- Author:
- pmr2.import <nobody@models.cellml.org>
- Date:
- 2009-06-17 15:36:39+12:00
- Desc:
- committing version01 of plant_1981
- Permanent Source URI:
- https://staging.physiomeproject.org/workspace/plant_1981/rawfile/2523327e4098a8139b32950c0d5c579217ed79b3/plant_1981.cellml
<?xml version='1.0' encoding='utf-8'?>
<!-- FILE : plant_model_1981.xml
CREATED : 23rd January 2003
LAST MODIFIED : 9th April 2003
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 16/01/2002 CellML Metadata 1.0 Specification.
DESCRIPTION : This file contains a CellML description of Plant's 1981 mathematical model of bursting nerve cells.
CHANGES:
09/04/2003 - AAC - Added publication date information.
--><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="plant_1981_version01" name="plant_1981_version01">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
<articleinfo>
<title>Parabolic Bursting in Neurons</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>
When exposed to a threshold concentration of glucose, pancreatic beta-cells from a wide range of species exhibit a complicated pattern of electrical activity. Bursts of action potential spikes (the "active" phase) are observed, separated by a "silent" phase of membrane repolarisation. At even higher glucose concentrations, continuous action potentials are seen. This electrical activity has two important physiological correlates: increased cytosolic Ca<superscript>2+</superscript> concentration ([Ca<superscript>2+</superscript>]<subscript>i</subscript>) and increased rate of insulin secretion during the active phase. It is generally accepted that the rise in [Ca<superscript>2+</superscript>]<subscript>i</subscript> plays a major role in insulin secretion and that the action potential spikes during a burst are responsible for the rise in [Ca<superscript>2+</superscript>]<subscript>i</subscript>.
</para>
<para>
Bursting in pancreatic beta-cells is a well studied phenomenon, and many mathematical models describing the process have been developed, including:
<itemizedlist>
<listitem>
<para>
<ulink url="${HTML_EXMPL_MITOCHONDRIAL_CA_HANDLING}">Magnus and Keizer, 1997</ulink>
</para>
</listitem>
<listitem>
<para>
<ulink url="${HTML_EXMPL_CHAY_MODEL97}">Chay, 1997</ulink>
</para>
</listitem>
<listitem>
<para>
<ulink url="${HTML_EXMPL_MAGNUS_MODEL}">Magnus and Keizer, 1999</ulink>
</para>
</listitem>
<listitem>
<para>
<ulink url="${HTML_EXMPL_GALL_MODEL}">Gall and Susa, 1999</ulink>
</para>
</listitem>
<listitem>
<para>
<ulink url="${HTML_EXMPL_BERTRAM_MODEL}">Bertram <emphasis>et al.</emphasis>, 2000</ulink>
</para>
</listitem>
</itemizedlist>
</para>
<para>
Another well studied example of busting is found in neurons (for example see the model by <ulink url="${HTML_EXMPL_FRIEL_MODEL}">Friel, 1995</ulink>). Analysis of a detailed mathematical model developed by Plant in 1981 reveals that the structure of this bursting oscillator is different from that in the beta-cell model. Where the beta-cell model has bistability, in Plant's model, oscillations arise from the autonomous activity of two slow variables. The bursting period is almost a parabolic function of time, which has lead to the name <emphasis>parabolic</emphasis> bursting.
</para>
<para>
Plant's model is similar to the beta-cell model in that it has a Ca<superscript>2+</superscript>-activated K<superscript>+</superscript> channel and a voltage-gated K<superscript>+</superscript> channel. However, it is distinct by having a voltage gated Na<superscript>+</superscript> channel and a slowly activating Ca<superscript>2+</superscript> current. The Na<superscript>+</superscript>, K<superscript>+</superscript>, and leak currents form the fast subsystem, while the Ca<superscript>2+</superscript> current forms the slow subsystem (see the figure below for a description of the model).
