<?xml version="1.0"?>
<model xmlns="http://www.cellml.org/cellml/1.1#" xmlns:cmeta="http://www.cellml.org/metadata/1.0#" cmeta:id="butera_single_cell_1999" name="butera_single_cell_1999">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
<articleinfo>
<title>Models of respiratory rhythm generation in the Pre-Botzinger complex. II. Populations of coupled pacemaker neurons</title>
<author>
<firstname>Catherine</firstname>
<surname>Lloyd</surname>
<affiliation>
<shortaffil>Auckland Bioengineering Institute, The University of Auckland</shortaffil>
</affiliation>
</author>
</articleinfo>
<section id="sec_status">
<title>Model Status</title>
<para>
This CellML model represents a single neuron in a network of cells. This cell model is based on "model 1" published in a preceding paper (Butera et al. 1999) which is also described in CellML as a stand alone CellML 1.0 model. To model inter-cellular coupling a synaptic input current I<subscript>syn_e</subscript> has been added to the model.
</para>
</section>
<sect1 id="sec_structure">
<title>Model Structure</title>
<informalfigure float="0" id="fig_reaction_diagram">
<mediaobject>
<imageobject>
<objectinfo>
<title>model diagram</title>
</objectinfo>
<imagedata fileref="butera_1999.png"/>
</imageobject>
</mediaobject>
<caption>The single cell neuron model is based on a single-compartment Hodgkin-Huxley type formalism. It is composed of five ionic currents across the plasma membrane: a fast sodium current, I<subscript>Na</subscript>; a delayed rectifier potassium current, I<subscript>K</subscript>; a persistent sodium current, I<subscript>NaP</subscript>; a passive leakage current, I<subscript>L</subscript>; and a tonic current, I<subscript>tonic_e</subscript> In addition for the multicellular model a synaptic current (I<subscript>syn_e</subscript>) has been added to connect the cells in the network.</caption>
</informalfigure>
<para>
The original paper reference is cited below:
</para>
<para>
Models of respiratory rhythm generation in the Pre-Botzinger complex. II. Populations Of coupled pacemaker neurons, Butera RJ Jr, Rinzel J, Smith JC, 1999, <emphasis>Journal of Neurophysiology</emphasis>, 82, 398-415. <ulink url="http://www.ncbi.nlm.nih.gov/pubmed/10400967">PubMed ID: 10400967</ulink>
</para>
</sect1>
</article>
</documentation>
<cmeta:comment xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" rdf:parseType="Resource">
<rdf:value>
This CellML model describes the single cell (neuron) of the Butera et al. 1999 model. This model is imported by the multicellular model 10 times.
</rdf:value>
<dc:creator xmlns:dc="http://purl.org/dc/elements/1.1/" rdf:parseType="Resource">
<vCard:FN xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Catherine Lloyd</vCard:FN>
</dc:creator>
</cmeta:comment>
<units name="millisecond">
<unit prefix="milli" units="second"/>
</units>
<units name="millivolt">
<unit prefix="milli" units="volt"/>
</units>
<units name="picoA">
<unit prefix="nano" units="ampere"/>
</units>
<units name="nanoS">
<unit prefix="nano" units="siemens"/>
</units>
<units name="picoF">
<unit prefix="pico" units="farad"/>
</units>
<cmeta:comment xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" rdf:parseType="Resource">
<rdf:value>
In order to facilitate the import of this single cell model by the multicellular model the current model is encapsulated in a top component called "single_neuron_model".
