"""
Implementation of a working memory model.
Literature:
Compte, A., Brunel, N., Goldman-Rakic, P. S., & Wang, X. J. (2000). Synaptic mechanisms and
network dynamics underlying spatial working memory in a cortical network model.
Cerebral Cortex, 10(9), 910-923.
Some parts of this implementation are inspired by material from
*Stanford University, BIOE 332: Large-Scale Neural Modeling, Kwabena Boahen & Tatiana Engel, 2013*,
online available.
Note: Most parameters differ from the original publication.
"""
# This file is part of the exercise code repository accompanying
# the book: Neuronal Dynamics (see http://neuronaldynamics.epfl.ch)
# located at http://github.com/EPFL-LCN/neuronaldynamics-exercises.
# This free software: you can redistribute it and/or modify it under
# the terms of the GNU General Public License 2.0 as published by the
# Free Software Foundation. You should have received a copy of the
# GNU General Public License along with the repository. If not,
# see http://www.gnu.org/licenses/.
# Should you reuse and publish the code for your own purposes,
# please cite the book or point to the webpage http://neuronaldynamics.epfl.ch.
# Wulfram Gerstner, Werner M. Kistler, Richard Naud, and Liam Paninski.
# Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition.
# Cambridge University Press, 2014.
import brian2 as b2
from brian2 import NeuronGroup, Synapses, PoissonInput, network_operation
from brian2.monitors import StateMonitor, SpikeMonitor, PopulationRateMonitor
from random import sample
from collections import deque
from neurodynex.tools import plot_tools
import numpy
import matplotlib.pyplot as plt
import math
from scipy.special import erf
from numpy.fft import rfft, irfft
b2.defaultclock.dt = 0.05 * b2.ms
[docs]def simulate_wm(
N_excitatory=1024, N_inhibitory=256,
N_extern_poisson=1000, poisson_firing_rate=1.4 * b2.Hz, weight_scaling_factor=2.,
sigma_weight_profile=20., Jpos_excit2excit=1.6,
stimulus_center_deg=180, stimulus_width_deg=40, stimulus_strength=0.07 * b2.namp,
t_stimulus_start=0 * b2.ms, t_stimulus_duration=0 * b2.ms,
distractor_center_deg=90, distractor_width_deg=40, distractor_strength=0.0 * b2.namp,
t_distractor_start=0 * b2.ms, t_distractor_duration=0 * b2.ms,
G_inhib2inhib=.35 * 1.024 * b2.nS,
G_inhib2excit=.35 * 1.336 * b2.nS,
G_excit2excit=.35 * 0.381 * b2.nS,
G_excit2inhib=.35 * 1.2 * 0.292 * b2.nS,
monitored_subset_size=1024, sim_time=800. * b2.ms):
"""
Args:
N_excitatory (int): Size of the excitatory population
N_inhibitory (int): Size of the inhibitory population
weight_scaling_factor (float): weight prefactor. When increasing the size of the populations,
the synaptic weights have to be decreased. Using the default values, we have
N_excitatory*weight_scaling_factor = 2048 and N_inhibitory*weight_scaling_factor=512
N_extern_poisson (int): Size of the external input population (Poisson input)
poisson_firing_rate (Quantity): Firing rate of the external population
sigma_weight_profile (float): standard deviation of the gaussian input profile in
the excitatory population.
Jpos_excit2excit (float): Strength of the recurrent input within the excitatory population.
Jneg_excit2excit is computed from sigma_weight_profile, Jpos_excit2excit and the normalization
condition.
stimulus_center_deg (float): Center of the stimulus in [0, 360]
stimulus_width_deg (float): width of the stimulus. All neurons in
stimulus_center_deg +\- (stimulus_width_deg/2) receive the same input current
stimulus_strength (Quantity): Input current to the neurons at stimulus_center_deg +\- (stimulus_width_deg/2)
t_stimulus_start (Quantity): time when the input stimulus is turned on
t_stimulus_duration (Quantity): duration of the stimulus.
distractor_center_deg (float): Center of the distractor in [0, 360]
distractor_width_deg (float): width of the distractor. All neurons in
distractor_center_deg +\- (distractor_width_deg/2) receive the same input current
distractor_strength (Quantity): Input current to the neurons at
distractor_center_deg +\- (distractor_width_deg/2)
t_distractor_start (Quantity): time when the distractor is turned on
t_distractor_duration (Quantity): duration of the distractor.
