Model definition syntax

Every line starting in # is taken to be a comment.

White lines are ignored.

Everything within a set of parentheses is read as one line. This can be used to create multi-line statements. Alternatively, code may be split over multiple lines by ending a line with

The first non-comment in a file must be a model header. The exact syntax of a model header is given below.

A model is divided into components. Components are declared by writing [component_name] on a new line (including the square brackets!). Each component lasts until the next component declaration.

Within a component, a variable is defined by writing its name on a new, unindented line. Variables can contain nested variables, not visible outside the scope of their parent variable. Nesting is indicated using indenting.

Variable names

Model, component and variable names follow standard programming conventions: each name must start with a letter and the remainder of the name can contain any ‘word’ character: lowercase letters, uppercase letters, numbers or an underscore.

Within a component, variables can simply be referred to using their name. To reference a variable from another component, its “fully qualified” name must be used. This consists of the component name followed by a period followed by the variable name: component_name.variable_name.

Model header

Each model must begin with a model header [[model]].

Meta-data can be added to a model using the syntax field: value

All state variables require an initial value to be specified in the model header using the syntax component.variable = value

Initial values can be numbers or expressions, as long as they don’t reference other variables.

Example:

[[model]]
name: This model's name
desc: Model description. Can be split over multiple lines using \
      backslash notation.
author: Identifies the author of the model implementation
membrane.V = -84
na_fast.m  = 0
na_fast.h  = 1.0
na_fast.j  = 1.0

Component syntax

A component starts with its name in square brackets, followed by any number of variables of meta-data properties

[component_1]
desc: This is the first component
x = <expression>
y = <expression>
dot(z) = <expression>

All component names must be unique.

Defining variables

Each variable must have a single defining equation of the form that describes either its value or its derivative. In the following example x is defined directly, whereas y is a state variable defined through its derivative:

x = 12
dot(y) = 10 * y

The dot() operator is the only operator that may appear on the left-hand side of an equation.

Meta-data and descriptions

Meta data can be specified by adding field: value pairs on the lines following a variable’s definition. To indicate they belong to the previous variable, these lines should be indented.

E_Na = 12
    desc: The Nernst potential of sodium
    latex: E_{Na^+}

All text to the right of a : sign is treated as plain text. Meta properties can be added at will.

The common meta property “desc” can be set using the following shorthand syntax:

x = 12 : A weight

This is equivalent to

x = 12
    desc: A weight

A model part cannot specify the same meta-data property twice. For example, a variable cannot have two properties ‘name’.

Meta-data properties can be grouped into namespaces using the syntax:

x = 123
    group1:property1: This is the first property in group 1
    group1:property2: This is the second property in group 1
    group2:property1: This is the first property in group 2

Units

Myokit has support for units in two ways: units attached to a variable and units attached to a literal value.

Variables can specify their units using the in keyword:

x = 12 : A weight
    in [kg]

This specifies that the variable x is in the unit kg, regardless of how x is defined: x = 12 or x = exp(cos(4)+2), we know that it’s in kg.

The second way of using units is by attaching them to a literal value. For example writing 5 [kg] instead of 5. This double specification can be used for unig checking, for example, if x is known to be invalid it makes no sense to assign it a value 7 [m/s].

For state variables, the in keyword refers to the variable, not its derivative. Thus:

dot(V) = 5
    in [mV]

specifies that V is in [mV]. Using [ms] as time unit, the expression dot(V) itself is expressed in [mV/ms].

Unit specifications use the following syntax:
  • A “simple unit” consists of a unit name (m, g, V etc) with an optional quantifier (mm, kg, etc). Not all unit names support quantifiers, a “centimile”, for example, will not be recognized.
  • Simple units can be exponentiated using ^. For example m^3 and s^-1
  • (Exponentiated) simple units can be strung together using multiplication (*) or division (/). For example kg/cm^2.
  • A full unit description is a string of (exponentiated) simple units wrapped in square brackets. For example [kg/cm^2].
  • An optional multiplication factor can be added. For example an inch can be written as [cm (2.54)] or [m (0.0254)].
  • Units with offsets (celsius and fahrenheit) are not supported.
Myokit supports at least the following units:
  • The seven base SI units kg, m, s, A, K, cd and mol
  • A number of derived SI units such as V, C, F and others
  • A number of non-si units such as M (molar) and L (liter)
  • Some alternative units such as lb, mile, day etc

A large number of predefined units are available in the module myokit.units.

