Views
Control of the pacemaker activity of the sinoatrial node by intracellular Ca2+. Experiments and modelling
The CellML code.
<!-- FILE : boyett_SAN_model_2001.xml
CREATED : 28th January 2002
LAST MODIFIED : 20th 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 16/01/2002 CellML Metadata 1.0 Specification.
DESCRIPTION : This file contains a CellML description of Boyett et al's 2001
mathematical model of control of the pacemaker activity of the sinoatrial node.
CHANGES:
25/02/2002 - CML - Corrected several equations.
19/07/2002 - CML - Added more metadata.
20/04/2005 - PJV - Updated syntax to conform with cellml 1.1 specs
-->
<model xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#" xmlns:dcterms="http://purl.org/dc/terms/" xmlns:cellml="http://www.cellml.org/cellml/1.0#" xmlns:bqs="http://www.cellml.org/bqs/1.0#" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:cmeta="http://www.cellml.org/metadata/1.0#" xmlns="http://www.cellml.org/cellml/1.0#" cmeta:id="boyett_zhang_garny_holden_2001_version01" name="boyett_zhang_garny_holden_2001_version01">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
<articleinfo>
<title>Modelling The Effect Of Intracellular Ca2+ On Sinoatrial Node Action Potentials, 2001</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 2000 H. Zhang <emphasis>et al</emphasis> published a mathematical model which describes the action potentials in the periphery and center of the rabbit sinoatrial node. The SA node is functionally, anatomically and electrophysiologically heterogeneous. In their model, Zhang <emphasis>et al</emphasis> accounted for this heterogeneity by defining two distinct models for the peripheral and the central cells of the SA node. These models have the same equations to define the ionic currents but they vary in their parameters.
</para>
<para>
The complete original paper reference is cited below:
</para>
<para>
<ulink url="http://ajpheart.physiology.org/cgi/content/abstract/279/1/H397">Mathematical models of action potentials in the periphery and center of the rabbit sinoatrial node</ulink>, H. Zhang, A.V. Holden, I. Honjo, M. Lei, T. Varghese and M.R. Boyett, 2000, <ulink url="http://ajpheart.physiology.org/">
<emphasis>Am. J. Physiol. Heart Circ. Physiol.</emphasis>
</ulink>, 279, H397-H421. (The <ulink url="http://ajpheart.physiology.org/cgi/content/full/279/1/H397">full text</ulink> of the article is available on the American Journal of Physiology website for Journal Members.) <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=10899081&dopt=Abstract">PubMed ID: 10899081</ulink>
</para>
<para>
In 2001, Boyett <emphasis>et al</emphasis> reviewed the results of several experiments which suggested that intracellular calcium controls a number of ionic currents in the SA node cells. This had not been considered in previous SA node cell models and Boyett <emphasis>et al</emphasis> incorporated an intracellular calcium handling element into the model of Zhang <emphasis>et al</emphasis> (2000). In addition, they included a sustained inward current based on the experiments and equations of Shinagawa <emphasis>et al</emphasis> (2000) (see the figure below). Like in Zhang et al's orginal model, differences between the central and peripheral SA node cells are taken into consideration by defining different parameters. The CellML description here is of the peripheral SA node cell model. To make it appropriate for the central SA node cells, certain parameters need to be changed (refer to the original paper reference which is cited below).
</para>
<para>
Control of the pacemaker activity of the sinoatrial node by intracellular Ca<superscript>2+</superscript>. Experiments and modelling, M.R. Boyett, H. Zhang, A. Garny and A.V. Holden, 2001, <ulink url="http://www.pubs.royalsoc.ac.uk/">
<emphasis>Phil. Trans. R. Soc. Lond. A.</emphasis>
</ulink>, 359, 1091-1110.
