- Author:
- pmr2.import <nobody@models.cellml.org>
- Date:
- 2006-09-04 03:08:16+12:00
- Desc:
- committing version01 of luo_rudy_1991
- Permanent Source URI:
- https://staging.physiomeproject.org/workspace/luo_rudy_1991/rawfile/dc5bf1c27c99098f98f4dc5c4d36e10553a53764/luo_rudy_1991.cellml
<?xml version='1.0' encoding='utf-8'?>
<!-- FILE : luo_rudy_I_model_1991.xml
CREATED : September 2001
LAST MODIFIED : 29th April 2005
AUTHOR : Catherine Lloyd
Department of Engineering Science
The University of Auckland
MODEL STATUS : This model conforms to the CellML 1.0 Specification released on
10th August 2001, and the CellML Metadata 1.0 Specification released on the 16th
January, 2002.
DESCRIPTION : This file contains a CellML description of the mammalian
ventricular action potential based on the Luo-Rudy I model, 1991. This model is
a significant update to the Beeler-Reuter mammalian ventricular model.
CHANGES:
19/10/2001 - CML - Removed document type definition as this is declared as
optional according to the W3C recommendation.
24/10/2001 - CML - Made changes to some of the metadata, bringing them up to
date with the most recent working draft (26th September) of
the Metadata specification.
07/12/2001 - CML - Changed equations after checking with mathml validator.
04/01/2002 - CML - Added an ionic_concentrations and an
intracellular_calcium_concentrations components and altered
the public interfaces and connections appropriately.
21/01/2002 - AAC - Updated metadata to conform to the 16/1/02 CellML Metadata
1.0 Specification.
18/02/2002 - AAC - Corrected the beta_K1_calculation eqn.
20/02/2002 - CML - Corrected the Cai differential eqn. 06/05/2002 - CML - Added some initial values.
18/07/2002 - CML - Added more metadata.
09/04/2003 - AAC - Added publication date information.
05/06/2003 - CML - Fixed MathML in a few components.
31/07/2003 - CML - Corrected equations.
29/04/2005 - PJV - Corrected value in equation.
--><model xmlns="http://www.cellml.org/cellml/1.0#" xmlns:cmeta="http://www.cellml.org/metadata/1.0#" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:bqs="http://www.cellml.org/bqs/1.0#" xmlns:cellml="http://www.cellml.org/cellml/1.0#" xmlns:dcterms="http://purl.org/dc/terms/" xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#" cmeta:id="luo_rudy_1991_version01" name="luo_rudy_1991_version01">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
<articleinfo>
<title>Luo-Rudy Mammalian Ventricular Model I 1991</title>
<author>
<firstname>Catherine</firstname>
<surname>Lloyd</surname>
<affiliation>
<shortaffil>Bioengineering Institute, University of Auckland</shortaffil>
</affiliation>
</author>
</articleinfo>
<section id="sec_status">
<title>Model Status</title>
<para>
This is the original unchecked version of the model imported from the previous
CellML model repository, 24-Jan-2006.
</para>
</section>
<sect1 id="sec_structure">
<title>Model Structure</title>
<para>
In 1991, Ching-hsing Luo and Yoram Rudy published a mathematical model of the ventricular cardiac action potential. This original model is the first of the two Luo-Rudy models, and it has subsequently come to be known as the Luo-Rudy I model. It is a significant update of the <ulink url="${HTML_EXMPL_BR_MODEL}">Beeler-Reuter mammalian ventricular model (1977)</ulink> (see <xref linkend="fig_cell_diagram"/> below), and like the the Beeler-Reuter model, the Luo-Rudy I model uses <ulink url="${HTML_EXMPL_HHSA_INTRO}">Hodgkin-Huxley type</ulink> equations to calculate ionic currents.
</para>
<para>
The complete original paper reference is cited below:
</para>
<para>
<ulink url="http://circres.ahajournals.org/cgi/content/abstract/68/6/1501">A Model of the Ventricular Cardiac Action Potential - Depolarisation, Repolarisation and Their Interaction</ulink>,, Ching-hsing Luo and Yoram Rudy, 1991 <ulink url="http://circres.ahajournals.org/">
<emphasis>Circulation Research</emphasis>
</ulink>, 68, 1501-1526. <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=1709839&dopt=Abstract">PubMed ID: 1709839</ulink>
</para>
<para>
The raw CellML description of the Luo-Rudy I model can be downloaded in various formats as described in <xref linkend="sec_download_this_model"/>. For an example of a more complete documentation for an electrophysiological model, see <ulink url="${HTML_EXMPL_HHSA_INTRO}">The Hodgkin-Huxley Squid Axon Model, 1952</ulink>.
</para>
<informalfigure float="0" id="fig_cell_diagram">
<mediaobject>
<imageobject>
<objectinfo>
<title>cell diagram of the LR-I model showing ionic currents, pumps and exchangers within the sarcolemma</title>
</objectinfo>
<imagedata fileref="luo_rudy_i_1991.png"/>
</imageobject>
</mediaobject>
<caption>A schematic diagram describing the current flows across the cell membrane that are captured in the LR-I model.</caption>
</informalfigure>
<informalfigure float="0" id="fig_cellml_rendering">
<mediaobject>
<imageobject>
<objectinfo>
<title>the cellml rendering of the Luo-Rudy I model</title>
</objectinfo>
<imagedata fileref="cellml_rendering.gif"/>
</imageobject>
</mediaobject>
<caption>The network defined in the CellML description of the Luo-Rudy I model. A key describing the significance of the shapes of the components and the colours of the connections between them is in the <ulink url="${HTML_EXMPL_GRAPHICAL_NOTATION}">notation guide</ulink>.</caption>
</informalfigure>
<para>
The membrane physically contains the currents, as indicated by the blue arrows in <xref linkend="fig_cellml_rendering"/>. The currents act independently and are not connected to each other. Four of the channels encapsulate <emphasis>and</emphasis> contain further components which represent activation and inactivation gates. The addition of an encapsulation relationship informs modellers and processing software that the gates are important parts of the current model. It also prevents any other components that aren't also encapsulated by the parent component from connecting to its gates, effectively hiding them from the rest of the model.
</para>
<para>
The breakdown of the model into components and the definition of encapsulation and containment relationships between them is somewhat arbitrary. When considering how a model should be broken into components, modellers are encouraged to consider which parts of a model might be re-used and how the physiological elements of the system being modelled are naturally bounded. Containment relationships should be used to provide simple rendering information for processing software (ideally, this will correspond to the layout of the physical system), and encapsulation should be used to group sets of components into sub-models.
</para>
</sect1>
</article>
</documentation>
<!--
Below, we define some additional units for association with variables and
constants within the model. The identifiers are fairly self-explanatory.
-->
<units name="millisecond">
<unit units="second" prefix="milli"/>
</units>
<units name="per_millisecond">
<unit units="second" prefix="milli" exponent="-1"/>
</units>
<units name="millivolt">
<unit units="volt" prefix="milli"/>
</units>
<units name="per_millivolt">
<unit units="volt" prefix="milli" exponent="-1"/>
</units>
<units name="per_millivolt_millisecond">
<unit units="millivolt" exponent="-1"/>
<unit units="millisecond" exponent="-1"/>
</units>
<units name="milliS_per_cm2">
<unit units="siemens" prefix="milli"/>
<unit units="metre" prefix="centi" exponent="-2"/>
</units>
<units name="microF_per_cm2">
<unit units="farad" prefix="micro"/>
<unit units="metre" prefix="centi" exponent="-2"/>
</units>
<units name="microA_per_cm2">
<unit units="ampere" prefix="micro"/>
<unit units="metre" prefix="centi" exponent="-2"/>
</units>
<units name="concentration_units">
<unit units="mole" prefix="milli"/>
<unit units="litre" exponent="-1"/>
</units>
<units name="joule_per_kilomole_kelvin">
<unit units="joule"/>
<unit units="mole" prefix="kilo" exponent="-1"/>
<unit units="kelvin" exponent="-1"/>
</units>
<units name="coulomb_per_mole">
<unit units="coulomb" exponent="-1"/>
<unit units="mole"/>
</units>
<!--
The "environment" component is used to declare variables that are used by
all or most of the other components, in this case just "time".
