Mathematics for Computational Neuroscience

What is Membrane Potential?

The membrane potential ($V_m$) is the voltage difference across a neuron’s membrane, defined as $V_m = V_i - V_e$, where $V_i$ is the intracellular potential and $V_e$ is the extracellular potential.

• It’s measured in millivolts (mV), with the inside of the cell typically being negative relative to the outside (e.g., ~-70 mV at rest).
• This potential arises from the asymmetric distribution of ions like Sodium (Na+), Potassium (K+), and Chloride (Cl-) across the membrane, a state maintained by active ion pumps.

Electrical Properties of the Membrane

To model the cell membrane, we can abstract its physical properties into an equivalent electrical circuit:

1. Capacitor (insulator): The lipid bilayer separates charges, acting like a capacitor that can store charge. This is crucial for generating action potentials.
2. Resistors (conductors): Ion channels allow ions to move across the membrane, functioning as resistors.
3. Voltage Source (Battery): The concentration gradients of ions create an electrochemical potential, which acts as a battery.

This leads to the simplest model of a cell membrane patch as an RC circuit:

---
title: Simplest Cellular Membrane Circuit
---
graph TD
    subgraph "Inside (V_i)"
        A --- C
    end
    subgraph "Outside (V_e)"
        B --- D
    end

    subgraph "Membrane"
        C -- C_m --> D
        C -- R_m --> E
        E -- E_L --> D
    end

    A -- V_m --- B

    linkStyle 3 stroke-width:0px,fill:none;

Governing Equations

• Nernst equation: Calculates the equilibrium potential for a single ion species.
• Goldman equation: An extension of the Nernst equation for multiple ion species, used to calculate the resting membrane potential.