On neural spike sorting with mixture models

ON NEURAL SPIKE SORTING WITH MIXTURE MODELS LI MENGXIN (B.Sc, University of Science and Technology of China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF STATISTICS AND APPLIED PROBABILITY NATIONAL UNIVERSITY OF SINGAPORE 2010 i Acknowledgements My thanks go to Department of Statistics and Applied Probability at NUS for providing both an admission and a financial aid to my study here in the graduate program. Otherwise perhaps I would have been lost on my road to pursue higher objectives. The department provided me an origin from which I started a fruitful journey. The most critical thanks are due to my advisor, Prof. Loh Wei Liem. Prof. Loh has been tremendously supportive throughout, providing encouragement and valuable advice. To name a few, his encouragement made me to choose a research area that has great scientific significance, and his advice guided me to think about mixture model with an extra comtamination class which is very new to the statistics field of Gaussian mixture. Moreover, the numerous conversations we had leveraged my thinking skills and shaped my research taste. It was also important for Visiting Prof. Chen Jiahua to teach the Advanced Statistics course here. My success in his course both inspired my statistical thinking and boosted my confidence in statistics. Besides research, there is always life. The friendships I built with my fellow students here contributed to my experience as a graduate student. The great number of chats with Wang Daqing were so pleasant that I could not forget. His sporty ACKNOWLEDGEMENTS ii spirit transformed my narrow viewpoint about excellence to a more broadened one. Jiang Binyan’s humors made us feel more joy on an arduous road to PhD. The computer games Liang Xuehua and I played together tunneled us through time to the happy childhood days. I would like to thank them all for providing an opportunity for me to learn from and stay with them for the years here. Finally I must acknowledge a great debt to the very many people who have formed a synergic environment both intellectual and social that I enjoyed during my graduate study at NUS. These include the professors, the graduate students and the administrative personnel from our department, and the undergraduate students from the university whom I had served for as a Teaching Assistant, and the friends from other departments. iii Contents Acknowledgements i Table of Contents iv Summary v List of Tables vii List of Figures viii Introduction 1.1 The Statistical and Neuroscience Problem . . . . . . . . . . . . . . 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Review of mixture models and methods . . . . . . . . . . . . 1.2.2 Review of neural spike sorting . . . . . . . . . . . . . . . . . 1.3 Preview of Our Work . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Isolated Spike Analysis 15 2.1 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Estimation of the Number of Neurons . . . . . . . . . . . . . . . . . 17 2.3 Estimation of the Spike Shapes . . . . . . . . . . . . . . . . . . . . 44 2.4 Convergence Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Isolated and Overlapped Spike Analysis 3.1 Estimation of the Number of Neurons Using Determinants . . . . . 54 54 3.1.1 The statistical modeling of the data . . . . . . . . . . . . . . 54 3.1.2 A method using determinant of moment matrix . . . . . . . 58 CONTENTS iv 3.1.3 A method using determinant of Toeplitz matrix . . . . . . . 78 3.1.4 Relaxation of the Gaussian assumption of the noise . . . . . 90 3.2 Simulations of the estimators using determinants 3.2.1 . . . . . . . . . . 98 Comparison of moment methods and trigonometric moment methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.2.2 Study of minimality condition of trigonometric moment method101 3.2.3 Finite sample simulation of trigonometric moment method . 103 3.2.4 Application of majority rule to trigonometric moment method106 3.3 Estimation of the Number of Neurons Using Eigenvalues . . . . . . 110 3.4 Estimation of Spike Shapes for Data with Overlapping Events: EM Algorithm and Simulation . . . . . . . . . . . . . . . . . . . . . . . 121 Bayesian Sorting of Isolated Spikes 4.1 4.2 131 Representation of Lewicki (1994) . . . . . . . . . . . . . . . . . . . 133 4.1.1 Definition of single-channel spikes . . . . . . . . . . . . . . . 133 4.1.2 Estimation of the spike shape under single neuron model . . 134 4.1.3 Estimation of the spike shapes for multiple neuron model . . 141 4.1.4 Estimation of the number of spike shapes . . . . . . . . . . . 