Point process filters have been applied successfully to decode neural signals and track neural dynamics. Traditionally these methods assume that multiunit spiking activity has already been correctly spike-sorted. As a result, these methods are not appropriate for situations where sorting cannot be performed with high precision, such as real-time decoding for brain-computer interfaces. Because the unsupervised spike-sorting problem remains unsolved, we took an alternative approach that takes advantage of recent insights into clusterless decoding. Here we present a new point process decoding algorithm that does not require multiunit signals to be sorted into individual units. We use the theory of marked point processes to construct a function that characterizes the relationship between a covariate of interest (in this case, the location of a rat on a track) and features of the spike waveforms. In our example, we use tetrode recordings, and the marks represent a four-dimensional vector of the maximum amplitudes of the spike waveform on each of the four electrodes. In general, the marks may represent any features of the spike waveform. We then use Bayes’s rule to estimate spatial location from hippocampal neural activity.
We validate our approach with a simulation study and experimental data recorded in the hippocampus of a rat moving through a linear environment. Our decoding algorithm accurately reconstructs the rat’s position from unsorted multiunit spiking activity. We then compare the quality of our decoding algorithm to that of a traditional spike-sorting and decoding algorithm. Our analyses show that the proposed decoding algorithm performs equivalent to or better than algorithms based on sorted single-unit activity. These results provide a path toward accurate real-time decoding of spiking patterns that could be used to carry out content-specific manipulations of population activity in hippocampus or elsewhere in the brain.
Neural systems encode information about external stimuli in temporal sequences of action potentials. Because action potentials have stereotyped impulse waveforms, they are most appropriately modeled as point processes (Brillinger, 1992). Neural systems are, moreover, dynamic in that the ensemble firing of populations of neurons, representing some biologically relevant variable, continually evolves. Decoding algorithms based on adaptive filters have been developed to study how the firing patterns maintain dynamic representations of relevant stimuli. More specifically, both discrete-time and continuous-time point process filter algorithms have been applied with great success to address problems of estimating a continuous state variable (Eden, Frank, Barbieri, Solo, & Brown, 2004; Smith & Brown, 2003; Smith et al., 2004), such as the location of an animal exploring its environment (Brown, Frank, Tang, Quirk, & Wilson, 1998; Huang, Brandon, Griffin, Hasselmo, & Eden, 2009; Koyama, Eden, Brown, & Kass, 2010).
A prerequisite for these increasingly efficient decoding methods is the application of spike sorting: the waveforms recorded extracellularly at electrodes must be clustered into putative single neurons. Therefore, the accuracy of the spike sorting has a critical impact on the accuracy of the decoding (Brown, Kass, & Mitra, 2004). Many algorithms for spike sorting, whether real time and automatic or offline and manual, have been developed since the 1960s (Lewicki, 1998; Wild, Prekopcsak, Sieger, Novak, & Jech, 2012). The majority of these algorithms are clustering-based methods, allocating each spike to a putative single cell based on the characteristics of spike waveforms. These types of pure waveform, hard boundary spike-sorting algorithms, suffer from many sources of error, such as nonstationary noises, nongaussian clusters, and synchrony (Lewicki, 1998; Harris, Henze, Csicsvari, Hirase, & Buzsaki, 2000; Quiroga, 2012). In addition, they have been shown to yield biased and inconsistent estimates for neural receptive fields (Ventura, 2009b). Another clustering method, soft or probabilistic spike assignment, has been incorporated into some decoding paradigms, and analyses have shown that these algorithms can yield unbiased estimates of stimulation parameters (Ventura, 2008, 2009a). Nonetheless, these algorithms, like most other hard sorting methods, are not well suited to real-time implementations because the algorithms are too computationally intensive and they rely on having the entire data set available for the clustering algorithm.
