Matterhorn

Technical Documentation

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1 General Introduction

Matterhorn is a novel general neuromorphic computing framework based on PyTorch.

2 Installation

This is the brief one, for complete version you can refer to the tutorial of Matterhorn's installation.

Environment

Python (>= 3.7 and <= 3.9)

PyTorch (>= 1.13.0)

TorchVision (>= 0.14.0)

Requirement Installation

For Windows version you may have to install GCC as well as G++ through Visual Studio (build tools) and MinGW.

Then execute:

git clone https://github.com/xjtuiair-cag/Matterhorn.git
cd Matterhorn
pip install -r requirements.txt

Don't forget to add sudo prefix if you are not the root user.

Install Matterhorn

By executing commands below to install from PyPI:

pip install matterhorn_pytorch

or by executing commands below to install from repository:

git clone https://github.com/xjtuiair-cag/Matterhorn.git
cd Matterhorn
python3 setup.py develop

Don't forget to add sudo if you are not the root user.

3 Module Explanation

For the references on specialized variables, functions or modules of SNNs, you can refer to the detailed version of the document.

Terms

ANNs - Artificial Neural Networks

SNNs - Spiking Neural Networks

Neurons in SNNs

As we're all known, the image below describes what ANNs look like.

Operation 1 is synapse function, which uses weights and bias to calculate those values from previous layer to current layer. Commonly used synapse functions are including full connection layer nn.Linear, convolution layer nn.Conv2D, etc.

We use an equation to describe the synapse function:

$$Y^{l}=synapse(X^{l-1})$$

Where $l$ here means the number of current layers.

Operation 2 is activation function, which filters information from synapses and sends the filtered information to next layer. Commonly used activation functions are including nn.ReLU, nn.Sigmoid, etc.

We use an equation to describe the activation function:

$$X^{l}=activation(Y^{l})$$

In conclusion, each of layers in ANNs has 2 functions. We can build our ANN model in PyTorch by the code below:

import torch.nn as nn

model = nn.Sequential(
    nn.Linear(28 * 28, 10),
    nn.ReLU()
)

This is a 1-layer MLP. It can take an image with the size of 28x28 as input and classify it into 10 classes. In this example, two equations of ANNs can be represented as below:

$$Y^{l}=W^{l}X^{l-1}+\vec{b}$$

$$X^{l}=ReLU(Y^{l})$$

In SNNs, the synapse equation is the same as that in ANNs. However, functions in soma are no longer like what is in ANNs. In the soma of SNNs, there exists a loop in time. The image below describes what SNNs look like.

Operation 1, the synapse function, calculates spikes from previous layer $O^{l-1}(t)$ thus generates the input potential $X^{l}(t)$.

We use an equation to describe the synapse function:

$$X^{l}(t)=synapse(O^{l-1}(t))$$

By operation 2, the input potential, with the history potential, is calculated based on a 1-order differential equation, thus generating the soma potential $U^{l}(t)$. We name it response function.

We use an equation to describe the response function:

$$U^{l}(t)=response(H^{l}(t-1),X^{l}(t))$$

Each spiking neuron model has its unique response differential equation.

For example, in a LIF neuron:

$$\tau \frac{du}{dt}=-(u-u_{rest})+RI$$

Discretizing it into a difference equation, we can get:

$$U^{l}(t)-H^{l}(t-1)=\frac{1}{\tau}[-[H^{l}(t-1)-u_{rest}]+X^{l}(t)]$$

Operation 3 uses Heaviside step function and threshold potential $u_{th}$ to decide whether to generate spikes $O^{l}(t)$. We name it firing function.

We use an equation to describe the firing function:

$$O^{l}(t)=spiking(U^{l}(t))$$

Generally, the firing function looks like this.

$$O^{l}(t)=Heaviside(U^{l}(t)-u_{th})$$

Where Heaviside step function returns 1 when input is greater than or equal to 0, returns 0 otherwise.

The aim of operation 4 is to set refractory time on neurons by output spikes $O^{l}(t)$. We name it reset function.

We use an equation to describe reset function:

$$H^{l}(t)=reset(U^{l}(t),O^{l}(t))$$

Under most occasions we use equation below to reset potential:

$$H^{l}(t)=U^{l}(t)[1-O^{l}(t)]+u_{rest}O^{l}(t)$$

In brief, we use 4 equations to describe spiking neurons. This is what SNN look like. The shape of a spiking neuron is like a trumpet. Its synapses transform those spikes from last neuron and pass the input response to soma, in which there is a time loop awaits.

By unfolding spiking neurons on temporal dimension, we can get the spatio-temporal topology network of SNNs.

