Connecting CellML models with ModelingToolkit equations

[termi-official]

First, thanks for the great ecosystem! ModelingToolkit really looks promising.

I think I run into a limitation with the current design of CellMLToolkit. From my understanding it simply parses the cellml document and directly parses and simplifies everything, followed by constructing the ODESystem. This fails if the cellml description is incomplete. However, I think it is a valid use case to connect the missing variables manually. To give a very simple example, let us consider a forced van der Pol oscillator.

<?xml version='1.0' encoding='UTF-8'?>
<model name="forced_oscillator" xmlns="http://www.cellml.org/cellml/1.1#" xmlns:cellml="http://www.cellml.org/cellml/1.1#">
    <units name="m_per_s">
        <unit units="meter"/>
        <unit exponent="-1" units="second"/>
    </units>
    <component name="system">
        <variable initial_value="0.0" name="time" units="second"/>
        <variable initial_value="1.0" name="mu" units="dimensionless"/>
        <variable name="f" public_interface="in" units="meter"/>
        <variable initial_value="0.0" name="x" public_interface="out" units="meter"/>
        <variable initial_value="0.0" name="v" public_interface="out" units="m_per_s"/>
        <math xmlns="http://www.w3.org/1998/Math/MathML">
            <apply>
                <eq/>
                <apply>
                    <diff/>
                    <bvar>
                        <ci>time</ci>
                    </bvar>
                    <ci>x</ci>
                </apply>
                <ci>v</ci>
            </apply>
            <apply>
                <eq/>
                <apply>
                    <diff/>
                    <bvar>
                        <ci>time</ci>
                    </bvar>
                    <ci>v</ci>
                </apply>
                <apply>
                    <plus/>
                    <apply>
                        <minus/>
                        <apply>
                            <times/>
                            <ci>mu</ci>
                            <apply>
                                <minus/>
                                <cn cellml:units="dimensionless">1</cn>
                                <apply>
                                    <times/>
                                    <ci>x</ci>
                                    <ci>x</ci>
                                </apply>
                            </apply>
                            <ci>v</ci>
                        </apply>
                        <ci>x</ci>
                    </apply>
                    <ci>f</ci>
                </apply>
            </apply>
        </math>
    </component>
</model>

Parsing this model fails, because f is not connected to anything:

using CellMLToolkit
ml = CellModel("forced_oscillator.cellml")
ERROR: value of system₊f is not found

However, it would be nice to have access to a partially parsed model, such that one could connect the missing components via ModelingToolkit.jl either directly by connecting the partial model with other partial models from other CellML files. To give a rough sketch on what I have in mind:

using ModelingToolkit, CellMLToolkit
mlc = CellModelComponent("forced_oscillator.cellml")
eqs = [
          0 ~ mlcsys.system₊f - sin(mlc.sys.system₊xˍtimetime)
      ]
ml = finalize(mlc, eqs)

[nickerso]

Apologies for the off-topic comment, but perhaps this might be useful in considering this kind of feature in CellMLToolkit..

In work we have been doing with libCellML and code generation with CellML 2.0 models, we generally expect the CellML model to be complete (giving f an initial value, for example) but then the user would flag f as an external variable so that in the generated code it would be treated as being defined outside the scope of the CellML model. We can then use that capability to add that new equation (or connect to another model or data).

In the very recent work (see cellml/libcellml#1077) the handling of these external variables is moved to the start of the model analysis/code generation. This could begin to support this use-case where such under-defined variables in the CellML model could be detected by the user and set external and so not flagged as an error as it would be taken out of the model analysis. This is an interesting use-case generally that I don't believe we have considered, and I often find myself setting those dummy initial values that I never intend to use and this could solve that problem...

[termi-official]

I was able to cook up a working example.

using CellMLToolkit, OrdinaryDiffEq, Plots

function custom_stim_model(filename; comp=:membrane, varname=:Istim)
    # Load model
    doc = CellMLToolkit.load_cellml(filename)
    CellMLToolkit.extract_mathml.(doc.xmls)
    CellMLToolkit.infer_iv(doc)
    
    # Classify variables as in the original
    class = CellMLToolkit.classify_variables(doc)
    
    # We now mark the stimulus as time-dependent
    class[CellMLToolkit.Var(comp, varname)] = true
    
    # Construct the systems
    systems = CellMLToolkit.subsystems(doc, class)
    
    # Define stimulus
    t = CellMLToolkit.get_ivₚ(doc)
    stimulus = getproperty(systems[comp], varname) ~ ifelse(t < 1.0, 0.5, 0.0)
    
    # Build the system
    odesys = ODESystem(
        [connections; stimulus],
        t,
        systems = collect(values(systems)),
        name = CellMLToolkit.gensym(:custom_stim_model)
    )

    # Simplify
    odesys_s = CellMLToolkit.structural_simplify(odesys)
    post_sub = CellMLToolkit.post_substitution(doc, systems)
    CellMLToolkit.@set! odesys_s.eqs = CellMLToolkit.substitute_eqs(equations(odesys_s), post_sub)
    
    # Update default values
    CellMLToolkit.@set! odesys_s.defaults = Dict(CellMLToolkit.find_list_value(doc, vcat(parameters(odesys_s), states(odesys_s))))
    return CellModel(doc, odesys_s)
end

br = custom_stim_model("models/beeler_reuter_1977_nostim.cellml.xml")
prob = ODEProblem(br, (0.0, 400.0))
sol = solve(prob, TRBDF2())
plot(sol, vars=[br.sys.membrane₊V])

Where the file is simply the provided Beeler-Reuter model without the stimulus protocol component, i.e.

models/beeler_reuter_1977_nostim.cellml.xml

<?xml version="1.0" encoding="utf-8"?>
<!--
This CellML file was generated on 07/05/2008 at 16:14:45 using:

COR (0.9.31.927)
Copyright 2002-2008 Dr Alan Garny
http://COR.physiol.ox.ac.uk/ - COR@physiol.ox.ac.uk

CellML 1.0 was used to generate this cellular model
http://www.CellML.org/
--><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#" xmlns:ns7="http://www.cellml.org/metadata/simulation/1.0#" cmeta:id="beeler_reuter_1977" name="beeler_reuter_1977_version06">
  
  
  <!--
    Below, we define some additional units for association with variables and
    constants within the model. The identifiers are fairly self-explanatory.
  -->
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
  <articleinfo>
  <title>Beeler-Reuter Mammalian Ventricular Model 1977</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 model has been curated by Penny Noble using Flavio Fenton's Java code as a reference (See http://thevirtualheart.org/ for Java applet rendering of model - Java code is available from Dr Fenton.) An artificial stimulus component has been added this model to allow it to reproduce the action potential simulation shown in Figure 4 of the publication. The model is known to run and integrate in the PCEnv and COR CellML environments. A PCEnv session file is also associated with this model.

