4. Neural Signal Recording and Processing

To obtain real spike waveforms, the signal amplification and acquisition system shown in Fig. 3.17 was set up as following. Set the model 3600 amplifier at a frequency range of 0.3 Hz–20 kHz, much larger than spike bandwidth. Set the PowerLab recorder at a sampling frequency of 100 kHz. For spike recordings, the low-frequency cutoff doesn't need to be as low as 0.3 Hz. However, during experiments, LFP signals often need to be recorded simultaneously, requiring the retention of low-frequency signals. Figure 4.2 displays a brief neural signal recorded in the rat hippocampal CA1 soma layer with the above settings. The first row shows the original signal, dominated by slow rhythms at about 0.7 Hz that overshadows small unit spikes. The second row reveals clear spikes after LFP removal by a 300–3000 Hz bandpass digital filter in the LabChart. The enlarged inset in the upper middle clearly shows high-frequency noises with a relatively large amplitude of about 50 μV under the frequency band up to 20 kHz. These noises are reduced significantly in the filtered signal shown below. The right insets are further enlarged in time scale, highlighting amplitude and waveform differences between the filtered and original spikes. Obviously, the filtering distorted spike waveforms simultaneously with noise removal. Next, we will obtain undistorted spikes using signal averaging.

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Signal averaging is widely used for denoising. When the noise is random and the wanted signal is consistent and repeatable, averaging can effectively eliminate the random noise while obtaining the deterministic signal with an improved signal-to-noise ratio (SNR). This process involves aligning repeated signals at a specific time point and then calculating their average. The increase in SNR is proportional to the square root of the number of averaged signals. For example, an auditory evoked potential (AEP) in a single trial of scalp EEG recording is typically undetectable, buried within other irrelevant EEG signals and noises. However, a clear AEP can emerge after averaging the EEG episodes from hundreds of repeated auditory stimulations (Enderle and Bronzino 2012).

Based on the “all-or-none” characteristic of action potentials, spikes from an identical neuron can be considered as repeated deterministic signals. Note that changes in burst spikes are excluded here. Figure 4.3A displays the spikes extracted from a wideband recording of 0.3 Hz to 20 kHz, sampled at 100 kHz as the recording shown on the upper row of Fig. 4.2, and detected by using a threshold on the 300–3000 Hz bandpass filtered signal. Each spike waveform included 200 sampling points, covering 1 ms before and after the negative peak respectively. Total 1500 spikes (green waves) were superimposed in Fig. 4.3A, aligned with their negative peaks. Due to the shift of low-frequency field potentials, each spike had a potential shift. Zero-mean spike waveforms were obtained by subtracting their mean from individual spikes (Fig. 4.3B). The high-frequency noise in each zero-mean spike became evident, which was removed by signal averaging (Fig. 4.3C–F). We quantified the denoising effect using the coefficient of variation (CV) among spike waveforms.

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Let x_ij be the jth sampling point of the ith spike waveform, M be the number of spikes, and x_j be the jth sampling point of the average waveform fromM spikes, then

Let A_PP represent the peak-to-peak amplitude of the average waveform (x_j), and N be the length of spike waveform—the number of sampling points per spike, then

The CV is the percentage ratio of the mean spike standard deviation to the A_PP, representing the variability among spike waveforms. Note that the definitions of x_ij and x_j may vary accordingly in different CV calculations.

Due to the shifts in low-frequency field potentials (Fig. 4.3A), the CV of the 1500 original spikes reached 278% (A_PP = 181 μV). After mean removal, the CV of zero-mean spikes dropped to 12% (Fig. 4.3B).

Using the CV index, we can examine how denoising efficiency changes with the number of averaged spikes. Averaging every 10 spikes from the total 1500 yielded 150 average spikes, reducing the CV to 3.6% (Fig. 4.3C). Additional noise reduction occurred when averaging every 100 and 500 spikes, respectively (Fig. 4.3D, E). The CV decreased rapidly as the number of averaged spikes increased (Fig. 4.3F). When more than 200 spikes were averaged, the CV fell below 1%, indicating that the obtained spikes can be treated as noise-free waveforms—real spikes. In the next section, we will use the wideband real spikes obtained through averaging to explore how frequency-band settings affect spike waveforms.

First, let's explore the selection of high-frequency cutoff (f_H). We set the amplifier's f_H respectively from high to low at 20, 10, 5, 3, and 1 kHz, while keeping the low-frequency cutoff (f_L) at 0.3 Hz and the sampling frequency at 100 kHz. Recordings were taken from the rat hippocampal CA1 pyramidal layer for about 2 min at each f_H setting. Figure 4.4A shows 10 ms recording samples and their corresponding spike SNR values under different f_H settings. The SNR was calculated as the ratio of the spike peak-to-peak amplitude (A_PP) to five times the noise standard deviation (Feng et al. 2012; Joshua et al. 2007):

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The high-frequency noise decreased with f_H decreasing, leading to an improved SNR. However, when f_H decreased to 1 kHz, the SNR dropped again due to a substantial A_PP reduction.

The noise-free spike waveforms under different f_H settings were calculated from 1300 to 1800 averaged spikes. Figure 4.4B shows the noise-free spike waveforms from an interneuron (left) and a pyramidal neuron (right) recorded at five different f_H values. No significant waveform loss occurred with an f_H above 5 kHz. As f_H decreased, the waveform loss increased, especially at the negative peak of spikes. However, even when f_H dropped to 1 kHz, the repolarization phase after the “inflection point” in the pyramidal neuron's spike waveforms (Gold et al. 2007)—denoted by a black arrow in the figure—retained mostly due to its relatively low frequency components.

The statistical results from eight rat experiments showed that the normalized spike amplitudes (A_PP) at different f_H were above 94% for f_H ≥ 5 kHz (Fig. 4.4C). However, at f_H = 1 kHz, the normalized A_PP dropped to ~ 52%, representing approximately half loss. Using the CV index to evaluate the waveform changes at different f_H settings with the 20 kHz waveform (the real spike) as a reference (Eq. (4.2)), for f_H ≥ 5 kHz, the CV remained below 2.5%, but increased to ~ 15% at f_H = 1 kHz (Fig. 4.4D). Notably, the spike SNR exhibited a non-monotonic change against f_H, peaking between 3 and 5 kHz (Fig. 4.4E). Note that the large standard deviations in SNRs (the error bars) resulted from substantial variations in spike A_PP values, which ranged from 90 to 205 μV across these rat experiments.

These results suggest that setting f_H at 5 kHz can maintain most high-frequency components in spike waveforms while achieving satisfactory spike SNR. Therefore, we typically set the amplifier's f_H at 5 kHz for spike recordings.

Second, let's explore the selection of the low-frequency cutoff (f_L). In wideband recordings, low-frequency field potentials can overshadow unit spikes (see Fig. 4.2, top left), making it difficult to detect spikes directly using a threshold method. Therefore, we typically filtered out low-frequency signals before spike detection. As shown in the bottom row of Fig. 4.2, a bandpass filter (300-3000 Hz) can remove low-frequency signals with the f_L = 300 Hz. Both the f_H and f_L cutoffs can be achieved by either analog filters in an amplifier or digital filters in software. We usually set f_H in the amplifier to eliminate high-frequency noise and avoid the need of a higher ADC sampling frequency, which could limit simultaneous multi-channel sampling. On the other hand, we typically set the low-frequency cutoff in the amplifier to 0.1 or 0.3 Hz to eliminate zero drift and record low-frequency LFPs. These settings allowed us to capture a wide range of neural electrical activity. After signal collection, we used a digital filter to remove low-frequency potentials for spike extraction. The following analysis thus focuses on how the f_L setting in digital filtering affects spike waveforms.

Again, we analyzed the spikes from both interneurons and pyramidal neurons in the rat hippocampal CA1 region. The amplifier was set to a frequency band of 0.3 Hz to 5 kHz. According to the Nyquist frequency principle, a 20 kHz sampling frequency would have been sufficient; however, we sampled the signals at 100 kHz to obtain smooth spike waveforms. Then, the 2-min recordings were high-pass filtered using the Kaiser window filter in LabChart software with a cutoff frequency (f_L) increasing from 100 to 900 Hz in 100 Hz increments to show how the f_L setting affected spike waveforms and their SNRs.

Figure 4.5A shows typical examples of noise-free spike waveforms for the two types of neurons, obtained by the averaging method described above. The wideband real waveforms are shown in red, while the other waveforms at nine different f_L settings are in blue. The waveforms at f_L = 900 Hz notably differed from the real waveforms. The superimposed plot of all the 10 waveforms shows how they changed as f_L increased: both their negative peak and peak-to-peak amplitudes gradually decreased, while a new positive peak emerged before the negative peak. In the pyramidal neuron spikes, the smooth posterior inflection (denoted by the black arrows in Fig. 4.5A) progressively sharpened into a positive peak. This change caused the spikes to lose their wide opening, making them increasingly similar to interneuron spikes. Both types of spikes showed a greater decrease in negative peak amplitude than in peak-to-peak amplitude, particularly at higher f_L values (Fig. 4.5B).

