# 1/*f* Neural Noise Reduction and Spike Feature Extraction Using a Subset of Informative Samples

## Abstract

This article describes a study on neural noise and neural signal feature extraction, targeting real-time spike sorting with miniaturized microchip implementation. Neuronal signature, noise shaping, and adaptive bandpass filtering are reported as the techniques to enhance the signal-to-noise ratio (SNR). A subset of informative samples of the waveforms is extracted as features for classification. Quantitative and comparative experiments with both synthesized and animal data are included to evaluate different feature extraction approaches. In addition, a preliminary hardware implementation has been realized using an integrated circuit.

### Keywords

Spike sorting Spike feature extraction Clustering Action potential## Introduction

Real-time extraction of information from composite neural recordings is a significant challenge in neural interfacing. Developing integrated circuit (IC) to enable portable and implantable systems is important to allow the study of complex behavior in neuroscience experiments, closed loop deep brain stimulation and cortical controlled neuromuscular prostheses. In order for a spike feature extraction algorithm to be functional as a small device with real-time low-latency processing and low-power operation, it must be efficient in both computation and IC implementation.

Implementing spike sorting before data telemetry offers many significant advantages. Spike feature extraction provides the necessary information required to sort spikes from raw sampled data. With this information, each spike event can be represented by its unique features and firing time, resulting in significant data compression. A data transceiver designed with the current semiconductor technology can simultaneously support a large number of recording channels for a microchip implementation to extract the spike feature.4,15 System integration using wireless power telemetry or a rechargeable battery as well as wireless data telemetry removes the need for tethering wires. As a result, a fully wireless operation would relieve the subjects’ overall stress factor and allow them to move freely in their natural environment.

Frequently used spike feature extraction algorithms include Principal Components Analysis (PCA),40,50 Bayesian algorithm,24 template matching,25,43,44,49 wavelets,23,30,31 Independent Component Analysis28,35,37, 38, 39 (ICA), and inter-spike interval-based algorithms.10,26,27 These all demand significant computation. Efforts to improve the efficiency of these algorithms have been reported; however, these approaches either rely on an over simplified functionality or use a hardware system that consumes too much power and space.

In part, complex algorithm procedures1,20,34 are applied to mediate the effects of noise and distortion in the recording process. Noise sources include ion channel noise,11 activity from distant neurons, low-frequency field potentials,36 thermal noise and circuit noise. Significant sampling distortion is also present since it is unrealistic to synchronize the sampling clock with individual spikes.

This article reports a new spike feature extraction algorithm which is suitable for real-time, low-latency spike sorting and enables IC implementation. “Overview of the Work” section gives overview of the work. “Similar Neurons, Noise, and Sampling Distortion” section describes spike waveform difference, noise, and distortion. “Sample Information” and “Enhancing SNR Using Noise Shaping Filter” sections focus on the selection of “informative samples.” “Experiments” section presents experimental results. “Conclusion” section concludes the work.

## Overview of the Work

### Variable Selection Techniques

As a complementary approach to dimensionality reduction algorithms, Jolliffe discussed a general feature extraction algorithm based on a subset of samples in a classic work.19 This concept requires only a subset of samples containing the necessary information to cluster the data, as opposed to using all of the samples. These informative samples are especially useful in the presence of single prominent sample set.

There are two challenges facing a sample selection algorithm. The first challenge is the computational burden to select informative samples. If the training procedure is as complicated as suggested in Jolliffe,19 it would prohibit microchip implementation for implant purposes. The power and area are the primary problems with the microchip implementation of other spike feature extraction algorithms. The second challenge is the availability of localized features. Improved performance compared to PCA is unlikely if localized features are not prominent.

### Our Approach

We have developed a spike feature extraction algorithm based on informative samples.48 The theoretical framework includes neuronal signature, noise shaping, and informative sample selection. By evaluating neuronal geometry signatures with the compartment model, we find that improved differentiation among similar spikes can be achieved by appropriately emphasizing signal spectrum.47 Studying the noise properties has revealed that a frequency shaping filter can be used to boost the signal-to-noise ratio (SNR).46 The sample selection technique using estimated entropy identifies informative samples for sorting spikes. In addition, a preliminary IC implementation of the algorithm has been recently fabricated6 and being integrated onto a multi-channel neural recording IC.5

## Similar Neurons, Noise, and Sampling Distortion

### Neuronal Signature

In this section, we briefly describe an analytical model for extracellular spike, based on which we study the spike waveform difference.