</para>
<para>
ValidateCellML detects unit inconsistency within this model.
</para>

<para>

          </para>
  </section>
  <sect1 id="sec_structure">
<title>Model Structure</title>

<para>
In contrast to the earlier Purkinje fibre ionic current models of <ulink url="${HTML_EXMPL_N_MODEL}">D. Noble (1962)</ulink> and <ulink url="${HTML_EXMPL_MNT_MODEL}">R.E. McAllister, D. Noble and R.W. Tsien (1975)</ulink>, the G.W. Beeler and H. Reuter 1977 model was developed to describe the mammalian ventricular action potential.  Not all the ionic currents of the Purkinje fibre model are present in ventricular tissue; therefore, this model is simpler than the MNT model.  The total ionic flux is divided into only four discrete, individual ionic currents (see <xref linkend="fig_cell_diagram"/> below).  The main additional feature of the Beeler-Reuter ionic current model is a representation of the intracellular calcium ion concentration.
</para>

<para>
The complete original paper reference is cited below:
</para>

<para>
Reconstruction of the action potential of ventricular myocardial fibres, Beeler, G.W. and Reuter, H. 1977 
            <emphasis>Journal of Physiology</emphasis>
          , 268, 177-210.  <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;list_uids=874889&amp;dopt=Abstract">PubMed ID: 874889</ulink>
</para>

<informalfigure float="0" id="fig_cell_diagram">
<mediaobject>
  <imageobject>
    <objectinfo>
      <title>cell diagram of the Beeler-Reuter model showing ionic currents across the cell surface membrane</title>
    </objectinfo>
    <imagedata fileref="beeler_reuter_1977.png"/>
  </imageobject>
</mediaobject>
<caption>A schematic diagram describing the current flows across the cell membrane that are captured in the BR model.</caption>
</informalfigure>

<informalfigure float="0" id="fig_cellml_rendering">
<mediaobject>
  <imageobject>
    <objectinfo>
      <title>the cellml rendering of the Beeler-Reuter model</title>
    </objectinfo>
    <imagedata fileref="cellml_rendering.gif"/>
  </imageobject>
</mediaobject>
<caption>The network defined in the CellML description of the Beeler-Reuter model. A key describing the significance of the shapes of the components and the colours of the connections between them is in the <ulink url="http://www.cellml.org/tutorial/notation/?searchterm=notation%20guide">notation guide</ulink>.  For simplicity, not all the variables are shown.</caption>
</informalfigure>

<para>
The membrane physically contains the currents as indicated by the blue arrows in <xref linkend="fig_cellml_rendering"/>. The currents act independently and are not connected to each other. Several of the channels encapsulate <emphasis>and</emphasis> contain further components which represent activation and inactivation gates. The addition of an encapsulation relationship informs modellers and processing software that the gates are important parts of the current model. It also prevents any other components that aren't also encapsulated by the parent component from connecting to its gates, effectively hiding them from the rest of the model.
</para>

<para>
The breakdown of the model into components and the definition of encapsulation and containment relationships between them is somewhat arbitrary. When considering how a model should be broken into components, modellers are encouraged to consider which parts of a model might be re-used and how the physiological elements of the system being modelled are naturally bounded. Containment relationships should be used to provide simple rendering information for processing software (ideally, this will correspond to the layout of the physical system), and encapsulation should be used to group sets of components into sub-models.
</para>