More recently, decoding methods without a spike-sorting step have been investigated (Luczak & Narayanan, 2005; Stark & Abeles, 2007; Fraser, Chase, Whitford, & Schwartz, 2009; Chen, Kloosterman, Layton, & Wilson, 2012; Kloosterman, Layton, Chen, & Wilson, 2014). Chen et al. (2012) and Kloosterman, Layton, Chen, and Wilson (2014) developed a spike feature decoding paradigm for unsorted spikes using a time-homogeneous spatiotemporal Poisson process. It incorporates a covariate-dependent method to estimate a nonparametric distribution of the animal’s position. This improves decoding performance by using information that is otherwise excluded by sorting spikes into discrete clusters. However, this method does not incorporate a model of the animal’s position trajectory, and therefore the decoding results can depend substantially on specific model parameters such as the choice of a discrete time bin width: if the timescale is too broad, the algorithm cannot track the stimuli fast enough; if too narrow, it cannot integrate information provided by spikes nearby in time. Additionally, this algorithm extracts information from spike intervals but does not optimally incorporate information from intervals that contain no spikes. Chen et al. (2012) suggested the possibility of applying a state-space framework with a temporal prior but do not provide a complete algorithm or an implementation of this method.
To address these issues, we generalize and extend this decoding paradigm for unsorted spikes to allow for general marked point process models. In doing so, we develop an iterative algorithm that solves the marked point process filter problem. In particular, this allows us to model neural activity that is dependent on the timing and mark values of previous spikes. In this letter, we propose a novel algorithm for adaptive decoding of spiking activity that avoids the clustering problem of spike sorting entirely by defining a joint model for the spike waveform and receptive field structure and uses a state-space model to incorporate knowledge of the properties of the signal to decode. The resulting algorithm is a general marked point process filter.
The goal of the proposed algorithm is to decode an adaptive state variable—in this case, the dynamic trajectory of the rat along a track—by computing the posterior distribution of the state process conditioned on the set of observations up until the current time. Our algorithm takes directly as inputs the recorded spikes where each spike is associated with a vector of characteristic features selected from the raw waveforms. Such inputs can be mathematically described by a marked point process where the points are the spike times and the marks are their corresponding waveform. In their seminal book on point processes in 1980, Cox and Isham had already explicitly suggested that “in the neurophysiological example mentioned the mark could be the magnitude of the peak signal at the point in question.”
In this letter, we first characterize the spiking activity of an ensemble of neurons using the conditional intensity function for marked point processes. Next, we construct a stochastic state marked point process filter to iteratively calculate the full posterior probability for the state variable. We illustrate our approach in a simulation study where the decoding algorithm is used to reconstruct an animal’s position from unsorted multiunit place cell spiking activity. We also apply the algorithm to experimental data recorded in the hippocampus of a rat navigating a linear environment. We then compare the quality of fit of our clusterless decoding algorithm to that of a traditional spike-sorting and decoding algorithm.
The goal of our decoding algorithm is to compute, at each time point, the posterior distribution of the state variable given observed marked spiking activity. To do this, we apply the theory of state-space adaptive filters (Haykin, 1996). Recursive filter equations can be solved in both discrete time and continuous time (Eden, Frank, Barbieri, Solo, & Brown, 2004; Eden & Brown, 2008). In this letter, we present an algorithm in discrete time.
Equation 2.4 has two components: , which is given by a state evolution model under the Markovian assumption, and , which is the posterior density from the last iteration step. We multiply this probability density function with the posterior distribution of the state variable at the previous time step and numerically integrate the product over all possible values of the previous state, . The resulting integral is a one-step prediction density at the current time tk.
We can interpret equation 2.5 by separating the product on the right-hand side into two terms. The term characterizes the distribution of firing spikes, such that the mark value of the ith spike in the interval is , where . If no spike occurs, that is, , this term equals 1. The term gives the probability of not firing any other spikes in the observation interval, where as defined in equation 2.2.