*j*

_{m}is the transmembrane current and

*σ*

_{e}is the conductivity of the tissue environment; \( \overrightarrow {{r_{0} }} \) and \( \overrightarrow {r} \) represent the locations of the point electrode and the active membrane segment, respectively.

^{−1}3,16,17), the recorded active membranes usually do not fire simultaneously. As a result, the detailed geometry of the underlying neuron may influence the shape of spikes. Following the computational model described in Greenberg

*et al.*,12 Rattay

*et al.*,32 Traub

*et al.*,41 Tuckwell,42 a neuron is modeled as compartment elements. An extracellular electrode only records those membrane segments within the recording radius (measured to be tens of micrometers2) and can be modeled as one or few compartments with uniform ion channel densities. Derived from Eq. (1), the spike waveform is expressed as the convolution of the transmembrane current profile and an implicit geometry kernel function

*W*(

*t*) is the geometry kernel function determined by geometry properties of the recorded membrane segments.

Equations (1)–(3) can be used to evaluate the spectrum properties of Δ*V*(*t*), which is helpful for designing an appropriate filter passing band. In the cases the ion channel populations are similar, Δ*V*(*t*) can have a useful spectrum at a higher frequency point, which helps to differentiate similar spike waveforms. In “Appendix” section, an analytical approach of exploring the spectrum of Δ*V*(*t*) is included.

### Noise

*f*

^{α}family noise. The frequency dependency of noise is dictated by multiple sources. Identified noise sources include 1/

*f*

^{x}neuron noise,9 electrode–electrolyte interface noise, and electronic noise, which are illustrated in Fig. 1 using lumped circuit model. Except tissue thermal noise that has a flatten spectrum, the rest ones show frequency dependency. Specifically, 1/

*f*

^{x}neuron noise that characterizes the superimposed background activities from distant neurons is debatably induced from stochastic variation of neuron’s activation.9 Numeric simulations based on simplified neuron models suggest that

*x*can vary in a range depending on parameters. For the electrode–electrolyte interface noise, non-faradaic type in particular, an effective distributed resistance (

*R*

_{ee}) can be defined and generates thermal noise that is attenuated quadratically to frequency by the interface capacitance (

*C*

_{ee}). For electronic noise, there are two major components: one is thermal noise (~

*kT*/

*g*

_{m}) and the other is called flicker noise (or 1/

*f*noise) that dominates at lower frequency range and is heavily dependent on fabrication process.

#### 1/*f*^{x} Neuron Noise

*V*

_{neu}(

*t*) represents the superimposed background activities of distant neurons;

*i*and

*t*

_{i,s}represent the object identification and its activation time, respectively, and

*V*

_{i.neu}is the spiking activity template of the

*i*th object. Based on Eq. (4), the power spectrum of

*V*

_{neu}is

*s*

_{0},

*P*{} is the spectrum operation,

*X*

_{i}(

*f*) is the Fourier transform of

*V*

_{i.neu}, and

*f*

_{i}is the frequency of spiking activity

*V*

_{i.neu}(the number of activations divided by a period of time). The spectrum of a delta function spike pulse train \( \left( {\sum\nolimits_{\text{s}} {\left\langle {\left. {e^{{2\pi jf(t_{{i,s_{0} + s}} - t_{{i,s_{0} }} )}} } \right\rangle } \right.} } \right), \) according to Davidsen and Schuster9, features a lower frequency and exhibits a 1/

*f*

^{α}frequency dependency. As this term multiplies |

*X*

_{i}(

*f*)|

^{2}, the unresolved spiking activities of distant neurons contribute a spectrum of 1/

*f*

^{x}within the signal spectrum.

#### Electrode Noise

*in vivo*recording environment that involves several different ionic particles, e.g. Na+, K+,…, the current flux of any

*i*th charged particle

*J*

_{i}(

*x*) at location

*x*assuming spatial concentration

*n*

_{i}(

*x*) is described by the Nernst equation

*D*

_{i}is the diffusion coefficient,

*Φ*electrical potential,

*z*

_{i}charge of the particle,

*Q*the charge of one electron,

*k*the Boltzmann constant,

*T*the temperature, and

*w*the convection coefficient. In a steady state,

*J*

_{i}(

*x*) is zero with the boundary condition of maintaining about 1 V drop from metal to electrolyte. In such a case, the electrode interface can be modeled as a lumped resistor

*R*

_{ee}in parallel with a lumped capacitor

*C*

_{ee}. This naturally forms a low-pass filter for the interface noise. As a result, the induced noise from

*R*

_{ee}at the input of the amplifier is

*C*

_{i}) is sufficiently small, introducing negligible waveform distortion, the integrated noise by electrode interface satisfies

Equation (8) suggests reducing electrode interface noise by increasing double layer capacitance (*C*_{ee}). Without increasing the size of electrodes, carbon-nanotube (CNT) coating21 can dramatically increase electrode surface area, thus, reducing the interface noise. “Enhancing SNR Using Noise Shaping Filter” section will compare conventional electrodes and CNT-coated electrodes from a noise point of view.

