Spiking Characteristics of Leaky Integrate-and-Fire Models

In this exercise, we will investigate the steady-state firing dynamics of LIF neurons under constant current injection (see Chapter 1, Section 3 for more information about LIF neurons). The resulting input-output response is graphically represented using the frequency-current (f-I) curve. The f-I curve serves multiple purposes in both, experimental and computational neuroscience. It provides insights into how neurons encode input current to output spikes.

The shape of the f-I curve also reflects the neuron’s type and its intrinsic properties. Specifically, type I neurons have continuous f-I curves whereas type II neurons have discontinuous f-I curves. This indicates that type I neurons can fire at low frequencies, while type II neurons cannot. For more information on neuron types, see Chapter 4, Section 4. You can also refer to Chapter 2, Section 2 for f-I curves of the biophysical model of Hodgkin and Huxley.

To realize this goal, we set up an experiment consisting of one neuron and inject a constant current onto its membrane. We will then investigate the effect of different neuron parameters on the shape of the f-I curve. Finally, we will determine the neuron type from the f-I curve and discuss models that yield different neuron types.

Experiment Setup

To prepare for the execution of experiments on BrainScaleS-2, we establish the connection to the system and import the required packages.

from _static.common.helpers import setup_hardware_client
setup_hardware_client()

%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
import quantities as pq

import pynn_brainscales.brainscales2 as pynn

Recording the f-I Curve

We deal here with a single neuron and inject a constant current onto its membrane. Neuron parameters in this exercise are in hardware units. To configure a constant current source, we use the following parameters in the neuron population definition: constant_current_enable (True) and constant_current_i_offset (range=0-1022). After running the experiment, we read-out the spikes and compute the firing rate. Finally, we visualize our results by plotting the firing rate versus the magnitude of injected current.

Exercises

  • Task 1: For ease of reproducibility, write a function fI_experiment that takes as arguments the neuron refractory period, leak conductance, array of injected currents, and runtime duration of each experiment. The rest of the neuron parameters can be hard-coded in the function. Your function should sweep over the current values and run an experiment for each value. In each experiment, record the spikes and compute the firing rate. You can reuse code from Parameters of a Leaky-Integrate-and-Fire Neuron Model. Your function should return the spike rate for plotting against the current range.

  • Task 2: Generate an f-I curve for a runtime duration of 1 ms, current range over (0-1000) with a step_size of 20, and refractory period of 10. Choose your neuron parameters such that you obtain a mean resting potential of 300, a membrane time constant of 10 us, and a threshold read of 350. You may use the values you determined in Parameters of a Leaky-Integrate-and-Fire Neuron Model. Observe the rheobase (current threshold for firing) and the slope of the f-I curve.

  • Task 3: Investigate the effect of the leak conductance on the f-I curve. Choose three values of the leak conductance with a difference of 100. Can you qualitatively explain the effects? Note, the range of the leak conductance is 0-1022.

  • Task 4: Investigate the effect of the refractory period on the f-I curve. Choose three values of the refractory period with minimal difference of 10. What is the maximum theoretical firing rate? Make sure the refractory period is consistent with the duration of experiment for high refractory periods. Note, the range of the refractory period is 10-1022.

  • Task 5: Examine the firing behavior of the neuron around the rheobase. Use your knowledge of the rheobase (current threshold) from task 2 to study the firing behavior of the neuron for a limited current range in the vicinity of the rheobase. For your f-I curve, sweep over a current range of (rheobase-20, rheobase+50) in a step-size of 1, a refractory period of 10, and an experiment duration of 1 ms. Comment on the type of the neuron following the LIF dynamics. Can you spot any imperfections in the plot? Which neuron models can result in different f-I curves?

  • Task 6: Finally, let’s discuss the significance of the f-I curve. What does the slope of the f-I curve tell about the encoding behavior? Can you forsee an advantage of f-I curves in computational neuroscience? Compare the f-I curve of the LIF model to the rectified linear unit (ReLU) activation function in machine learning.