }_{\mathcal{K}}\lambda }}(t,\kappa |x_t,{\mathcal{H}}_t)\text{d}\kappa \text{d}t\right){\displaystyle \prod _{i=1}^n\lambda }(t_i,{\kappa }_i|x_{t_i},{\mathcal{H}}_{t_i})$$

$$=\exp \left(-{\displaystyle {\int }_0^T\Lambda }(t|x_t,{\mathcal{H}}_t)\text{d}t\right){\displaystyle \prod _{i=1}^n\lambda }(t_i,{\kappa }_i|x_{t_i},{\mathcal{H}}_{t_i})\mathrm{.}$$ (8)

This equation can be interpreted as follows. First, we divide (0,T] into intervals so that the length of each interval is Δ. For a sufficiently small Δ, the probability of an event with a mark κ occurring in (t, t + Δ] is $\lambda (t,\kappa |x_t,{\mathcal{H}}_t)\Delta$, whereas the probability of the event not occurring is $1-{\displaystyle {\int }_{\mathcal{K}}\lambda }(t,\kappa |x_t,{\mathcal{H}}_t)\Delta$. Multiplying this probability for all intervals and taking Δ → 0, we get Eq. 8.

Proposed Method

As mentioned in the introduction to this article, we estimate an intensity function using infinite mixture models. In this section, we review this estimation procedure and present our model formulation.

Similarly to the method proposed by Kloosterman et al. (2014), we assume that the spike train is generated from a marked inhomogeneous Poisson process modulated by x_t:

$$\lambda \left(t,\kappa |x_t,{\mathcal{H}}_t,\xi \right)=\lambda (\kappa |x_t,\xi ),$$

$$\Lambda \left(t|x_t,{\mathcal{H}}_t,\xi \right)=\Lambda \left(x_t,\xi \right)={\displaystyle {\int }_{\mathcal{K}}\lambda }(\kappa |x_t,\xi )\text{d}\kappa ,$$ (9)

where λ(κ|x,ξ) is an unknown function with parameter ξ.

Using marked point processes does not require spike sorting beforehand. Hence, this approach avoids many problems due to spike sorting and greatly improves decoding accuracy. However, since modeling a spike train as an marked point process has been introduced only recently, there is not yet sufficient knowledge of how to estimate λ(·). Kloosterman et al. (2014) used kernel density estimation to construct an estimator

$$\widehat{\lambda }(\kappa |x)=\frac{n}{T}·\frac{\frac{1}{n}{\displaystyle \sum _{i=1}^n{K}_{H_{\kappa x}}}\left(\left(\begin{array}{c}\kappa \\ x\end{array}\right)-\left(\begin{array}{c}{\kappa }_i\\ x_i\end{array}\right)\right)}{\frac{1}{R}{\displaystyle \sum _{r=1}^R{K}_{{\tilde{H}}_x}}\left(x-x_{r\Delta }\right)}$$ (10)

for λ(·). Here, $K_{H_{\kappa x}}$ and $K_{{\tilde{H}}_x}$ are nonnegative kernel functions with band matrices H_κx and ${\tilde{H}}_x$, respectively. Using kernel density estimation, we can represent the complex structure of the relation between covariates and marks without any prior information. Although this approach is pioneering, using kernel density estimation for intensity estimation could cause several problems. First, there is no sufficient method for determining the band matrices H_xκ and ${\tilde{H}}_x$. When we use a maximum-likelihood estimator for these values, we need to calculate likelihood for a large number of band matrices along the likelihood maximization procedure. However, the computational cost of likelihood calculation is proportional to the square of the number of spikes, so that maximization takes an excessive amount of time. Moreover, there is the possibility of converging to a bad local solution. Kloosterman et al. (2014) also referred to the difficulty of determining an optimal full multivariate bandwidth matrix and suggested the use of simplified parametrization, for example, restriction of this parameter to the diagonal matrix. However, the appropriate type of simplification is not known beforehand. Second, the accuracy can become poor in some situations. When the dimensions of the marks are high, the accuracy of kernel density estimation becomes poor, with the result that decoding accuracy also becomes poor. Moreover, since this procedure takes the ratio of two separately estimated density functions, the estimated intensity function $\widehat{\lambda }(\kappa |x)$ sometimes takes unstable values. This problem has been recognized in the field of statistics, e.g., Ripley (2008, Chapter 6), and alternative methods to estimate the conditional probability or the density ratio directly have been investigated in the field of statistics and machine learning. Third, since the computational costs for calculating $\widehat{\lambda }(\kappa |x)$ are proportional to the data size, the existing method has a trade-off between decoding accuracy and computational cost.

Now, we choose an appropriate parametric model for λ(κ|x,ξ). As mentioned above, we generally assume that each neuron responds to specific values of external stimuli and generates action potentials with characteristic features. Under this assumption, the intensity function defined on the product space of mark and covariate is multimodal, with peaks corresponding to each neuron. Hence, mixture models are suitable for λ(κ|x,ξ). Here, we consider the infinite mixture model in a nonparametric Bayes context, as proposed by Ishwaran and James (2004).

We assume that λ(κ|x,ξ) is expressed by a nonnegative kernel function k that has a parameter θ ∈ Θ and a finite measure ξ defined on Θ as

$$\lambda (\kappa |x,\xi )={\displaystyle {\int }_{\Theta }k}(\kappa ,x|\theta )\xi (\text{d}\theta )\mathrm{.}$$ (11)

Under this assumption, the likelihood of ξ given (t_i,κ_i), i = 1, …, n is expressed as

$$\mathcal{L}(\xi )=\exp \left(-{\displaystyle {\int }_0^T{\displaystyle {\int }_{\mathcal{K}}\lambda }}(\kappa |x_t,\xi )\text{d}\kappa \text{d}t\right){\displaystyle \prod _{i=1}^n\lambda }({\kappa }_i|x_{t_i},\xi )$$

$$=\exp \left(-{\displaystyle {\int }_{\Theta }{\displaystyle {\int }_0^T{\displaystyle {\int }_{\mathcal{K}}k\left(\kappa ,x_t|\theta \right)}}}\mathrm{d}\kappa \mathrm{d}t\xi \left(\mathrm{d}\theta \right)\right)·{\displaystyle \prod _{i\text{=}1}^n{\displaystyle \int k}}\left({\kappa }_i,x_i|\theta \right)\xi \left(\mathrm{d}\theta \right)\mathrm{.}$$ (12)

All we need to do is to estimate ξ from data.