</para>
<informalfigure float="0" id="fig_cell_diagram">
<mediaobject>
<imageobject>
<objectinfo>
<title>cell diagram of the Boyett et al SAN model showing ionic currents, pumps and exchangers within the sarcolemma and the sarcoplasmic reticulum</title>
</objectinfo>
<imagedata fileref="../images/B_SAN_model_2001/boyett_2001.png" />
</imageobject>
</mediaobject>
<caption>A schematic diagram describing the current flows across the cell membrane that are captured in the Boyett <emphasis>et al</emphasis> 2001 model of the action potentials in the peripheral cells of the SA node.</caption>
</informalfigure>
<informalfigure float="0" id="fig_cellml_rendering">
<mediaobject>
<imageobject>
<objectinfo>
<title>the cellml rendering of the Boyett et al SAN model</title>
</objectinfo>
<imagedata fileref="../images/B_SAN_model_2001/cellml_rendering.gif" />
</imageobject>
</mediaobject>
<caption>The network defined in the CellML description of the Boyett <emphasis>et al</emphasis> 2001 model. For simplicity, not all the variables are shown.</caption>
</informalfigure>
</sect1>
</article>
</documentation>
<!--
Below, are defined some additional units for association with variables and
constants within the model. The identifiers are fairly self-explanatory.
-->
<units name="per_second">
<unit units="second" 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_second">
<unit units="millivolt" exponent="-1" />
<unit units="second" exponent="-1" />
</units>
<units name="microS">
<unit units="siemens" prefix="micro" />
</units>
<units name="picoF">
<unit units="farad" prefix="pico" />
</units>
<units name="nanoA">
<unit units="ampere" prefix="nano" />
</units>
<units name="millimolar">
<unit units="mole" prefix="milli" />
<unit units="litre" exponent="-1" />
</units>
<units name="per_second_millimolar">
<unit units="second" exponent="-1" />
<unit units="millimolar" exponent="-1" />
</units>
<units name="millimole_per_second">
<unit units="mole" prefix="milli" />
<unit units="second" exponent="-1" />
</units>
<units name="millijoule_per_mole_kelvin">
<unit units="joule" prefix="milli" />
<unit units="mole" exponent="-1" />
<unit units="kelvin" exponent="-1" />
</units>
<units name="coulomb_per_mole">
<unit units="coulomb" exponent="-1" />
<unit units="mole" />
</units>
<units name="microlitre">
<unit units="litre" prefix="micro" />
</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="second" 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">
<variable units="millivolt" public_interface="out" name="V" initial_value="-63.25" />
<variable units="millijoule_per_mole_kelvin" public_interface="out" name="R" initial_value="8314.0" />
<variable units="kelvin" public_interface="out" name="T" initial_value="310.0" />
<variable units="coulomb_per_mole" public_interface="out" name="F" initial_value="96845.0" />
<variable units="picoF" name="C" initial_value="65.0" />
<variable units="second" public_interface="in" name="time" />
<variable units="nanoA" public_interface="in" name="i_Na" />
<variable units="nanoA" public_interface="in" name="i_Ca_L" />
<variable units="nanoA" public_interface="in" name="i_Ca_T" />
<variable units="nanoA" public_interface="in" name="i_to" />
<variable units="nanoA" public_interface="in" name="i_sus" />
<variable units="nanoA" public_interface="in" name="i_K_r" />
<variable units="nanoA" public_interface="in" name="i_K_s" />
<variable units="nanoA" public_interface="in" name="i_f" />
<variable units="nanoA" public_interface="in" name="i_b_Na" />
<variable units="nanoA" public_interface="in" name="i_b_Ca" />
<variable units="nanoA" public_interface="in" name="i_b_K" />
<variable units="nanoA" public_interface="in" name="i_NaCa" />
<variable units="nanoA" public_interface="in" name="i_p" />
<variable units="nanoA" public_interface="in" name="i_st" />
<variable units="nanoA" public_interface="in" name="i_Ca_P" />
<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>
<divide />
<cn cellml:units="dimensionless"> -1.0 </cn>
