Online Neural Sequence Detection with Hierarchical Dirichlet Point Processes
Anonymous Author(s)
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Abstract
Neural sequence detection plays an important role in neuroscience research.
Recentimpressive works utilize convolutive nonnegative matrix factorization and NeymanScott process to solve this problems.
However, they still face two limitations.
Firstly, they accommodate the entire dataset into memory and perform iterativeupdates of multiple passes, which can be inefficient when dataset is large or growsfrequently.
Secondly, they rely on the prior knowledge of the number of sequencetypes, which can be impractical with real data when the future situation is unknown.
To tackle these limitations, we propose a hierarchical Dirichlet point processesmodel for efficient neural sequence detection.
Instead of computing the entire data,our model can sequentially detect sequences in an online unsupervised mannerwith Particle filter.
Besides, the Dirichlet prior enables our model to automaticallyintroduce new sequence types on the fly as needed, thus avoiding specifying thenumber of types in advance.
We manifest these advantages on synthetic data andreal-world recordings from songbird higher vocal center and rodent hippocampus.
Introduction
Neural sequences are a set of sequential firing neurons occurring repeatedly over time, and they playcrucial roles in the brain activities of working memory [6, 10], motor production [9], and memoryreplay [4].
In general, it’s hard to directly observe neural sequences from high-dimensional neuralrecordings due to the unsorted neurons, and methods that use linear dimensionality reduction ofneural populations may be inappropriate to handle such sequences since they form a high-dimensionalstructure and cannot be efficiently summarized by a low-dimensional representation [20, 15].
Finally,the relationship between neural sequences and behavior is not always so regular and predictable [27],thus an unsupervised method to identify neural sequences is required.
While a great process in neural sequence detection has been made by recent works [19, 15, 25], theyhave two drawbacks when it comes to the contexts of online learning.
Firstly, they are ill-suited toaccommodate new data or handle large datasets.
Many real-world neural recordings involve plentyof hours’ or days’ signals that sometimes may be hard to accommodate entirely in the memory.
The inference algorithms of these works (e.g., coordinate descent, Gibbs sampling, etc.) maintainthe entire data and perform iterative updates of multiple passes, which may be a waste of bothcomputational and storage resources.
Secondly, these works require the number of sequence types tobe specified in advance, which is inappropriate when a future situation is unknown.
To address these problems, we propose a hierarchical Dirichlet point process (HDPP) with an efficientonline inference algorithm.
Briefly, we use a two-level structure in our approach. At a lower level,we model the set of neural sequences by a Dirichlet nonhomogeneous Poisson process: The observedspikes are considered as a mixture of neural sequences, and those spikes belonging to differentsequences are modeled via different nonhomogeneous Poisson processes, for which the intensity of each process varies across neurons and times.
At a higher level, we model the types of sequencesby a Dirichlet Hawkes process[5]: The detected sequences are partitioned into an automaticallyinferred number of types.
The temporal dynamics of these sequences are captured by several Hawkesprocesses [11], where each process’s distinctive temporal relation will help our model to disambiguatedifferent types of sequences with spatial overlapping neurons.
Besides, the prior of each sequencetype is given by a Dirichlet distribution, so either the number of sequences or the number of sequencetypes could vary during inference.
Note that our model has only one Dirichlet point process at thelower level, which is different from the classical hierarchical Dirichlet process (HDP) [22].
We derive a Particle filter [2] to learn the HDPP in an online manner: (1) learn the model with a singlepass of the data; (2) make the decisions regularly without requiring future data [1].
In particular, weuse a collection of particles to sequentially approximate the posterior, taking in one spike at a timeto make the update.
We also develop specific mechanisms to merge similar sequences and pruneinaccurate ones during inference, thus further improving the performance.
We evaluate the proposedmodel on synthetic data as well as real-world data from songbird higher vocal center [15] and rodenthippocampus [3, 7, 8].
Results show empirically the proposed model can properly identify the neuralsequence and be robust to noise in a single pass over data.
The contributions of our work include: (1) We propose a novel hierarchical Dirichlet Point Processfor neural sequence detection, and different from prior works [14, 15, 25], our model does not specifythe number of sequence type in advance, while determines adaptively during inference.
(2) Ourmodel with an efficient online inference algorithm performs comparable to the state-of-the-art [25]with a significantly lower time cost.