</sect1>
</article>
</documentation>
   <units name="ms">
      <unit units="second" prefix="milli"/>
   </units>
   <units name="per_ms">
      <unit units="second" prefix="milli" exponent="-1"/>
   </units>
   <units name="mV">
      <unit units="volt" prefix="milli"/>
   </units>
   <units name="per_mV">
      <unit units="volt" prefix="milli" exponent="-1"/>
   </units>
   <units name="per_mV_ms">
      <unit units="mV" exponent="-1"/>
      <unit units="ms" exponent="-1"/>
   </units>
   <units name="mS_per_mm2">
      <unit units="siemens" prefix="milli"/>
      <unit units="metre" prefix="milli" exponent="-2"/>
   </units>
   <units name="uF_per_mm2">
      <unit units="farad" prefix="micro"/>
      <unit units="metre" prefix="milli" exponent="-2"/>
   </units>
   <units name="uA_per_mm2">
      <unit units="ampere" prefix="micro"/>
      <unit units="metre" prefix="milli" exponent="-2"/>
   </units>
   <units name="concentration_units">
      <unit units="mole" prefix="nano"/>
      <unit units="metre" prefix="milli" exponent="-3"/>
   </units>
   <units name="per_concentration_units">
      <unit units="concentration_units" exponent="-1"/>
   </units>
   <units name="coulomb_per_mole">
      <unit units="coulomb"/>
      <unit units="mole" exponent="-1"/>
   </units>
   <units name="per_mm">
      <unit units="metre" prefix="-3" exponent="-1"/>
   </units>
   <component name="environment">
      <variable units="ms" public_interface="out" name="time" cmeta:id="environment_time"/>
   </component>
   <component name="membrane">
      <variable units="mV" public_interface="out" cmeta:id="membrane_V" name="V" initial_value="-84.624"/>
      <variable units="uF_per_mm2" cmeta:id="membrane_C" name="C" initial_value="0.01"/>
      <variable units="ms" public_interface="in" name="time"/>
      <variable units="uA_per_mm2" public_interface="in" name="i_Na"/>
      <variable units="uA_per_mm2" public_interface="in" name="i_s"/>
      <variable units="uA_per_mm2" public_interface="in" name="i_x1"/>
      <variable units="uA_per_mm2" public_interface="in" name="i_K1"/>
      <variable units="uA_per_mm2" public_interface="in" name="Istim"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="membrane_voltage_diff_eq">
         <apply>
            <eq/>
            <apply>
               <diff/>
               <bvar>
                  <ci>time</ci>
               </bvar>
               <ci>V</ci>
            </apply>
            <apply>
               <divide/>
               <apply>
                  <minus/>
                  <ci>Istim</ci>
                  <apply>
                     <plus/>
                     <ci>i_Na</ci>
                     <ci>i_s</ci>
                     <ci>i_x1</ci>
                     <ci>i_K1</ci>
                  </apply>
               </apply>
               <ci>C</ci>
            </apply>
         </apply>
      </math>
   </component>
   <component name="sodium_current">
      <variable units="uA_per_mm2" public_interface="out" name="i_Na" cmeta:id="sodium_current_i_Na"/>
      <variable units="mS_per_mm2" name="g_Na" initial_value="4e-2"/>
      <variable units="mV" name="E_Na" initial_value="50"/>
      <variable units="mS_per_mm2" name="g_Nac" initial_value="3e-5"/>
      <variable units="ms" public_interface="in" private_interface="out" name="time"/>
      <variable units="mV" public_interface="in" private_interface="out" name="V"/>
      <variable units="dimensionless" private_interface="in" name="m"/>
      <variable units="dimensionless" private_interface="in" name="h"/>
      <variable units="dimensionless" private_interface="in" name="j"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <ci>i_Na</ci>
            <apply>
               <times/>
               <apply>
                  <plus/>
                  <apply>
                     <times/>
                     <ci>g_Na</ci>
                     <apply>
                        <power/>
                        <ci>m</ci>
                        <cn cellml:units="dimensionless">3</cn>
                     </apply>
                     <ci>h</ci>
                     <ci>j</ci>
                  </apply>
                  <ci>g_Nac</ci>
               </apply>
               <apply>
                  <minus/>
                  <ci>V</ci>
                  <ci>E_Na</ci>
               </apply>
            </apply>
         </apply>
      </math>
   </component>
   <component name="sodium_current_m_gate">
      <variable units="dimensionless" public_interface="out" name="m" initial_value="0.011"/>
      <variable units="per_ms" name="alpha_m"/>
      <variable units="per_ms" name="beta_m"/>
      <variable units="mV" public_interface="in" name="V"/>
      <variable units="ms" public_interface="in" name="time"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <ci>alpha_m</ci>
            <apply>
               <divide/>
               <apply>
                  <times/>
                  <apply>
                     <minus/>
                     <cn cellml:units="per_mV_ms">1</cn>
                  </apply>
                  <apply>
                     <plus/>
                     <ci>V</ci>
                     <cn cellml:units="mV">47</cn>
                  </apply>
               </apply>
               <apply>
                  <minus/>
                  <apply>
                     <exp/>
                     <apply>
                        <times/>
                        <apply>
                           <minus/>
                           <cn cellml:units="per_mV">0.1</cn>
                        </apply>
                        <apply>
                           <plus/>
                           <ci>V</ci>
                           <cn cellml:units="mV">47</cn>
                        </apply>
                     </apply>
                  </apply>
                  <cn cellml:units="dimensionless">1</cn>
               </apply>
            </apply>
         </apply>
         <apply>
            <eq/>
            <ci>beta_m</ci>
            <apply>
               <times/>
               <cn cellml:units="per_ms">40</cn>
               <apply>
                  <exp/>
                  <apply>
                     <times/>
                     <apply>
                        <minus/>
                        <cn cellml:units="per_mV">0.056</cn>
                     </apply>
                     <apply>
                        <plus/>
                        <ci>V</ci>
                        <cn cellml:units="mV">72</cn>
                     </apply>
                  </apply>
               </apply>
            </apply>
         </apply>
         <apply>
            <eq/>
            <apply>
               <diff/>
               <bvar>
                  <ci>time</ci>
               </bvar>
               <ci>m</ci>
            </apply>
            <apply>
               <minus/>
               <apply>
                  <times/>
                  <ci>alpha_m</ci>
                  <apply>
                     <minus/>
                     <cn cellml:units="dimensionless">1</cn>
                     <ci>m</ci>
                  </apply>
               </apply>
               <apply>
                  <times/>
                  <ci>beta_m</ci>
                  <ci>m</ci>
               </apply>
            </apply>
         </apply>
      </math>
   </component>
   <component name="sodium_current_h_gate">
      <variable units="dimensionless" public_interface="out" name="h" initial_value="0.988"/>
      <variable units="per_ms" name="alpha_h"/>
      <variable units="per_ms" name="beta_h"/>
      <variable units="mV" public_interface="in" name="V"/>
      <variable units="ms" public_interface="in" name="time"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <ci>alpha_h</ci>
            <apply>
               <times/>
               <cn cellml:units="per_ms">0.126</cn>
               <apply>
                  <exp/>
                  <apply>
                     <times/>
                     <apply>
                        <minus/>
                        <cn cellml:units="per_mV">0.25</cn>
                     </apply>
                     <apply>
                        <plus/>
                        <ci>V</ci>
                        <cn cellml:units="mV">77</cn>
