Data Reduction Techniques in Neural Recording Microsystems

Nowadays, implantable devices developed for electrically interfacing to the brain are of great interest. Such devices, also known as brain-machine interfaces (BMI), are expected to revolu‐ tionalize so many aspects of the human life, such as the way we interface with the external world, and how we cure deseases and disabilities such as the Parkinson’s desease, paralysis, and blindness. General concept of intra-cortical neural recording using implantable microsys‐ tems along with an example of such systems is illustrated in Figure 1. In a wide variety of applications for such systems, there is a need for recording neural activities from a certain region of the brain with enough spatial resolution. To be able to come up with meaningful information from the region of interest in the brain, implantable neural recording devices are typically designed to record from tens to hundreds of recording sites [1-3].


Introduction
Nowadays, implantable devices developed for electrically interfacing to the brain are of great interest. Such devices, also known as brain-machine interfaces (BMI), are expected to revolutionalize so many aspects of the human life, such as the way we interface with the external world, and how we cure deseases and disabilities such as the Parkinson's desease, paralysis, and blindness. General concept of intra-cortical neural recording using implantable microsystems along with an example of such systems is illustrated in Figure 1. In a wide variety of applications for such systems, there is a need for recording neural activities from a certain region of the brain with enough spatial resolution. To be able to come up with meaningful information from the region of interest in the brain, implantable neural recording devices are typically designed to record from tens to hundreds of recording sites [1][2][3].
• wireless interface module; This module is used for data exchange with the external setup and in some cases for supporting power telemetry from the outside to the implant.

Challenges in the development of high-density neural recording microsystems
As the number of recording channels for a wireless neural recording microsystem increases, many aspects of the design of the system will be challenging. From among the more important design challenges, one can point to the low power consumption and small physical dimensions of the system. For tens to hundreds of recording channels, transferring a huge amount of neural -wireless interface module; This module is used for data exchange with the external setup and in some cases for supporting power telemetry from the outside to the implant. (a) Intra-cortical neural recording using an implantable microsystem [6] (b) a 64-channel neural recording microsystem developed at the University of Michigan [1] data through a wireless link is also a design bottleneck. This is simply due to the fact that an implantable neural recording device needs to transmit the recorded neural information to the external world through wireless connection, and the frequency band used for wireless communication is not unlimited. One of the efficient ways to overcome this problem is to either compress the data being telemetered or at least to extract and transmit only the useful information needed for the target application.

Spike reporting
An intra-cortically-recorded neural signal, in general, comprises three major components: action potentials (also known as neural spikes or simply spikes), local field potential (LFP), and background noise. It is believed that most of the important information in neural signals is reflected in the occurrence rate of neural spikes. As a result, in some applications (e.g., prosthetic applications) only the occurrence of spikes is detected and reported to the external world. In some other applications (e.g., neuroscientific research), however, researchers and scientists need more information on how or where the neural activities occur.
Recording the entire neural signal (action potentials superposed with background noise) is the maximum function expected from a general neural recording system, which allows for studying different components of a neural signal including the background noise. For multichannel wireless neural recording implants, because of the limited bandwidth available for transmitting the neural data, the number of recording channels will be limited if the entire signal is intended to be telemetered. In many applications, the rate of spike occurrence is the most important information that is expected from a neural recording system to provide. Hence, it will be much more bandwidth-efficient if the spikes are detected by the implanted recording system and only the occurrences of the spikes are reported to the external host rather than transmitting the entire neural signal.
-neural signal processing module; This is where most of the signal handling and signal processing tasks take place, e.g., analog signal processing, analog/digital conversion, and digital signal processing.

-wireless interface module;
This module is used for data exchange with the external setup and in some cases for supporting power telemetry from the outside to the implant.

Challenges in the development of high-density neural recording microsystems
As the number of recording channels for a wireless neural recording microsystem increases, many aspects of the design of the system will be challenging. From among the more important design challenges, one can point to the low power consumption and small physical dimensions of the system. For tens to hundreds of recording channels, transferring a huge amount of neural data through a wireless link is also a design bottleneck. This is simply due to the fact that an implantable neural recording device needs to transmit the recorded neural information to the external world through wireless connection, and the frequency band used for wireless communication is not unlimited. One of the efficient ways to overcome this problem is to either compress the data being telemetered or at least to extract and transmit only the useful information needed for the target application.