Quantifiers such as “k” for kilo, “m” for milli etc. can be added for all base SI units, derived SI units and a couple of non-SI ones (notably mL and mM). The available quantifiers are:

y yocto 1e-24
z zepto 1e-21
a atto 1e-18
f femto 1e-15
p pico 1e-12
n nano 1e-9
u micro 1e-6
m milli 1e-3
c centi 1e-2
d deci 1e-1
h hecto 1e2
k kilo 1e3
M mega 1e6
G giga 1e9
T tera 1e12
E exa 1e15
Z zetta 1e18
Y yotta 1e21

Note the omission of “deca” (da) and the use of “u” for micro.

Some examples of valid unit declarations are:

F = [C/mol]
R = 8314 [mJ/mol/K]
T = 310 [K]

length = 0.01 [cm]
radius = 0.0011 [cm] : Cell radius
volume = 3.14 * 1000 * radius * radius * length
    in [uL]
    desc: Cell volume
v_cyt = volume * 0.678
    in [uL]

Foreign variables

Variables from other components can be addressed using the syntax component_name.variable_name.

[membrane]
dot(V) = expression

[other]
x = 5 * exp(membrane.V)

Local aliases

Within a component, it is possible to define an alias for commonly used variables from different components:

[membrane]
dot(V) = expression

[other]
use membrane.V as Vm
x = 5 * exp(Vm)

If no name is specified with “as”, the original variable name is used. In the following example the [other] component is equivalent to the one given above:

[other]
use membrane.V
x = 5 * exp(V)

Alias definitions can be chained together with commas:

[other]
use membrane.V, comp.var1 as v1, comp.var2 as v2
x = 5 * exp(V) + v1 * v2

Nested variables

Many electrophysiological equations contain repeated terms or terms with a conceptual meaning that are not used by any other equations within the system. To separate these “sub-equations”, myokit allows nesting of variables.

Nested variables can be added to a variable definition by writing them indented on the subsequent line:

dot(m) = a * (1 - m) + b * m
    a = 5 * exp(3)
    b = 10 * 1 / exp(V + 40)

In this example, m is said to be the parent of a and b. Variables with the same parent are referred to as siblings.

Myokit allows multi-level nesting:

dot(m) = a * (1 - m) + b * m
    a = 5 * exp(3)
    b = c + 14
        c = 5

Here, the set of m and b are refered to as c‘s ancestors.

Scope and naming

Using an unqualified name, a variable can always access its own child variables or a child of any of its ancestors. Access to children of any other variables is not allowed.

Using a qualified name (component.variable), a variable can access non-nested variables in any component.

This is reflected in the naming scope rules: when adding a variable to a component or another variable the naming rules are checked to ensure names are unique with each variable’s scope.

Multi-line expressions

Variable expressions spanning multiple lines can be created by ending a line in \ or by wrapping the expression in parentheses:

[membrane]
dot(V) = 1 / C * ( I_one
             + I_two
             + I_three)
I_one = g * (V - E)   \
      + a + b + c

Multi-line metadata

Multi-line metadata values can be entered by wrapping them in triple quotes:

R = 8314
    desc: """
          This is a very
          very
          long description
          """

The line breaks in multi-line values are maintained, all whitespace is trimmed from the right-hand side. On the left, whitespace corresponding to the lowest indentation level is trimmed.

Expression syntax

The following operators are provided:

+ Addition 1 + 1 = 2
- Subtraction 2 - 1 = 1
* Multiplication 4 * 2 = 8
/ Division 8 / 4 = 2
// Integer division / Quotient 11 // 3 = 3
% Modulo / Remainder 11 % 3  = 2
^ Exponentiation / Power 3 ^ 2 = 9

In addition, + and - can be used to indicate signs: +5+-2=3

Parts of expressions can be grouped using parentheses 5 * (4 - 2) = 10

The following conditional operators are defined:

== Equality
!= Inequality
> Greater than
< Less than
>= Greater than or equal
<= Less than or equal

Conditions can be strung together using and and or, or negated with not.

