Generating Nerve Impulses in Hodgkin-Huxley Model with R: A Comprehensive Modeling Approach (Part 2)

Generating Nerve Impulses in Hodgkin-Huxley Model with R: A Comprehensive Modeling Approach (Part 2)

Introduction:

Introduction:
In this part, we will discuss a numerical method for solving the partial differential equation (PDE) system that governs the propagation of action potential. To simulate the model, we will utilize the R-Packages deSolve and ReacTran. The underlying Hodgkin-Huxley model forms the basis for our simulation, which is based on the telegraph equations. Unlike standard models, we will not neglect the inductance factor in the plasma membrane of nerves. We will also take into account the ionic currents through the cylindrical membrane, using the Resistance-Inductance-Capacitance (RLC) electric circuit analogy.

Full Article: Generating Nerve Impulses in Hodgkin-Huxley Model with R: A Comprehensive Modeling Approach (Part 2)

Propagation of Action Potential in Neural Axons: A Numerical Method

Introduction

In this second part, we will present a numerical method for solving the Partial Differential Equation (PDE) system that describes the propagation of action potential. We will simulate the model using the R-Packages deSolve and ReacTran. The model is based on the Hodgkin-Huxley model, which is a telegraph equation-based model that takes into account the self conductance of the axon. This model considers the Resistance-Inductance-Capacitance (RLC) electric circuit analogue and includes ionic currents through the cylindrical membrane.

Mathematical Equation for Propagation of Action Potential

The mathematical equation that describes the propagation of action potential along a neural axon in space and time is given by:

𝜕²𝑉𝑚/𝜕𝑥² – 𝐿𝐶𝑎𝜕²𝑉𝑚/𝜕𝑡² = (2/𝑎)𝑅𝐶𝑎(𝜕𝑉𝑚/𝜕t) + (2/𝑎)𝐿(𝜕𝐼𝑖𝑜𝑛/𝜕t) + (2/𝑎)𝑅𝐼𝑖𝑜𝑛

In this equation:
– 𝑉𝑚 is the potential difference across the membrane (dependent variable, depends on 𝑥 and 𝑡).
– 𝑥 is the independent variable representing one dimension of three-dimensional space.
– 𝑡 is the independent variable representing time.
– 𝐿 is the axon-specific self-inductance.
– 𝑅 is the specific resistance of an axon.
– 𝐶𝑎 is the axon self-capacitance per unit area per unit length.
– 𝐼𝑖𝑜𝑛 is the sum of ion currents.
– 𝑎 is the axon radius.

To derive this equation, we used the RLC electric circuit analogue for the axon represented by the 𝑅𝐿𝐶 circuit. Note that neglecting the induction in the system yields a non-linear cable equation, which cannot be solved analytically.

Hodgkin-Huxley Model and Ion Current

In the Hodgkin-Huxley model, the ion current (𝐼𝑖𝑜𝑛) is defined as the sum of the potassium current (𝐼𝐾), sodium current (𝐼𝑁𝑎), and the leakage current (𝐼𝐿), as given by the equation:

𝐼𝑖𝑜𝑛 = 𝐼𝐾 + 𝐼𝑁𝑎 + 𝐼𝐿 = 𝑔𝐾(𝑉𝑚 – 𝑉𝐾) + 𝑔𝑁𝑎(𝑉𝑚 – 𝑉𝑁𝑎) + 𝑔𝐿(𝑉𝑚 – 𝑉𝐿)

In this equation:
– 𝑔𝐾, 𝑔𝑁𝑎, and 𝑔𝐿 are conductances of potassium, sodium, and leakage, respectively.
– 𝑉𝑚, 𝑉𝐾, 𝑉𝑁𝑎, and 𝑉𝐿 are reversal potentials for potassium, sodium, and leakage, respectively.

The Simulation

To solve the equation described above, we

Summary: Generating Nerve Impulses in Hodgkin-Huxley Model with R: A Comprehensive Modeling Approach (Part 2)

In this second part, we present a numerical method for solving the PDE system that describes the propagation of an action potential. We use the R-Packages deSolve and ReacTran to simulate the model. The Hodgkin-Huxley model is used, which is based on the telegraph equations and includes the self conductance of the axon. The mathematical equation for the propagation of the action potential is given, and we derive the final nerve propagation equation for the Hodgkin & Huxley model. We then provide the code for the numeric solution using the R-Packages deSolve and ReacTran. Finally, we plot the results and show the time evolution of the membrane potential of the H&H neuron at various distances along the axon.

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