(3) We use a group of Hawkes processes to capture the temporaldynamics of different types of sequences, which enables our model to distinguish sequences withspatial overlapping neurons.
Related works
Previous studies about neural sequence detection can be divided into three
Supervised methods. Averaging spikes times over multiple trials is the simplest and most usedsupervised method to identify neural sequences.
This approach requires simultaneous sensory cuesor behavioral actions recording and uses a template matching procedure to discover exact timestampsof each sequence [18, 12].
However, as previoulys mentioned, the relationship between neuralactivity and behavior is not always reliable [27], thus this approach may be useless under certaincircumstances.
Bottom-up unsupervised methods. Bottom-up unsupervised methods [20, 21, 23] usually identifyneural sequences via a variety of statistics between pairs of neurons (e.g., cross-correlations).
Thesemethods may be beneficial when involving small numbers of neurons [29], but they have highcomputational complexity and low statistical power when dealing with large number of neurons.
Top-down unsupervised methods. Top-down unsupervised methods can be defined as a kind ofmethod that discover a sequence from the perspective of the full population and is robust to noise atthe level of individual neurons [29].
Maboudi et al.
[14] identifies neural sequences by visualizingthe transition matrix of Hidden Markov Models.
Peter et al.
[19] and Mackevicius et al.
[15] useconvolutive nonnegative matrix factorization (ConvNMF) with different regularization to factorizethe neural data into a couple of neural factors and temporal factors, where each neural factor encodesa type of neural sequence and each temporal factor indicates the times of sequences.
Williams etal.
[25] models neural sequences as a Neyman-Scott process (PP-Seq), which is a spatial-temporalpoint process originally designed to discover clusters in the galaxy.
Each sequence in this modelcorresponds to a latent cause that generates spikes as the offspring events.
Background
A basic probabilistic model of continuous-time event stream is temporal point process whose realization can be represented as an event sequences { x i } N 1 = { t 1 , t 2 , ... } with times of occurrencet i ∈ [0 , T ] .
A way to characterize temporal point processes is via intensity function λ ( t ) , for whichλ ( t ) equals the expected instantaneous rate of happening next event given the history. The functional uθ 1n q 1 =( ґ 1 , m 1 =1) б 2N uθ 1n q 2 =( ґ 2 , m 2 =2) q 3 =( ґ 3 , m 3 =1) ґ ґґґґґ Neuron
Neuron n Neuron N ... uб 1N б 1N θ 2n Three detected sequences Raw spikes
Temporal dynamics of sequences [Figure] sequences form of λ ( t ) varies among different phenomena of interests. In a homogeneous Poisson process (PP),λ ( t ) is assumed to be a constant over time, while in a nonhomogeneous Poisson process, λ ( t ) istime-varying and independent of the history.
Hawkes process. A Hawkes process (HP) [11] is a self-exciting nonhomogeneous Poisson process inwhich past events in history have influenced the intensity of current and future events.
Such influencesare positive but decay exponentially with time:
[Equation]
(1) where λ 0 is the base intensity independent of the history, α is the kernel parameter, and δ is the decayrate of that influence.
An important extension of the Hawkes process is to incorporate additionalinformation about each event, for example, an associated event "label" k i with each event x
i =
( t i , k i ) .
As a consequence, we can define a Hawkes process as a sum of independent Hawkes processes fromdifferent types based on the superposition principle: λ ( t ) =
(cid:80)
Kk
=1 λ k ( t ) , where λ k ( t ) is the intensityfunction from k -th category.
Dirichlet Hawkes process. Dirichlet Hawkes process (DHP) [5, 16] is a method to naturally handleinfinite mixture clusters with temporal dependency.
As opposed to traditional parametric models,
DHP allows the number of clusters to vary over time.
It uses a Hawkes process to model the intensityof events (e.g., the times of neural sequences), while the Dirichlet process captures the diversity ofevent types (e.g., the clusters of neural sequences).
Similar to the Dirichlet process, a DHP, typicallydenoted by DHP ( λ 0 , θ ) is characterized by a concentration parameter λ 0 and a base distributionθ .
It uses the samples θ 1: K from base distribution θ and a collection of triggering kernel functionsϕ θ i ( t, t i ) as the parameters for modeling K different event types.
Then, the events ( t i , θ i ) from
DHP ( λ 0 , θ ) can be simulated as follows:
Sample t 1 from PP( λ 0 ) and θ 1 from θ .