James (2003) showed that when likelihood is expressed as Eq. 12, a weighted gamma process is a conjugate prior for ξ and the posterior of ξ is also a weighted gamma process.

A weighted gamma process is a random variable taking values on a space of finite measures on Θ. Let α be a finite measure and β a positive integrable function with respect to α defined on Θ. We call γ_α a gamma process with shape parameter α if for set $A\hspace{0.17em}\subset \hspace{0.17em}\Theta$,

$${\gamma }_{\alpha }(A)~\text{Gamma}(\alpha (A))$$ (13)

where Gamma(α(A)) is a gamma distribution with shape parameter α(A) and scale parameter 1, whose density is

$$\frac{x^{\alpha (A)-1}}{\Gamma (\alpha (A))}{\text{e}}^{-x},x>0.$$ (14)

Furthermore, we call ξ_α,β a weighted gamma process with shape parameter α and scale parameter β, if for set $A\hspace{0.17em}\subset \hspace{0.17em}\Theta$,

$${\xi }_{\alpha ,\beta }(A)={\displaystyle {\int }_A\beta }(\theta ){\gamma }_{\alpha }(\text{d}\theta )\mathrm{.}$$ (15)

We denote a distribution of ξ_α,β by $\mathcal{G}(\text{d}\xi |\alpha ,\beta )$. It is known that with probability 1, weighted gamma processes take values on the space of finite discrete measure defined on Θ. In other words, one sample from a weighted gamma process can be expressed by an infinite number of atoms and weights. Therefore, with probability 1, ξ and λ(κ|x,ξ) in Eq. 11 can be expressed as

$$\xi ={\displaystyle \sum _{j=1}^{\infty }{\pi }_j}{\delta }_{\theta (j),}$$

$$\lambda (\kappa |x,\xi )={\displaystyle \sum _{j=1}^{\infty }{\pi }_j}k(\kappa ,x|\theta (j))$$ (16)

where {θ(j)} are atoms of ξ and {π_j} are weights assigned to these atoms. This is why this model is called an extension of finite mixture models. Intuition of our parametric expression for λ(·) when using Gaussian kernel N(κ,x|μ,Λ) for kernel function is shown in Fig. 1.

Fig. 1.

Intuition of our parametric expression for intensity function when using Gaussian kernel N(κ,x|μ,Λ) for kernel function. In this case, θ corresponds to the pair of mean vector μ and precision matrix Λ and Θ represents the whole of these elements. There is 1-to-1 correspondence between discrete measure ξ and intensity function λ(·).

Let θ_i ∈ Θ, i = 1, …, n be augmented variables for (t_i, κ_i), i = 1, …, n, where n is the number of spikes, and let θ = (θ₁, …, θ_n). The parameter θ corresponding to the neuron-generated ith spike is θ_i. Mathematically speaking, each θ_i indicates the parameter of the kernel function k that generates (t_i, κ_i). Then, the likelihood of ξ given (t_i, κ_i, θ_i), i = 1, …, n is

$$\mathcal{L}(\xi )=\exp \left(-{\displaystyle {\int }_{\Theta }{\displaystyle {\int }_0^T{\displaystyle {\int }_{\mathcal{K}}k\left(\kappa ,x_t|\theta \right)}}}\mathrm{d}\kappa \mathrm{d}t\xi \left(\mathrm{d}\theta \right)\right)·{\displaystyle \prod _{i\text{=}1}^n{\displaystyle \int k}}\left({\kappa }_i,x_i|{\theta }_i\right)\xi \left(\mathrm{d}{\theta }_i\right)\mathrm{.}$$ (17)

We set the prior for ξ as a weighted gamma process such as

$$\mathcal{G}(\text{d}\xi |\alpha ,\beta )$$ (18)

where α and β are hyperparameters.

Now, we summarize our model formulation explicitly. We define kernel function k as

$$k(\kappa ,x|\theta )=\text{N}(\kappa |{\mu }^{\kappa },{\Lambda }^{\kappa })\text{N}(x|{\mu }^x,{\Lambda }^x)$$ (19)

where θ = (μ^κ, Λ^κ, μ^x, Λ^x) and N(·|·) is the Gaussian kernel

$$\text{N}(x|\mu ,\Lambda )=\frac{{\left|\Lambda \right|}^{\frac{1}{2}}}{{(2\pi )}^{\frac{d}{2}}}\exp \left(-\frac{1}{2}{(x-\mu )}^{\text{T}}\Lambda (x-\mu )\right)\mathrm{.}$$ (20)

From the viewpoint of sampling efficiency, we define hyperparameters α, β as

$$\alpha ({\mu }^{\kappa },{\Lambda }^{\kappa },{\mu }^x,{\Lambda }^x)={\alpha }_0{G}_0^{\kappa }({\mu }^{\kappa },{\Lambda }^{\kappa })G_0^x({\mu }^x,{\Lambda }^x),$$

$$\beta ({\mu }^{\kappa },{\Lambda }^{\kappa },{\mu }^x,{\Lambda }^x)={\beta }_0,$$

$$G_0^{\kappa }({\mu }^{\kappa },{\Lambda }^{\kappa })=\text{N}\left({\mu }^{\kappa }|m^{\kappa },{\rho }^{\kappa }{\Lambda }^{\kappa }\right)·\text{W}\left({\Lambda }^{\kappa }|{\nu }^{\kappa },S^{\kappa }\right),$$

$$G_0^x({\mu }^x,{\Lambda }^x)=\text{N}\left({\mu }^x|m^x,{\rho }^x{\Lambda }^x\right)·\text{W}\left({\Lambda }^x|{\nu }^x,S^x\right)\mathrm{.}$$ (21)

Here, α₀, β₀ > 0 are scalars,

$$\text{W}(\Lambda |\nu ,S)=\frac{{\left|\Lambda \right|}^{\frac{\nu -d-1}{2}}}{|S{|}^{\frac{\nu }{2}}{2}^{\frac{\nu d}{2}}{\pi }^{\frac{d(d-1)}{4}}{\displaystyle \prod _{i=1}^d\Gamma }\left(\frac{\nu +1-i}{2}\right)}\exp \left(-\frac{1}{2}\text{tr}\left(S^{-1}\Lambda \right)\right)$$ (22)

is the probability density function of the Wishart distribution, m^κ, m^x are location vectors, S^κ, S^x are scale matrices, ρ^κ, ρ^x > 0 are scalars, and ν^κ > dim(κ) − 1, ν^x > dim(x) − 1 are also scalars. These parameters are hyperparameters whose values must be determined in advance.