</rdf:value>
<dc:creator xmlns:dc="http://purl.org/dc/elements/1.1/" rdf:parseType="Resource">
<vCard:FN xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Catherine Lloyd</vCard:FN>
</dc:creator>
</cmeta:comment>
<component cmeta:id="single_neuron_model" name="single_neuron_model">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/0P/PP-rdf-syntax-ns#">
<rdf:Description rdf:about="#single_neuron_model">
<dc:title xmlns:dc="http://purl.org/dc/elements/1.1/">single_neuron_model</dc:title>
<dcterms:alternative xmlns:dcterms="http://purl.org/dc/terms/">top component which encapsulates the single neuron cell model</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable name="s" private_interface="in" public_interface="out" units="dimensionless"/>
<variable name="sum_g_syn_e_s" private_interface="out" public_interface="in" units="nanoS"/>
</component>
<connection>
<map_components component_1="single_neuron_model" component_2="synaptic_input"/>
<map_variables variable_1="sum_g_syn_e_s" variable_2="sum_g_syn_e_s"/>
<map_variables variable_1="s" variable_2="s"/>
</connection>
<component name="environment">
<variable name="time" public_interface="out" units="millisecond"/>
</component>
<component name="membrane">
<variable initial_value="-50.0" name="V" public_interface="out" units="millivolt"/>
<variable initial_value="21.0" name="C" units="picoF"/>
<variable initial_value="0.0" name="i_app" units="picoA"/>
<variable name="time" public_interface="in" units="millisecond"/>
<variable name="i_NaP" public_interface="in" units="picoA"/>
<variable name="i_Na" public_interface="in" units="picoA"/>
<variable name="i_K" public_interface="in" units="picoA"/>
<variable name="i_L" public_interface="in" units="picoA"/>
<variable name="i_tonic_e" public_interface="in" units="picoA"/>
<variable name="i_syn_e" public_interface="in" units="picoA"/>
<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>
<plus/>
<apply>
<minus/>
<apply>
<plus/>
<ci> i_NaP </ci>
<ci> i_Na </ci>
<ci> i_K </ci>
<ci> i_L </ci>
<ci> i_tonic_e </ci>
<ci> i_syn_e </ci>
</apply>
</apply>
<ci> i_app </ci>
</apply>
<ci> C </ci>
</apply>
</apply>
</math>
</component>
<component name="fast_sodium_current">
<variable name="i_Na" public_interface="out" units="picoA"/>
<variable initial_value="50.0" name="E_Na" public_interface="out" units="millivolt"/>
<variable initial_value="28.0" name="g_Na" units="nanoS"/>
<variable name="time" private_interface="out" public_interface="in" units="millisecond"/>
<variable name="V" private_interface="out" public_interface="in" units="millivolt"/>
<variable name="m_infinity" private_interface="in" units="dimensionless"/>
<variable name="n" private_interface="in" units="dimensionless"/>
<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 xmlns:cellml="http://www.cellml.org/cellml/1.1#" cellml:units="dimensionless"> 3.0 </cn>
</apply>
<apply>
<minus/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.1#" cellml:units="dimensionless"> 1.0 </cn>
<ci> n </ci>
</apply>
<apply>
<minus/>
<ci> V </ci>
<ci> E_Na </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="fast_sodium_current_m_gate">
<variable name="m_infinity" public_interface="out" units="dimensionless"/>
<variable initial_value="-34.0" name="theta_m" units="millivolt"/>
<variable initial_value="-5.0" name="sigma_m" units="millivolt"/>
<variable name="V" public_interface="in" units="millivolt"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="fast_sodium_current_m_gate_m_infinity_calculation">
<eq/>
<ci> m_infinity </ci>
<apply>
<divide/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.1#" cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.1#" cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<ci> V </ci>
<ci> theta_m </ci>
</apply>
<ci> sigma_m </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="fast_sodium_current_n_gate">
<variable initial_value="0.01" name="n" public_interface="out" units="dimensionless"/>
<variable name="n_infinity" units="dimensionless"/>
<variable name="tau_n" units="millisecond"/>
<variable initial_value="10.0" name="tau_n_max" units="millisecond"/>
<variable initial_value="-29.0" name="theta_n" units="millivolt"/>
<variable initial_value="-4.0" name="sigma_n" units="millivolt"/>
<variable name="V" public_interface="in" units="millivolt"/>
<variable name="time" public_interface="in" units="millisecond"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="fast_sodium_current_n_gate_n_diff_eq">
<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="fast_sodium_current_n_gate_n_infinity_calculation">
<eq/>
<ci> n_infinity </ci>
<apply>
<divide/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.1#" cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.1#" cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<ci> V </ci>
<ci> theta_n </ci>
</apply>
<ci> sigma_n </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply id="fast_sodium_current_n_gate_tau_n_calculation">