For n > 1 , sample t n from HP( λ 0 + (cid:80) Kk =1 λ k ( t n ) ) • Sample a new θ K +
from θ with probability: λ 0 / ( λ 0 + (cid:80) Kk
=1 λ k ( t n ))
• Reuse previous θ k with probability: λ k ( t n ) / ( λ 0 + (cid:80)
Kk
=1 λ k ( t n ))
where λ k ( t n ) = (cid:80) n − 1 i =1 ϕ
θ
i
t n , t i ) I
( θ i = θ k ) is the intensity function of a Hawkes process for pastevents with type k .
Hierarchical Dirichlet Point Process of Neural Sequences
We model the neural sequences as a hierarchical Dirichlet point process (HDPP).
We use a Dirichletnonhomogeneous Poisson process as the prior for observed spikes, while using a Dirichlet Hawkes process as the prior for neural sequences.
As in Fig. 1, the goal of HDPP is to sequentially identifysequences from observed spikes, then partition these sequences into different types based on bothtemporal dynamics and included neurons.
Consider a continuous-time spike train data with N neurons, its observed spikes can be denotedby a set of S labeled events x s =
( t s , n s , k s ) referring to the spike time, t s ∈ [0 , T ] , neuron,n s ∈ { 1 , ..., N } and indicator to the sequence that x s belongs to, k s ∈ { 1 , ..., K } .
Note that theindicator k s remains unknown until we infer the sequences from the data.
Similarly, the inferredneural sequences can also be represented as a collection of tuples q k = ( τ k , m k ) referring to the timeτ k ∈ [0 , T ] and indicator for sequence type m k ∈ { 1 , .., M } .
According to the definition of the Dirichlet Hawkes process, for a spike x s occurring at time t s itsprobability of belonging to the k -th neural sequence is:
[Equation]
where g 0 is the concentration parameter, g noise is the firing rate for background spikes, and g k ( t ) isthe intensity function of sequence q k , controlling the temporal dynamics of related spikes.
Likewise,the probability of assigning spike x s to either background noise or a new sequence is given by:
[Equation]
After a neural sequence q k> 0 is chosen, we assign it to an existing type m k or create a new type onthe fly as needed with probability:
[Equation]
where λ m ( τ ) is the intensity function that captures temporal dynamics of sequences in type m .
Notethat we assume g k ( t ) and λ m ( τ ) take different functional forms, and g kn ( t ) can be viewed as thefiring rate of neuron n caused by sequence q k .
Following the work in [25], we represent g kn ( t ) via a
Gaussian form as:
[Equation]
Here θ m k n , µ m k n and σ m k n are parameters of each sequence type m k , indicating the weight, offset,and width of neuron n ’s firing rate.
β k is the amplitude of sequence q k , denoting the expected numberof spikes induced by q k [25].
For λ m ( τ ) , we opt the similar form as (1), with parameter α m > 0controlling the self-excitation of previous sequences, hyperparameter δ > 0 controls the decay andhyperparameter η denotes the expected time interval of sequences:
[Equation]
Based on the definition of g k ( t ) and λ m ( τ ) , we regard that the temporal dynamics of spikes andneural sequences are captured by a nonhomogeneous Poisson process and a self-exciting Hawkesprocess, respectively.
For sequence type parameters α m , { θ mn } Nn =1 and { ( µ mn , σ mn ) }
Nn =1 , weassume they follow a Gamma distribution, a Dirichlet distribution, and a normal-inverse-chi-squareddistribution [17, 25] for every neuron and type.
For amplitude β k and background spike rate g noise , weassume they obey a Gamma distribution.
Then, we can describe the proposed hierarchical Dirichletpoint process in a generative way as:
For neural sequence q k , τ k ∼ HP( λ 0 + (cid:80) Mm =1 λ m ( τ k − 1 ) ), β k ∼ Gamma ( a 1 , b 1 ) .
Sample sequence type m k using (4).
• If m k
= M + 1 : θ
M +1 ∼ Dir ( θ 0 ) , ( µ M +1 , σ M +1 ) ∼ NI χ 2 ( µ 0 , κ 0 , υ 0 , σ 0 ) , α M +1 ∼ Gamma ( a 2 , b 2 ) and M = M + 1 .