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The CV values of the spike waveforms at each f_L were calculated by using the real waveforms as a reference. As shown in Fig. 4.5C, the CV gradually increased as f_L increased. Notably, pyramidal neurons showed significantly higher CV values than interneurons across the f_L range of 100–900 Hz. This difference resulted from the more low-frequency components in pyramidal neuron spikes. These CV values were calculated using spike waveforms spanning 3 ms for interneurons and 4 ms for pyramidal neurons, as shown in Fig. 4.5A, which were different from the 1 ms span shown in Fig. 4.4B.

Figure 4.5D shows how the SNR values of the two types of spikes changed with f_L. Spikes of pyramidal neurons had higher SNRs in the f_L range of 400-600 Hz, while those of interneurons had higher SNRs in the f_L range of 500–800 Hz. The standard deviations of these SNRs were similar to those shown in Fig. 4.4E, influenced by the various peak-to-peak amplitudes of spikes. For clearly displaying the SNR trends, the standard deviation error bars were omitted in Fig. 4.5D.

These results showed that the f_L affected the spike waveforms of pyramidal neurons more significantly than those of interneurons. Thef_L range for maximum SNR was also different between the two types of spikes. Notably, waveform distortions were already evident at the f_L ranges that produced maximum SNRs. For instance, at f_L = 600 Hz, the pyramidal neuron spikes showed a negative peak amplitude of only ~ 60% of the real spike value due to the loss of low-frequency components, while the peak-to-peak amplitude was ~ 85%. At f_L = 800 Hz, the interneuron spikes maintained higher fidelity, with the negative peak amplitude at ~ 80% of the real spike value and the peak-to-peak amplitude at ~ 90% (Fig. 4.5B).

For any spike signals, a high-pass filter with a higher f_L can produce oscillations immediately before and after the spikes (Wiltschko et al. 2008). Figure 4.6 shows a burst from a pyramidal neuron. While the interval potentials between spikes were relatively flat in the original wideband signal, they became as oscillations after high-pass filtering with f_L = 500 Hz. It is important to distinguish these filter-produced oscillations from real neural rhythms in hippocampal LFP—such as ripples at 140–200 Hz or even 300–600 Hz (Buzsáki 2006).

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A proper frequency band setting is essential for accurate spike recording and detection. In theory, a higher f_H in amplifiers is preferable for recording complete spike waveforms. However, the magnitude of electron thermal noise in electrodes and recording instruments is directly proportional to the frequency bandwidth (Bretschneider and de Weille 2006). When the high-frequency band is widened, it can at least increase by several kilohertz, causing a dramatic increase in noise. Our findings showed that at f_H = 5 kHz, the loss in spike peak-to-peak amplitude was only ~ 5%, the waveform CV was only ~ 2%, and the spike SNR reached its maximum (Fig. 4.4C–E). At f_H = 3 kHz, although waveform distortion increased, the SNR remained comparable to that at 5 kHz. Therefore, for detecting spikes to obtain neuronal firing sequences (Lewicki 1998), setting the f_H (the low-pass cutoff) between 3 and 5 kHz is suitable. However, for analyzing subtle features of spike waveforms (Henze et al. 2000) or using them for spatial localization of cell current sources (Lee et al. 2007), a higher f_H can be required to retain higher frequency components. Actually, in such cases, both the high and low-frequency components of the spikes are required to preserve as much as possible.

A high SNR is crucial for accurate spike detection using methods like threshold detection (Musial et al. 2002), where spike distortion is a secondary concern. Due to the neuronal morphology and the formation mechanism of action potentials (Gold et al. 2007), pyramidal neuron (Pyr) spikes contain more low-frequency components than interneurons (IN) spikes. This leads to greater peak loss and SNR decrease at higher f_L values in Pyr spikes. As a result, Pyr spikes showed maximum SNR in an f_L range of 400–600 Hz, while IN spikes had a different f_L range of 500–800 Hz (Fig. 4.5D). For detecting both types of spikes, an optimal f_L can be set at 500–600 Hz.

Each recording can typically contain spikes from multiple neurons, making spike sorting necessary after obtaining spike waveforms (refer to Sect. 4.1.4). This process relies on the prerequisite that spikes from the same neuron have similar waveforms, while those from different neurons are distinct—with greater differences being preferable. Sorting accuracy depends on factors such as spike SNR and waveform fidelity. Because noise can cause variations in spike waveforms from an identical neuron, spikes with a low SNR create a more scattered distribution in the clustering space, making accurate sorting difficult. Similarly, waveform distortions—such as an increased similarity between different spikes caused by high-pass filtering (Fig. 4.5A)—can also complicate the sorting process (Feng et al. 2012).

In summary, this section explores the changes in spike waveforms from two types of neurons under different frequency cutoffs. The choice of frequency range depends on factors including the features of spike waveforms and the purpose of spike usage. In our studies on neuromodulation by electrical stimulations—covered in Part II of this book, we needed to accurately detect spikes, classify them, and distinguish their neuron types. We also needed to simultaneously record local field potentials and stimulus-evoked potentials, such as APS and OPS. For these purposes, we configured the amplifier to a wideband of 0.3 Hz to 5 kHz, used a 20 kHz sampling frequency to collect signals, and then applied digital high-pass filter in LabChart to remove low-frequency components (with an f_L typically set at 500 Hz), which produced clear spikes for subsequent detection and sorting.

Additionally, neuron types can be distinguished based on their spike waveforms and firing patterns. Our method for distinguishing pyramidal neurons from interneurons in the hippocampal region was based on the rising phase width of spike waveforms from the original wideband signals. Spikes with a rising phase wider than 0.7 ms were classified as from pyramidal neurons (Fig. 4.1C), while those narrower than 0.4 ms were considered from interneurons (Barthó et al. 2004). Figure 4.5A demonstrates that high-pass filtering, commonly used in spike detection, can distort spike rising phases—particularly narrowing Pyr spikes. Therefore, when identifying neuron types, we should use the original spike waveforms from wideband signals. Furthermore, as shown in Fig. 4.3, using averaged, noise-free waveforms can produce more accurate measurements.

Modern data acquisition systems for electrophysiological signals rarely use analog devices such as magnetic tape recorders. Instead, they perform real-time ADC to sample amplified signals and store them digitally. ADC has two key specifications: sampling frequency (or sampling rate) and resolution. The sampling frequency (f_s) determines the time accuracy, which must be at least twice the highest frequency in the signal (the Nyquist frequency). In practice, an f_s of about four times or higher is typically used. Neuronal spikes require a high f_s due to their brief duration (~ 1 ms) and high frequency components.

ADC resolution (or quantization level) determines the voltage accuracy in the sampled signal. Spikes require high ADC resolution due to their small amplitudes, typically at tens to hundreds of microvolts. This resolution becomes especially critical when simultaneously recording other signals with larger amplitudes, such as population spikes (PS) above 10 mV and slow waves of spreading depression (Fig. 3.16). A 16-bit ADC converter provides 65,536 quantization levels. At a recording range of ± 20 mV, the ADC resolution is 625nV, allowing a 50 μV spike to occupy about 82 quantization levels. If the recording range is increased to ± 100 mV, the ADC resolution decreases to 3.125 μV, reducing the quantization levels for the same spike to only 16.

Let's examine the spike waveform from a pyramidal neuron shown in Fig. 4.7. The upper left plot shows the waveform with dense data points, captured at f_s = 100 kHz with a 0.3 Hz–5 kHz wideband. The remaining plots in the top row, enclosed in dashed boxes, display the same waveform sampled at progressively decreased f_s of 20, 10, and 5 kHz. The corresponding plots in the bottom row overlay these waveforms on the 100 kHz sampling waveform (in grey) to highlight differences. At 20 and 10 kHz, the spike waveforms are well-preserved. However, at 5 kHz, the spike negative peak degrades significantly. The steep falling phase of this spike lasts only about 0.3 ms, as shown by the shadow in the upper left plot. At 20 kHz, six data points are captured in the falling phase, while at 10 and 5 kHz, only 3 and 2 data points are captured, respectively. In the bottom left plot, even at f_s = 100 kHz, reducing the resolution to 10 μV results in the entire waveform to occupy only 18 quantization levels—appearing as “ladders” with a significant loss of details. Yet for spike detection, where the main goal is to determine spike timings, a lower resolution usually doesn't impact results significantly. For comparison, the resolution of the other waveforms in Fig. 4.7 is 625nV, allowing the spike to span approximately 280 quantization levels.

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Spike detection is to extract individual spikes from a recording signal, while spike sorting is to determine which neuron generated each spike. Each electrode contact can typically capture spikes from multiple nearby neurons. To determine the firing sequences of distinct neurons, we need to identify the origin of individual spikes based on their unique waveforms—a process known as spike sorting. Both detection and sorting are essential for obtaining the neuronal firing sequences (or neural coding information) in response to external stimuli. An electrode contact with a larger area can record more neurons simultaneously, but this creates challenges for spike analysis. When multiple neurons emit action potentials simultaneously, their spikes overlap, making extraction difficult. The more neurons recorded and the higher their firing rates, the greater the probability of producing overlapped spikes. Conversely, with fewer neurons recorded, overlapped spikes become minimal and negligible. Additionally, the contact surface acts as an equipotential conductor. If its area is too large, it can create a short-circuiting effect on the surrounding electrical fields, including those formed by spike currents. This makes it difficult to measure large spikes with such electrodes. Given these reasons, an electrode with a smaller contact is more suitable for recording spikes (refer to Sect. 3.4.1).

