# 2. Leaky-integrate-and-fire model¶

**Book chapters**

See Chapter 1 Section 3 on general information about leaky-integrate-and-fire models.

**Python classes**

The `leaky_integrate_and_fire.LIF`

module contains all code required for this exercise.
At the beginning of your exercise solutions, import the contained functions by running

```
from neurodynex.leaky_integrate_and_fire.LIF import *
```

You can then simply run the exercise functions by executing

```
LIF_Step() # example Step
LIF_Sinus() # example Sinus
```

## 2.1. Exercise¶

Use the function `LIF_Step()`

to simulate a Leaky Integrate-And-Fire
neuron stimulated by a current step of a given amplitude. The goal of
this exercise is to modify the provided python functions and use the
`numpy`

and `matplotlib`

packages to answer the following questions.

### 2.1.1. Question¶

What is the minimum current step amplitude `I_amp`

to elicit a spike
with model parameters as given in `LIF_Step()`

? Plot the injected
values of current step amplitude against the frequency of the spiking
response (you can use the inter-spike interval to calculate this – let
the frequency be \(0Hz\) if the model does not spike, or emits only
a single spike) during a \(500ms\) current step.

## 2.2. Exercise¶

Use the function `LIF_Sinus()`

to simulate a Leaky Integrate-And-Fire
neuron stimulated by a sinusoidal current of a given frequency. The goal
of this exercise is to modify the provided python functions and use the
`numpy`

and `matplotlib`

packages to plot the amplitude and frequency
gain and phase of the voltage oscillations as a function of the input
current frequency.

### 2.2.1. Question¶

For input frequencies between \(0.1kHz\) and \(1.kHz\), plot the
input frequency against the resulting *amplitude of subthreshold
oscillations* of the membrane potential. If your neuron emits spikes at
high stimulation frequencies, decrease the amplitude of the input
current.

### 2.2.2. Question¶

For input frequencies between \(0.1kHz\) and \(1.kHz\), plot the
input frequency against the resulting *frequency and phase of
subthreshold oscillations* of the membrane potential. Again, keep your
input amplitude in a regime, where the neuron does not fire action
potentials.