<?xml version='1.0' encoding='utf-8'?>

<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#" name="butera_1999" cmeta:id="butera_1999">

<documentation xmlns="http://cellml.org/tmp-documentation">
<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>Auckland Bioengineering Institute, The University of Auckland</shortaffil>
    </affiliation>
  </author>
</articleinfo>
  <section id="sec_status">
    <title>Model Status</title>
    <para>
          This CellML model runs in OpenCell and COR to reproduce the published results (Figure 5 A3 where E_L = -50 mv).  Please note that the model has to be run for a duration of 10000 ms with a step size of 0.01 ms and a high point density of 100000 points/graph.  This model represents model 2 from the published paper (which includes a slow potassium current).
          </para>
  </section>
  <sect1 id="sec_structure">
<title>Model Structure</title>

<para>
ABSTRACT: A network of oscillatory bursting neurons with excitatory coupling is hypothesized to define the primary kernel for respiratory rhythm generation in the pre-Botzinger complex (pre-BotC) in mammals. Two minimal models of these neurons are proposed. In model 1, bursting arises via fast activation and slow inactivation of a persistent Na+ current INaP-h. In model 2, bursting arises via a fast-activating persistent Na+ current INaP and slow activation of a K+ current IKS. In both models, action potentials are generated via fast Na+ and K+ currents. The two models have few differences in parameters to facilitate a rigorous comparison of the two different burst-generating mechanisms. Both models are consistent with many of the dynamic features of electrophysiological recordings from pre-BotC oscillatory bursting neurons in vitro, including voltage-dependent activity modes (silence, bursting, and beating), a voltage-dependent burst frequency that can vary from 0.05 to greater than 1 Hz, and a decaying spike frequency during bursting. These results are robust and persist across a wide range of parameter values for both models. However, the dynamics of model 1 are more consistent with experimental data in that the burst duration decreases as the baseline membrane potential is depolarized and the model has a relatively flat membrane potential trajectory during the interburst interval. We propose 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>
Models of Respiratory Rhythm Generation in the Pre-Botzinger Complex. I. Bursting Pacemaker Neurons, Robert J. Butera, Jr., John Rinzel and Jeffrey C. Smith, 1999, <emphasis>Journal of Neurophysiology</emphasis>, 81, 382-397. <ulink url="http://www.ncbi.nlm.nih.gov/pubmed/10400966">PubMed ID: 10400966</ulink> 
</para>

<informalfigure float="0" id="fig_cell_diagram1">
<mediaobject>
  <imageobject>
    <objectinfo>
      <title>diagram of the first model</title>
    </objectinfo>
    <imagedata fileref="butera_1999a.png"/>
  </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="butera_1999b.png"/>
  </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>
  
  
 
  
  <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>
  
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    <variable units="millisecond" public_interface="out" name="time"/>
  </component>
  
  <component name="membrane">
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    <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_KS"/>
    <variable units="picoA" public_interface="in" name="i_L"/>
    <variable units="picoA" public_interface="in" name="i_tonic_e"/>
     
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				<ci> i_Na </ci>
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                <ci> i_L </ci>
                <ci> i_tonic_e </ci>
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    <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"/>
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    <variable units="dimensionless" private_interface="in" name="n"/>
    
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        <ci> i_Na </ci>
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          <ci> g_Na </ci>
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  <component name="fast_sodium_current_m_gate">
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    <variable units="millivolt" name="sigma_m" initial_value="-5.0"/>
     
    <variable units="millivolt" public_interface="in" name="V"/>
    
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                </apply>
                <ci> sigma_m </ci>
              </apply>
            </apply>    
          </apply>
        </apply>
      </apply>
    </math>
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    <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="sigma_n" initial_value="-4.0"/>
     
    <variable units="millivolt" public_interface="in" name="V"/>
    <variable units="millisecond" public_interface="in" name="time"/>
   
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          <ci> n </ci>
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            <ci> n </ci>
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          <ci> tau_n </ci>
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      <apply id="fast_sodium_current_n_gate_n_infinity_calculation">
        <eq/>
        <ci> n_infinity </ci>
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      </apply>
      
      <apply id="fast_sodium_current_n_gate_tau_n_calculation">
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            </apply>
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  <component name="potassium_current">
    <variable units="picoA" public_interface="out" name="i_K"/>
    <variable units="millivolt" public_interface="out" name="E_K" initial_value="-85.0"/>
    
    <variable units="nanoS" name="g_K" initial_value="11.2"/>   
    
    <variable units="millisecond" public_interface="in" private_interface="out" name="time"/>
    <variable units="millivolt" public_interface="in" private_interface="out" name="V"/>
    
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    <math xmlns="http://www.w3.org/1998/Math/MathML">
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        <eq/>
        <ci> i_K </ci>
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          <times/>
          <ci> g_K </ci>
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            <power/>
            <ci> n </ci>
            <cn cellml:units="dimensionless"> 4.0 </cn>
          </apply>
          <apply>
            <minus/>
            <ci> V </ci>
            <ci> E_K </ci>
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        </apply>
      </apply>
    </math>
  </component>
  
  <component name="potassium_current_n_gate">
    <variable units="dimensionless" public_interface="out" name="n" initial_value="0.01"/>
    
    <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="sigma_n" initial_value="-4.0"/>
     
    <variable units="millivolt" public_interface="in" name="V"/>
    <variable units="millisecond" public_interface="in" name="time"/>
   
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          </apply>
          <ci> tau_n </ci>
        </apply>
      </apply>
      
