Neuromorphic Computing - Algorithms for Spiking Neural Networks (SNNs)

Spiking Neural Networks (SNNs) are the third generation of neural networks that use discrete electrical impulses called spikes to process, learn, and store information. Several algorithms have been developed to train and optimize SNNs. In this section, we will explore key algorithms Spike-Time-Dependent Plasticity (STDP) and Spike-Based Backpropagation, along with their examples and applications in neuromorphic systems.

Spike-Time-Dependent Plasticity (STDP)

Spike-Time-Dependent Plasticity (STDP) is an unsupervised learning algorithm based on Hebbian learning principles. It can be summarized by the rule: "neurons that fire together, wire together." The timing of spikes plays a crucial role in determining whether the synapse between two neurons is strengthened (long-term potentiation, LTP) or weakened (long-term depression, LTD). If a presynaptic spike precedes a postsynaptic spike, the synapse is strengthened; if the opposite occurs, the synapse is weakened.

STDP Algorithm Example (Python)

The Python code below simulates the STDP learning rule, by modifying the strength of a synapse based on the timing of pre and post synaptic spikes. The code calculates the time difference between spikes, then updates the synaptic weight using the STDP rule.

import numpy as np

# Define parameters for STDP
tau_pre = 20  # Time constant for presynaptic spike
tau_post = 20  # Time constant for postsynaptic spike
A_plus = 0.005  # Weight increase factor
A_minus = 0.005  # Weight decrease factor

# Initialize weight and spike timings
weight = 0.5  # Initial synaptic weight
pre_spike_time = np.random.uniform(0, 100)  
post_spike_time = np.random.uniform(0, 100)  

# Calculate the time difference between spikes
delta_t = post_spike_time - pre_spike_time

# STDP update rule
if delta_t > 0:
    weight += A_plus * np.exp(-delta_t / tau_pre)
elif delta_t 

In this example, the synaptic weight is updated based on the relative timing of pre- and post-synaptic spikes. The weight is increased if the presynaptic spike occurs before the postsynaptic spike and decreased if the reverse happens.

Spike-Based Backpropagation Algorithm
 
Spike-based backpropagation make slight changes to traditional backpropagation algorithm used in artificial neural networks (ANNs) to work with SNNs. It adjusts the synaptic weights based on the timing of spikes and the error gradients in the network. However, applying backpropagation directly to SNNs is challenging due to the non-differentiable nature of spiking neurons. To overcome this, approximations of the derivative can be made around key points such as the spiking time, membrane potential, or by using ReLU-like activations.
 

Example of Spike-Based Backpropagation
 
The algorithm works by adjusting synaptic weights based on the occurrence of spikes and the networks output error. Here we used surrogate gradient to approximate the derivative of the spike function. By doing this, spike-based backpropagation can be applied to adjust synaptic weights based on the error signal.
 

import numpy as np

# Spike function with a surrogate gradient
def spike_function(v, threshold=1.0):
    return np.heaviside(v - threshold, 0)

# Approximation of the derivative of the spike function
def surrogate_gradient(v, threshold=1.0):
    return 1 / (1 + np.exp(-v + threshold))

# Forward pass
def forward_pass(inputs, weights):
    membrane_potential = np.dot(inputs, weights)
    spikes = spike_function(membrane_potential)
    return spikes, membrane_potential

# Backward pass (Weight update)
def backward_pass(spikes, membrane_potential, error, learning_rate=0.01):
    grad = surrogate_gradient(membrane_potential) * error
    weight_update = learning_rate * grad
    return weight_update

# Example inputs and weights
inputs = np.array([0.1, 0.4, 0.5])  # Input spikes
weights = np.array([0.2, 0.6, 0.3])  # Synaptic weights
error = np.array([0.01, -0.02, 0.05])  # Error signal

# Forward and backward pass
spikes, membrane_potential = forward_pass(inputs, weights)
weight_update = backward_pass(spikes, membrane_potential, error)

# Update weights
weights += weight_update
print(f"Updated Weights: {weights}")



Other Algorithms for SNNs
In addition to STDP and Spike-Based Backpropagation, several other algorithms are used to train SNNs, including:

• Reinforcement Learning for SNNs: Reinforcement learning methods can be applied to SNNs, where synaptic weights are updated based on reward signals. This is similar to how animals learn by trial and error.
• Evolutionary Algorithms: These algorithms optimize the structure and parameters of SNNs by simulating the process of natural selection.
• Surrogate Gradient Methods: These methods are used to overcome the non-differentiability problem in SNNs by approximating the gradient of the spike function during the learning process.