# -*- coding: utf-8 -*- # # brunel_siegert_nest.py # # This file is part of NEST. # # Copyright (C) 2004 The NEST Initiative # # NEST is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 2 of the License, or # (at your option) any later version. # # NEST is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # You should have received a copy of the GNU General Public License # along with NEST. If not, see . """ 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] 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 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.resolution = dt nest.print_time = True nest.overwrite_files = True print("Building network") ############################################################################### # 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", params=neuron_params) siegert_in = nest.Create("siegert_neuron", params=neuron_params) ############################################################################### # 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", 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, "synapse_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, "synapse_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, "synapse_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 = multimeter.events rates_ex = data["rate"][numpy.where(data["senders"] == siegert_ex.global_id)] rates_in = data["rate"][numpy.where(data["senders"] == siegert_in.global_id)] times = data["times"][numpy.where(data["senders"] == siegert_in.global_id)] print(f"Excitatory rate : {rates_ex[-1]:.2f} spks/s") print(f"Inhibitory rate : {rates_in[-1]:.2f} spks/s")