                     </apply>
                  </apply>
               </apply>
            </apply>
         </apply>
         <apply>
            <eq/>
            <ci>beta_h</ci>
            <apply>
               <divide/>
               <cn cellml:units="per_ms">1.7</cn>
               <apply>
                  <plus/>
                  <apply>
                     <exp/>
                     <apply>
                        <times/>
                        <apply>
                           <minus/>
                           <cn cellml:units="per_mV">0.082</cn>
                        </apply>
                        <apply>
                           <plus/>
                           <ci>V</ci>
                           <cn cellml:units="mV">22.5</cn>
                        </apply>
                     </apply>
                  </apply>
                  <cn cellml:units="dimensionless">1</cn>
               </apply>
            </apply>
         </apply>
         <apply>
            <eq/>
            <apply>
               <diff/>
               <bvar>
                  <ci>time</ci>
               </bvar>
               <ci>h</ci>
            </apply>
            <apply>
               <minus/>
               <apply>
                  <times/>
                  <ci>alpha_h</ci>
                  <apply>
                     <minus/>
                     <cn cellml:units="dimensionless">1</cn>
                     <ci>h</ci>
                  </apply>
               </apply>
               <apply>
                  <times/>
                  <ci>beta_h</ci>
                  <ci>h</ci>
               </apply>
            </apply>
         </apply>
      </math>
   </component>
   <component name="sodium_current_j_gate">
      <variable units="dimensionless" public_interface="out" name="j" initial_value="0.975"/>
      <variable units="per_ms" name="alpha_j"/>
      <variable units="per_ms" name="beta_j"/>
      <variable units="mV" public_interface="in" name="V"/>
      <variable units="ms" public_interface="in" name="time"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <ci>alpha_j</ci>
            <apply>
               <divide/>
               <apply>
                  <times/>
                  <cn cellml:units="per_ms">0.055</cn>
                  <apply>
                     <exp/>
                     <apply>
                        <times/>
                        <apply>
                           <minus/>
                           <cn cellml:units="per_mV">0.25</cn>
                        </apply>
                        <apply>
                           <plus/>
                           <ci>V</ci>
                           <cn cellml:units="mV">78</cn>
                        </apply>
                     </apply>
                  </apply>
               </apply>
               <apply>
                  <plus/>
                  <apply>
                     <exp/>
                     <apply>
                        <times/>
                        <apply>
                           <minus/>
                           <cn cellml:units="per_mV">0.2</cn>
                        </apply>
                        <apply>
                           <plus/>
                           <ci>V</ci>
                           <cn cellml:units="mV">78</cn>
                        </apply>
                     </apply>
                  </apply>
                  <cn cellml:units="dimensionless">1</cn>
               </apply>
            </apply>
         </apply>
         <apply>
            <eq/>
            <ci>beta_j</ci>
            <apply>
               <divide/>
               <cn cellml:units="per_ms">0.3</cn>
               <apply>
                  <plus/>
                  <apply>
                     <exp/>
                     <apply>
                        <times/>
                        <apply>
                           <minus/>
                           <cn cellml:units="per_mV">0.1</cn>
                        </apply>
                        <apply>
                           <plus/>
                           <ci>V</ci>
                           <cn cellml:units="mV">32</cn>
                        </apply>
                     </apply>
                  </apply>
                  <cn cellml:units="dimensionless">1</cn>
               </apply>
            </apply>
         </apply>
         <apply>
            <eq/>
            <apply>
               <diff/>
               <bvar>
                  <ci>time</ci>
               </bvar>
               <ci>j</ci>
            </apply>
            <apply>
               <minus/>
               <apply>
                  <times/>
                  <ci>alpha_j</ci>
                  <apply>
                     <minus/>
                     <cn cellml:units="dimensionless">1</cn>
                     <ci>j</ci>
                  </apply>
               </apply>
               <apply>
                  <times/>
                  <ci>beta_j</ci>
                  <ci>j</ci>
               </apply>
            </apply>
         </apply>
      </math>
   </component>
   <component name="slow_inward_current">
      <variable units="uA_per_mm2" public_interface="out" name="i_s" cmeta:id="slow_inward_current_i_s"/>
      <variable units="mS_per_mm2" name="g_s" initial_value="9e-4"/>
      <variable units="mV" name="E_s"/>
      <variable units="concentration_units" name="Cai" initial_value="1e-4"/>
      <variable units="ms" public_interface="in" private_interface="out" name="time"/>
      <variable units="mV" public_interface="in" private_interface="out" name="V"/>
      <variable units="dimensionless" private_interface="in" name="d"/>
      <variable units="dimensionless" private_interface="in" name="f"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <ci>E_s</ci>
            <apply>
               <minus/>
               <apply>
                  <minus/>
                  <cn cellml:units="mV">82.3</cn>
               </apply>
               <apply>
                  <times/>
                  <cn cellml:units="mV">13.0287</cn>
                  <apply>
                     <ln/>
                     <apply>
                        <times/>
                        <ci>Cai</ci>
                        <cn cellml:units="per_concentration_units">0.001</cn>
                     </apply>
                  </apply>
               </apply>
            </apply>
         </apply>
         <apply>
            <eq/>
            <ci>i_s</ci>
            <apply>
               <times/>
               <ci>g_s</ci>
               <ci>d</ci>
               <ci>f</ci>
               <apply>
                  <minus/>
                  <ci>V</ci>
                  <ci>E_s</ci>
               </apply>
            </apply>
         </apply>
         <apply>
            <eq/>
            <apply>
               <diff/>
               <bvar>
                  <ci>time</ci>
               </bvar>
               <ci>Cai</ci>
            </apply>
            <apply>
               <plus/>
               <apply>
                  <divide/>
                  <apply>
                     <times/>
                     <apply>
                        <minus/>
                        <cn cellml:units="per_mm">0.01</cn>
                     </apply>
                     <ci>i_s</ci>
                  </apply>
                  <cn cellml:units="coulomb_per_mole">1</cn>
               </apply>
               <apply>
                  <times/>
                  <cn cellml:units="per_ms">0.07</cn>
                  <apply>
                     <minus/>
                     <cn cellml:units="concentration_units">0.0001</cn>
                     <ci>Cai</ci>
                  </apply>
               </apply>
            </apply>
         </apply>
      </math>
   </component>
   <component name="slow_inward_current_d_gate">
      <variable units="dimensionless" public_interface="out" name="d" initial_value="0.003"/>
      <variable units="per_ms" name="alpha_d"/>
      <variable units="per_ms" name="beta_d"/>
      <variable units="mV" public_interface="in" name="V"/>
      <variable units="ms" public_interface="in" name="time"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <ci>alpha_d</ci>
            <apply>
               <divide/>
               <apply>
