## Pulse packet example

This script compares the average and individual membrane potential excursions in response to a single pulse packet with an analytically acquired voltage trace (see: Diesmann M. (2002) **Conditions for Stable Propagation of Synchronous Spiking in Cortical Neural Networks, Single Neuron Dynamics and Network Properties**, http://d-nb.info/968772781/34).

A pulse packet is a transient spike volley with a Gaussian rate profile. The user can specify the neural parameters, the parameters of the pulse-packet and the number of trials.

First, we import all necessary modules for simulation, analysis and plotting.

```
import nest
import numpy
import pylab
import array
```

Second, parameters are assigned to variables.

Properties of pulse packet:

```
a = 100 # number of spikes in one pulse packet
sdev = 10. # width of pulse packet (ms)
weight = 0.1 # PSP amplitude (mV)
pulsetime = 500. # occurrence time (center) of pulse-packet (ms)
```

Network and neuron characteristics:

```
n_neurons = 100 # number of neurons
cm = 200. # membrane capacitance (pF)
tau_s = 0.5 # synaptic time constant (ms)
tau_m = 20. # membrane time constant (ms)
V0 = 0.0 # resting potential (mV)
Vth = numpy.inf # firing threshold, high value to avoid spiking
```

Simulation and analysis parameters:

```
simtime = 1000. # how long we simulate (ms)
simulation_resolution = 0.1 # (ms)
sampling_resolution = 1. # for voltmeter (ms)
convolution_resolution = 1. # for the analytics (ms)
```

Some parameters in base units.

```
Cm = cm * 1e-12 # convert to Farad
Weight = weight * 1e-12 # convert to Ampere
Tau_s = tau_s * 1e-3 # convert to sec
Tau_m = tau_m * 1e-3 # convert to sec
Sdev = sdev * 1e-3 # convert to sec
Convolution_resolution = convolution_resolution * 1e-3 # convert to sec
def make_psp(Time, Tau_s, Tau_m, Cm, Weight):
```

This function calculates the membrane potential excursion in response to a single input spike (the equation is given for example in Diesmann 2002, eq.2.3). It expects: Time: a time array or a single time point (in sec) Tau_s and Tau_m: the synaptic and the membrane time constant (in sec) Cm: the membrane capacity (in Farad) Weight: the synaptic weight (in Ampere) It returns the provoked membrane potential (in mV)

```
term1 = (1 / (Tau_s) - 1 / (Tau_m))
term2 = numpy.exp(-Time / (Tau_s))
term3 = numpy.exp(-Time / (Tau_m))
PSP = (Weight / Cm * numpy.exp(1) / Tau_s *
(((-Time * term2) / term1) + (term3 - term2) / term1 ** 2))
return PSP * 1e3
def find_loc_pspmax(tau_s, tau_m):
```

This function finds the exact location of the maximum of the PSP caused by a single input spike. The location is obtained by setting the first derivative of the equation for the PSP (see `make_psp()`

) to zero. The resulting equation can be expressed ain terms of a LambertW function. This function is implemented in nest as a .sli file. In order to access this function in pynest we called the function 'nest.sli_func()'. This function expects: Tau_s and Tau_m: the synaptic and membrane time constant (in sec) It returns the location of the maximum (in sec)

```
var = tau_m / tau_s
lam = nest.sli_func('LambertWm1', -numpy.exp(-1 / var) / var)
t_maxpsp = (-var * lam - 1) / var / (1 / tau_s - 1 / tau_m) * 1e-3
return t_maxpsp
```

- Analytically acquired voltage trace.

1a) First, we construct a Gaussian kernel for a given standard derivation (`sig`

) and mean value (`mu`

). In this case the standard derivation is the width of the pulse packet (see Diesmann 2002).

```
sig = Sdev
mu = 0.0
x = numpy.arange(-4 * sig, 4 * sig, Convolution_resolution)
term1 = 1 / (sig * numpy.sqrt(2 * numpy.pi))
term2 = numpy.exp(-(x - mu) ** 2 / (sig ** 2 * 2))
gauss = term1 * term2 * Convolution_resolution
```

1b) Second, we calculate the PSP of a neuron due to a single spiking input. (see Diesmann 2002, eq. 2.3) Since we do that in discrete time steps, we first construct an array (`t_psp`

) that contains the time points we want to consider. Then, the function make_psp() (that creates the PSP) takes the time array as its first argument.

```
t_psp = numpy.arange(0, 10 * (Tau_m + Tau_s), Convolution_resolution)
psp = make_psp(t_psp, Tau_s, Tau_m, Cm, Weight)
```

1c) Now, we want to normalized the PSP amplitude to one. We therefore have to divide the PSP by its maximum [Diesmann 2002, sec 6.1]. The function `find_loc_pspmax()`

returns the exact time point (`t_pspmax`

) when we expect the maximum to occur. The function `make_psp()`

calculates the corresponding PSP value, which is our PSP amplitude (`psp_ampl`

).

```
t_pspmax = find_loc_pspmax(Tau_s, Tau_m)
psp_amp = make_psp(t_pspmax, Tau_s, Tau_m, Cm, Weight)
psp_norm = psp / psp_amp
```