The discrete-time likelihood or observation distribution defined by equation 2.5 assumes that within a small time step , conditional on the history Hk and the current value of the state vector xk, spiking activity for the neural ensemble is independent. However, the spiking activity at a given time step can explicitly depend on the history of activity from the entire population, including dependencies between neurons.
3 Simulation Study
We first tested this approach on simulated data corresponding to the activity of two place cells firing as an animal traverses a linear track. Place cells are neurons in the hippocampus that are activated primarily when an animal is located in a certain portion of its environment (O’Keefe & Dostrovsky, 1971; O’Keefe, 1979). A substantial amount of information about the position carried by place cells has been reliably quantified with a formal statistical algorithm and used to reconstruct the trajectory and predict the future position of the rat (Muller & Kubie, 1989; Wilson & McNaughton, 1993; Zhang, Ginzburg, McNaughton, & Sejnowski, 1998; Brown et al., 1998).
In this section, we apply the clusterless decoding algorithm to the problem of decoding a one-dimensional position of a rat on a linear track using a marked point process arising from two hippocampal place cells, where the one-dimensional mark represents the peak height of the spike waveform. Here we note that accurate decoding of position based on place cell activity normally requires many cells. Our goal here is therefore not to test the accuracy of decoding, but rather to provide intuition about how our approach works in a simple case. First, we simulate the rat’s trajectory using an autoregressive process and the joint mark intensity function of the two cells using a mixture of two bivariate gaussian distributions. Then we use a marked point process filter to reconstruct the location of the rat at each time step. We show that the filter yields exact full posterior densities that are often multimodal. We also reconstruct the trajectory using a traditional algorithm in which decoding was applied after spike sorting. Last, we compare the performance of the clusterless decoding algorithm to the traditional decoding algorithm.
3.1 Data Simulation
The parameters for this model include for , which controls the maximum in-field firing rate for the cth neuron. is the animal’s linearized position at time t, and is the one-dimensional mark value. is the center of the place field, and is the mean of the density function for marks for neuron c. and are the standard deviations of the place field and the mark space for neuron c, respectively. In this illustrative example, we set the centers of the place field at , and , respectively, with a variance. We set the maximum in-field firing rate of both neurons to spikes per second. The one-dimensional mark space for the two neurons is centered at and , respectively, with standard deviations varying between 0.01 and 5.
By plugging the simulated position trajectories, , into equation 3.2, we computed the instantaneous intensity at each time step. The underlying unmarked spike trains are generated under an inhomogeneous Poisson process model with rate given by equation 2.2. The marks associated with the spike trains are sampled from the probability density .
Figure 1 displays the results of a representative 1 second trial where the spiking activity of two neurons with moderate mark space overlap was simulated. Figure 1A shows the simulated trajectory of the animal with the linear location on the y-axis. Figure 1B plots the simulated spike train with the mark value on the y-axis. Visually, we cannot identify a clear clustering of the mark values. Figure 1D shows the true joint mark intensity function used to simulate the marked spike train in Figure 1B. The two simulated neurons overlap over a moderate amount of mark space, but are fairly localized in space, with little overlap of their place fields.
3.2 Decoding Results
In this section, we present the decoding results of the simulated data described in section 3.1 using the clusterless decoding algorithm described in the section 2. When calculating the observation distribution defined in equation 2.5, we used the true joint mark intensity function specified in equation 3.2. In real data, we need to estimate an encoding model for the joint mark intensity function. However, this simulation study affords us the opportunity to explore errors due purely to the decoding algorithms.
Figure 2A shows the posterior density for the position of the animal during a 1 second trial as a function of time, using the clusterless decoding algorithm. The blue line shows the true position of the animal, and the red region is the estimated posterior density at that time step. We can see that the estimate tracks the true position relatively closely, and the region of high posterior density covers the true position most of the time.
Figure 1C plots the spike raster of the two simulated neurons sorted using a linear discriminant function on their marks. Figure 1E shows the true condition intensity function for each neuron c, that is, the place field of each neuron.