<ci> C </ci>
</apply>
<apply>
<plus />
<ci> i_Na </ci>
<ci> i_Ca_L </ci>
<ci> i_Ca_T </ci>
<ci> i_to </ci>
<ci> i_sus </ci>
<ci> i_K_r </ci>
<ci> i_K_s </ci>
<ci> i_f </ci>
<ci> i_b_Na </ci>
<ci> i_b_Ca </ci>
<ci> i_b_K </ci>
<ci> i_NaCa </ci>
<ci> i_p </ci>
<ci> i_st </ci>
<ci> i_Ca_P </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component cmeta:id="sodium_current" name="sodium_current">
<variable units="nanoA" public_interface="out" name="i_Na" />
<variable units="microS" name="g_Na" initial_value="0.0000012" />
<variable units="millivolt" public_interface="in" name="E_Na" />
<variable units="millimolar" public_interface="in" name="Na_o" />
<variable units="millijoule_per_mole_kelvin" public_interface="in" name="R" />
<variable units="coulomb_per_mole" public_interface="in" name="F" />
<variable units="kelvin" public_interface="in" name="T" />
<variable units="second" public_interface="in" private_interface="out" name="time" />
<variable units="millivolt" public_interface="in" private_interface="out" name="V" />
<variable units="dimensionless" private_interface="in" name="m" />
<variable units="dimensionless" private_interface="in" name="h" />
<math xmlns="http://www.w3.org/1998/Math/MathML">
<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> Na_o </ci>
<apply>
<divide />
<apply>
<power />
<ci> F </ci>
<cn cellml:units="dimensionless"> 2.0 </cn>
</apply>
<apply>
<times />
<ci> R </ci>
<ci> T </ci>
</apply>
</apply>
<apply>
<divide />
<apply>
<minus />
<apply>
<exp />
<apply>
<divide />
<apply>
<times />
<apply>
<minus />
<ci> V </ci>
<ci> E_Na </ci>
</apply>
<ci> F </ci>
</apply>
<apply>
<times />
<ci> R </ci>
<ci> T </ci>
</apply>
</apply>
</apply>
<cn cellml:units="dimensionless"> 1.0 </cn>
</apply>
<apply>
<minus />
<apply>
<exp />
<apply>
<divide />
<apply>
<times />
<ci> V </ci>
<ci> F </ci>
</apply>
<apply>
<times />
<ci> R </ci>
<ci> T </ci>
</apply>
</apply>
</apply>
<cn cellml:units="dimensionless"> 1.0 </cn>
</apply>
</apply>
<ci> V </ci>
</apply>
</apply>
</math>
</component>
<component name="sodium_current_m_gate">
<variable units="dimensionless" public_interface="out" name="m" initial_value="0.135" />
<variable units="dimensionless" name="m_infinity" />
<variable units="second" name="tau_m" />
<variable units="millivolt" public_interface="in" name="V" />
<variable units="second" public_interface="in" name="time" />
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="m_diff_eq">
<eq />
<apply>
<diff />
<bvar>
<ci> time </ci>
</bvar>
<ci> m </ci>
</apply>
<apply>
<divide />
<apply>
<minus />
<ci> m_infinity </ci>
<ci> m </ci>
</apply>
<ci> tau_m </ci>
</apply>
</apply>
<apply id="m_infinity_calculation">
<eq />
<ci> m_infinity </ci>
<apply>
<power />
<apply>
<divide />
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus />
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp />
<apply>
<divide />
<apply>
<minus />
<ci> V </ci>
</apply>
<cn cellml:units="millivolt"> 5.46 </cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<divide />
<cn cellml:units="dimensionless"> 1.0 </cn>
<cn cellml:units="dimensionless"> 3.0 </cn>
</apply>
</apply>
</apply>
<apply id="tau_m_calculation">
<eq />
<ci> tau_m </ci>
<apply>
<plus />
<apply>
<divide />
<cn cellml:units="second"> 0.6247e-3 </cn>
<apply>
<plus />
<apply>
<times />
<cn cellml:units="dimensionless"> 0.832 </cn>
<apply>
<exp />
<apply>
<times />
<cn cellml:units="dimensionless"> -0.335 </cn>
<apply>
<plus />
<ci> V </ci>
<cn cellml:units="millivolt"> 56.7 </cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<times />
<cn cellml:units="dimensionless"> 0.627 </cn>
<apply>
<exp />
<apply>
<times />
<cn cellml:units="dimensionless"> 0.082 </cn>
<apply>
<plus />
<ci> V </ci>
<cn cellml:units="millivolt"> 65.01 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
<cn cellml:units="second"> 0.00004 </cn>
</apply>
</apply>
</math>
</component>
<component name="sodium_current_h_gate">