-->
<component name="environment">
<variable units="millisecond" public_interface="out" name="time"/>
</component>
<!--
The "membrane" component is really the `root' node of our model.
It defines the action potential variable "V" among other things.
-->
<component name="membrane">
<!-- These variables are defined here and used in other components. -->
<variable units="millivolt" public_interface="out" name="V" initial_value="-60.0"/>
<variable units="joule_per_kilomole_kelvin" public_interface="out" name="R" initial_value="8314"/>
<variable units="kelvin" public_interface="out" name="T" initial_value="310.0"/>
<variable units="coulomb_per_mole" public_interface="out" name="F" initial_value="96484.6"/>
<!-- These variables are defined here and only used internally. -->
<variable units="microF_per_cm2" name="C" initial_value="1.0"/>
<variable units="microA_per_cm2" name="I_stim"/>
<!-- These variables are imported from other components. -->
<variable units="millisecond" public_interface="in" name="time"/>
<variable units="microA_per_cm2" public_interface="in" name="i_Na"/>
<variable units="microA_per_cm2" public_interface="in" name="i_si"/>
<variable units="microA_per_cm2" public_interface="in" name="i_K"/>
<variable units="microA_per_cm2" public_interface="in" name="i_K1"/>
<variable units="microA_per_cm2" public_interface="in" name="i_Kp"/>
<variable units="microA_per_cm2" public_interface="in" name="i_b"/>
<!--
The membrane voltage (V) is calculated as an ordinary
differential equation in terms of the currents.
-->
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="membrane_voltage_diff_eq">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> V </ci>
</apply>
<apply>
<times/>
<apply>
<minus/>
<apply>
<divide/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<ci> C </ci>
</apply>
</apply>
<apply>
<plus/>
<ci> I_stim </ci>
<ci> i_Na </ci>
<ci> i_si </ci>
<ci> i_K </ci>
<ci> i_K1 </ci>
<ci> i_Kp </ci>
<ci> i_b </ci>
</apply>
</apply>
</apply>
</math>
</component>
<!--
The "fast_sodium_current" component contains the differential equations
governing the influx of sodium ions through the cell surface membrane
into the cell. Note that no initial values are needed on many
variables as they are all directly dependent on membrane voltage.
-->
<component name="fast_sodium_current">
<!-- This variable is defined here and used in other components. -->
<variable units="microA_per_cm2" public_interface="out" name="i_Na"/>
<!-- These variables are defined here and only used internally. -->
<variable units="milliS_per_cm2" name="g_Na" initial_value="23.0"/>
<variable units="millivolt" name="E_Na" initial_value="54.4"/>
<!--
Time is imported from the "environment", and membrane potential is
imported from the "membrane" component. These variables are used in the
"sodium_current" parent component, which also acts as an interface,
passing the variables to its encapsulated gate components.
-->
<variable units="millisecond" public_interface="in" private_interface="out" name="time"/>
<variable units="millivolt" public_interface="in" private_interface="out" name="V"/>
<!-- These variables are imported from encapsulated components. -->
<variable units="dimensionless" private_interface="in" name="m"/>
<variable units="dimensionless" private_interface="in" name="h"/>
<variable units="dimensionless" private_interface="in" name="j"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<!--
The following equation calculates the sodium current in terms
of the conductance, the membrane voltage, and the gate variables.
-->
<apply id="i_Na_calculation">
<eq/>
<ci> i_Na </ci>
<apply>
<times/>
<ci> g_Na </ci>
<apply>
<power/>
<ci> m </ci>
<cn cellml:units="dimensionless"> 3.0 </cn>
</apply>
<ci> h </ci>
<ci> j </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> E_Na </ci>
</apply>
</apply>
</apply>
</math>
</component>
<!--
The "fast_sodium_current_m_gate" is the m gate encapsulated inside the fast
sodium current.
-->
<component name="fast_sodium_current_m_gate">
<!-- This variable is defined here and used in other components. -->
<variable units="dimensionless" public_interface="out" name="m"/>
<!-- These variables are defined here and only used internally. -->
<variable units="per_millisecond" name="alpha_m"/>
<variable units="per_millisecond" name="beta_m"/>
<!--
These variables are imported from the "environment" and the "membrane" via
the "fast_sodium_current" component.
-->
<variable units="millivolt" public_interface="in" name="V"/>
<variable units="millisecond" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<!--
The rate constants on the m, h and j gates are functions
of membrane voltage.
-->
<apply id="alpha_m_calculation">
<eq/>
<ci> alpha_m </ci>
<apply>
<divide/>
<apply>
<times/>
<cn cellml:units="per_millivolt_millisecond"> 0.32 </cn>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 47.13 </cn>
</apply>
</apply>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> -0.1 </cn>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 47.13 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply id="beta_m_calculation">
<eq/>
<ci> beta_m </ci>
<apply>
<times/>
<cn cellml:units="per_millisecond"> 0.08 </cn>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<ci> V </ci>
</apply>
<cn cellml:units="millivolt"> 11.0 </cn>
</apply>
</apply>
</apply>
</apply>
<apply id="dm_dt">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> m </ci>
</apply>
<apply>
<minus/>
<apply>
<times/>
<ci> alpha_m </ci>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<ci> m </ci>
</apply>
</apply>
<apply>
<times/>
<ci> beta_m </ci>
<ci> m </ci>
</apply>
</apply>
</apply>
</math>
</component>
<!--
The "fast_sodium_current_h_gate" component is the h gate encapsulated in
the fast sodium current.
-->
<component name="fast_sodium_current_h_gate">
<!-- This variable is defined here and used in other components. -->
<variable units="dimensionless" public_interface="out" name="h"/>
<!-- These variables are defined here and only used internally. -->
<variable units="per_millisecond" name="alpha_h"/>
<variable units="per_millisecond" name="beta_h"/>
<!--
These variables are imported from the "environment" and the "membrane" via
the "fast_sodium_current" component.
-->
<variable units="millivolt" public_interface="in" name="V"/>
<variable units="millisecond" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<!--
The equations for the rate constant coefficients of the inactivation
gates h and j are dependent on the value of the membrane potential.