One common method used to extract spikes from recordings is the threshold method. Other methods include the window method and template matching method.

Figure 4.8 shows an original wideband recording and its spike signal obtained by removing low-frequency potentials by digital high-pass filtering (with a cutoff frequency of 500 Hz). This filtered signal contains spikes from multiple nearby neurons, known as multiple unit activity (MUA). The spikes in MUA are clearly distinguishable. Most spikes have a negative peak greater than their positive peak. Therefore, a negative threshold can be set to capture spikes with negative peaks below it. However, setting an appropriate threshold is practically challenging. If the threshold (absolute value) is too small, noises may be mistaken as spikes. Conversely, if the threshold is too large, small spikes may be missed. It is critical to set a threshold that can minimize both errors. A common algorithm for automatic spike detection is to set the threshold at 3–5 times the standard deviation of noise or the MUA signals, sometimes with manual adjustments. Additionally, some spikes can have positive peaks greater than negative peaks. To reduce missed detections, both positive and negative thresholds can be set simultaneously.

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A window method enables direct spike detection in original wideband recordings that contain low-frequency potentials. The method can be performed using a peak-to-peak amplitude threshold to extract spikes. To do it, a time window with a fixed width moves along the recording signal. The difference between the maximum and minimum values within the window at each position is calculated. When this difference exceeds the pre-set threshold, a putative spike is captured and its time span is then determined. The width of time window can be set to ~ 1 ms, matching typical spike widths. Using the peak-to-peak threshold, this method can evaluate both amplitude and slope changes within the signal segment in each narrow window. This effectively excludes low-frequency waves that have large amplitudes but small slopes, enabling direct spike detection from wideband signals.

When using a window method, it is critical to avoid duplicate spike detections. If the division between two adjacent window falls in the middle of a spike waveform, the peak-to-peak values in both windows may reach the threshold, causing a single spike to be mistakenly detected twice. To prevent this, a minimum time interval threshold between adjacent spike peaks can be set to eliminate any duplicate detections that occur too close together.

Figure 4.9A shows a recording during antidromic high-frequency stimulation (A-HFS) on the alveus in the rat hippocampal CA1 region. The A-HFS was a biphasic pulse train with an intensity of only 0.05 mA and a pulse interval of 10 ms (corresponding to 100 Hz). Each pulse triggered a population spike (APS). Upon the rising phase of many APSs, an interneuron spike appeared superimposed (indicated by a blue dot). To detect these spikes using a threshold method, the stimulus artifacts were first removed from the original signal (Fig. 4.9B, see Sect. 4.5.2 for details). Then, the low-frequency potentials were removed using a high-pass filter to obtain the MUA signal shown in Fig. 4.9C. Although spikes during non-stimulation periods were readily detected by a threshold method, the high-pass filtering produced oscillations around the APS waveforms during stimulation, obscuring the following interneuron spikes and making detection difficult. To solve this problem, either remove the APS waveforms before high-pass filtering as reported (Feng et al. 2013), or detect spikes directly using a window method on the original wideband signal (Fig. 4.9A), as described in Sect. 4.1.5.

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A template matching method can detect and classify spikes in the recording signal. For details, please refer to the spike sorting section below.

Threshold and window methods for spike detection only provide approximate time spans of spike waveforms. Before proceeding to spike sorting, the entire spike waveforms must be extracted by determining their start and end points. This process is known as spike alignment, which requires a reference point—typically either the negative peak or the first data point that crosses the detection threshold.

When aligning based on negative peaks, deviations may occur in the extracted spike waveforms due to differences between sampled peaks and actual peaks caused by limited sampling frequency. This issue also occurs when aligning based on threshold points. The actual intersection between the spike waveform and the threshold line may fall between two sampled points rather than exactly on one. Therefore, aligning based on the first sampled point crossing the threshold can cause deviations in extracted waveforms due to differences between the sampled points and the threshold line. To improve accuracy, the precise intersection point between each spike waveform and the threshold line can be calculated by an interpolation algorithm before alignment. Nevertheless, these alignment methods—based solely on a single sampled point—are inevitably affected by noise and limited sampling frequency. An alternative approach uses multiple data points from the major spike portion to calculate a “gravity center” point as the alignment reference, which can reduce deviations.

Spike sorting is to classify spikes to their source neurons based on the fact that spikes from the same neuron have similar waveforms. The two main spike sorting algorithms are the parameter method, which uses waveform parameters for sorting, and the template matching method, which compares each waveform with pre-set spike templates.

The waveform parameters commonly used in spike sorting include amplitude measurements, such as peak-to-peak amplitude, positive peak amplitude, and negative peak amplitude (Fig. 4.10A), as well as time parameters like spike width. Figure 4.10B shows a sorting example using a single parameter—peak-to-peak amplitude (A_pp). TheA_pp data from all spikes are plotted in a histogram with its vertical axis showing the count of spikes falling in each small A_pp bin. The peaks in the histogram represent spikes from different neurons. When the detection threshold is set low, noise can be mistakenly detected as spikes, forming a “noise” cluster in the small amplitude range (Fig. 4.10B). These noise artifacts can be easily removed during the classification.

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Parameter selection is crucial when sorting spikes based on their waveform parameters. Let's take the MUA signal shown in Fig. 4.11A as an example. It contains four distinct types of spikes. However, creating a histogram using positive peak amplitude alone yields only a single peak and fails to provide proper classification (Fig. 4.11B). This occurs because the positive peak amplitudes of the four spike types are too similar to be distinguished from one another. A histogram using peak-to-peak amplitude alone can classify only three types of spikes (Fig. 4.11C), as the amplitudes of the third and fourth spike types overlap considerably. Using multiple parameters together can improve the classifications. Figure 4.11D shows a two-dimensional scatter plot using two parameters: peak-to-peak amplitude and negative peak amplitude. It clearly distinguishes all four types of spikes (encircled by red dashed ellipses). Furthermore, a three-dimensional scatter plot incorporating three parameters: peak-to-peak amplitude, negative peak amplitude, and positive peak amplitude, provides a clearer visualization of the differences among these spike types (Fig. 4.11E).

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Although intuitive parameters like amplitude and width are common choices, they are not always optimal. Using all sampling points from spike waveforms as high-dimensional data for classification can yield better sorting results—though this approach requires intensive computation. Principal component analysis (PCA) offers a solution by transforming this high-dimensional data into a lower-dimensional one while preserving essential information. PCA can arrange the principal components by their variances, which correspond to component's energy and amplitude. Using the first few principal components with highest variances can achieve effective spike sorting.

Spike sorting by template matching involves two main steps. First, waveform templates are created for each spike type. Second, the recorded signals are scanned and compared with each template, while the root mean square (RMS) of differences between signal sampling points in the scan window and template points is calculated at each scan movement. When the RMS value falls below a preset threshold, the signal in the scan window is identified as a spike matching the template type. This method can achieve both spike detection and sorting simultaneously on either MUA signals or original wideband signals. It can also be used on extracted spike waveforms to reduce computational cost.

A template matching method can achieve real-time spike detection and sorting during signal collection (Schaffer et al. 2021), although it requires substantial computational resources, precise spike templates, and an appropriate preset fitting threshold. In addition, the method can, to some extent, classify overlapped spikes occurring almost simultaneously from different neurons (Mokri et al. 2017). Furthermore, it can improve the detection and sorting of spikes with gradually varying amplitudes, such as in bursts (Shabestari et al. 2021).

It is difficult to correctly achieve spike sorting based solely on a single channel recording signal. Each electrode contact can simultaneously capture spikes from multiple nearby neurons. For a same type of neurons at similar distances from a contact, the amplitudes and waveforms of their recorded spikes are similar. It becomes difficult to determine whether these spikes originate from one or multiple neurons based solely on the single recording. Therefore, spikes recorded simultaneously across multiple channels on an array microelectrode should be used to determine spike origins.

Figure 4.12A shows three channel recordings (Ch1-3) from an array microelectrode (A2 × 2-tet-3 mm-150–150-312, NeuroNexus) with a diamond-shape arrangement of groups of four contacts (Fig. 3.12B). The eight spikes recorded in Ch1 are too similar to distinguish when viewed alone. However, when examining all three channel recordings together, we can clearly identify that these spikes come from two distinct neurons (indicated by purple and red triangles above the signals). The spikes with red triangles show large amplitudes across all three channels, while those with purple triangles appear small on both Ch2 and Ch3, indicating that this neuron was close to contact 1 but far from the other two contacts.

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Figure 4.12B shows another example of 4-channel MUA signals from a dual column array (A1 × 16-Poly2-5mm-50s-177, NeuroNexus). Colored triangles and blue dots above the signals indicate three types of spikes obtained through four-channel spike sorting for the adjacent channels (Ch4-7). When using Ch4 alone, only two spike types can be distinguished, as it cannot differentiate between the two types of triangular-marked spikes that show substantial differences on Ch6. The spikes marked by red triangles appear larger on Ch6, suggesting that the neuron was closer to the contact 6 on the array. Conversely, the neuron producing spikes with purple triangles was far from the contact 6 because its spikes hardly appear on Ch6. Obviously, the three spike types can be effectively sorted using just the data of a single parameter—the peak-to-peak amplitudes from the four channel recordings.