<variable units="dimensionless" public_interface="out" name="h" />
<variable units="dimensionless" name="F_Na" />
<variable units="dimensionless" name="h1" initial_value="0.03123" />
<variable units="dimensionless" name="h2" initial_value="4.894" />
<variable units="dimensionless" name="h1_infinity" />
<variable units="dimensionless" name="h2_infinity" />
<variable units="second" name="tau_h1" />
<variable units="second" name="tau_h2" />
<variable units="millivolt" public_interface="in" name="V" />
<variable units="second" public_interface="in" name="time" />
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="h_calculation">
<eq />
<ci> h </ci>
<apply>
<plus />
<apply>
<times />
<apply>
<minus />
<cn cellml:units="dimensionless"> 1.0 </cn>
<ci> F_Na </ci>
</apply>
<ci> h1 </ci>
</apply>
<apply>
<times />
<ci> F_Na </ci>
<ci> h2 </ci>
</apply>
</apply>
</apply>
<apply id="h1_diff_eq">
<eq />
<apply>
<diff />
<bvar>
<ci> time </ci>
</bvar>
<ci> h1 </ci>
</apply>
<apply>
<divide />
<apply>
<minus />
<ci> h1_infinity </ci>
<ci> h1 </ci>
</apply>
<ci> tau_h1 </ci>
</apply>
</apply>
<apply id="h2_diff_eq">
<eq />
<apply>
<diff />
<bvar>
<ci> time </ci>
</bvar>
<ci> h2 </ci>
</apply>
<apply>
<divide />
<apply>
<minus />
<ci> h2_infinity </ci>
<ci> h2 </ci>
</apply>
<ci> tau_h2 </ci>
</apply>
</apply>
<apply id="h1_infinity_calculation">
<eq />
<ci> h1_infinity </ci>
<apply>
<divide />
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus />
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp />
<apply>
<divide />
<apply>
<plus />
<ci> V </ci>
<cn cellml:units="millivolt"> 66.1 </cn>
</apply>
<cn cellml:units="millivolt"> 6.4 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply id="h2_infinity_calculation">
<eq />
<ci> h2_infinity </ci>
<ci> h1_infinity </ci>
</apply>
<apply id="tau_h1_calculation">
<eq />
<ci> tau_h1 </ci>
<apply>
<plus />
<apply>
<divide />
<apply>
<times />
<cn cellml:units="second"> 0.000003171 </cn>
<apply>
<exp />
<apply>
<times />
<cn cellml:units="dimensionless"> -0.2815 </cn>
<apply>
<plus />
<ci> V </ci>
<cn cellml:units="millivolt"> 17.11 </cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<plus />
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<times />
<cn cellml:units="second"> 0.003732 </cn>
<apply>
<exp />
<apply>
<times />
<cn cellml:units="dimensionless"> -0.3426 </cn>
<apply>
<plus />
<ci> V </ci>
<cn cellml:units="millivolt"> 37.76 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
<cn cellml:units="second"> 0.0005977 </cn>
</apply>
</apply>
<apply id="tau_h2_calculation">
<eq />
<ci> tau_h2 </ci>
<apply>
<plus />
<apply>
<divide />
<apply>
<times />
<cn cellml:units="second"> 0.00000003186 </cn>
<apply>
<exp />
<apply>
<times />
<cn cellml:units="dimensionless"> -0.6219 </cn>
<apply>
<plus />
<ci> V </ci>
<cn cellml:units="millivolt"> 18.8 </cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<plus />
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<times />
<cn cellml:units="second"> 0.00007189 </cn>
<apply>
<exp />
<apply>
<times />
<cn cellml:units="dimensionless"> -0.6683 </cn>
<apply>
<plus />
<ci> V </ci>
<cn cellml:units="millivolt"> 34.07 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
<cn cellml:units="second"> 0.003556 </cn>
</apply>
</apply>
</math>
</component>
<component cmeta:id="L_type_Ca_channel" name="L_type_Ca_channel">
<variable units="nanoA" public_interface="out" name="i_Ca_L" />
<variable units="microS" name="g_Ca_L" initial_value="0.0659" />
<variable units="millivolt" name="E_Ca_L" initial_value="46.4" />
<variable units="second" public_interface="in" private_interface="out" name="time" />
<variable units="millivolt" public_interface="in" private_interface="out" name="V" />
<variable units="dimensionless" private_interface="in" name="d_L" />
<variable units="dimensionless" private_interface="
CREATED : 28th January 2002
LAST MODIFIED : 20th 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 16/01/2002 CellML Metadata 1.0 Specification.
