Campbell & Siegert approximation example

Example script that applies Campbell's theorem and Siegert's rate approximation to and integrate-and-fire neuron. This script calculates the firing rate of an integrate-and-fire neuron in response to a series of Poisson generators, each specified with a rate and a synaptic weight. The calculated rate is compared with a simulation using the iaf_psc_alpha model Sven Schrader, Nov 2008, Siegert implementation by Tom Tetzlaff
from scipy.special import erf
from scipy.optimize import fmin

import numpy as np

import nest

We first set the parameters of neurons, noise and the simulation. First settings are with a single Poisson source, second is with two Poisson sources with half the rate of the single source. Both should lead to the same results.
weights = [0.1]    # (mV) psp amplitudes
rates = [10000.]   # (1/s) rate of Poisson sources


C_m = 250.0       # (pF) capacitance
E_L = -70.0       # (mV) resting potential
I_e = 0.0         # (nA) external current
V_reset = -70.0   # (mV) reset potential
V_th = -55.0      # (mV) firing threshold
t_ref = 2.0       # (ms) refractory period
tau_m = 10.0      # (ms) membrane time constant
tau_syn_ex = .5   # (ms) excitatory synaptic time constant
tau_syn_in = 2.0  # (ms) inhibitory synaptic time constant

simtime = 20000  # (ms) duration of simulation
n_neurons = 10   # number of simulated neurons

# weights = [0.1, 0.1] # (mV) psp amplitudes# rates = [5000., 5000.] # (1/s) rate of Poisson sources For convenience we define some units.
pF = 1e-12
ms = 1e-3
pA = 1e-12
mV = 1e-3

mu = 0.0
sigma2 = 0.0
J = []

assert(len(weights) == len(rates))

In the following we analytically compute the firing rate of the neuron based on Campbell's theorem [1] and Siegerts approximation [2].References: [1] Papoulis A (1991) Probability, Random Variables, and Stochastic Processes, McGraw-Hill [2] Siegert AJ (1951) On the first passage time probability problem, Phys Rev 81: 617–623
for rate, weight in zip(rates, weights):

    if weight > 0:
        tau_syn = tau_syn_ex
    else:
        tau_syn = tau_syn_in

    t_psp = np.arange(0., 10. * (tau_m * ms + tau_syn * ms), 0.0001)

We define the form of a single PSP, which allows us to match the maximal value to or chosen weight.
    psp = lambda x: -(C_m * pF) / (tau_syn * ms) * (1 / (C_m * pF)) *
        (np.exp(1) / (tau_syn * ms)) *
        (((-x * np.exp(-x/(tau_syn * ms))) /
          (1 / (tau_syn * ms) - 1 / (tau_m * ms))) +
         (np.exp(-x / (tau_m * ms)) - np.exp(-x / (tau_syn * ms))) /
         ((1 / (tau_syn * ms) - 1 / (tau_m * ms))**2))

    min_result = fmin(psp, [0], full_output=1, disp=0)

We need to calculate the PSC amplitude (i.e., the weight we set in NEST) from the PSP amplitude, that we have specified above.
    fudge = -1./min_result[1]
    J.append(C_m * weight / (tau_syn) * fudge)

We now use Campbell's theorem to calculate mean and variance of the input due to the Poisson sources. The mean and variance add up for each Poisson source.
    mu += (rate * (J[-1] * pA) * (tau_syn * ms) *
           np.exp(1) * (tau_m * ms) / (C_m * pF))

    sigma2 += rate * (2 * tau_m * ms + tau_syn * ms) * 
        (J[-1] * pA * tau_syn * ms * np.exp(1) * tau_m * ms /
         (2 * (C_m * pF) * (tau_m * ms + tau_syn * ms))) ** 2

mu += (E_L * mV)
sigma = np.sqrt(sigma2)

Having calculate mean and variance of the input, we can now employ Siegert's rate approximation.
num_iterations = 100
upper = (V_th * mV - mu) / sigma / np.sqrt(2)
lower = (E_L * mV - mu) / sigma / np.sqrt(2)
interval = (upper - lower) / num_iterations
tmpsum = 0.0
for cu in range(0, num_iterations+1):
    u = lower + cu * interval
    f = np.exp(u**2) * (1 + erf(u))
    tmpsum += interval * np.sqrt(np.pi) * f
r = 1. / (t_ref*ms + tau_m*ms * tmpsum)