One common type of traditional recording array is the tetrode, which can be assembled by four microwires glued together (Fig. 4.13). Each wire is insulated except for the measuring contact at its tip. The amplitudes of spikes from a same neuron can vary across the four recording channels due to different distances between the neuron and the contacts. Using MUA signals from all four channels recorded simultaneously can substantially improve spike detection and sorting (Buzsáki 2004). The Neuronexus planar array electrode shown in Fig. 4.12A imitates this tetrode contact arrangement. The manufacturer, therefore, names it “tetrode” (see website: http://neuronexus.com/). The accuracy of spike sorting can be improved when multiple contacts with sufficient spatial density are used. In our laboratory, we commonly used dual column array electrodes (Fig. 4.12B). With such an electrode, MUA signals from any four adjacent channels can be used to classify spikes. Additionally, with its vertically linear arrangement spanning up to 350 μm, this type of electrodes can simultaneously record evoked potentials at different hippocampal layers, such as population spikes at the soma layer and postsynaptic potentials at the dendritic layer (Fig. 3.6).

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The following are the main steps for spike detection and sorting by using multi-channel signals. First, apply a threshold method to each channel individually to detect spikes. The detection results are then merged since the number of detected spikes can vary across channels. As spikes from a same neuron can have different amplitudes across channels, spikes missed on channels with smaller amplitudes may still be detected on channels with larger amplitudes. To avoid noise, you can set a higher threshold based on the channel with larger spike amplitudes—for example, a threshold of 5 times or above the signal standard deviation. This higher threshold can reduce false positives without increasing missed detections. By combining all detected spikes from the channels and removing repeated detections based on spike timing, you can maximize spike detection. As shown in Fig. 4.12B, the spikes marked by red triangles can be detected in Ch4, Ch5, and Ch6. A merging algorithm is needed to remove repeated spikes.

After detection, proper parameters of each spike waveform are extracted for spike sorting. For instance, we employed a custom MATLAB program to extract the first principal components and the peak-to-peak amplitudes of spike waveforms from four adjacent channels to combine into eight parameters for sorting. The sorting was performed in SpikeSort3D, an open-source platform from Neuralynx Inc. (www.Neuralynx.com), which integrates the KlustaKwik clustering program (http://klustakwik.sourceforge.net/)—a widely used spike sorting tool that can also be used independently (Harris et al. 2000; Wild et al. 2012). Figure 4.14 shows an example of our sorting results in SpikeSort3D. The four sections in the middle of the figure display the overlapped spike waveforms in four channels respectively. This SpikeSort3D user interface allows for manual adjustments to improve sorting accuracy.

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In our laboratory, after collecting neural signals from the rat hippocampus using a PowerLab recorder (see Sect. 3.5.3), we selected four adjacent channels for spike detection and sorting. The main process includes: (1) Obtain MUA signals. High-pass filter the original wideband signals to produce MUA signals in LabChart software, with a cutoff frequency typically set to 500 Hz. Export the four-channel MUA signals into a MATLAB data file (.mat). (2) Detect spikes and extract parameters of spike waveforms. Run our custom MATLAB program to read the MUA data file, apply a threshold method to detect spikes, and extract the amplitudes and the first PCA components of each spike waveform across four channels. Generate a Spikesort3D data file (.ntt) consisting of these 8 parameters from all spikes. (3) Perform spike sorting. Open the.ntt file in Spikesort3D, adjust settings accordingly, and select the Klustakwik program for classification (Fig. 4.14). The classification can yield the spike sequences of individual neurons. Export these sequences from Spikesort3D into a text file for subsequent analyses. (4) Verify the spike sorting results. Combine the spike sequences with the original recording signals and MUA signals to form a LabChart file (.bin). Opening this file in LabChart as shown in Fig. 4.15 to visually check the sorting results. The left window in this figure spans ~ 4.12 s, while the enlarged right window shows ~ 0.28 s. The top four rows show the original recording signals from the four adjacent channels on an array electrode as shown in Fig. 4.12B. Spikes appear riding on ~ 4 Hz rhythmic field potentials in the left window and become clearer in the enlarged signals in the right window. The middle four rows show the MUA signals with distinct spikes. The bottom four rows show parts of the sorting results—the spike sequences of four neurons, in which each vertical line (with a constant height) represents a spike whose occurrence timing aligns to the upper channels. Using the scalable display in LabChart, it is easy to examine the details of spike waveforms to verify the sorting results.

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In addition to the spike detection and sorting methods described above, we have developed other approaches, including the window-based spike detection introduced in the next section.

Neuronal unit spikes are typically much smaller than the low-frequency field potentials. Thus, the field potentials are usually filtered out by a high-pass filter before using a threshold to detect spikes. However, a challenge can arise when the original recording contains other large potentials with a spectrum overlapping with that of spikes, such as population spikes (PS) from synchronized neuronal firing. In such case, filtering cannot remove PS waveforms without losing unit spikes, and may even introduce additional interference (see Fig. 4.9C). To solve this problem, we developed a window-based method for detecting spikes in the presence of PSs during high-frequency stimulation (HFS) (Wang et al. 2016). Its algorithm steps are as follows.

Figure 4.16A and B show recording examples during O-HFS and A-HFS in the rat hippocampal CA1 region. Removing low-frequency field potentials with a high-pass filter can cause oscillations near the evoked PS waveforms, which obscure nearby spikes (as indicated by the red dots in the figure). This makes spike detection difficult using a threshold method. Additionally, the filtering-produced oscillations may cross the threshold and be misidentified as spikes.

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As shown in Fig. 4.17, the window-based algorithm enables direct spike detection on wideband signals after removing stimulus artifacts. The blue dashed lines represent window segmentations. In each window, the minimum data is initially assumed as the negative peak point of a putative spike. Then, the thresholds A_L and A_H filter out non-spike signals. The two red circles mark putative spikes with an A_0.3 between these thresholds, while the two red boxes mark a field potential with a small A_0.3 belowA_L and a PS falling phase with a large A_0.3 above A_H.

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The window-based algorithm can directly detect spikes in wideband signals without the need to filter out low-frequency field potentials while effectively eliminating interference from large PSs. For spike detection during HFS periods, removing stimulation artifacts beforehand is optional. When running the algorithm directly on signals containing artifacts, the artifacts can be eliminated by properly setting the high threshold (A_H), since artifacts are typically much larger than spikes. Even if artifacts are occasionally misidentified as spikes, they can be easily excluded during spike sorting due to their narrow waveforms from ~ 100μs pulses commonly used in brain stimulations. This window-based algorithm is thus useful for detecting spikes and studying individual neuron activity, especially in recordings containing PS waveforms during HFS or epileptiform discharges. As detailed in Sect. 5.4, this method was used to analyze interneuron firing during A-HFS.

Many algorithms have been developed for spike detection and sorting (Pachitariu et al. 2024; Meyer et al. 2023; Zhang et al. 2023; Carlson and Carin 2019; Lefebvre et al. 2016). We also developed an algorithm specifically for four-channel spike sorting. It combined the spikes from all four channels to create composite waveforms, which were then used to calculate PCA components for spike clustering (Wang and Feng 2009), as described below.

Since determining the true number and waveforms of various spikes in real recordings is challenging, to verify the algorithm, we created four-channel simulated data by combining spike waveforms with Gaussian white noise. The spike waveform v_s(t) was produced using a combination of sinc and exponential functions:

By adjusting the constants a, b and c, and setting the range of time variable t, we created spike waveforms matching real ones. Figure 4.18 shows the two distinct types of waveforms designed to simulate spikes from two neurons. We then constructed four-channel spike signals by adding Gaussian white noise. Each channel contained spikes from both neurons, but with different amplitudes.

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We created ten sets of four-channel simulated data with different SNR values based on the SNR range in real recordings. Figure 4.19 shows one set of the simulated data, including spikes from Neuron-1 (Sp1) and Neuron-2 (Sp2). Sp1 had larger amplitudes in Ch1 and Ch3, while Sp2 had smaller amplitudes across all four channels, particularly in Ch1 and Ch3. This made Sp2 difficult to detect in Ch1 and Ch3. The amplitudes of Sp1 and Sp2 were similar in Ch2 and Ch4, making them difficult to distinguish in these channels. Although the amplitude difference between the two spike types existed in Ch3, using the Ch3 signal alone for spike sorting was not feasible due to noise, as shown in the lower right of Fig. 4.19. This scatter plot was created using the first three PCA components (pc1, pc2 and pc3) of Ch3 spikes. All the 173 detected spikes were scattered and not able to cluster.

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For the same data set, using all four-channel signals was able to detect an additional 34 spikes, increasing the total spike count to 207. We then performed PCA analysis on their composite spikes (as shown in Fig. 4.18). The scatter plot of the first three PCA components clearly differentiated between the two spike types (Fig. 4.19, upper right). For visualization, both scatter plots in Fig. 4.19 show only the first three PCA components. However, we actually used more PCA components— achieving a cumulative contribution rate over 90%—to perform the spike sorting.

