Wavelet methodology to improve single unit isolation in primary motor cortex cells

Abstract

The proper isolation of action potentials recorded extracellularly from neural tissue is an active area of research in the fields of neuroscience and biomedical signal processing. This paper presents an isolation methodology for neural recordings using the wavelet transform (WT), a statistical thresholding scheme, and the principal component analysis (PCA) algorithm. The effectiveness of five different mother wavelets was investigated: biorthogonal, Daubachies, discrete Meyer, symmetric, and Coifman; along with three different wavelet coefficient thresholding schemes: fixed form threshold, Stein’s unbiased estimate of risk, and minimax; and two different thresholding rules: soft and hard thresholding. The signal quality was evaluated using three different statistical measures: mean-squared error, root-mean squared, and signal to noise ratio. The clustering quality was evaluated using two different statistical measures: isolation distance, and L-ratio. This research shows that the selection of the mother wavelet has a strong influence on the clustering and isolation of single unit neural activity, with the Daubachies 4 wavelet and minimax thresholding scheme performing the best.

1. Introduction

One of the challenges facing the burgeoning field of neuro-science is the reliable recording of electrophysiological signals from live tissues. In addition to the challenges of reaching the desired region (cortex, basal ganglia, reticular formation, etc.) there is also the challenge of isolating and properly identifying action potentials from single cells in a noisy and crowded environment. Oftentimes the recording shows similar signal features among multiple neurons, high background noise, as well as various background waveforms from neighboring cells. Improving the interpretation of the recording based on these factors is of great importance for cell identification in various applications such as brain computer interface (BCI) (Ortiz-Rosario and Adeli, 2013) and understanding and diagnosis of various neurological disorders (Florin et al., 2013). To that end, approaches have been proposed such as tetrode electrode arrangements (Gray et al., 1995), spectral transformations (Luczak and Narayanan, 2005), template matching (Kaneko et al., 1999), and various approaches using wavelet transform (WT) (Pavlov et al., 2007; Wiltschko et al., 2008; Shalchyan et al., 2012; Lai et al., 2011; Chan et al., 2010; Geng and Hu, 2012).

WT is known primarily as a signal denoising and compression tool (Katicha et al., 2013). In recent years it has been used widely in different applications such as feature extraction (Hsu, 2013; Kodogiannis et al., 2013; Amini et al., 2013), image compression (Tao et al., 2012), vibration control (Amini and Zabihi-Samani, 2014), system identification (Su et al., 2014), and most recently for the automated diagnosis of neurological disorders such as epilepsy (Adeli et al., 2003; Ghosh-Dastidar and Adeli, 2007; Ghosh-Dastidar et al., 2007; Adeli and Ghosh-Dastidar, 2010), Alzheimer’s disease (Adeli et al., 2005a, b, 2008; Sankari et al., 2011, 2012; Sankari and Adeli, 2011), autism spectrum disorder (Ahmadlou et al., 2012b, d, 2010), ADHD (Ahmadlou and Adeli, 2010b, 2011; Ahmadlou et al., 2012c), and Major Depressive Disorder (MDD) (Ahmadlou et al., 2012a).

The power of WT stems from its ability to provide a multi-resolution time–frequency analysis that allows detection of rapid frequency variations, discontinuities, and other features invisible in time domain. Unlike Fourier transform, which uses cosine and sine functions, wavelets use a predefined family of functions that can be scaled to achieve different goals depending on the data set. The scaling of WT allows a signal to be decomposed into approximation (low frequency, high amplitude) and detail (high frequency, low amplitude) signals. This decomposition allows signal processing to be performed in different levels of detail, extracting with each layer more information out of the signal; hence the name multi-resolution (Fig. 1). Each layer of the decomposition yields two down-sampled signals, one representing the approximation, and the other representing the detail, with subsequent layers extracting additional detail signals from the approximation signal.

Fig. 1.

WT decomposition of a signal S into approximation coefficients (cA) and detail coefficients (cD). The figure shows a standard WT two-level signal decomposition into an approximation signal and a detail signal. In each subsequent level of decomposition additional detail coefficients are extracted from the approximation signal.

In the context of spike sorting, researchers have used WT as a means of feature extraction, as well as denoising. Pavlov et al. (2007) used WT as a feature extraction, using the WT coefficients as features, and comparing it against principal component analysis (PCA). Wiltschko et al. (2008) compared the performance of denoising using WT versus the Butterworth bandpass filter and observed that with the removal of the approximation coefficients (the low frequency components of the signal), the cell isolation was improved. Shalchyan et al. (2012) utilized WT as a means of filtering and feature extraction in the context of a BCI, as the proper isolation of cells is of great interest in the operation of a BCI device.