Pre-defined Functions

The following functions are defined:

sqrt(x) Square root
sin(x) Sine (all trigonomic functions work with radians)
cos(x) Cosine
tan(x) Tangent
asin(x) Inverse sine
acos(x) Inverse cosine
atan(x) Inverse tangent
exp(x) Returns e to the power of x
log(x) Returns the natural logarithm (also known as ln) of x
log(x, b) Returns the base-b logarithm of x
log10(x) Returns the base-10 logarithm of x
floor(x) Returns the largest integer less than or equal to x
ceil(x) Returns the smallest integer greater than or equal to x
abs(x) Returns the absolute value of x

In addition, the expression dot(x) can be used to reference the time derivative of state variable x.

Conditional statements

Conditional statements can be made using the if function:

x = if(V < -50,
    0.2 * exp((V - 12) / 4.7),
    0.5 * exp((V + 19) / 1.2))

Which should be read as:

if V < -50 then
    x = 0.2 * exp((V - 12) / 4.7)
else
    x = 0.5 * exp((V + 19) / 1.2)

Advanced conditional statements

Conditional statements with more than 1 branch can be made using the piecewise construct:

x = piecewise(
    V < -50, 0.2 * exp((V - 12) / 4.7),
    V <   0, 0.5 * exp((V + 19) / 1.2),
    0)

Which should be read as:

if V < -50 then
    x = 0.2 * exp((V - 12) / 4.7)
else if V < 0 then
    x = 0.5 * exp((V + 19) / 1.2)
else
    x = 0

The final “else” part is not optional. If conditions overlap, only the first condition that evaluates to true will be used.

Very often, like in the example above, piecewise statements are used to define a piecewise continuous function over some range. For example f(x) = f0(x) for x < 0, f(x) = f1(x) for 0 <= x < 10 and f(x) = f2(x) otherwise. Such statements can benefit from (1) a shorter syntax and (2) an optimised implementation (for example a switch() statement or a binary decision tree). For such cases, the opiecewise function (short for “ordered piecewise”) can be used:

x = piecewise(
    V < -50, 0.2 * exp((V - 12) / 4.7),
    V <   0, 0.5 * exp((V + 19) / 1.2),
    0)

can be written as:

x = opiecewise(V,
    -50, 0.2 * exp((V - 12) / 4.7),
      0, 0.5 * exp((V + 19) / 1.2),
      0)

When using opiecewise the different switching points (-50 and 0 in the example) should be specified in increasing order.

Splines & Polynomials

The Spline construct, specified as spline can be used to define a piecewise polynomial function where each polynomial has the same degree. Polynomials can be written in myokit using the polynomial function. For example, the following:

p = polynomial(x, 4, 2, 3)

is mathematically equivalent to:

p = 4 + 2*x + 3*x^2

Splines follow the same syntax as opiecewise, but each piece must be written using polynomial:

s = spline(x,
    0.5, polynomial(x, 1, 2, 3),
    1.5, polynomial(x, 2, 3, 4),
    3.5, polynomial(x, 5, 6, 7))

User defined functions

A user may define template functions by adding them to the header. User functions may reference each other but not themselves. The syntax is shown in the following example:

[[model]]
sigmoid(V, Vh, s, lo, hi) = lo + (hi - lo) / (1 + exp((Vh - V) / s))

Interfacing with the outside world

In many cases, not all variables of interest are contained within the model. For example if a simulation engine is used to drive the model this engine may provide a variable time. Other examples of external variables include a pacing or driving variable or an input current derived from neighbouring cells.

The mmt syntax allows variables to be bound to an external value using the bind keyword:

[environment]
t = 5 bind time

In this example, the variable t is defined and given the value 5. However, when the model is passed to a simulation or export routine that provides the external source “time”, it will know to replace t’s value with the appropriate value (in this case the simulation time) on every iteration. If the routine doesn’t provide a suitable “time” it can simply revert to the default value 5. This way, a model can be made suitable for use with different simulation routines.

Bindings are unique: two variables in the same model cannot be bound to the same input.

