<?xml version='1.0' encoding='utf-8'?>
<!-- FILE : miftakhov_model_1999.xml
CREATED : 11th July 2002
LAST MODIFIED : 27th April 2005
AUTHOR : Catherine Lloyd
The Bioengineering Institute
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 16th
January, 2002.
DESCRIPTION : This file contains a CellML description of Miftakhov et al's 1999 mathematical model of the electrical activity of a smooth muscle cell from the small intestine.
CHANGES:
18/07/2002 - CML - Added more metadata
09/04/2003 - AAC - Added publication date information.
27/04/2005 - PJV - Changed initial values to make them valid cellml.
--><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="miftakhov_abdusheva_christensen_1999_version01" name="miftakhov_abdusheva_christensen_1999_version01">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
<articleinfo>
<title>Motility Patterns of the Small Bowel</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>
Intestinal motility describes the rhythmic, propulsive contractile activity of the distal stomach, intestine and colon. Peristaltic waves are generated through interplay between the enteric nervous system, smooth muscle cells and interstitial cells of Cajal (ICC). ICC are critical for initiating slow wave activity, making ICC the <emphasis>pacemakers</emphasis> of the gut. The slow wave membrane depolarisations, generated by ICC, determine the maximum frequency and the propagation characteristics of the intestinal rhythmic contractions.
</para>
<para>
Smooth muscle contraction is the fundamental event in gastrointestinal motion. Smooth muscle cells respond to the slow wave activity imposed by ICC pacemaker cells. Although many of the biochemical mechanisms underlying the excitation-contraction coupling are not yet defined, it is known that cytosolic Ca<superscript>2+</superscript> is the essential component in the coupling phenomenon. Slow wave membrane depolarisation activates smooth muscle contraction, mainly by triggering an L-type Ca<superscript>2+</superscript> current. This is the main source of Ca<superscript>2+</superscript> for contraction.
</para>
<para>
Since the late 1960s, slow wave propagation in the gastrointestinal tract has been the subject of mathematical models. In 1968 Nelson and Becker proposed that a chain of relaxation oscillators could simulate the electromechanical activity in the small intestine. For the next two decades this idea persisted. However, more recently core conductor models, which consider the ionic fluxes and the electrical activity of individual cells, have been developed.
</para>
<para>
In 1999, Miftakhov <emphasis>et al.</emphasis> published a mathematical model, which aimed to simulate motility patterns of the small bowel. Within the functional unit of an electromyogenic syncytium are equations that represent the dynamics of ion channels in smooth muscle cells. These include voltage-gated Ca<superscript>2+</superscript> channels (L and T-type), a voltage-gated K<superscript>+</superscript> channel, a calcium-activated K<superscript>+</superscript> channel, and a leak chloride current. The kinetics of these ion channels determine the electrical activity of the whole functional unit.
</para>
<para>
The complete original paper reference is cited below:
</para>
<para>
<ulink url="http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WMD-45FSB7Y-53&_coverDate=03%2F07%2F1999&_alid=187124419&_rdoc=1&_fmt=&_orig=search&_qd=1&_cdi=6932&_sort=d&view=c&_acct=C000011498&_version=1&_urlVersion=0&_userid=140507&md5=abeef0b8b2b3eb46ad90e8680570122c">Numerical Simulation of Motility Patterns of the Small Bowel. 1. Formulation of a Mathematical Model</ulink>, R. N. Miftakhov, G. R. Abdusheva and J. Christensen, 1999, <ulink url="http://www.sciencedirect.com/science?_ob=JournalURL&_cdi=6932&_auth=y&_acct=C000011498&_version=1&_urlVersion=0&_userid=140507&md5=c8e64d20ff203cda2255f4b6b7affe4f">
<emphasis>Journal of Theoretical Biology</emphasis>
</ulink>, 197, 89-112. (A PDF version of the article is available to subscribers on the <emphasis>Science Direct</emphasis>.) <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=10036210&dopt=Abstract">PubMed ID: 10036210</ulink>
</para>
<para>
The raw CellML descriptions of the model can be downloaded in various formats as described in <xref linkend="sec_download_this_model"/>.