A key decision in successful application of WT in electrophysiological recordings is the selection of the mother wavelet. Most researchers in the field have used the Daubechies wavelets with a few using other types of wavelets such as symmetric wavelet (Diedrich et al., 2003), Coifman wavelet (Kim and Kim, 2003), and biorthogonal wavelet (Nenadic and Burdick, 2005).

In this paper, a methodology is presented for the improvement of neuronal cell isolation using wavelet transform for denoising. The study compared various mother wavelets, four thresholding schemes, two thresholding rules, and three thresholding scaling, and then employed PCA for feature extraction (Meraoumia et al., 2013). Identifying the mother wavelet and thresholding schemes that yielded the best results can help develop a more effective approach for the processing of neurophysiological data. Two data sets were used in this research: simulated single unit activity (SUA) recordings using Neurocube (Camunas-Mesa and Quiroga, 2013), and real SUA recorded from an awake behaving primate. Statistical measures were used to evaluate both signal and clustering quality.

2. Materials and methods

2.1. Subject and task

The data used in this research came from an old-world male macaque monkey (Macaca fascicularis). He was trained on a reach task with the goal of mimicking a natural reach motion bilaterally. This was part of a larger study designed to reveal neural activity associated with control of reaching. The subject was trained to sit in a primate chair and reach to touch targets displayed on a computer monitor. The task required the subject to reach with one arm while holding the other on a switch on a desk-like surface at waist level. Both arms were used for reaching, with the computer software controlling the task (Tempo, Reflective Computing, St. Louis, MO, USA) set to vary the arm and target across trials. A trial began with the subject placing both hands on pressure sensitive plates. This action triggered the display of an indicator light for each switch to show that the hand was in position. Then a target was displayed on one side of the screen or the other, a box at eye level. Through operant conditioning, the subject learned that green targets had to be touched with the left hand and red targets with the right. After the target was shown for about 1 s a flashing circular image in the center of the screen (noted as a circle in Fig. 2) cued the subject to reach for the correct target. Thus, there were four possible movement combinations: left hand/left target (LHLT), left hand/right target (LHRT), right hand/left target (RHLT), and right hand/right target (RHRT). Once the target was reached with the proper hand, the subject was rewarded with a banana pellet dropped into the food well.

Fig. 2.

Task scheme with four possible movements: left hand/left target (LHLT), left hand/right target (LHRT), right hand/left target (RHLT), and right hand/right target (RHRT).

It took about 6 months to train the subject to perform the task with an accuracy of over 90%, deemed to be sufficient for the purpose of this research. The subject was implanted recording chambers and subcutaneous EMG electrodes following a surgical procedure described previously (Davidson and Buford, 2004; Herbert et al., 2010). Analysis of EMGs was not part of the present analysis. The recording chamber was inserted and fixed in a craniotomy made in the left parietal bone and was immobilized using bone screws and dental acrylic. The chamber was angled 10° to avoid major blood vessels. The experiment was compliant with NIH Guide for the Care and Use of Laboratory Animals and was approved by animal care protocols established by The Ohio State University.

2.2. Data acquisition

Three different types of invasive brain signals are typically used in neuroscience research: local field potentials (LFP), multiple unit activity (MUA), and single unit activity (SUA). The LFP is an electrophysiological signal that reflects the electrical current of nearby dendritic synapses from neurons in proximity of the recording device and are typically low-passed (typically below 500 Hz) (Donoghue et al., 2007). MUAs are a collection of spikes (or units) recorded over a grid, typically a square array for cortical recordings (Donoghue et al., 2007) or axial array for deeper linear recordings (Woolley et al., 2013). The spikes recorded in MUA cannot be typically sorted due to their large quantity, and are taken to evaluate firing frequencies only, which is why they are plotted using raster plots instead of voltage plots. Lastly, SUA is the recording of neuronal activity at close proximity to a neuron or group of neurons using a single tip electrode. An SUA has enough resolution to identify individual spikes within the electrophysiological signal. Oftentimes, spikes are discretized from the SUA signal by means of amplitude triggers reaching a voltage considered high enough for an action potential. The spikes are then sampled in a limited time window (~1 ms) known to be wide enough to observe the entire ionic exchange (or shape) of the waveform. The present analysis was designed to detect SUA.

