The N1 Codec

An implementation of 24+ lossless compression algorithms from dictionary-based codecs to specialized audio codecs tailored for Neuralink's N1 Implant, achieving compression ratios up to a high of 3.11x.

The Neuralink Compression Challenge is a challenge announced by Bliss Chapman aimed at drastically reducing the data bandwidth requirements for the N1 implant without losing a single sample. This implant, situated in the motor cortex of a non-human primate, generates around 200 Mbps of data from 1024 electrodes, each sampling at 20 kHz with 10-bit resolution.

To transmit this data wirelessly in real-time, a compression ratio greater than 200x is necessary, all while operating under stringent conditions of less than 1 ms latency and power consumption below 10 mW.

The goal of this project was to deterministically get a sense of how well-established lossless compression techniques could effectively compress this unique type of data.

[!IMPORTANT]

There were 5 groups that these algorithms were categorized in based on their core methodologies and performance characteristics:

1. Dictionary-Based Codecs (zstd, lz4, lz4hc, gzip, zlib, bz2 etc.)
2. Predictive Codecs (delta etc.)
3. Entropy Codecs (huffman, arithmetic etc.)
4. Specialized Audio Codecs (wavpack etc.)
5. Blosc Hybrid Codecs

Despite the common application of these algorithms in various data types, little work has been done to investigate how these compression techniques perform specifically on electrophysiology data.

Holistic Overview of Algorithms Deployed:

24+ lossless compression algorithms were tested & benchmarked against each other.

Table of Contents

• Build Instructions
  • Repository Structure
  • How to Run
  • Setup Instructions
  • Execution Commands
• Data Analysis
  • Dataset Characteristics
  • Desired Compression Ratio of 200x
  • Compression Tactics
  • Dataset Visualization
• The Suite of Lossless Algorithms
  • Benchmarking Metrics
  • Results
• B2Z Algorithm
  • Compressing Data with BZ2
  • Burrows-Wheeler Transform (BWT)
  • Move-to-Front (MTF) Transform
  • Huffman Coding
  • Decompressing Data with BZ2
  • Compression Level Impact
• Delta-Huffman Algorithm
  • Building the Huffman Tree
  • Generating Huffman Codes
  • Compressing Data with Canonical Huffman Codes
  • Decompressing Data
  • Serialization of Huffman Codes
• Next Steps
• Open for Pull Requests (PRs) 👋

Build Instructions

Repository Structure

• Files:
  • encode.py: Script for encoding the N1 wav files using the compression algorithm you pick.
  • decode.py: Script for decoding the compressed audio files to validate lossless compression.
  • eval.sh: Bash script to automate the compression and decompression process, and to calculate compression ratios.
  • grid_search.sh: Bash script to sweep through all the compression algorithms of your choice and record data to the ~/tests/compression_results.csv file
  • Makefile:Used to build and manage symbolic links for Python encode/decode scripts, and to run evaluation scripts.
• Directories:
  • ~/algorithms: Holds all the algorithm logic. If you wanted to add your own algorithm, you could add it in this directory and call it from encode.py.
  • ~/data: Holds all the data (743 wav files).
  • ~/tests: CSV & Notebooks for unit testing.

How to Run

Setup Instructions

To install all dependencies, simply run:

make install

[!NOTE]

For WavPack, simply run:
1) Install the WavPack Library
a) brew install wavpack (Mac)
b) sudo apt-get install wavpack (Debian Based Systems) 2) Go to Home directory in repo & cd /n1-codec/misc/wavpack-numcodecs 3) Install directory: pip install .

Execution Commands

To Evaluate a Single Algorithm's Compression:

1. Ensure that mode = insert algo mode here for both encode.py and decode.py
2. Run make encode to make the encode executable
3. Run make decode to make the decode executable
4. Run the eval.sh bash script by running make eval

To Evaluate a Series of Algorithm's Compression:

1. Ensure that mode = sys.argv[3] for both encode.py and decode.py
2. Run make encode to make the encode executable
3. Run make decode to make the decode executable
4. Run the grid_results.sh bash script by running make grid

Data Analysis

The fundamental characteristics of this dataset are as follows:

• High Throughput: The N1 implant generates approximately 200 Mbps of data, stemming from 1024 electrodes.
• High Sampling Rate: Each electrode samples neural signals at 20 kHz.
• Resolution: The data from each electrode is quantized at 10-bit resolution.

