## 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

First, we import all necessary modules for simulation and analysis. Scipy should be imported before nest.

```
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
```

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.

```
def psp(x):
return - ((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))
```