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Models of Respiratory Rhythm Generation in the Pre-Botzinger Complex. I. Bursting Pacemaker Neurons
The CellML code.
<!-- FILE : butera_model_1999.xml
CREATED : 9th May 2002
LAST MODIFIED : 20th April 2005
AUTHOR : Catherine Lloyd
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 Butera et al's first 1999 mathematical model of respiratory rhythm generation in the pre-Botzinger complex in bursting pacemaker neurons.
CHANGES:
18/07/2002 - CML - Added more metadata.
09/04/2003 - AAC - Added publication date information.
20/04/2005 - PJV - Made MathML id's unique
-->
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<article>
<articleinfo>
<title>Models Of Respiratory Rhythm Generation In The Pre-Botzinger Complex. I. Bursting Pacemaker 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>
Inhalation and exhalation movements associated with respiration in mammals are generated by networks of neurons in the lower brain stem that produce a rhythmic pattern of electrical activity. The principal neuronal kernel for rhythm generation has been located in the pre-Botzinger complex, a subregion of the ventro-lateral medulla. In order to understand respiratory rhythm generation, it is necessary to consider mechanisms incorporating intrinsic cellular pacemaker properties. These mechanisms are captured by a hybrid pacemaker model (Smith 1997; Smith <emphasis>et al</emphasis>. 1995), in which a rhythm arises from the dynamic interactions of both intrinsic and synaptic properties within a bilaterally distributed population of coupled bursting pacemaker neurons. ("Bursting" refers to a complicated pattern of electrical activity. Bursts of action potential spikes (the "active" phase) are observed, separated by a "silent" phase of membrane repolarisation).
</para>
<para>
In their 1999 paper, Robert J. Butera, J<subscript>R</subscript>., John Rinzel and Jeffrey C. Smith present two computational versions of this hybrid pacemaker-network model (see <xref linkend="fig_cell_diagram1" /> and <xref linkend="fig_cell_diagram2" /> below). In the first model, bursting arises via fast activation and slow inactivation of a persistent Na<superscript>+</superscript> current, I<subscript>NaP</subscript>. In the second model, bursting arises via a fast-activating persistent Na<superscript>+</superscript> current, I<subscript>NaP</subscript>,(the inactivation term "h" has been removed) and slow activation of a K<superscript>+</superscript> current, I<subscript>KS</subscript>. In both models, action potentials are generated via fast Na<superscript>+</superscript> and K<superscript>+</superscript> currents. Both models are consistent with experimental data, and the authors suggest several experimental tests to demonstrate the validity of either model and to differentiate between the two mechanisms.
</para>
<para>
The complete original paper reference is cited below:
</para>
<para>
<ulink url="http://jn.physiology.org/cgi/content/abstract/82/1/382">Models of Respiratory Rhythm Generation in the Pre-Bötzinger Complex. I. Bursting Pacemaker Neurons</ulink>, Robert J. Butera, Jr., John Rinzel and Jeffrey C. Smith, 1999, <ulink url="http://jn.physiology.org/">
<emphasis>Journal of Neurophysiology</emphasis>
</ulink>, 81, 382-397. (<ulink url="http://jn.physiology.org/cgi/content/full/82/1/382">Full text</ulink> and <ulink url="http://jn.physiology.org/cgi/reprint/82/1/382.pdf">PDF</ulink> versions of the article are available for Journal Members on the JN website.) <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=10400966&dopt=Abstract">PubMed ID: 10400966</ulink>
</para>
<para>
The raw CellML description of the model can be downloaded in various formats as described in <xref linkend="sec_download_this_model" />
</para>
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<title>diagram of the first model</title>
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<caption>The first mathematical 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> (although this last current is considered to be inactive in these models).</caption>
</informalfigure>
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<title>diagram of the first model</title>
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<imagedata fileref="../images/butera_model_1999/cell_diagram2.gif" />
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<caption>The second model appears identical to the first except with the addition of a slow K<superscript>+</superscript> current, I<subscript>KS</subscript>. (The removal of the inactivation term "h" from I<subscript>NaP</subscript> is not visible in the model diagram.)</caption>
</informalfigure>
</sect1>
</article>
</documentation>
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-->
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CREATED : 9th May 2002
LAST MODIFIED : 20th April 2005
AUTHOR : Catherine Lloyd
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 Butera et al's first 1999 mathematical model of respiratory rhythm generation in the pre-Botzinger complex in bursting pacemaker neurons.
CHANGES:
18/07/2002 - CML - Added more metadata.
09/04/2003 - AAC - Added publication date information.