</para>
<informalfigure float="0" id="fig_cell_diagram">
<mediaobject>
<imageobject>
<objectinfo>
<title>cell_diagram</title>
</objectinfo>
<imagedata fileref="miftakhov_1999.png"/>
</imageobject>
</mediaobject>
<caption>The myoelectrical activity of the gastrointestinal smooth muscle cell is governed by the dynamics of voltage-dependent Ca<superscript>2+</superscript> L and T-type channels (<emphasis>I<subscript>Ca,L</subscript> and I<subscript>Ca,T</subscript>
</emphasis>), a voltage-gated K<superscript>+</superscript> channel (<emphasis>I<subscript>K</subscript>
</emphasis>), a calcium-activated K<superscript>+</superscript> channel (<emphasis>I<subscript>Ca-K</subscript>
</emphasis>), and a leak chloride current (<emphasis>I<subscript>Cl</subscript>
</emphasis>).</caption>
</informalfigure>
</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="millimolar">
<unit units="mole" prefix="milli"/>
<unit units="litre" exponent="-1"/>
</units>
<units name="microA_per_cm2">
<unit units="ampere" prefix="micro"/>
<unit units="metre" prefix="centi" exponent="-2"/>
</units>
<component name="environment">
<variable units="millisecond" public_interface="out" name="time"/>
</component>
<component name="model_constants">
<variable units="dimensionless" public_interface="out" name="alpha" initial_value="0.12"/>
<variable units="dimensionless" public_interface="out" name="lamda" initial_value="12.5"/>
</component>
<component name="membrane">
<variable units="millivolt" public_interface="out" name="V" initial_value="-55.0"/>
<variable units="microF_per_cm2" name="Cm" initial_value="2.5"/>
<variable units="millisecond" public_interface="in" name="time"/>
<variable units="dimensionless" public_interface="in" name="alpha"/>
<variable units="microA_per_cm2" public_interface="in" name="i_Ca_T"/>
<variable units="microA_per_cm2" public_interface="in" name="i_Ca_L"/>
<variable units="microA_per_cm2" public_interface="in" name="i_Ca_K"/>
<variable units="microA_per_cm2" public_interface="in" name="i_K"/>
<variable units="microA_per_cm2" public_interface="in" name="i_Cl"/>
<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>
<apply>
<times/>
<ci> Cm </ci>
<ci> alpha </ci>
</apply>
</apply>
</apply>
<apply>
<plus/>
<ci> i_Ca_T </ci>
<ci> i_Ca_L </ci>
<ci> i_Ca_K </ci>
<ci> i_K </ci>
<ci> i_Cl </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="T_type_calcium_current">
<variable units="microA_per_cm2" public_interface="out" name="i_Ca_T"/>
<variable units="millivolt" public_interface="out" name="E_Ca" initial_value="80.0"/>
<variable units="milliS_per_cm2" name="g_Ca_T" initial_value="0.51"/>
<variable units="millisecond" public_interface="in" private_interface="out" name="time"/>
<variable units="millivolt" public_interface="in" private_interface="out" name="V"/>
<variable units="dimensionless" public_interface="in" private_interface="out" name="alpha"/>
<variable units="dimensionless" public_interface="in" private_interface="out" name="lamda"/>
<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_Ca_T_calculation">
<eq/>
<ci> i_Ca_T </ci>
<apply>
<times/>
<ci> g_Ca_T </ci>
<apply>
<power/>
<ci> m </ci>
<cn cellml:units="dimensionless"> 3.0 </cn>
</apply>
<ci> h </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> E_Ca </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="T_type_calcium_current_m_gate">
<variable units="dimensionless" public_interface="out" name="m"/>
<variable units="per_millisecond" name="alpha_m"/>
<variable units="per_millisecond" name="beta_m"/>
<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_m_calculation">
<eq/>
<ci> alpha_m </ci>
<apply>
<divide/>
<apply>
<times/>
<cn cellml:units="per_millivolt_millisecond"> 0.1 </cn>
<apply>
<minus/>
<cn cellml:units="millivolt"> 50.0 </cn>
<ci> V </ci>
</apply>
</apply>
<apply>
<minus/>
<apply>
<exp/>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 5.0 </cn>
<apply>
<times/>
<ci> V </ci>
<cn cellml:units="dimensionless"> 0.1 </cn>
</apply>
</apply>
</apply>
<cn cellml:units="dimensionless"> 1.0 </cn>
</apply>
</apply>
</apply>
<apply id="beta_m_calculation">
<eq/>
<ci> beta_m </ci>
<apply>
<times/>
<cn cellml:units="per_millisecond"> 4.0 </cn>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<cn cellml:units="millivolt"> 25.0 </cn>
<ci> V </ci>
</apply>
<cn cellml:units="millivolt"> 18.0 </cn>
</apply>
</apply>
</apply>
</apply>
<apply id="dm_dt">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> m </ci>
</apply>
<apply>
<divide/>
<ci> alpha_m </ci>
<apply>
<plus/>
<ci> alpha_m </ci>