2.2.1. Simulated recordings

The Neurocube (Camunas-Mesa and Quiroga, 2013) was used to create a realistic simulation of SUA in the brain. The simulation creates a custom cube of tissue which is randomly populated by neurons, each with its own firing frequency and waveform. Multiple options are customizable, such as size of the cube, neuronal density, rate of activity, and duration of the recording. For this work, the model was used to create a 1-mm³ cube with 300,000 neurons/mm³, and 7% rate of activity. A recording of 30 s was used which was considered computationally manageable for both simulation as well as analysis. Other customizable options include firing rates, amount of single spikes, as well as the size of the electrode tip and its configuration (single or tetrode). The options were left in their default (exponential firing rates, automatic single unit count, and single electrode tip configuration), but the electrode tip size was changed to 7 μm from a default value of 40 μm to properly simulate the electrodes used in the lab. A total of four signals were generated to test and validate the models. The first signal, shown in Fig. 4, was used to choose which model configuration yielded the best results. Fig. 4A displays the most prevalent spikes (69 in total) with their respective amplitudes. Fig. 4B displays the same spikes modified by the inverse of the distance squared, which enhances those closest to the electrode tip. Fig. 4C portrays all 69 cells in their default position noted by circles with the closest cells to the tip identified with solid circles. The relevant cells in the simulation are identified by numbers 1–69. The other three signals were created, with varying rates of activity (8%, 9%, and 10%), to validate the selected model and compare results in noisier environments. The simulated signals and the resulting waveform were converted into discrete spikes using Spike2 software (Cambridge Electronic Design, Cambridge, UK).

Fig. 4.

Simulation used for model selection. (A) Displays the most prevalent spikes (69 in total) with their respective amplitudes. (B) Displays the same spikes modified by the inverse of the distance squared, which enhances those closest to the electrode tip. (C) Portrays all 69 cells in their default position noted by circles with the closest cells to the tip identified with solid circles.

2.2.2. In vivo recordings

An additional in vivo recording was used to validate the model. An awake behaving primate was used to record this date from the motor cortex. The data was acquired using Spike2 software and a 1401 data acquisition unit (Cambridge Electronic Design, Cambridge, UK) which recorded the neural data sampled at 50 kHz, and amplified 1000 times. The neural data was bandpass-filtered in the 300–4000 Hz band. During the recording, the trigger threshold in Spike2 was set to record spikes crossing −0.75 mV (the inverted action potential had to be lower than the set value to trigger). A glass coated tungsten microelectrode was used to record the neuron action potentials (Alpha Omega, Alpharetta, GA). The recording session consists of 30 s of SUA in the primary motor cortex. The recording had a total of 6295 spikes and three suspected neurons.

2.3. Wavelet filtering

Some researchers have suggested that the choice of the mother wavelet should be based on similarities between the signal and the wavelet ψ function (Rafiee et al., 2009; Shalchyan et al., 2012; Pavlov et al., 2007). Based on similarity with the shapes of action potentials, five different mother wavelet functions and multiple orders were investigated in this research: Daubachies (db1–db5), symmetric (sym2–sym6), Coifman (coif1–coif 5), Meyer (mey), and biorthogonal (bior1.1, bior1.3, bior1.5, bior2.2). Multiple wavelet orders were chosen to test the properties that change with each, including the vanishing moments, which relates to the ability for the wavelet to represent polynomial behavior and/or obtain different information in a signal. Fig. 5 presents the scaling function (φ) on the left and the wavelet function (ψ) the right for each type of wavelet. It is the latter that presents similarity to extra-cellular action potentials. Fig. 6 displays a recorded SUA where an action potential along with different inverted wavelet ψ functions where the similarity can be observed more easily. The shape of the action potentials is affected by the distances, orientation, and impedance of the recording device, as well as other surrounding factors such as proximate cells and overall background activity. A few researchers (Bestel et al., 2012; Lai et al., 2011) have used Haar wavelet, the oldest mother wavelet, in their analysis of spike sorting. This wavelet was included in this research in the form of the first order Daubachies (db1 = haar) even though it is highly dissimilar to a spike and displays poor performance in other signal processing applications (Cao et al., 2003).

Fig. 5.

Scaling (φ) and wavelet (ψ) functions for five different mother wavelets: Daubachies 4 (db4), symmetric 4 (sym4), and Coifman 4 (coif4), discrete Meyer (dmey), and biorthogonal 1.5 (bior1.5).

Fig. 6.