SPIKE REPORTING
An intra-cortically-recorded neural signal, in general, comprises three major components: action potentials (also known as neural spikes or simply spikes), local field potential (LFP), and background noise. It is believed that most of the important information in neural signals is reflected in the occurrence rate of neural spikes. As a result, in some applications (e.g., prosthetic applications) only the occurrence of spikes is detected and reported to the external world. In some other applications (e.g., neuroscientific research), however, researchers and scientists need more information on how or where the neural activities occur.
Recording the entire neural signal (action potentials superposed with background noise) is the maximum function expected from a general neural recording system, which allows for studying different components of a neural signal including the background noise. For multi-channel wireless neural recording implants, because of the limited bandwidth available for transmitting the neural data, the number of recording channels will be limited if the entire signal is intended to be telemetered. In many

Spike detection
In addition to the small action potentials with the amplitude of around 100~500μV, a neural signal contains background noise and probably low-frequency baseline variations. To prepare the neural signal for spike detection, it is amplified with a gain of around 40~60dB and also its low-frequency (below 1~10Hz) and high-frequency (above 7~10kHz) contents are filtered out [7], [2]. Then, this preconditioned signal is delivered to a spike detector.
There are various spike detection approaches that can be classified into two major categories: feature-based spike detection methods, and spike detection by hard thresholding. In the former, a preprocessor searches the input neural signal for certain features of a spike to occur, while in the latter, a threshold level is defined and a spike is detected when the neural signal goes beyond the threshold.
Feature-Based Approaches. Only a few years after artificial neural networks (ANN's) were introduced as an efficient tool to implement artificial intelligence, due to the feature extraction capability that certain types of ANN's had, they showed to be attractive candidates for automatic spike detection either by themselves or in conjunction with preprocessors [8][9]. Kohonen and Grossberg networks with unsupervised learning, and Multi-Layered Perceptron network with "error back-propagation" as a supervised learning algorithm have been used to perform spike detection. Although there are cases where the raw neural signal is fed to the ANN for spike detection [9][10], it is mostly preferred to use a preprocessor for extracting certain spike features first, and then use an ANN for processing them and detecting the spikes, as illustrated in Figure 3. Because of their relatively large area-and power-consuming electronic implementations, this class of spike processors has never been used in implantable neural recording microsystems.  Spike Detection Based on Nonlinear Energy Operator (NEO). Traditional spike detectors (explained above) usually need prior information about action potentials, which is usually not available before recording the neural signals in real applications. In contrast with these methods, which are mostly based on the amplitude or time-domain features of the neural signal, detection of action potentials, i.e., spikes, can also be performed based on the energy content of the signal. Direct square of the signal, absolute value operator, and variance estimator are the energy-based operators commonly used for the detection of bio-potentials. The nonlinear energy operator (NEO), also called the Teager energy operator (TEO) is one of the unsupervised action potential detectors exhibiting satisfactory performance in the case of low signal-to-noise ratio (SNR) for the neural signal and also convincing speed for the alignment of spikes in the real time. In its original form, continuous-time NEO is defined as: which is sensitive to signal with short time interval and at a high-frequency band [11]. An efficient hardware implementation for an NEO neural signal processor employing custom OTA 1 -C analog circuits was reported in [12].