                  <times/>
                  <cn cellml:units="per_ms">0.095</cn>
                  <apply>
                     <exp/>
                     <apply>
                        <divide/>
                        <apply>
                           <minus/>
                           <apply>
                              <minus/>
                              <ci>V</ci>
                              <cn cellml:units="mV">5</cn>
                           </apply>
                        </apply>
                        <cn cellml:units="mV">100</cn>
                     </apply>
                  </apply>
               </apply>
               <apply>
                  <plus/>
                  <cn cellml:units="dimensionless">1</cn>
                  <apply>
                     <exp/>
                     <apply>
                        <divide/>
                        <apply>
                           <minus/>
                           <apply>
                              <minus/>
                              <ci>V</ci>
                              <cn cellml:units="mV">5</cn>
                           </apply>
                        </apply>
                        <cn cellml:units="mV">13.89</cn>
                     </apply>
                  </apply>
               </apply>
            </apply>
         </apply>
         <apply>
            <eq/>
            <ci>beta_d</ci>
            <apply>
               <divide/>
               <apply>
                  <times/>
                  <cn cellml:units="per_ms">0.07</cn>
                  <apply>
                     <exp/>
                     <apply>
                        <divide/>
                        <apply>
                           <minus/>
                           <apply>
                              <plus/>
                              <ci>V</ci>
                              <cn cellml:units="mV">44</cn>
                           </apply>
                        </apply>
                        <cn cellml:units="mV">59</cn>
                     </apply>
                  </apply>
               </apply>
               <apply>
                  <plus/>
                  <cn cellml:units="dimensionless">1</cn>
                  <apply>
                     <exp/>
                     <apply>
                        <divide/>
                        <apply>
                           <plus/>
                           <ci>V</ci>
                           <cn cellml:units="mV">44</cn>
                        </apply>
                        <cn cellml:units="mV">20</cn>
                     </apply>
                  </apply>
               </apply>
            </apply>
         </apply>
         <apply>
            <eq/>
            <apply>
               <diff/>
               <bvar>
                  <ci>time</ci>
               </bvar>
               <ci>d</ci>
            </apply>
            <apply>
               <minus/>
               <apply>
                  <times/>
                  <ci>alpha_d</ci>
                  <apply>
                     <minus/>
                     <cn cellml:units="dimensionless">1</cn>
                     <ci>d</ci>
                  </apply>
               </apply>
               <apply>
                  <times/>
                  <ci>beta_d</ci>
                  <ci>d</ci>
               </apply>
            </apply>
         </apply>
      </math>
   </component>
   <component name="slow_inward_current_f_gate">
      <variable units="dimensionless" public_interface="out" name="f" initial_value="0.994"/>
      <variable units="per_ms" name="alpha_f"/>
      <variable units="per_ms" name="beta_f"/>
      <variable units="mV" public_interface="in" name="V"/>
      <variable units="ms" public_interface="in" name="time"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <ci>alpha_f</ci>
            <apply>
               <divide/>
               <apply>
                  <times/>
                  <cn cellml:units="per_ms">0.012</cn>
                  <apply>
                     <exp/>
                     <apply>
                        <divide/>
                        <apply>
                           <minus/>
                           <apply>
                              <plus/>
                              <ci>V</ci>
                              <cn cellml:units="mV">28</cn>
                           </apply>
                        </apply>
                        <cn cellml:units="mV">125</cn>
                     </apply>
                  </apply>
               </apply>
               <apply>
                  <plus/>
                  <cn cellml:units="dimensionless">1</cn>
                  <apply>
                     <exp/>
                     <apply>
                        <divide/>
                        <apply>
                           <plus/>
                           <ci>V</ci>
                           <cn cellml:units="mV">28</cn>
                        </apply>
                        <cn cellml:units="mV">6.67</cn>
                     </apply>
                  </apply>
               </apply>
            </apply>
         </apply>
         <apply>
            <eq/>
            <ci>beta_f</ci>
            <apply>
               <divide/>
               <apply>
                  <times/>
                  <cn cellml:units="per_ms">0.0065</cn>
                  <apply>
                     <exp/>
                     <apply>
                        <divide/>
                        <apply>
                           <minus/>
                           <apply>
                              <plus/>
                              <ci>V</ci>
                              <cn cellml:units="mV">30</cn>
                           </apply>
                        </apply>
                        <cn cellml:units="mV">50</cn>
                     </apply>
                  </apply>
               </apply>
               <apply>
                  <plus/>
                  <cn cellml:units="dimensionless">1</cn>
                  <apply>
                     <exp/>
                     <apply>
                        <divide/>
                        <apply>
                           <minus/>
                           <apply>
                              <plus/>
                              <ci>V</ci>
                              <cn cellml:units="mV">30</cn>
                           </apply>
                        </apply>
                        <cn cellml:units="mV">5</cn>
                     </apply>
                  </apply>
               </apply>
            </apply>
         </apply>
         <apply>
            <eq/>
            <apply>
               <diff/>
               <bvar>
                  <ci>time</ci>
               </bvar>
               <ci>f</ci>
            </apply>
            <apply>
               <minus/>
               <apply>
                  <times/>
                  <ci>alpha_f</ci>
                  <apply>
                     <minus/>
                     <cn cellml:units="dimensionless">1</cn>
                     <ci>f</ci>
                  </apply>
               </apply>
               <apply>
                  <times/>
                  <ci>beta_f</ci>
                  <ci>f</ci>
               </apply>
            </apply>
         </apply>
      </math>
   </component>
   <component name="time_dependent_outward_current" cmeta:id="time_dependent_outward_current">
      <variable units="uA_per_mm2" public_interface="out" name="i_x1" cmeta:id="time_dependent_outward_current_i_x1"/>
      <variable units="ms" public_interface="in" private_interface="out" name="time"/>
      <variable units="mV" public_interface="in" private_interface="out" name="V"/>
      <variable units="dimensionless" private_interface="in" name="x1"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <ci>i_x1</ci>
            <apply>
               <divide/>
               <apply>
                  <times/>
                  <ci>x1</ci>
                  <cn type="e-notation" cellml:units="uA_per_mm2">8<sep/>-3</cn>
                  <apply>
                     <minus/>
                     <apply>
                        <exp/>
                        <apply>
                           <times/>