Like building ANNs in PyTorch, we can build our SNN model in Matterhorn by the code below:

import torch
import matterhorn_pytorch.snn as snn

snn_model = snn.Temporal(
    snn.Spatial(
        snn.Linear(28 * 28, 10),
        snn.LIF()
    )
)

In the code, Spatial is one of Matterhorn's containers to represent sequential SNN layers on spatial dimension, and Temporal is another Matterhorn's container to repeat calculating potential and spikes on temporal dimension. By using Spatial and Temporal, an spatio-temporal topology network is built and thus used for training and evaluating.

The built network takes an $n+1$ dimensional torch.Tensor as input spike train. It will take the first dimension as time steps, thus calculating through each time step. after that, it will generate a torch.Tensor as output spike train, just like what ANNs takes and generates in PyTorch. The only difference, which is also a key point, is that we should encode our information into spike train and decode the output spike train.

Encoding and Decoding

A spike train is a set of Dirac impulse functions on the axis of time.

$$O(t)=\sum_{t_{i}}δ(t-t_{i})$$

In other words, there will only be 0s and 1s in discrete spike train. Therefore, we can use an $n+1$ dimensional tensor to represent our spike train. For example, if neurons are flattened into a 1-dimensional vector, we can use another dimension to represent time, thus let it be a 2-dimensional matrix to represent the spike train through space and time.

$$ \begin{matrix} & →s \\ ↓t & \begin{bmatrix} 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 \\ \end{bmatrix} \end{matrix} $$

The matrix above shows what a spike train looks like. It has 4 rows, representing 4 time steps. Besides, it has 8 columns, representing 8 output neurons.

To transform our traditional binary information (images, sounds, etc.) into spike train, an encoder is needed. The most commonly used encoder for non-event data is Poisson encoder, which is a kind of rate coding encoder. It sees intensity of a pixel as probability to fire a spike.

You can use Poisson encoder in Matterhorn by the code below:

import torch
import matterhorn_pytorch.snn as snn

encoder = snn.PoissonEncoder(
    time_steps = 32
)

Then, you can use it by the code below:

spike_train = encoder(image)

An image with the shape of [H, W, C] would be encoded into a spike train with the shape of [T, H, W, C]. For example, a MNIST image which shape is [28, 28] would be encoded (T=32) into a spike train with the shape of [32, 28, 28].

After encoding and processing, the network would generate an output spike train. To get the information, we need to decode. A commonly used decoding method is to count average spikes each output neuron has generated.

$$o_{i}=\frac{1}{T}\sum_{t=1}^{T}{O_{i}^{K}(t)}$$

You can use average decoder in Matterhorn by the code below:

import torch
import matterhorn_pytorch.snn as snn

decoder = snn.AvgSpikeDecoder()

It will take first dimension as temporal dimension, and generate statistical results as output. The output can be transported into ANNs for further processes.

Matterhorn provides a convenient container matterhorn_pytorch.snn.Sequential to connect all your SNN and ANN models.

import torch
import matterhorn_pytorch.snn as snn

model = snn.Sequential(
    snn.PoissonEncoder(
        time_steps = time_steps,
    ),
    snn.Flatten(),
    snn.Linear(28 * 28, 10, bias = False),
    snn.LIF(tau_m = tau, trainable = True),
    snn.AvgSpikeDecoder()
).multi_step_mode_()

You should notice that .multi_step_mode_() is applied on snn.Sequential module. Step mode is switched by multi_step_mode() ( and single_step_mode(), which equals to multi_step_mode(False) ) functions. In single step mode, every time step you should input the spike tensors with the same shape as feature maps ([B, ...]), while in multi step mode, the spikes are calculated at once, and you should input the spike tensors with another dimension T as the first dimension ([T, B, ...]).

By now, you have experienced what SNNs look like and how to build it by Matterhorn. For further experience, you can refer to examples/1_starting.py.

cd Matterhorn
python3 examples/1_starting.py

Why Should We Need Surrogate Gradient

In spiking neurons, we usually use Heaviside step function $u(t)$ to decide whether to generate a spike:

$$O^{l}(t)=u(U^{l}(t)-u_{th})$$

However, Heaviside step function has a derivative that can make everyone headache. Its derivative is Dirac impulse function $\delta (t)$. Dirac impulse function is infinity when x equals to 0, and 0 otherwise. If it is directly used for backpropagation, the gradient must be all damned.

Therefore, some functions must be there to replace Dirac impulse function to join the backpropagation. We call those functions surrogate gradients.

One of the most common surrogate gradients is rectangular function. It is a positive constant when absolute value of x is small enough, and 0 otherwise.

Also, functions suitable for surrogate gradient include the derivative of sigmoidal function, Gaussian function, etc.

You can inspect all provided surrogate gradient functions in matterhorn_pytorch.snn.firing.

Learning: BPTT Vs. STDP

Training SNNs could be as easy as training ANNs after gradient problem of Heaviside step function is solved. After we unfold SNNs into a spatio-temporal network, backpropagation through time (BPTT) could be used in SNNs. On spatial dimension, gradients can be propagated through firing function and synapse function, thus neurons of previous layer would receive the gradient; On temporal dimension, the gradient of the next time step can be propagated through firing function and response function, thus soma of previous time would receive the gradient.