</para>
<para>
The complete original paper reference is cited below:
</para>
<para>
Bifurcation and resonance in a model for bursting nerve cells, R.E. Plant, 1981, <ulink url="http://link.springer.de/link/service/journals/00285/index.htm">
<emphasis>Journal of Mathematical Biology</emphasis>
</ulink>, 11, 15-32. <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=7252375&dopt=Abstract">PubMed ID: 7252375</ulink>
</para>
<informalfigure float="0" id="fig_cell_diagram">
<mediaobject>
<imageobject>
<objectinfo>
<title>diagram of the model</title>
</objectinfo>
<imagedata fileref="plant_1981.png"/>
</imageobject>
</mediaobject>
<caption>A schematic representation of the transmebrane ionic currents described by the Plant 1981 model of a bursting neuron. The model includes a voltage dependent sodium current, I<subscript>Na</subscript>; a slowly activating calcium current, I<subscript>Ca</subscript>; a voltage gated potassium current, I<subscript>K</subscript>; a calcium activated potassium current, I<subscript>K,Ca</subscript>; and a leak current, I<subscript>L</subscript>.</caption>
</informalfigure>
</sect1>
</article>
</documentation>
<!--
Below, we define some additional units for association with variables and
constants within the model.
-->
<units name="millivolt">
<unit units="volt" prefix="milli"/>
</units>
<units name="per_millivolt">
<unit units="volt" prefix="milli" exponent="-1"/>
</units>
<units name="millisecond">
<unit units="second" prefix="milli"/>
</units>
<units name="per_millivolt_millisecond">
<unit units="volt" prefix="milli" exponent="-1"/>
<unit units="millisecond" exponent="-1"/>
</units>
<units name="per_millisecond">
<unit units="millisecond" exponent="-1"/>
</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>
<units name="microF_per_cm2">
<unit units="farad" prefix="micro"/>
<unit units="metre" prefix="centi" exponent="-2"/>
</units>
<units name="milliS_per_cm2">
<unit units="siemens" prefix="milli"/>
<unit units="metre" prefix="centi" exponent="-2"/>
</units>
<component name="environment">
<variable units="millisecond" public_interface="out" name="time"/>
</component>
<component name="membrane">
<variable units="millivolt" public_interface="out" name="V"/>
<variable units="microF_per_cm2" name="Cm" initial_value="1.0"/>
<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_Ca"/>
<variable units="microA_per_cm2" public_interface="in" name="i_K"/>
<variable units="microA_per_cm2" public_interface="in" name="i_K_Ca"/>
<variable units="microA_per_cm2" public_interface="in" name="i_L"/>
<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>
<divide/>
<apply>
<minus/>
<apply>
<plus/>
<ci> i_Na </ci>
<ci> i_Ca </ci>
<ci> i_K </ci>
<ci> i_K_Ca </ci>
<ci> i_L </ci>
</apply>
</apply>
<ci> Cm </ci>
</apply>
</apply>
</math>
</component>
<component name="sodium_current">
<variable units="microA_per_cm2" public_interface="out" name="i_Na"/>
<variable units="milliS_per_cm2" name="g_Na" initial_value="4.0"/>
<variable units="millivolt" name="V_Na" initial_value="30.0"/>
<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" private_interface="in" name="m_infinity"/>
<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_infinity </ci>
<cn cellml:units="dimensionless"> 3.0 </cn>
</apply>
<ci> h </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> V_Na </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="sodium_current_m_gate">
<variable units="dimensionless" public_interface="out" name="m_infinity"/>
<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/>
<ci> V </ci>
<cn cellml:units="millivolt"> 50.0 </cn>
</apply>
</apply>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<times/>
<cn cellml:units="dimensionless"> -1.0 </cn>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 50.0 </cn>
</apply>
<cn cellml:units="millivolt"> -10.0 </cn>
</apply>
</apply>
</apply>
</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/>