This dataset presents substantial challenges due to its size and complexity, necessitating sophisticated compression strategies to reduce bandwidth effectively while preserving the integrity and quality of the neural signals.

Desired Compression Ratio of 200x ....

The Neuralink Compression Challenge set a formidable goal of achieving a 200x lossless compression ratio. This target, while theoretically appealing, presents substantial practical challenges given the inherent noise and complexity of electrophysiological data.

In an article a friend and I wrote last year reviewing all of Neuralink's patents and papers, we delved deep into what Neuralink had deployed in order to get the most important features from their data.

Here's a long but insightful excerpt on the compression tactics (w. loss) that Neuralink has deployed:

Compression strategies predominantly involve applying thresholds to detect spikes in a specific range, summary statistics like channel-wise averages, and/or event-based triggers off-chip. Alternatively, information-theoretic lossless compression techniques like PNG, TIFF, or ZIP may be used. In some examples, the reduction in bandwidth from the compression engine can exceed 1,000 times fewer data.

These thresholds may be set on the voltage of the signal or the frequency of the signal. Low-frequency and high-frequency signals may not be valuable to the recorder and can be filtered out by the compression engine. Non-spike signals are discarded, essentially reducing the size of the data packets, and compressing the signal. For voltage-based thresholds, a technique called non-linear energy operator (NEO) may be used to automatically find a threshold that accurately detects spikes.

Briefly reviewing NEO, it essentially filters the signals for the periods at which there are fast frequency and amplitude changes of spikes, which can be seen as short peaks in the NEO filtered output.

$$ \psi[x(n)] = |x(n) \cdot x(n)| - |x(n-1) \cdot x(n+1)| $$

NEO, represented by 𝝍[x(n)], of a signal x(n) can be computed as shown above. It simply compares the deviation between the signal at n time step and the signal at n-1 and n+1 time steps.

$$ \text{Thr} = C \times \frac{1}{N} \sum_{n=1}^{N} \psi[x(n)] $$

Furthermore, a threshold for NEO detection can be calculated as the mean of the NEO filtered output multiplied by a factor C. In this equation, N is the number of samples of the signal. C is found empirically and should be tested on several neural datasets beforehand to achieve the best results.

Both the compression engine and controller play a crucial role in throttling the amount of data being generated by each chip. Throttling allows for power and performance efficiency improvements for the N1 system.

Alternatively, during the Neuralink launch event, DJ Seo introduced a novel on-chip spike detection algorithm that involved directly characterizing the shape of a spike. This method is able to compress neural data by more than 200x and only takes 900 nanoseconds to compute, which is faster than the time it takes for the brain to realize it happened. This technique even allows for identifying different neurons from the same electrode based on shape.

Dataset Visualization

1/ the data is inconsistent & noisy

The global amplitude statistics for the WAV files are as follows:

• Mean: 975.1664476018635
• Median: 992.0
• Standard Deviation: 3154.619387966767
• Skewness: -0.009132604033709467
• Kurtosis: 59.332425608861676
  

The data exhibits substantial variability, highlighted by a large standard deviation indicating wide fluctuations in signal amplitude, and a leptokurtic distribution with a high kurtosis value suggesting data points are densely clustered around the mean with frequent extreme values. Despite a skewness value near zero indicating symmetry, the mode significantly diverges from the mean and median, underscoring the presence of notable outliers. This combination of statistics suggests a highly variable dataset with a complex, outlier-influenced structure.

2/ mid-range spectral entropy of the data -- yet also extremely variable across files

The spectral entropy of the electrophysiology data averages at 4.88, indicating a moderately complex and unpredictable spectral distribution. The standard deviation of 1.16 points to significant variability across files, which complicates achieving consistent compression ratios.