143 4.1.5 Decomposition of overlapped spikes . . . . . . . . . . . . . . 144 Bayesian Clustering of Multichannel Isolated Spikes using Smoothness Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.2.1 Definition of multi-channel Spikes . . . . . . . . . . . . . . . 146 4.2.2 Detection of multi-channel spikes . . . . . . . . . . . . . . . 147 4.2.3 Estimation of the spike shape for one neuron . . . . . . . . . 148 4.2.4 Estimation of the spike shapes of multiple neurons . . . . . . 155 4.2.5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . 158 Conclusion 161 Bibliography 163 v Summary While the nature of physics is to understand matter, the nature of neuroscience is perhaps to understand brain. With the advent of neural data collecting hardware, from single electrode tip to electrode array, there is a need to analyze these huge amount of neural data. The analysis of these data will require new developments in the inferential and statistical tools. This thesis attempts to develop a new set of statistical mixture models and methods and apply them to the neural data analysis. The problem we are trying to solve is called neural spike sorting in literature. There are three basic objectives of spike sorting. The first is to estimate the number of neurons which contribute to the recorded neural data. The second is to identify the spikes, i.e. the little curves in the recorded neural data, with the neurons. The third is to find the characteristic spike shape of each neuron. Spike sorting can not be formulated in standard terms of multivariate clustering. Because a spike can originate from simultaneous activity of multiple neurons, and is called an overlapped spike. These overlapped spikes not belong to any of the available clusters. Therefore new model can be developed. This thesis attempts to sort spikes either when there are no overlapped spikes or when there are, while providing a new set of statistical mixture models and SUMMARY vi methods. To estimate the number of neurons, we extend the current statistical mixture models to allow a contamination class of mixture components, and extend the current statistical mixture methods to estimate the number of mixture components, i.e. the number of neurons. To estimate the characteristic shape and identify the spikes with the neurons, we extend the current statistical mixture models and methods to allow a sparse set of mixture components which model the overlapped spikes. Lastly we also develop a multivariate extension of Lewicki (1994) to sort spikes from multiple electrode tips. vii List of Tables 2.1 frequency of accurate estimation of ν0 = . . . . . . . . . . . . . . 52 2.2 frequency of accurate estimation of ν0 = . . . . . . . . . . . . . . 52 2.3 frequency of accurate estimation of ν0 = . . . . . . . . . . . . . . 52 2.4 frequency of accurate estimation of ν0 = . . . . . . . . . . . . . . 53 3.1 percentage accuracy of estimation of ν0 using determinant of moment matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.2 percentage accuracy of estimation of ν0 = using determinant of moment matrices when n = 100, 000 . . . . . . . . . . . . . . . . . . 99 3.3 percentage accuracy of estimation of ν0 using determinant of Toeplitz matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.4 percentage accuracy of estimation of ν0 = using determinant of Toeplitz matrices when n = 100, 000 . . . . . . . . . . . . . . . . . . 99 3.5 Minimal γ with precision 0.01 for subspace of parameter space Ω . . 102 3.6 Frequency (%) of νˆ1 = ν with standard error in parenthesis . . . . . 104 3.7 7-dimensional spike shapes . . . . . . . . . . . . . . . . . . . . . . . 107 3.8 Frequency (%) of νˆ1,k = ν, k = 1, . . . , 7, with standard error in parenthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.9 Frequency (%) of νˆmaj = ν with standard error in parenthesis . . . . 110 4.1 average number of classification errors . . . . . . . . . . . . . . . . 159 viii List of Figures 1.1 four clusters of sample spikes, each cluster from a distinct neuron . 1.2 a sample recording of an electrode . . . . . . . . . . . . . . . . . . . 3.1 true spike shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.2 initialized spike shapes . . . . . . . . . . . . . . . . . . . . . . . . . 128 3.3 estimated spike shapes . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.1 true multi-channel spike shapes . . . . . . . . . . . . . . . . . . . . 