The external sources provided by each simulation engine or export are listed in their documentation.

Time dependence and pacing

Explicit time dependence is discouraged, but possible in many simulations using the external source time.

In principle, this variable can be used to pace the model, but there are a number of problems with this:

  1. Conceptually, it makes sense to apply different protocols to the same cell model.
  2. Pacing tends to be applied in block pulses. Because these are discontinuous, there is nothing in their derivatives that indicates to an ODE solver that something interesting is about to happen. As a result, the solver may skip over the - typically very short - stimuli.

To remedy this, the standard myokit simulation engine has an event-driven pacing mechanism that can be accessed through the variable pace:

[stimulus]
level = 0 bind pace
amplitude = -25
istim = level * amplitude

For information on defining a pacing protocol, see the section Pacing protocol syntax.

Labelling special variables

Some variables in a model have a special meaning that may be relevant to simulation engines. These can be marked using the label keyword. For example, a multi-cell simulation might need to know the membrane potential to determine the appropriate input current from one cell to the next or a single cell simulation may wish to calculate the maximum dV/dt.

A typical label is “membrane_potential”:

[membrane]
dot(V) = -(I_K + I_Na + I_Ca + I_stim)
    label membrane_potential

A quick syntax for the label construct is provided:

[membrane]
dot(V) = -(I_K + I_Na + I_Ca + I_stim) label membrane_potential

Like bindings, label names are unique: a label can only be applied to one variable per model. In addition, bindings and labels share the same namespace: the names of labels and bindings cannot overlap.

The labels and bindings supported by simulation engines or exports are listed in their documentation.

Namespaces and ontologies

At the time of writing, Myokit does not define any ontology providing the names of labels and bindings. Instead, each simulation engine or experiment specifies the labels and binds it uses in its documentation.

However, the following two constraints are imposed:

  1. Names of bindings and labels follow the same naming rules as unqualified variable names in Myokit.
  2. Labels and bindings share a namespace: The names of external inputs (bindings) and labels can not overlap.

References, solvability

The order in which variables are specified doesn’t matter. However, cycles in the variables’ dependencies are not allowed. For the sake of modelling, it is often nice to have a non-cyclical graph of component dependencies, but no such requirements are made by myokit.

Shorthand syntax

Variable units, bindings, labels and descriptions can be written in a shorthand syntax on the same line as the variable definition. If multiple shorthands are used, their order is important. The correct order is:

x = 15 in [ms] bind time label special : comment

Example: Luo-Rudy 1991

What follows is an adaptation of the 1991 Luo-Rudy model for the ventricular myocyte:

[[model]]
name: Luo-Rudy model 1991 (LR91)
desc: """
      Test implementation of the Luo-Rudy model for the ventricular
      myocyte.
      The original model can be downloaded from http://rudylab.wustl.edu
      """
# Template functions
sig(V, Vstar, a, b) = exp(a * (Vstar - V)) / (1 + exp(b * (Vstar - V)))
# Initial conditions
membrane.V         = -84.4
na_fast.m          = 0.0017
na_fast.h          = 0.98
na_fast.j          = 0.99
ca_slow_inward.d   = 0.003
ca_slow_inward.f   = 0.999
k_time_dependent.x = 0.042
ca_slow_inward.Cai = 0.00018

[engine]
time = 0 bind time
pace = 0 bind pace

[phys]
R = 8314 [J/kmol/K] : Gas constant
T = 310 [K] : The cell temperature
F = 96484.6 [C/mol] : Faraday's constant
RTF = R * T / F

[membrane]
C = 1 [uF/cm^2]
stim_amplitude = -25.5 [uA/cm^2]
I_stim = engine.pace * stim_amplitude
dot(V) = (-1 / C) * (
         I_stim +
         na_fast.i_Na +
         ca_slow_inward.i_si +
         k_time_dependent.i_K +
         k_time_independent.i_K1 +
         k_plateau.i_Kp +
         background_current.i_b )
    label membrane_potential
    desc: The membrane potential
    in [mV]

[ions]
Nao = 140 [mmol/L] : External Na+ concentration
Nai = 18  [mmol/L] : Internal Na+ concentration
Ki  = 145 [mmol/L] : Internal K+ concentration
Ko  = 5.4 [mmol/L] : External K+ concentration