G_inhib2inhib (Quantity): projections from inhibitory to inhibitory population (later
rescaled by weight_scaling_factor)
G_inhib2excit (Quantity): projections from inhibitory to excitatory population (later
rescaled by weight_scaling_factor)
G_excit2excit (Quantity): projections from excitatory to excitatory population (later
rescaled by weight_scaling_factor)
G_excit2inhib (Quantity): projections from excitatory to inhibitory population (later
rescaled by weight_scaling_factor)
monitored_subset_size (int): nr of neurons for which a Spike- and Voltage monitor
is registered.
sim_time (Quantity): simulation time
Returns:
results (tuple):
rate_monitor_excit (Brian2 PopulationRateMonitor for the excitatory population),
spike_monitor_excit, voltage_monitor_excit, idx_monitored_neurons_excit,\
rate_monitor_inhib, spike_monitor_inhib, voltage_monitor_inhib, idx_monitored_neurons_inhib,\
weight_profile_45 (The weights profile for the neuron with preferred direction = 45deg).
"""
# specify the excitatory pyramidal cells:
Cm_excit = 0.5 * b2.nF # membrane capacitance of excitatory neurons
G_leak_excit = 25.0 * b2.nS # leak conductance
E_leak_excit = -70.0 * b2.mV # reversal potential
v_firing_threshold_excit = -50.0 * b2.mV # spike condition
v_reset_excit = -60.0 * b2.mV # reset voltage after spike
t_abs_refract_excit = 2.0 * b2.ms # absolute refractory period
# specify the weight profile in the recurrent population
# std-dev of the gaussian weight profile around the prefered direction
# sigma_weight_profile = 12.0 # std-dev of the gaussian weight profile around the prefered direction
#
# Jneg_excit2excit = 0
# specify the inhibitory interneurons:
Cm_inhib = 0.2 * b2.nF
G_leak_inhib = 20.0 * b2.nS
E_leak_inhib = -70.0 * b2.mV
v_firing_threshold_inhib = -50.0 * b2.mV
v_reset_inhib = -60.0 * b2.mV
t_abs_refract_inhib = 1.0 * b2.ms
# specify the AMPA synapses
E_AMPA = 0.0 * b2.mV
tau_AMPA = .9 * 2.0 * b2.ms
# specify the GABA synapses
E_GABA = -70.0 * b2.mV
tau_GABA = 10.0 * b2.ms
# specify the NMDA synapses
E_NMDA = 0.0 * b2.mV
tau_NMDA_s = .65 * 100.0 * b2.ms # orig: 100
tau_NMDA_x = .94 * 2.0 * b2.ms
alpha_NMDA = 0.5 * b2.kHz
# projections from the external population
G_extern2inhib = 2.38 * b2.nS
G_extern2excit = 3.1 * b2.nS
# projectsions from the inhibitory populations
G_inhib2inhib *= weight_scaling_factor
G_inhib2excit *= weight_scaling_factor
# projections from the excitatory population
G_excit2excit *= weight_scaling_factor
G_excit2inhib *= weight_scaling_factor # todo: verify this scaling
t_stimulus_end = t_stimulus_start + t_stimulus_duration
t_distractor_end = t_distractor_start + t_distractor_duration
# compute the simulus index
stim_center_idx = int(round(N_excitatory / 360. * stimulus_center_deg))
stim_width_idx = int(round(N_excitatory / 360. * stimulus_width_deg / 2))
stim_target_idx = [idx % N_excitatory
for idx in range(stim_center_idx - stim_width_idx, stim_center_idx + stim_width_idx + 1)]
# compute the distractor index
distr_center_idx = int(round(N_excitatory / 360. * distractor_center_deg))
distr_width_idx = int(round(N_excitatory / 360. * distractor_width_deg / 2))
distr_target_idx = [idx % N_excitatory for idx in range(distr_center_idx - distr_width_idx,
distr_center_idx + distr_width_idx + 1)]
# precompute the weight profile for the recurrent population
tmp = math.sqrt(2. * math.pi) * sigma_weight_profile * erf(180. / math.sqrt(2.) / sigma_weight_profile) / 360.