<ci> V </ci>
<cn cellml:units="millivolt"> 25.0 </cn>
</apply>
<cn cellml:units="millivolt"> -18.0 </cn>
</apply>
</apply>
</apply>
</apply>
<apply id="m_infinity_calculation">
<eq/>
<ci> m_infinity </ci>
<apply>
<divide/>
<apply>
<times/>
<ci> alpha_m </ci>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> 1.21 </cn>
<ci> V </ci>
</apply>
<cn cellml:units="dimensionless"> 78.71 </cn>
</apply>
</apply>
<apply>
<plus/>
<apply>
<times/>
<ci> alpha_m </ci>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> 1.21 </cn>
<ci> V </ci>
</apply>
<cn cellml:units="dimensionless"> 78.71 </cn>
</apply>
</apply>
<apply>
<times/>
<ci> beta_m </ci>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> 1.21 </cn>
<ci> V </ci>
</apply>
<cn cellml:units="dimensionless"> 78.71 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="sodium_current_h_gate">
<variable units="dimensionless" public_interface="out" name="h"/>
<variable units="dimensionless" name="h_infinity"/>
<variable units="per_millisecond" name="alpha_h"/>
<variable units="per_millisecond" name="beta_h"/>
<variable units="millisecond" name="tau_h"/>
<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/>
<ci> V </ci>
<cn cellml:units="millivolt"> 25.0 </cn>
</apply>
<cn cellml:units="millivolt"> -20.0 </cn>
</apply>
</apply>
</apply>
</apply>
<apply id="beta_h_calculation">
<eq/>
<ci> beta_h </ci>
<apply>
<divide/>
<apply>
<times/>
<cn cellml:units="per_millisecond"> 0.07 </cn>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 55.0 </cn>
</apply>
<cn cellml:units="millivolt"> 10.0 </cn>
</apply>
</apply>
</apply>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 55.0 </cn>
</apply>
<cn cellml:units="millivolt"> 10.0 </cn>
</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/>
<ci> h_infinity </ci>
<ci> h </ci>
</apply>
<ci> tau_h </ci>
</apply>
</apply>
<apply id="h_infinity_calculation">
<eq/>
<ci> h_infinity </ci>
<apply>
<divide/>
<apply>
<times/>
<ci> alpha_h </ci>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> 1.21 </cn>
<ci> V </ci>
</apply>
<cn cellml:units="dimensionless"> 78.71 </cn>
</apply>
</apply>
<apply>
<plus/>
<apply>
<times/>
<ci> alpha_h </ci>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> 1.21 </cn>
<ci> V </ci>
</apply>
<cn cellml:units="dimensionless"> 78.71 </cn>
</apply>
</apply>
<apply>
<times/>
<ci> beta_h </ci>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> 1.21 </cn>
<ci> V </ci>
</apply>
<cn cellml:units="dimensionless"> 78.71 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply id="tau_h_calculation">
<eq/>
<ci> tau_h </ci>
<apply>
<divide/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<times/>
<cn cellml:units="dimensionless"> 0.08 </cn>
<apply>
<plus/>
<apply>
<times/>
<ci> alpha_h </ci>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> 1.21 </cn>
<ci> V </ci>
</apply>
<cn cellml:units="dimensionless"> 78.71 </cn>
</apply>
</apply>
<apply>
<times/>
<ci> beta_h </ci>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> 1.21 </cn>
<ci> V </ci>
</apply>
<cn cellml:units="dimensionless"> 78.71 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="calcium_current">
<variable units="microA_per_cm2" public_interface="out" name="i_Ca"/>
<variable units="millimolar" public_interface="out" name="c"/>
<variable units="millivolt" name="V_Ca" initial_value="140.0"/>
<variable units="milliS_per_cm2" name="g_Ca" initial_value="0.004"/>
<variable units="per_millisecond" name="f" initial_value="0.0003"/>
<variable units="per_millivolt" name="k1" initial_value="0.0085"/>
<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" private_interface="in" name="x"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="i_Ca_calculation">
<eq/>
<ci> i_Ca </ci>
<apply>
<times/>
<ci> g_Ca </ci>
<ci> x </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> V_Ca </ci>
</apply>
</apply>
</apply>
