Add an example of a homeostatic, inhibitory STDP rule with a global f…

…actor

Contribution requested here:
https://brian.discourse.group/t/valueerror-equations-of-type-parameter-cannot-have-a-flag-linked-only-the-following-flags-are-allowed-constant-shared/1373/4

examples/synapses/homeostatic_stdp_at_inhibitory_synapes.py

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+#!/usr/bin/env python
+
+"""This script implements a homeostatic STDP rule for inhibitory
+synapses onto excitatory neurons presented in Zenke et al. (2015) and
+explained in detail in the accompanying supplementary information.
+
+In essence, the magnitude and sign of the STDP kernel at inhibitory
+synapses targeting excitatory neurons is modulated by a global factor
+G, which corresponds to the difference between the average firing rate
+of the excitatory postsynaptic neurons and a target firing rate
+$\gamma$. If the average firing rate of excitatory neurons exceeds the
+target firing rate, pre/post pairings (and post/pre pairings) at
+inhibitory synapses connecting to excitatory neurons result in
+potentiation; depression of the inhibitory synapses occurs when the
+excitatory neurons are unable to match the desired target rate.
+
+The script emulates classic STDP protocols, simulating a connected
+presynaptic and postsynaptic neuron pair with precisely controlled
+spike times, and a large, independently firing population of
+excitatory neurons.
+
+Paul Brodersen, 2025
+
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+import brian2 as b2
+
+b2.start_scope()
+
+net = b2.Network(b2.collect())
+
+# excitatory neuron population
+epoch_length = 1 * b2.second
+spike_rate = np.arange(0, 12.5, 2.5)
+total_epochs = len(spike_rate)
+population_spike_rate = b2.TimedArray(spike_rate * b2.Hz, dt=epoch_length)
+population_size = 1000
+spike_generators = b2.PoissonGroup(population_size, rates="population_spike_rate(t)")
+net.add(spike_generators)
+
+# global factor G
+gamma = 5 * b2.Hz
+tau_H = 0.1 * b2.second # NB: value in Zenke et al (2015) is 10 seconds; reduced here to allow for shorter simulations
+global_factor_model = """
+G = H - gamma    : Hz
+dH/dt = -H/tau_H : Hz
+"""
+global_factor = b2.NeuronGroup(1, model=global_factor_model)
+net.add(global_factor)
+spike_aggregator = b2.Synapses(spike_generators, global_factor, on_pre="H_post += 1/(tau_H * population_size)")
+spike_aggregator.connect()
+net.add(spike_aggregator)
+
+# pre-synaptic neuron
+spike_times = epoch_length * (np.arange(total_epochs) + 0.5)
+spike_indices = np.zeros_like(spike_times / b2.second)
+presynaptic_neuron = b2.SpikeGeneratorGroup(1, spike_indices, spike_times)
+net.add(presynaptic_neuron)
+
+# postsynaptic neuron
+delay = 1 * b2.msecond
+postsynaptic_neuron = b2.SpikeGeneratorGroup(1, spike_indices, spike_times + delay)
+net.add(postsynaptic_neuron)
+
+# synapse
+tau_stdp = 20. * b2.msecond
+eta = 1. * b2.nsiemens * b2.second
+synapse_model = """
+wij : siemens
+dzi/dt = -zi / tau_stdp : 1 (event-driven)
+dzj/dt = -zj / tau_stdp : 1 (event-driven)
+"""
+on_pre = """
+zi += 1.
+wij += eta * G * zj
+"""
+on_post = """
+zj += 1.
+wij += eta * G * (zi + 1)
+"""
+synapse = b2.Synapses(
+    presynaptic_neuron, postsynaptic_neuron,
+    model=synapse_model, on_pre=on_pre, on_post=on_post
+)
+synapse.connect(i=0, j=0)
+# Synapses currently don't support linked variables.
+# The following syntax hence results in an error:
+# synapses.G = b2.linked_var(global_factor, "G")
+# Instead we the reference G as shown below.
+# See also:
+# https://brian.discourse.group/t/valueerror-equations-of-type-parameter-cannot-have-a-flag-linked-only-the-following-flags-are-allowed-constant-shared/1373/2
+synapse.variables.add_reference("G", group=global_factor, index='0')
+synapse.wij = 0 * b2.nsiemens
+net.add(synapse)
+
+# monitor
+monitor = b2.StateMonitor(synapse, ["G", "wij"], record=True, dt=1*b2.msecond)
+net.add(monitor)
+
+# run
+net.run(total_epochs * epoch_length)
+
+# analysis
+G = np.squeeze(monitor.G / b2.Hz)
+wij = np.squeeze(monitor.wij / b2.nsiemens)
+time = np.squeeze(monitor.t / b2.ms)
+spike_times = spike_times / b2.ms
+
+# visualize experimental protocol
+fig, axes = plt.subplots(2, 1, sharex=True)
+fig.suptitle("Experimental protocol")
+axes[0].plot(time, G)
+axes[0].set_ylabel("G [Hz]")
+axes[1].plot(time, wij)
+axes[1].set_ylabel("Weight [nS]")
+for ax in axes:
+    ax.vlines(spike_times, *ax.get_ylim(), color="gray", linestyle=":", label="Pre/postt spike pairings")
+axes[0].legend(loc="upper left")
+axes[-1].set_xlabel("Time [ms]")
+
+# plot relationship between G and \Delta wij
+g = np.zeros_like(spike_times)
+dw = np.zeros_like(spike_times)
+for ii, spike_time in enumerate(spike_times):
+    g[ii] = G[np.argmin(np.abs(time - spike_time))]
+    w0 = wij[np.argmin(np.abs(time - spike_time - 10))]
+    w1 = wij[np.argmin(np.abs(time - spike_time + 10))]
+    dw[ii] = w1 - w0
+
+fig, ax = plt.subplots()
+ax.plot(g, dw, 'o')
+ax.set_xlabel("G [Hz]")
+ax.set_ylabel(r"$\Delta$weight [nS]")
+
+plt.show()