<?xml version="1.0"?>
<!-- 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
--><model xmlns="http://www.cellml.org/cellml/1.0#" xmlns:cmeta="http://www.cellml.org/metadata/1.0#" cmeta:id="butera_1999" name="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 4 A3 where E_L = -57.5 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 1 from the published paper (which does not include 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 >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 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>
<component name="environment">
<variable cmeta:id="environment_time" name="time" public_interface="out" units="millisecond"/>
</component>
<component name="membrane">
<variable cmeta:id="membrane_V" 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"/>
<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 cmeta:id="fast_sodium_current_i_Na" 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/1.0#" cellml:units="dimensionless"> 3.0 </cn>
</apply>
<apply>
<minus/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" 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/1.0#" cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" 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/1.0#" cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" 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/1.0#" cellml:units="dimensionless"> 2.0 </cn>
<ci> sigma_n </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="potassium_current">
<variable cmeta:id="potassium_current_i_K" 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/1.0#" 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/1.0#" cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" 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/1.0#" cellml:units="dimensionless"> 2.0 </cn>
<ci> sigma_n </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="persistent_sodium_current">
<variable cmeta:id="persistent_sodium_current_i_NaP" 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/1.0#" cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" 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/1.0#" cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" 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/1.0#" cellml:units="dimensionless"> 2.0 </cn>
<ci> sigma_h </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="leakage_current">
<variable cmeta:id="leakage_current_i_L" 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 cmeta:id="tonic_current_i_tonic_e" name="i_tonic_e" public_interface="out" units="picoA"/>
<variable cmeta:id="tonic_current_g_tonic_e" initial_value="0.0" name="g_tonic_e" units="nanoS"/>
<variable initial_value="0.0" name="E_syn_e" units="millivolt"/>
<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>
<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="persistent_sodium_current">
<component_ref component="persistent_sodium_current_m_gate"/>
<component_ref component="persistent_sodium_current_h_gate"/>
</component_ref>
</group>
<connection>
<map_components component_1="membrane" component_2="environment"/>
<map_variables variable_1="time" variable_2="time"/>
</connection>
<connection>
<map_components component_1="fast_sodium_current" component_2="environment"/>
<map_variables variable_1="time" variable_2="time"/>
</connection>
<connection>
<map_components component_1="potassium_current" component_2="environment"/>
<map_variables variable_1="time" variable_2="time"/>
</connection>
<connection>
<map_components component_1="persistent_sodium_current" component_2="environment"/>
<map_variables variable_1="time" variable_2="time"/>
</connection>
<connection>
<map_components component_1="fast_sodium_current" component_2="membrane"/>
<map_variables variable_1="V" variable_2="V"/>
<map_variables variable_1="i_Na" variable_2="i_Na"/>
</connection>
<connection>
<map_components component_1="potassium_current" component_2="membrane"/>
<map_variables variable_1="V" variable_2="V"/>
<map_variables variable_1="i_K" variable_2="i_K"/>
</connection>
<connection>
<map_components component_1="persistent_sodium_current" component_2="membrane"/>
<map_variables variable_1="V" variable_2="V"/>
<map_variables variable_1="i_NaP" variable_2="i_NaP"/>
</connection>
<connection>
<map_components component_1="leakage_current" component_2="membrane"/>
<map_variables variable_1="V" variable_2="V"/>
<map_variables variable_1="i_L" variable_2="i_L"/>
</connection>
<connection>
<map_components component_1="tonic_current" component_2="membrane"/>
<map_variables variable_1="V" variable_2="V"/>
<map_variables variable_1="i_tonic_e" variable_2="i_tonic_e"/>
</connection>
<connection>
<map_components component_1="fast_sodium_current" component_2="persistent_sodium_current"/>
<map_variables variable_1="E_Na" variable_2="E_Na"/>
</connection>
<connection>
<map_components component_1="fast_sodium_current" component_2="fast_sodium_current_m_gate"/>
<map_variables variable_1="m_infinity" variable_2="m_infinity"/>
<map_variables variable_1="V" variable_2="V"/>
</connection>
<connection>
<map_components component_1="fast_sodium_current" component_2="fast_sodium_current_n_gate"/>
<map_variables variable_1="n" variable_2="n"/>
<map_variables variable_1="V" variable_2="V"/>
<map_variables variable_1="time" variable_2="time"/>
</connection>
<connection>
<map_components component_1="potassium_current" component_2="potassium_current_n_gate"/>
<map_variables variable_1="n" variable_2="n"/>
<map_variables variable_1="V" variable_2="V"/>
<map_variables variable_1="time" variable_2="time"/>
</connection>
<connection>
<map_components component_1="persistent_sodium_current" component_2="persistent_sodium_current_m_gate"/>
<map_variables variable_1="m_infinity" variable_2="m_infinity"/>
<map_variables variable_1="V" variable_2="V"/>
</connection>
<connection>
<map_components component_1="persistent_sodium_current" component_2="persistent_sodium_current_h_gate"/>
<map_variables variable_1="h" variable_2="h"/>
<map_variables variable_1="V" variable_2="V"/>
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