Besides BPTT, there is another simple way to train locally in each neuron without supervision, which we call spike-timing-dependent plasticity (STDP). STDP uses precise time differences between input and output spikes to calculate the weight increment.

STDP follows equation below:

$$Δw_{ij}=\sum_{t_{j}}{\sum_{t_{i}}W(t_{i}-t_{j})}$$

where the weight function $W(x)$ is:

$$ W(x)= \begin{aligned} A_{+}e^{-\frac{x}{τ_{+}}},x>0 \\\ 0,x=0 \\\ -A_{-}e^{\frac{x}{τ_{-}}},x<0 \end{aligned} $$

By setting parameters $A_{+}$, $τ_{+}$, $A_{-}$ and $τ_{-}$, we can easily train SNNs unsupervised. For further experience, you can refer to examples/2_using_stdp.py.

cd Matterhorn
python3 examples/2_using_stdp.py

Note: Please make sure you have installed matterhorn_cpp_extensions (or matterhorn_cuda_extensions if you have CUDA), otherwise it will be extremely slow.

cd matterhorn_cpp_extensions
python3 setup.py develop

if you have CUDA, you can install CUDA version:

cd matterhorn_cuda_extensions
python3 setup.py develop

Neuromorphic Datasets

Matterhorn provides several neuromorphic datasets for training SNNs. You can experience provided neuromorphic dataset in Matterhorn by example examples/3_convolution_and_event_datasets.py.

cd Matterhorn
python3 examples/3_convolution_and_event_datasets.py

NMNIST

We know MNIST. MNIST dataset is for training image classification, consisting of a set of 28x28 pixel grayscale images of handwritten digits (0-9). NMNIST is like MNIST, which is different is that it distorts images and record them into events. The shape of events in NMNIST Dataset is [T, 2, 34, 34].

You can use NMNIST dataset in Matterhorn by the code below:

from matterhorn_pytorch.data import NMNIST

time_steps = 128

train_dataset = NMNIST(
    root = "your/data/path",
    train = True,
    download = True,
    time_steps = time_steps
)
test_dataset = NMNIST(
    root = "your/data/path",
    train = False,
    download = True,
    time_steps = time_steps
)

CIFAR10-DVS

CIFAR10-DVS dataset records distorted CIFAR-10 image by a DVS camera. The shape of events in CIFAR10-DVS Dataset is [T, 2, 128, 128] .

You can use CIFAR10-DVS Dataset in Matterhorn by the code below:

from matterhorn_pytorch.data import CIFAR10DVS

time_steps = 128

train_dataset = CIFAR10DVS(
    root = "your/data/path",
    train = True,
    download = True,
    time_steps = time_steps
)
test_dataset = CIFAR10DVS(
    root = "your/data/path",
    train = False,
    download = True,
    time_steps = time_steps
)

DVS128 Gesture

DVS128 Gesture dataset records gestures from 29 different people under 3 different illuminating conditions by DVS camera. The shape of events in DVS128 Gesture dataset is [T, 2, 128, 128] .

You can use DVS128 Gesture dataset in Matterhorn by the code below:

from matterhorn_pytorch.data import DVS128Gesture

time_steps = 128

train_dataset = DVS128Gesture(
    root = "your/data/path",
    train = True,
    download = True,
    time_steps = time_steps
)
test_dataset = DVS128Gesture(
    root = "your/data/path",
    train = False,
    download = True,
    time_steps = time_steps
)

Spiking Heidelberg Digits (SHD)

SHD dataset records vocal number from 1 to 10 in both English and German and turns them into events. The shape of events in SHD dataset is [T, 700].

You can use SHD dataset in Matterhorn by the code below:

from matterhorn_pytorch.data import SpikingHeidelbergDigits

time_steps = 128

train_dataset = SpikingHeidelbergDigits(
    root = "your/data/path",
    train = True,
    download = True,
    time_steps = time_steps
)
test_dataset = SpikingHeidelbergDigits(
    root = "your/data/path",
    train = False,
    download = True,
    time_steps = time_steps
)

4 Neuromorphic Hardware Support

Will come out soon, but not today. Sorry.

References (and Special Thanks to)

[1] Fang W, Chen Y, Ding J, et al. SpikingJelly: An open-source machine learning infrastructure platform for spike-based intelligence[J]. Science Advances, 2023, 9(40): eadi1480.

[2] Fang W, Yu Z, Chen Y, et al. Deep residual learning in spiking neural networks[J]. Advances in Neural Information Processing Systems, 2021, 34: 21056-21069.

[3] Yao M, Gao H, Zhao G, et al. Temporal-wise attention spiking neural networks for event streams classification[C]//Proceedings of the IEEE/CVF International Conference on Computer Vision. 2021: 10221-10230.