<eq/>
<ci> tau_n </ci>
<apply>
<divide/>
<ci> tau_n_max </ci>
<apply>
<cosh/>
<apply>
<divide/>
<apply>
<minus/>
<ci> V </ci>
<ci> theta_n </ci>
</apply>
<apply>
<times/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.1#" cellml:units="dimensionless"> 2.0 </cn>
<ci> sigma_n </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="potassium_current">
<variable name="i_K" public_interface="out" units="picoA"/>
<variable initial_value="11.2" name="g_K" units="nanoS"/>
<variable initial_value="-85.0" name="E_K" units="millivolt"/>
<variable name="time" private_interface="out" public_interface="in" units="millisecond"/>
<variable name="V" private_interface="out" public_interface="in" units="millivolt"/>
<variable name="n" private_interface="in" units="dimensionless"/>
<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 xmlns:cellml="http://www.cellml.org/cellml/1.1#" 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 initial_value="0.01" name="n" public_interface="out" units="dimensionless"/>
<variable name="n_infinity" units="dimensionless"/>
<variable name="tau_n" units="millisecond"/>
<variable initial_value="10.0" name="tau_n_max" units="millisecond"/>
<variable initial_value="-29.0" name="theta_n" units="millivolt"/>
<variable initial_value="-4.0" name="sigma_n" units="millivolt"/>
<variable name="V" public_interface="in" units="millivolt"/>
<variable name="time" public_interface="in" units="millisecond"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="potassium_current_n_gate_n_diff_eq">
<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="potassium_current_n_gate_n_infinity_calculation">
<eq/>
<ci> n_infinity </ci>
<apply>
<divide/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.1#" cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.1#" cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<ci> V </ci>
<ci> theta_n </ci>
</apply>
<ci> sigma_n </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply id="potassium_current_n_gate_tau_n_calculation">
<eq/>
<ci> tau_n </ci>
<apply>
<divide/>
<ci> tau_n_max </ci>
<apply>
<cosh/>
<apply>
<divide/>
<apply>
<minus/>
<ci> V </ci>
<ci> theta_n </ci>
</apply>
<apply>
<times/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.1#" cellml:units="dimensionless"> 2.0 </cn>
<ci> sigma_n </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="persistent_sodium_current">
<variable name="i_NaP" public_interface="out" units="picoA"/>
<variable initial_value="2.8" name="g_NaP" units="nanoS"/>
<variable name="time" private_interface="out" public_interface="in" units="millisecond"/>
<variable name="V" private_interface="out" public_interface="in" units="millivolt"/>
<variable name="E_Na" public_interface="in" units="millivolt"/>
<variable name="m_infinity" private_interface="in" units="dimensionless"/>
<variable name="h" private_interface="in" units="dimensionless"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="i_NaP_calculation">
<eq/>
<ci> i_NaP </ci>
<apply>
<times/>
<ci> g_NaP </ci>
<ci> m_infinity </ci>
<ci> h </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> E_Na </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="persistent_sodium_current_m_gate">
<variable name="m_infinity" public_interface="out" units="dimensionless"/>
<variable initial_value="-40.0" name="theta_m" units="millivolt"/>
<variable initial_value="-6.0" name="sigma_m" units="millivolt"/>
<variable name="V" public_interface="in" units="millivolt"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="persistent_sodium_current_m_gate_m_infinity_calculation">
<eq/>
<ci> m_infinity </ci>
<apply>
<divide/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.1#" cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.1#" cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<ci> V </ci>
<ci> theta_m </ci>
</apply>
<ci> sigma_m </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="persistent_sodium_current_h_gate">
<variable initial_value="0.46" name="h" public_interface="out" units="dimensionless"/>
<variable name="h_infinity" units="dimensionless"/>
<variable name="tau_h" units="millisecond"/>
<variable initial_value="10000.0" name="tau_h_max" units="millisecond"/>
<variable initial_value="-48.0" name="theta_h" units="millivolt"/>
<variable initial_value="6.0" name="sigma_h" units="millivolt"/>
<variable name="V" public_interface="in" units="millivolt"/>
<variable name="time" public_interface="in" units="millisecond"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="h_diff_eq">
<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/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.1#" cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.1#" cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<ci> V </ci>
<ci> theta_h </ci>
</apply>
<ci> sigma_h </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply id="tau_h_calculation">
<eq/>
<ci> tau_h </ci>
<apply>
<divide/>
<ci> tau_h_max </ci>
<apply>
<cosh/>
<apply>