<apply id="dc_dt">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> c </ci>
</apply>
<apply>
<times/>
<ci> f </ci>
<apply>
<minus/>
<apply>
<times/>
<ci> k1 </ci>
<ci> x </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> V_Ca </ci>
</apply>
</apply>
<ci> c </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="calcium_current_x_gate">
<variable units="dimensionless" public_interface="out" name="x"/>
<variable units="dimensionless" name="x_infinity"/>
<variable units="millisecond" name="tau_x" initial_value="235.0"/>
<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_dt">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> x </ci>
</apply>
<apply>
<divide/>
<apply>
<minus/>
<ci> x_infinity </ci>
<ci> x </ci>
</apply>
<ci> tau_x </ci>
</apply>
</apply>
<apply id="x_infinity_calculation">
<eq/>
<ci> x_infinity </ci>
<apply>
<divide/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus/>
<apply>
<exp/>
<apply>
<times/>
<cn cellml:units="dimensionless"> -0.15 </cn>
<apply>
<plus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 50.0 </cn>
</apply>
</apply>
</apply>
<cn cellml:units="dimensionless"> 1.0 </cn>
</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="V_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" 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/>
<apply>
<times/>
<ci> g_K </ci>
<apply>
<power/>
<ci> n </ci>
<cn cellml:units="dimensionless"> 4.0 </cn>
</apply>
</apply>
<apply>
<minus/>
<ci> V </ci>
<ci> V_K </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="potassium_current_n_gate">
<variable units="dimensionless" public_interface="out" name="n"/>
<variable units="dimensionless" name="n_infinity"/>
<variable units="per_millisecond" name="alpha_n"/>
<variable units="per_millisecond" name="beta_n"/>
<variable units="millisecond" name="tau_n"/>
<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_millisecond"> 0.07 </cn>
<apply>
<minus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 55.0 </cn>
</apply>
</apply>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<ci> V </ci>
<cn cellml:units="millivolt"> 55.0 </cn>
</apply>
<cn cellml:units="millivolt"> -10.0 </cn>
</apply>
</apply>
</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/>
<ci> V </ci>
<cn cellml:units="millivolt"> 45.0 </cn>
</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/>
<ci> n_infinity </ci>
<ci> n </ci>
</apply>
<ci> tau_n </ci>
</apply>
</apply>
<apply id="n_infinity_calculation">
<eq/>
<ci> n_infinity </ci>
<apply>
<divide/>
<apply>
<times/>
<ci> alpha_n </ci>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> 1.21 </cn>
<ci> V </ci>
</apply>
<cn cellml:units="dimensionless"> 78.71 </cn>
</apply>
</apply>
<apply>
<plus/>
<apply>
<times/>
<ci> alpha_n </ci>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> 1.21 </cn>
<ci> V </ci>
</apply>
<cn cellml:units="dimensionless"> 78.71 </cn>
</apply>
</apply>
<apply>
<times/>
<ci> beta_n </ci>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> 1.21 </cn>
<ci> V </ci>
</apply>
<cn cellml:units="dimensionless"> 78.71 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply id="tau_n_calculation">
<eq/>
<ci> tau_n </ci>
<apply>
<divide/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<times/>
<cn cellml:units="dimensionless"> 0.08 </cn>
<apply>
<plus/>
<apply>
<times/>
<ci> alpha_n </ci>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="per_millivolt"> 1.21 </cn>
<ci> V </ci>
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This is the CellML description of Plant's 1981 mathematical model of
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Plant's 1981 mathematical model of bursting nerve cells.
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Bifurcation and resonance in a model for bursting nerve cells
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The University of Auckland, Bioengineering Institute
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