{ "cells": [ { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "%matplotlib inline" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Mean-field theory for random balanced network\n\nThis script performs a mean-field analysis of the spiking network of\nexcitatory and an inhibitory population of leaky-integrate-and-fire neurons\nsimulated in ``brunel_delta_nest.py``. We refer to this spiking network of LIF\nneurons with 'SLIFN'.\n\nThe self-consistent equation for the population-averaged firing rates\n(eq.27 in [1]_, [2]_) is solved by integrating a pseudo-time dynamics\n(eq.30 in [1]_). The latter constitutes a network of rate neurons, which is\nsimulated here. The asymptotic rates, i.e., the fixed points of the\ndynamics (eq.30), are the prediction for the population and\ntime-averaged from the spiking simulation.\n\n## References\n\n.. [1] Hahne J, Dahmen D, Schuecker J, Frommer A, Bolten M,\n Helias M and Diesmann M. (2017). Integration of continuous-time\n dynamics in a spiking neural network simulator. Front. Neuroinform.\n 11:34. doi: 10.3389/fninf.2017.00034\n\n.. [2] Schuecker J, Schmidt M, van Albada SJ, Diesmann M.\n and Helias, M. (2017). Fundamental activity constraints lead\n to specific interpretations of the connectome.\n PLOS Computational Biology 13(2): e1005179.\n https://doi.org/10.1371/journal.pcbi.1005179\n\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "import nest\nimport numpy\n\nnest.ResetKernel()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Assigning the simulation parameters to variables.\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "dt = 0.1 # the resolution in ms\nsimtime = 50.0 # Simulation time in ms" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Definition of the network parameters in the SLIFN\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "g = 5.0 # ratio inhibitory weight/excitatory weight\neta = 2.0 # external rate relative to threshold rate\nepsilon = 0.1 # connection probability" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Definition of the number of neurons and connections in the SLIFN, needed\nfor the connection strength in the Siegert neuron network\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "order = 2500\nNE = 4 * order # number of excitatory neurons\nNI = 1 * order # number of inhibitory neurons\nCE = int(epsilon * NE) # number of excitatory synapses per neuron\nCI = int(epsilon * NI) # number of inhibitory synapses per neuron\nC_tot = int(CI + CE) # total number of synapses per neuron" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Initialization of the parameters of the Siegert neuron and the connection\nstrength. The parameter are equivalent to the LIF-neurons in the SLIFN.\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "tauMem = 20.0 # time constant of membrane potential in ms\ntheta = 20.0 # membrane threshold potential in mV\nneuron_params = {'tau_m': tauMem,\n 't_ref': 2.0,\n 'theta': theta,\n 'V_reset': 0.0,\n }\n\nJ = 0.1 # postsynaptic amplitude in mV in the SLIFN\nJ_ex = J # amplitude of excitatory postsynaptic potential\nJ_in = -g * J_ex # amplitude of inhibitory postsynaptic potential\n# drift_factor in diffusion connections (see [1], eq. 28) for external\n# drive, excitatory and inhibitory neurons\ndrift_factor_ext = tauMem * 1e-3 * J_ex\ndrift_factor_ex = tauMem * 1e-3 * CE * J_ex\ndrift_factor_in = tauMem * 1e-3 * CI * J_in\n# diffusion_factor for diffusion connections (see [1], eq. 29)\ndiffusion_factor_ext = tauMem * 1e-3 * J_ex ** 2\ndiffusion_factor_ex = tauMem * 1e-3 * CE * J_ex ** 2\ndiffusion_factor_in = tauMem * 1e-3 * CI * J_in ** 2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "External drive, this is equivalent to the drive in the SLIFN\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "nu_th = theta / (J * CE * tauMem)\nnu_ex = eta * nu_th\np_rate = 1000.0 * nu_ex * CE" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Configuration of the simulation kernel by the previously defined time\nresolution used in the simulation. Setting ``print_time`` to `True` prints the\nalready processed simulation time as well as its percentage of the total\nsimulation time.\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "nest.SetKernelStatus({\"resolution\": dt, \"print_time\": True,\n \"overwrite_files\": True})\n\nprint(\"Building network\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Configuration of the model ``siegert_neuron`` using ``SetDefaults``.\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "nest.SetDefaults(\"siegert_neuron\", neuron_params)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Creation of the nodes using ``Create``. One rate neuron represents the\nexcitatory population of LIF-neurons in the SLIFN and one the inhibitory\npopulation assuming homogeneity of the populations.\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "siegert_ex = nest.Create(\"siegert_neuron\", 1)\nsiegert_in = nest.Create(\"siegert_neuron\", 1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Poisson drive in the SLIFN is replaced by a driving rate neuron,\nwhich does not receive input from other neurons. The activity of the rate\nneuron is controlled by setting ``mean`` to the rate of the corresponding\npoisson generator in the SLIFN.\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "siegert_drive = nest.Create('siegert_neuron', 1, params={'mean': p_rate})" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To record from the rate neurons a multimeter is created and the parameter\n``record_from`` is set to `rate` as well as the recording interval to `dt`\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "multimeter = nest.Create(\n 'multimeter', params={'record_from': ['rate'], 'interval': dt})" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Connections between ``Siegert neurons`` are realized with the synapse model\n``diffusion_connection``. These two parameters reflect the prefactors in\nfront of the rate variable in eq. 27-29 in [1].\n\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Connections originating from the driving neuron\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "syn_dict = {'drift_factor': drift_factor_ext,\n 'diffusion_factor': diffusion_factor_ext,\n 'synapse_model': 'diffusion_connection'}\n\nnest.Connect(\n siegert_drive, siegert_ex + siegert_in, 'all_to_all', syn_dict)\nnest.Connect(multimeter, siegert_ex + siegert_in)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Connections originating from the excitatory neuron\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "syn_dict = {'drift_factor': drift_factor_ex, 'diffusion_factor':\n diffusion_factor_ex, 'synapse_model': 'diffusion_connection'}\nnest.Connect(siegert_ex, siegert_ex + siegert_in, 'all_to_all', syn_dict)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Connections originating from the inhibitory neuron\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "syn_dict = {'drift_factor': drift_factor_in, 'diffusion_factor':\n diffusion_factor_in, 'synapse_model': 'diffusion_connection'}\nnest.Connect(siegert_in, siegert_ex + siegert_in, 'all_to_all', syn_dict)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Simulate the network\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "nest.Simulate(simtime)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Analyze the activity data. The asymptotic rate of the Siegert neuron\ncorresponds to the population- and time-averaged activity in the SLIFN.\nFor the symmetric network setup used here, the excitatory and inhibitory\nrates are identical. For comparison execute the example ``brunel_delta_nest.py``.\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "data = multimeter.events\nrates_ex = data['rate'][numpy.where(data['senders'] == siegert_ex.global_id)]\nrates_in = data['rate'][numpy.where(data['senders'] == siegert_in.global_id)]\ntimes = data['times'][numpy.where(data['senders'] == siegert_in.global_id)]\nprint(\"Excitatory rate : %.2f Hz\" % rates_ex[-1])\nprint(\"Inhibitory rate : %.2f Hz\" % rates_in[-1])" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.12" } }, "nbformat": 4, "nbformat_minor": 0 }