<divide/>
<apply>
<minus/>
<ci> V </ci>
<ci> theta_h </ci>
</apply>
<apply>
<times/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.1#" cellml:units="dimensionless"> 2.0 </cn>
<ci> sigma_h </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="leakage_current">
<variable name="i_L" public_interface="out" units="picoA"/>
<variable initial_value="2.8" name="g_L" units="nanoS"/>
<variable initial_value="-57.5" name="E_L" units="millivolt"/>
<variable name="V" public_interface="in" units="millivolt"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="i_L_calculation">
<eq/>
<ci> i_L </ci>
<apply>
<times/>
<ci> g_L </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> E_L </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="tonic_current">
<variable name="i_tonic_e" public_interface="out" units="picoA"/>
<variable initial_value="0.0" name="E_syn_e" public_interface="out" units="millivolt"/>
<variable initial_value="0.0" name="g_tonic_e" units="nanoS"/>
<variable name="V" public_interface="in" units="millivolt"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="i_tonic_e_calculation">
<eq/>
<ci> i_tonic_e </ci>
<apply>
<times/>
<ci> g_tonic_e </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> E_syn_e </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="synaptic_input">
<variable name="i_syn_e" public_interface="out" units="picoA"/>
<variable initial_value="1" name="s" private_interface="out" public_interface="out" units="dimensionless"/>
<variable name="s_infinity" units="dimensionless"/>
<variable initial_value="1.0" name="kr" units="dimensionless"/>
<variable initial_value="5.0" name="tau_s" units="millisecond"/>
<variable initial_value="-5.0" name="sigma_s" units="millivolt"/>
<variable initial_value="-10.0" name="theta_s" units="millivolt"/>
<variable name="E_syn_e" public_interface="in" units="millivolt"/>
<variable name="sum_g_syn_e_s" public_interface="in" units="nanoS"/>
<variable name="V" public_interface="in" units="millivolt"/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><apply id="i_syn_e_calculation"><eq/><ci>i_syn_e</ci><apply><times/><ci>sum_g_syn_e_s</ci><apply><minus/><ci>V</ci><ci>E_syn_e</ci></apply></apply></apply><apply id="s_infinity_calculation"><eq/><ci>s_infinity</ci><apply><divide/><cn xmlns:cellml="http://www.cellml.org/cellml/1.1#" cellml:units="dimensionless">1</cn><apply><plus/><cn xmlns:cellml="http://www.cellml.org/cellml/1.1#" cellml:units="dimensionless">1</cn><apply><exp/><apply><divide/><apply><minus/><ci>V</ci><ci>theta_s</ci></apply><ci>sigma_s</ci></apply></apply></apply></apply></apply><apply id="s_calculation"><eq/><apply><diff/><bvar><ci>time</ci></bvar><ci>s</ci></apply><apply><divide/><apply><minus/><apply><times/><apply><minus/><cn xmlns:cellml="http://www.cellml.org/cellml/1.1#" cellml:units="dimensionless">1</cn><ci>s</ci></apply><ci>s_infinity</ci></apply><apply><minus/><apply><times/><ci>kr</ci><ci>s</ci></apply></apply></apply><ci>tau_s</ci></apply></apply></math>
<variable name="time" public_interface="in" units="millisecond"/></component>
<component cmeta:id="synaptic_coupling" name="synaptic_coupling">
<variable name="sum_g_syn_e_s" public_interface="out" units="nanoS"/>
<variable initial_value="0.10" name="g_syn_e_1_2" units="nanoS"/>
<variable initial_value="0.10" name="g_syn_e_1_3" units="nanoS"/>
<variable initial_value="0.10" name="g_syn_e_1_4" units="nanoS"/>
<variable initial_value="0.10" name="g_syn_e_1_5" units="nanoS"/>
<variable name="s1" public_interface="in" units="dimensionless"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci>sum_g_syn_e_s</ci>
<apply>
<plus/>
<apply>
<times/>
<ci>g_syn_e_1_2</ci>
<ci>s1</ci>
</apply>
<apply>
<times/>
<ci>g_syn_e_1_3</ci>
<ci>s1</ci>
</apply>
<apply>
<times/>
<ci>g_syn_e_1_4</ci>
<ci>s1</ci>
</apply>
<apply>
<times/>
<ci>g_syn_e_1_5</ci>
<ci>s1</ci>
</apply>
</apply>
</apply>
</math>
</component>
<group>
<relationship_ref relationship="encapsulation"/>
<component_ref component="single_neuron_model">
<component_ref component="environment"/>
<component_ref component="membrane"/>
<component_ref component="fast_sodium_current">
<component_ref component="fast_sodium_current_m_gate"/>
<component_ref component="fast_sodium_current_n_gate"/>
</component_ref>
<component_ref component="potassium_current">
<component_ref component="potassium_current_n_gate"/>
</component_ref>
<component_ref component="persistent_sodium_current">
<component_ref component="persistent_sodium_current_m_gate"/>
<component_ref component="persistent_sodium_current_h_gate"/>
</component_ref>
<component_ref component="leakage_current"/>
<component_ref component="tonic_current"/>
<component_ref component="synaptic_input"/>
</component_ref>
</group>
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Models of Respiratory Rhythm Generation in the Pre-Botzinger Complex. II. Populations of Coupled Pacemaker Neurons
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