-->
<apply id="alpha_h_calculation">
<eq/>
<ci> alpha_h </ci>
<piecewise>
<piece>
<apply>
<times/>
<cn cellml:units="per_millisecond"> 0.135 </cn>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<plus/>
<cn cellml:units="millivolt"> 80.0 </cn>
<ci> V </ci>
</apply>
<cn cellml:units="millivolt"> -6.8 </cn>
</apply>
</apply>
</apply>
<apply>
<lt/>
<ci> V </ci>
<cn cellml:units="millivolt"> -40.0 </cn>
</apply>
</piece>
<otherwise>
<cn cellml:units="per_millisecond"> 0.0 </cn>
</otherwise>
</piecewise>
</apply>
<apply id="beta_h_calculation">
<eq/>
<ci> beta_h </ci>
<piecewise>
<piece>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="per_millisecond"> 3.56 </cn>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="millivolt"> 0.079 </cn>
<ci> V </ci>
</apply>
</apply>
</apply>
<apply>
<times/>
<cn cellml:units="per_millisecond"> 310000.0 </cn>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> 0.35 </cn>
<ci> V </ci>
</apply>
</apply>
</apply>
</apply>
<apply>
<lt/>
<ci> V </ci>
<cn cellml:units="millivolt"> -40.0 </cn>
</apply>
</piece>
<otherwise>
<apply>
<divide/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<times/>
<cn cellml:units="millisecond"> 0.13 </cn>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 10.66 </cn>
</apply>
<cn cellml:units="millivolt"> -11.1 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</otherwise>
</piecewise>
</apply>
<apply id="dh_dt">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> h </ci>
</apply>
<apply>
<minus/>
<apply>
<times/>
<ci> alpha_h </ci>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<ci> h </ci>
</apply>
</apply>
<apply>
<times/>
<ci> beta_h </ci>
<ci> h </ci>
</apply>
</apply>
</apply>
</math>
</component>
<!--
The "fast_sodium_current_j_gate" component is the j gate encapsulated in
the fast sodium current.
-->
<component name="fast_sodium_current_j_gate">
<!-- This variable is defined here and used in other components. -->
<variable units="dimensionless" public_interface="out" name="j"/>
<!-- These variables are defined here and only used internally. -->
<variable units="per_millisecond" name="alpha_j"/>
<variable units="per_millisecond" name="beta_j"/>
<!--
These variables are imported from the "environment" and the "membrane" via
the "fast_sodium_current" component.
-->
<variable units="millivolt" public_interface="in" name="V"/>
<variable units="millisecond" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="alpha_j_calculation">
<eq/>
<ci> alpha_j </ci>
<piecewise>
<piece>
<apply>
<times/>
<apply>
<minus/>
<apply>
<times/>
<cn cellml:units="per_millivolt_millisecond"> -127140.0
</cn>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> 0.2444 </cn>
<ci> V </ci>
</apply>
</apply>
</apply>
<apply>
<times/>
<cn cellml:units="per_millivolt_millisecond"> 0.00003474 </cn>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> -0.04391 </cn>
<ci> V </ci>
</apply>
</apply>
</apply>
</apply>
<apply>
<divide/>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 37.78 </cn>
</apply>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> 0.311 </cn>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 79.23 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply>
<lt/>
<ci> V </ci>
<cn cellml:units="millivolt"> -40.0 </cn>
</apply>
</piece>
<otherwise>
<cn cellml:units="per_millisecond"> 0.0 </cn>
</otherwise>
</piecewise>
</apply>
<apply id="beta_j_calculation">
<eq/>
<ci> beta_j </ci>
<piecewise>
<piece>
<apply>
<times/>
<cn cellml:units="per_millisecond"> 0.1212 </cn>
<apply>
<divide/>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> -0.01052 </cn>
<ci> V </ci>
</apply>
</apply>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> -0.1378 </cn>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 40.14 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply>
<lt/>
<ci> V </ci>
<cn cellml:units="millivolt"> -40.0 </cn>
</apply>
</piece>
<otherwise>
<apply>
<times/>
<cn cellml:units="per_millisecond"> 0.3 </cn>
<apply>
<divide/>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> -0.0000002535 </cn>
<ci> V </ci>
</apply>
</apply>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> -0.1 </cn>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 32.0 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
</otherwise>
</piecewise>
</apply>
<apply id="dj_dt">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> j </ci>
</apply>
<apply>
<minus/>
<apply>
<times/>
<ci> alpha_j </ci>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<ci> j </ci>
</apply>
</apply>
<apply>
<times/>
<ci> beta_j </ci>
<ci> j </ci>
</apply>
</apply>
</apply>
</math>
</component>
<!--
The "slow_inward_current " component contains the differential
equations governing the transient inward ionic current.It has two
encapsulated gates; an activation d gate and an inactivation f gate.
-->
<component name="slow_inward_current">
<!-- This variable is defined here and used in other components. -->
<variable units="microA_per_cm2" public_interface="out" name="i_si"/>
<!-- These variables are defined here and only used internally. -->
<variable units="millivolt" name="E_si"/>
<!--
Time is imported from the "environment", and membrane potential is
imported from the "membrane" component. These variables are used in the
"slow_inward_current" parent component, which also acts as an interface,
passing the variables to its encapsulated gate component.
-->
<variable units="millisecond" public_interface="in" private_interface="out" name="time"/>
<variable units="millivolt" public_interface="in" private_interface="out" name="V"/>
<variable units="concentration_units" public_interface="in" name="Cai"/>
<!-- These variables are imported from encapsulated components. -->
<variable units="dimensionless" private_interface="in" name="d"/>
<variable units="dimensionless" private_interface="in" name="f"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<!--
The following equation determines the reversal potential of the
slow inward current.
-->
<apply id="E_si_calculation">
<eq/>
<ci> E_si </ci>
<apply>
<minus/>
<cn cellml:units="millivolt"> 7.7 </cn>
<apply>
<times/>
<cn cellml:units="millivolt"> 13.0287 </cn>
<apply>
<ln/>
<ci> Cai </ci>
</apply>
</apply>
</apply>
</apply>
<!--
The following equation calculates the slow inward current in terms of
the conductance, the membrane voltage and the gate variables.
-->
<apply id="i_si_calculation">
<eq/>
<ci> i_si </ci>
<apply>
<times/>
<cn cellml:units="milliS_per_cm2"> 0.09 </cn>
<ci> d </ci>
<ci> f </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> E_si </ci>
</apply>
</apply>
</apply>
</math>
</component>
<!--
The "slow_inward_current_d_gate" component is the d gate encapsulated in
the slow inward current.
-->
<component name="slow_inward_current_d_gate">
<!-- This variable is defined here and used in other components. -->
<variable units="dimensionless" public_interface="out" name="d"/>
<!-- These variables are defined here and only used internally. -->
<variable units="per_millisecond" name="alpha_d"/>
<variable units="per_millisecond" name="beta_d"/>
<!--
These variables are imported from the "environment" and the "membrane" via
the "slow_inward_current" component.
-->
<variable units="millivolt" public_interface="in" name="V"/>
<variable units="millisecond" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<!--
The rate constants on the d gate are functions of membrane voltage.
-->
<apply id="alpha_d_calculation">
<eq/>
<ci> alpha_d </ci>
<apply>
<divide/>
<apply>
<times/>
<cn cellml:units="per_millisecond"> 0.095 </cn>
<apply>
<exp/>
<apply>
<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 0.01 </cn>
</apply>
<apply>
<minus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 5.0 </cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 0.072 </cn>
</apply>
<apply>
<minus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 5.0 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply id="beta_d_calculation">
<eq/>
<ci> beta_d </ci>
<apply>
<divide/>
<apply>
<times/>
<cn cellml:units="per_millisecond"> 0.07 </cn>
<apply>
<exp/>
<apply>
<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 0.017 </cn>
</apply>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 44.0 </cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="dimensionless"> 0.05 </cn>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 44.0 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply id="dd_dt">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> d </ci>
</apply>
<apply>
<minus/>
<apply>
<times/>
<ci> alpha_d </ci>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<ci> d </ci>
</apply>
</apply>
<apply>
<times/>
<ci> beta_d </ci>
<ci> d </ci>
</apply>
</apply>
</apply>
</math>
</component>
<!--
The "slow_inward_current_f_gate" component is the f gate encapsulated in
the slow inward current.
