## Example for the quantal_stp_synapse

The quantal_stp_synapse is a stochastic version of the Tsodys-Markram model for synaptic short term plasticity (STP). This script compares the two variants of the Tsodyks/Markram synapse in NEST.

This synapse model implements synaptic short-term depression and short-term facilitation according to the quantal release model described by Fuhrmann et al. [1] and Loebel et al. [2].

Each presynaptic spike will stochastically activate a fraction of the available release sites. This fraction is binomialy distributed and the release probability per site is governed by the Fuhrmann et al. (2002) model. The solution of the differential equations is taken from Maass and Markram 2002 [3].

The connection weight is interpreted as the maximal weight that can be obtained if all n release sites are activated.

Parameters: The following parameters can be set in the status dictionary: U double - Maximal fraction of available resources [0,1], default=0.5 u double - available fraction of resources [0,1], default=0.5 p double - probability that a vesicle is available, default = 1.0 n long - total number of release sites, default = 1 a long - number of available release sites, default = n tau_rec double - time constant for depression in ms, default=800 ms tau_rec double - time constant for facilitation in ms, default=0 (off)

References: [1] Fuhrmann, G., Segev, I., Markram, H., & Tsodyks, M. V. (2002). Coding of information by activity-dependent synapses. Journal of Neurophysiology. [2] Loebel, A., Silberberg, G., Helbig, D., Markram, H., Tsodyks, M. V, & Richardson, M. J. E. (2009). Multiquantal release underlies the distribution of synaptic efficacies in the neocortex. Frontiers in Computational Neuroscience, 3:27. doi:10.3389/neuro.10.027. [3] Maass, W., & Markram, H. (2002). Synapses as dynamic memory buffers.

```
import nest
import nest.voltage_trace
import numpy
import pylab
nest.ResetKernel()
```

On average, the quantal_stp_synapse converges to the tsodyks2_synapse, so we can compare the two by running multiple trials.

First we define the number of trials as well as the number of release sites.

```
n_syn = 10.0 # number of synapses in a connection
n_trials = 100 # number of measurement trials
```

Next, we define parameter sets for facilitation

```
fac_params = {"U": 0.02, "u": 0.02, "tau_fac": 500.,
"tau_rec": 200., "weight": 1.}
```

Then, we assign the parameter set to the synapse models

```
t1_params = fac_params # for tsodyks2_synapse
t2_params = t1_params.copy() # for quantal_stp_synapse
t1_params['x'] = t1_params['U']
t2_params['n'] = n_syn
```

To make the responses comparable, we have to scale the weight by the number of synapses.

`t2_params['weight'] = 1. / n_syn`

Next, we chage the defaults of the various models to our parameters.

```
nest.SetDefaults("tsodyks2_synapse", t1_params)
nest.SetDefaults("quantal_stp_synapse", t2_params)
nest.SetDefaults("iaf_psc_exp", {"tau_syn_ex": 3.})
```

We create three different neurons. Neuron one is the sender, the two other neurons receive the synapses.

`neuron = nest.Create("iaf_psc_exp", 3)`

The connection from neuron 1 to neuron 2 is a deterministic synapse.

`nest.Connect([neuron[0]], [neuron[1]], syn_spec="tsodyks2_synapse")`

The connection from neuron 1 to neuron 3 has a stochastic quantal_stp_synapse.

`nest.Connect([neuron[0]], [neuron[2]], syn_spec="quantal_stp_synapse")`

The voltmeter will show us the synaptic responses in neurons 2 and 3.

```
voltmeter = nest.Create("voltmeter", 2)
nest.SetStatus(voltmeter, {"withgid": True, "withtime": True})
```

One dry run to bring all synapses into their rest state. The default initialization does not achieve this. In large network simulations this problem does not show, but in small simulations like this, we would see it.

```
nest.SetStatus([neuron[0]], "I_e", 376.0)
nest.Simulate(500.0)
nest.SetStatus([neuron[0]], "I_e", 0.0)
nest.Simulate(1000.0)
```

Only now do we connect the voltmeter to the neurons.

```
nest.Connect([voltmeter[0]], [neuron[1]])
nest.Connect([voltmeter[1]], [neuron[2]])
```

This loop runs over the n_trials trials and performs a standard protocol of a high-rate response, followed by a pause and then a recovery response.

```
for t in range(n_trials):
nest.SetStatus([neuron[0]], "I_e", 376.0)
nest.Simulate(500.0)
nest.SetStatus([neuron[0]], "I_e", 0.0)
nest.Simulate(1000.0)
```

Flush the last voltmeter events from the queue by simulating one time-step.

`nest.Simulate(.1)`

Extract the reference trace.

```
vm = numpy.array(nest.GetStatus([voltmeter[1]], 'events')[0]['V_m'])
vm_reference = numpy.array(nest.GetStatus([voltmeter[0]], 'events')[0]['V_m'])
vm.shape = (n_trials, 1500)
vm_reference.shape = (n_trials, 1500)
```

Now compute the mean of all trials and plot agains trials and references.

```
vm_mean = numpy.array([numpy.mean(vm[:, i]) for (i, j) in enumerate(vm[0, :])])
vm_ref_mean = numpy.array([numpy.mean(vm_reference[:, i])
for (i, j) in enumerate(vm_reference[0, :])])
pylab.plot(vm_mean)
pylab.plot(vm_ref_mean)
```

Finally, print the mean-suqared error between the trial-average and the reference trace. The value should be < 10^-9.

`print(numpy.mean((vm_ref_mean - vm_mean) ** 2))`