Jneg_excit2excit = (1. - Jpos_excit2excit * tmp) / (1. - tmp)
presyn_weight_kernel = \
[(Jneg_excit2excit +
(Jpos_excit2excit - Jneg_excit2excit) *
math.exp(-.5 * (360. * min(j, N_excitatory - j) / N_excitatory) ** 2 / sigma_weight_profile ** 2))
for j in range(N_excitatory)]
# validate the normalization condition: (360./N_excitatory)*sum(presyn_weight_kernel)/360.
fft_presyn_weight_kernel = rfft(presyn_weight_kernel)
weight_profile_45 = deque(presyn_weight_kernel)
rot_dist = int(round(len(weight_profile_45) / 8))
weight_profile_45.rotate(rot_dist)
# define the inhibitory population
inhib_lif_dynamics = """
s_NMDA_total : 1 # the post synaptic sum of s. compare with s_NMDA_presyn
dv/dt = (
- G_leak_inhib * (v-E_leak_inhib)
- G_extern2inhib * s_AMPA * (v-E_AMPA)
- G_inhib2inhib * s_GABA * (v-E_GABA)
- G_excit2inhib * s_NMDA_total * (v-E_NMDA)/(1.0+1.0*exp(-0.062*v/volt)/3.57)
)/Cm_inhib : volt (unless refractory)
ds_AMPA/dt = -s_AMPA/tau_AMPA : 1
ds_GABA/dt = -s_GABA/tau_GABA : 1
"""
inhib_pop = NeuronGroup(
N_inhibitory, model=inhib_lif_dynamics,
threshold="v>v_firing_threshold_inhib", reset="v=v_reset_inhib", refractory=t_abs_refract_inhib,
method="rk2")
# initialize with random voltages:
inhib_pop.v = numpy.random.uniform(v_reset_inhib / b2.mV, high=v_firing_threshold_inhib / b2.mV,
size=N_inhibitory) * b2.mV
# set the connections: inhib2inhib
syn_inhib2inhib = Synapses(inhib_pop, target=inhib_pop, on_pre="s_GABA += 1.0", delay=0.0 * b2.ms)
syn_inhib2inhib.connect(condition="i!=j", p=1.0)
# set the connections: extern2inhib
input_ext2inhib = PoissonInput(target=inhib_pop, target_var="s_AMPA",
N=N_extern_poisson, rate=poisson_firing_rate, weight=1.0)
# specify the excitatory population:
excit_lif_dynamics = """
I_stim : amp
s_NMDA_total : 1 # the post synaptic sum of s. compare with s_NMDA_presyn
dv/dt = (
- G_leak_excit * (v-E_leak_excit)
- G_extern2excit * s_AMPA * (v-E_AMPA)
- G_inhib2excit * s_GABA * (v-E_GABA)
- G_excit2excit * s_NMDA_total * (v-E_NMDA)/(1.0+1.0*exp(-0.062*v/volt)/3.57)
+ I_stim
)/Cm_excit : volt (unless refractory)
ds_AMPA/dt = -s_AMPA/tau_AMPA : 1
ds_GABA/dt = -s_GABA/tau_GABA : 1
ds_NMDA/dt = -s_NMDA/tau_NMDA_s + alpha_NMDA * x * (1-s_NMDA) : 1
dx/dt = -x/tau_NMDA_x : 1
"""
excit_pop = NeuronGroup(N_excitatory, model=excit_lif_dynamics,
threshold="v>v_firing_threshold_excit", reset="v=v_reset_excit; x+=1.0",
refractory=t_abs_refract_excit, method="rk2")
# initialize with random voltages:
excit_pop.v = numpy.random.uniform(v_reset_excit / b2.mV, high=v_firing_threshold_excit / b2.mV,
size=N_excitatory) * b2.mV
excit_pop.I_stim = 0. * b2.namp
# set the connections: extern2excit
input_ext2excit = PoissonInput(target=excit_pop, target_var="s_AMPA",
N=N_extern_poisson, rate=poisson_firing_rate, weight=1.0)
# set the connections: inhibitory to excitatory
syn_inhib2excit = Synapses(inhib_pop, target=excit_pop, on_pre="s_GABA += 1.0")
syn_inhib2excit.connect(p=1.0)
# set the connections: excitatory to inhibitory NMDA connections
syn_excit2inhib = Synapses(excit_pop, inhib_pop,
model="s_NMDA_total_post = s_NMDA_pre : 1 (summed)", method="rk2")
syn_excit2inhib.connect(p=1.0)
