## Structural Plasticity example

This example shows a simple network of two populations where structural plasticity is used. The network has 1000 neurons, 80% excitatory and 20% inhibitory. The simulation starts without any connectivity. A set of homeostatic rules are defined, according to which structural plasticity will create and delete synapses dynamically during the simulation until a desired level of electrical activity is reached. The model of structural plasticity used here corresponds to the formulation presented in Butz, M., & van Ooyen, A. (2013). A simple rule for dendritic spine and axonal bouton formation can account for cortical reorganization after focal retinal lesions. PLoS Comput. Biol. 9 (10), e1003259.

At the end of the simulation, a plot of the evolution of the connectivity in the network and the average calcium concentration in the neurons is created.

import nest
import numpy
import matplotlib.pyplot as pl
import sys

First, we have import all necessary modules.

class StructralPlasticityExample:
def __init__(self):

We define general simulation parameters

        # simulated time (ms)
self.t_sim = 200000.0
# simulation step (ms).
self.dt = 0.1
self.number_excitatory_neurons = 800
self.number_inhibitory_neurons = 200

# Structural_plasticity properties
self.update_interval = 1000
self.record_interval = 1000.0
# rate of background Poisson input
self.bg_rate = 10000.0
self.neuron_model = 'iaf_psc_exp'

In this implementation of structural plasticity, neurons grow connection points called synaptic elements. Synapses can be created between compatible synaptic elements. The growth of these elements is guided by homeostatic rules, defined as growth curves. Here we specify the growth curves for synaptic elements of excitatory and inhibitory neurons.

        # Excitatory synaptic elements of excitatory neurons
self.growth_curve_e_e = {
'growth_curve': "gaussian",
'growth_rate': 0.0001,  # (elements/ms)
'continuous': False,
'eta': 0.0,  # Ca2+
'eps': 0.05,  # Ca2+
}

# Inhibitory synaptic elements of excitatory neurons
self.growth_curve_e_i = {
'growth_curve': "gaussian",
'growth_rate': 0.0001,  # (elements/ms)
'continuous': False,
'eta': 0.0,  # Ca2+
'eps': self.growth_curve_e_e['eps'],  # Ca2+
}

# Excitatory synaptic elements of inhibitory neurons
self.growth_curve_i_e = {
'growth_curve': "gaussian",
'growth_rate': 0.0004,  # (elements/ms)
'continuous': False,
'eta': 0.0,  # Ca2+
'eps': 0.2,  # Ca2+
}

# Inhibitory synaptic elements of inhibitory neurons
self.growth_curve_i_i = {
'growth_curve': "gaussian",
'growth_rate': 0.0001,  # (elements/ms)
'continuous': False,
'eta': 0.0,  # Ca2+
'eps': self.growth_curve_i_e['eps']  # Ca2+
}

Now we specify the neuron model.

        self.model_params = {'tau_m': 10.0,  # membrane time constant (ms)
# excitatory synaptic time constant (ms)
'tau_syn_ex': 0.5,
# inhibitory synaptic time constant (ms)
'tau_syn_in': 0.5,
't_ref': 2.0,  # absolute refractory period (ms)
'E_L': -65.0,  # resting membrane potential (mV)
'V_th': -50.0,  # spike threshold (mV)
'C_m': 250.0,  # membrane capacitance (pF)
'V_reset': -65.0  # reset potential (mV)
}

self.nodes_e = None
self.nodes_i = None
self.mean_ca_e = []
self.mean_ca_i = []
self.total_connections_e = []
self.total_connections_i = []

We initialize variables for the post-synaptic currents of the excitatory, inhibitory and external synapses. These values were calculated from a PSP amplitude of 1 for excitatory synapses, -1 for inhibitory synapses and 0.11 for external synapses.

        self.psc_e = 585.0
self.psc_i = -585.0
self.psc_ext = 6.2

def prepare_simulation(self):
nest.ResetKernel()
nest.set_verbosity('M_ERROR')

We set global kernel parameters. Here we define the resolution for the simulation, which is also the time resolution for the update of the synaptic elements.

        nest.SetKernelStatus(
{
'resolution': self.dt
}
)

Set Structural Plasticity synaptic update interval which is how often the connectivity will be updated inside the network. It is important to notice that synaptic elements and connections change on different time scales.

        nest.SetStructuralPlasticityStatus({
'structural_plasticity_update_interval': self.update_interval,
})

Now we define Structural Plasticity synapses. In this example we create two synapse models, one for excitatory and one for inhibitory synapses. Then we define that excitatory synapses can only be created between a pre synaptic element called 'Axon_ex' and a post synaptic element called Den_ex. In a similar manner, synaptic elements for inhibitory synapses are defined.

        nest.CopyModel('static_synapse', 'synapse_ex')
nest.SetDefaults('synapse_ex', {'weight': self.psc_e, 'delay': 1.0})
nest.CopyModel('static_synapse', 'synapse_in')
nest.SetDefaults('synapse_in', {'weight': self.psc_i, 'delay': 1.0})
nest.SetStructuralPlasticityStatus({
'structural_plasticity_synapses': {
'synapse_ex': {
'model': 'synapse_ex',
'post_synaptic_element': 'Den_ex',
'pre_synaptic_element': 'Axon_ex',
},
'synapse_in': {
'model': 'synapse_in',
'post_synaptic_element': 'Den_in',
'pre_synaptic_element': 'Axon_in',
},
}
})

def create_nodes(self):