-->
<component name="slow_inward_current_f_gate">
<!-- This variable is defined here and used in other components. -->
<variable units="dimensionless" public_interface="out" name="f"/>
<!-- These variables are defined here and only used internally. -->
<variable units="per_millisecond" name="alpha_f"/>
<variable units="per_millisecond" name="beta_f"/>
<!--
These variables are imported from the "environment" and the "membrane" via
the "slow_inward_current" component.
-->
<variable units="millivolt" public_interface="in" name="V"/>
<variable units="millisecond" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<!--
The rate constants on the f gate are functions of membrane voltage.
-->
<apply id="alpha_f_calculation">
<eq/>
<ci> alpha_f </ci>
<apply>
<divide/>
<apply>
<times/>
<cn cellml:units="per_millisecond"> 0.012 </cn>
<apply>
<exp/>
<apply>
<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 0.008 </cn>
</apply>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 28.0 </cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="dimensionless"> 0.15 </cn>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 28.0 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply id="beta_f_calculation">
<eq/>
<ci> beta_f </ci>
<apply>
<divide/>
<apply>
<times/>
<cn cellml:units="per_millisecond"> 0.0065 </cn>
<apply>
<exp/>
<apply>
<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 0.02 </cn>
</apply>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 30.0 </cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 0.2 </cn>
</apply>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 30.0 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply id="df_dt">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> f </ci>
</apply>
<apply>
<minus/>
<apply>
<times/>
<ci> alpha_f </ci>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<ci> f </ci>
</apply>
</apply>
<apply>
<times/>
<ci> beta_f </ci>
<ci> f </ci>
</apply>
</apply>
</apply>
</math>
</component>
<!--
The "time_dependent_potassium_current" component contains a
time-dependent activation gate (X) and a time-independent inactivation
gate (Xi), neither of which are dependent on the extracellular
concentration of potassium ions ([K]o).
-->
<component name="time_dependent_potassium_current">
<!-- This variable is defined here and used in other components. -->
<variable units="microA_per_cm2" public_interface="out" name="i_K"/>
<!-- These variables are defined here and only used internally. -->
<variable units="milliS_per_cm2" name="g_K" initial_value="0.282"/>
<variable units="volt" name="E_K"/>
<variable units="dimensionless" name="PR_NaK" initial_value="0.01833"/>
<!--
Time is imported from the "environment", and membrane potential and
extracellular potassium concentration are imported from the membrane
component. These variables are used in the
"time_dependent_potassium_current" parent component, which also acts as an
interface, passing the variables to its encapsulated gate components.
-->
<variable units="millisecond" public_interface="in" private_interface="out" name="time"/>
<variable units="millivolt" public_interface="in" private_interface="out" name="V"/>
<variable units="joule_per_kilomole_kelvin" public_interface="in" name="R"/>
<variable units="kelvin" public_interface="in" name="T"/>
<variable units="coulomb_per_mole" public_interface="in" name="F"/>
<variable units="concentration_units" public_interface="in" name="Ko"/>
<variable units="concentration_units" public_interface="in" name="Ki"/>
<variable units="concentration_units" public_interface="in" name="Nao"/>
<variable units="concentration_units" public_interface="in" name="Nai"/>
<!-- These variables are imported from encapsulated components. -->
<variable units="dimensionless" private_interface="in" name="X"/>
<variable units="dimensionless" private_interface="in" name="Xi"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<!--
The following equation calculates the maximum conductance of the
potassium channel (gK).
-->
<apply id="g_K_calculation">
<eq/>
<ci> g_K </ci>
<apply>
<times/>
<cn cellml:units="milliS_per_cm2"> 0.282 </cn>
<apply>
<root/>
<apply>
<divide/>
<ci> Ko </ci>
<cn cellml:units="concentration_units"> 5.4 </cn>
</apply>
</apply>
</apply>
</apply>
<apply id="E_K_calculation">
<eq/>
<ci> E_K </ci>
<apply>
<times/>
<apply>
<divide/>
<apply>
<times/>
<ci> R </ci>
<ci> T </ci>
</apply>
<ci> F </ci>
</apply>
<apply>
<ln/>
<apply>
<divide/>
<apply>
<plus/>
<ci> Ko </ci>
<apply>
<times/>
<ci> PR_NaK </ci>
<ci> Nao </ci>
</apply>
</apply>
<apply>
<plus/>
<ci> Ki </ci>
<apply>
<times/>
<ci> PR_NaK </ci>
<ci> Nai </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
<!--
The following equation calculates the time-dependent potassium
current in terms of the conductance, the membrane voltage and the
gate variables.
-->
<apply id="i_K_calculation">
<eq/>
<ci> i_K </ci>
<apply>
<times/>
<ci> g_K </ci>
<ci> X </ci>
<ci> Xi </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> E_K </ci>
</apply>
</apply>
</apply>
</math>
</component>
<!--
The "time_dependent_potassium_current_X_gate" component is the
time-dependent activation gate encapsulated in the time-dependent
potassium current.
-->
<component name="time_dependent_potassium_current_X_gate">
<!-- This variable is defined here and used in other components. -->
<variable units="dimensionless" public_interface="out" name="X"/>
<!-- These variables are defined here and only used internally. -->
<variable units="per_millisecond" name="alpha_X"/>
<variable units="per_millisecond" name="beta_X"/>
<!--
These variables are imported from the "environment" and the "membrane" via
the "time_dependent_potassium_current" component.
-->
<variable units="millivolt" public_interface="in" name="V"/>
<variable units="millisecond" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<!-- The rate constants for X are governed by the following equations. -->
<apply id="alpha_X_calculation">
<eq/>
<ci> alpha_X </ci>
<apply>
<times/>
<cn cellml:units="per_millisecond"> 0.0005 </cn>
<apply>
<divide/>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="dimensionless"> 0.083 </cn>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 50.0 </cn>
</apply>
</apply>
</apply>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="dimensionless"> 0.057 </cn>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 50.0 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply id="beta_X_calculation">
<eq/>
<ci> beta_X </ci>
<apply>
<times/>
<cn cellml:units="per_millisecond"> 0.0013 </cn>
<apply>
<divide/>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="dimensionless"> -0.06 </cn>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 20.0 </cn>
</apply>
</apply>
</apply>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="dimensionless"> -0.04 </cn>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 20.0 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply id="dx_dt">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> X </ci>
</apply>
<apply>
<minus/>
<apply>
<times/>
<ci> alpha_X </ci>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<ci> X </ci>
</apply>
</apply>
<apply>
<times/>
<ci> beta_X </ci>
<ci> X </ci>
</apply>
</apply>
</apply>
</math>
</component>
<!--
The "time_dependent_potassium_current_Xi_gate" component is the
time-independent inactivation gate encapsulated in the time-dependent
potassium current.
-->
<component name="time_dependent_potassium_current_Xi_gate">
<!-- This variable is defined here and used in other components. -->
<variable units="dimensionless" public_interface="out" name="Xi"/>
<!--
These variables are imported from the "environment" and the "membrane" via
the "time_dependent_potassium_current" component.
-->
<variable units="millivolt" public_interface="in" name="V"/>
<variable units="millisecond" public_interface="in" name="time"/>
<!--
Xi is the inward rectification parameter and is given by the following
equation.