Figures 2B and 2D are results using decoding with spike sorting where the decoder is applied after classifying the spikes into two clusters. Figure 2B shows the posterior density using decoding with spike sorting. Visually, we obtained comparable tracking for the animal’s trajectory.
To illustrate the advantage of using a clusterless decoding algorithm when clear clustering of neurons is difficult, we zoomed in on the decoding results at times between 400 ms and 600 ms, shown in figures 2C and 2D. From Figures 1B and 1C, we can see that even though the rat’s position is positive, and therefore only neuron 2 is spiking, the spike waveforms cannot be perfectly resolved. This leads to incorrect decoding results using the presorted spiking activity. However, the marked point process decoder results in a bimodal posterior distribution that accurately reflects the uncertainty due to the waveform overlap. Figure 2C demonstrates the multimodality of the posterior estimated by a clusterless decoder. For example, at time around 430 ms when a new spike arrives, the posterior density splits into two modes, one near a linearized position of −1 and another around a linearized position at 1.5. The posterior density is slightly higher in the region correctly predicting the animal’s position in the negative regime. For the decoding with spike-sorting results shown in Figure 2D, the algorithm has some trouble tracking the trajectory after new spikes from both neurons arrive because the posterior density in this case is not multimodal. For example, at time around 430 ms, the posterior density incorrectly estimates the animal’s position at a positive value near 1.
3.3 Goodness-of-Fit Analysis
In order to compare the quality of fit between the two decoding algorithms, we simulated 100 trials of spiking activity for different degrees of overlap between the two neurons in the mark space. We evaluate the ability of these two algorithms to track the location of the animal when the overlaps in the mark space between the joint mark intensity functions of the simulated neural ensemble vary from low () to high () in units of the number of standard deviations, using two different measures: the root-mean-squared error (rMSE) between the true positions of the animal and their estimated values and the percentage of the time that the true position values were covered by the 99% highest posterior density (HPD) region (Casella & Berger, 2011).
Figure 3A shows the rMSEs between the true positions of the animal and their estimated values averaged across 100 trials as a function of . Error bars represent two standard deviations from the mean rMSE. The dotted line represents the performance of the clusterless decoding method and the solid line the performance of the decoding with the spike-sorting method. When the standard deviations of both mark spaces exceed 1, the clusterless decoder consistently gives the lower mean rMSE and there is no overlap between the error bars for the two algorithms.
The comparison of the rMSEs shows that as the overlap between the marks for the individual neurons increases and clustering of the spike waveforms becomes more difficult, the clusterless decoding algorithm provides a consistently more accurate estimate of the rat’s position than the decoding algorithm using presorted spikes.
Figure 3B shows the fraction of time that the true position values were covered by the 99% HPD region averaged across 100 trials as a function of , which measures how well each algorithm characterizes their uncertainty. The flat dotted line with narrow error bars illustrates that regardless of the degree of the overlap between mark spaces, the true trajectory stays in the 99% HPD region of the clusterless decoding method around 99% of the time with very little variance about this percentage of time across the 100 repeated trials of the simulation. The descending solid line with widening error bars illustrates that for decoding with spike sorting, as the degree of the overlap between mark spaces increases, the fraction of time that the true trajectory stays in the 99% HPD region is decreasing from 99% to 80% with an increasing variance.
The comparison of the 99% HPD region illustrates two advantages of the clusterless decoding algorithm regarding the uncertainty of the state estimates. First, the width of a 99% HPD describes the degree of uncertainty in the estimates. As the overlap between the mark spaces increases, the width of the 99% HPD region of the decoding algorithm using presorted spikes increases, indicating decreasing certainty in the estimated position. However, the clusterless decoding algorithm provides a narrower 99% HPD region regardless of the mark space overlap. This means that the clusterless decoding algorithm provides more estimates of the rat’s position and is less influenced by the degree of overlap in the mark space. Second, the percentage of time within the 99% HPD represents the accuracy of uncertainty in the estimates. As the overlap in the mark distribution between the neurons increases, the decoding with presorted spikes estimates the uncertainty in position with less accuracy, while the estimated certainty of the clusterless decoding algorithm remains correct at around 99%. Therefore, the estimated certainty for the clusterless decoding algorithm is both higher and more accurate than the estimated certainty for decoding with presorted spikes.