      <apply id="potassium_current_n_gate_n_infinity_calculation">
        <eq/>
        <ci> n_infinity </ci>
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        </apply>
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      <apply id="potassium_current_n_gate_tau_n_calculation">
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            </apply>
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        </apply>
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  <component name="slow_potassium_current">
    <variable units="picoA" public_interface="out" name="i_KS"/>
    
    <variable units="nanoS" name="g_KS" initial_value="5.6"/>   
    
	<variable units="millivolt" public_interface="in" name="E_K"/>
    <variable units="millisecond" public_interface="in" private_interface="out" name="time"/>
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    <variable units="dimensionless" private_interface="in" name="k"/>
    
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    <variable units="millisecond" name="tau_k"/>
    <variable units="millisecond" name="tau_k_max" initial_value="10000.0"/>
    <variable units="millivolt" name="theta_k" initial_value="-38.0"/>
    <variable units="millivolt" name="sigma_k" initial_value="-6.0"/>
     
    <variable units="millivolt" public_interface="in" name="V"/>
    <variable units="millisecond" public_interface="in" name="time"/>
   
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    <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"/>
    
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          <apply>
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            <ci> E_Na </ci>
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        </apply>
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    </math>
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  <component name="persistent_sodium_current_m_gate">
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    <variable units="millivolt" name="theta_m" initial_value="-40.0"/>
    <variable units="millivolt" name="sigma_m" initial_value="-6.0"/>
     
    <variable units="millivolt" public_interface="in" name="V"/>
    
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                <ci> sigma_m </ci>
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        </apply>
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    </math>
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  <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="-50.0"/> 
    
    <variable units="millivolt" public_interface="in" name="V"/>
    
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            <ci> E_L </ci>
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        </apply>
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  <component name="tonic_current">
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    <variable units="nanoS" name="g_tonic_e" initial_value="0.0"/>
    <variable units="millivolt" name="E_syn_e" initial_value="0.0"/> 
    
    <variable units="millivolt" public_interface="in" name="V"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
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            <ci> E_syn_e </ci>
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        </apply>
      </apply>
    </math>
  </component>
  
  <group>
    <relationship_ref relationship="encapsulation"/>
      <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="slow_potassium_current">
        <component_ref component="slow_potassium_current_k_gate"/>
      </component_ref>
      <component_ref component="persistent_sodium_current">
        <component_ref component="persistent_sodium_current_m_gate"/>
      </component_ref>
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  <connection>
    <map_components component_2="environment" component_1="membrane"/>
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    <map_components component_2="environment" component_1="fast_sodium_current"/>
    <map_variables variable_2="time" variable_1="time"/>
  </connection>
  
  <connection>
    <map_components component_2="environment" component_1="potassium_current"/>
    <map_variables variable_2="time" variable_1="time"/>
  </connection>
  
  <connection>
    <map_components component_2="environment" component_1="slow_potassium_current"/>
    <map_variables variable_2="time" variable_1="time"/>
  </connection>
  
  <connection>
    <map_components component_2="environment" component_1="persistent_sodium_current"/>
    <map_variables variable_2="time" variable_1="time"/>
  </connection>
  
  <connection>
    <map_components component_2="membrane" component_1="fast_sodium_current"/>
    <map_variables variable_2="V" variable_1="V"/>
    <map_variables variable_2="i_Na" variable_1="i_Na"/>
  </connection>
  
  <connection>
    <map_components component_2="membrane" component_1="potassium_current"/>
    <map_variables variable_2="V" variable_1="V"/>
    <map_variables variable_2="i_K" variable_1="i_K"/>
  </connection>
  
  <connection>
    <map_components component_2="membrane" component_1="slow_potassium_current"/>
    <map_variables variable_2="V" variable_1="V"/>
    <map_variables variable_2="i_KS" variable_1="i_KS"/>
  </connection>
  
  <connection>
    <map_components component_2="membrane" component_1="persistent_sodium_current"/>
    <map_variables variable_2="V" variable_1="V"/>
    <map_variables variable_2="i_NaP" variable_1="i_NaP"/>
  </connection>
  
  <connection>
    <map_components component_2="membrane" component_1="leakage_current"/>
    <map_variables variable_2="V" variable_1="V"/>
    <map_variables variable_2="i_L" variable_1="i_L"/>
  </connection>
  
  <connection>
    <map_components component_2="membrane" component_1="tonic_current"/>
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    <map_variables variable_2="i_tonic_e" variable_1="i_tonic_e"/>
  </connection>
  
  <connection>
    <map_components component_2="persistent_sodium_current" component_1="fast_sodium_current"/>
    <map_variables variable_2="E_Na" variable_1="E_Na"/>
  </connection>
  
  <connection>
    <map_components component_2="potassium_current" component_1="slow_potassium_current"/>
    <map_variables variable_2="E_K" variable_1="E_K"/>
  </connection>
  
  <connection>
    <map_components component_2="fast_sodium_current_m_gate" component_1="fast_sodium_current"/>
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    <map_variables variable_2="V" variable_1="V"/>
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  <connection>
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    <map_variables variable_2="V" variable_1="V"/>
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  </connection>
  
  <connection>
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    <map_variables variable_2="n" variable_1="n"/>
    <map_variables variable_2="V" variable_1="V"/>
    <map_variables variable_2="time" variable_1="time"/>
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  <connection>
    <map_components component_2="slow_potassium_current_k_gate" component_1="slow_potassium_current"/>
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    <map_variables variable_2="time" variable_1="time"/>
  </connection>
  
  <connection>
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    <map_variables variable_2="m_infinity" variable_1="m_infinity"/>
    <map_variables variable_2="V" variable_1="V"/>
  </connection>
  
  
  
  
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            Models of Respiratory Rhythm Generation in the Pre-Botzinger Complex. I. Bursting Pacemaker Neurons
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