-->
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="Xi_calculation">
<eq/>
<ci> Xi </ci>
<piecewise>
<piece>
<apply>
<divide/>
<apply>
<times/>
<cn cellml:units="per_millivolt_millisecond"> 2.837 </cn>
<apply>
<minus/>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> 0.04 </cn>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 77.0 </cn>
</apply>
</apply>
</apply>
<cn cellml:units="dimensionless"> 1.0 </cn>
</apply>
</apply>
<apply>
<times/>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 77.0 </cn>
</apply>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> 0.04 </cn>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 35.0 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply>
<gt/>
<ci> V </ci>
<cn cellml:units="millivolt"> -100.0 </cn>
</apply>
</piece>
<otherwise>
<cn cellml:units="per_millisecond"> 1.0 </cn>
</otherwise>
</piecewise>
</apply>
</math>
</component>
<!--
The "time_independent_potassium_current" component contains a single
inactivation gate (K1) whose time constant is small enough that it
can be approximated by its steady-state value K1_infinity.
-->
<component name="time_independent_potassium_current">
<!-- These variables are defined here and used in other components. -->
<variable units="microA_per_cm2" public_interface="out" name="i_K1"/>
<variable units="millivolt" public_interface="out" private_interface="out" name="E_K1"/>
<!-- This variable is defined here and only used internally. -->
<variable units="milliS_per_cm2" name="g_K1"/>
<!--
These variables are imported from other components. They are all used in
the parent "time_independent_potassium_current" component and some are
also passed via this interface to the encapsulated gates.
-->
<variable units="millisecond" public_interface="in" private_interface="out" name="time"/>
<variable units="millivolt" public_interface="in" private_interface="out" name="V"/>
<variable units="concentration_units" public_interface="in" name="Ko"/>
<variable units="concentration_units" public_interface="in" name="Ki"/>
<variable units="joule_per_kilomole_kelvin" public_interface="in" name="R"/>
<variable units="kelvin" public_interface="in" name="T"/>
<variable units="coulomb_per_mole" public_interface="in" name="F"/>
<!-- This variable is imported from an encapsulated component. -->
<variable units="dimensionless" private_interface="in" name="K1_infinity"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<!--
The following equation determines the maximum conductance of the
potassium channel.
-->
<apply id="g_K1_calculation">
<eq/>
<ci> g_K1 </ci>
<apply>
<times/>
<cn cellml:units="milliS_per_cm2"> 0.6047 </cn>
<apply>
<root/>
<apply>
<divide/>
<ci> Ko </ci>
<cn cellml:units="concentration_units"> 5.4 </cn>
</apply>
</apply>
</apply>
</apply>
<!--
The following equation calculates the reversal potential of the
time-independent potassium current.
-->
<apply id="E_K1_calculation">
<eq/>
<ci> E_K1 </ci>
<apply>
<times/>
<apply>
<divide/>
<apply>
<times/>
<ci> R </ci>
<ci> T </ci>
</apply>
<ci> F </ci>
</apply>
<apply>
<ln/>
<apply>
<divide/>
<ci> Ko </ci>
<ci> Ki </ci>
</apply>
</apply>
</apply>
</apply>
<!--
The following equation calculates the time-independent potassium
current in terms of the conductance, the membrane voltage and the
gate variables.
-->
<apply id="i_K1_calculation">
<eq/>
<ci> i_K1 </ci>
<apply>
<times/>
<ci> g_K1 </ci>
<ci> K1_infinity </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> E_K1 </ci>
</apply>
</apply>
</apply>
</math>
</component>
<!--
The "time_independent_potassium_current_K1_gate" component is the K1 gate
encapsulated in the time-independent potassium current.
-->
<component name="time_independent_potassium_current_K1_gate">
<!-- This variable is defined here and used in other components. -->
<variable units="dimensionless" public_interface="out" name="K1_infinity"/>
<!-- These variables are defined here and only used internally. -->
<variable units="per_millisecond" name="alpha_K1"/>
<variable units="per_millisecond" name="beta_K1"/>
<!--
These variables are imported from the "environment", "membrane" and
"time_independent_potassium_current" components.
-->
<variable units="millivolt" public_interface="in" name="V"/>
<variable units="millisecond" public_interface="in" name="time"/>
<variable units="millivolt" public_interface="in" name="E_K1"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<!--
The rate constants on the K1 gate are functions of membrane voltage.
-->
<apply id="alpha_K1_calculation">
<eq/>
<ci> alpha_K1 </ci>
<apply>
<divide/>
<cn cellml:units="per_millisecond"> 1.02 </cn>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> 0.2385 </cn>
<apply>
<minus/>
<apply>
<minus/>
<ci> V </ci>
<ci> E_K1 </ci>
</apply>
<cn cellml:units="millivolt"> 59.215 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply id="beta_K1_calculation">
<eq/>
<ci> beta_K1 </ci>
<apply>
<divide/>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="per_millisecond"> 0.49124 </cn>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="dimensionless"> 0.08032 </cn>
<apply>
<minus/>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 5.476 </cn>
</apply>
<ci> E_K1 </ci>
</apply>
</apply>
</apply>
</apply>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="per_millisecond"> 0.06175 </cn>
<apply>
<minus/>
<ci> V </ci>
<apply>
<plus/>
<ci> E_K1 </ci>
<cn cellml:units="millivolt"> 594.31 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> -0.5143 </cn>
<apply>
<plus/>
<apply>
<minus/>
<ci> V </ci>
<ci> E_K1 </ci>
</apply>
<cn cellml:units="millivolt"> 4.753 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
<!--
The steady-state value K1_infinity is given by the following
equation.
-->
<apply id="K1_infinity_calculation">
<eq/>
<ci> K1_infinity </ci>
<apply>
<divide/>
<ci> alpha_K1 </ci>
<apply>
<plus/>
<ci> alpha_K1 </ci>
<ci> beta_K1 </ci>
</apply>
</apply>
</apply>
</math>
</component>
<!--
The "plateau_potassium_current" component contains the equations
which describe the contribution of a time independent
[K]o-insensitive channel at plateau potentials.
-->
<component name="plateau_potassium_current">
<!-- This variable is defined here and used in other components. -->
<variable units="microA_per_cm2" public_interface="out" name="i_Kp"/>
<!-- These variables are defined here and only used internally. -->
<variable units="millivolt" name="E_Kp"/>
<variable units="milliS_per_cm2" name="g_Kp" initial_value="0.0183"/>
<variable units="dimensionless" name="Kp"/>
<!-- These variables are imported from other components. -->
<variable units="millisecond" public_interface="in" name="time"/>
<variable units="millivolt" public_interface="in" name="V"/>
<variable units="millivolt" public_interface="in" name="E_K1"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<!--
The following equation calculates the plateau potassium current
reversal potential.
-->
<apply id="E_Kp_calculation">
<eq/>
<ci> E_Kp </ci>
<ci> E_K1 </ci>
</apply>
<!-- The following equation calculates the constant Kp. -->
<apply id="Kp_calculation">
<eq/>
<ci> Kp </ci>
<apply>
<divide/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<cn cellml:units="millivolt"> 7.488 </cn>
<ci> V </ci>
</apply>
<cn cellml:units="millivolt"> 5.98 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
<!--
The following equation determines the plateau potassium current in terms of the membrane voltage.
-->
<apply id="i_Kp_calculation">
<eq/>
<ci> i_Kp </ci>
<apply>
<times/>
<ci> g_Kp </ci>
<ci> Kp </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> E_Kp </ci>
</apply>
</apply>
</apply>
</math>
</component>
<!--
The "background_current" component describes an additional current
which contributes at plateau potentials.