4 An Application to Position Decoding from Multiunit Activity in Rat Hippocampus
We also applied the clusterless decoding algorithm to experimental data recorded using a multielectrode array in the hippocampus of a rat running back and forth on a linear track. The data used in this analysis were recorded from five tetrodes in the dorsal CA1 and CA2 regions of the hippocampus. Manual clustering of spike waveforms above a 40 V threshold during the linear track session yielded 15 distinct unit clusters. Four units with fewer than 100 spikes during the 840 second recording session were excluded. One of the remaining units was a putative interneuron, identified by a firing rate exceeding 7 Hz and an exceptionally narrow spikewidth. In the clusterless decoding, all of the thresholded spikes above the 40 V threshold, including this putative interneuron, were included. Details of the experimental preparation, data acquisition, and choice of spike-sorting method can be found in the appendix.
We then compare the results to decoding based on sorted data. These data included 11 active place cells across an 840-second recording session. To assess whether the models generalize well to unobserved data as well as to limit overfitting, both algorithms are carried out using a five-fold cross-validation.
Figure 4 illustrates the decoding results from signals recorded from five tetrodes in the CA1 and CA2 regions of the hippocampus using the two different algorithms. Here we display the first 125 seconds of decoding results in both panels. Figure 4A is decoding results using the clusterless decoding algorithm. Figure 4B is decoding results where the decoder is applied after the spikes have been manually sorted into clusters. The blue line shows the true position of the animal, and the red region shows the estimated posterior density at that time step. We can see that the estimate tracks the true position closely, and the region of highest posterior density covers the true position most of the time. In contrast, as shown in Figure 4B, the posterior density using decoding with spike sorting has some trouble tracking the true trajectory, for example, between time 6250 seconds and 6260 seconds.
To assess quality of fit, we calculated the root-mean-square error (rMSE) in centimeters and the percentage of the time that the true position values fall within the 99% HPD region. We compared the rMSE and percentage of time within the 99% HPD region for the decoding results on the entire 840 seconds of recording. The posterior density computed through the clusterless decoding algorithm has an rMSE of 14.3 cm and remains within the 99% HPD region 74.25% of the time. The size of the 99% HPD region for the clusterless decoding algorithm has a mean of 10.5 cm and a standard deviation of 6.96 cm. The posterior density computed by first spike sorting and then decoding has an rMSE of 26.0 cm and remains within the 99% HPD region 70.76% of the time. The size of the 99% HPD region for the decoding with sorted spikes has a mean of 17.2 cm and a standard deviation of 13.14 cm.
The clusterless algorithm has a narrower 99% HPD region, more accurately reflecting the uncertainty of the decoding estimates. The algorithm that uses spike sorting has a wider 99% HPD region, suggesting less certainty in the estimates. Therefore, the estimated certainty for the clusterless decoding method is both higher and more accurate than the estimated certainty for the spike-sorting-based method. Both visually and numerically, we showed that our proposed clusterless algorithm performs as well as or better than the algorithm based on sorted spikes.
We previously used point process theory to develop efficient decoding algorithms based on spike train observations (Brown et al., 1998; Eden et al., 2004; Huang et al., 2009; Koyama et al., 2010). In each of these cases, a key assumption is that the signals have been accurately sorted into single units before the decoding algorithm is applied. Although new spike-sorting algorithms are being developed, spike sorting remains a time-consuming, difficult task; suffers from many sources of errors; and likely provides biased estimates (Lewicki, 1998; Harris et al., 2000; Quiroga, 2012). In this letter, we have proposed a novel clusterless decoding algorithm that maintains the accuracy of previous methods, but avoids spike sorting.
