<?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>
<component name="environment">
<variable units="millisecond" public_interface="out" name="time"/>
</component>
<component name="membrane">
<variable units="millivolt" public_interface="out" name="V" initial_value="-55.0"/>
<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"/>
<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_KS </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>
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<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="sigma_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>
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<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
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<apply>
<minus/>
<ci> V </ci>
<ci> theta_m </ci>
</apply>
<ci> sigma_m </ci>
</apply>
</apply>
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<component name="fast_sodium_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"/>
<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> 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 cellml:units="dimensionless"> 2.0 </cn>
<ci> sigma_n </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<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"/>
<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" 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"/>
<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> 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 cellml:units="dimensionless"> 2.0 </cn>
<ci> sigma_n </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<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"/>
<variable units="millivolt" public_interface="in" private_interface="out" name="V"/>
<variable units="dimensionless" private_interface="in" name="k"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="i_KS_calculation">
<eq/>
<ci> i_KS </ci>
<apply>
<times/>
<ci> g_KS </ci>
<ci> k </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> E_K </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="slow_potassium_current_k_gate">
<variable units="dimensionless" public_interface="out" name="k" initial_value="0.22"/>
<variable units="dimensionless" name="k_infinity"/>
<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"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="slow_potassium_current_k_gate_k_diff_eq">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> k </ci>
</apply>
<apply>
<divide/>
<apply>
<minus/>
<ci> k_infinity </ci>
<ci> k </ci>
</apply>
<ci> tau_k </ci>
</apply>
</apply>
<apply id="slow_potassium_current_k_gate_k_infinity_calculation">
<eq/>
<ci> k_infinity </ci>
<apply>
<divide/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
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<apply>
<minus/>
<ci> V </ci>
<ci> theta_k </ci>
</apply>
<ci> sigma_k </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply id="slow_potassium_current_k_gate_tau_k_calculation">
<eq/>
<ci> tau_k </ci>
<apply>
<divide/>
<ci> tau_k_max </ci>
<apply>
<cosh/>
<apply>
<divide/>
<apply>
<minus/>
<ci> V </ci>
<ci> theta_k </ci>
</apply>
<apply>
<times/>
<cn cellml:units="dimensionless"> 2.0 </cn>
<ci> sigma_k </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"/>
<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>
<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="sigma_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> sigma_m </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</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="-50.0"/>
<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="millivolt" public_interface="in" name="V"/>
<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>
<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>
</group>
<connection>
<map_components component_2="environment" component_1="membrane"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<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"/>
<map_variables variable_2="V" variable_1="V"/>
<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"/>
<map_variables variable_2="m_infinity" variable_1="m_infinity"/>
<map_variables variable_2="V" variable_1="V"/>
</connection>
<connection>
<map_components component_2="fast_sodium_current_n_gate" component_1="fast_sodium_current"/>
<map_variables variable_2="n" variable_1="n"/>
<map_variables variable_2="V" variable_1="V"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="potassium_current_n_gate" component_1="potassium_current"/>
<map_variables variable_2="n" variable_1="n"/>
<map_variables variable_2="V" variable_1="V"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="slow_potassium_current_k_gate" component_1="slow_potassium_current"/>
<map_variables variable_2="k" variable_1="k"/>
<map_variables variable_2="V" variable_1="V"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="persistent_sodium_current_m_gate" component_1="persistent_sodium_current"/>
<map_variables variable_2="m_infinity" variable_1="m_infinity"/>
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