DESCRIPTION : This file contains a CellML description of Boyett et al's 2001
mathematical model of control of the pacemaker activity of the sinoatrial node.
CHANGES:
25/02/2002 - CML - Corrected several equations.
19/07/2002 - CML - Added more metadata.
20/04/2005 - PJV - Updated syntax to conform with cellml 1.1 specs
-->
<model xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#" xmlns:dcterms="http://purl.org/dc/terms/" xmlns:cellml="http://www.cellml.org/cellml/1.0#" xmlns:bqs="http://www.cellml.org/bqs/1.0#" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:cmeta="http://www.cellml.org/metadata/1.0#" xmlns="http://www.cellml.org/cellml/1.0#" cmeta:id="boyett_zhang_garny_holden_2001_version01" name="boyett_zhang_garny_holden_2001_version01">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
<articleinfo>
<title>Modelling The Effect Of Intracellular Ca2+ On Sinoatrial Node Action Potentials, 2001</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 2000 H. Zhang <emphasis>et al</emphasis> published a mathematical model which describes the action potentials in the periphery and center of the rabbit sinoatrial node. The SA node is functionally, anatomically and electrophysiologically heterogeneous. In their model, Zhang <emphasis>et al</emphasis> accounted for this heterogeneity by defining two distinct models for the peripheral and the central cells of the SA node. These models have the same equations to define the ionic currents but they vary in their parameters.
</para>
<para>
The complete original paper reference is cited below:
</para>
<para>
<ulink url="http://ajpheart.physiology.org/cgi/content/abstract/279/1/H397">Mathematical models of action potentials in the periphery and center of the rabbit sinoatrial node</ulink>, H. Zhang, A.V. Holden, I. Honjo, M. Lei, T. Varghese and M.R. Boyett, 2000, <ulink url="http://ajpheart.physiology.org/">
<emphasis>Am. J. Physiol. Heart Circ. Physiol.</emphasis>
</ulink>, 279, H397-H421. (The <ulink url="http://ajpheart.physiology.org/cgi/content/full/279/1/H397">full text</ulink> of the article is available on the American Journal of Physiology website for Journal Members.) <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=10899081&dopt=Abstract">PubMed ID: 10899081</ulink>
</para>
<para>
In 2001, Boyett <emphasis>et al</emphasis> reviewed the results of several experiments which suggested that intracellular calcium controls a number of ionic currents in the SA node cells. This had not been considered in previous SA node cell models and Boyett <emphasis>et al</emphasis> incorporated an intracellular calcium handling element into the model of Zhang <emphasis>et al</emphasis> (2000). In addition, they included a sustained inward current based on the experiments and equations of Shinagawa <emphasis>et al</emphasis> (2000) (see the figure below). Like in Zhang et al's orginal model, differences between the central and peripheral SA node cells are taken into consideration by defining different parameters. The CellML description here is of the peripheral SA node cell model. To make it appropriate for the central SA node cells, certain parameters need to be changed (refer to the original paper reference which is cited below).
</para>
<para>
Control of the pacemaker activity of the sinoatrial node by intracellular Ca<superscript>2+</superscript>. Experiments and modelling, M.R. Boyett, H. Zhang, A. Garny and A.V. Holden, 2001, <ulink url="http://www.pubs.royalsoc.ac.uk/">
<emphasis>Phil. Trans. R. Soc. Lond. A.</emphasis>
</ulink>, 359, 1091-1110.