-->
<component name="background_current">
<!-- This variable is defined here and used in other components. -->
<variable units="microA_per_cm2" public_interface="out" name="i_b"/>
<!-- These variables are defined here and only used internally. -->
<variable units="millivolt" name="E_b" initial_value="-59.87"/>
<variable units="milliS_per_cm2" name="g_b" initial_value="0.03921"/>
<!--
Time and membrane potential are imported from the "environment" and
"membrane" components.
-->
<variable units="millisecond" public_interface="in" name="time"/>
<variable units="millivolt" public_interface="in" name="V"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<!--
The folowing equation calculates the background current in terms
of the membrane voltage.
-->
<apply id="i_b_calculation">
<eq/>
<ci> i_b </ci>
<apply>
<times/>
<ci> g_b </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> E_b </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component cmeta:id="ionic_concentrations" name="ionic_concentrations">
<!-- These variables are defined here and used in other components -->
<variable units="concentration_units" public_interface="out" name="Nao" initial_value="140.0"/>
<variable units="concentration_units" public_interface="out" name="Nai" initial_value="18.0"/>
<variable units="concentration_units" name="Cao" initial_value="1.8"/>
<variable units="concentration_units" public_interface="out" name="Ki" initial_value="145.0"/>
<variable units="concentration_units" public_interface="out" name="Ko" initial_value="5.4"/>
</component>
<component name="intracellular_calcium_concentration">
<!-- This variable is defined here and used in other components. -->
<variable units="concentration_units" public_interface="out" name="Cai" initial_value="0.0002"/>
<!-- These variables are defined here and only used internally. -->
<variable units="millivolt" name="E_b" initial_value="-59.87"/>
<variable units="milliS_per_cm2" name="g_b" initial_value="0.03921"/>
<!-- These variables are imported from other components. -->
<variable units="millisecond" public_interface="in" name="time"/>
<variable units="microA_per_cm2" public_interface="in" name="i_si"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<!--
Change in [Ca]i (intracellular calcium concentration) is given in terms
of the slow inward current as:
-->
<apply id="dCai_dt">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> Cai </ci>
</apply>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="dimensionless"> -0.0001 </cn>
<ci> i_si </ci>
</apply>
<apply>
<times/>
<cn cellml:units="dimensionless"> 0.07 </cn>
<apply>
<minus/>
<cn cellml:units="concentration_units"> 0.0001 </cn>
<ci> Cai </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<!--
The following <group> element specifies a single containment hierarchy
that encompasses all of the components in the model, with the exception of
the "environment" component. The "containment" relationship is used to
describe geometric hierarchies - or how components are physically arranged
in relation to eachother.
-->
<group>
<relationship_ref relationship="containment"/>
<component_ref component="membrane">
<component_ref component="fast_sodium_current">
<component_ref component="fast_sodium_current_m_gate"/>
<component_ref component="fast_sodium_current_h_gate"/>
<component_ref component="fast_sodium_current_j_gate"/>
</component_ref>
<component_ref component="slow_inward_current">
<component_ref component="slow_inward_current_d_gate"/>
<component_ref component="slow_inward_current_f_gate"/>
</component_ref>
<component_ref component="time_dependent_potassium_current">
<component_ref component="time_dependent_potassium_current_X_gate"/>
<component_ref component="time_dependent_potassium_current_Xi_gate"/>
</component_ref>
<component_ref component="time_independent_potassium_current">
<component_ref component="time_independent_potassium_current_K1_gate"/>
</component_ref>
<component_ref component="plateau_potassium_current"/>
<component_ref component="background_current"/>
<component_ref component="ionic_concentrations"/>
<component_ref component="intracellular_calcium_concentration"/>
</component_ref>
</group>
<!--
The following <group> element specifies how the components
representing activation and inactivation coefficients are
encapsulated inside the sodium and potassium channel components.
Encapsulation describes the logical hierarchy of components in a network,
and may or may not reflect their physical arrangement.
-->
<group>
<relationship_ref relationship="encapsulation"/>
<component_ref component="fast_sodium_current">
<component_ref component="fast_sodium_current_m_gate"/>
<component_ref component="fast_sodium_current_h_gate"/>
<component_ref component="fast_sodium_current_j_gate"/>
</component_ref>
<component_ref component="slow_inward_current">
<component_ref component="slow_inward_current_d_gate"/>
<component_ref component="slow_inward_current_f_gate"/>
</component_ref>
<component_ref component="time_dependent_potassium_current">
<component_ref component="time_dependent_potassium_current_X_gate"/>
<component_ref component="time_dependent_potassium_current_Xi_gate"/>
</component_ref>
<component_ref component="time_independent_potassium_current">
<component_ref component="time_independent_potassium_current_K1_gate"/>
</component_ref>
</group>
<!--
The connections hold the actual mappings between variables declared in
different components. When more than one variable is mapped between two
components, all variable mappings must be listed in the same connection
element (there can only be one connection between two components). The
variables are assumed to be listed in the same order within the two
<component_ref> elements. The modeler should NOT rely on matching the
names of the variables in the two components.
-->
<!--
"time" is passed from the "environment" component into the
"membrane" and current components.
-->
<connection>
<map_components component_2="environment" component_1="membrane"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="fast_sodium_current"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="slow_inward_current"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="time_dependent_potassium_current"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="time_independent_potassium_current"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="plateau_potassium_current"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="background_current"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="intracellular_calcium_concentration"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<!--
A number of variables are passed between the "membrane" and its six
sub-components, the "fast_sodium_current", "slow_inward_current",
"time_dependent_potassium_current", time_independent_potassium_current",
"plateau_potassium_current" and the "background_current".
-->
<connection>
<map_components component_2="fast_sodium_current" component_1="membrane"/>
<map_variables variable_2="V" variable_1="V"/>
<map_variables variable_2="i_Na" variable_1="i_Na"/>
</connection>
<connection>
<map_components component_2="slow_inward_current" component_1="membrane"/>
<map_variables variable_2="V" variable_1="V"/>
<map_variables variable_2="i_si" variable_1="i_si"/>
</connection>
<connection>
<map_components component_2="time_dependent_potassium_current" component_1="membrane"/>
<map_variables variable_2="V" variable_1="V"/>
<map_variables variable_2="i_K" variable_1="i_K"/>
<map_variables variable_2="R" variable_1="R"/>
<map_variables variable_2="T" variable_1="T"/>
<map_variables variable_2="F" variable_1="F"/>
</connection>
<connection>
<map_components component_2="time_independent_potassium_current" component_1="membrane"/>
<map_variables variable_2="V" variable_1="V"/>
<map_variables variable_2="i_K1" variable_1="i_K1"/>
<map_variables variable_2="R" variable_1="R"/>
<map_variables variable_2="T" variable_1="T"/>
<map_variables variable_2="F" variable_1="F"/>
</connection>
<connection>
<map_components component_2="plateau_potassium_current" component_1="membrane"/>
<map_variables variable_2="V" variable_1="V"/>
<map_variables variable_2="i_Kp" variable_1="i_Kp"/>
</connection>
<connection>
<map_components component_2="background_current" component_1="membrane"/>
<map_variables variable_2="V" variable_1="V"/>
<map_variables variable_2="i_b" variable_1="i_b"/>
</connection>
<!-- A few variables are passed between the other components. -->
<connection>
<map_components component_2="intracellular_calcium_concentration" component_1="slow_inward_current"/>
<map_variables variable_2="i_si" variable_1="i_si"/>
<map_variables variable_2="Cai" variable_1="Cai"/>
</connection>
<connection>
<map_components component_2="ionic_concentrations" component_1="time_dependent_potassium_current"/>
<map_variables variable_2="Ko" variable_1="Ko"/>
<map_variables variable_2="Ki" variable_1="Ki"/>
<map_variables variable_2="Nao" variable_1="Nao"/>
<map_variables variable_2="Nai" variable_1="Nai"/>
</connection>
<connection>
<map_components component_2="plateau_potassium_current" component_1="time_independent_potassium_current"/>
<map_variables variable_2="E_K1" variable_1="E_K1"/>
</connection>
<connection>
<map_components component_2="ionic_concentrations" component_1="time_independent_potassium_current"/>
<map_variables variable_2="Ko" variable_1="Ko"/>
<map_variables variable_2="Ki" variable_1="Ki"/>
</connection>
<!--
Various gating variables are passed between the "fast_sodium_current",
"slow_inward_current", "time_dependent_potassium_current" and the
"time_independent_potassium_current" components and the encapsulated "gate"
components.