158 Chapter Introduction In electrophysiological experiments to record neural signals, the firing of the neurons usually shows up as a voltage waveform on the electrode tip. This shortduration waveform, called a neural spike, is commonly modeled as a spike shape contaminated with noise. If a spike involves a single neuron firing, the spike is called an isolated spike. Otherwise, the spike involves multiple neuron firings, the spike is called an overlapped spike. Sometimes an overlapped spike is easy to spot by eye. For example two close peaks in one spike may be the evidence that multiple neurons involved. Figure 1.1 illustrates a number of isolated spikes from four neurons, which are aligned, i.e. the peaks are along the same vertical line. It is spike detection algorithm that aligns spikes properly for later clustering. The statistical methods in this thesis assume the spikes have been properly aligned, thus are essentially clustering algorithms. Most algorithms available so far not consider overlapped spikes, or consider them as rare outliers. The principle component analysis method (Lewicki (1998)) and the wavelet method (Quian (2004)) are in this category. Other algorithms go further to decompose overlapped spikes. For example, the algorithm in Lewicki (1994) uses search tree to decompose overlapped spikes and Bayesian clustering to CHAPTER 4. BAYESIAN SORTING OF ISOLATED SPIKES 156 and ∫ ZV (Σω,k ) = exp[− ∑ r ∏ √ Σ−1 ω,k ⃗vk (r)T ⃗vk (r)]d⃗vk = ( 2π)JR ( R j √ j 2σω,k R ) . R This choice of prior would penalize the probability of “rugged” spike shapes and prefer the “smooth” spike shapes. We start by investigating different forms of preference of the smoothness of spike waveform. When Σω,k does not vary over k, it means our preference of smoothness does not vary over different spike models. When Σω,k is a diagonal matrix, it means our preferences of smoothness of different channels are “independent”. Since the preference is something we can choose, in the following context I would like to assume Σω = Diag(Σω,1 , ., Σω,K ). Then the joint prior representing the preference of smoothness is P (⃗v1:K | Σω , M ) = K ∏ P (⃗vk | Σω,k , M ). k=1 In Lewicki (1994), the latent variables of mixture model, i.e. the classification of observations to the mixture components, are not explicitly dealt with. The likelihood used is thus marginal distribution which integrates out the latent variables. In our extension this section, we use full likelihood, which does not integrate out the latent variables. Let cn = k denotes that the n-th spike comes from the k-th ⃗ {n:cn =k} denote the spikes which come from the k-th mixture component. Let D mixture component. Assume the classification are c1 , . . . , cN , the hyper-parameters are Σω and Ση , ⃗ {n:cn =k} , the covariance matrix of multichannel the data of the kth spike model are D CHAPTER 4. BAYESIAN SORTING OF ISOLATED SPIKES 157 noise is Ση = diag(ση,1 , . . . , ση,J ), and the spike shape is ⃗sk or the first difference of it is ⃗sk (·), i.e. ⃗vk . Then the probability of the multi-channel spikes is ⃗ {n:cn =k} | ⃗vk , Ση , Mk ) P (D ∑ ∑ ⃗ Σ−1 ⃗ sk (r))] sk (r))T 2η (D exp[− {n:cn =k} r (D n (r) − ⃗ n (r) − ⃗ , (4.16) ⃗ ⃗ {n:cn =k} , Σω,k , Ση ) = P (D{n:cn =k} | ⃗vk , Ση )P (⃗vk | Σω,k ) P (⃗vk | D ⃗ {n:cn =k} | Ση , Σω ) P (D (4.17) = in which ZD (Ση ) )|{n:cn =k}| (√ , ZD (Ση ) = ( 2π)RJ |Ση |−R/2 where |{n : cn = k}| is the set size of {n : cn = k}. The posterior for ⃗vk is, = exp[−M (⃗vk )] , ZM (Ση , Σω,k ) (4.18) in which M (⃗vk ) = T ∑ vk,j ∑ ∑ vk,j Σ−1 ⃗ n (r) − ⃗s(r)) + ⃗ n (r) − ⃗s(r))T η (D (D j 2(Rσω,k )2 r j {n:cn =k} and ∫ ZM (Ση , Σω,k ) = exp[−M (⃗vk )]d⃗vk Rearrange the posterior equation (4.17), we have ⃗ {n:cn =k} | Ση , Σω ) = P (D ⃗ {n:cn =k} | ⃗vk , Ση )P (⃗vk | Σω,k ) P (D . ⃗ {n:cn =k} , Σω,k , Ση ) P (⃗vk | D (4.19) CHAPTER 4. BAYESIAN SORTING OF ISOLATED SPIKES −1.0 0.0 1.0 channel two potential −1.0 0.0 potential 1.0 channel one 158 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 relative time relative time Figure 4.1: true multi-channel spike shapes Apply (4.15),(4.16),(4.18) to the rearranged equation (4.19), we have ⃗ {n:cn =k} | Ση , Σω,k ) = P (D ZM (Ση , Σω,k ) . ZW (Σω,k )ZD (Ση ) This splitting of data into classes which are independent gives chance to calculate the most probable estimate of parameters “independently” conditioning on the classifications c1 , . . . , cN . Then the classification can be re-estimated by using the obtained most probable estimate of parameters based on the posterior probabilities of a spike coming from one of the set of mixture components (just classify the spike to the mixture component with the maximum probability mass). 