<ci> beta_m </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="T_type_calcium_current_h_gate">
<variable units="dimensionless" public_interface="out" name="h"/>
<variable units="per_millisecond" name="alpha_h"/>
<variable units="per_millisecond" name="beta_h"/>
<variable units="dimensionless" public_interface="in" name="alpha"/>
<variable units="dimensionless" public_interface="in" name="lamda"/>
<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_h_calculation">
<eq/>
<ci> alpha_h </ci>
<apply>
<times/>
<cn cellml:units="per_millisecond"> 0.07 </cn>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<cn cellml:units="millivolt"> 25.0 </cn>
<ci> V </ci>
</apply>
<cn cellml:units="millivolt"> 20.0 </cn>
</apply>
</apply>
</apply>
</apply>
<apply id="beta_h_calculation">
<eq/>
<ci> beta_h </ci>
<apply>
<divide/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus/>
<cn cellml:units="millisecond"> 1.0 </cn>
<apply>
<exp/>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 5.5 </cn>
<apply>
<times/>
<ci> V </ci>
<cn cellml:units="dimensionless"> 0.1 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply id="dh_dt">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> h </ci>
</apply>
<apply>
<divide/>
<apply>
<minus/>
<apply>
<times/>
<ci> alpha_h </ci>
<apply>
<minus/>
<cn cellml:units="per_millivolt"> 1.0 </cn>
<ci> h </ci>
</apply>
</apply>
<apply>
<times/>
<ci> beta_h </ci>
<ci> h </ci>
</apply>
</apply>
<apply>
<times/>
<ci> alpha </ci>
<ci> lamda </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="L_type_calcium_current">
<variable units="microA_per_cm2" public_interface="out" name="i_Ca_L"/>
<variable units="milliS_per_cm2" name="g_Ca_L" initial_value="0.004"/>
<variable units="millivolt" public_interface="in" name="E_Ca"/>
<variable units="dimensionless" public_interface="in" private_interface="out" name="alpha"/>
<variable units="millisecond" public_interface="in" private_interface="out" name="time"/>
<variable units="millivolt" public_interface="in" private_interface="out" name="V"/>
<variable units="dimensionless" public_interface="out" private_interface="in" name="x_Ca"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="i_Ca_L_calculation">
<eq/>
<ci> i_Ca_L </ci>
<apply>
<times/>
<ci> g_Ca_L </ci>
<ci> x_Ca </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> E_Ca </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="L_type_calcium_current_x_Ca_gate">
<variable units="dimensionless" public_interface="out" name="x_Ca"/>
<variable units="millisecond" name="tau_x_Ca" initial_value="500.0"/>
<variable units="dimensionless" public_interface="in" name="alpha"/>
<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="dx_Ca_dt">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> x_Ca </ci>
</apply>
<apply>
<divide/>
<apply>
<minus/>
<apply>
<divide/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="dimensionless"> 0.15 </cn>
<apply>
<minus/>
<apply>
<minus/>
<ci> V </ci>
</apply>
<cn cellml:units="dimensionless"> 50.0 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
<ci> x_Ca </ci>
</apply>
<apply>
<times/>
<ci> alpha </ci>
<ci> tau_x_Ca </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="potassium_current">
<variable units="microA_per_cm2" public_interface="out" name="i_K"/>
<variable units="millivolt" public_interface="out" name="E_K" initial_value="-75.0"/>
<variable units="milliS_per_cm2" name="g_K" initial_value="0.3"/>
<variable units="millisecond" public_interface="in" private_interface="out" name="time"/>
<variable units="millivolt" public_interface="in" private_interface="out" name="V"/>
<variable units="dimensionless" public_interface="in" private_interface="out" name="alpha"/>
<variable units="dimensionless" public_interface="in" private_interface="out" name="lamda"/>
<variable units="dimensionless" private_interface="in" name="n"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="i_K_calculation">
<eq/>
<ci> i_K </ci>
<apply>
<times/>
<ci> g_K </ci>
<apply>
<power/>
<ci> n </ci>
<cn cellml:units="dimensionless"> 4.0 </cn>
</apply>
<apply>
<minus/>
<ci> V </ci>
<ci> E_K </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="potassium_current_n_gate">
<variable units="dimensionless" public_interface="out" name="n"/>
<variable units="per_millisecond" name="alpha_n"/>
<variable units="per_millisecond" name="beta_n"/>
<variable units="dimensionless" public_interface="in" name="alpha"/>
<variable units="dimensionless" public_interface="in" name="lamda"/>
<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_n_calculation">