For (μ^κ, Λ^κ) and (μ^x, Λ^x), we adopt normal-Wishart distributions as priors. The normal-Wishart distribution is the conjugate prior of a multivariate normal distribution with unknown mean and unknown precision matrix. In the proposed method, selecting the normal-Wishart distribution as the prior simplifies sampling procedures of μ^κ, Λ^κ. However, there is no conjugate prior for μ^x, Λ^x. Hence, there is no special reason for selecting the normal-Wishart distribution as the prior for μ^x, Λ^x.

Since x_t is not observed continuously, we approximate ${\displaystyle {\int }_0^{T}k}(\kappa ,x_t|\theta )\text{d}t$ using x_rΔ, r = 1, …, R as ${\displaystyle \sum _{r=1}^{R}k}(\kappa ,x_{r\Delta }|\theta )\Delta$, where Δ = T/R.

For notational simplicity, we use a general notation for k, α, β in the remainder of this section. In the above situation, the posterior of (θ,ξ) given ${\{(t_i,{\kappa }_i)\}}_{i=1}^n$, denoted as $\pi (\text{d}\theta ,\text{d}\xi |{\{(t_i,{\kappa }_i)\}}_{i=1}^n)$, is expressed as

$$\pi \left(\mathrm{d}\mathbf{\theta },\mathrm{d}\xi |{\left\{\left(t_i,{\kappa }_i\right)\right\}}_{i\text{=}1}^n\right)\text{=}\mathcal{G}\left(\mathrm{d}\xi |\alpha \text{+}{\displaystyle \sum _{i\text{=}1}^n{\delta }_{{\theta }_i}},\beta *\right)\pi \left(\mathrm{d}\theta |{\left\{\left(t_i,{\kappa }_i\right)\right\}}_{i\text{=}1}^n\right)$$ (23)

where

$$\beta *\left(\text{θ}\right)\text{=}\frac{\beta \left(\theta \right)}{1\text{+}\beta \left(\text{θ}\right)f\left(\theta \right)},$$

$$f\left(\theta \right)\text{=}{\displaystyle {\int }_0^T{\displaystyle {\int }_{\mathcal{K}}k\left(\kappa \text{,}{x}_t|\theta \right)}}\text{d}\kappa \text{d}t,$$

$$\pi (\text{d}\mathbf{\theta }|{\{(t_i,{\kappa }_i)\}}_{i=1}^n)\propto m(\text{d}\mathbf{\theta })\left({\displaystyle \prod _{i=1}^n{\beta }^{*}}({\theta }_i)k({\kappa }_i,x_i|{\theta }_i)\right),$$

$$m(\text{d}\mathbf{\theta })={\displaystyle \int {\displaystyle \prod _{i=1}^n\xi }}(\text{d}{\theta }_i)\mathcal{G}(\text{d}\xi |\alpha ,1)={\displaystyle \prod _{i=1}^n\left(\alpha +{\displaystyle \sum _{j=1}^{i-1}{\delta }_{{\theta }_j}}\right)}(\text{d}{\theta }_i),$$ (24)

(see James 2003).

Therefore, when we want to know the posterior of (θ,ξ), it is enough to consider the right-hand side of Eq. 23. Moreover, since m(dθ) in Eq. 24 is the nonnormalized Pólya urn density, the conditional density of θ_i given θ₁, …, θ_i-1, θ_i+1, …, θ_n and ${\{(t_i,{\kappa }_i)\}}_{i=1}^n$ can be calculated. Hence, we can sample θ from $\pi (\text{d}\mathbf{\theta }|{\{(t_i,{\kappa }_i)\}}_{i=1}^n)$ by a Pólya urn sampler that is frequently used in Dirichlet process problems.

However, a Pólya urn sampler gives strongly correlated samples, with the result that the mixing rate is poor. Moreover, since $\mathcal{G}(·|\alpha ,\beta )$ is defined on an infinite-dimensional space, we need some sort of approximation to sample ξ. To address these issues, we use a blocked Gibbs sampler (Ishwaran and James 2004). The principal idea of a blocked Gibbs sampler is to approximate the prior $\mathcal{G}(·|\alpha ,\beta )$ with a mixture of gamma processes ${\mathcal{G}}_N(·|\alpha ,\beta )$. Details of the theoretical derivation of the blocked Gibbs sampler and the sampling procedure in our problem settings are provided in appendix a.

Using the blocked Gibbs sampler, we generate samples from the posterior of ξ given data. We then calculate the posterior mean λ(κ|x, ξ), using these samples. In the decoding step, we use this posterior mean as the estimator of λ(·).

RESULTS

Simulation Study

In this section, we apply the proposed and existing methods to simulated data and compare their performance.

Consider that the activity of five neurons is observed simultaneously. Let κ be a feature of each spike and assume that the spike train of the kth neuron is generated from an inhomogeneous marked Poisson process modulated by external stimuli x_t whose intensity is

$${\alpha }_k\hspace{0.17em}\text{exp}\left(-\frac{{(\kappa -{\mu }_k^{\kappa })}^2}{2{\sigma }_k^{\kappa }{}^2}-\frac{{(x_t-{\mu }_k^x)}^2}{2{\sigma }_k^x{}^2}\right)\mathrm{.}$$ (25)

Here, α_k is a maximum rate, ${\mu }_k^{\kappa }$ is the mean of the density for κ, ${\mu }_k^x$ is a stimulus value to which this neuron reacts most extremely, and ${\sigma }_k^{\kappa }$, ${\sigma }_k^x$ are scale parameters for κ and x, respectively. We also assume that the spike train of each neuron is independent given x_t. Hence, the overall intensity function is expressed as

$$\lambda (\kappa |x_t)={\displaystyle \sum _{k=1}^5{\alpha }_k}\hspace{0.17em}\text{exp}\left(-\frac{{(\kappa -{\mu }_k^{\kappa })}^2}{2{\sigma }_k^{\kappa }{}^2}-\frac{{(x_t-{\mu }_k^x)}^2}{2{\sigma }_k^x{}^2}\right)\mathrm{.}$$ (26)

The parameter settings in Eq. 26 are shown in Table 1. We generated spike trains for 150 s in this situation. A summary of the data set is shown in Fig. 2.