                           <cn cellml:units="per_mV">0.04</cn>
                           <apply>
                              <plus/>
                              <ci>V</ci>
                              <cn cellml:units="mV">77</cn>
                           </apply>
                        </apply>
                     </apply>
                     <cn cellml:units="dimensionless">1</cn>
                  </apply>
               </apply>
               <apply>
                  <exp/>
                  <apply>
                     <times/>
                     <cn cellml:units="per_mV">0.04</cn>
                     <apply>
                        <plus/>
                        <ci>V</ci>
                        <cn cellml:units="mV">35</cn>
                     </apply>
                  </apply>
               </apply>
            </apply>
         </apply>
      </math>
   </component>
   <component name="time_dependent_outward_current_x1_gate">
      <variable units="dimensionless" public_interface="out" name="x1" initial_value="0.0001"/>
      <variable units="per_ms" name="alpha_x1"/>
      <variable units="per_ms" name="beta_x1"/>
      <variable units="mV" public_interface="in" name="V"/>
      <variable units="ms" public_interface="in" name="time"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <ci>alpha_x1</ci>
            <apply>
               <divide/>
               <apply>
                  <times/>
                  <cn type="e-notation" cellml:units="per_ms">5<sep/>-4</cn>
                  <apply>
                     <exp/>
                     <apply>
                        <divide/>
                        <apply>
                           <plus/>
                           <ci>V</ci>
                           <cn cellml:units="mV">50</cn>
                        </apply>
                        <cn cellml:units="mV">12.1</cn>
                     </apply>
                  </apply>
               </apply>
               <apply>
                  <plus/>
                  <cn cellml:units="dimensionless">1</cn>
                  <apply>
                     <exp/>
                     <apply>
                        <divide/>
                        <apply>
                           <plus/>
                           <ci>V</ci>
                           <cn cellml:units="mV">50</cn>
                        </apply>
                        <cn cellml:units="mV">17.5</cn>
                     </apply>
                  </apply>
               </apply>
            </apply>
         </apply>
         <apply>
            <eq/>
            <ci>beta_x1</ci>
            <apply>
               <divide/>
               <apply>
                  <times/>
                  <cn cellml:units="per_ms">0.0013</cn>
                  <apply>
                     <exp/>
                     <apply>
                        <divide/>
                        <apply>
                           <minus/>
                           <apply>
                              <plus/>
                              <ci>V</ci>
                              <cn cellml:units="mV">20</cn>
                           </apply>
                        </apply>
                        <cn cellml:units="mV">16.67</cn>
                     </apply>
                  </apply>
               </apply>
               <apply>
                  <plus/>
                  <cn cellml:units="dimensionless">1</cn>
                  <apply>
                     <exp/>
                     <apply>
                        <divide/>
                        <apply>
                           <minus/>
                           <apply>
                              <plus/>
                              <ci>V</ci>
                              <cn cellml:units="mV">20</cn>
                           </apply>
                        </apply>
                        <cn cellml:units="mV">25</cn>
                     </apply>
                  </apply>
               </apply>
            </apply>
         </apply>
         <apply>
            <eq/>
            <apply>
               <diff/>
               <bvar>
                  <ci>time</ci>
               </bvar>
               <ci>x1</ci>
            </apply>
            <apply>
               <minus/>
               <apply>
                  <times/>
                  <ci>alpha_x1</ci>
                  <apply>
                     <minus/>
                     <cn cellml:units="dimensionless">1</cn>
                     <ci>x1</ci>
                  </apply>
               </apply>
               <apply>
                  <times/>
                  <ci>beta_x1</ci>
                  <ci>x1</ci>
               </apply>
            </apply>
         </apply>
      </math>
   </component>
   <component name="time_independent_outward_current">
      <variable units="uA_per_mm2" public_interface="out" name="i_K1" cmeta:id="time_independent_outward_current_i_K1"/>
      <variable units="ms" public_interface="in" name="time"/>
      <variable units="mV" public_interface="in" name="V"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <ci>i_K1</ci>
            <apply>
               <times/>
               <cn cellml:units="uA_per_mm2">0.0035</cn>
               <apply>
                  <plus/>
                  <apply>
                     <divide/>
                     <apply>
                        <times/>
                        <cn cellml:units="dimensionless">4</cn>
                        <apply>
                           <minus/>
                           <apply>
                              <exp/>
                              <apply>
                                 <times/>
                                 <cn cellml:units="per_mV">0.04</cn>
                                 <apply>
                                    <plus/>
                                    <ci>V</ci>
                                    <cn cellml:units="mV">85</cn>
                                 </apply>
                              </apply>
                           </apply>
                           <cn cellml:units="dimensionless">1</cn>
                        </apply>
                     </apply>
                     <apply>
                        <plus/>
                        <apply>
                           <exp/>
                           <apply>
                              <times/>
                              <cn cellml:units="per_mV">0.08</cn>
                              <apply>
                                 <plus/>
                                 <ci>V</ci>
                                 <cn cellml:units="mV">53</cn>
                              </apply>
                           </apply>
                        </apply>
                        <apply>
                           <exp/>
                           <apply>
                              <times/>
                              <cn cellml:units="per_mV">0.04</cn>
                              <apply>
                                 <plus/>
                                 <ci>V</ci>
                                 <cn cellml:units="mV">53</cn>
                              </apply>
                           </apply>
                        </apply>
                     </apply>
                  </apply>
                  <apply>
                     <divide/>
                     <apply>
                        <times/>
                        <cn cellml:units="per_mV">0.2</cn>
                        <apply>
                           <plus/>
                           <ci>V</ci>
                           <cn cellml:units="mV">23</cn>
                        </apply>
                     </apply>
                     <apply>
                        <minus/>
                        <cn cellml:units="dimensionless">1</cn>
                        <apply>
                           <exp/>
                           <apply>
                              <times/>
                              <apply>
                                 <minus/>
                                 <cn cellml:units="per_mV">0.04</cn>
                              </apply>
                              <apply>
                                 <plus/>
                                 <ci>V</ci>