<eq/>
<ci> alpha_n </ci>
<apply>
<divide/>
<apply>
<times/>
<cn cellml:units="per_millivolt_millisecond"> 0.01 </cn>
<apply>
<minus/>
<cn cellml:units="millivolt"> 55.0 </cn>
<ci> V </ci>
</apply>
</apply>
<apply>
<minus/>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<cn cellml:units="millivolt"> 55.0 </cn>
<ci> V </ci>
</apply>
<cn cellml:units="millivolt"> 10.0 </cn>
</apply>
</apply>
<cn cellml:units="dimensionless"> 1.0 </cn>
</apply>
</apply>
</apply>
<apply id="beta_n_calculation">
<eq/>
<ci> beta_n </ci>
<apply>
<times/>
<cn cellml:units="per_millisecond"> 0.125 </cn>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<cn cellml:units="millivolt"> 45.0 </cn>
<ci> V </ci>
</apply>
<cn cellml:units="millivolt"> 80.0 </cn>
</apply>
</apply>
</apply>
</apply>
<apply id="dn_dt">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> n </ci>
</apply>
<apply>
<divide/>
<apply>
<minus/>
<apply>
<times/>
<ci> alpha_n </ci>
<apply>
<minus/>
<cn cellml:units="per_millivolt"> 1.0 </cn>
<ci> n </ci>
</apply>
</apply>
<apply>
<times/>
<ci> beta_n </ci>
<ci> n </ci>
</apply>
</apply>
<apply>
<times/>
<ci> alpha </ci>
<ci> lamda </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="calcium_activated_potassium_current">
<variable units="microA_per_cm2" public_interface="out" name="i_Ca_K"/>
<variable units="millimolar" name="Ca" initial_value="1E-4"/>
<variable units="milliS_per_cm2" name="g_Ca_K" initial_value="0.03"/>
<variable units="per_millisecond" name="rho" initial_value="0.125E3"/>
<variable units="per_millivolt" name="K_c" initial_value="425.0E-5"/>
<variable units="dimensionless" public_interface="in" name="alpha"/>
<variable units="millivolt" public_interface="in" name="E_K"/>
<variable units="millivolt" public_interface="in" name="E_Ca"/>
<variable units="millisecond" public_interface="in" name="time"/>
<variable units="millivolt" public_interface="in" name="V"/>
<variable units="dimensionless" public_interface="in" name="x_Ca"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="i_Ca_K_calculation">
<eq/>
<ci> i_Ca_K </ci>
<apply>
<divide/>
<apply>
<times/>
<ci> g_Ca_K </ci>
<ci> Ca </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> E_K </ci>
</apply>
</apply>
<apply>
<plus/>
<cn cellml:units="millimolar"> 0.5 </cn>
<ci> Ca </ci>
</apply>
</apply>
</apply>
<apply id="dCa_dt">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> Ca </ci>
</apply>
<apply>
<times/>
<apply>
<divide/>
<ci> rho </ci>
<ci> alpha </ci>
</apply>
<apply>
<minus/>
<apply>
<times/>
<ci> K_c </ci>
<ci> x_Ca </ci>
<apply>
<minus/>
<ci> E_Ca </ci>
<ci> V </ci>
</apply>
</apply>
<ci> Ca </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="leak_chloride_current">
<variable units="microA_per_cm2" public_interface="out" name="i_Cl"/>
<variable units="milliS_per_cm2" name="g_Cl" initial_value="0.003"/>
<variable units="millivolt" name="E_Cl" initial_value="-40.0"/>
<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">
<apply id="i_Cl_calculation">
<eq/>
<ci> i_Cl </ci>
<apply>
<times/>
<ci> g_Cl </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> E_Cl </ci>
</apply>
</apply>
</apply>
</math>
</component>
<group>
<relationship_ref relationship="containment"/>
<component_ref component="membrane">
<component_ref component="T_type_calcium_current">
<component_ref component="T_type_calcium_current_m_gate"/>
<component_ref component="T_type_calcium_current_h_gate"/>
</component_ref>
<component_ref component="L_type_calcium_current">
<component_ref component="L_type_calcium_current_x_Ca_gate"/>
</component_ref>
<component_ref component="potassium_current">
<component_ref component="potassium_current_n_gate"/>
</component_ref>
<component_ref component="calcium_activated_potassium_current"/>
<component_ref component="leak_chloride_current"/>
</component_ref>
</group>
<group>
<relationship_ref relationship="encapsulation"/>
<component_ref component="T_type_calcium_current">
<component_ref component="T_type_calcium_current_m_gate"/>
<component_ref component="T_type_calcium_current_h_gate"/>
</component_ref>
<component_ref component="L_type_calcium_current">
<component_ref component="L_type_calcium_current_x_Ca_gate"/>
</component_ref>
<component_ref component="potassium_current">
<component_ref component="potassium_current_n_gate"/>
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Numerical Simulation of Motility Patterns of the Small Bowel. I. Formulation of a Mathematical Model
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