*n*

_{i}(

*x*) = 0 results in a flattened noise spectrum. Here, we use a lumped bulk resistance

*R*

_{b}in series with the double-layer interface for modeling noise

*R*

_{b}is the bulk resistance,

*ρ*

_{tissue}is the electrolyte resistivity,

*r*

_{s}is the radius of the electrode, and

*χ*is a constant that relates to the electrode geometry. As given in Wiley and Webster45, χ ≈ 0.5 for a plate electrode.

#### Electronic Noise

*N*

_{cthermal}is the circuit thermal noise,

*N*

_{cflicker}the flicker noise,

*g*

_{m}the transconductance of the amplifier (∂

*i*

_{out}/∂

*v*

_{in}),

*γ*a circuit architecture-dependent constant typically <4,

*K*a process-dependent constant on the order of 10

^{−25}V

^{2},33

*C*

_{ox}the transistor gate capacitance density, and

*W*and

*L*the transistor width and length, respectively. Thermal noise can be reduced by increasing transconductance (

*g*

_{m}), which is linearly proportional to power consumption. Flicker noise can be reduced using design techniques such as large size input transistors and chopper modulations. In a sense, circuit noise can be used to trade off circuit power and area.

#### Total Noise

*N*

_{neu}), electrode–electrolyte interface noise (

*N*

_{ee}), thermal noise from the electrolyte bulk (

*N*

_{eb}) and active circuitry (

*N*

_{cthermal}), and flicker noise (

*N*

_{cflicker}). The noise spectrum is empirically fitted by

*N*

_{1}/

*f*

^{x}and

*N*

_{0}represent the frequency dependent and flat terms, respectively. Equation (11) describes a combination of both colored noise (1/

*f*

^{x}) and broad band noise.

### Sampling Distortion

*t*) and the red traces are sin(

*t*+ Δ

*t*) with random deviation Δ

*t*∈ [−1/16π, 1/16π]. Using the black square curve to represent an ideal neural spike, a red trace is then a recorded spike which deviates from the ideal spike. Such deviation referring as sampling distortion introduces spike sorting errors, especially when sampling distortion is comparable to the spike difference from different neurons.

Intuitive methods to reduce sampling distortion include increasing the sampling frequency of the ADC or performing interpolation and template matching in the digital domain.

Both approaches require additional power, computation, and storage space, which are not favorable to microchip implementation. As shown in Fig. 2, the amount of distortion varies at different samples, which is intuitive since sampling distortion is related to the waveform slope. Inspired by this property, we use a density function-based procedure to select good samples, from which the spike features are derived. The algorithm for selecting samples is efficient in both computation and storage space, and targeting hardware implementation. The details are shown in sections “Sample Information” and “Enhancing SNR Using Noise Shaping Filter.”

## Sample Information

Methods to quantify information carried by individual spike samples are discussed in this section. Intuitively, a sample is considered to be informative if the superimposed spikes can be classified into multiple clusters by evaluating that sample alone. The method used to quantify the sample information is outlined below.

**Algorithm**

Sample Information Estimation

- Input:
*M*peak aligned spike segments {*v*_{i},*i*= (1,2,…,*M*)} with*N*samples for each segment- Output:
Information

*info*_{j}carried by spike samples {*v*_{i}(*j*),*i*= (1,2,…,*M*)}

*j*= 1, construct one-dimensional data set*X*= {*v*_{i}(*j*),*i*= (1,2,…,*M*)}Obtain a nested cluster configuration based on X

Estimate the probability

*p*_{q}that a sample being partitioned into the*q*th cluster. Use the entropy to estimate the information*info*_{j}= −Σ*p*_{q}>*p*_{0}*p*_{q}ln(*p*_{q}), where*p*_{0}is a threshold of the cluster size.Repeat the procedures to a different sample, e.g.

*j*=*j*+ 1.