Sample spikes induced by sequence q k : s k
=
Poisson( β k ), { n s } s k s =1 ∼ Categorical ( θ m k )
and { t s } s k s =1 ∼ N ( τ k + µ m k n s , σ m k n s )
s
=
Poisson( g noise · ∆ t ) { n } s 0 s = ( θ noise ) t s } s 0 s =1 ∼ PP( g noise )
.where θ 0 is a hyperparameter parameter for Dirichlet prior, a 0 , b 0 , a 1 , b 1 , a 2 , b 2 are hyperparameterparameters for Gamma prior, and µ 0 , κ 0 , υ 0 , σ 0 are hyperparameter parameters for normal-inversechi-squared prior.
Inference
Given spikes { x s = ( t s , n s ) } Ss =1 , as previously discussed, the model aims to sample neural sequencesindicator k s for each spike.
Using this insight, we can use a set of particles to sequentially approximateposterior P ( k 1: s | t 1: s , n 1: s ) , while sampling indicator m k for each sequence.
We derive a Particle filter approach for our proposed model.
This inference algorithm exploitsthe temporal dependencies in the observed spikes to sequentially sample the latent variables.
Toefficiently sample from the posterior distribution, a Particle filter keeps track of an approximationof P ( k 1: s − 1 | t 1: s − 1 , n 1: s − 1 ) , and updates it to have an approximation for P ( k 1: s | t 1: s , n 1: s ) when 168 receiving a newly observed spike.
The approximation is evaluated by a set of weighted particles{ p } P , where the weight of each particle reflects how well the observed spikes are explained.
For each particle p , its weight is defined as the ratio between true posterior and importance distribution:w ps = P ( k 1: s | t 1: s ,n 1: s ) Q ( k 1: s | t 1: s ,n 1: s ) .
As the work in [1, 5] did, we choose P ( k s | k 1: s − 1 , t 1: s , n 1: s ) to be theimportance distribution.
Next, the unnormalized weight w ps can be updated as:w
[Equation]
[Equation]
Here, let C m ks \ x s , C m ks \ x s n s denote the spike counts of sequence type m k s and spike counts ofneuron n s in sequence type m k s , both excluding x s .
The likelihood P ( n s | k 1: s , n 1: s − 1 ) is given by a
Dirichlet-Categorical conjugate relation [16]:
[Equation]
After sampling indicator k s from posterior P ( k s | t s , n s , rest ) for each spike, we then sample the typeindicators m k from posterior P ( m k | τ k , X k , rest ) .
The posteriors of both indicators are determined bythe likelihood of the spike/sequence assigned to a given sequence/sequence type and by the temporalprior under Dirichlet nonhomogeneous Poisson process (2)(3) or Dirichlet Hawkes process (4):
[Equation]
[Equation]
where rest represents all other latent variables with x 1: s − 1 , X k = { n s : k s = k } is the neurons ink -th sequence.
The exact form of the two posteriors is given in Supplement A.
Updating parameters. There are three groups of parameters that need to be updated after attributingeach spike to a neural sequence: sequence type parameters Θ m =
( θ m , α m , { µ mn } Nn =1 , { σ mn } Nn =1 ) ,sequence parameters
Θ k = { τ k , β k } and background parameters Θ ∅ = g noise .
Following the literaturein Particle filter devoted to the estimation of an online parameter [2], for most parameters, we performa single Gibbs sampling step to sample from their closed-form posterior under conjugate relation.
Details about derivations are in Supplement B.
Merge and Prune. Generally speaking, neural sequences created during this sequential inferringprocedure are often truly needed, as the decisions of adding new sequences are based on the accumulated knowledge from the past.
However, there is still a chance that some sequences generated atearly samples would become incorrect and that multiple sequences may be close in time.
Thus, weintroduce a mechanism to merge similar sequences and prune incorrect ones.
The similarity of two sequences q k and q k ′ can be defined in terms of the time difference betweenτ k and τ k ′ , as d ( k, k ′ )
= | τ k − τ k ′ | .
We will merge sequences q k , q k ′ and their associated sufficientstatistics when d ( k, k ′ ) <
∆ d .
We also check the variance of firing times for each neuron in every sequence.
That is, we reserve the spikes which fire consecutively in the same neuron n :
Var ( { t s } s n s =1 ) <
ϵ , while removing those that appear to be sparsely distributed.
Besides, we willremove an entire sequence if it has too few spikes after pruning.