Hard thresholding
Four possible ways of spike detection by hard thresholding are illustrated in Figure 4. Spike detection is mostly performed to detect either positive (Figure 4 (a)) or negative ( Figure 4 (b)) spikes [7], [2], [13]. Having such a fixed pre-assumption for the polarity of the spikes limits the operation of the system. Recognition of both positive and negative spikes, i.e., bi-phasic spike detection, can be realized in two major ways, illustrated in Figure 4 (c) and (d). In Figure  4 (c), the spike detector returns a logical "1" on the Spike Occurrence (S.O.) output upon the detection of a spike, no matter if it is positive or negative. This is a bandwidth-efficient way of bi-phasic spike detection, which requires almost the same bandwidth as the uni-phasic methods, but pays the price by losing the spike polarity. The simplest realization of this idea is to filter out the DC component of the input signal, find its absolute value, and then detect the spikes using one comparator and one threshold [14] as shown in Figure 5(a). Aside from the need for a precise full-wave rectifier in this realization, the fact that both the positive and the negative spikes are compared with the same threshold level might be considered as a drawback. Figure 5 (b) shows another realization of bi-phasic spike detection with no polarity, which uses two comparators and an OR gate and also allows for comparing positive and negative spikes with separate threshold levels. This approach is used in [7], [13] with positive and negative thresholds, V TH,P and V TH,N , defined by a threshold value (THR) and a threshold offset (ThrOS), as shown in Figure 6.
The circuit shown in Figure 5 (b) can also be used with minor modifications to realize the complete bi-phasic spike detection method illustrated in Figure 4   There is a variety of methods for generating the thresholds required for spike detection. The threshold can be either statically defined by the user [2], [13] or automatically set by the internal circuitry. There is a variety of methods for generating the thresholds required for spike detection. The threshold can be either statically defined by the user [2], [13] or automatically set by the internal circuitry.
Automatic Threshold Setting. The 32-channel spike detector ASIC reported by [7] uses a straightforward approach for automatic threshold generation. In this method, the average (AVG) and the standard deviation (SD) of the neural signal is calculated, and then the two thresholds required for bi-phasic spike detection are set above and below the average value as: where k is a constant. Typical value for k varies from 3 to 7 depending on the signal-to-noise ratio (SNR) of the recorded neural signal. Functional block diagram of this spike processor implementing the above method in digital domain is shown in Figure. Thirty two channels of preconditioned neural signals, which are already time-division multiplexed on four lines in analog domain by a recording front-end (not shown), are delivered to the spike detector ASIC. The four multiplexed inputs each carrying 8 channels of neural signals are first converted to digital by four A/D converters simultaneously. The Sample Distributor, which is synchronized with the time-division multiplexer on the recording front-end, demultiplexes the amplitude samples into 32 digital neural channels. Then, the digital spike detector computes the averages and standard deviations of the 32 channels separately and accordingly calculates their threshold values. After the Threshold Calculation program is executed, the Spike Detection program is run. The amplitude sample of each channel is compared to the associated threshold, and if it is beyond the threshold level, it is considered as a detected spike. As long as a channel is active, its amplitude samples are tagged with the associated channel address, and put in a buffer to be sent to a wireless interface.
where k is a constant. Typical value for k varies from 3 to 7 depending on the signal-to-noise ratio (SNR) of the recorded neural signal. Functional block diagram of this spike processor implementing the above method in digital domain is shown in Figure . Thirty two channels of preconditioned neural signals, which are already time-division multiplexed on four lines in analog domain by a recording frontend (not shown), are delivered to the spike detector ASIC. The four multiplexed inputs each carrying 8 channels of neural signals are first converted to digital by four A/D converters simultaneously. The Sample Distributor, which is synchronized with the time-division multiplexer on the recording front-end, demultiplexes the amplitude samples into 32 digital neural channels. Then, the digital spike detector computes the averages and standard deviations of the 32 channels separately and accordingly calculates In [15], a spike detector circuit is