### Probability Function Estimation

The probability that a sample being partitioned into the *q*th cluster, *p*_{q}, is obtained by locating peaks and valleys of an estimated density function of variable *v*_{i}. In this section, a density function estimation method using convex kernels is introduced.

*x*

_{1},

*x*

_{2},…,

*x*

_{M}are independent and identically distributed samples of a random variable, the kernel density estimate to approximate the probability density function is

*G*(

*x*) is an arbitrary isotropic kernel with a convex profile

*g*(

*x*), i.e., it satisfies

*G*(

*x*) =

*g*(|

*x*|

^{2}) and

*g*(

*x*

_{i}) −

*g*(

*x*

_{j}) ≥

*g*′(

*x*

_{j})(

*x*

_{i}−

*x*

_{j}),

*d*is the dimension of the data (“1” here),

*h*is the kernel bandwidth, and

*d*is the dimension of the data space. Compared with the histogram that also approximates a probability density function, the kernel density estimate defined in Eq. (12) is a smoothed one avoiding artificial peaks/valleys due to insufficient samples. As an example, density functions of spike samples approximated by the histogram and convex kernels are displayed in Fig. 3. It is intuitive to see that kernel scope

*h*is a sensitive parameter that affects the estimated density function. In this study, we use the local kernel bandwidth scheme, which is reported to be robust by many authors.8,13

### Discussions on Sample Selection

*M*= 300, the information scores approximately settle to the expected values, as shown in Fig. 4.

Investigation of informative samples in noisy spikes has been carried out. Results using synthesized spikes with recordings from neocortex and basal ganglia31 are shown in Fig. 4. There are two clear observations. First, the amount of information carried by each sample varies, indicating a non-uniform signal-to-noise plus distortion ratio. Second, it is necessary to create informative samples if due to severe noise, distortion and similarity of spike clusters, few of the samples is informative. As a constraint to create informative samples, the computation and storage space have to be feasible for microchip implementation.

## Enhancing SNR Using Noise Shaping Filter

As mentioned in “Similar Neurons, Noise, and Sampling Distortion” section, a noise shaping filter can be used to enhance SNR. The fundamentals of noise shaping are straightforward. Instead of equally amplifying the spectrum, a noise shaping filter allocates more weight to high-SNR regions while reducing weight at low-SNR regions. This results in an increased ratio of the integrated signal power over the noise power. In this section, we use derivative operation as an example to illustrate the usefulness of the frequency shaping filter and further demonstrate that the filter creates additional informative samples.

*f*

_{s}is the sampling frequency of the ADC.

*f*

^{−2}noise profile for illustration, the filter’s influence on noise could be quantified by

*λ*

*f*

_{c1}and

*f*

_{c2}are the lower and higher corner frequencies of the digital filter, respectively. In case

*λ*is less than 1, SNR further increases, which favors spike sorting from the noise perspective.

The sampling distortion distribution among samples is altered after taking the derivative. In the original waveforms, samples close to peaks suffer less distortion compared with those in transition. After taking the derivative, samples initially suffering from large distortion become less distorted, because *V*″(*t*) has at least one zero crossing point during the transition. Quantitative experiments to demonstrate the creation of informative samples have been carried out and shown in Fig. 4. In these data, the black solid lines represent information carried by the samples from spikes and the dotted red lines represent the derivatives. The spike data are eight challenging sequences from Quian Quiroga *et al.*31 They are compiled from recordings in the neocortex and basal ganglia with superimposed noise. All eight sequences contain three neuronal sources.

The corresponding feature extraction results using the most informative samples from spikes as well as their derivatives are shown in Figs. 6a–h, which clearly presents a 3-cluster configuration.

## Spike Detection and Feature Extraction Hardware Implementation

*f*

_{c1}and

*f*

_{c2}. Second, the filter outputs the derivative of the spike waveforms to identify neurons’ kernel signatures. To handle a variety of noise profiles and spike widths, 32 filter coefficients are programmable to perform different orders of noise shaping (>30 dB attenuation of low-frequency field potential and 60 Hz noise) and achieve a flexible

*f*

_{c2}in kHz range. When sampling frequency is below 25 kHz, the filter can induce additional 30 dB out-of-band rejection at frequency close to DC through high-pass filtering. As shown in Fig. 6a,

*f*

_{s}= 25 kHz,

*f*

_{c1}≈ 600 Hz, and in band ripple <1 dB. It is worth mentioning that an increase of the sampling frequency

*f*

_{s}would increase

*f*

_{c1}proportionally (unless using a higher order filter). An example of applying the filter to a spike sequence recorded at 40 kHz is shown in Fig. 6b, where near-DC field potentials, 60 Hz noise, and its harmonics are severely attenuated to be less than the integrated signal power.