SUA signal (~3 ms) displayed along with different inverted mother wavelet ψ functions. The functions were inverted to enhance the resemblance to the action potential.

The recorded data was exported from Spike2 into Matlab (Math-works, MA, USA) for processing using the Wavelet Toolbox (Misiti et al., 1996). Discrete wavelet transform (DWT) was used to analyze the entire recording file. For each mother wavelet, wavelet decomposition was carried up to level 7; additional levels of decomposition did not improve the accuracy. Fig. 7 shows the filtering and isolation procedure used in this research schematically.

Fig. 7.

Schematic representation of the methodology used. The first step is decomposition of the recorded signal using five different mother wavelets. Then, the approximation and detail signals are processed using different threshold algorithms (SQRT, SURE, MIMA) with different parameters (soft/hard thresholding and coefficient scaling). Next, they are reconstructed into a filtered signal. To extract the spikes from the signal a window of 1 ms and a trigger of −0.75 mV are used. Then, the spikes are plotted using the PCA algorithm and clustered using Gaussian mixture algorithm. Finally, mother wavelets are compared using different measures of the signal and the clustering quality.

2.4. Thresholding

After the signals were decomposed using WT, a thresholding scheme was used to remove the noise in the resulting detail and approximation signals. Thresholding is composed of three steps: thresholding scheme, thresholding scaling, and thresholding rule.

2.4.1. Thresholding scheme

Four different strategies were used to determine which amplitude threshold level to apply: fixed form threshold using squared log of the signal (SQRT), minimax (MIMA), rigorous Stein’s unbiased estimate of risk (R-SURE), and heuristic Stein’s unbiased estimate of risk (H-SURE) (Cao et al., 2003). They each determine a level at which the values of the different detail/approximation coefficients must cross in order to be selected for removal. This way, lower amplitude noise, among other things, can be safely removed from the signal while the relevant information is kept. The following formula was used to set a threshold level using SQRT (Donoho and Johnstone, 1994):

$$t=\sqrt{2\times log(N)}$$ (1)

where N is the number of samples of each individual detail/approximation signal.

R-SURE was used to approximate in an unbiased manner the accuracy of an estimator through the indirect measure of the mean square error (MSE). In the case of wavelets, the coefficients of each approximation/detail signal can be considered an estimator of the signal. R-SURE’s indirect relation to MSE can help minimize the risk associated with squared errors. For the case of R-SURE the threshold level was computed from the following equation (Donoho and Johnstone, 1995):

$$t=\sqrt{2ln[N{log}_2(N)]}$$ (2)

H-SURE is a heuristic variation of R-SURE which combines SQRT and R-SURE. H-SURE evaluates the signal-to-noise ratio, if it is very small, it avoids SURE’s noisy estimation and instead uses SQRT.

MIMA, used in decision theory as well as statistics, is intended to minimize the risk in the worst case scenario (maximum loss) (El Habiby and Sideris, 2006; Antoniadis and Fan, 2001; Donoho and Johnstone, 1998). For wavelet coefficients, MIMA estimates the minimum MSE among the maximum (worst-case) set of coefficients found when reconstructing the signal (Cao et al., 2003). The following MIMA formula from the Matlab’s thresholding was used in this research:

$$t=0.3936+0.1829\times \left(\frac{lnN}{ln2}\right)$$ (3)

2.4.2. Thresholding scaling

After the threshold levels are estimated by the thresholding scheme they are typically scaled depending on the expected type of noise in the signal. These scaling approaches take into account different sources of noise that may be imbedded in the signal, and adjust the thresholding level accordingly in order to ensure a better noise filtering. Three scaling approaches were evaluated in this research: no-scale (NONE), white noise (SINGLE), and non-white noise (MULTIPLE). In the case of NONE, the threshold levels were not modified in any way, leaving them as estimated by the equations above (1–4). For the case of SINGLE, all the threshold levels were scaled as a factor of the standard deviation of the single highest frequency detail signal, which is often a robust approximation of white noise; this ensures the removal of white noise in the signal. On the other hand, if the signal is to be filtered of non-white noise, the threshold can be scaled level-by-level (MULTIPLE). This can be accomplished by multiplying the threshold level with the standard deviation of each decomposed signal (detail or approximation). By scaling the threshold level in each signal, this approach ensures the identification of noise within each detail or approximation signal.

