# Mean-field theory for random balanced network¶

This script performs a mean-field analysis of the spiking network of excitatory and an inhibitory population of leaky-integrate-and-fire neurons simulated in brunel_delta_nest.py. We refer to this spiking network of LIF neurons with ‘SLIFN’.

The self-consistent equation for the population-averaged firing rates (eq.27 in 1, 2) is solved by integrating a pseudo-time dynamics (eq.30 in 1). The latter constitutes a network of rate neurons, which is simulated here. The asymptotic rates, i.e., the fixed points of the dynamics (eq.30), are the prediction for the population and time-averaged from the spiking simulation.

## References¶

1(1,2)

Hahne J, Dahmen D, Schuecker J, Frommer A, Bolten M, Helias M and Diesmann M. (2017). Integration of continuous-time dynamics in a spiking neural network simulator. Front. Neuroinform. 11:34. doi: 10.3389/fninf.2017.00034

2

Schuecker J, Schmidt M, van Albada SJ, Diesmann M. and Helias, M. (2017). Fundamental activity constraints lead to specific interpretations of the connectome. PLOS Computational Biology 13(2): e1005179. https://doi.org/10.1371/journal.pcbi.1005179

import nest
import pylab
import numpy

nest.ResetKernel()


Assigning the simulation parameters to variables.

dt = 0.1  # the resolution in ms
simtime = 50.0  # Simulation time in ms


Definition of the network parameters in the SLIFN

g = 5.0  # ratio inhibitory weight/excitatory weight
eta = 2.0  # external rate relative to threshold rate
epsilon = 0.1  # connection probability


Definition of the number of neurons and connections in the SLIFN, needed for the connection strength in the siegert neuron network

order = 2500
NE = 4 * order  # number of excitatory neurons
NI = 1 * order  # number of inhibitory neurons
CE = int(epsilon * NE)  # number of excitatory synapses per neuron
CI = int(epsilon * NI)  # number of inhibitory synapses per neuron
C_tot = int(CI + CE)  # total number of synapses per neuron


Initialization of the parameters of the siegert neuron and the connection strength. The parameter are equivalent to the LIF-neurons in the SLIFN.

tauMem = 20.0  # time constant of membrane potential in ms
theta = 20.0  # membrane threshold potential in mV
neuron_params = {'tau_m': tauMem,
't_ref': 2.0,
'theta': theta,
'V_reset': 0.0,
}

J = 0.1  # postsynaptic amplitude in mV in the SLIFN
J_ex = J  # amplitude of excitatory postsynaptic potential
J_in = -g * J_ex  # amplitude of inhibitory postsynaptic potential
# drift_factor in diffusion connections (see [1], eq. 28) for external
# drive, excitatory and inhibitory neurons
drift_factor_ext = tauMem * 1e-3 * J_ex
drift_factor_ex = tauMem * 1e-3 * CE * J_ex
drift_factor_in = tauMem * 1e-3 * CI * J_in
# diffusion_factor for diffusion connections (see [1], eq. 29)
diffusion_factor_ext = tauMem * 1e-3 * J_ex ** 2
diffusion_factor_ex = tauMem * 1e-3 * CE * J_ex ** 2
diffusion_factor_in = tauMem * 1e-3 * CI * J_in ** 2


External drive, this is equivalent to the drive in the SLIFN

nu_th = theta / (J * CE * tauMem)
nu_ex = eta * nu_th
p_rate = 1000.0 * nu_ex * CE


Configuration of the simulation kernel by the previously defined time resolution used in the simulation. Setting print_time to True prints the already processed simulation time as well as its percentage of the total simulation time.

nest.SetKernelStatus({"resolution": dt, "print_time": True,
"overwrite_files": True})

print("Building network")


Configuration of the model siegert_neuron using SetDefaults.

nest.SetDefaults("siegert_neuron", neuron_params)


Creation of the nodes using Create. One rate neuron represents the excitatory population of LIF-neurons in the SLIFN and one the inhibitory population assuming homogeneity of the populations.

siegert_ex = nest.Create("siegert_neuron", 1)
siegert_in = nest.Create("siegert_neuron", 1)


The Poisson drive in the SLIFN is replaced by a driving rate neuron, which does not receive input from other neurons. The activity of the rate neuron is controlled by setting mean to the rate of the corresponding poisson generator in the SLIFN.

siegert_drive = nest.Create('siegert_neuron', 1, params={'mean': p_rate})


To record from the rate neurons a multimeter is created and the parameter record_from is set to rate as well as the recording interval to dt

multimeter = nest.Create(
'multimeter', params={'record_from': ['rate'], 'interval': dt})


Connections between siegert neurons are realized with the synapse model diffusion_connection. These two parameters reflect the prefactors in front of the rate variable in eq. 27-29 in [1].

Connections originating from the driving neuron

syn_dict = {'drift_factor': drift_factor_ext,
'diffusion_factor': diffusion_factor_ext,
'model': 'diffusion_connection'}

nest.Connect(
siegert_drive, siegert_ex + siegert_in, 'all_to_all', syn_dict)
nest.Connect(multimeter, siegert_ex + siegert_in)


Connections originating from the excitatory neuron

syn_dict = {'drift_factor': drift_factor_ex, 'diffusion_factor':
diffusion_factor_ex, 'model': 'diffusion_connection'}
nest.Connect(siegert_ex, siegert_ex + siegert_in, 'all_to_all', syn_dict)


Connections originating from the inhibitory neuron

syn_dict = {'drift_factor': drift_factor_in, 'diffusion_factor':
diffusion_factor_in, 'model': 'diffusion_connection'}
nest.Connect(siegert_in, siegert_ex + siegert_in, 'all_to_all', syn_dict)


Simulate the network

nest.Simulate(simtime)


Analyze the activity data. The asymptotic rate of the siegert neuron corresponds to the population- and time-averaged activity in the SLIFN. For the symmetric network setup used here, the excitatory and inhibitory rates are identical. For comparison execute the example brunel_delta_nest.py.

data = nest.GetStatus(multimeter)[0]['events']
rates_ex = data['rate'][numpy.where(data['senders'] == siegert_ex)]
rates_in = data['rate'][numpy.where(data['senders'] == siegert_in)]
times = data['times'][numpy.where(data['senders'] == siegert_in)]
print("Excitatory rate   : %.2f Hz" % rates_ex[-1])
print("Inhibitory rate   : %.2f Hz" % rates_in[-1])


Total running time of the script: ( 0 minutes 0.000 seconds)

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