Single Cell Scale

For electrophysiological simulation, the basic unit of the model is the cellular activity. Cellular models of cardiac electrical activity were first constructed over 50 years ago, and today, these models have been refined and are routinely put into organ scale simulations. The modelling components are described below.

Ionic models refer to the electrical representation of the cell based on the movement of ions across the cell membrane, resulting in a change in the voltage across the membrane. Schematically, a capacitor is placed in parallel with a number of ionic transport mechanisms. In general, an ionic model is of the form

 I_{\rm m} = C_m \frac{d V_{\rm m}}{d t} + \Sigma_\chi I_\chi  \nonumber

where V_{\rm m} is the voltage across the cell membrane, I_{\rm m} is the net current across the membrane, C_{\rm m} is the membrane capacitance, and I_{\chi} is a particular transport mechanism, either a channel, pump or exchanger. Additional equations may track ionic concentrations and the processes which affect them, like calcium-induced release from the sarcoplasmic reticulum. By convention, outward current, i.e., positive ions leaving the cell, is defined as being positive, while inward currents are negative.

Channels are represented as resistors in series with a battery. The battery represents the Nernst Potential resulting from the electrical field developed by the ion concentration difference across the membrane. It is given by

    E_S = \frac{RT}{zF}\ln\frac{[S]_i}{[S]_e}  \nonumber

where R is the gas constant, T is the temperature, F is Faraday’s constant and z is the valence of the ion species S. Channels are dynamical systems which open and close in response to various conditions like transmembrane voltage, stretch, ligands, and other factors. The opening and closing rates of the channels vary by several orders of magnitudes, and are generally, nonlinear in nature. Thus, the equivalent electrical resistance of a channel is a time dependent quantity.

The first representation of a cardiac cell was that produced by D. Noble of a Purkinje cell, based on modification of the Hodgkin-Huxley nerve action potential. Since then, hundreds of models have been developed for many reasons. Ionic models need to be developed to match experimental procedures if the models are to be predictive and offer insight into mechanistic workings. In mammals, action potential durations range from tens of milliseconds for mice to several hundreds of milliseconds for large animals. Atrial myocyte protein expression is quite different from that of ventricular myocytes, resulting in different action potential durations and shapes. Even nearby cells exhibit action potential differences due to slightly different levels of channel protein expression. The complexity of ionic models has been steadily growing as more knowledge is gained through better experimental techniques, equipment and specific blockers of transport mechanisms. As more mechanisms are identified as affecting electrophysiology, either directly or indirectly, they are incorporated into ionic models. Selection of the model to use is not always obvious as different models may be available for the same species and heart location, which may yield quite different behaviour. Regardless, identifying the strengths and weaknesses of an ionic model for a particular application is important.

Ionic models may be biophysically detailed or phenomenological. Biophysically detailed models attempt to discretely depict important currents and processes within the cell. Early models had approximately ten equations depicting only sodium and potassium channels, while present models have hundreds of equations taking into account not only membrane ion transporters, but intracellular calcium handling, mitochondrial function, and biochemical signalling pathways. Conversely, phenomenological ionic models use a set of currents which faithfully reproduce whole cell behaviour in terms of action potential shape and duration, and restitution properties. The currents in the phenomenological model are not physiological but can be considered as amalgamations of known currents. While these models are computationally much simpler and easier to tune, they lose the direct correspondence with the biophysics which makes implementing drug effects or cellular pathologies challenging. Finally, cellular automata models have also been used, and can be considered as a type of phenomenological model. These models do not use differential equations but have a set of rules which dictate transitions between discrete states of the model. As such, these models are computationally light, and can be as detailed as required. However, behaviour may not be as rich as differential equations. The choice of model, biophysical or phenomenological, depends on the nature of the problem being considered, and the availability of computational resources for the problem size.


Fig. 34 Ionic models A: Myocyte schematic showing the membrane (blue) which contains channels (yellow cylinders), pumps (yellow circles) and exchangers (blue circles). The sarcoplasmic reticulum is divided into the junctional (JSR) and network (NSR) regions and channels and pumps control ion movement into and out of it. Figure taken from Michailova et al. B: Electrical representation of a cell showing a limited number of currents: fast sodium (I_{\rm {Na}}), inward rectifier (I_{\rm {K1}}), fast and slow repolarizing currents (I_{\mathrm{Kr}} and I_{\mathrm{Ks}}) L-type calcium (I_{\mathrm{Ca,L}}}, the sodium-potassium pump (I_{\mathrm{NaK}}) and the sodium-calcium exchanger (I_{\mathrm{NCX}}). Nernst potentials for the ionic species are indicated (E_{\rm S}) as is the membrane capacitance (C_{\rm m}). C: Effect of applying a suprathreshold current at time 0 to the ten Tuscher ionic model of the human ventricular myocyte. Top: Major ionic currents. Note that the following currents are clipped: I_{\mathrm {Na}} (-46.4), I_{\mathrm{to}} (15.3), and I_{\mathrm{Ca,L}} (-9.6). Bottom: the resulting action potential (V_{\rm m}) and myoplasmic calcium transient ([Ca_{\rm i}]). }

When considering the mathematical description of ionic channels, there are two main approaches which are detailed below.

