The mathematical approach; Ionic movement producing electrical signals along the axon of a neuron

Our nervous system consists of billions of neurons, designed to carry messages through an electrochemical process with approximately 100 billion neurons located only in the brain. As viewed from the picture below, a neuron cell consists of a cell body, dendrites, a nucleus, an axon covered in Schwann cells as well as oligodendrocytes known as myelin sheaths, with node of ranvier, axon terminal and synapses.

A neuron is voltage dependent. The resting potential of a neuron is negative, due to the inside of the axon being filled with negative ions, including potassium. A stimuli i.e. on the surface of the skin gets detected by the voltage sensors on the neurons. This leads to the activation of the Sodium channels to open and an electrochemical gradient of sodium ions to enter the axon through their specific ion channels, leading to the depolarisation of the neuron and the start of an action potential.

As a result, potassium channels open, leading to the exiting of the potassium ions from the axon. Furthermore, as more potassium ions leave the cell, the system undergoes Repolarization also leading to Hyperpolarization at which voltage gated potassium channels open Potassium ions continue leaving the axon until after the potential has passed the resting potential, known as the refractory period. The resting potential is later restored and in summation, the neuron propagates the electrical signal, making an action potential an all or nothing action.

A mathematical approach to the electrical potential generated across the membrane at electrochemical equilibrium, the equilibrium potential can be predicted using the Nernst Equation. However, this particular equation only predicts the potential, if only one ion is involved;

The R is the gas constant, T the absolute temperature, z the valence charge of the ion, F the Faraday constant and the X the amount of moles of the ion on each side of the membrane.

With more than one ion involved, the Goldman equation can be used;

The V is the voltage across the membrane and Px indicates the permeability of each ion of interest.

As seen in the graph above, the permeability of Pk is higher at first, since more potassium is found inside the neuron in contrast to the sodium. However, as the neuron depolarises, PNa increases and the complete action potential system is then mathematically proven through this equation.

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Image 3-7: The book "Neuroscience 6th edition - Dale Purves"

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