# # set the connections: UNSTRUCTURED excitatory to excitatory
# syn_excit2excit = Synapses(excit_pop, excit_pop,
# model= "s_NMDA_total_post = s_NMDA_pre : 1 (summed)", method="rk2")
# syn_excit2excit.connect(condition="i!=j", p=1.)
# set the STRUCTURED recurrent input. use a network_operation
@network_operation()
def update_nmda_sum():
fft_s_NMDA = rfft(excit_pop.s_NMDA)
fft_s_NMDA_total = numpy.multiply(fft_presyn_weight_kernel, fft_s_NMDA)
s_NMDA_tot = irfft(fft_s_NMDA_total)
excit_pop.s_NMDA_total_ = s_NMDA_tot
@network_operation(dt=1 * b2.ms)
def stimulate_network(t):
if t >= t_stimulus_start and t < t_stimulus_end:
# excit_pop[stim_start_i - 15:stim_start_i + 15].I_stim = 0.25 * b2.namp
# Todo: review indexing
# print("stim on")
excit_pop.I_stim[stim_target_idx] = stimulus_strength
else:
# print("stim off")
excit_pop.I_stim = 0. * b2.namp
# add distractor
if t >= t_distractor_start and t < t_distractor_end:
excit_pop.I_stim[distr_target_idx] = distractor_strength
def get_monitors(pop, nr_monitored, N):
nr_monitored = min(nr_monitored, (N))
idx_monitored_neurons = \
[int(math.ceil(k))
for k in numpy.linspace(0, N - 1, nr_monitored + 2)][1:-1] # sample(range(N), nr_monitored)
rate_monitor = PopulationRateMonitor(pop)
# record= some_list is not supported? :-(
spike_monitor = SpikeMonitor(pop, record=idx_monitored_neurons)
voltage_monitor = StateMonitor(pop, "v", record=idx_monitored_neurons)
return rate_monitor, spike_monitor, voltage_monitor, idx_monitored_neurons
# collect data of a subset of neurons:
rate_monitor_inhib, spike_monitor_inhib, voltage_monitor_inhib, idx_monitored_neurons_inhib = \
get_monitors(inhib_pop, monitored_subset_size, N_inhibitory)
rate_monitor_excit, spike_monitor_excit, voltage_monitor_excit, idx_monitored_neurons_excit = \
get_monitors(excit_pop, monitored_subset_size, N_excitatory)
b2.run(sim_time)
return \
rate_monitor_excit, spike_monitor_excit, voltage_monitor_excit, idx_monitored_neurons_excit,\
rate_monitor_inhib, spike_monitor_inhib, voltage_monitor_inhib, idx_monitored_neurons_inhib,\
weight_profile_45
[docs]def getting_started():
b2.defaultclock.dt = 0.1 * b2.ms
rate_monitor_excit, spike_monitor_excit, voltage_monitor_excit, idx_monitored_neurons_excit,\
rate_monitor_inhib, spike_monitor_inhib, voltage_monitor_inhib, idx_monitored_neurons_inhib,\
weight_profile\
= simulate_wm(N_excitatory=256, N_inhibitory=64, weight_scaling_factor=8., sim_time=500. * b2.ms,
stimulus_center_deg=120, t_stimulus_start=100 * b2.ms, t_stimulus_duration=200 * b2.ms,
stimulus_strength=.07 * b2.namp)
plot_tools.plot_network_activity(rate_monitor_excit, spike_monitor_excit, voltage_monitor_excit,
t_min=0. * b2.ms)
plt.show()
if __name__ == "__main__":
getting_started()