2.4.3. Soft or hard thresholding rules

Two different types of thresholding rules are used once the threshold levels are calculated and scaled. The first, hard thresholding, often called “keep or kill” (Aboufadel and Schlicker, 1999), removes coefficients that are below the threshold level assigned by the selection rule described above. The coefficient elimination follows the following formula:

$$\widehat{c}(x)=\{\begin{array}{ll}x\hfill & \mid x\mid \ge t\hfill \\ 0\hfill & \mid x\mid <t\hfill \end{array}$$ (4)

where the new coefficients ĉ(x) are kept only if they are above the threshold level t. This approach proves effective when measured against MSE.

The other thresholding approach is called soft thresholding which yields smoother denoised signal, compared to hard thresholding. This approach is based on the following formula:

$$\widehat{c}(x)=\{\begin{array}{ll}x-t\hfill & x>t\hfill \\ 0\hfill & \mid x\mid \le \mid t\mid \hfill \\ x+t\hfill & -x<-t\hfill \end{array}$$ (5)

where the new coefficients ĉ(x) are eliminated if below threshold t. Unlike hard thresholding, the coefficients are modified by t (added or subtracted) if above the thresholding level. This method provides smoothness, and some statistical benefits over hard thresholding (Donoho and Johnstone, 1995; Ghael et al., 1997). This method is often called wavelet shrinkage, as the resulting signal is shrunk by the threshold level adjustment.

2.4.4. Model building

While testing all the possible combinations of model, their names were combined to achieve a unique identifier. The notation scheme–thresholding–scaling format was used for testing the models with a fixed mother wavelet, and the notation scheme–thresholding–wavelet was used for testing the wavelets with the best performing models. For example: selecting the scheme MIMA, with hard thresholding, and single scaling would result in the name mima-h-s. For the wavelets, the different mother wavelets were encoded as following: Daubachies (d#), symlet (s#), Coiflet (c#), bior (b#), and Meyer (dy). The pound sign (#) represents the model order, which for the case of the discrete Meyer wavelet a letter y was presented as this mother wavelet has only one order. This notation should aid the presentation of models from here on.

2.5. Feature extraction and classification

2.5.1. SU Windowing

Before performing the PCA for feature extraction, the SUA must be converted into discrete spikes to ensure the analysis is not biased by non-relevant features found in the SUA signal. The two parameters that delimit a spike are its trigger (identified by the horizontal dashed line in Fig. 3B) and time window (identified by two vertical dashed lines in Fig. 3B). The trigger equals to the amplitude that a peak must cross to be considered a relevant spike. In the context of extracellular recordings, the trigger is negative because the cell depolarization/hyperpolarization is recorded as negative by being outside the cell. The time window is the temporal length of the spike to be recorded. This window is considered default at 1 ms which is enough time to capture the whole cycle of activation from the cell. Other time windows have been recommended (for example, 0.5 ms) (Wiltschko et al., 2008), but since the scope of this research is to investigate the impact of wavelet type on the signal, such parameters were kept at their default setting and not changed.

Fig. 3.

Different types of signals typically used in invasive brain research: multi-unit activity (MUA) and single unit action potentials (SUs). (A) MUAs represent a collection of units, with more focus in the firing frequency than any resulting shape of the action potential. (B) SUA represents a voltage due to electrical synaptic discharges. (C) Spikes are individual action potentials and can be acquired individually or extracted from SUA (dashed lines indicate the time window and trigger).

2.5.2. Feature extraction

PCA is used for data reduction and feature extraction (Bestel et al., 2012; Ghosh-Dastidar et al., 2008). PCA operates by deconstructing a data set into linearly uncorrelated variables called principal components. Through eigenvalue decomposition and orthogonal transformations, each principal component contains the largest possible variance with the limitation of being orthogonal to the previous component. This way, as the principal components are consecutively constructed, the remaining variance is allocated to each component. For the case of extracellular recordings in neurons, typically the first three components possess enough variance to aid in differentiating cells (Letelier and Weber, 2000). By plotting the first three components in a three dimensional space, this allows the identification of different neurons in the form of linearly separable clusters; each neuron’s spike contributes differently to the overarching variance in the data set.

2.5.3. Feature clustering and classification

After feature extraction by the PCA a Gaussian mixture algorithm (Kim, 2006) was used to cluster the data into classes which closely relate to individual neurons. The algorithm requires the number of classes, K, to be defined. In this research, K, was determined by visually inspecting the clustered data in a three-dimensional space.