3/ very noisy, random spectogram

The Suite of Lossless Algorithms

[!NOTE]

As a reminder, there were 5 groups that these algorithms were categorized in based on their core methodologies and performance characteristics:

1. Dictionary-Based Codecs (zstd, lz4, lz4hc, gzip, zlib, bz2 etc.)
2. Predictive Codecs (delta etc.)
3. Entropy Codecs (huffman, arithmetic etc.)
4. Specialized Audio Codecs (wavpack etc.)
5. Blosc Hybrid Codecs

Benchmarking Metrics

The following variables were measured in order to effectively benchmark the algorithms against one another:

• Compression Ratio (CR), computed by dividing the number of bytes in the original data file by the number of bytes written to disk after compression. The higher the compression ratio, the lower disk space the compressed data will occupy. For example, a CR of 2 implies that the compressed data occupies half of the disk space (50%) compared to uncompressed data, a CR of 4 occupies 25% of the disk space, and so on.
• Compression speed (in units of xRT, or “times real-time”), computed by dividing the time needed to compress an entire recording divided by the original recording duration.
• Decompression speed (in xRT), computed by dividing the time needed to decompress the data in a 5s time range by 5.

Results

insights/thoughts:

• bz2 achieved the highest compression ratio (3.11x) primarily because it effectively exploits the local redundancy within electrophysiology data. The Burrows-Wheeler Transform rearranges the data into runs of similar characters, which are highly compressible, especially in data with repetitive neural spike patterns.
• Delta encoding showed significant improvements when used as a preprocessing step (e.g., delta_zstd, delta_huff). It reduces data variability by encoding differences between consecutive samples, which is effective in neural data where sequential readings often differ by small amounts. This reduction in variance enhances the efficiency of subsequent entropy or dictionary-based compression.
• lzma, despite its computational intensity, provided strong compression (2.75x) due to its use of a very large dictionary size, which is more capable of capturing the longer, complex repeating patterns that simpler algorithms might miss in high-resolution neural data.
• WavPack, designed for audio, did not perform optimally, highlighting that the stochastic and non-periodic nature of neural spikes contrasts significantly with the more predictable and periodic nature of audio waveforms.
• Huffman coding alone performed relatively well (2.53x) because electrophysiological signals, despite their complexity, still exhibit certain predictable frequency distributions that Huffman's method can capitalize on by assigning shorter codes to more frequent patterns.

B2Z Algorithm:

Compressing Data with BZ2

The BZ2 compressor utilizes the Burrows-Wheeler Transform (BWT) followed by the Move-to-Front (MTF) Transform and Huffman coding. Here's how each step contributes to the compression process:

Burrows-Wheeler Transform (BWT):

• Objective: Transform the data to make it more compressible.
• Process: Rearranges the data into runs of similar characters. This is particularly effective for data with repetitive sequences such as neural spikes.
• Equation: Given a string ( S ), BWT produces a matrix of all cyclic permutations of ( S ) sorted lexicographically. The last column of this matrix, often more repetitive, is the output.

$$ \text{BWT}(S) = \text{last column of sorted cyclic permutations of } S $$

Move-to-Front (MTF) Transform:

• Objective: Convert the output of BWT into a form that is more amenable to entropy encoding.
• Process: Records the position of each character, moving the most recently used character to the front of a list. This reduces the entropy by exploiting the locality of reference.
• Behavior: The transform typically results in a sequence where the most frequent characters are represented by smaller integers, which are easier to compress using Huffman coding.

Huffman Coding:

• Objective: Encode the output from MTF using variable-length codes.
• Process: Assigns shorter codes to more frequently occurring characters.
• Efficiency: Huffman coding is optimal for a known set of probabilities and typically works best when there are clear patterns or frequency biases in the data, which is common in structured neural recordings.

