8.1 Dynamics of the firing rate of single compartmental cells

Introduction

We seek to understand how to reduce complicated neuronal models into simplified versions that still capture essential features of the neuron. It has been observed through the detailed Hodgkin-Huxley model that the firing rate of a cell depends on current input. This module examines the dynamics of an active single-compartment cell and verifies the existence of a linear relationship between the firing rate and the injected current. It will reproduce the results found in Section 2.1 of “Rate Models for Conductance-Based Cortical Neuronal Networks," by Shriki et al [1]. In particular, it verifies Figures 1 and 2 of the Shriki paper through MATLAB simulations .

Dynamics of a single-compartment cell

The Hodgkin-Huxley model of a single-compartment cell obeys

with the following definitions:

If $I^{app}$ were kept constant in time and sufficiently large, then the cell will fire at a rate $f$ . There is a simple relationship between $f$ and $I^{app}$ , namely, $f=F(I^{app},g_L),$ called the f-I curve . It was noted in the Shriki paper that in many cortical neurons, the f-I curve is approximately linear if $I$ is above threshold. Our goal is to reproduce the f-I curves shown in Figures 1 and 2 of Shriki et al. through simulation of the Hodgkin-Huxley model on a single-compartment cell.

Active ionic currents,

In order to generate f-I curves, we must stimulate the cell with varying current input (with the current kept constant across each simulation), and count the number of times it fired each time. But first, we need to understand the active ionic currents, an integral part of .

There are three types of active currents, Na, K, and A. The first two currents are gated by sodium and potassium channels respectively. They obey the following equations:

Here, $E_{Na}$ and $E_K$ are the reversal potentials of Na and K respectively. The A-current is a slow current gated by a type of potassium channel. It obeys

The A-current is introduced to linearize the f-I curve, as we shall later see in Figure 1.

Adding these currents, we get