2.6. Measuring signal quality

Two measures are used to evaluate the performance of the wavelet denoising: MSE, and root-mean squared (RMS). The purpose of MSE is to compare the reconstructed signal with the unprocessed original signal as a measure of information loss in the reconstruction. The RMS measures the effectiveness of the denoising; the smaller the RMS value the better the denoising (Cao et al., 2003).

2.7. Measuring clustering quality

The quality of the clustering is evaluated using three measurements: isolation distance, L-ratio, signal-to-noise ratio (SNR), and firing frequency correlation index (or correlation index). The isolation distance and L-ratio are used jointly to measure cluster quality. The isolation distance, in general, measures the separation of a cluster from neighboring clusters (a measure of false positives). It is defined through the minimum squared Mahalanobis distance (Schmitzer-Torbert et al., 2005):

$$D_{i,c}^2={(x_i-{\mu }_c)}^T{\sum }_c^{-1}(x_i-{\mu }_c)$$ (6)

where D represents the distance of each event (or SU) (x_i) to the mean feature vector (or cluster center) (μ_i). Σ_c represents the covariance matrix (Σ_c = X^TX) of the SUs in the desired cluster. The isolation distance is the smallest Mahalanobis distance between the cluster and outside cluster points. In contrast to the Euclidean distance, the Mahalanobis distance compensates for the elliptical shape of clusters resulting from their multivariate nature, which means that one direction possesses a different distribution than the other. When all points are enclosed in the same distribution, the Mahalanobis distance becomes equal to the Euclidean distance.

L-ratio, a measure of false negatives and compactness of clusters, is defined as

$$L_{\mathit{ration}}(c)=\frac{{\sum }_{i\notin c}1-C{D}_{{\chi }_{df}^2}(D_{i,c}^2)}{n_c}$$ (7)

where ${CD}_{{\chi }_{df}^2}$ represents the cumulative probability value of a χ² (Chi-squared) distribution for each of the Mahalanobis distances ( $D_{i,c}^2$) of non-clustered spikes in the feature space and the number of spikes in the evaluated cluster (n_c). The cumulative probability is evaluated using the degrees of freedom (df) equal to the number of the features used (which is equal to 3 in this research for every principal component used). In general, the bigger the Mahalanobis distance, the higher the cumulative probability. By subtracting from one, the remainder probability is used; thus, the smaller the number, the lesser the risk of classifying a spike from an outside cluster. The higher the isolation distance and the lower the L-ratio, the clearer the clusters will be in the feature space.

In this research, SNR is defined as the ratio of the average variances of the spikes; spikes inside the cluster versus spikes outside the cluster:

$$\text{SNR}=\frac{{\sigma }_{\mathit{signal}}^2}{{\sigma }_{\mathit{noise}}^2}$$ (8)

Finally, using Pearson’s linear correlation, the binned firing frequency of each classified clustered was compared to the firing frequency of all available cells (69) sampled in 0.025 s bins. This approach was selected over direct computation of the percentage of spikes based on the observation that with the discretization of the simulated signal the fire times were only slightly offset, and some cells even possessed the same fire times. In all instances the correlation between all simulated cells was not greater than 0.6 (this value was used as the lowest boundary for accuracy).

2.8. Statistical analysis

A multiple linear regression analysis was performed to determine whether each individual model factor (dependent variables: thresholding scheme, scaling, rule, and wavelet) had any statistically significant impact on each of the signal results (explanatory variables: MSE, RMS) and cluster results (explanatory variables: correlation index, isolation distance, L-ratio, and SNR). This analysis allows the statistical evaluation of multiple factors per response compared which is not obtained from a standard ANOVA.

3. Results

3.1. Simulated data

3.1.1. Selecting a thresholding scheme, scale, and rule

Using the db4 wavelet, a multiple linear regression analysis was performed to evaluate the impact of thresholding scheme, scale, and rule on all the statistical measures. The analysis led to the conclusion that the firing frequency correlation index (p < 0.001), isolation distance (p < 0.001), SNR (p < 0.001), MSE (p < 0.001), and RMS (p < v.001) were affected by the combinations of thresholding scheme, scale, and rule, while the L-ratio (p = 0.092) was found not to be significantly affected by the factors.