Note that there is no need to performsuch merge or prune at every iteration.
Since the operation of merge sequences takes O ( K ) andcomputing variance for each sequence takes O ( s n · N ) , we propose to examine similarity and varianceat a fixed interval O ( i · T ) and only for the current sampled sequence.
The overall pseudocode for inference is presented in Algorithm 1, and its time complexity is
O ( SP ( K + M )) , where S is the number of spikes, P is the number of particles, K is the number ofsequences and M is the number of types.
Initialize do end ; if then
Resample particles end end
Experiments6.1 Synthetic Data
We created three kinds of synthetic data to investigate the robustness to noise, the effectiveness oftemporal dynamics, and the ability to infer the number of types of sequences, respectively, of HDPP:(1) Two types of sequences with background noise.
(2) Two types of sequences occur side-by-sidewith spatial overlapping neurons.
(3) A large set of sequence types with spatial overlapping neurons,where they are much uncertainty in sequence types.
Robustness to background noise.
Figure 2(a) presents the first case of synthetic data, whichdemonstrates the robustness of HDPP.
This data has N = 60 neurons and T = 30 seconds.
The rawspikes (Figure 2(a) left panel) show no visible neural sequences.
However, sequences are revealed bysorting the neurons with the learned weight, type, and offset of this sequence.
Besides, backgroundspikes can also be correctly identified as noise.
Distinguishing overlapping sequences.
Figure 2(b) shows a much more complex case where twosequences with overlap populations of neurons occur side-by-side.
This data has N = 60 neurons and T = 20 seconds.
Usually, it’s hard to distinguish these sequences from the perspective of neurons.
But with the help of temporal dynamics, HDPP could be able to capture sequences’ distinctiveintensity functions (Figure 2(b) top panel), and disambiguate them to different types.
Note that theintensities at the very beginning are missing since HDPP resamples a type indicator for the firstsequence whenever it receives a spike and those empty types are dropped as well as their recordedintensities.
Inferring the types of sequences from data.
Figure 2(c) shows the comparison of HDPP and PP-Seq, demonstrating that HDPP can infer the number of types of sequences from data.
This datahas N = 60 neurons and T = 400 seconds, where there are 11 different types of sequences: 6 typesof smaller sequences involving 10 neurons, while other 5 types of larger sequences involving 20neurons. Note that the larger sequences and smaller sequences have spatial overlapping neurons. HDPP could be able to distinguish the sequences with overlapping neurons without prior information [Figure]
Noise Rate A r ea und e r R O C HDPP PP-Seq ConvNMF
Time
N e u r on s
N e u r on s
Time
N e u r on s
[Figure]
PP
[Figure]
figure: 6,
[Figure]
figure:
Time
Time
[Figure]
s
Time [Figure] on the number of types (Figure 2(c), left panel). However, PP-Seq can only identify the largersequences as a new type when setting the number of types of sequences to 11 (Figure 2(c),
Further, we also tested the robustness of HDPP quantitatively with the area under reciever operatingcharacteriztic (ROC) curves as a metric.
To keep in line with previous work[25], we simulated adataset with M = 1 sequence type and N = 100 neurons, where we varied the background spikesg noise to quantitatively measure the robustness.
From the results of the area under ROC in Fig. 2(d), HDPP achieves promising results as we increase the noise levels.
6.2 Performance with Neural Recordings
6.2.1 Songbird Higher Vocal Center Recording
For real-world data, we first applied our model to the deconvolved spike train from functional imagingdata recorded in songbird HVC during singing [15] 1 .
One of the core features of this data is the highvariability of the song, thus making it difficult to identify neural sequences by a template matchingprocedure.
Identifying neural sequences from Songbird Recording.
Figure 3(a) visualizes the results from HDPP and PP-Seq 2 , where we chose the neurons whose weight { θ mn } Nn =1 was above a threshold,and sorted them based on each sequence’s type m k and offset { µ mn } Nn =1 .
Figure 3(b)-(c) visualizethe learned offset and weight from HDPP.
For offset { µ mn } Nn =1 , there is a clear time-shift amongdifferent neurons ( ± 0.25s for type 1 and ± 0.3s for type 2), and for weight { θ mn } Nn =1 , the twosequence types have their own preference for neurons (neuron 27~62 for type 1 and neuron 1~26for type 2).