We now simulate neurons receiving Poisson spike trains as input, and compare the theoretical result to the empirical value.
nest.ResetKernel()
nest.set_verbosity('M_WARNING')
neurondict = {'V_th': V_th, 'tau_m': tau_m, 'tau_syn_ex': tau_syn_ex,
              'tau_syn_in': tau_syn_in, 'C_m': C_m, 'E_L': E_L, 't_ref': t_ref,
              'V_m':  E_L, 'V_reset': E_L}

Neurons and devices are instantiated. We set a high threshold as we want free membrane potential. In addition we choose a small resolution for recording the membrane to collect good statistics.
nest.SetDefaults('iaf_psc_alpha', neurondict)
n = nest.Create('iaf_psc_alpha', n_neurons)
n_free = nest.Create('iaf_psc_alpha', 1, [{'V_th': 1e12}])
pg = nest.Create('poisson_generator', len(rates),
                 [{'rate': float(rate_i)} for rate_i in rates])
vm = nest.Create('voltmeter', 1, [{'interval': .1}])
sd = nest.Create('spike_detector', 1)

We connect devices and neurons and start the simulation.
for i, currentpg in enumerate(pg):
    nest.Connect([currentpg], n,
                 syn_spec={'weight': float(J[i]), 'delay': 0.1})
    nest.Connect([currentpg], n_free,
                 syn_spec={'weight': J[i]})

nest.Connect(vm, n_free)
nest.Connect(n, sd)

nest.Simulate(simtime)

Here we read out the recorded membrane potential. The first 500 steps are omitted so initial transients do not perturb our results. We then print the results from theory and simulation.
              (n_neurons * simtime * ms), r))
  CampbellSiegert
                                # -*- coding: utf-8 -*-
#
# CampbellSiegert.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 <http://www.gnu.org/licenses/>.

'''
Campbell & Siegert approximation example
----------------------------------------

Example script that applies Campbell's theorem and Siegert's rate
approximation to and integrate-and-fire neuron.

This script calculates the firing rate of an integrate-and-fire neuron
in response to a series of Poisson generators, each specified with a
rate and a synaptic weight. The calculated rate is compared with a
simulation using the iaf_psc_alpha model

Sven Schrader, Nov 2008, Siegert implementation by Tom Tetzlaff
'''

from scipy.special import erf
from scipy.optimize import fmin

import numpy as np

import nest

'''
We first set the parameters of neurons, noise and the simulation.
First settings are with a single Poisson source, second is with two
Poisson sources with half the rate of the single source. Both should
lead to the same results.
'''
weights = [0.1]    # (mV) psp amplitudes
rates = [10000.]   # (1/s) rate of Poisson sources

# weights = [0.1, 0.1]    # (mV) psp amplitudes
# rates = [5000., 5000.]  # (1/s) rate of Poisson sources

C_m = 250.0       # (pF) capacitance
E_L = -70.0       # (mV) resting potential
I_e = 0.0         # (nA) external current
V_reset = -70.0   # (mV) reset potential
V_th = -55.0      # (mV) firing threshold
t_ref = 2.0       # (ms) refractory period
tau_m = 10.0      # (ms) membrane time constant
tau_syn_ex = .5   # (ms) excitatory synaptic time constant
tau_syn_in = 2.0  # (ms) inhibitory synaptic time constant

simtime = 20000  # (ms) duration of simulation
n_neurons = 10   # number of simulated neurons

'''
For convenience we define some units.
'''
pF = 1e-12
ms = 1e-3
pA = 1e-12
mV = 1e-3

mu = 0.0
sigma2 = 0.0
J = []

assert(len(weights) == len(rates))

'''
In the following we analytically compute the firing rate of the
neuron based on Campbell's theorem [1] and Siegerts approximation [2].