Table 1.

True parameter setting for generating a data set in simulation study

k	αk	${\mu }_k^{\kappa }$	${\sigma }_k^{\kappa }{}^2$	${\mu }_k^x$	${\sigma }_k^x{}^2$
1	10	4	2	−40	180
2	10	1	1	−10	100
3	10	−3	5	0	240
4	10	−5	1	0	1,000
5	10	−2	1	20	90

Fig. 2.

Summary of the data set. A: covariate x_t used for generating data. Time interval Δ is 0.01 s. B: histogram of x_rΔ, r = 1, …, R. C: spike train from 0 to 5 s. D: 2-dimensional histogram of (κ_i, x_i), i = 1, …, n.

We estimated the intensity function using the interval from 0 to 100 s. In the proposed method, we set hyperparameters in Eq. 21, as follows:

$${\alpha }_0=10,{\beta }_0={10}^5,$$

$$m^{\kappa }=0,{\rho }^{\kappa }=1,{\nu }^{\kappa }=1,S^{\kappa }=1,$$

$$m^x=0,{\rho }^x=1,{\nu }^x=1,S^x=0.1.$$ (27)

Under these settings, we generated 2,000 samples, using a blocked Gibbs sampler with N = 10. We then selected 50 samples from the last 1,000 samples and calculated the posterior mean of the intensity function from these samples as

$$\widehat{\lambda }(\kappa |x)=\frac{1}{50}{\displaystyle \sum _{j=1}^{50}\lambda }(\kappa |x,{\xi }^{(j)})$$ (28)

where ξ^(j) is the jth sample of ξ. In the existing method, we selected Gaussian kernels as kernel functions $K_{H_{\kappa x}},K_{{\tilde{H}}_x}$. As mentioned above, optimizing the band matrices H_κx, ${\tilde{H}}_x$ is difficult in actual application scenes. Hence, we determined band matrices using Scott’s rule (Scott 1979). With Scott’s rule, band matrix H is calculated as

$$H=n^{-1/(d+4)}\tilde{\mathrm{\Sigma }}$$ (29)

where n is a sample size, d is a dimension and $\tilde{\Sigma }$ is a sample covariance matrix.

The results of intensity estimation are shown in Fig. 3. The figure shows that the intensity estimated with the proposed method is similar to the true intensity. On the other hand, the intensity estimated with the existing method is wavy, and some peaks are connected.

Fig. 3.

Results of intensity estimation. A: true intensity. B: proposed method. C: existing method.

Using the estimated intensity, we reconstructed x_t for the interval ranging from 100 to 150 s. Since the observation model (Eq. 1) is non-Gaussian, the one-step predictive and posterior distributions in Eq. 3 and Eq. 4 cannot be expressed in closed form. Hence, we need to approximate these distributions at each step. In previous research, Laplace approximation or a particle filter has commonly been used for this approximation (Brown et al. 1998; Eden et al. 2004; Koyama et al. 2010). Here, we used a particle filter with a particle population of 1,000. We assume that Δ = 0.01 and that the system model p(x_rΔ|x_(r−1)Δ) is AR(1), that is,

$$x_{r\Delta }=a{x}_{(r-1)\Delta }+b{ϵ}_r,{ϵ}_r~\text{Normal}(0,1)\mathrm{.}$$ (30)

We calculated maximum-likelihood estimates (MLEs) for a and b using x_t for the interval ranging from 0 to 100 s, and we reset a and b as this MLE of a and 10 times the MLE of b, respectively.

Decoding results are shown in Fig. 4. As a reference, we also show the results of the traditional decoding approach using spike sorting that was proposed by Brown et al. (1998). Details of this decoding method are given in appendix b. Figure 4 shows that use of the marked point process reduces decoding error and that the accuracy of the proposed method is superior to that of the existing method. To compare their performance quantitatively, we calculated two measures of prediction accuracy for the decoding results. One was the root-mean-squared error, and the other was the percentage of time that true positions were covered by the prediction intervals. These values are presented in Table 2. This table also demonstrates the superiority of the proposed method.

Fig. 4.

Decoding results for simulated data. Black line represents true values, and blue line represents predicted values. Light blue region represents the 95% prediction interval. A: proposed method. B: existing method. C: sorted decoding.

Table 2.

Summary of decoding performance

	Proposed Method	Existing Method	Sorted Decoding
Root-mean-squared error	4.206	7.021	11.44
% of time within 95% prediction interval	96.08	83.32	46.90

Values are root-mean-squared error between true positions and estimated values and percentage of time that true positions were covered by 95% prediction intervals.

Application to Position Decoding from Place Cells’ Activity

In this section, we apply the proposed method to spike trains recorded in a rat hippocampus, using data from Mizuseki et al. (2013). These data contain multiunit recordings in the hippocampus of a rat running back and forth on a linear track. Details of this data are provided in appendix c. We have also applied our method to other data sets such as Grosmark et al. (2016) and confirmed the effectiveness of our method; we do not include details here.

In principle, we should also apply the existing method to these data and compare performance. However, the excessive computational cost of the existing method precludes its application here. Hence, we only show the results of the proposed method in this section. A discussion of computational complexity is included in discussion.