We also compared single-channel and four-channel spike sorting for experimental recordings from the rat hippocampal CA1 region. Figure 4.20 shows an example of MUA signals recorded by four contacts on a tetrode array (Fig. 4.20A). Figure 4.20B shows the superimposed spikes identified through each single-channel analysis, while Fig. 4.20C shows the results of four-channel analysis using composite spikes and their PCA components.

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Through careful visual inspection (see Fig. 4.15 for methodology), we identified a total of 912 spikes in the four-channel signals. These spikes originated from five distinct neurons, labeled Sp1–Sp5 with the following counts: 625 (Sp1), 133 (Sp2), 140 (Sp3), 9 (Sp4), and 5 (Sp5). Notably, Sp5 might be an overlapped waveform resulting from simultaneous firing of two neurons. Using these manual analysis results as reference, we compared single-channel and four-channel automatic analyses with a detection threshold of five times the standard deviation of the MUA signal. Table 4.1 shows the detection accuracy, classification accuracy, and number of identified neurons for both methods.

In the single-channel analyses, using Ch1 alone missed spikes across all five neurons, particularly Sp3 spikes with small amplitudes. Only three spike types were identified, with the remaining two types merged into others (Fig. 4.20B). However, these merged types contained few spikes, minimally impacting classification accuracy. Using Ch2 alone resulted in a high detection accuracy, but failed to distinguish between Sp1 and Sp3 due to their similar amplitudes and waveforms. Sp2 was incorrectly divided into two types, resulting in low classification accuracy. Using Ch3 alone yielded the lowest detection accuracy, because Sp3 amplitudes were too small to detect and many Sp2 spikes were also missed. Approximately one-quarter of the total spikes were undetected. Although the classification accuracy was not low, the Ch3 analysis identified only four spike types, with one being misclassified and another partially confused. Using Ch4 alone resulted in a high detection accuracy, but similar amplitudes across the five spike types led to significant confusion, resulting in the lowest classification accuracy.

The multi-channel analysis achieved the highest detection accuracy, missing only 3 out of the total 912 spikes. These missed spikes were all Sp3 type with small amplitudes across all four channels. The analysis successfully identified all five spike types and achieved the highest classification accuracy. Note: the classification accuracy here referred to the percentage of correctly classified spikes among detected spikes, excluding missed ones.

The multi-channel method for spike detection and sorting can be used with signals from two or more channels. To record spikes simultaneously from same neurons across multiple channels, electrode contacts must be closely spaced. When a contact is within 50 μm distance from a neuronal soma, the measured spike amplitude can exceed 60 μV, facilitating detection. Contacts placed 200 μm or more apart can rarely record spikes from a same neuron (Buzsáki 2004; Blanche et al. 2005). As shown in Fig. 4.12, high-density array electrodes with closely arranged contacts can effectively record spikes from same neurons. By combining waveforms from multiple channels into composite spikes, even small or absent spikes in some channels can contribute to the distinctive features of these composite waveforms. As a result, the multi-channel sorting using PCA components can improve classification accuracy, making the method significantly superior to single-channel methods.

Note that the experimental spike waveforms in the above example were extracted from the high-frequency filtered MUA signals (Fig. 4.20A), not directly from the original wideband recordings. The filtering can distort spikes (Fig. 4.5), which may affect classification. Using original spike waveforms may improve spike sorting.

An inter-spike-interval (ISI) histogram can help verify spike sorting results and determine neuron types, such as pyramidal neurons and interneurons. Gerstein and colleagues defined the ISI histogram mathematically as follows (Gerstein and Kiang 1960; Rodieck et al. 1962).

The spike sequence can be represented as the sum of impulse functions with a unit area:where \(\delta (t)\) is the unit impulse function, t represents time, t_k represents the occurrence time of kth spike, and N represents the total number of spikes in the sequence. The ISI histogram, ISI(τ), is then represented as

Discretize the continuous time variable \(\tau\) into \(\tau_{j}\), where j = 1, 2, 3,…. Any \(\tau\) falling within \(\tau_{j} \le \tau < \tau_{j + 1}\) is measured as \(\tau_{j}\). The bin size \(\tau_{j + 1} - \tau_{j} \equiv \Delta \tau\) determines the time resolution of the histogram. In this context, ISI(τ) represents the number of intervals (t_k − t_k–1) falling within the \(\tau_{j}\).

Figure 4.21A shows two episodes of spike sequences respectively from a pyramidal neuron and an interneuron in the rat hippocampus. Each small bar in these raster plots represents a spike. Figure 4.21B, C are the ISI histograms revealing their firing features: (1) The action potential's refractory period appears as an absence of spikes near zero time. An absence of this gap suggests incorrect spike sorting—spikes from multiple neurons may have been grouped together. (2) These histograms show distinct firing patterns between pyramidal neurons and interneurons. The histogram of pyramidal neuron has peaks around ± 3 ms (Fig. 4.21B), signifying a burst pattern—multiple spikes firing rapidly in clusters (see the spikes indicated by red triangles in Fig. 4.12A, B). In contrast, the histogram of interneuron is relatively flat without sharp peaks (Fig. 4.21C), indicating a random and isolated firing pattern (see the spikes indicated by blue dots in Fig. 4.12B).

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For principal neurons involved in signal transmission, burst firing can enhance reliability by increasing neurotransmitter release. Isolated spikes can be treated as noise in some brain regions through a synaptic “filtering” effect, while burst spikes act as transmissible signals carrying information (Lisman 1997). ISI histograms can also feature other firing patterns, such as those induced by high-frequency stimulations with a phase-locked feature (see Sect. 5.3).

An ISI histogram is essentially a component in the autocorrelation function of a spike sequence. For the spike sequence described by Eq. (4.5), its autocorrelation function is (Gerstein and Kiang 1960):

This double sum can be decomposed intowhere each term corresponds to let l = k, l = k − 1, l = k − 2, … in Eq. (4.7). Therefore, the autocorrelation function is still a sequence of δ functions. Discretize the continuous time τ into \(\tau_{j}\), j = 1, 2, 3, …. All τ values that satisfy \(\tau_{j} \le \tau < \tau_{j + 1}\) are measured as \(\tau_{j}\). Thus \(\tau_{j + 1} - \tau_{j} \equiv \Delta \tau\) defines a time bin that determines the resolution. Consequently, all δ functions of \(\varphi_{ff}\) falling between \(\tau_{j}\) and \(\tau_{j + 1}\) are summed to yield the value of the autocorrelation histogram at \(\tau_{j}\).

The first term (where l = k) in Eq. (4.8) represents the constant N at \(\tau = 0\), which is the total number of spikes in the sequence. When the sequence duration is known, this term reflects the average spike firing rate. The second term (where l = k − 1) represents the intervals between two adjacent spikes—the kth and (k − 1)th, which forms the ISI histogram represented by Eq. (4.6). The third term (where l = k − 2) is another interval histogram measuring intervals between every other spikes—the k-th and (k − 2)th (see Fig. 4.22). It is known as the second-order interval histogram. All subsequent terms represent interval histograms for higher-order intervals in the spike sequence.

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The autocorrelation function of a spike sequence can reveal both the refractory period and rhythms in neuron firing. For example, when a hippocampal neuron fires at θ rhythms, the autocorrelation function of its spike sequence can exhibit θ rhythms.

The relationship between spike sequences of two neurons, or between a spike sequence and an external event sequence, can be analyzed using a cross-correlation function. The definition of this function resembles the above-mentioned autocorrelation function. Let the spike sequence of neuron-A beand that of neuron-B be

Then, the cross-correlation function between the two neurons is (Gerstein and Kiang 1960):

It is still a sequence of δ functions. By discretizing the continuous time τ into small bins \(\tau_{j}\) (j = 1, 2, 3, …), we can create a cross-correlation histogram. As shown in Fig. 4.23, using neuron-A sequence as the reference, count the intervals between neuron-B spikes and each reference spike that fall within each ∆τ to form the histogram. Its x-axis represents time, while its y-axis represents spike count. Dividing the spike count by the total spike number yields neuron-B firing probability against τ, producing a histogram showing how neuron-B fires in relation to neuron-A's firing.

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When the spike sequences of both neurons are independent, the expected value for each ∆τ of the cross-correlation histogram is E = N_AN_B∆τ/T, where N_A and N_B represent the total spike numbers of the two neurons, and T represents the recording duration.

A cross-correlation histogram of spike sequences can reveal direct synaptic connections between a neuron pair (Barthó et al. 2004). When a connection exists, the firing of the presynaptic neuron can either excite or inhibit the postsynaptic neuron after a delay of a few milliseconds caused by axonal conduction and synaptic transmission. For example, an excitatory connection from a presynaptic neuron (reference) to a postsynaptic neuron (target) can produce a histogram with a prominent peak around 2 ms which can represent an interneuron's firing triggered by a pyramidal neuron. In contrast, an inhibitory connection can result in a distinct gap after time zero in the histogram which indicates that the target neuron's firing is suppressed immediately following the reference neuron's firing. In addition, bidirectional synaptic connections between two neurons—where an interneuron (reference) inhibits a pyramidal neuron (target) while the latter excites the former—can show a peak to the left and a gap to the right of time zero in the histogram. This pattern reflects mutual synaptic connections, such as the local feedback inhibitory circuits in the hippocampus (see Sect. 2.3).