                                 <cn cellml:units="mV">23</cn>
                              </apply>
                           </apply>
                        </apply>
                     </apply>
                  </apply>
               </apply>
            </apply>
         </apply>
      </math>
   </component>
   <group>
      <relationship_ref relationship="containment"/>
      <component_ref component="membrane">
         <component_ref component="sodium_current">
            <component_ref component="sodium_current_m_gate"/>
            <component_ref component="sodium_current_h_gate"/>
            <component_ref component="sodium_current_j_gate"/>
         </component_ref>
         <component_ref component="slow_inward_current">
            <component_ref component="slow_inward_current_d_gate"/>
            <component_ref component="slow_inward_current_f_gate"/>
         </component_ref>
         <component_ref component="time_dependent_outward_current">
            <component_ref component="time_dependent_outward_current_x1_gate"/>
         </component_ref>
         <component_ref component="time_independent_outward_current"/>
      </component_ref>
   </group>
   <group>
      <relationship_ref relationship="encapsulation"/>
      <component_ref component="sodium_current">
         <component_ref component="sodium_current_m_gate"/>
         <component_ref component="sodium_current_h_gate"/>
         <component_ref component="sodium_current_j_gate"/>
      </component_ref>
      <component_ref component="slow_inward_current">
         <component_ref component="slow_inward_current_d_gate"/>
         <component_ref component="slow_inward_current_f_gate"/>
      </component_ref>
      <component_ref component="time_dependent_outward_current">
         <component_ref component="time_dependent_outward_current_x1_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="sodium_current"/>
      <map_variables variable_2="time" variable_1="time"/>
   </connection>
   <connection>
      <map_components component_2="environment" component_1="slow_inward_current"/>
      <map_variables variable_2="time" variable_1="time"/>
   </connection>
   <connection>
      <map_components component_2="environment" component_1="time_dependent_outward_current"/>
      <map_variables variable_2="time" variable_1="time"/>
   </connection>
   <connection>
      <map_components component_2="environment" component_1="time_independent_outward_current"/>
      <map_variables variable_2="time" variable_1="time"/>
   </connection>
   <connection>
      <map_components component_2="sodium_current" component_1="membrane"/>
      <map_variables variable_2="V" variable_1="V"/>
      <map_variables variable_2="i_Na" variable_1="i_Na"/>
   </connection>
   <connection>
      <map_components component_2="slow_inward_current" component_1="membrane"/>
      <map_variables variable_2="V" variable_1="V"/>
      <map_variables variable_2="i_s" variable_1="i_s"/>
   </connection>
   <connection>
      <map_components component_2="time_dependent_outward_current" component_1="membrane"/>
      <map_variables variable_2="V" variable_1="V"/>
      <map_variables variable_2="i_x1" variable_1="i_x1"/>
   </connection>
   <connection>
      <map_components component_2="time_independent_outward_current" component_1="membrane"/>
      <map_variables variable_2="V" variable_1="V"/>
      <map_variables variable_2="i_K1" variable_1="i_K1"/>
   </connection>
   <connection>
      <map_components component_2="sodium_current_m_gate" component_1="sodium_current"/>
      <map_variables variable_2="m" variable_1="m"/>
      <map_variables variable_2="time" variable_1="time"/>
      <map_variables variable_2="V" variable_1="V"/>
   </connection>
   <connection>
      <map_components component_2="sodium_current_h_gate" component_1="sodium_current"/>
      <map_variables variable_2="h" variable_1="h"/>
      <map_variables variable_2="time" variable_1="time"/>
      <map_variables variable_2="V" variable_1="V"/>
   </connection>
   <connection>
      <map_components component_2="sodium_current_j_gate" component_1="sodium_current"/>
      <map_variables variable_2="j" variable_1="j"/>
      <map_variables variable_2="time" variable_1="time"/>
      <map_variables variable_2="V" variable_1="V"/>
   </connection>
   <connection>
      <map_components component_2="time_dependent_outward_current_x1_gate" component_1="time_dependent_outward_current"/>
      <map_variables variable_2="x1" variable_1="x1"/>
      <map_variables variable_2="time" variable_1="time"/>
      <map_variables variable_2="V" variable_1="V"/>
   </connection>
   <connection>
      <map_components component_2="slow_inward_current_d_gate" component_1="slow_inward_current"/>
      <map_variables variable_2="d" variable_1="d"/>
      <map_variables variable_2="time" variable_1="time"/>
      <map_variables variable_2="V" variable_1="V"/>
   </connection>
   <connection>
      <map_components component_2="slow_inward_current_f_gate" component_1="slow_inward_current"/>
      <map_variables variable_2="f" variable_1="f"/>
      <map_variables variable_2="time" variable_1="time"/>
      <map_variables variable_2="V" variable_1="V"/>
   </connection>
<rdf:RDF>
  <rdf:Bag rdf:about="rdf:#9e1ab371-cc1e-490c-ae79-95a6988c4736">
    <rdf:li>cardiac</rdf:li>
    <rdf:li>cardiac electrophysiology</rdf:li>
    <rdf:li>electrophysiology</rdf:li>
    <rdf:li>ventricular myocyte</rdf:li>
    <rdf:li>electrophysiological</rdf:li>
  </rdf:Bag>
  <rdf:Seq rdf:about="rdf:#9667b556-1be7-429b-b568-9a2989ce274f">
    <rdf:li rdf:resource="rdf:#87aa2d90-f40e-4ec3-9f6a-876c04eba648"/>
    <rdf:li rdf:resource="rdf:#c3d555f3-8681-41b7-a04c-e8232bc1628c"/>
  </rdf:Seq>
  <rdf:Description rdf:about="rdf:#46dea9a4-f216-4183-8dec-e37c44430f80">
    <dcterms:modified rdf:resource="rdf:#98f2fb7a-ae19-422e-bf73-26e592cfc801"/>
    <rdf:value>Changed model cmeta:id from beeler_reuter_1977_version06 to beeler_reuter_1977</rdf:value>
    <cmeta:modifier rdf:resource="rdf:#25a69a5a-c684-4da8-b278-1dc132a0d1d2"/>
  </rdf:Description>
  <rdf:Description rdf:about="">
    <dc:publisher/>
    <cmeta:comment rdf:resource="rdf:#43885da7-17c7-47e4-83a2-9680d72b78db"/>
    <dcterms:created rdf:resource="rdf:#a33cc46e-5ead-468d-9b18-0cd58caf6527"/>
    <dc:creator rdf:resource="rdf:#88adedca-bf60-46b7-94d4-d2fadfea04f6"/>
    <cmeta:modification rdf:resource="rdf:#46dea9a4-f216-4183-8dec-e37c44430f80"/>
    <cmeta:modification rdf:resource="rdf:#4a58e4ff-222b-4ee4-89ba-ab5b6b51306c"/>
    <cmeta:modification rdf:resource="rdf:#77f98dda-c979-49f1-9bcf-da4944906f91"/>
    <cmeta:modification rdf:resource="rdf:#fc128820-8eb0-45dd-9403-858fa9abecce"/>
  </rdf:Description>
  <rdf:Description rdf:about="rdf:#baef8605-6d17-47bc-b214-261f23273934">
    <vCard:N rdf:resource="rdf:#ef9ea29b-ae5e-4db0-975e-42fe8f0092ed"/>
  </rdf:Description>
  <rdf:Description rdf:about="rdf:#1fe62786-b9ed-4025-849c-9363a40a8179">
    <bqs:subject_type>keyword</bqs:subject_type>
    <rdf:value rdf:resource="rdf:#9e1ab371-cc1e-490c-ae79-95a6988c4736"/>
  </rdf:Description>
  <rdf:Description rdf:about="rdf:#0d7eb214-b680-484f-950d-994248dbaf49">
    <vCard:Given>James</vCard:Given>
    <vCard:Family>Lawson</vCard:Family>
    <vCard:Other>Richard</vCard:Other>
  </rdf:Description>
  <rdf:Description rdf:about="rdf:#ef9ea29b-ae5e-4db0-975e-42fe8f0092ed">
    <vCard:Given>Penny</vCard:Given>
    <vCard:Family>Noble</vCard:Family>
  </rdf:Description>
  <rdf:Description rdf:about="rdf:#$orjp43">
    <rdf:rest rdf:resource="http://www.w3.org/1999/02/22-rdf-syntax-ns#nil"/>
    <rdf:first rdf:resource="rdf:#$prjp43"/>
  </rdf:Description>
  <rdf:Description rdf:about="#beeler_reuter_1977">
    <ns7:simulation rdf:resource="rdf:#$nrjp43"/>
    <bqs:reference rdf:resource="rdf:#6891ea32-aff0-429b-8dde-ca1296c3234a"/>
    <bqs:reference rdf:resource="rdf:#6da04ee6-84f7-4f63-a253-c15568cfcf77"/>
    <cmeta:comment rdf:resource="rdf:#aeb90a45-e2e6-4579-a33b-356cc62b1dc4"/>
  </rdf:Description>
  <rdf:Description rdf:about="rdf:#77f98dda-c979-49f1-9bcf-da4944906f91">
    <dcterms:modified rdf:resource="rdf:#e7527bce-f615-4263-afc0-c075d70b6e23"/>
    <rdf:value>Added an intial value for X1 to enable the model to run.</rdf:value>
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</model>

