Computational Study of Neural Action Potentials Using the Hodgkin-Huxley Model

Oct 2024 - Present

Introduction

My project plans to investigate the dynamics of nerve impulses using simulations of the Hodgkin-Huxley model. This model describes how action potentials are initiated and then propagated in neurons by the opening and closing of ion channels. The model is a system of four coupled differential equations that describe the time evolution of the membrane potential and the probabilities of the ion channels being open. In this project, we plan to explore both steady-state behavior and dynamic responses to various stimuli.

Objectives

  • Implement the Hodgkin-Huxley differential equations in Python using numerical integration techniques
  • Study the steady-state behavior of ion channel gates
  • Analyze the membrane's response to different current stimuli
  • Investigate threshold phenomena and spike train dynamics

Technical Approach

We will utilize Python's scientific computing libraries (NumPy, SciPy, Matplotlib) to implement:

  • Steady-state analysis of gate variables (n, m, h) as functions of membrane potential
  • Visualization tools for steady-state and time-dependent behavior
  • Parameter sweep to investigate the effects of varying parameters on the system

Steady-State Analysis

Plot n, m, and h gates as functions of membrane potential V. Then, analyze operating characteristics of each gate type.

Membrane Dynamics

We will simulate and plot V(t) using the Euler-Cromer method under zero external current conditions, serving as a baseline for understanding the system's behavior.

Single Spike Analysis

We will study the system's response to current pulses by implementing a 1 ms stimulus with a 7 µA/cm² current. This will determine the threshold current required for generating action potentials.

Spike Train Dynamics

We will examine how constant current inputs generate spike trains in action potentials. We will vary the stimulus amplitude to determine the threshold for repetitive firing and analyze how the spike intervals change with varying stimulus amplitude.

Step Response

Finally, we will implement a two-phase current with an initial current I₁ applied for 20 ms followed by a step to I₂ = I₁ + δI. We will examine four cases:

  • [I₁ = 2 µA, δI = 1.5 µA]
  • [I₁ = 2 µA, δI = 5.0 µA]
  • [I₁ = 7 µA, δI = 1.0 µA]
  • [I₁ = 7 µA, δI = 4.0 µA]

We will then classify different response types and map the behavior across parameter space.