A Hodgkin-Huxley Model Exhibiting Bursting Oscillations

Model Status

This is the original unchecked version of the model imported from the previous CellML model repository, 24-Jan-2006.

Model Structure

Like many other types of excitable cell, pituitary corticotrophs display bursting behaviour. These electrical bursts consist of Ca²⁺-carrying action potentials, alternating with silent phases of repolarisation. Spikes representing sudden membrane depolarisation are often accompanied by oscillations in cytosolic Ca²⁺ concentration ([Ca²⁺]_i), and also in corticotrophs, they are followed by small oscillations in the membrane potential.

The mechanisms underlying this bursting behaviour have been the subject of several studies. In this study described here, Paul R. Shorten and David J.N. Wall develop a Hodgkin-Huxley type mathematical model (for the original model description see The Hodgkin-Huxley Squid Axon Model, 1952) of pituitary corticotrophs. The model includes the major plasma membrane ionic currents and the associated intracellular Ca²⁺ dynamics (see the figure below). The bursting process is driven by the slow modulation of the endoplasmic reticulum (ER) Ca²⁺ concentration ([Ca²⁺]_er), giving rise to a slow component in [Ca²⁺]_i. This then gives rise to the electrical bursting via a Ca²⁺-activated potassium current (I_K-Ca ). Model simulations showed that bursting frequency is dependent on the ER Ca²⁺ storage capacity, the Ca²⁺ transport mechanisms, and the activation of a Ca²⁺-activated K⁺ current.

Excitable cells display a wide range of different types of bursting behaviours. Shorten and Wall discovered that their model exhibits a novel form of bursting due to bistability between two stable oscillatory solutions. Due to the bifurcations involved, this type of bursting is called 'fold cycle/fold cycle' bursting. In their paper, which is fully referenced below, Shorten and Wall aim to highlight interesting modes of bursting in Hodgkin-Huxley type mathematical models and elucidate their underlying mechanisms. They especially emphasise how small parameter changes can cause large changes in the model behaviour.

A Hodgkin-Huxley Model Exhibiting Bursting Oscillations, Paul R. Shorten and David J.N. Wall, 2000, Bulletin of Mathematical Biology , 62, 695-715. (A PDF version of the article is available for Journal Members on the Bulletin of Mathematical Biology journal website.) PubMed ID: 10938629

Schematic diagram of a pituitary corticotroph cell showing the transmembrane ionic currents and the intracellular Ca²⁺ dynamics captured by the mathematical model. Arrows indicate ionic channels and pumps. I_Ca-L represents an L-type Ca²⁺ current responsible for most of the Ca²⁺ influx during an action potential. I_Ca-T is a T-type voltage-sensitive Ca²⁺ current. A voltage-sensitive K⁺ current, I_K-DR , is mainly responsible for action potential repolarisation. A Ca²⁺-activated K⁺ current, I_K-Ca , is essential for bursting behaviour. The remaining leak current, I_leak , represents all other ionic fluxes across the plasma membrane which are not specifically described by the model. J_eff and J_up are the ER and plasma membrane Ca²⁺-ATPase pumps, and J_rel represents the ER Ca²⁺ leakage term. Within the ER and the cytosol, significant portions of Ca²⁺ are bound to buffers, denoted by B_er and B_c respectively.