This also works for the forced oscillator above and we can connect multiple models with this approach.

However, this example heavily relies on the internals of this package. Hence, I would ask if we could add some of the internals to the public API and keep them at least somewhat stable between versions.

I am also not sure if the new equation is constructed as intended (stimulus = getproperty(systems[comp], varname) ~ ifelse(t < 1.0, 0.5, 0.0)).

[anandijain]

However, this example heavily relies on the internals of this package. Hence, I would ask if we could add some of the internals to the public API and keep them at least somewhat stable between versions.

We love PRs! So we can definitely add stuff to the public API

This has been weighing on me for a while now: we may want to think about redoing this package to use libcellml with a jll. Then we can merge efforts with the libcellml people instead of having a bunch of slightly different implementations.

I tried to do a binary builder thing JuliaPackaging/Yggdrasil#4947 but there was funky stuff in the build that I couldn't quite get right.

I know this isn't exactly relevant to the problem you're solving but since you're actively doing CellMLTk stuff, I figured I'd let you know in case you want to pick it up.

[termi-official]

Sorry I could not really find time yet to tackle this. Just wanting to check if someone has dumped time into the rework. If not, what is the general idea here?

Related to this, my vision would be that we could try to extract components from the CellML file into MTK components. I have especially the idea in mind that we should have the possibility to e.g. extract the individual ion channels in their respective formulation (hodgkin-huxley, markov chain,...) or the calcium model or the sarcomere model, if we can either discover them automatically or if they are baked into the CellML model.

[ChrisRackauckas]

No one has done a substantial rework yet. Extracting this into the component based structure of MTK would be the right thing to do.

[shahriariravanian]

In JuliaCon, we talked about identifying CellML components and using Conductor.jl as a bridge to MTK. Of course, the problem is that CellML files do not include the relevant metadata. It should be possible to use pattern-matching to do the job (at least partially); however, it is not trivial and only worth it if there is enough interest.

[termi-official]

Indeed, if we have no meta information at all, then pattern matching (and normal forms) become something useful, but I need to think more about the exact patterns here. However, maybe a first step could be to extract components which are actually defined, as e.g. in this cellml tutorial https://models.cellml.org/e/e1/tutorial . Still, not straight forward and I have to check libcellml again on how to do it.