20/04/2005 - PJV - Made MathML id's unique
-->
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<article>
<articleinfo>
<title>Models Of Respiratory Rhythm Generation In The Pre-Botzinger Complex. I. Bursting Pacemaker 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>
Inhalation and exhalation movements associated with respiration in mammals are generated by networks of neurons in the lower brain stem that produce a rhythmic pattern of electrical activity. The principal neuronal kernel for rhythm generation has been located in the pre-Botzinger complex, a subregion of the ventro-lateral medulla. In order to understand respiratory rhythm generation, it is necessary to consider mechanisms incorporating intrinsic cellular pacemaker properties. These mechanisms are captured by a hybrid pacemaker model (Smith 1997; Smith <emphasis>et al</emphasis>. 1995), in which a rhythm arises from the dynamic interactions of both intrinsic and synaptic properties within a bilaterally distributed population of coupled bursting pacemaker neurons. ("Bursting" refers to a complicated pattern of electrical activity. Bursts of action potential spikes (the "active" phase) are observed, separated by a "silent" phase of membrane repolarisation).
</para>
<para>
In their 1999 paper, Robert J. Butera, J<subscript>R</subscript>., John Rinzel and Jeffrey C. Smith present two computational versions of this hybrid pacemaker-network model (see <xref linkend="fig_cell_diagram1" /> and <xref linkend="fig_cell_diagram2" /> below). In the first model, bursting arises via fast activation and slow inactivation of a persistent Na<superscript>+</superscript> current, I<subscript>NaP</subscript>. In the second model, bursting arises via a fast-activating persistent Na<superscript>+</superscript> current, I<subscript>NaP</subscript>,(the inactivation term "h" has been removed) and slow activation of a K<superscript>+</superscript> current, I<subscript>KS</subscript>. In both models, action potentials are generated via fast Na<superscript>+</superscript> and K<superscript>+</superscript> currents. Both models are consistent with experimental data, and the authors suggest several experimental tests to demonstrate the validity of either model and to differentiate between the two mechanisms.
</para>
<para>
The complete original paper reference is cited below:
</para>
<para>
<ulink url="http://jn.physiology.org/cgi/content/abstract/82/1/382">Models of Respiratory Rhythm Generation in the Pre-Bötzinger Complex. I. Bursting Pacemaker Neurons</ulink>, Robert J. Butera, Jr., John Rinzel and Jeffrey C. Smith, 1999, <ulink url="http://jn.physiology.org/">
<emphasis>Journal of Neurophysiology</emphasis>
</ulink>, 81, 382-397. (<ulink url="http://jn.physiology.org/cgi/content/full/82/1/382">Full text</ulink> and <ulink url="http://jn.physiology.org/cgi/reprint/82/1/382.pdf">PDF</ulink> versions of the article are available for Journal Members on the JN website.) <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=10400966&dopt=Abstract">PubMed ID: 10400966</ulink>
</para>
<para>
The raw CellML description 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_diagram1">
<mediaobject>
<imageobject>
<objectinfo>
<title>diagram of the first model</title>
</objectinfo>
<imagedata fileref="../images/butera_model_1999/cell_diagram1.gif" />
</imageobject>
</mediaobject>
<caption>The first mathematical 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> (although this last current is considered to be inactive in these models).</caption>
</informalfigure>
<informalfigure float="0" id="fig_cell_diagram2">
<mediaobject>
<imageobject>
<objectinfo>
<title>diagram of the first model</title>
</objectinfo>
<imagedata fileref="../images/butera_model_1999/cell_diagram2.gif" />
</imageobject>
</mediaobject>
<caption>The second model appears identical to the first except with the addition of a slow K<superscript>+</superscript> current, I<subscript>KS</subscript>. (The removal of the inactivation term "h" from I<subscript>NaP</subscript> is not visible in the model diagram.)</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="millivolt">
<unit units="volt" prefix="milli" />
</units>
<units name="picoA">
<unit units="ampere" prefix="nano" />
</units>
<units name="nanoS">
<unit units="siemens" prefix="nano" />
</units>
<units name="picoF">
<unit units="farad" prefix="pico" />
</units>
<!--
The "environment" component is used to declare variables that are used by
all or most of the other components, in this case just "time".