Figure 3(d) indicates HDPP converges to similar parameter ranges: The total numberof detected sequences over five independent runs (different random seeds for initiation) are close toeach other. Finally, Figure 3(e) shows the log-likelihood comparison of HDPP and PP-Seq averagingover different runs. [Figure]
Neurons O ff se t Neurons Sequence Type 1 Sequence Type
HDPP PP-Seq [Figure]
TrainingTesting
HDPP {+PP-Seq 1 10 19 28 37 46 55 64 73 0. W e i gh t (b) (c) (d)    
 Th e nu m b e r o f se qu e n ces (e) 
+}
LogL i ke li hood s
6.2.2 Rodent Hippocampal Recording
We then tested our model on a more complex spike train data from a rat’s hippocampal recordingwhen it performs repeating runs down a wooden 1.6m linear track [3, 8, 7] 3 .
The session we used is"Achilles 10252013", which has N = 120 neurons and T = 2068 seconds.
As in [8], we expect toobserve two sequences encoding the two opposite running directions on this liner track.
To have agood knowledge of hyperparameter settings, we performed cross-validation via a "speckled holdoutpattern" strategy, which was previously used in PCA [30], and has recently been applied to neuraldata [25, 26].
We then evaluated the performance of HDPP by calculating log-likelihood on theheld-out set.
Identifying neural sequences from Hippocampal Recording.
Figure 4(a)-(c) shows the raw spikes,sorted results, and two types of sequences annotated by HDPP, where we can clearly observe twogroups of place coding neurons revealed by the sequences.
Figure 4(d) presents the validationlog-likelihoods of four key hyperparameters in HDPP, while other parameters have less impact onperformance or are easy to locate the optimal range.
Not surprisingly, without taking into account ofefficiency, a larger number of particles has a positive effect on the performance of the model, but theimprovement is not obvious during a certain range (e.g. 20~40).
What’s more, HDPP on this dataprefers larger sequence amplitude β k , and higher prior rate b 0 in gamma distribution, which insteadmeans smaller noise firing rate g noise .
6.3 Scalability to Large Dataset
To scale with large recordings, HDPP is supposed to process each spike in an expected constant time.
In other words, the time cost of either calculating intensity functions or updating parameters shouldnot grow with the number of sequences detected so far.
In HDPP, we can note that the influencesof past sequences decay exponentially, using this insight, we can reduce the computational loadby ignoring those past sequences far away from the current time.
More specifically, given a timewindow ∆ t , we maintain an active list of sequences { q k } Kk =1 where t − τ k ≤ ∆ t .
Similarly, usingtemporal windows is also acceptable for PP-Seq by limiting the max length of each sequence, whichis supposed to reduce its loop operations.
Figure 4(e) shows the performance comparison between HDPP and PP-Seq on rodent hippocampal recording. For HDPP, we set the particle number to 20,and for PP-Seq, we adopt parallel MCMC to parallelize the computation. The results indicate that
HDPP is faster than parallel PP-Seq even with 20 seconds as its max sequence length. Figure 4(f)presents the time cost as a function of detected sequences, which empirically verifies that HDPP hasa constant time cost per spike after settling in. (b) (c) (d) (a) [Figure]
[Figure]
Discussion
We present a hierarchical Dirichlet point process (HDPP) for neural sequence detection with anefficient online inference algorithm.
The proposed model follows the framework of the Dirichlet pointprocess, where spike firing rates are modeled as a nonhomogeneous Poisson process and temporaldynamic of different sequences is captured via a Hawkes process.
Unlike previous works, HDPPlearns the number of sequence types adaptively during inference and takes in data in a streamingfashion, thus can be easily adapted to new data or large neural recordings.
In the experiments,favorable results from both synthetic data and real-world spike recordings have demonstrated that HDPP can reliably and efficiently identify neural sequences in a single pass.
However, there are some conditions our model may fail: (1) Time warping noise.
Our model does notmodel the variability in sequence, which is said to be a common phenomenon in neural data [25, 28].(2)
A Low firing rate of sequences.
Generally speaking, a high firing rate makes it easier to identifysequence from background noise.
As the firing rate becomes lower, the performance of our modeland existing methods decreased (supplement material Figure 1).
Furthermore, it is an exciting futuredirection to investigate how we can learn HDPP via online variational inference without truncatedapproximation of the stick-breaking construction [13, 24].
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