reported that computes the detection threshold in analog dom of this circuit is shown in Figure 8. One of the advantages of this circuit is that unlike the spike shold is computed in real time. The circuit assumes that the input signal has already been amp tered, and the background noise has a Gaussian distribution. Since the input signal is assumed nent, the noise can be described by its RMS value, V1σ , which is equivalent to its standard dev well above the noise level, the threshold voltage is set to VKσ =K.V1σ , with K set to 5 in [15]. Alth the low-frequency baseline variations of the input signal have been already filtered out, further shows that the detection threshold can adaptively follow the baseline variations. There is, howe the frequency-amplitude product of the baseline variations that the adaptive threshold is capable er dissipation of the implementation of this approach is very low, in the range of microwatt and con area. The drawbacks of this method include the circuit's sensitivity to the absolute value of ments, which are usually subject to relatively large fabrication tolerances, and the difficulty in frequency low-pass filter required in the RMS block. Figure shows another idea in analog spike detection, in which two OTA-based low-pass filters w quencies play the key role [17]. One filter has a higher cut-off frequency to remove high frequen has a lower cut-off frequency to make a local average. The difference between the high-pass filt average is provided by an OTA, and is recognized by another OTA as a detected spike when exc value (Vref). This method is robust against changes in both noise level and the input signal's DC o likely to happen in long-term neural recording. The OTAs operate in subthreshold region to re voltages are set off chip to enable adjustment of the cut-off frequencies after fabrication. In this tion threshold level is set by Vbias, which along with the other bias and reference voltages shou probably fine-tuned for long-term recordings.  Figure 7. Functional block diagram of the 32-channel digital spike detector ASIC reported in [7] In [15], a spike detector circuit is reported that computes the detection threshold in analog domain. Functional diagram of this circuit is shown in Figure 8. One of the advantages of this circuit is that unlike the spike detector in [7], the threshold is computed in real time. The circuit assumes that the input signal has already been amplified and band-pass filtered, and the background noise has a Gaussian distribution. Since the input signal is assumed to have no DC component, the noise can be described by its RMS value, V 1σ, which is equivalent to its standard deviation, σ. In order to be well above the noise level, the threshold voltage is set to V Kσ =K.V 1σ, with K set to 5 in [15]. Although it is assumed that the low-frequency baseline variations of the input signal have been already filtered out, further analysis of this method shows that the detection threshold can adaptively follow the baseline variations. There is, however, an upper bound on the frequency-amplitude product of the baseline variations that the adaptive threshold is capable of following [16]. Power dissipation of the implementation of this approach is very low, in the range of microwatt and occupies very small silicon area. The drawbacks of this method include the circuit's sensitivity to the absolute value of some of the circuit elements, which are usually subject to relatively large fabrication tolerances, and the difficulty in implementing the low-frequency low-pass filter required in the RMS block.  Figure 9. shows another idea in analog spike detection, in which two OTA-based low-pass filters with different cut-off frequencies play the key role [17]. One filter has a higher cut-off frequency to remove high frequency noise, and the other has a lower cut-off frequency to make a local average. The difference between the high-pass filtered signal and its local average is provided by an OTA, and is recognized by another OTA as a detected spike when exceeds a certain reference value (V ref ). This method is robust against changes in both noise level and the input signal's DC offsets, both of which are likely to happen in long-term neural recording. The OTAs operate in subthreshold region to reduce power. The τ bias voltages are set off chip to enable adjustment of the cut-off frequencies after fabrication. In this circuit, the spike detection threshold level is set by V bias , which along with the other bias and reference voltages should be properly set, and probably fine-tuned for long-term recordings. Figure 9. Schematic diagram of the analog spike detector of [16] In both of the above analog approaches, there are device and circuit parameters that should be set by the user for proper operation, which make these circuits inappropriate for implantable applications.