</para>
<informalfigure float="0" id="fig_cell_diagram">
<mediaobject>
<imageobject>
<objectinfo>
<title>cell diagram of the Boyett et al SAN model showing ionic currents, pumps and exchangers within the sarcolemma and the sarcoplasmic reticulum</title>
</objectinfo>
<imagedata fileref="../images/B_SAN_model_2001/boyett_2001.png" />
</imageobject>
</mediaobject>
<caption>A schematic diagram describing the current flows across the cell membrane that are captured in the Boyett <emphasis>et al</emphasis> 2001 model of the action potentials in the peripheral cells of the SA node.</caption>
</informalfigure>
<informalfigure float="0" id="fig_cellml_rendering">
<mediaobject>
<imageobject>
<objectinfo>
<title>the cellml rendering of the Boyett et al SAN model</title>
</objectinfo>
<imagedata fileref="../images/B_SAN_model_2001/cellml_rendering.gif" />
</imageobject>
</mediaobject>
<caption>The network defined in the CellML description of the Boyett <emphasis>et al</emphasis> 2001 model. For simplicity, not all the variables are shown.</caption>
</informalfigure>
</sect1>
</article>
</documentation>
<!--
Below, are defined some additional units for association with variables and
constants within the model. The identifiers are fairly self-explanatory.
-->
<units name="per_second">
<unit units="second" 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_second">
<unit units="millivolt" exponent="-1" />
<unit units="second" exponent="-1" />
</units>
<units name="microS">
<unit units="siemens" prefix="micro" />
</units>
<units name="picoF">
<unit units="farad" prefix="pico" />
</units>
<units name="nanoA">
<unit units="ampere" prefix="nano" />
</units>
<units name="millimolar">
<unit units="mole" prefix="milli" />
<unit units="litre" exponent="-1" />
</units>
<units name="per_second_millimolar">
<unit units="second" exponent="-1" />
<unit units="millimolar" exponent="-1" />
</units>
<units name="millimole_per_second">
<unit units="mole" prefix="milli" />
<unit units="second" exponent="-1" />
</units>
<units name="millijoule_per_mole_kelvin">
<unit units="joule" prefix="milli" />
<unit units="mole" exponent="-1" />
<unit units="kelvin" exponent="-1" />
</units>
<units name="coulomb_per_mole">
<unit units="coulomb" exponent="-1" />
<unit units="mole" />
</units>
<units name="microlitre">
<unit units="litre" prefix="micro" />
</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="second" 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">
<variable units="millivolt" public_interface="out" name="V" initial_value="-63.25" />
<variable units="millijoule_per_mole_kelvin" public_interface="out" name="R" initial_value="8314.0" />
<variable units="kelvin" public_interface="out" name="T" initial_value="310.0" />
<variable units="coulomb_per_mole" public_interface="out" name="F" initial_value="96845.0" />
<variable units="picoF" name="C" initial_value="65.0" />
<variable units="second" public_interface="in" name="time" />
<variable units="nanoA" public_interface="in" name="i_Na" />
<variable units="nanoA" public_interface="in" name="i_Ca_L" />
<variable units="nanoA" public_interface="in" name="i_Ca_T" />
<variable units="nanoA" public_interface="in" name="i_to" />
<variable units="nanoA" public_interface="in" name="i_sus" />
<variable units="nanoA" public_interface="in" name="i_K_r" />
<variable units="nanoA" public_interface="in" name="i_K_s" />
<variable units="nanoA" public_interface="in" name="i_f" />
<variable units="nanoA" public_interface="in" name="i_b_Na" />
<variable units="nanoA" public_interface="in" name="i_b_Ca" />
<variable units="nanoA" public_interface="in" name="i_b_K" />
<variable units="nanoA" public_interface="in" name="i_NaCa" />
<variable units="nanoA" public_interface="in" name="i_p" />
<variable units="nanoA" public_interface="in" name="i_st" />
<variable units="nanoA" public_interface="in" name="i_Ca_P" />
<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>