Nevertheless, in paired neurons with direct synaptic connections, not every spike of the presynaptic neuron can trigger or suppress the firing in the postsynaptic neuron. This is because each neuron typically has thousands of synapses receiving inputs from many other neurons. To produce an action potential, it typically requires simultaneous excitatory inputs from multiple synapses. The activation from a single presynaptic neuron can only increase or decrease the firing probability of the postsynaptic neuron. This probabilistic nature of synaptic transmission results in peaks or gaps in cross-correlation histograms.

If the reference sequence \(f(t)\) is not a neuronal spike sequence but rather a series of external stimuli (e.g., a pulse train), the relationship between the spike sequence \(g(t)\) and the external events \(f(t)\) can also be analyzed using the cross-correlation function—Eq. (4.10). The equation can be rewritten as

On the right side of this equation, the first term in the square brackets represents the cumulative count of intervals between each spike in \(g(t)\) and its nearest preceding stimulus in \(f(t)\) that fall within each ∆τ, i.e., \(t_{k} < t_{l} < t_{k + 1}\). The second term represents the cumulative count of intervals between each spike and all other stimuli that fall within each ∆τ, i.e., \(t_{l} \ge t_{k + 1}\) or \(t_{l} \le t_{k}\).

When the time interval between adjacent applied stimuli is sufficiently long, the effect of one stimulus on neuronal firing can completely dissipate before the next stimulus arrives. In this case, the first term in Eq. (4.11) represents the post-stimulus time histogram (PSTH), also known as the “peri-stimulus time histogram”, which is the sum of spike histograms generated by N_A repeated stimuli. A clear peak in the PSTH indicates that the neuron fires in relation to the stimulation. Conversely, a flat and uniform PSTH suggests that the neuronal firing occurs randomly, unrelated to the stimulation. The PSTH is one of the principal methods for analyzing neuronal responses to external stimuli which will be introduced in the next section, along with other methods including joint peri-stimulus scatter diagram (JPSSD) and joint peri-stimulus time histogram (JPSTH).

As defined by the first term of Eq. (4.11), the PSTH calculation involves two main steps: (1) Divide the time axis (x-axis) evenly into small bins with a resolution of ∆τ, using the stimulus timing as zero; (2) Count the spikes that fall into each ∆τ bin across all stimulation cycles to create the histogram (Rolls et al. 1989). Figure 4.24 illustrates the responses of a neuron to 50 cycles of a 100 Hz pulse stimulation. It shows both the raster plot of neuronal firing (upper) and its PSTH (bottom). The red arrow marks the zero time point when each pulse occurred. Using a bin size of Δτ = 0.5 ms, this PSTH reveals the temporal relationship between the neuronal spikes and the stimulation pulses, with a peak appeared at 7–7.5 ms, indicating the response latency. Notably, not every pulse triggered a spike. The PSTH can also be presented as a curve instead of a bar plot as in this figure. Additionally, the y-axis can display either spike count or firing probability—known as the post-stimulus probability distribution.

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When the firing of Neuron-1 and -2 are both related to a same stimulation, their relationship can be described using a joint peri-stimulus scatter diagram (JPSSD). Figure 4.25 shows the definitions of two interval variables—I_N1S and I_N2S (left), and the creation of JPSSD as a two-dimensional scatter plot using these variables (right). The I_N1S (y-axis) represents the intervals between spikes of Neuron-1 and stimulus-S, while the I_N2S (x-axis) represents the intervals between spikes of Neuron-2 and stimulus-S. With the stimulus time set as t = 0, when Neuron-1 has a spike at time t_N1 and Neuron-2 has a spike at time t_N2, draw a dot at position (t_N1, t_N2) on the two-dimensional plot. Draw all dots sequentially within one stimulus cycle and then repeat for other cycles. The collection of all these dots forms the JPSSD.

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The following JPSSD examples show various relationships between the stimulation S and the spike sequences of two neurons (N1 and N2). The JPSSDs can be analyzed along with the cross-correlation histograms of the two neurons and their respective PSTHs (Gerstein and Perkel 1972).

In summary, the JPSSD of two neurons can reveal the effects from both external stimulation and neuronal interconnections. High-density bands parallel to the coordinate axes represent excitatory stimulation effects on neurons, while a high-density diagonal band parallel to main diagonal represents synchronous firing of both neurons. A narrow diagonal band indicates a direct excitatory synapse connection between the two neurons, whereas a wide and sparse diagonal band indicates both neurons being affected by a third source.

Both cross-correlation histogram and JPSSD can reveal the relationships between the spike sequences of two neurons. However, a cross-correlation histogram only shows the average result of entire spike sequences. Although it can reveal whether one neuron's firing leads or lags another's, it cannot pinpoint when this relationship occurs—whether throughout the entire sequence or only during specific periods. A joint peri-stimulus time histogram (JPSTH) based on JPSSD can offer more details of dynamic changes in the neuronal relationships and in the effects of external stimulation (Gerstein et al. 1989).

Let x(n) be the discrete signal produced by ADC with a sampling frequency of f_s and a corresponding sampling period of ΔT = 1/f_s. As a sequence of N samples, x(n) can be written as x(1), x(2), …, x(N). Its discrete Fourier transform (DFT) is:

The frequency range of X(f) is -f_s/2 ≤ f ≤ f_s/2, with a resolution of Δf = 1/(NΔt). The X(f) is a discrete sequence of N data points: X(1), X(2), …, X(N), evenly spaced by Δf. The plot of X(f) mode (|X(f)|) against frequency is known as amplitude spectrum. Squaring |X(f)| produces the energy spectrum. Then, dividing the energy by the signal duration NΔt produces the power spectrum. Finally, power spectral density (PSD) is the power per unit frequency. Thus, a spectrum can be expressed in four forms: amplitude, energy, power, and PSD. For a real sequence x(n) (e.g., an actual sampled signal), the negative half of its frequency spectrum mirrors the positive half, making it sufficient to calculate only the positive half.

For a random signal, estimating its spectrum from a single signal episode can yield unreliable results due to high variance. Fortunately, periodogram averaging—also known as the Bartlett approach—can effectively reduce the variance. This method divides a signal of N sampled data into K segments of length M, calculates the spectrum of each segment separately, and then averages the results. This process can reduce the variance of spectrum estimation to 1/K (Shiavi 2007).

The duration of a signal used to calculate the discrete Fourier transform determines the spectral frequency resolution. When the signal is segmented, it becomes shorter, resulting in a decreased spectrum resolution. Therefore, it is crucial to compromise on both spectral variance and resolution when setting the segment number. To enhance spectral resolution, segments can be overlapped and reused to yield longer segments, resulting in the modified Welch method. It divides the signal into segments with M sampled data, where the starting points of adjacent segments are apart by only D sampled data, less than M. The number of segments becomes K = (N − M)/D + 1, as shown in Fig. 4.26. The overlap ratio is (M-D)/M × 100%. Increasing this ratio can either raise the segment number K (with a fixed M) or extend the segment length M (with a fixed K), or increase bothK and M. However, an excessively high overlap ratio cannot further improve the spectrum variance. Typically, the overlap ratio is set at about 50%.

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A sampled signal has resolution and range (or duration) in both its frequency and time domains. Understanding these parameters and their relationships is essential. In frequency domain, the frequency resolution of spectrum is determined by the duration of the signal used to calculate the Fourier transform in the time domain (i.e., the signal after segmentation). This resolution is the reciprocal of the signal duration. For example, a 10-s signal gives a frequency resolution of 0.1 Hz, while a 0.1-s signal only achieves 10 Hz—waveforms below 10 Hz cannot be accurately captured in such a short time window. Additionally, the spectrum repeats periodically with a period equal to the sampling frequency (f_s). For real signals, the basic positive frequency range is from 0 to f_s/2. For instance, with f_s = 1000 Hz, the spectrum upper limit is 500 Hz, as waveforms with a frequency beyond this limit cannot be fully captured. In the time domain, the signal resolution (sampling period) is ΔT = 1/f_s—the reciprocal of the spectral period, while the signal duration is the reciprocal of the frequency resolution. These relationships between time and frequency domains exhibit symmetry.

Spectral leakage is another issue to consider when using the periodogram method for spectrum estimation. Extracting a segment from a recorded signal is equivalent to applying a rectangular window to the signal. The spectrum of this window has large sidelobes, leading to leakage and thus deviation in the estimated spectrum. To reduce the leakage, a common approach is to multiply each signal segment by a non-rectangular window with smaller sidelobes—such as Hamming or Hanning windows—before calculating the spectrum.

Popular signal processing software like MATLAB can easily calculate spectra of brain signals such as EEG and LFP. For example, MATLAB function “pwelch” can compute the PSD using the Welch periodogram method. The function's calling format is:

In the parameter list on the right side, “x” represents the sampled signal, “window” is the chosen window with an equal length to the signal segment, “noverlap” is the overlap ratio between adjacent segments, “nfft” is the number of data points used for the Fourier transform, and “fs” is the sampling frequency of signal x. This pwelch function uses the Fast Fourier Transform (FFT) algorithm to speed up computation when “nfft” is a power of 2. Note that the window length must not exceed “nfft” to avoid errors. In the return list on the left side, “Pxx” represents the PSD of signal x, while “f” is the corresponding frequency vector. Additionally, the pwelch can provide confidence intervals for the spectrum estimation. For more details, refer to the MATLAB help documentation.

