Li, Rinzel, 1994
Model Status
This is the original unchecked version of the model imported from the previous CellML model repository, 24Jan2006.
Model Structure
Cytosolic calcium ([Ca^{2+}]_{i}) shows complex spatiotemporal oscillatory profiles in a large number of cell types. A number of mathematical models have been published that attempt to model the temporal [Ca^{2+}]_{i} oscillations. Some, such as the Keizer and De Young model (see The De YoungKeizer model, 1992) assume that IP_{3} is necessarily oscillatory. They constructed a model of the IP_{3} receptor/channel by assuming that three equivalent and independent subunits are involved in Ca^{2+} conduction. Each subunit has three binding sites: one for IP_{3}, one for Ca^{2+} activation, and one for Ca^{2+} inactivation. Thus, each subunit may exist in eight states with transitions governed by secondorder (association) and firstorder (dissociation) rate constants (see below). All three subunits must be in the state S_{110} (one IP3 and one activating Ca^{2+} bound) for the channel to be open and conducting.
In 1994, YueXian Li and John Rinzel analysed the ninevariable De YoungKeizer model (1992) and reduced it to a twovariable system. Their reduced system is analogous in form to the classic HodgkinHuxley equations (see The HodgkinHuxley Squid Axon Model, 1952) for plasma membrane electrical excitability. [Ca^{2+}]_{i} dynamics in this model thus involve endoplasmic reticulum (ER) membraneassociated excitability. Assuming further that the binging of IP_{3} does not depend on Ca^{2+} occupancy at the inactivation site, Li and Rinzel obtain a "minimal" model which still retains the ability to reproduce experimental observations.
The complete original paper reference is cited below:
Equations for InsP_{3} Receptormediated [Ca^{2+}]_{i} Oscillations Derived from a Detailed Kinetic Model: A HodgkinHuxley Like Formalism, YueXian Li and John Rinzel, 1994, Journal of Theoretical Biology , 166, 461473. (A PDF of the article is available to subscribers on the IDEAL website.) PubMed ID: 8176949
In addition to the original version of the Li and Rinzel mathematical model, the system described in equation nine of the paper has also been implemented in CellML as a separate model (see for the downloads of both models in various formats). This is a set of three coupled differential equations, namely:

One equation for describing the calcium dynamics;

One equation which represents the gateshutting variable h; and

One equation which describes the influx of extracellular calcium into the cell cytosol through the plasma membrane
The advantage of reducing the model to this set of three equations is that it more simple, yet it still retains the ability to explain oscillatory behaviours caused by the calcium dynamics. All the model parameters have been set according to those which are provided in Figure 5(a) of the paper.
The raw CellML descriptions of the Li and Rinzel 1994 model can be downloaded in various formats as described in .
A schematic diagram of the kinetics of an IP_{3} receptor/channel subunit. 