The proposed clusterless decoding algorithm has several important advantages. First, it does not require that the multiunit activity has been accurately sorted into single units. Instead, by using the theory of marked point processes, the algorithm characterizes directly the relationship between a desired variable and features of the spike waveforms. We bypass spike sorting by modeling the spiking activity as a joint function of the state variable to decode and of features of the spike waveforms. Therefore, this new algorithm can incorporate information from spikes that in previous decoders that relied on spike sorting may have been thrown out because of difficulties in clustering or were misclassified. Indeed, in a recent analysis of the effects of spike-sorting schemes on decoding performance, Todorova, Sadtler, Batista, Chase, and Ventura (2014) confirmed that discarding waveforms that do not match any template—the “hash”—systematically degrades decoding, consistent with previous studies (Stark & Abeles, 2007; Fraser et al., 2009).
Another advantage of the marked point process approach is that the joint intensity mark function defines a population-level place receptive field structure, which will typically have multimodal structure. That is, the joint intensity mark function is expected to have multiple peaks in the joint place and mark space. Previous work by Huang et al. (2009) has shown that decoding algorithms that allow for highly nongaussian and multimodal posterior densities perform better at reconstructing the animal’s trajectory and predicting future decisions.
Finally, formulating the decoding problem within a state-space paradigm offers a number of specific benefits. It allows us to incorporate knowledge about the system we are decoding—in this case, the rat’s position trajectory. This also imposes an implicit continuity constraint, preventing large fluctuations in the state estimates. This allows us to track the state variable smoothly without high dependence on the choice of time step during decoding. Moreover, due to the state-space approach, the resulting decoding algorithm incorporates information from both spike intervals and nonspike intervals.
We tested the new decoding algorithm on tetrode recordings from the hippocampus of a rat running back and forth on a linear track. We showed that the clusterless decoding algorithm provided a slightly improved accuracy than that of the decoding with a manual spike-sorting approach. The intention of this example is to demonstrate comparable decoding accuracy without the additional time-consuming spike-sorting step, thus making this algorithm a suitable candidate for real-time application.
There are a number of directions in which this work may be extended. The numerical integration we used to compute the filtering is efficient when decoding low-dimensional signals. When the desired variable is high dimensional, alternative adaptive algorithms can be used to ensure efficiency, such as sequential Monte Carlo methods or gaussian approximate filters.
In the application example, we presented a nonparametric encoding model to estimate the joint mark intensity function based on kernel methods. We have found that our results remain consistent for a wide range of bandwidths. Therefore, in real-time decoding, it is reasonable to set the smoothing parameter and bandwidth matrix beforehand. We also recognize that for nonparametric kernel-based encoding, the computation time increases with the total number of spikes. In the interest of real-time applications, it is possible to explore other more efficient models as well. Within the family of nonparametric encoding model, one can include a time dimension with receding horizon to reduce the number of spikes encoded at any given time.
Another choice of encoding model is semiparametric models, for example, a multidimension grid interpolated for the joint distribution of mark and place field. This grid can serve as a kind of look-up table for the estimated joint mark intensity function. By calculating this grid beforehand, we can perform real-time clusterless decoding without any computation time spent in encoding. A more general approach to reduce encoding time is to perform dimension reduction on the multidimensional mark space before encoding.
Finally, as we move away from the prerequisite of spike sorting, multivariate marked point process models can be developed to describe coupling between neurons (Ba, Temereanca, & Brown, 2014).