Figure 4.27 shows examples of PSD calculations for EEG signals collected from the central (C3 and C4) and occipital (O1 and O2) areas of an awake, resting adult subject with closed eyes. The EEG signals were 10 s in length with a sampling frequency of 250 Hz. Their spectra were estimated using the Welch periodogram method, with a segment length of 256 data points, a Hanning window, and a 50% overlap. When the subject was awake and quiet (Fig. 4.27A), the EEG signals showed prominent α rhythms in the 8–13 Hz range, which produced clear peaks in their power spectra (Fig. 4.27C). In contrast, when the subject performed complex mental arithmetic (Fig. 4.27B), the α rhythms were suppressed, resulting in a significant decrease in the power of α frequency range (Fig. 4.27D).

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After obtaining the PSD, the mean power in each rhythmic range can be calculated based on the definitions of rhythmic frequency ranges for EEG and LFP (Feng and Zheng 2004a). For example, we can estimate the power of δ (0.5–4 Hz), θ (4–8 Hz), α (8–13 Hz), and β (14–30 Hz) rhythms. From this, the amplitude of individual rhythms can be determined. An alternative approach is to calculate the ratio of power in each rhythm range to the total signal power. This method can eliminate individual variations, facilitating comparisons among different subjects.

Long recordings of spontaneous EEG and LFP signals are typically non-stationary, changing with the behaviors and mental states of humans or animals. The above-mentioned spectral estimations are only suitable for stationary signals. Time–frequency analyses—such as short-time Fourier transform (STFT) and wavelet transform (WT)—are more suitable for non-stationary signals. STFT analyzes a signal using a shifting window with a fixed length, exhibiting time–frequency features in a two-dimensional plot. However, STFT has a trade-off: shorter signals result in poorer frequency resolution, making it impossible to achieve high resolutions in both time and frequency domains simultaneously. Unlike STFT, WT can decompose a signal using basic wavelets with varying displacement and scale factors. By using larger windows for lower frequency components and shorter ones for higher frequency components, WT can meet resolution requirements in both time and frequency domains (Clark et al. 1995; Feng 2003; Feng and Zheng 2004b). In MATLAB, the “spectrogram” function can calculate STFT, while its Wavelet Toolbox provides various functions for performing wavelet transforms.

We used a PowerLab recorder to collect signals in the rat brain (see Sect. 3.5.3). This recorder comes with the LabChart software, which offers spectrum analysis using the periodogram method. It employs FFT to calculate the discrete Fourier transform (Eq. (4.12)). FFT requires the number of data points to be a power of 2. As shown in Fig. 4.28, the “FFT size” option in the LabChart settings dialog provides options including 32, 64… 1 K (1024), 2 K (2048)…to 128 K (131,072) in the drop-down menu, all powers of 2. This FFT size corresponds to the data length M shown in Fig. 4.26. Note that this M is the length of divided signal segments, not the total signal length. The M is crucial as it determines the spectral frequency resolution. For example, using the default FFT size of 1 K (1024 sampled data points) for a signal sampled at a 20 kHz frequency results in a spectral resolution of only 20000 Hz/1024 = 19.531 Hz. This means that the spectrum of low-frequency field potentials below ~ 20 Hz cannot be obtained. Increasing the FFT size to 32 K (32,768) or 64 K (65,536) can improve the spectral resolution to 0.610 Hz or 0.305 Hz.

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The “Data window” option in the settings dialog allows user to choose a window applied to each signal segment to reduce leakage error. Available windows include Bartlett, Hamming, and Hanning windows. The “Window overlap” represents the overlap ratio between adjacent signal segments—as shown in Fig. 4.26. The “Mode” option sets the spectrum form, with options including amplitude, power, and power density.

In the cortex and hippocampus of a urethane-anesthetized rat, the recorded LFP signals typically exhibit large-amplitude slow waveforms of δ rhythms. However, when a small clip is used to slightly clamp the rat's tail, the LFPs can quickly shift to a signal dominated by small-amplitude θ rhythms (Buzsáki et al. 1983). Here, I use such a recording as an example to demonstrate LabChart spectral analysis. The upper row of Fig. 4.29A shows an LFP signal from the rat hippocampus, which spanning ~ 867 s and containing two tail-clamping events. The expanded view below clearly shows the LFP change caused by the first tail clamping.

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Before entering the PowerLab recorder for sampling, the signal was magnified 100-fold by the pre-amplifier (Model 3600) with a frequency range of 0.3–5000 Hz (see Sect. 3.5 for details). Therefore, a ± 1 mV signal appears as ± 100 mV in Fig. 4.29A.

To perform the STFT spectral analysis in LabChart, set the FFT size to 128 K—M = 131,072. With a sampling frequency of 20 kHz, this FFT size produces a signal segment of 6.5536 s, yielding a spectral frequency resolution of 0.153 Hz. Using a Hamming window with 50% overlap can produce a time resolution of 6.5536 s/2 = 3.2768 s for the time–frequency spectrum. Figure 4.29B shows the PSD of the entire signal shown in Fig. 4.29A, demonstrating that both tail clamping events shifted the LFP from δ rhythm (~ 0.76 Hz) to θ rhythm (~ 3.5 Hz) dominance. The expanded view below focuses on the period from 143 to 182 s, clearly showing discrete spectral data with a time resolution of about 3.3 s. The large-amplitude δ rhythms (yellow) before tail clamping and the small-amplitude θ rhythms (green) during tail clamping are distinctly visible.

Figure 4.29C shows the mean PSDs of the two signals: one before tail clamping (~153 s) and the other during tail clamping (~ 262 s), marked by ① and ② in Figure 4.29A, B. The pre-clamping PSD (orange), calculated from 47 signal segments (K = 47), shows a δ peak at 0.76 Hz. The during-clamping PSD (green), calculated from 78 signal segments (K = 78), shows a θ peak at 3.1 Hz. The δ peak (0.19mV²/Hz) is approximately 14 times that of the θ peak (0.014mV²/Hz), indicating a 3.7:1 amplitude ratio in the original recordings. Note: the θ rhythm frequency in rat LFPs under urethane anesthesia is about 2–5 Hz, lower than the 6-9 Hz observed during awake state (Buzsáki 2002).

We have used the power spectrum analysis to study various neurophysiological phenomena in rats, including exploring changes in cortical electroencephalogram (ECoG) across different sleep stages and at various depths of anesthesia (Feng 2003; Feng and Zheng 2004a), as well as analyzing changes in hippocampal field potentials in response to high-frequency pulse stimulations (Yu et al. 2016; Ma et al. 2019).

To investigate how the waveforms of stimulation artifacts change with the relative position and distance between stimulation and recording electrodes, we designed a simple setup to imitate stimulation and recording in rat brain. As shown in Fig. 4.31A, we filled a glass dish (75 mm diameter, 15 mm height) with physiological saline and inserted two electrodes: a bipolar concentric stimulation electrode (CBCSG75) and a 16-channel array recording electrode (A1 × 16-Poly2-5 mm-50 s-177). We connected a 50 Hz sinusoidal signal with a 5 V peak-to-peak amplitude from a signal generator (Model DG1000Z, RIGOL Technologies) to a Model 2200 analog stimulator with electrical isolation (see Sect. 3.5.4). Then, the stimulator's output—current sine waves—was connected to the stimulation electrode, with its inner pole serving as anode and outer pole as cathode.

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We set the stimulator output to a 5μA peak-to-peak amplitude and placed the recording array near the stimulation electrode, allowing the recording contacts to span both poles of the stimulation electrode (Fig. 4.31A). The recorded signals closely matched the sine waves from both the signal generator and stimulator outputs, without significant distortions (Fig. 4.31B). The phase of recorded sine waves on each channel aligned with its nearer stimulation pole. We then increased the stimulator output from 5 to 50 μA peak-to-peak sine waves and moved the recording array about 1 mm away and above the stimulation electrode. As all the recording channels were now closer to the cathode pole, the recorded sine waves aligned with the cathode pole phase, opposing to the original stimulator output signal. Additionally, the recorded waveforms became distorted.

We also used this experimental setup to explore artifact waveforms from pulse stimulations, though details are omitted here. Note that this saline model cannot fully replicate the electrical properties of brain tissue. Stimulation artifacts recorded in-vivo in rat hippocampal regions can be more complex.

Figure 4.32 shows a recording with a 2 Hz sinusoidal current stimulation (1-min, 40 μA peak-to-peak amplitude) applied to the Schaffer collaterals of rat hippocampal CA1 region. The signal was recorded in the CA1 pyramidal layer downstream of the stimulation site. The electrode placements were same as shown in Fig. 4.30A. The enlarged insets in Fig. 4.32 show that the recording during stimulation contained clear 2 Hz sinusoidal artifacts which were absent during baseline period before stimulation. These artifacts were removed by a 2 Hz notch filter, as shown in the figure. The sinusoidal stimulation produced a clear 2 Hz rhythm in neuronal firing, particularly visible in the MUA signal produced after 500 Hz high-pass filtering (Fig. 4.32, lower right), demonstrating the effect of stimulation modulation.