-->
<component name="environment">
<variable units="millisecond" public_interface="out" name="time" />
</component>
<component name="membrane">
<variable units="millivolt" public_interface="out" name="V" />
<variable units="picoF" name="C" initial_value="21.0" />
<variable units="picoA" name="i_app" initial_value="0.0" />
<variable units="millisecond" public_interface="in" name="time" />
<variable units="picoA" public_interface="in" name="i_NaP" />
<variable units="picoA" public_interface="in" name="i_Na" />
<variable units="picoA" public_interface="in" name="i_K" />
<variable units="picoA" public_interface="in" name="i_L" />
<variable units="picoA" public_interface="in" name="i_tonic_e" />
<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>
</apply>
</apply>
<ci> i_app </ci>
</apply>
<ci> C </ci>
</apply>
</apply>
</math>
</component>
<component name="fast_sodium_current">
<variable units="picoA" public_interface="out" name="i_Na" />
<variable units="millivolt" public_interface="out" name="E_Na" initial_value="50.0" />
<variable units="nanoS" name="g_Na" initial_value="28.0" />
<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="n" />
<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>
<apply>
<minus />
<cn 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 units="dimensionless" public_interface="out" name="m_infinity" />
<variable units="millivolt" name="theta_m" initial_value="-34.0" />
<variable units="millivolt" name="omega_m" initial_value="-5.0" />
<variable units="millivolt" public_interface="in" name="V" />
<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 cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus />
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp />
<apply>
<divide />
<apply>
<minus />
<ci> V </ci>
<ci> theta_m </ci>
</apply>
<ci> omega_m </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="fast_sodium_current_n_gate">
<variable units="dimensionless" public_interface="out" name="n" />
<variable units="dimensionless" name="n_infinity" />
<variable units="millisecond" name="tau_n" />
<variable units="millisecond" name="tau_n_max" initial_value="10.0" />
<variable units="millivolt" name="theta_n" initial_value="-29.0" />
<variable units="millivolt" name="omega_n" initial_value="-4.0" />
<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="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 cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus />
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp />
<apply>
<divide />
<apply>
<minus />
<ci> V </ci>
<ci> theta_n </ci>
</apply>
<ci> omega_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 cellml:units="dimensionless"> 2.0 </cn>
<ci> omega_n </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="potassium_current">
<variable units="picoA" public_interface="out" name="i_K" />
<variable units="nanoS" name="g_K" initial_value="11.2" />
<variable units="millivolt" name="E_K" initial_value="-85.0" />
<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="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="dimensionless" name="n_infinity" />
<variable units="millisecond" name="tau_n" />
<variable units="millisecond" name="tau_n_max" initial_value="10.0" />
<variable units="millivolt" name="theta_n" initial_value="-29.0" />
<variable units="millivolt" name="omega_n" initial_value="-4.0" />
<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="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 cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus />
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp />
<apply>
<divide />
<apply>
<minus />
<ci> V </ci>
<ci> theta_n </ci>
</apply>
<ci> omega_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 cellml:units="dimensionless"> 2.0 </cn>
<ci> omega_n </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="persistent_sodium_current">
<variable units="picoA" public_interface="out" name="i_NaP" />
<variable units="nanoS" name="g_NaP" initial_value="2.8" />
<variable units="millisecond" public_interface="in" private_interface="out" name="time" />
<variable units="millivolt" public_interface="in" private_interface="out" name="V" />
<variable units="millivolt" public_interface="in" name="E_Na" />
<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_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 units="dimensionless" public_interface="out" name="m_infinity" />
<variable units="millivolt" name="theta_m" initial_value="-40.0" />
<variable units="millivolt" name="omega_m" initial_value="-6.0" />
<variable units="millivolt" public_interface="in" name="V" />
<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 cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus />
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp />
<apply>
<divide />
<apply>
<minus />
<ci> V </ci>
<ci> theta_m </ci>
</apply>
<ci> omega_m </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="persistent_sodium_current_h_gate">
<variable units="dimensionless" public_interface="out" name="h" />
<variable units="dimensionless" name="h_infinity" />
<variable units="millisecond" name="tau_h" />
<variable units="millisecond" name="tau_h_max" initial_value="10000.0" />
<variable units="millivolt" name="theta_h" initial_value="-48.0" />
<variable units="millivolt" name="omega_h" initial_value="6.0" />
<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="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 cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus />
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp />
<apply>
<divide />
<apply>
<minus />
<ci> V </ci>
<ci> theta_h </ci>
</apply>
<ci> omega_h </ci>
</apply>
</apply>
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</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 cellml:units="dimensionless"> 2.0 </cn>
<ci> omega_h </ci>
</apply>
</apply>
</apply>
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</component>
<component name="leakage_current">
<variable units="picoA" public_interface="out" name="i_L" />
<variable units="nanoS" name="g_L" initial_value="2.8" />
<variable units="millivolt" name="E_L" initial_value="-65.0" />
<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_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 units="picoA" public_interface="out" name="i_tonic_e" />
<variable units="nanoS" name="g_tonic_e" initial_value="0.0" />
<variable units="millivolt" name="E_syn_e" initial_value="0.0" />
<variable units="millisecond" public_interface="in" name="time" />
<variable units="millivolt" public_interface="in" name="V" />
<math xmlns="http://www.w3.org/1998/Math/MathMtonic_e">
<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>
<group>
<relationship_ref relationship="containment" />
<component_ref component="membrane">
<component_ref component="fast_sodium_current">
<component_ref component="fast_sodium_current_m_gate" />