4.2.5 Simulation results When the number of mixture components is unknown, we may use the methods developed in Chapter and to estimate it. Therefore in this simulation we assume the number of mixture components is known to be ν = 4. The C programming CHAPTER 4. BAYESIAN SORTING OF ISOLATED SPIKES 159 Table 4.1: average number of classification errors sd n=20 0.04 0.06 0.08 3.2 0.10 4.8 n=100 0.1 0.8 language implementation of spike sorting software (see Lewicki (1994)) outputs four estimated spike shapes when analyzing one of the dataset distributed with the software. We use them as the four true spike shapes in channel one. Lewicki’s software can be obtained from Lewicki’s website http://cnl.salk.edu/˜lewicki/. The four spike shapes in channel two are chosen to be the spike shapes in channel one scaled by 0.7 for convenience. Figure 4.1 shows the four spike shapes in channel one on the left panel and the four spike shapes in channel two on the right panel. In Table 4.1 n is the number of multi-channel spikes. Since we choose to generate the same number of multi-channel spikes from each neuron, each multichannel spike shape is associated with n/ν multi-channel spikes. The sd is the standard deviation of noise, i.e. the noise level. These noise makes multi-channel spikes different from multi-channel spike shape. The heights of the peaks of the four spike shapes in channel one are 0.1349, 0.6385, 0.3692, 0.2047. The ones in channel two are 0.09442, 0.4469, 0.2584, 0.1433. These height values are compared with sd to let us have a rough idea about the signal noise ratio of simulated data. The entries in Table 4.1 are the average number of classification errors out of n multichannel spikes for certain sd and n. It is an average of the number of classification errors of 10 repetitions, therefore some entries are fractional numbers. When sd = 0.10 and n = 100, that is, the noise level is 0.10 and the number of multichannel spikes is 100, and the signal to noise ratio is approximately 1, the classification error rate is averagely out of 100, or 3%. The algorithm achieves CHAPTER 4. BAYESIAN SORTING OF ISOLATED SPIKES 160 this small error rate under signal to noise ratio approximately (harsh condition for a single random variable) because it pools all the information in the time series together to the clustering. With more multichannel spikes while fixing the number of clusters, we expect the error rate is even lower. The column n = 100 in Table 4.1 shows that the smaller the noise level, or the higher the signal noise ratio, the lower the error rate. When sd = 0.10 and n = 20, that is, the noise level is 0.10 and the number of multichannel spike is only 20, the average classification error rate is 4.8 out of 20, or 24%, which is quite high. The interpretation is that when signal noise ratio is low, approximately indeed, and the number of multichannel spikes in each cluster is small, about 20/4 = 5, there are not enough data to estimate the spike shapes well, therefore the error rate is higher. When we look at sd = 0.04 and n = 20, that is, the signal noise ratio is higher, approximately indeed, the spike shapes are estimated well, therefore, classification error rate is low, indeed. Note the here means no error classification occurred in our simulation, but does not imply no error classification in infinite number of simulations. 161 Chapter Conclusion In this thesis we have developed a new set of statistical mixture models and methods and applied them to analyze simulated neural data. Our work contribute to both the statistics and the neuroscience. On the statistics side, we have developed statistical mixture models to include contamination mixture components and statistical mixture methods to estimate the number of “major” mixture components of such mixture models, and we have also developed statistical mixture models to include extra structure among the set of means of mixture components and an EM algorithm to find MLE of such mixture models. On the neuroscience side, we have attempted to sort both isolated and overlapped spikes using frequentist approach, and we have also attempted to sort spikes from multiple electrode tips recording using Bayesian approach. Our frequentist approach has the robustness that the estimators of the number of neuron have good convergence rate regardless of the nuisance overlapped spikes. Our Bayesian approach pools information from multiple channel thus possibly gives better accuracy. Also like all Bayesian approach, many times only very small size of data is needed to achieve reasonable results. 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[...]