[na_fast]
use membrane.V
g_Na = 23 [mS/cm^2]
E_Na = phys.RTF * log(ions.Nao / ions.Nai)
    desc: Na+ Nernst potential
    in [uF/cm^2]
i_Na = g_Na * m^3 * h * j * (V - E_Na)
dot(m) = alpha * (1 - m) - beta * m : m-gate of the fast sodium channel
    alpha = 0.32 * (V + 47.13) / (1 - exp(-0.1 * (V + 47.13)))
    beta = 0.08 * exp(-V / 11)
dot(h) = alpha * (1 - h) - beta * h : h-gate of the fast sodium channel
    alpha = piecewise(V < -40,
        0.135 * exp((80 + V) / -6.8),
        0
        )
    beta = piecewise(
        V < -40,
        3.56 * exp(0.079 * V) + 310000 * exp(0.35 * V),
        1 / (0.13 * (1 + exp((V + 10.66) / -11.1)))
        )
dot(j) = alpha * (1 - j) - beta * j : j-gate of the fast sodium channel
    alpha = piecewise(V < -40,
        (-127140 * exp(0.2444 * V) - 0.00003474 * exp(-0.04391 * V))
         * (V + 37.78) / (1 + exp(0.311 * (V + 79.23))),
        0
        )
    beta = piecewise(V < -40,
        0.1212 * exp(-0.01052 * V) / (1 + exp(-0.1378 * (V + 40.14))),
        0.3 * exp(-0.0000002535 * V) / (1 + exp(-0.1 * (V + 32)))
        )

[ca_slow_inward]
use membrane.V
E_si = 7.7 - 13.0287 * log(Cai)
i_si = 0.09 * d * f * (V - E_si)
dot(d) = alpha * (1 - d) - beta * d
    alpha = 0.095 * sig(V, 5, 0.01, 0.072)
    beta  = 0.07 * sig(V, -44, 0.017, -0.05)
dot(f) = alpha * (1 - f) - beta * f
    alpha = 0.012 * sig(V, -28, 0.008, -0.15)
    beta = 0.0065 * sig(V, -30, 0.02, 0.2)
dot(Cai) = -0.0001 * i_si + 0.07 * (0.0001 - Cai)

[k_time_dependent]
use membrane.V
use ions.Ko, ions.Nao, ions.Ki, ions.Nai
PR_NaK = 0.01833
g_K = 0.282 * sqrt(ions.Ko / 5.4)
E_K = phys.RTF * log((Ko + PR_NaK * Nao) / (Ki + PR_NaK * Nai))
xi = piecewise(V > -100,
        2.837 * (exp(0.04*(V + 77)) - 1) / ((V + 77)*exp(0.04 * (V + 35))),
        1)
i_K = g_K * x * xi * (V - E_K)
dot(x) = alpha * (1 - x) - beta * x
    alpha = 0.0005 * sig(V, -50, -0.083, -0.057)
    beta  = 0.0013 * sig(V, -20, 0.06,  0.04)

[k_time_independent]
use membrane.V
E_K1 = phys.RTF * log(ions.Ko / ions.Ki)
g_K1 = 0.6047 * sqrt(ions.Ko / 5.4)
i_K1 = g_K1 * (alpha / (alpha + beta)) * (V - E_K1)
    alpha = 1.02 / (1 + exp(0.2385 * (V - E_K1 - 59.215)))
    beta = (0.49124 * exp(0.08032 * (V - E_K1 + 5.476))
            + exp(0.06175 * (V - E_K1 - 594.31))
           ) / (1 + exp(-0.5143 * (V - E_K1 + 4.753)))

[k_plateau]
g_Kp = 0.0183 [mS/cm^2]
E_Kp = k_time_independent.E_K1
i_Kp = g_Kp * Kp * (membrane.V - E_Kp)
    Kp = 1 / (1 + exp((7.488 - membrane.V) / 5.98))

[background_current]
E_b = -59.87 [mV]
g_b = 0.03921 [mS/cm^2]
i_b = g_b * (membrane.V - E_b)