Mathematical approaches
Mathematical transforms are among the most common methods of data compression. Recently, the Discrete Wavelet Transform (DWT) has been successfully employed in neural recording microsystems to compress the neural information, while preserving the wave shape of action potentials [18][19][20][21][22][23][24][25][26]. The DWT transforms discrete signals from the time domain into the time-frequency domain. One-level DWT for a given signal is achieved by convolving the signal samples through low-pass and high-pass decomposition filters [19,21]. The filtering is then followed by sub sampling to obtain the approximation and detail coefficients. For multilevel DWT, approximation coefficients should be fed to the same decomposition filters recursively [27]. Characteristics of the filters are determined by the wavelet basis. For neural signal compression, the optimal choice is a wavelet function that can approximate the action potentials waveform with minimum DWT coefficients and error. It has been shown that by proper selection of the wavelet basis, most of the spike energy is concentrated in a few large coefficients, while many small coefficients carry insignificant information and are mainly attributed to noise [19]. Therefore, in order to achieve higher data reduction rates, the DWT coefficients are passed through a thresholding stage. In this block, data coefficients smaller than desired certain threshold level are set to zero, while others are left unchanged. It is obvious that the threshold value plays an important role in the overall data compression rate, and also in the quality of the reconstructed signal. Hence, the threshold level should be set carefully based on the requirements of the target application.
Due to power and size constrains in biomedical implants, VLSI implementation of DWT is of great importance. In [20], it has been shown that from a signal compression standpoint, symmlet4 wavelet basis is advantageous over other wavelet functions for neural recording applications. It is believed that this is mainly because of the similarity of this function to the general wave shape of action potentials. For hardware implementation of symmlet4 function, lifting method is proposed in [21]. Furthermore, two different circuit designs, pipelined and sequential, are presented and compared for the lifting scheme. It is demonstrated that for single-level single-channel integer DWT, the pipelined approach consumes lower power, but occupies more silicon area compared with the sequential implementation. On the other hand, 4-level multi-channel implementation of the two designs indicates that the sequential approach requires significantly smaller chip area, while the power consumption of both is almost the same. As a result, the sequential execution architecture is employed in [23] to design a complete 32-channel compression system based on the 4-level symmlet4 DWT. The chip consumes 3 mW of power and occupies only 5.75 mm 2 in a 0.5-μm CMOS technology. Also, with a sampling rate of 25 KSample/Sec per channel and 10-bits data samples, the system provides data compression of more than 20 times, resulting in a total output bit rate of less than 370 kbps.
In [26], a neural signal compression method based on the Discrete Haar Wavelet Transforms (DHWT) is proposed. From the standpoint of data compression, Haar basis function may not perform as efficient as complex functions such as high-order Daubechies and symmlet, but due to its simple hardware implementation, it can be easily used for large number of neural channels with less concern about power and area. As discussed in [26] for two-point DHWT, data coefficients can be calculated by only a buffer, an adder, and a subtractor. Moreover, in order to compare Haar and symmlet4 basis functions, they have been both designed for processing a single channel with 8-bits data samples. Results indicate that before the thresholding stage, relative error (between the original signal and the corresponding reconstructed signal) for the DHWT is only 0.01% larger than the symmlet4 case, which is obviously negligible.
On the other hand, hardware implementation of the DHWT shows around 83% saving in number of transistors, and more than 90% in the occupied silicon area, when physically laid out in a 0.13-μm CMOS technology. The complete 64-channel DHWT-based neural compressor achieves a compression rate of 112 with an error of 2.22%. Additionally, the compressor circuit consumes as low as 0.12 mW @1.2 V supply voltage, and occupies less than 0.1 mm 2 in a 0.13-μm CMOS technology. Therefore, it can be said that with this architecture, the gain in circuit simplicity and the bit-rate improvement are much more significant than the penalty paid by the noise added to the signal. However, it is worth mentioning that, in general, the appropriate architecture should be selected based on the application.