-->
<connection>
<map_components component_2="fast_sodium_current_m_gate" component_1="fast_sodium_current"/>
<map_variables variable_2="m" variable_1="m"/>
<map_variables variable_2="time" variable_1="time"/>
<map_variables variable_2="V" variable_1="V"/>
</connection>
<connection>
<map_components component_2="fast_sodium_current_h_gate" component_1="fast_sodium_current"/>
<map_variables variable_2="h" variable_1="h"/>
<map_variables variable_2="time" variable_1="time"/>
<map_variables variable_2="V" variable_1="V"/>
</connection>
<connection>
<map_components component_2="fast_sodium_current_j_gate" component_1="fast_sodium_current"/>
<map_variables variable_2="j" variable_1="j"/>
<map_variables variable_2="time" variable_1="time"/>
<map_variables variable_2="V" variable_1="V"/>
</connection>
<connection>
<map_components component_2="slow_inward_current_d_gate" component_1="slow_inward_current"/>
<map_variables variable_2="d" variable_1="d"/>
<map_variables variable_2="time" variable_1="time"/>
<map_variables variable_2="V" variable_1="V"/>
</connection>
<connection>
<map_components component_2="slow_inward_current_f_gate" component_1="slow_inward_current"/>
<map_variables variable_2="f" variable_1="f"/>
<map_variables variable_2="time" variable_1="time"/>
<map_variables variable_2="V" variable_1="V"/>
</connection>
<connection>
<map_components component_2="time_dependent_potassium_current_X_gate" component_1="time_dependent_potassium_current"/>
<map_variables variable_2="X" variable_1="X"/>
<map_variables variable_2="time" variable_1="time"/>
<map_variables variable_2="V" variable_1="V"/>
</connection>
<connection>
<map_components component_2="time_dependent_potassium_current_Xi_gate" component_1="time_dependent_potassium_current"/>
<map_variables variable_2="Xi" variable_1="Xi"/>
<map_variables variable_2="time" variable_1="time"/>
<map_variables variable_2="V" variable_1="V"/>
</connection>
<connection>
<map_components component_2="time_independent_potassium_current_K1_gate" component_1="time_independent_potassium_current"/>
<map_variables variable_2="K1_infinity" variable_1="K1_infinity"/>
<map_variables variable_2="time" variable_1="time"/>
<map_variables variable_2="V" variable_1="V"/>
<map_variables variable_2="E_K1" variable_1="E_K1"/>
</connection>
<rdf:RDF>
<rdf:Bag rdf:about="rdf:#f4b6d7a7-a25a-47b2-aee4-d34e9eaf7e26">
<rdf:li>Ventricular Myocyte</rdf:li>
<rdf:li>cardiac</rdf:li>
<rdf:li>electrophysiology</rdf:li>
</rdf:Bag>
<rdf:Seq rdf:about="rdf:#citationAuthorsSeq">
<rdf:li rdf:resource="rdf:#author1Vcard"/>
<rdf:li rdf:resource="rdf:#author2Vcard"/>
</rdf:Seq>
<rdf:Description rdf:about="rdf:#b36c8180-7ef5-480c-8f2b-44a45517c2f2">
<dcterms:modified rdf:resource="rdf:#561c1c6a-8d7d-40a0-aae3-f14dedd42924"/>
<rdf:value>
Corrected equations: alpha_j_calculation and beta_j_calculation in
fast_sodium_current_j_gate, alpha_d_calculation and beta_d_calculation in slow_inward_current_d_gate, alpha_f_calculation and
beta_f_calculation in slow_inward_current_f_gate, alpha_X_calculation
and beta_X_calculation in time_dependent_potassium_current_X_gate, and beta_K1_calculation in time_independent_potassium_current_K1_gate.
</rdf:value>
<cmeta:modifier rdf:resource="rdf:#b3d7c16c-f8cc-453a-9795-6050ccac1034"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#cdf4eed3-ba6f-4f31-bf83-2b9c090df7f2">
<vCard:N rdf:resource="rdf:#e1c3f52a-394b-4d0c-9525-1d94da86605a"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#55d7156a-a822-48ce-93d8-6e641ed091f1">
<dcterms:W3CDTF>2001-09-15</dcterms:W3CDTF>
</rdf:Description>
<rdf:Description rdf:about="rdf:#e43a579d-ab15-4143-8b0c-850d6d2ca70d">
<vCard:N rdf:resource="rdf:#91d3f35b-5e69-4f34-9739-04419757ada8"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#author2Vcard">
<rdf:type rdf:resource="http://www.cellml.org/bqs/1.0#Person"/>
<vCard:N rdf:resource="rdf:#author2VcardN"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#author1Vcard">
<rdf:type rdf:resource="http://www.cellml.org/bqs/1.0#Person"/>
<vCard:N rdf:resource="rdf:#author1VcardN"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#df14288d-be66-44d0-9b5e-8b3ab9ef71c8">
<dcterms:W3CDTF>1991-06-01</dcterms:W3CDTF>
</rdf:Description>
<rdf:Description rdf:about="rdf:#fd8bb2fd-03a9-4726-aa05-e40e7e27c3a9">
<dcterms:modified rdf:resource="rdf:#4e3c1707-4bc8-4e8c-a86b-1432004169e7"/>
<rdf:value>
Added some initial values from Penny Noble's documentation.