References:
[1] Papoulis A (1991) **Probability, Random Variables, and
Stochastic Processes**, *McGraw-Hill*
[2] Siegert AJ (1951) **On the first passage time probability problem**,
*Phys Rev 81: 617–623*
'''
for rate, weight in zip(rates, weights):

    if weight > 0:
        tau_syn = tau_syn_ex
    else:
        tau_syn = tau_syn_in

    t_psp = np.arange(0., 10. * (tau_m * ms + tau_syn * ms), 0.0001)

    '''
    We define the form of a single PSP, which allows us to match the
    maximal value to or chosen weight.
    '''
    psp = lambda x: -(C_m * pF) / (tau_syn * ms) * (1 / (C_m * pF)) *
        (np.exp(1) / (tau_syn * ms)) *
        (((-x * np.exp(-x/(tau_syn * ms))) /
          (1 / (tau_syn * ms) - 1 / (tau_m * ms))) +
         (np.exp(-x / (tau_m * ms)) - np.exp(-x / (tau_syn * ms))) /
         ((1 / (tau_syn * ms) - 1 / (tau_m * ms))**2))

    min_result = fmin(psp, [0], full_output=1, disp=0)

    '''
    We need to calculate the PSC amplitude (i.e., the weight we set in NEST)
    from the PSP amplitude, that we have specified above.
    '''
    fudge = -1./min_result[1]
    J.append(C_m * weight / (tau_syn) * fudge)

    '''
    We now use Campbell's theorem to calculate mean and variance of
    the input due to the Poisson sources. The mean and variance add up
    for each Poisson source.
    '''
    mu += (rate * (J[-1] * pA) * (tau_syn * ms) *
           np.exp(1) * (tau_m * ms) / (C_m * pF))

    sigma2 += rate * (2 * tau_m * ms + tau_syn * ms) * 
        (J[-1] * pA * tau_syn * ms * np.exp(1) * tau_m * ms /
         (2 * (C_m * pF) * (tau_m * ms + tau_syn * ms))) ** 2

mu += (E_L * mV)
sigma = np.sqrt(sigma2)

'''
Having calculate mean and variance of the input, we can now employ
Siegert's rate approximation.
'''
num_iterations = 100
upper = (V_th * mV - mu) / sigma / np.sqrt(2)
lower = (E_L * mV - mu) / sigma / np.sqrt(2)
interval = (upper - lower) / num_iterations
tmpsum = 0.0
for cu in range(0, num_iterations+1):
    u = lower + cu * interval
    f = np.exp(u**2) * (1 + erf(u))
    tmpsum += interval * np.sqrt(np.pi) * f
r = 1. / (t_ref*ms + tau_m*ms * tmpsum)


'''
We now simulate neurons receiving Poisson spike trains as input,
and compare the theoretical result to the empirical value.
'''
nest.ResetKernel()
nest.set_verbosity('M_WARNING')
neurondict = {'V_th': V_th, 'tau_m': tau_m, 'tau_syn_ex': tau_syn_ex,
              'tau_syn_in': tau_syn_in, 'C_m': C_m, 'E_L': E_L, 't_ref': t_ref,
              'V_m':  E_L, 'V_reset': E_L}

'''
Neurons and devices are instantiated. We set a high threshold as
we want free membrane potential. In addition we choose a small
resolution for recording the membrane to collect good statistics.
'''
nest.SetDefaults('iaf_psc_alpha', neurondict)
n = nest.Create('iaf_psc_alpha', n_neurons)
n_free = nest.Create('iaf_psc_alpha', 1, [{'V_th': 1e12}])
pg = nest.Create('poisson_generator', len(rates),
                 [{'rate': float(rate_i)} for rate_i in rates])
vm = nest.Create('voltmeter', 1, [{'interval': .1}])
sd = nest.Create('spike_detector', 1)

'''
We connect devices and neurons and start the simulation.
'''
for i, currentpg in enumerate(pg):
    nest.Connect([currentpg], n,
                 syn_spec={'weight': float(J[i]), 'delay': 0.1})
    nest.Connect([currentpg], n_free,
                 syn_spec={'weight': J[i]})

nest.Connect(vm, n_free)
nest.Connect(n, sd)

nest.Simulate(simtime)

'''
Here we read out the recorded membrane potential. The first 500
steps are omitted so initial transients do not perturb our results.
We then print the results from theory and simulation.
'''
v_free = nest.GetStatus(vm, 'events')[0]['V_m'][500:-1]
print('mean membrane potential (actual / calculated): {0} / {1}'
      .format(np.mean(v_free), mu * 1000))
print('variance (actual / calculated): {0} / {1}'
      .format(np.var(v_free), sigma2 * 1e6))
print('firing rate (actual / calculated): {0} / {1}'
      .format(nest.GetStatus(sd, 'n_events')[0] /
              (n_neurons * simtime * ms), r))