Spike trains in this data are recorded with silicon probes with eight shanks. Similarly to Deng et al. (2015), we assume that each shank records spike trains from different neurons. In other words, we assume that spikes in ((r − 1)Δ,rΔ] at each shank are independent given x_rΔ, that is, the joint likelihood is

$$p(\text{spikes\hspace{0.17em}in\hspace{0.17em}}((r-1)\Delta ,r\Delta ]|x_{r\Delta })$$

$$={\displaystyle \prod _{l=1}^{8}p}(\text{spikes\hspace{0.17em}in\hspace{0.17em}}((r-1)\Delta ,r\Delta ]\text{\hspace{0.17em}at\hspace{0.17em}}l\text{th\hspace{0.17em}shank}|x_{r\Delta })\mathrm{.}$$ (31)

We estimated the intensity function with the proposed method, using data from the interval ranging from 100 to 700 s. We used the maximum and minimum amplitudes of waveforms as marks in this experiment. It should be noted that, if there are other characteristic values such as waveform width or principal components, the proposed method can use such information. For priors, we set the parameters in Eq. 21 as

$$\begin{array}{c}{\alpha }_0=100,{\beta }_0=10,\\ m^{\kappa }=\left(\begin{array}{c}0\\ 0\end{array}\right),{\rho }^{\kappa }=1,{\nu }^{\kappa }=2,S^{\kappa }=\left(\begin{array}{cc}0.5& 0\\ 0& 0.5\end{array}\right),\\ m^x=0,{\rho }^x=1,{\nu }^x=1,S^x=\mathrm{0.1.}\end{array}$$ (32)

Under these settings, we generated 20,000 samples, using a blocked Gibbs sampler with N = 20. We then selected 50 samples from the last 5,000 samples and calculated the posterior mean of the intensity function from these samples. Using the estimated intensity function, we reconstructed the rat movement for the interval ranging from 700 to 750 s and the interval ranging from 750 to 800 s, respectively. Similarly to what we described in Simulation Study, we used Δ = 0.0256 and the AR(1) model for the system model. We calculated MLEs for a and b using x_t for the interval ranging from 100 to 700 s, and we reset a and b as this MLE of a and 20 times the MLE of b. In the decoding step, we used a particle filter with a particle population of 10,000.

Decoding results are shown in Fig. 5. The root-mean-squared error was 78.72 cm for t ∈ (700, 750) and 72.40 cm for t ∈ (750, 800). Although the proposed method missed the true positions in some intervals, such as t ∈ (712, 718), so that the root-mean-squared error was large, on the whole it can predict true values accurately.

Fig. 5.

Decoding results for experimental data. Black line represents true values, and blue line represents predicted values. Light blue region represents the 95% prediction interval. A: t ∈ (700, 750). B: t ∈ (750, 800).

The following hypotheses may explain why the method misses some true values.

The proposed method is too simple to express all of the characteristics of a place cell.

For example, the proposed method cannot consider the directional dependence of a place cell. McNaughton et al. (1983) suggested that when a rat is running back and forth on a linear track, place cells spike differently depending on the direction of movement of the rat. Therefore, if we only observe spikes from neurons that react to movement in one direction, predicted values may miss the true values when the rat moves in an opposite direction. In this regard, Fig. 5 shows that when the rat returns (from moving in a positive to a negative direction), predicted values veer away from the true values. It may be possible to prevent such a sudden change of direction by using AR(2) model for the system model instead of AR(1).

The particle filter failed to calculate the true posterior.

Degeneration of particles may have occurred around t ≈ 715, 728, 743, 758, 770, 788.

Spiking activities of some place cells whose information is necessary for reconstructing rat trajectory could not be observed during the measurement phase.

Consider that measuring devices failed to measure the activity of neurons that spikes when x_t ∈ (−100, −50). Then, the predicted values miss the true values because of the loss of information around this area.

There was a difference between the position recognized by the rat and the true position.

The trajectory calculated from the spike train is the trajectory recognized by the rat. Hence, if the rat is confused about its position along the trial, predicted values differ from true values even if the method is correct.

It should be noted that these problems are not specific to the proposed method and can also occur when we use the existing method.

As a reference, we also reconstructed rat movements from the same data set with a spike sorting approach. We used the spike assignment calculated by Mizuseki et al. (2013). The root-mean-squared error of this approach was 147.82 cm for t ∈ (700, 750), which is larger than the root-mean-squared error of the proposed method. This result shows the advantage of using a marked point process in experimental application. We omit other details of this result.

DISCUSSION

Kloosterman et al. (2014) proposed a new decoding method using marked point processes. In their method, they estimate intensity functions with kernel density estimation. However, kernel density estimation can result in problems such as low decoding accuracy and is associated with a high computational cost. To overcome these problems, we propose a new decoding method using infinite mixture models. The proposed method performs better than the existing method in terms of decoding accuracy and efficiency. We applied the proposed method to simulated and experimental data. The proposed method predicted covariate values accurately for both of them.

Here, we discuss the computational complexity of these methods. In the estimation step, the proposed method calculated the posterior with a blocked Gibbs sampler. Let L be the number of sampling steps in a blocked Gibbs sampler, n be the number of spikes in the estimation step, R be the number of sampling times of x_t, and N be the number of Gaussian kernels. The bottleneck of the computational cost when using a blocked Gibbs sampler comes from the calculation of Dirichlet variable W. At each step, calculation of W costs O((n + R) N). Over L steps, the time complexity is O((n + R) NL). In Application to Position Decoding from Place Cells’ Activity, we implemented a blocked Gibbs sampler in Julia, which was processed with a 2 × 2.93 GHz 6-Core Intel Xeon with 64 GB RAM. In this environment, observable run times were ∼4 h for 10 min worth of data, where n ≈ 30,000, R ≈ 30,000, N = 20, and L = 20,000. These run times can be shortened by implementing our method in parallel. On the other hand, the existing method does not require any calculation for the estimation step.

The computational cost of a blocked Gibbs sampler is much lower than that of a particle filter in the proposed method. Therefore, it is sufficient to consider the computational cost required for the decoding step in practical application scenes. In the decoding step, the difference in computational costs between the proposed method and the existing method comes from calculation of the intensity function. In our setting, these costs are proportional to the number of Gaussian kernels that are used to express the intensity function. Let J be the number of selected samples for calculating the posterior of the intensity function. Then calculation of the intensity function for one particle costs O(NJ) in the proposed method, while this is O(n) in the existing method. Hence, the proposed method is more efficient than the existing method when NJ ≪ n. In Application to Position Decoding from Place Cells’ Activity, we set N = 20, J = 50, and the number of spikes n occurring in the interval ranging from 100 to 700 s at one shank was roughly 30,000. Therefore, the computational cost of the existing method is ~30 times higher than that of the proposed method.