To properly evaluate the impact of thresholding scheme, scale, and rule individually, an average line plot was performed for all the measures for the first simulated signal. The first simulated signal is used to build and understand the thresholding models. Fig. 8A shows the cluster performance of all methods for the proximate cells found in the recording identified by their correlation index. The results displayed in the left column of Fig. 8A are the means across four thresholding scheme. The results displayed in the middle column of Fig. 8A are the means across three thresholding scalings. The results displayed in the right column of Fig. 8A are the means across two thresholding rules. Three additional simulated signals and the in vivo signal are used to validate the effectiveness of the selected models. These cells (out of the 69) are #11, #15, #17, and #49. All four were identified in most models, where others, such as #38 or #51, did not consistently show up and produced low correlation (#38, r = 0.620; #51, r = 0.608) compared to the main four (#11, r = 0.941; #15, r = 0.907; #17, r = 0.821; #49, r = 0.721), for this reason these cells were not displayed. The clustering measures clearly show schemes H-SURE and R-SURE to outperform MIMA and SQRT in isolation distance, L-ratio, and SNR. H-SURE performed slightly better than R-SURE in the combined isolation of cells #11 and #15. Soft thresholding achieved lower isolation distances compared to hard thresholding, and generally lower L-ratio with the exception of cell #15. For the scaling, single (white-noise) scaling outperformed the other two types of scaling in isolation distance and L-ratio, with unnoticeable difference in SNR. Fig. 8B displays results of signal performance after denoising. H-SURE and R-SURE outperformed MIMA and SQRT with lower MSE, and higher RMS. As expected and mentioned in Section 2, hard thresholding outperformed soft thresholding in MSE but not in RMS. SINGLE scaling outperformed MULTIPLE and NONE with significantly lower MSE, and higher RMS of the three.

Fig. 8.

Summary of results for cluster and signal measures: (A) cluster measure’s means based on scheme, scaling, and rule from first simulation; (B) signal measure’s means based on thresholding scheme, scaling, and rule with their respective confidence intervals (indicated by a vertical I bar); (C) relative cluster size and total spikes after denoising. The original file had a total of 2454 spikes.

An analysis was performed on cluster/total spikes by scheme/scale/rule. The results are presented in Fig. 8C which shows resulting cluster sizes and total spikes after processing with averaged over across scheme, scaling, and rule. The cluster sizes were normalized by the number of spikes generated by Neurocube. It is observed that cell # 15 and #17, identified by squares and diamonds, respectively, generally attained more spikes than simulated. Specifically schemes MIMA and SQRT affected the classification allowing for more events to be accepted, while HSURE and RSURE did not. It is observed that cell #11 was not affected in terms of size. Finally, all methods achieved significantly lower spikes compared to the raw recording, which had 2454 events. This observation points to the reduction of spikes without the impact on clustering/signal quality.

Focusing on isolation distance, Fig. 9 shows the summary for the results and best performing clusters. Fig. 9A displays a bar graph with isolation distance (gray bars) and correlation index (lines) of the best performing models compared to the original recording (far right). Cell #15’s isolation was improved only by three models (mima-h-s, hsure-s-s, and hsure-h-s). The other cells (#11 and #17) were improved by more models, five and 12, respectively. Fig. 9B shows the clusters from the best performing models. The model mima-h-s achieved a significantly higher isolation of cell #15 compared to the other two. This model achieved significant improvement only for cell #15, while not affecting cell #11 and distorting #17 beyond recognition. The other two models (hsure-s-s and hsure-h-s) achieved general improvements which can be observed in the reduction of noise (noted with capital N) and slight separation of the clusters. An example is visible where cell #17 is clearly separated from the noise in both hsure-s-s and hsure-h-s.

Fig. 9.

Specific cluster performance on models which outperformed the original recording. (A) Graph of isolation distance and correlation index sorted in decreasing order with original recording last. (B) Clusters of the top three models compared with original in two views X–Y (1–2) and X–Z (1–3).

3.1.2. Selecting a mother wavelet

Based on the results from the thresholding scheme, scaling, and rule four models were selected out of a total of 24 based on their performance on cell #15 (MIMA-H-S, MIMA-S-S, HSURE-S-S, and HSURE-H-S). To assess their performance with different mother wavelets, the same simulated recording was processed with all 20 mother wavelets (db1–db5, coif1–coif5, sym2–sym6, bior1.1, bior1.3, bior1.5, bior2.2, and dmey).

Fig. 10 displays the isolation distance and Pearson’s correlation index on firing frequency. The models and wavelet types plotted were selected based on higher isolation distances as well as higher correlation index than the original recording. The db4 model improved the isolation of all three cells consistently, whereas MIMA achieved higher isolation distances only for cell #15 and cell #11, and H-SURE achieved overall improved performance for all cells.

Fig. 10.

The isolation distance and Pearson’s correlation index on firing frequency. The bold-faced models are the db4 models used previously to build the models.