Hardware approaches
To avoid adding extra power-and area-hungry signal processing blocks for data reduction, and at the same time preserving important information of the neural signals, there is a different category of data reduction techniques, known as hardware approaches. These are the approaches focused on modifying the hardware of the recording system in such a way that considerable bit-rate reduction can be achieved. Obvious advantages of these approaches are smaller silicon area and power consumption as compared with the mathematical approaches explained in the previous section.
To benefit from the advantages associated with digital signal processing and also digital data communication (as opposed to their analog counterparts), neural recording devices are commonly designed to convert neural signals into digital as the first step. As a result, analogto-digital converters (ADCs) are known as one of the key building blocks in such systems.
Recently some efforts are put on designing application-specific ADCs to efficiently utilize the bandwidth allocated for wireless data telemetry. In this section, an efficient method for analogto-digital (A/D) conversion of neural signals is discussed. This method results in significant reduction of data-rate for multi-channel cortical neural recording microsystems.

Anti-logarithmc quantization
Although linear ADCs are typically used to digitize neural signals in neural recording microsystems, it is beneficial to design a nonlinear ADC for such specific signals. Choosing the best-suited nonlinearity function for a specific signal requires recognition of the concentration of information along the signal amplitude range. As illustrated in the left side of Figure  10, in general, signals can be categorized into three types according to how the information they carry is distributed along the amplitude range.
Type-I signals are named "Signals with Non-Concentrated Information (NCI)" due to their almost uniform distribution of information concentration. Important information for Type-II signals is concentrated at the lower side of the amplitude range. Audio signals are of this type, referred to as "signals with Information Concentration at Low Amplitudes (ICLA)". Conversely, for "signals with Information Concentration at High Amplitudes (ICHA)", i.e., Type-III signals, more information is present at higher side of the amplitude range, with neural signals as examples. amplitudes, where more information is concentrated. For example, compressing/expanding (companding) of audio signals in communications systems is based on logarithmic quantization, which increases the dynamic range and improves the SNR [28]. Whereas quantization functions with increasing slope along the input amplitude range, such as the exponential function, put more resolution in the quantization of the larger amplitudes and are preferred for ICHA signals.    Figure 11, in time domain a typical intracortically-recorded extracellular neural signal can be divided into two parts: action potentials (APs) and background noise (B-Noise). In probability density function (PDF) domain, APs are concentrated at large amplitudes while B-Noise is concentrated at small amplitudes. In a wide variety of neuroscientific and neurophysiological studies, as well as in many neuroprosthetic applications, it is the APs that carry the useful information embedded in neural signals. As illustrated in Figure  12, in implantable neural recording microdevices, neural signals are usually digitized using linear ADCs, i.e., ADCs with linear quantization characteristics. This means that the non-useful B-Noise is digitized with the same resolution as the useful APs are. In other words, when telemetering a digitized neural signal, part of the outgoing bit-rate is wasted to carry the noise content present in the neural signal. In [29] the idea of digitizing neural signals using an ADC with non-uniform quantization steps has been proposed. According to the classification presented in the previous section, neural signals are categorized under Type-III (i.e., ICHA). Hence, the best type of nonlinearity function for the quantization of neural signals is signals with increasing slopes such as parabolic and exponential functions.
Digitizing neural signals using exponential ADC (exp-ADC) helps saving the bandwidth in wireless data telemetry between the implanted device and the external host. Data reduction for an 8-bit exp-ADC is 24% as compared with its linear counterpart. Along with data reduction, anti-logarithmic quantization of neural signals significantly reduces the power consumption of the ADC, comparing with a standard linear ADC. This is due to less number of digital code transitions for the exp-ADC. Moreover, anti-logarithmic quantization increases the SNR of neural signal by reducing its noise content.
Converting Back to Analog. The transfer characteristics for conventional linear analog-todigital-to-analog (A/D/A) conversion process is a linear function, i.e., analog input signal is digitized by a linear ADC and then is converted back to the analog domain using a linear DAC.
On the other hand, in a nonlinear A/D/A conversion process, analog input signal is digitized by a nonlinear quantization function. As shown in Figure 13 to convert back to the analog domain, digital signal should be passed through a nonlinear DAC with exact inverse characteristic. The resulted A/D/A conversion transfer characteristic is similar to that of a linear A/D/A conversion process, except that the quantization steps are non-uniform. In the case of anti-logarithmic A/D/A conversion, quantization steps are decreasing in length along the input amplitude range.     Assuming that the neural signal is preamplified a baseline level, as illustrated in Figure 14, the nonlinear quantization function needs to b form around the baseline of the signal. Therefore, the nonlinear ADC needed to cover th range ADC (FR-ADC), is realized using two complimentary half-range ADCs (HR-ADCs) signal range. Hence, assuming that the basic nonlinear quantization function is used (UHR-ADC), fUHR(x), the quantization function used for the lower half-range ADC (LHR-A

Delete Replace with
The nonlinear ADC discussed hereafter is assumed to be the ADC that covers the upper i.e., the UHR-ADC, unless otherwise stated.
Dummy Text Half-Range Characteristic Function. The input-output relationship for an N quantization function is: Covering the Full Range. Assuming that the neural signal is preamplified and positioned around a certain baseline level, as illustrated in Figure 14, the nonlinear quantization function needs to be defined in an odd symmetric form around the baseline of the signal. Therefore, the nonlinear ADC needed to cover the entire input signal range, full-range ADC (FR-ADC), is realized using two complimentary half-range ADCs (HR-ADCs) each covering half of the input signal range. Hence, assuming that the basic nonlinear quantization function is used for the upper half-range ADC (UHR-ADC), f UHR (x), the quantization function used for the lower halfrange ADC (LHR-ADC) will be: The nonlinear ADC discussed hereafter is assumed to be the ADC that covers the upper half of the input signal range, i.e., the UHR-ADC, unless otherwise stated.