In the traditional decoding method using spike sorting, we have to assign each spike to at least one cluster even if its waveform matches multiple clusters or does not match any cluster at all. Therefore, misassignment of spikes often occurs, and this affects decoding accuracy. Using a marked point process can address such uncertainty so we can reduce the decoding error. Moreover, since this approach avoids spike sorting that involves batch processing, we can construct an online decoder in a real sense. Hence, this approach could have a broader impact on laboratory experimentation and the development of brain-machine interfaces.

However, the proposed method has some problems that must be solved for practical use. First, it requires high computational cost for Gibbs sampling. In particular, since we use a Metropolis-Hastings step, the mixing rate becomes poor. This necessitates finding more efficient sampling procedures. Second, there is arbitrariness in determining hyperparameters. In particular, since α₀ and β₀ control the number of components, it is important to determine these values from data properly. Hence, we need to consider another sampling procedure that can sample α₀, β₀ simultaneously or a variational inference approach that can determine these values in an empirical Bayes context.

In terms of model validity, the proposed method fails to express some features of neuron activity. We made an assumption that different neurons’ activities are independent given the external stimuli. However, this assumption often does not hold. Even in the case of place cells, there are oscillations that span the hippocampus: overall firing rate fluctuates even if the rat stays in the same position. Introducing the correlation structure to the proposed method is also our future work.

The proposed method can be extended to other types of data, not only those obtained by multiunit recording. One important example is calcium imaging. When we use calcium imaging data for neural decoding, we have to extract the information associated with each neuron, such as shape and spike time, from a raw movie beforehand. In a similar way to spike sorting, this preprocessing causes errors and loss of information. Hence treating a calcium movie as a marked point process, we may be able to construct an accurate and efficient decoding method for calcium imaging data. The proposed method is appropriate for this purpose, whereas the existing method cannot solve such a high-dimensional problem.

GRANTS

This work was supported by MEXT KAKENHI Grant 16H06533.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

R.S. and F.K. conceived and designed research; R.S. analyzed data; R.S. drafted manuscript; R.S. and F.K. edited and revised manuscript; R.S. and F.K. approved final version of manuscript.

APPENDIX A: DETAILS OF GIBBS SAMPLING PROCEDURE IN PROPOSED METHOD

Derivation of Blocked Gibbs Sampler

Here, we give an outline of the derivation of the blocked Gibbs sampler (Ishwaran and James 2004). A more detailed, intuitive derivation is given in the Supplemental Material (available at the Journal website).

In the blocked Gibbs sampler, we approximate the prior rather than the posterior. Let U₁, …, U_N be independently and identically distributed according to H(·) = α(·)/α(Θ). Let U = (U₁, …, U_N) and denote the joint distribution by H^N. Let ${\mathcal{G}}_N(·|\alpha ,\beta )$ be defined by

$${\mathcal{G}}_N(·|\alpha ,\beta )={\displaystyle \int \mathcal{G}}\left(·|{\alpha }_N,\beta \right)H^N(\text{d}\mathit{U})$$ (33)

where

$${\alpha }_N(·)=\frac{\alpha (\Theta )}{N}{\displaystyle \sum _{k=1}^N{\delta }_{U_k}}(·)\mathrm{.}$$ (34)

Since ${\mathcal{G}}_N(·|\alpha ,\beta )$ weakly converges to $\mathcal{G}(·|\alpha ,\beta )$ as N → ∞ (Ishwaran and James 2004), $\mathcal{G}(·|\alpha ,\beta )$ can be approximated by ${\mathcal{G}}_N(·|\alpha ,\beta )$. Hence, for sufficiently large N, we can assume that the posterior calculated from ${\mathcal{G}}_N(·|\alpha ,\beta )$ approximates the true posterior well.

Let ${\pi }_N(\text{d}\theta ,\text{\hspace{0.17em}d}\xi |{\{(t_i,{\kappa }_i)\}}_{i=1}^n)$ be the posterior when we set ${\mathcal{G}}_N(·|\alpha ,\beta )$ as the prior for ξ. Then ${\pi }_N(\text{d}\theta ,\hspace{0.17em}\text{d}\xi |{\{(t_i,{\kappa }_i)\}}_{i=1}^n)$ is expressed as

$${\pi }_N\left(\mathrm{d}\mathbf{\theta },\mathrm{d}\xi |{\left\{\left(t_i,{\kappa }_i\right)\right\}}_{i\text{=}1}^n\right)\text{=}{\displaystyle \int \mathcal{G}}\left(\mathrm{d}\xi |{\alpha }_N\text{+}{\displaystyle \sum _{i\text{=}1}^n{\delta }_{{\theta }_i}},\beta *\right){\pi }_N\left(\mathrm{d}\mathbf{\theta }\text{,d}\mathit{U}|{\left\{\left(t_i,{\kappa }_i\right)\right\}}_{i\text{=}1}^n\right).$$ (35)

Comparing Eq. 35 with Eq. 23, we see that the shape parameter of the weighted gamma process in Eq. 35 is a finite discrete measure whose support consists of a finite number of atoms. Hence, the measure sampling from this weighted gamma process has mass on a finite number of atoms. Consequently, given θ₁, …, θ_n we can sample ξ from this weighted gamma process exactly.

Here, ${\pi }_N(\text{d}\theta ,\hspace{0.17em}\text{d}\mathit{U}|{\{(t_i,{\kappa }_i)\}}_{i=1}^n)$ in Eq. 35 is expressed as

$${\pi }_N(\text{d}\mathbf{\theta },\text{d}\mathit{U}|{\{(t_i,{\kappa }_i)\}}_{i=1}^n)$$

$$\propto \left({\displaystyle \int {\displaystyle \prod _{i=1}^{n}P}}(\text{d}{\theta }_i)\mathcal{P}(\text{d}P|{\alpha }_N)\right)H^N(\text{d}\mathit{U})D_N(\mathit{\text{U}})\left({\displaystyle \prod _{i=1}^n{\beta }^{*}}({\theta }_i)k({\kappa }_i,x_i|{\theta }_i)\right)$$ (36)

where

$$D_N(\mathit{\text{U}})=\exp \left(-\frac{\alpha (\Theta )}{N}{\displaystyle \sum _{k=1}^N\log }\left(1+\beta (U_k)f(U_k)\right)\right)$$ (37)

and P is a random measure distributed to $\mathcal{P}(·|{\alpha }_N)$ that is expressed as

$$P(·)={\displaystyle \sum _{k=1}^N{W}_k}{\delta }_{U_k}(·)$$ (38)

where W = (W₁, …, W_N) is a random variable distributed according to a Dirichlet distribution π_W(·) whose density is

$$\frac{\Gamma (\alpha (\Theta ))}{\Gamma {\left(\frac{\alpha (\Theta )}{N}\right)}^N}{\displaystyle \prod _{k=1}^N{w}_k^{\frac{\alpha (\Theta )}{N}-1}}\mathrm{.}$$ (39)