1d) Now we have all ingredients to compute the membrane potential excursion (`U`

) This calculation implies a convolution of the Gaussian with the normalized PSP (see Diesmann 2002, eq. 6.9). In order to avoid an offset in the convolution we need to add a pad of zeros on the left side of the normalized PSP. Later on we want to compare our analytical results with the simulation outcome. Therefore we need a time vector (`t_U`

) with the correct temporal resolution, that places the excursion of the potential at the correct time.

```
tmp = numpy.zeros(2 * len(psp_norm))
tmp[len(psp_norm) - 1:-1] += psp_norm
psp_norm = tmp
del tmp
U = a * psp_amp * pylab.convolve(gauss, psp_norm)
l = len(U)
t_U = (convolution_resolution * numpy.linspace(-l / 2., l / 2., l) +
pulsetime + 1.)
```

- Simulation. In this section we simulate a network of multiple neurons. All these neurons receive an individual pulse packet that is drawn from a Gaussian distribution.

2.1 We reset the Kernel, define the simulation resolution and set the verbosity using `set_verbosity()`

to suppress info messages.

```
nest.ResetKernel()
nest.SetStatus([0], [{'resolution': simulation_resolution}])
nest.set_verbosity("M_WARNING")
```

2.1 Afterwards we create several neurons, the same amount of pulse-packet-generators and a voltmeter. All these nodes/devices have specific properties that are specified in device specific dictionaries (here: `neuron_pars`

for the neurons, `ppg_pars'' for the and pulse-packet-generators and`

vm_pars`` for the voltmeter).

```
neuron_pars = {
'V_th': Vth,
'tau_m': tau_m,
'tau_syn_ex': tau_s,
'C_m': cm,
'E_L': V0,
'V_reset': V0,
'V_m': V0
}
neurons = nest.Create('iaf_psc_alpha', n_neurons, neuron_pars)
ppg_pars = {
'pulse_times': [pulsetime],
'activity': a,
'sdev': sdev
}
ppgs = nest.Create('pulsepacket_generator', n_neurons, ppg_pars)
vm_pars = {
'record_to': ['memory'],
'withtime': True,
'withgid': True,
'interval': sampling_resolution
}
vm = nest.Create('voltmeter', 1, vm_pars)
```

2.2 Now, we connect each pulse generator to one neuron via static synapses. We want to keep all properties of the static synapse constant except the synaptic weight. Therefore we change the weight with the help of the command `SetDefaults()`

. The command `Connect`

connects all kinds of nodes/devices. Since multiple nodes/devices can be connected in different ways e.g. each source connects to all targets, each source connects to a subset of targets or each source connects to exactly one target, we have to specify the connection. In our case we use the `one-to-one`

connection routine since we connect one pulse generator (source) to one neuron (target). In addition we also connect the voltmeter to the neurons'

```
nest.SetDefaults('static_synapse', {'weight': weight})
nest.Connect(ppgs, neurons, 'one_to_one')
nest.Connect(vm, neurons)
```

2.3. In the next step we run the simulation for a given duration ( `simtime`

, unit is ms).

`nest.Simulate(simtime)`

2.4 Finally, we record the membrane potential, when it occurred and to which neuron it belongs. We obtain this information using the command 'nest.GetStatus(vm, 'events')[0]'). The sender and the time point of a voltage data point at position x in the voltage array (`V_m`

), can be found at the same position x in the sender (`senders`

) and the time array (`times`

).

```
Vm = nest.GetStatus(vm, 'events')[0]['V_m']
times = nest.GetStatus(vm, 'events')[0]['times']
senders = nest.GetStatus(vm, 'events')[0]['senders']
```

- Plotting section. Here we plot the membrane potential derived from theory and from simulation. Since we simulate multiple neurons that received slightly different pulse packets, we plot the individual and the averaged membrane potentials.

Here we plot the analytical solution U (the resting potential V0 shifts the membrane potential up- or downwards).

`pylab.plot(t_U, U + V0, 'r', lw=2, zorder=3, label='analytical solution')`

Here we plot all individual membrane potentials. The time axes is the range of the simulation time in steps of ms.

```
Vm_single = [Vm[senders == ii] for ii in neurons]
simtimes = numpy.arange(1, simtime)
for idn in range(n_neurons):
if idn == 0:
pylab.plot(simtimes, Vm_single[idn], 'gray',
zorder=1, label='single potentials')
else:
pylab.plot(simtimes, Vm_single[idn], 'gray', zorder=1)
```

Here we plot the averaged membrane potential.

```
Vm_average = numpy.mean(Vm_single, axis=0)
pylab.plot(simtimes, Vm_average, 'b', lw=4,
zorder=2, label='averaged potential')
pylab.legend()
pylab.xlabel('time (ms)')
pylab.ylabel('membrane potential (mV)')
pylab.xlim((-5 * (tau_m + tau_s) + pulsetime,
10 * (tau_m + tau_s) + pulsetime))
```