<divide />
<cn cellml:units="dimensionless"> -1.0 </cn>
<ci> C </ci>
</apply>
<apply>
<plus />
<ci> i_Na </ci>
<ci> i_Ca_L </ci>
<ci> i_Ca_T </ci>
<ci> i_to </ci>
<ci> i_sus </ci>
<ci> i_K_r </ci>
<ci> i_K_s </ci>
<ci> i_f </ci>
<ci> i_b_Na </ci>
<ci> i_b_Ca </ci>
<ci> i_b_K </ci>
<ci> i_NaCa </ci>
<ci> i_p </ci>
<ci> i_st </ci>
<ci> i_Ca_P </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component cmeta:id="sodium_current" name="sodium_current">
<variable units="nanoA" public_interface="out" name="i_Na" />
<variable units="microS" name="g_Na" initial_value="0.0000012" />
<variable units="millivolt" public_interface="in" name="E_Na" />
<variable units="millimolar" public_interface="in" name="Na_o" />
<variable units="millijoule_per_mole_kelvin" public_interface="in" name="R" />
<variable units="coulomb_per_mole" public_interface="in" name="F" />
<variable units="kelvin" public_interface="in" name="T" />
<variable units="second" public_interface="in" private_interface="out" name="time" />
<variable units="millivolt" public_interface="in" private_interface="out" name="V" />
<variable units="dimensionless" private_interface="in" name="m" />
<variable units="dimensionless" private_interface="in" name="h" />
<math xmlns="http://www.w3.org/1998/Math/MathML">
<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> Na_o </ci>
<apply>
<divide />
<apply>
<power />
<ci> F </ci>
<cn cellml:units="dimensionless"> 2.0 </cn>
</apply>
<apply>
<times />
<ci> R </ci>
<ci> T </ci>
</apply>
</apply>
<apply>
<divide />
<apply>
<minus />
<apply>
<exp />
<apply>
<divide />
<apply>
<times />
<apply>
<minus />
<ci> V </ci>
<ci> E_Na </ci>
</apply>
<ci> F </ci>
</apply>
<apply>
<times />
<ci> R </ci>
<ci> T </ci>
</apply>
</apply>
</apply>
<cn cellml:units="dimensionless"> 1.0 </cn>
</apply>
<apply>
<minus />
<apply>
<exp />
<apply>
<divide />
<apply>
<times />
<ci> V </ci>
<ci> F </ci>
</apply>
<apply>
<times />
<ci> R </ci>
<ci> T </ci>
</apply>
</apply>
</apply>
<cn cellml:units="dimensionless"> 1.0 </cn>
</apply>
</apply>
<ci> V </ci>
</apply>
</apply>
</math>
</component>
<component name="sodium_current_m_gate">
<variable units="dimensionless" public_interface="out" name="m" initial_value="0.135" />
<variable units="dimensionless" name="m_infinity" />
<variable units="second" name="tau_m" />
<variable units="millivolt" public_interface="in" name="V" />
<variable units="second" public_interface="in" name="time" />
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="m_diff_eq">
<eq />
<apply>
<diff />
<bvar>
<ci> time </ci>
</bvar>
<ci> m </ci>
</apply>
<apply>
<divide />
<apply>
<minus />
<ci> m_infinity </ci>
<ci> m </ci>
</apply>
<ci> tau_m </ci>
</apply>
</apply>
<apply id="m_infinity_calculation">
<eq />
<ci> m_infinity </ci>
<apply>
<power />
<apply>
<divide />
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus />
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp />
<apply>
<divide />
<apply>
<minus />
<ci> V </ci>
</apply>
<cn cellml:units="millivolt"> 5.46 </cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<divide />
<cn cellml:units="dimensionless"> 1.0 </cn>
<cn cellml:units="dimensionless"> 3.0 </cn>
</apply>
</apply>
</apply>
<apply id="tau_m_calculation">
<eq />
<ci> tau_m </ci>
<apply>
<plus />
<apply>
<divide />
<cn cellml:units="second"> 0.6247e-3 </cn>
<apply>
<plus />
<apply>
<times />
<cn cellml:units="dimensionless"> 0.832 </cn>
<apply>
<exp />
<apply>
<times />
<cn cellml:units="dimensionless"> -0.335 </cn>
<apply>
<plus />
<ci> V </ci>
<cn cellml:units="millivolt"> 56.7 </cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<times />
<cn cellml:units="dimensionless"> 0.627 </cn>
<apply>
<exp />
<apply>
<times />
<cn cellml:units="dimensionless"> 0.082 </cn>
<apply>
<plus />
<ci> V </ci>
<cn cellml:units="millivolt"> 65.01 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
<cn cellml:units="second"> 0.00004 </cn>