</rdf:value>
<cmeta:modifier rdf:resource="rdf:#36d1c422-6b3d-4b35-8fb1-fd7291f228d4"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#e1c3f52a-394b-4d0c-9525-1d94da86605a">
<vCard:Given>Catherine</vCard:Given>
<vCard:Family>Lloyd</vCard:Family>
<vCard:Other>May</vCard:Other>
</rdf:Description>
<rdf:Description rdf:about="rdf:#ba1b1e28-88b8-40a8-b758-b03358e4c3a4">
<rdf:type rdf:resource="http://purl.org/dc/terms/W3CDTF"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#acea85af-954b-4aad-843f-5a432b62b1ae">
<dcterms:W3CDTF>2002-02-20</dcterms:W3CDTF>
</rdf:Description>
<rdf:Description rdf:about="rdf:#d2ca5fcb-cb7b-4ef1-9d41-37954ab619af">
<vCard:N rdf:resource="rdf:#139af924-81b1-43ae-b98c-cb40e9718629"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#d8e0616d-dea0-49d6-8786-181b6476af23">
<vCard:FN>Catherine Lloyd</vCard:FN>
</rdf:Description>
<rdf:Description rdf:about="rdf:#871ca6d8-1684-4453-9884-a3db63cc7824">
<dc:creator rdf:resource="rdf:#d8e0616d-dea0-49d6-8786-181b6476af23"/>
<rdf:value>This is the CellML description of Luo and Rudy's mathematical model of the membrane action potential of the mammalian ventricular cell. It describes six ionic currents and it is a development of the Beeler-Reuter 1977 mammalian ventricular model, using Hodgkin-Huxley type equations.</rdf:value>
</rdf:Description>
<rdf:Description rdf:about="#luo_rudy_1991_version01">
<dc:title>
The Luo-Rudy I Model of Mammalian Ventricular Cardiac Action Potentials,
1991
</dc:title>
<cmeta:bio_entity>Ventricular Myocyte</cmeta:bio_entity>
<cmeta:comment rdf:resource="rdf:#871ca6d8-1684-4453-9884-a3db63cc7824"/>
<bqs:reference rdf:resource="rdf:#2db5b00f-ff87-4c4c-b2d8-b97ac9bb09bf"/>
<bqs:reference rdf:resource="rdf:#4ea6e853-8e8c-4dc3-afed-9264a76f61b2"/>
<cmeta:species>Mammalia</cmeta:species>
</rdf:Description>
<rdf:Description rdf:about="rdf:#926ee723-7ffe-49d8-a995-83d4cdda166b">
<dc:title>Circulation Research</dc:title>
</rdf:Description>
<rdf:Description rdf:about="rdf:#23acf7c9-2b15-4346-8b97-fe8bf28951fe">
<dcterms:modified rdf:resource="rdf:#d37710ee-955e-45e2-ac0c-727c7c027901"/>
<rdf:value>
Removed document type definition as this is declared as optional
according to the W3C recommendation.
</rdf:value>
<cmeta:modifier rdf:resource="rdf:#58760822-ca0f-41eb-aa58-60d2af1cc0ef"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#435cf6cf-abf4-44bf-ad73-e8d64879af8e">
<vCard:Given>Catherine</vCard:Given>
<vCard:Family>Lloyd</vCard:Family>
<vCard:Other>May</vCard:Other>
</rdf:Description>
<rdf:Description rdf:about="rdf:#4ea6e853-8e8c-4dc3-afed-9264a76f61b2">
<bqs:Pubmed_id>1709839</bqs:Pubmed_id>
<bqs:JournalArticle rdf:resource="rdf:#ee6717e9-776e-4639-af3f-af14625bccc3"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#29c23b6f-d742-483d-b9f6-e838f15de36d">
<dcterms:modified rdf:resource="rdf:#701d2fb7-7259-41fb-8978-b7d0cd3145ac"/>
<rdf:value>
Added an ionic_concentrations and an
intracellular_calcium_concentrations components and altered the public
interfaces and connections appropriately.
</rdf:value>
<cmeta:modifier rdf:resource="rdf:#e43a579d-ab15-4143-8b0c-850d6d2ca70d"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#be6bfe6d-8241-4957-9221-45d729aea2e4">
<vCard:Given>Catherine</vCard:Given>
<vCard:Family>Lloyd</vCard:Family>
<vCard:Other>May</vCard:Other>
</rdf:Description>
<rdf:Description rdf:about="rdf:#739989f8-2e17-4029-a929-2e570c4fc969">
<dcterms:W3CDTF>2002-02-18</dcterms:W3CDTF>
</rdf:Description>
<rdf:Description rdf:about="rdf:#4e3c1707-4bc8-4e8c-a86b-1432004169e7">
<dcterms:W3CDTF>2002-05-06</dcterms:W3CDTF>
</rdf:Description>
<rdf:Description rdf:about="rdf:#abe012ee-6785-4ef0-86bc-c4376e093fb6">
<vCard:Given>Autumn</vCard:Given>
<vCard:Family>Cuellar</vCard:Family>
<vCard:Other>A.</vCard:Other>
</rdf:Description>
<rdf:Description rdf:about="rdf:#6dd52b07-e06e-45cd-b192-9b765cf99a6d">
<rdf:type rdf:resource="http://imc.org/vCard/3.0#internet"/>
<rdf:value>c.lloyd@auckland.ac.nz</rdf:value>
</rdf:Description>
<rdf:Description rdf:about="rdf:#91d3f35b-5e69-4f34-9739-04419757ada8">
<vCard:Given>Catherine</vCard:Given>
<vCard:Family>Lloyd</vCard:Family>
<vCard:Other>May</vCard:Other>
</rdf:Description>
<rdf:Description rdf:about="rdf:#36d1c422-6b3d-4b35-8fb1-fd7291f228d4">
<vCard:N rdf:resource="rdf:#223ca6fe-89d6-4570-afc3-4becd0770b49"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#author2VcardN">
<vCard:Given>Yoram</vCard:Given>
<vCard:Family>Rudy</vCard:Family>
</rdf:Description>
<rdf:Description rdf:about="rdf:#ee6717e9-776e-4639-af3f-af14625bccc3">
<dc:creator rdf:resource="rdf:#citationAuthorsSeq"/>
<dc:title>A Model of the Ventricular Cardiac Action Potential</dc:title>
<bqs:volume>68</bqs:volume>
<bqs:first_page>1501</bqs:first_page>
<bqs:Journal rdf:resource="rdf:#926ee723-7ffe-49d8-a995-83d4cdda166b"/>
<dcterms:issued rdf:resource="rdf:#df14288d-be66-44d0-9b5e-8b3ab9ef71c8"/>
<bqs:last_page>1526</bqs:last_page>
</rdf:Description>
<rdf:Description rdf:about="rdf:#0600eb0c-78ca-49c5-a36f-4f093abd4174">
<dcterms:W3CDTF>2005-04-29</dcterms:W3CDTF>
</rdf:Description>
<rdf:Description rdf:about="rdf:#edabffdc-f335-43fd-ac45-5c49b149960d">
<vCard:N rdf:resource="rdf:#abe012ee-6785-4ef0-86bc-c4376e093fb6"/>
</rdf:Description>
<rdf:Description rdf:about="#ionic_concentrations">
<cmeta:comment rdf:resource="rdf:#707e500a-bc06-4ca2-b884-c4ca8c40fb71"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#707e500a-bc06-4ca2-b884-c4ca8c40fb71">
<rdf:value>
In their 1991 model for most ions, Luo and Rudy do not include
equations to calculate changes in their concentrations. Instead
they fix the variables and specify their values, as shown in the
component below. The only exception is for intracellular calcium
concentrations which is defined by an equation. Extracellular
calcium concentration is defined here but is not included in any of
the ionic flux calculations. Therefore it has no public_interface.
</rdf:value>
</rdf:Description>
<rdf:Description rdf:about="rdf:#c9e994b8-2292-446f-a435-b8782e2c4d39">
<rdf:type rdf:resource="http://www.w3.org/2001/vcard-rdf/3.0#N"/>
<vCard:Given>Catherine</vCard:Given>
<vCard:Family>Lloyd</vCard:Family>
<vCard:Other>May</vCard:Other>
</rdf:Description>
<rdf:Description rdf:about="rdf:#b3d7c16c-f8cc-453a-9795-6050ccac1034">
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Corrected the Cai differential eqn.
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Corrected value in equation.
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Changed equations after checking with mathml validator.
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Corrected the beta_K1_calculation eqn.
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Corrected equations.
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