STDP - single synapse

#!pip install ANNarchy

This notebook demonstrates the online implementation of the spike time-dependent plasticity (STDP) rule for a pair of neurons.

import numpy as np
import ANNarchy as ann
import matplotlib.pyplot as plt

ANNarchy 4.8 (4.8.2) on darwin (posix).

The STDP learning rule maintains exponentially-decaying traces for the pre-synaptic and post-synaptic spikes.

$$\tau^+ \, \frac{d x(t)}{dt} = -x (t)$$

$$\tau^- \, \frac{d y(t)}{dt} = -x (t)$$

LTP and LTD occur at spike times depending on the corresponding traces.

• When a pre-synaptic spike occurs, $x(t)$ is incremented and LTD is applied proportionally to $y(t)$.
• When a post-synaptic spike occurs, $y(t)$ is incremented and LTP is applied proportionally to $x(t)$.

STDP = ann.Synapse(
    parameters = """
        tau_plus = 20.0 : projection ; tau_minus = 20.0 : projection
        A_plus = 0.01 : projection   ; A_minus = 0.01 : projection
        w_min = 0.0 : projection     ; w_max = 2.0 : projection
    """,
    equations = """
    
        tau_plus * dx/dt = -x : event-driven # pre-synaptic trace
    
        tau_minus * dy/dt = -y : event-driven # post-synaptic trace
    
    """,
    pre_spike="""
        
        g_target += w
        
        x += A_plus * w_max
        
        w = clip(w - y, w_min , w_max) # LTD
    """,
    post_spike="""
        
        y += A_minus * w_max
        
        w = clip(w + x, w_min , w_max) # LTP
    """
)

We create two dummy populations with one neuron each, whose spike times we can control.

pre = ann.SpikeSourceArray([[0.]])
post = ann.SpikeSourceArray([[50.]])

We connect the population using a STDP synapse.

proj = ann.Projection(pre, post, 'exc', STDP)
proj.connect_all_to_all(1.0)

<ANNarchy.core.Projection.Projection at 0x1291c8ec0>

ann.compile()

Compiling ...  OK 

The presynaptic neuron will fire at avrious times between 0 and 100 ms, while the postsynaptic neuron keeps firing at 50 ms.

pre_times = np.linspace(100.0, 0.0, 101)

weight_changes = []
for t_pre in pre_times:
    
    # Reset the populations
    pre.clear()
    post.clear()
    pre.spike_times = [[t_pre]]
    post.spike_times = [[50.0]]
    
    # Reset the traces
    proj.x = 0.0
    proj.y = 0.0
    
    # Weight before the simulation
    w_before = proj[0].w[0]
    
    # Simulate long enough
    ann.simulate(105.0)
    
    # Record weight change
    delta_w = proj[0].w[0] - w_before
    weight_changes.append(delta_w)

We can now plot the classical STDP figure:

plt.figure(figsize=(10, 8))
plt.plot(50. - pre_times, weight_changes, "*")
plt.plot([-50, 50], [0, 0], 'k')
plt.plot([0, 0], [min(weight_changes), max(weight_changes)], 'k')
plt.xlabel("t_post - t_pre")
plt.ylabel("delta_w")
plt.show()