Since θ₁, …, θ_n can be treated as independently and identically distributed samples from P, each θ_i can be expressed as ${\theta }_i=U_{K_i}$ where K_i ∈ {1, …, N} is a categorical variable whose distribution is

$$\text{Pr}\left(K_i|\mathit{\text{W}}\right)={\displaystyle \sum _{k=1}^N{W}_k}{\delta }_k(K_i)\mathrm{.}$$ (40)

Let K = (K₁, …, K_n).

Here, instead of (θ,U), let us consider sampling (K,W,U). The posterior distribution of (K,W,U) is proportional to

$$\left({\displaystyle \prod _{i=1}^n\left({\displaystyle \sum _{k=1}^N{W}_k}{\delta }_k(K_i)\right)}\right){\pi }_{\text{W}}(\text{d}\mathit{W})H^N(\text{d}\mathit{U})D_N(\text{U})\times \left({\displaystyle \prod _{i=1}^n{\beta }^{*}}(U_{K_i})k({\kappa }_i,x_i|U_{K_i})\right)\mathrm{.}$$ (41)

From this equation, we can construct a Gibbs sampler that updates parameters in blocks such as K → W → U → ξ → K → …. For each block, the sampling distribution is given by

$$\mathit{\text{K}}|\mathit{\text{W}},\mathit{\text{U}}~{\displaystyle \prod _{i=1}^n\left({\displaystyle \sum _{k=1}^N{W}_k}{\delta }_k(K_i)\right)}{\beta }^{*}(U_{K_i})k({\kappa }_i,x_i|U_{K_i}),$$

$$\mathit{\text{W}}|\mathit{\text{K}},\mathit{\text{U}}~\left({\displaystyle \prod _{i=1}^n\left({\displaystyle \sum _{k=1}^N{W}_k}{\delta }_k(K_i)\right)}\right){\pi }_{\text{W}}(\text{d}\mathit{W}),$$

$$\mathit{\text{U}}|\mathit{\text{K}},\mathit{\text{W}}~H^N(\text{d}\mathit{U})D_N(\mathit{\text{U}})\left({\displaystyle \prod _{i=1}^n{\beta }^{*}}(U_{K_i})k({\kappa }_i,x_i|U_{K_i})\right),$$

$$\xi |\mathit{\text{K}},\mathit{\text{W}},\mathit{\text{U}}~\mathcal{G}\left(\text{d}\xi |{\alpha }_N+{\displaystyle \sum _{i=1}^n{\delta }_{U_{K_i}}},{\beta }^{*}\right)\mathrm{.}$$ (42)

For details of the general sampling procedure, see Ishwaran and James (2004).

Sampling Procedure in the Proposed Method

In this section, we show details of the Gibbs sampling procedure in the proposed method. To run the blocked Gibbs sampler, cycle the following steps.

Conditional draw for K.

Sample K_i, i = 1, …, n independently from

$${\displaystyle \sum _{k=1}^N{W}_{k,i}}{\delta }_k(·)$$ (43)

where

$$(W_{1,i},\mathrm{...},W_{N,i})$$

$$\propto \left(\frac{W_1\text{N}({\kappa }_i|{\mu }_1^{\kappa },{\Lambda }_1^{\kappa })\text{N}(x_i|{\mu }_1^x,{\Lambda }_1^x)}{1+\frac{{\beta }_{0}T}{R}{\displaystyle \sum _{r=1}^R\text{N}}(x_{r\Delta }|{\mu }_1^x,{\Lambda }_1^x)},\mathrm{...},\frac{W_N\text{N}({\kappa }_i|{\mu }_N^{\kappa },{\Lambda }_N^{\kappa })\text{N}(x_i|{\mu }_N^x,{\Lambda }_N^x)}{1+\frac{{\beta }_{0}T}{R}{\displaystyle \sum _{r=1}^R\text{N}}(x_{r\Delta }|{\mu }_N^x,{\Lambda }_N^x)}\right)\mathrm{.}$$ (44)

Conditional draw for W.

Sample W_k, k = 1, …, N from a Dirichlet distribution whose density is

$$\frac{\Gamma ({\alpha }_0+n)}{{\displaystyle \prod _{k=1}^N\Gamma }\left(\frac{{\alpha }_0}{N}+M_k\right)}{\displaystyle \prod _{k=1}^N{w}_k^{\frac{{\alpha }_0}{N}+M_k-1}}$$ (45)

where M_k = #{i|K_i = k}.

Conditional draw for (μ^κ,Λ^κ,μ^x,Λ^x).

Sample (μ^κ,Λ^κ,μ^x,Λ^x), k = 1, …, N. Let $\{K_1^{*},\hspace{0.17em}\mathrm{...},\hspace{0.17em}{K}_{N_0}^{*}\}$ be a set that contains unique values of K_i,i = 1, …, n.

for k $\notin \hspace{0.17em}\{K_1^{*},\hspace{0.17em}\mathrm{...},\hspace{0.17em}{K}_{N_0}^{*}\}$.