</apply>
</apply>
</math>
</component>
<component name="sodium_current_h_gate">
<variable units="dimensionless" public_interface="out" name="h" />
<variable units="dimensionless" name="F_Na" />
<variable units="dimensionless" name="h1" initial_value="0.03123" />
<variable units="dimensionless" name="h2" initial_value="4.894" />
<variable units="dimensionless" name="h1_infinity" />
<variable units="dimensionless" name="h2_infinity" />
<variable units="second" name="tau_h1" />
<variable units="second" name="tau_h2" />
<variable units="millivolt" public_interface="in" name="V" />
<variable units="second" public_interface="in" name="time" />
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="h_calculation">
<eq />
<ci> h </ci>
<apply>
<plus />
<apply>
<times />
<apply>
<minus />
<cn cellml:units="dimensionless"> 1.0 </cn>
<ci> F_Na </ci>
</apply>
<ci> h1 </ci>
</apply>
<apply>
<times />
<ci> F_Na </ci>
<ci> h2 </ci>
</apply>
</apply>
</apply>
<apply id="h1_diff_eq">
<eq />
<apply>
<diff />
<bvar>
<ci> time </ci>
</bvar>
<ci> h1 </ci>
</apply>
<apply>
<divide />
<apply>
<minus />
<ci> h1_infinity </ci>
<ci> h1 </ci>
</apply>
<ci> tau_h1 </ci>
</apply>
</apply>
<apply id="h2_diff_eq">
<eq />
<apply>
<diff />
<bvar>
<ci> time </ci>
</bvar>
<ci> h2 </ci>
</apply>
<apply>
<divide />
<apply>
<minus />
<ci> h2_infinity </ci>
<ci> h2 </ci>
</apply>
<ci> tau_h2 </ci>
</apply>
</apply>
<apply id="h1_infinity_calculation">
<eq />
<ci> h1_infinity </ci>
<apply>
<divide />
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus />
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp />
<apply>
<divide />
<apply>
<plus />
<ci> V </ci>
<cn cellml:units="millivolt"> 66.1 </cn>
</apply>
<cn cellml:units="millivolt"> 6.4 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply id="h2_infinity_calculation">
<eq />
<ci> h2_infinity </ci>
<ci> h1_infinity </ci>
</apply>
<apply id="tau_h1_calculation">
<eq />
<ci> tau_h1 </ci>
<apply>
<plus />
<apply>
<divide />
<apply>
<times />
<cn cellml:units="second"> 0.000003171 </cn>
<apply>
<exp />
<apply>
<times />
<cn cellml:units="dimensionless"> -0.2815 </cn>
<apply>
<plus />
<ci> V </ci>
<cn cellml:units="millivolt"> 17.11 </cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<plus />
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<times />
<cn cellml:units="second"> 0.003732 </cn>
<apply>
<exp />
<apply>
<times />
<cn cellml:units="dimensionless"> -0.3426 </cn>
<apply>
<plus />
<ci> V </ci>
<cn cellml:units="millivolt"> 37.76 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
<cn cellml:units="second"> 0.0005977 </cn>
</apply>
</apply>
<apply id="tau_h2_calculation">
<eq />
<ci> tau_h2 </ci>
<apply>
<plus />
<apply>
<divide />
<apply>
<times />
<cn cellml:units="second"> 0.00000003186 </cn>
<apply>
<exp />
<apply>
<times />
<cn cellml:units="dimensionless"> -0.6219 </cn>
<apply>
<plus />
<ci> V </ci>
<cn cellml:units="millivolt"> 18.8 </cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<plus />
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<times />
<cn cellml:units="second"> 0.00007189 </cn>
<apply>
<exp />
<apply>
<times />
<cn cellml:units="dimensionless"> -0.6683 </cn>
<apply>
<plus />
<ci> V </ci>
<cn cellml:units="millivolt"> 34.07 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
<cn cellml:units="second"> 0.003556 </cn>
</apply>
</apply>
</math>
</component>
<component cmeta:id="L_type_Ca_channel" name="L_type_Ca_channel">
<variable units="nanoA" public_interface="out" name="i_Ca_L" />
<variable units="microS" name="g_Ca_L" initial_value="0.0659" />
<variable units="millivolt" name="E_Ca_L" initial_value="46.4" />
<variable units="second" public_interface="in" private_interface="out" name="time" />
<variable units="millivolt" public_interface="in" private_interface="out" name="V" />
<variable units="dimensionless" private_interface="in" name="d_L" />
<variable units="dimensionless" private_interface="