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The sinusoidal artifact appeared as a continuous signal coexisting with neuronal responses throughout the stimulation period. In contrast, a transient narrow pulse can trigger neuronal responses with a time delay (latency), as shown by the single-pulse and paired-pulse evoked population spikes in Sect. 2.2. Even though stimulation pulses and evoked responses occur at different times, artifact removal can still be required for detecting certain neuronal signals, such as unit spikes (see Fig. 4.9). Unlike sinusoidal artifacts—which can be removed using a notch filter, pulse artifacts have a broader spectrum with high-frequency components that overlap with spike spectrum. Pulse artifacts cannot be simply filtered out (Heffer and Fallon 2008; Wichmann 2000). The next section will introduce the artifacts of narrow pulses and their removal methods.

Many neuromodulation therapies, including deep brain stimulation (DBS), use pulse electrical stimulations, which often creating large-amplitude artifacts in recordings. The waveform and amplitude of these artifacts depend on multiple factors, including the structure of stimulation electrodes, the pulse waveform, intensity and frequency (rate), the relative positions of recording and stimulation electrodes, as well as the features and settings of amplifiers and stimulators. Electrical isolation between stimulation and recording circuits can greatly reduce the artifacts. With proper isolation, most stimulation current can loop through the stimulation circuit, with minimal invasion into the recording circuit. Additionally, using a bipolar stimulation electrode with two closely spaced poles (millimeter or sub-millimeter) can confine the stimulation current to a small brain area. This can also minimize stimulation artifacts in recordings, provided the stimulation site is sufficiently distant from the recording site. However, stimulation artifacts recorded by nearby electrodes can still disturb neuronal signals. This interference is particularly problematic when detecting unit spikes, which typically have only tens to hundreds of microvolts in amplitude—much smaller than stimulation artifacts. The magnitude difference between stimulation artifacts and neuronal signals can reach 4–5 orders (Wagenaar and Potter 2002).

For example, in our in-vivo experiments of electrical stimulations in rat hippocampal regions, the recording and stimulation sites were about 1–2 mm apart. Using a Model 3800 stimulator with isolators and a Model 3600 amplifier (see Sect. 3.5 for details), we applied narrow current pulses with a width of 100 μs and intensity of ~ 0.3 mA. This stimulation usually produced artifacts with several millivolts in amplitude—far greater than both unit spikes and normal field potentials. The artifacts were sometimes even larger than the population spikes (PS) generated by synchronized neuronal firing.

Large-amplitude stimulation artifacts can interfere with both neural signal display and signal extraction. Especially during high-frequency stimulation (HFS), the dense artifacts can obscure neural signals. Methods have been designed to eliminate pulse artifacts, including template subtraction, waveform fitting, and linear interpolation (Erez et al. 2010; Wagenaar and Potter 2002; Hashimoto et al. 2002). The template subtraction method involves obtaining an artifact template, which is then subtracted from the recorded signal wherever artifacts are identified. This method can recover neural signals superimposed on artifacts, provided that the recording is not truncated by amplifier saturations. In other words, the amplitudes of recorded signals with artifacts must stay within the amplifier range.

In addition, the template method requires artifact waveforms to remain constant, which is challenging in practice. First, the electrical properties of brain tissue—particularly impedance values—can change, especially during sustained HFS. This can lead to variations in artifact waveforms. Second, for narrow pulses, additional factors affect the recorded artifact waveforms, including the limitation of ADC sampling rate and the clock timing difference between the recorder ADC and stimulator.

Figure 4.33 illustrates the impact of a clock difference by an example of sampling a 10 kHz sine wave (0.1 ms cycles, the orange curve). Based on the sampling theorem, the signal is sampled at 20 kHz, yielding two samples per cycle. Assume that the clock controlling the ADC for sampling does not exactly synchronize with the clock of digital signal generator that produces the original sine wave. Taking the clock of signal generator as reference, let the actual sampling rate of sampler be 20.83 kHz rather than exact 20 kHz. This discrepancy can result in oscillation waves in the sampled data with amplitudes varying periodically at 0.83 kHz (black dots and lines in Fig. 4.33). This example exaggerates the clock difference to illustrate potential changes in sampled artifacts. In practice, clock differences between instruments are typically very small and negligible. Nevertheless, in long recording, even small differences can lead to significant changes in sampled stimulation artifacts, as shown in the following example.

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We used a Model 3800 stimulator to generate HFS pulses applied into the rat brain. A PowerLab recorder sampled the signals collected by a recording electrode (see Fig. 3.17). Figure 4.34A shows a signal recorded in the hippocampal CA1 soma layer during a 1-min HFS of biphasic current pulses applied on the alveus. The pulses had an intensity of 0.3 mA, width of 100 μs per phase, and frequency of 200 Hz. We set the amplifier frequency band at 0.3 Hz–5 kHz and used a 20 kHz sampling rate. The dense stimulation artifacts in the original recording obscured most of the neural signal, except for the large APSs at the onset of HFS (Fig. 4.34A). The artifacts created a periodic oscillation envelope due to the clock difference between the 3800 stimulator and PowerLab sampler. This envelope had a period of about 4 s, corresponding a frequency difference of 0.25 Hz—a minuscule 0.00125% of the 20 kHz sampling rate. However, this small difference caused significant changes in pulse artifacts. With the 20 kHz sampling rate and 100 μs pulse width, the recorder captured only a few samples per pulse artifact. In the enlarged insets in Fig. 4.34B, the sampled artifact data are shown in sparse red dots. Due to the clock difference, the sampling positions on the artifacts constantly shifted, causing varying artifact waveforms and amplitudes.

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By the way, the LabChart software of PowerLab recorder can display sampled signals in two modes: “Join Points with Line” and “Show Points as Dots”. Figure 4.34A used the former, while the upper plot of Fig. 4.34B used the latter. The dot display mode can allow clear visualization of neural signals and on-line observation of changes in evoked potentials (APSs) during HFS. Even with HFS at a 200 Hz pulse frequency as in this example, the biphasic pulse artifacts (with a total width of ~ 0.2 ms) occupy only a small fraction of time. The artifacts contain far fewer sample data than neural signals, enabling the dot display to effectively reveal the neural signals.

These results show that sampled artifacts of narrow pulses can vary over time, even when the actual stimulation artifacts remain constant. This variation makes artifact removal using template methods difficult. To address artifact interference, researchers have had to compromise by either reducing stimulation intensity or inferring neuronal activity during stimulation by examining signals immediately after stimulation ends (Rosa et al. 2012).

We used bipolar stimulation electrodes to confine the stimulation area in our rat experiments. However, large stimulation artifacts were still captured by nearby recording electrodes, affecting the detection of unit spikes. Figure 4.35A shows a brief segment of the original wideband recording (0.3–5000 Hz) with 100 Hz 0.3 mA HFS pulses. The pulse artifacts had a large amplitude of about 15 mV. Unit spikes with an amplitude of ~ 0.2 mV (indicated by blue dots) were only visible in the enlarged signal. These spikes were superimposed on low-frequency field potentials which needed to be removed by a high-pass filter before using a threshold method to detect the spikes. However, while filtering removed low-frequency potentials, it also altered the artifacts, making it difficult to identify spikes near the artifacts, as shown in the bottom of Fig. 4.35A. In Fig. 4.35B, each artifact segment in the original signal was replaced with a straight line connecting its endpoints before high-pass filtering was applied. This process resulted in a MUA signal with clear spikes. Figure 4.35C shows overlaid plots of the filtered signals both with and without artifacts, highlighting their differences. High-pass filtering with artifacts caused oscillations around the artifacts and interfered with spike detection. The artifact-free signal in Fig. 4.34C was also produced by this artifact removal process. The method is suitable for HFS with narrow pulses, as the artifact duration is minimal. We used it to process recordings with various high-frequency pulse stimulations reported in Part II of this book (Chaps. 5–8).

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Removing an artifact by replacing it with a straight line can be accomplished by a linear interpolation algorithm, which can be implemented using a MATLAB program. Its algorithm consists of two main steps: where \(x_{{0}}\) and \(x_{{N - {1}}}\) represent the sampled data at the interpolation starting and ending points, N is the number of data on the interpolation line, and \(x_{n}\) represents the n-th interpolated data, n = 1, 2, …, N − 2. The length of the interpolation segment (N) needs to be determined from the actual recorded signal. Besides the above-mentioned sampling limitations that can result in pulse artifacts different completely from the applied pulses (as shown in Figs. 4.34B and 4.36), additional factors can distort pulse artifacts and extend their duration, including transient amplifier saturation caused by excessive artifact amplitudes and the volume conductor effect of neural tissue. Consequently, the interpolation segment for artifact removal can be slightly longer than the width of applied pulse, but typically narrower than spikes (Fig. 4.35) and evoked population potentials (Fig. 4.36).

The linear interpolation method for artifact removal has a disadvantage: it can lead to neural signal loss during the artifact-occupied period. However, when the pulse amplitude exceeds the amplifier's range—causing saturation, the signal during the artifact is already truncated by the amplifier. In this case, artifact removal will not worsen the existing signal loss. Such signal truncations commonly occurred during pulse periods in our experimental recordings. High-quality neural signal recordings require a high ADC resolution (see Sect. 3.5.3). This means that the amplifier range should be set to match the maximum amplitude of neural signals, which can inevitably lead to larger stimulation artifacts being truncated (see Fig. 3.25).