Half-Range Characteristic Function.
The input-output relationship for an N-bit HR-ADC with exponential quantization function is: where (b N-1 ...b 1 b 0 ) is digital representation of analog input, v in , and V FS is the full-scale input range for the UHR-ADC. To satisfy the boundary conditions for minimum and maximum values of v in it can be shown that: Parameter a sets the curvature of the characteristic function. The smaller this parameter is, the more rapid the exponential input-output relationship will be. Quantization steps along the input range are known, in general, as least significant bits (LSB), and are calculated as: for i=0,1,...,2 N -1. The largest and the smallest quantization steps, LSB max and LSB min , are calculated using eq.(6) for i=0 and 2 N -1, respectively, as: and ,min 1 ln 1 .
In general, dynamic range (DR) of a nonlinear ADC is defined to be the ratio of the full-scale input voltage to the smallest resolvable signal, V LSB,min [32]. DR in the case of exp-ADC is achieved as: ,min The choice of the largest quantization step, V LSB,max , is perhaps the most critical decision in forming the quantization function for the exp-ADC. This is because of its key role in the reduction of the noise content of the neural signal. The largest LSB is responsible for the largest quantization error. It is along the V LSB,max that variations of the input signal are intentionally not seen and replaced with 0 (the baseline level in our design). By using this method, not only the quantization error is not a disturbing phenomenon for the signal, but it also plays a denoising role as it replaces the B-Noise around the baseline with 0. To achieve significant reduction in the B-Noise power, V LSB,max is set to 3σ, where σ is the standard deviation of the B-Noise PDF. This way, most of the B-Noise will be intentionally removed from the neural signal during the digitization process.
Noise Analysis. PDF of quantization noise (Q-Noise) for linear ADC is uniform along the input amplitude range. This can be shown by eq. (10) which formulates PDF of the Q-Noise associated with code n in an N-bit linear HR-ADC [30]: Uniform distribution of Q-Noise along the input amplitude makes linear ADCs suitable for digitizing NCI signals. For specific signals an NLADC might be useful in terms of SNR improvement. This advantage comes from the fact that NLADCs exhibit non-uniform Q-Noise distribution. In a Logarithmic ADC, Q-Noise energy is shifted to large amplitudes. As a result, logarithmic ADC has widely been used in digitizing of ICLA signals, such as audio. PDF of Q-noise associated with code n for an N-bit logarithmic HR-ADC is formulated in eq. (11) and is illustrated in Figure 15(a):   a n a n k n P N N HR (11) In the case of the anti-logarithmic N-bit HR-ADC, the Q-Noise PDF associated with code n is derived as: Uniform distribution of Q-Noise along the input amplitude makes linear ADCs suitable for digitizing NCI signals. For specific signals an NLADC might be useful in terms of SNR improvement. This advantage comes from the fact that NLADCs exhibit non-uniform Q-Noise distribution. In a Logarithmic ADC, Q-Noise energy is shifted to large amplitudes. As a result, logarithmic ADC has widely been used in digitizing of ICLA signals, such as audio. PDF of Q-noise associated with code n for an N-bit logarithmic HR-ADC is formulated in eq. (11) and is illustrated in Figure 15  In the case of the anti-logarithmic N-bit HR-ADC, the Q-Noise PDF associated with code n is derived as: As shown in Figure 15(b), exp-ADC shapes the Q-Noise in such a way that most of its energy concentrates at small amplitudes, making it suitable for digitizing ICHA signals. As an example, in neural signals, APs with large amplitudes are quantized with higher SQNR as opposed to the B-Noise with rather small amplitudes. Figure 15(c) illustrates that most of the noise content of the signal (B-Noise) lies within the very first LSBs. The interesting point here is that since it is some of the noise content of the neural signal that is lost during the quantization process, not only the associated quantization error is not undesirable, but it is also welcomed as it leads to noise-content reduction and consequently to significant improvement in the SNR of the neural signal being digitized.
Noise-Content-Reduction Ratio, NCRR, is a measure of capability of an ADC in reducing the noise content of the neural signal being digitized, and can be defined as [29]: 10 average noise power @ input NCRR (dB) 10 log average noise power @ output where the average noise power at the input of the ADC is calculated as: In this equation, P ni (n i ) is the probability density function for noise content of the neural signal, n i (t). Similarly, the average noise power of the signal at the output of the ADC is derived to be:

SAR & Switch Control Logic
In-Segment Digitization Segment Selection Digital Output Binary to Thermometer Converter   Figure 17. Circuit schematic for the two-step SAR ADC designed to implement exp-ADC and its simplified timing diagram Circuit Design. To realize the proposed exp-ADC with reasonable power and silicon area, successive-approximation register (SAR) architecture was chosen [31]. In order to facilitate the realization of the exponential quantization function needed for the ADC, a piecewise-linear (PWL) approximation of the required function was implemented. As shown in the timing diagram of Figure 17, the proposed ADC operates in three phases: sign detection (SD), offset cancelation (OC), and conversion. In the SD phase, the analog input voltage, v in , is compared with a certain threshold voltage, V TH , which is temporarily set to the baseline voltage, V BL . The result of this comparison determines the half range in which the input voltage is located. In the OC phase, an ordinary offset cancellation technique is applied on the comparator and buffers. In the conversion phase, successive approximation algorithm first finds the segment of interest, encoded by b 6 b 5 b 4 , in 3 clock cycles. These 3 bits are converted to a 7-bit thermometer code, T 7~T1 , which will be used in the Segment Selection block to generate two analog voltages associated with the endpoints of the segment of interest. An in-segment linear A/D conversion process is then performed to determine the remained 4 LSBs, b 3 b 2 b 1 b 0 , in 4 clock cycles. Finally, an end of conversion signal is generated to reset the ADC and prepare for the next conversion cycle. A low-voltage band-gap reference (BGR) was designed to generate the required baseline, reference and threshold voltages for the exp-ADC. Figure 17. Circuit schematic for the two-step SAR ADC designed to implement exp-AD Dummy Text Circuit Design. To realize the proposed exp-ADC with rea approximation register (SAR) architecture was chosen [31]. In order to facilit zation function needed for the ADC, a piecewise-linear (PWL) approximatio As shown in the timing diagram of Figure 17, the proposed ADC operate cancelation (OC), and conversion. In the SD phase, the analog input voltag voltage, VTH, which is temporarily set to the baseline voltage, VBL. The re range in which the input voltage is located. In the OC phase, an ordinary o comparator and buffers. In the conversion phase, successive approximation encoded by b6b5b4, in 3 clock cycles. These 3 bits are converted to a 7-bit the the Segment Selection block to generate two analog voltages associated with in-segment linear A/D conversion process is then performed to determin cycles. Finally, an end of conversion signal is generated to reset the ADC a low-voltage band-gap reference (BGR) was designed to generate the require for the exp-ADC.  Figure 18, in which the chip occupies a total size of 220 ×230 μm 2 . The measured worst-case differential non-linearity (DNL) and integral nonlinearity (INL) are +0.8/-0.9 LSB and +4.3/-2.1 LSB, respectively. Table 1 summarizes the specifications of the NLADC and compares it with some of the nonlinear and linear ADCs reported in the literature.
A proof-of-concept prototype of a 4-channel neural recording system based on anti-logarithmic quantization was reported in [6]. As shown in the block diagram of Figure 19, a time-domain multiplexer (TDM) shares an anti-logarithmic ADC (AL-ADC) between 4 channels (each sampled at 20kSps). The output digital codes are then packed by the data packaging block to be transmitted to the outside world via a wireless link. At the external host, received signal is first recovered and then converted back to analog via a PC software. This inverse conversion is performed using a logarithmic DAC. Quantization characteristic function for the DAC is an exact inverse for that of the NLADC used in the recording system. To evaluate the operation of the system, neural signals recorded from the auditory cortex of a Guinea pig are used for in-vitro tests. Figure 20 shows the input signal to one of the neural channels before entering the nonlinear quantization process on the implant side along with the associated signal on the external setup after reconstruction. To verify the concept of the noise reduction caused by overlapping PDFs for B-Noise and Q-Noise, distributions of the measured quantization error along the input amplitude are depicted in Figure 21.

Conclusion
To overcome bandwidth limitation in the wireless telemetry of recorded neural data, a wide variety of data reduction techniques has been reported. These techniques range from spike reporting approaches such as spike detection and spike sorting techniques to mathematical approaches such as the discrete wavelet transform. Although it is proven that spike reporting approaches contain enough information to actuate prosthetic devices [33], in some other applications, such as neuroscientific studies, they are not satisfactory due to considerable loss of important information, e.g., spike wave shapes. Mathematical approaches, on the other hand, have been successful from the standpoint of data compression, while preserving wave shape of the spikes. Nonetheless, increasing the number of recording channels can result in the potential problems of these approaches: large silicon area and high power consumption.
In contrast with all of the mentioned techniques, hardware approaches are capable of data reduction without adding any extra block to the microsystem. This is achieved by modifying the present hardware of the implant. An implementation of one of these approaches focusing on the ADC circuit of the system was presented and discussed in detail. The proposed method results in considerable reduction of bit-rate in multi-channel neural recording microsystems. Thus, efficient design of application-specific circuits for building blocks of neural implants should be taken into account as an appropriate method of data reduction.