3.2. Validation

Since both thresholding schemes (MIMA and H-SURE) achieved good results in clusters, further preprocessing was performed using three additional signals simulated in Neurocube and a 30 s In Vivo recording. To ensure that the methodology was noise-insensitive, the rate of the firing activity was increased to 8%, 9%, and 10%. Fig. 11 displays the results for isolation distance and relative (or normalized) size averaged across thresholding scheme or rule while scaling was fixed. In general, the four models improved isolation distances, with MIMA-H providing the most consistent improvement even though in some cases it classified more spikes than simulated by Neurocube. For the simulated signals of 8% and 9% rate of activity, MIMA scored a higher isolation distance than SURE and soft thresholding rule achieved better scores than hard thresholding rule. The resulting clusters are presented in Fig. 12 where it is observed that in general all models provide some improvement in the isolation of the clusters, with MIMA providing the most improvement.

Fig. 11.

Summary of isolation distance and relative size for each signal used in validation.

Fig. 12.

Original PCA versus MIMA PCA and HSURE PCA.

4. Discussion

The basis of this work was founded on the idea of finding a signal processing methodology that would filter noise and smooth the signal to improve cluster isolation in a PCA model. The work in this paper presents an alternative pre-processing methodology for spike sorting applications by using wavelet transform, statistical thresholding, and principal component analysis. This methodology is capable of improving the isolation distance of clusters representing live neurons. To the authors’ best knowledge this is the first attempt to provide a working model, as well as data on the performance of various mother wavelet functions and statistical thresholding approaches on a spike sorting application. Based on the results obtained in this research, it is suggested using the mother wavelet Daubachies order 4 (db4), with MIMA, soft thresholding rule, and single-level scaling.

Other methodologies used as preprocessing for isolation distance improvement depend on completely removing low frequency components entirely from the signal (Wiltschko et al., 2008) who wrote judiciously “First, a more subtle wavelet denoising technique might be used. In this paper we destroyed all the approximation coefficients, and, by not computing higher-level, lower-frequency detail coefficients, we effectively destroyed them as well.” The methodology introduced in this paper sets the thresholding levels properly to fine-tune which coefficients are to be kept and which to be removed, thus overcoming the aforementioned challenge. By doing this it is possible to remove low amplitude spikes or noise while improving the isolation distance of the cluster of high amplitude cells.

A downfall of this methodology is that in PCA the number of clusters is unknown. With a good recording, it is possible to have linearly separable clusters, yet some recordings require an expert decision on separation of clusters containing separable information. Other classification methodologies such as support vector machines (Jacob et al., 2004) and enhanced probabilistic neural networks with local decision circles groups (Ahmadlou and Adeli, 2010a) can be explored to improve the clustering and classification. Another possible improvement would be utilizing the wavelet packets transform (WPT), a generalized approach to WT, which enables better resolution analysis on a signal.

5. Conclusion

With numerous applications in the neurosciences, from neuronal population studies and neural prosthesis (Shalchyan et al., 2012), to the guiding of electrodes for deep brain stimulation (Starr et al., 2006), the sorting and classification of neurons in electrophysiological signals continues to be an active area of research in the fields of neuroscience and biomedical signal processing. This paper presented an effective neuron cell isolation methodology using the wavelet transform (WT), a statistical thresholding scheme, and the principal component analysis (PCA) algorithm.

A key decision in the application of WT is the selection of the most appropriate mother wavelet. This research shows that the selection of mother wavelet has a strong impact on the clustering and isolation of neurons from extracellular microelectrode recordings. In some fields such as engineering fault analysis, researchers have attempted heuristic approaches to select the most suitable mother wavelet by comparing features of the signal and the mother wavelet to model the signal (Rafiee et al., 2009). In other words, the selection of the mother wavelet is often inspired by the features extracted from the signal. Yet, in this research, no correlation was observed between signal features and the wavelet. Researchers reporting the use of WT in the context of spike sorting often select a popular wavelet somewhat arbitrarily. The present study shows that Daubachies performs best and presents a comprehensive approach to WT for denoising of extracellular recording.

HIGHLIGHTS.

• We present a neuron cell isolation methodology.

• Wavelets investigated: biorthogonal, Daubachies 4, Meyer, symmetric, and Coifman.

• Effectiveness of 3 wavelet coefficient thresholding schemes is investigated.

• The signal quality is evaluated using three different statistical measures.

• Selection of mother wavelet matters with symmetric 4 wavelet performing the best.