... assess the accuracy On the other hand, we may also apply statistical models including finite mixture to spike sorting, which have theoretical justification about the estimation when the model assumptions are verified In this thesis we develop statistical mixture models with an eye on its application to spike sorting In Chapter 2, we extend the estimation of the order of univariate mixture in Dacunha-Castelle... simulation results In Chapter 3, we reconsider the estimation of the order of finite mixture by using determinants We found that we can estimate the order of finite mixture even when there are contaminations in the finite mixture provided the proportion of contamination is reasonably and non-asymptotically small and the distribution of outliers satisfies a mild condition This idea of modeling overlapped spikes... we shall develop some new mixture models and methods and apply these models and methods to the neuroscience problem, i.e spike sorting It is important to look at the state-of-the-art developments in mixture models and methods and neural spike sorting 1.2.1 Review of mixture models and methods The number of mixture components is also called the order of mixture As the order of mixture increases, the number... giving exponential convergence rate This idea of allowing contaminations in mixture model was originally suggested by Sahani (1999) In Sahani (1999) the distribution of contaminations is assumed to be uniform In our work the distribution of contaminations is only required to satisfy a mild condition In Chapter 3, we also provide a method to estimate the spike shapes given the number of neurons The data... all spikes for the convenience of analysis A spike can be an isolated spike when the number of neurons which are detected as fired during this subinterval is only one, or an overlapped spike when the number is more than one Throughout this chapter we will focus on the analysis of isolated spikes, so we use spike as a short for isolated spike Assume we have n0 number of spikes recorded, with the nth spike. .. spikes with peak aligned isolated spikes The second step of spike sorting is an optional dimension reduction When we do not use dimension reduction, all the measurements in the waveform of spikes are being used for clustering This is associated with template method mentioned in Lewicki (1994) Since the dimension is very high in this case, it is difficult to do clustering The alternative is to use dimension... These little curves with noise are also called action potentials, or neural spikes, or simply spikes The neuroscience problem, which is called spike sorting, is to determine the number of neurons that have fired during the recording of the time series, assign the spikes to the associated neurons, and estimate the characteristic curve without noise associated with individual neuron In order to do this,... mixture components, or in the context of isolated spike analysis, the number of neurons 2.2 Estimation of the Number of Neurons As we can see from (2.1), the problem fits well into a finite mixture of multivariate normal distributions Each mixture component is corresponding to a distinct neuron Thus the number of mixture components is exactly the number of neurons In Dacunha-Castelle and Gassiat (1997) the... probable estimate of the spike shapes Shoham et al (2003) used mixture of t-distribution to lessen the effect of outliers on estimation in Gaussian mixture It used a penalty term based on minimum message length criterion to form a penalized maximum likelihood method to estimate the number of neurons But no theoretical convergence to the CHAPTER 1 INTRODUCTION 12 true number of neurons was proven Other approaches... characteristic spike shapes For example, Lewicki (1994) used continuous piece-wise linear spline to model the spike shape, and finite Gaussian mixture to model random noise and the uncertainty that a spike could be from one of a set of neurons The parameters of the splines represent the characteristic spike shapes of the neurons A critical problem in spike sorting is the clustering of overlapped spikes An . waveform on the electrode tip. This short- duration waveform, called a neural spike, is commonly modeled as a spike shape contaminated with noise. If a spike involves a single neuron firing, the spike. integrate these overlapp ed spikes with peak aligned isolated spikes. The second step of spike sorting is an optional dimension reduction. When we do not use dimension reduction, all the measurements. of clusters. Second, we may use clustering algorithms based on statistical models, usually mixture models. Most applications of mixture model to spike sorting used Gaussian distribution to describe