Decompressing Data with BZ2

Decompression reverses the compression steps:

• Huffman Decoding: Converts the Huffman encoded data back to the MTF encoded sequence.
• Inverse Move-to-Front Transform: Restores the original order post-BWT transformation.
• Inverse Burrows-Wheeler Transform: Recreates the original data from the BWT output.

Compression Level Impact

The compression level in BZ2 can be adjusted, typically ranging from 1 to 9. A higher compression level increases the compression ratio but also the computational expense:

• Compression Level Equation:
  • Given a compression level ( L ), the number of iterations or the thoroughness of the search for redundancies increases, which can be represented as:

$$ \text{Compression Time} \propto L \times \text{Data Complexity} $$

• Higher levels are best used when compression ratio is prioritized over speed, such as in long-term storage of large datasets where reading frequency is low.

Delta-Huffman Algorithm:

Delta Huffman Coding combines the principles of delta encoding and Huffman coding to effectively compress data, especially beneficial for time-series or signal data where consecutive samples are often similar.

Building the Huffman Tree

• Objective: Construct a binary tree with optimal prefix codes for efficient data representation based on frequency.
• Process:
  • Heap Queue: Utilizes a min-heap to build the Huffman tree. Symbols are inserted into a priority queue, which is always ordered by frequency.
  • Tree Construction: The two nodes with the lowest frequency are repeatedly removed from the heap, combined into a new node (summing their frequencies), and this new node is reinserted into the heap. This process continues until only one node remains, representing the root of the Huffman tree.

Generating Huffman Codes

• Objective: Assign binary codes to each symbol such that frequently occurring symbols use shorter codes.
• Process:
  • Starting from the root of the Huffman tree, traverse to each leaf node. Assign '0' for left moves and '1' for right moves, accumulating the path as the code for the symbol at each leaf.
  • Canonical Huffman Codes: Standardize code lengths and order them lexicographically to further optimize the code table, facilitating easier encoding and decoding.

Compressing Data with Canonical Huffman Codes

• Objective: Convert input data into a compact binary representation using the generated Huffman codes.
• Process:
  • Encoding: Each symbol in the input data is replaced by its corresponding Huffman code.
  • Padding: Since the total length of the encoded data may not be a multiple of 8, padding is added to align it to byte boundaries.
  • Compression Equation:

$$ \text{Encoded Data Length} = \sum (\text{Code Length of Symbol} \times \text{Frequency of Symbol}) $$

Decompressing Data

• Objective: Reconstruct the original data from its encoded form.
• Process:
  • Binary Conversion: The encoded byte data is converted back into a binary string.
  • Traversal: Starting from the root of the Huffman tree, traverse according to the binary digits until a leaf node (symbol) is reached, then restart from the root.

Serialization of Huffman Codes

• Objective: Store the Huffman codes in a compact format that can be easily shared or stored.
• Process:
  • JSON Serialization: The Huffman codes are converted into a JSON string, mapping symbols to their respective binary codes, which can be used to reconstruct the Huffman tree during decompression.

Next Steps

• [ ] Explore Adaptive Compression Techniques: Investigate adaptive compression methods that dynamically adjust to the varying characteristics of electrophysiology data to optimize compression ratios and speeds in real-time.

• [ ] Implement Linear Predictive Coding (LPC): Evaluate the use of LPC for lossless compression, leveraging its ability to model and predict signal values based on past samples to enhance compression efficiency.

• [ ] Expand Algorithm Permutations: Utilize the modular interface to run evaluations on new permutations of existing algorithms, enabling the discovery of optimized combinations for specific data patterns.

• [ ] Integrate Machine Learning Models: Explore integrating machine learning models that can learn and predict data patterns for improved compression, ensuring the process remains lossless.

Open for Pull Requests (PRs) 👋

Feel free to contribute to the N1 Codec project! If you have ideas or improvements, just submit a PR. Here are some areas where your contributions could really help:

• New Compression Algorithms: Implement and evaluate new lossless compression algorithms tailored for electrophysiology data.
• Optimization of Existing Algorithms: Optimize current algorithms for better performance in terms of speed and compression ratio.