Now we assign the growth curves to the corresponding synaptic elements

        synaptic_elements = {
'Den_ex': self.growth_curve_e_e,
'Den_in': self.growth_curve_e_i,
'Axon_ex': self.growth_curve_e_e,
}

synaptic_elements_i = {
'Den_ex': self.growth_curve_i_e,
'Den_in': self.growth_curve_i_i,
'Axon_in': self.growth_curve_i_i,
}

Then it is time to create a population with 80% of the total network size excitatory neurons and another one with 20% of the total network size of inhibitory neurons.

        self.nodes_e = nest.Create('iaf_psc_alpha',
self.number_excitatory_neurons,
{'synaptic_elements': synaptic_elements})

self.nodes_i = nest.Create('iaf_psc_alpha',
self.number_inhibitory_neurons,
{'synaptic_elements': synaptic_elements_i})
nest.SetStatus(self.nodes_e, 'synaptic_elements', synaptic_elements)
nest.SetStatus(self.nodes_i, 'synaptic_elements', synaptic_elements_i)

def connect_external_input(self):

We create and connect the Poisson generator for external input

        noise = nest.Create('poisson_generator')
nest.SetStatus(noise, {"rate": self.bg_rate})
nest.Connect(noise, self.nodes_e, 'all_to_all',
{'weight': self.psc_ext, 'delay': 1.0})
nest.Connect(noise, self.nodes_i, 'all_to_all',
{'weight': self.psc_ext, 'delay': 1.0})

In order to save the amount of average calcium concentration in each population through time we create the function record_ca. Here we use the GetStatus function to retrieve the value of Ca for every neuron in the network and then store the average.

    def record_ca(self):
ca_e = nest.GetStatus(self.nodes_e, 'Ca'),  # Calcium concentration
self.mean_ca_e.append(numpy.mean(ca_e))

ca_i = nest.GetStatus(self.nodes_i, 'Ca'),  # Calcium concentration
self.mean_ca_i.append(numpy.mean(ca_i))

In order to save the state of the connectivity in the network through time we create the function record_connectivity. Here we use the GetStatus function to retrieve the number of connected pre synaptic elements of each neuron. The total amount of excitatory connections is equal to the total amount of connected excitatory pre synaptic elements. The same applies for inhibitory connections.

    def record_connectivity(self):
syn_elems_e = nest.GetStatus(self.nodes_e, 'synaptic_elements')
syn_elems_i = nest.GetStatus(self.nodes_i, 'synaptic_elements')
self.total_connections_e.append(sum(neuron['Axon_ex']['z_connected']
for neuron in syn_elems_e))
self.total_connections_i.append(sum(neuron['Axon_in']['z_connected']
for neuron in syn_elems_i))

We define a function to plot the recorded values at the end of the simulation.

    def plot_data(self):
fig, ax1 = pl.subplots()
ax1.axhline(self.growth_curve_e_e['eps'],
linewidth=4.0, color='#9999FF')
ax1.plot(self.mean_ca_e, 'b',
label='Ca Concentration Excitatory Neurons', linewidth=2.0)
ax1.axhline(self.growth_curve_i_e['eps'],
linewidth=4.0, color='#FF9999')
ax1.plot(self.mean_ca_i, 'r',
label='Ca Concentration Inhibitory Neurons', linewidth=2.0)
ax1.set_ylim([0, 0.275])
ax1.set_xlabel("Time in [s]")
ax1.set_ylabel("Ca concentration")
ax2 = ax1.twinx()
ax2.plot(self.total_connections_e, 'm',
label='Excitatory connections', linewidth=2.0, linestyle='--')
ax2.plot(self.total_connections_i, 'k',
label='Inhibitory connections', linewidth=2.0, linestyle='--')
ax2.set_ylim([0, 2500])
ax2.set_ylabel("Connections")
ax1.legend(loc=1)
ax2.legend(loc=4)
pl.savefig('StructuralPlasticityExample.eps', format='eps')

It is time to specify how we want to perform the simulation. In this function we first enable structural plasticity in the network and then we simulate in steps. On each step we record the calcium concentration and the connectivity. At the end of the simulation, the plot of connections and calcium concentration through time is generated.

    def simulate(self):
if nest.NumProcesses() > 1:
sys.exit("For simplicity, this example only works " +
"for a single process.")
nest.EnableStructuralPlasticity()
print("Starting simulation")
sim_steps = numpy.arange(0, self.t_sim, self.record_interval)
for i, step in enumerate(sim_steps):
nest.Simulate(self.record_interval)
self.record_ca()
self.record_connectivity()
if i % 20 == 0:
print("Progress: " + str(i / 2) + "%")
print("Simulation finished successfully")

Finally we take all the functions that we have defined and create the sequence for our example. We prepare the simulation, create the nodes for the network, connect the external input and then simulate. Please note that as we are simulating 200 biological seconds in this example, it will take a few minutes to complete.

if __name__ == '__main__':
example = StructralPlasticityExample()
# Prepare simulation
example.prepare_simulation()
example.create_nodes()
example.connect_external_input()
# Start simulation
example.simulate()
example.plot_data()