In this implementation, samples including the positive and negative peaks of the spike derivative and spike heights are the features chosen for classification. The choice of this subset is made due to the small cost on computation and storage space. A NEO-based spike detector, noise shaping filter, feature extractor, the corresponding storing device, and control units described in Fig. 5 are implemented with a custom digital IC with 0.35 μm CMOS process, which consumes 1.62 × 1.62 mm^{2} and 93 μW.

## Experiments

### Comparative Results on Synthesized Data

Accuracy comparison of using different spike feature extraction algorithms

Sequence Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

Informative Sample | 97.8% | 97.8% | 97.8% | 97.0% | 98.0% | 99.2% | 96.6% | 92.0% |

Hardware | 97.6% | 97.6% | 97.4% | 95.4% | 98.2% | 98.4% | 93.2% | 91.0% |

PCA | 97.8% | 89.0% | 60.4% | 55.2% | 97.6% | 77.8% | 80.2% | 68.8% |

Wavelets | 92.4% | 91.0% | 81.8% | 57.4% | 97.4% | 68.2% | 51.0% | 49.4% |

Spike Peaks | 34.2% | 33.8% | 35.4% | 34.0% | 36.2% | 37.8% | 35.6% | 36.0% |

### Comparative Results on *In Vivo* Data

In this subsection, data recorded from *in vivo* preparations are used to evaluate the performance of PCA and the proposed algorithm.

*In Vivo* Data Testing A—From a Monkey Preparation

As a comparison, spike sorting results using the informative sample selection and noise shaping are shown in Figs. 8c–f. Specifically, in Figs. 8c and f, the histograms of sample 20 and sample 28 from noise-shaped spike waveforms are plotted. Figure 8d displays detected spikes after noise shaping. A classification is color-coded and based on the feature space in (e). Figure 8e displays the feature extraction results using the two informative samples, sample 20 and sample 28. When tested with PCA, the feature space shows a one-cluster configuration (Fig. 8b). When tested with the proposed algorithm, the feature space shows a two-clustering configuration (Fig. 8e), suggesting improved feature space isolation.

*In Vivo* Data Testing B—Simultaneous Intra- and Extra-Cellular Recording in Anesthetized Rats

*et al.*14 In this subsection, comparative testing results on PCA and the proposed algorithm are reported in Fig. 9.

Figure 9a display detected spikes. Figure 9b extracted features using PCA. Figures 9c–i display comparative results using informative sample set. Figures 9c and f display unprocessed histograms of sample 10 and sample 18 after noise shaping. Figure 9d displays detected spikes after noise shaping. Figure 9e displays the feature extraction results using the two informative samples, sample 10 and sample 18. Compared with PCA-based feature extraction (Fig. 9b), the proposed algorithm gives a clear 3-cluster configuration (Fig. 9e). To visually examine the validity of the 3-cluster configuration, classified spike templates are individually plotted in Figs. 9g–i.

*In Vivo* Data Testing C—From a Cat Preparation

## Conclusion

A sample selection-based spike feature extraction algorithm is reported in this article. The theoretical framework includes neuronal signature, frequency shaping filter, and informative sample selection. Unlike PCA which uses correlated features, the sample selection algorithm focuses on localized and uncorrelated features which are strengthened by the frequency shaping filter. With simulated spike waveforms from a public database, the algorithm demonstrates an improved sorting accuracy compared with many competing algorithms. The algorithm is designed for integrated microchip implementation and performs real-time spike sorting. A preliminary hardware implementation has been realized using an IC chip interfaced with a personal computer.

## Notes

### Acknowledgments

The authors acknowledge the founding provided by the USA National Science Foundation through BMES-ERC and UC Lab Fee Program. The authors acknowledge the start-up grant provided by National University of Singapore. The authors are grateful to Dr. Victor Pikov, Eric Basham, Plexon, and BMES-ERC Cortical Testbed for providing *in vivo* neural data and suggestions. The authors acknowledge the *in vivo* database contributed by Gyorgy Buzsáki lab (http://crcns.org/data-sets/hc/) and synthesized database contributed by Quian Quiroga (http://www.vis.caltech.edu/~rodri/Wave_clus/Wave_clus_home.htm).

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