Rendering of the source text

<?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>
				<para>
				The MATLAB script (<ulink url="buteraModel.m">buteraModel.m</ulink>) should be used to generate a CellML file for multi-cell models.  The generated model will import butera_single_cell_1999.cellml repeatedly, once for each of the multiple cells, and it will generate values from a random normal distribution for the required connectivity coefficients between the cells.  The main model for the exposure (butera_ten_cell_1999.cellml)  was generated from the MATLAB script.
				</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"/>
        <variable name="V" private_interface="in" public_interface="out" units="millivolt"/>
    <variable name="time" private_interface="in" public_interface="out" units="millisecond"/></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 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"/>
        <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/>
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        <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"/>
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                <component_ref component="persistent_sodium_current_h_gate"/>
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            <component_ref component="leakage_current"/>
            <component_ref component="tonic_current"/>
            <component_ref component="synaptic_input"/>
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    <connection>
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    <connection>
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    <connection>
        <map_components component_1="tonic_current" component_2="membrane"/>
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        <map_components component_1="fast_sodium_current" component_2="persistent_sodium_current"/>
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    <connection>
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    <connection>
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    <connection>
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    <connection>
        <map_components component_1="single_neuron_model" component_2="synaptic_input"/>
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