Another future role for these methods is in the development of new types of closed-loop experiments. Traditionally, experiments designed to study the role of specific spike patterns in stimulus-response tasks take one of two forms: observational studies that characterize statistical properties of neural activity during such tasks or interventional studies that broadly alter neural activity over an entire neural population or brain region. However, new closed-loop experiments designed based on the content of neural signals aim to characterize causal relationships between neural activity and the biological and behavioral signals they encode. The proposed algorithm can allow investigators to manipulate millisecond-timescale spike patterns in a content-specific way, altering spiking activity related to certain neural patterns while leaving activity related to other patterns intact.
In conclusion, we develop a novel method for modeling neural response properties and decoding biological and behavioral signals by expanding the class of neural models to incorporate marked point processes. We found that the resulting decoding signals were estimated with higher accuracy and more confidence than traditional spike-sorting-based methods. We believe this work has broad implications, allowing for better neural coding models across a wide range of brain areas and neural systems.
A.1 Hippocampal Data Collection and Preprocessing
The hippocampal data in this paper are from a single male Long-Evans rat (500–600 g) trained to alternate in a linear track for liquid reward (condensed milk). A microdrive array containing 21 independently movable tetrodes was implanted targeting the hippocampal cell layers according to University of California San Francisco Institutional Animal Care and Use Committee and U.S. National Institutes of Health guidelines. All neural signals were recorded relative to a reference tetrode in the corpus callosum. Following data collection, electrode locations were verified histologically to localize to the CA1–CA2 region of the hippocampus.
Data were collected using the NSpike data acquisition system (L.M.F. and J. MacArthur, Harvard Instrumentation Design Laboratory). An infrared diode array with a large and a small cluster of diodes was attached to the preamps during recording. Following recording, the rat’s position on the track was reconstructed using a semiautomated analysis of digital video of the experiment. Individual units (putative single neurons) were identified by clustering spikes using peak amplitude, principal components, and spike width as variables (MatClust, M. Karlsson) (Karlsson & Frank, 2008).
A.2 Gaussian Approximation to the Posterior Density
Because in this letter the state variable is low-dimensional, we used a simple Riemann sum to perform numerical integration over the state space in order to solve the one-step prediction density in equation 2.4. If the state variable is high dimensional and the posterior density can be assumed to be approximately gaussian, a stochastic state point process filter (SSPPF) can be used. The SSPPF is constructed using a gaussian approximation to the posterior density. Detailed derivation of approximate gaussian filters for temporal point processes can be found in Brown et al. (1998) and Eden et al. (2004). Here we provide an extension of the SSPPF to the marked case.
A.3 Sequential Monte Carlo Decoding Algorithm
When the state variable is high dimensional and the posterior density cannot be assumed to be approximately gaussian, another computationally efficient alternative is a sequential Monte Carlo algorithm. Point process adaptive filters using sequential Monte Carlo approximations to the posterior density have been developed in previous literature (Ergun et al., 2007; Meng, Kramer, & Eden, 2011). Here we provide a pseudocode description of the algorithm with extension to marked point processes. This is a bootstrap filter, so the proposal distribution is based on the one-step prediction density from the previous time step.
At each time step t, the algorithm produces a collection of weighed samples, or particles, each containing proposed values for the state variable xt. We construct estimates for the state variable by computing their sample means over all the particles and construct approximate 95% confidence intervals.
Initialization: Set and for particles, draw the initial states and parameters from an initial probability distribution, and set the importance weight of the ith particle for all i. Set .
- Importance sampling: Using the particles from the previous step that represent the one-step prediction density defined in equation 2.4 as the sampling distribution, update all of the states xt. Evaluate the importance weight of the ith particle, is computed by equation 2.5 or 4.1. Normalize the importance weights:
Repeat steps 2 and 3.
We thank Mikio Aoi for helpful comments on the manuscript. This research was supported in part by a grant from the Simons Foundation (SCGB 320135 to L.M.F. and SCGB 337036 to U.T.E.), NSF grant IIS-0643995 and NINDS grant R01 NS073118 to U.T.E., NIH grant R01 MH0901188 to L.M.F., and NSF GRFP grant 1144247 to D.F.L.