## Balanced neuron example

This script simulates a neuron driven by an excitatory and an inhibitory population of neurons firing Poisson spike trains. The aim is to find a firing rate for the inhibitory population that will make the neuron fire at the same rate as the excitatory population.

Optimization is performed using the `bisection`

method from Scipy, simulating the network repeatedly.

This example is also shown in the article Eppler et al. (2009) **PyNEST: A convenient interface to the NEST simulator**, *Front. Neuroinform.* http://dx.doi.org/10.3389/neuro.11.012.2008

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

```
from scipy.optimize import bisect
import nest
import nest.voltage_trace
```

Additionally, we set the verbosity using `set_verbosity`

to suppress info messages.

```
nest.set_verbosity("M_WARNING")
nest.ResetKernel()
```

Second, the simulation parameters are assigned to variables.

```
t_sim = 25000.0 # how long we simulate
n_ex = 16000 # size of the excitatory population
n_in = 4000 # size of the inhibitory population
r_ex = 5.0 # mean rate of the excitatory population
r_in = 20.5 # initial rate of the inhibitory population
epsc = 45.0 # peak amplitude of excitatory synaptic currents
ipsc = -45.0 # peak amplitude of inhibitory synaptic currents
d = 1.0 # synaptic delay
lower = 15.0 # lower bound of the search interval
upper = 25.0 # upper bound of the search interval
prec = 0.01 # how close need the excitatory rates be
```

Third, the nodes are created using `Create`

. We store the returned handles in variables for later reference.

```
neuron = nest.Create("iaf_psc_alpha")
noise = nest.Create("poisson_generator", 2)
voltmeter = nest.Create("voltmeter")
spikedetector = nest.Create("spike_detector")
```

Fourth, the excitatory `poisson_generator`

(`noise[0]`

) and the `voltmeter`

are configured using `SetStatus`

, which expects a list of node handles and a list of parameter dictionaries. The rate of the inhibitory Poisson generator is set later. Note that we need not set parameters for the neuron and the spike detector, since they have satisfactory defaults.

```
nest.SetStatus(noise, [{"rate": n_ex * r_ex}, {"rate": n_in * r_in}])
nest.SetStatus(voltmeter, {"withgid": True, "withtime": True})
```

Fifth, the `iaf_psc_alpha`

is connected to the `spike_detector`

and the `voltmeter`

, as are the two Poisson generators to the neuron. The command `Connect`

has different variants. Plain `Connect`

just takes the handles of pre- and post-synaptic nodes and uses the default values for weight and delay. It can also be called with a list of weights, as in the connection of the noise below. Note that the connection direction for the `voltmeter`

is reversed compared to the `spike_detector`

, because it observes the neuron instead of receiving events from it. Thus, `Connect`

reflects the direction of signal flow in the simulation kernel rather than the physical process of inserting an electrode into the neuron. The latter semantics is presently not available in NEST.

```
nest.Connect(neuron, spikedetector)
nest.Connect(voltmeter, neuron)
nest.Connect(noise, neuron, syn_spec={'weight': [[epsc, ipsc]], 'delay': 1.0})
```

To determine the optimal rate of the neurons in the inhibitory population, the network is simulated several times for different values of the inhibitory rate while measuring the rate of the target neuron. This is done by calling `Simulate`

until the rate of the target neuron matches the rate of the neurons in the excitatory population with a certain accuracy. The algorithm is implemented in two steps:

First, the function `output_rate`

is defined to measure the firing rate of the target neuron for a given rate of the inhibitory neurons.

```
def output_rate(guess):
print("Inhibitory rate estimate: %5.2f Hz" % guess)
rate = float(abs(n_in * guess))
nest.SetStatus([noise[1]], "rate", rate)
nest.SetStatus(spikedetector, "n_events", 0)
nest.Simulate(t_sim)
out = nest.GetStatus(spikedetector, "n_events")[0] * 1000.0 / t_sim
print(" -> Neuron rate: %6.2f Hz (goal: %4.2f Hz)" % (out, r_ex))
return out
```

The function takes the firing rate of the inhibitory neurons as an argument. It scales the rate with the size of the inhibitory population and configures the inhibitory Poisson generator (`noise[1]`

) accordingly. Then, the spike counter of the `spike_detector`

is reset to zero. The network is simulated using `Simulate`

, which takes the desired simulation time in milliseconds and advances the network state by this amount of time. During simulation, the `spike_detector`

counts the spikes of the target neuron and the total number is read out at the end of the simulation period. The return value of `output_rate()`

is the firing rate of the target neuron in Hz.

Second, the scipy function `bisect`

is used to determine the optimal firing rate of the neurons of the inhibitory population.

```
in_rate = bisect(lambda x: output_rate(x) - r_ex, lower, upper, xtol=prec)
print("Optimal rate for the inhibitory population: %.2f Hz" % in_rate)
```

The function `bisect`

takes four arguments: first a function whose zero crossing is to be determined. Here, the firing rate of the target neuron should equal the firing rate of the neurons of the excitatory population. Thus we define an anonymous function (using `lambda`

) that returns the difference between the actual rate of the target neuron and the rate of the excitatory Poisson generator, given a rate for the inhibitory neurons. The next two arguments are the lower and upper bound of the interval in which to search for the zero crossing. The fourth argument of `bisect`

is the desired relative precision of the zero crossing.

Finally, we plot the target neuron's membrane potential as a function of time.

`nest.voltage_trace.from_device(voltmeter)`