Sample $({\mu }_k^{\kappa },{\Lambda }_k^{\kappa })$ from

$$\text{N}\left({\mu }_k^{\kappa }|m^{\kappa },{\rho }^{\kappa }{\Lambda }_k^{\kappa }\right)·\text{W}\left({\Lambda }_k^{\kappa }|{\nu }^{\kappa },S^{\kappa }\right)\mathrm{.}$$ (46)

In addition, sample $({\mu }_k^x,{\Lambda }_k^x)$ from a distribution whose density is proportional to

$$\text{N}\left({\mu }_k^x|m^x,{\rho }^x{\Lambda }_k^x\right)\text{W}\left({\Lambda }_k^x|{\nu }^x,S^x\right){\left(1+\frac{{\beta }_{0}T}{R}{\displaystyle \sum _{r=1}^R\text{N}}(x_{r\Delta }|{\mu }_k^x,{\Lambda }_k^x)\right)}^{-\frac{{\alpha }_0}{N}}\mathrm{.}$$ (47)

This distribution cannot be expressed in the closed form. Hence, we have to use the Metropolis-Hastings step.

for k $\in \hspace{0.17em}\{K_1^{*},\hspace{0.17em}\mathrm{...},\hspace{0.17em}{K}_{N_0}^{*}\}$.

Sample $({\text{μ}}_k^{\text{κ}},{\Lambda }_k^{\kappa })$ from

$$\mathrm{N}\left({\mu }_k^{\kappa }|\mu ,\Lambda \right)\mathrm{W}\left({\Lambda }_k^{\kappa }|\nu ,S\right)$$ (48)

where μ, Λ, ν, S are

$$\overline{\kappa }\text{=}\frac{1}{M_k}{\displaystyle \sum _{i\text{:}{K}_i=k}{\kappa }_i,}$$

$$\nu ={\nu }^{\kappa }+M_k,$$

$$S^{-1}={(S^{\kappa })}^{-1}+{\displaystyle \sum _{i:K_i=k}({\kappa }_i-\overline{\kappa }){({\kappa }_i-\overline{\kappa })}^{\text{T}}}+\frac{M_k{\rho }^{\kappa }}{M_k+{\rho }^{\kappa }}(\overline{\kappa }-m^{\kappa }){(\overline{\kappa }-m^{\kappa })}^{\text{T}},$$

$$\mu =\frac{M_k\overline{\kappa }+{\rho }^{\kappa }{m}^{\kappa }}{M_k+{\rho }^{\kappa }},$$

$$\Lambda =(M_k+{\rho }^{\kappa }){\Lambda }_k^{\kappa }\mathrm{.}$$ (49)

In addition, sample $({\mu }_k^x,{\Lambda }_k^x)$ from a distribution whose density is proportional to

$$\text{N}\left({\mu }_k^x|m^x,{\rho }^x{\Lambda }_k^x\right)\text{W}\left({\Lambda }_k^x|{\nu }^x,S^x\right){\left(1+\frac{{\beta }_{0}T}{R}{\displaystyle \sum _{r=1}^R\text{N}}(x_{r\Delta }|{\mu }_k^x,{\Lambda }_k^x)\right)}^{-\frac{{\alpha }_0}{N}}$$

$$·{\displaystyle \prod _{i:K_i=k}\frac{\text{N}(x_i|{\mu }_k^x,{\Lambda }_k^x)}{1+\frac{{\beta }_{0}T}{R}{\displaystyle \sum _{r=1}^R\text{N}}(x_{r\Delta }|{\mu }_k^x,{\Lambda }_k^x).}}$$ (50)

This distribution cannot be expressed in closed form. Hence, we must use a Metropolis-Hastings step.

Conditional draw for ξ.

Sample ξ as follows:

$$\xi (·)={\displaystyle \sum _{k=1}^N\frac{G_k{\beta }_0}{1+\frac{{\beta }_{0}T}{R}{\displaystyle \sum _{r=1}^R\text{N}}(x_{r\Delta }|{\mu }_k^x,{\Lambda }_k^x)}}{\delta }_{({\mu }_k^{\kappa },{\Lambda }_k^{\kappa },{\mu }_k^x,{\Lambda }_k^x)}(·)$$ (51)

where

$$G_k~\text{Gamma}\left(\frac{{\alpha }_0}{N}+M_k\right).$$ (52)

In the Metropolis-Hastings step for sampling (μ^x,Λ^x), we use

$$q({\mu }_{\text{new}}^x,{\Lambda }_{\text{new}}^x|{\mu }^x,{\Lambda }^x)=\text{N}\left({\mu }_{\text{new}}^x|{\mu }^x,0.5I\right)\text{W}\left({\Lambda }_{\text{new}}^x|12,0.1{\Lambda }^x\right)$$ (53)

as the proposal distribution.

APPENDIX B: DETAILS OF THE DECODING METHOD USING SPIKE SORTING

In this appendix, we explain the traditional decoding method used in Simulation study. In this method, we first carry out spike sorting using κ as a waveform feature and then estimate the intensity function of unmarked point processes for individual neurons.

We used KlustaKwik (Kadir et al. 2014) for spike sorting. As a result of spike sorting, the number of neurons was judged to be three. Similar to Brown et al. (1998), we considered that a spike train is generated from an unmarked point process with intensity

$${\lambda }_c(x_t|{\alpha }_c,{\mu }_c,{\sigma }_c)=\exp \left({\alpha }_c-\frac{{(x_t-{\mu }_c)}^2}{2{\sigma }_c^2}\right),c=1,2,3.$$ (54)

Under this assumption, we calculated MLEs for (α_c, μ_c, σ_c), c = 1, 2, 3 and used these for decoding.

APPENDIX C: DESCRIPTION OF MIZUSEKI ET AL. (2013)

In this section, we provide a description of the data set recorded by Mizuseki et al. (2013). This data set is used in Diba and Buzsáki (2008) and Mizuseki et al. (2009). For more information about this data set, see those articles and Mizuseki et al. (2013).

This data set contains multiunit recordings in the hippocampal area in Long-Evans rats while the rats are performing tasks. Mizuseki et al. (2013) recorded various layers and tasks; we use a recording in CA1 of a rat running on a linear track.

In the recording session, the rats run back and forth on a 250-cm linear track. Light-emitting diodes attached to a rat’s head are used to detect its position. The rat is also implanted with a silicon probe with eight shanks in the CA1 layer of its right hippocampus. Each shank has seven or eight recording sites that simultaneously record electrical potentials. A threshold value is calculated with these potentials at each shank. Spikes are detected with these threshold values. The waveform of each detected spike is defined by potentials recorded in a particular length interval whose center is the detected time. Feature values of spikes, such as maximum amplitude, minimum amplitude, and principal components, are calculated from these waveforms.