Hodgkin-Huxley Dynamics

The Hodgkin-Huxley approach is named after the Nobel laureates who were the first to develop a mathematical model of the neural action potential. Single channel activity recordings show that channels have a small set of discrete conductance states, with the channel stochastically transitioning between closed states and open states. Short time analysis of a single channel is very difficult to interpret but ensemble averaging clearly reveals smooth kinetics in changes of channel conductance. In the Hodgkin-Huxley formulation, channel conductance is assumed to be controlled by gates which take on values between 0 and unity, representing the portion of the cells in one state. Since cells have hundreds of ion channels if not more, this approximation holds well. Current flow produced by ion species X passing through a channel is then described by

    I_{\rm S} = \overline{g}_S \prod_n \eta_n ( V_{\rm m} - E_{\rm S} )  \nonumber

where \overline{g}_{\rm S} is the maximum conductance of the channel and \eta_{\rm n} is a gating variable. Often the gating variable is assumed to follow first order dynamics so

    \frac{d\eta}{dt} & = \alpha(V_{\rm m})(1-\eta) - \beta(V_{\rm m})\eta \nonumber \\
                     & = \frac{\eta_{\infty}(V_{\rm m}) - \eta}{\tau_{\eta}(V_{\rm m})} \nonumber

where \alpha and \beta are rates which can be cast into an equivalent form of a steady state value (\eta_{\infty}) and a rate of change (\tau_\eta). The advantage of this latter formulation is that mathematically, the update of the gating variables can be performed by a numercial integration method, the Rush-Larsen technique, which is guaranteed to unconditionally keep the gating variable bounded within the range [0,1] while allowing a large time step.

Markov Models

Markov models describe the channel as a set off states, such as the conductance states seen in single channel recordings, as well as the non conducting states through which a channel must pass to reach them. Transitions between states are described by rate constants which can be functions of concentrations or voltages, or fixed. These is a variables for each state which represent the portion of channels in a cell which are in that state. As such, the sum of state variables is unity. A Hodgkin-Huxley made can be converted to a Markov model by considering a gating variable as a two state model. However, the advantage of the Markov representation is its ability to model drug interaction. Essentially, drug binding doubles the number of possible states in an ionic model, augmenting the original states with drug-bound versions. One may easily limit drug binding to a particular subset of channel states, and more finely control channel kinetics.

bench - Single Cell Modeling Tool

The single cell experimentation tool bench is quite versatile and can be used to replicate a wider range of single cell experimental protocols. A tutorial on how to use bench is given here. The following sections demonstrate how one explores all available bench functions and explains the command line arguments available.

Getting Help

The --help, --detailed-help or --full-help arguments provide a list of all available arguments with increasing levels of detail.

bench --help


Use the IMP library for single cell experiments. All times in ms; all voltages
in mV;  currents in uA/cm^2

Usage: LIMPET [-h|--help] [--detailed-help] [--full-help] [-V|--version]
         [-cFLOAT|--stim-curr=FLOAT] [--stim-volt=FLOAT] [--stim-file=STRING]
         [-aDOUBLE|--duration=DOUBLE] [-DDOUBLE|--dt=DOUBLE] [-nINT|--num=INT]
         [--ext-vm-update] [--rseed=INT] [-TDOUBLE|--stim-dur=DOUBLE]
         [-A|--stim-assign] [--stim-species=STRING] [--stim-ratios=STRING]
         [--resistance=DOUBLE] [-ISTRING|--imp=STRING]
         [-pSTRING|--imp-par=STRING] [-PSTRING|--plug-in=STRING]
         [-mSTRING|--plug-par=STRING] [--load-module=STRING]
         [-lDOUBLE|--clamp=DOUBLE] [-LDOUBLE|--clamp-dur=DOUBLE]
         [--clamp-start=DOUBLE] [--clamp-SVs=STRING] [--SV-clamp-files=STRING]
         [--SV-I-trigger] [--AP-clamp-file=STRING] [-NDOUBLE|--strain=DOUBLE]
         [-tDOUBLE|--strain-time=DOUBLE] [-yFLOAT|--strain-dur=FLOAT]
         [--strain-rate=DOUBLE] [-oDOUBLE|--dt-out=DOUBLE]
         [-OSTRING|--fout=STRING] [--no-trace] [--trace-no=INT] [-B|--bin]
         [-uSTRING|--imp-sv-dump=STRING] [-gSTRING|--plug-sv-dump=STRING]
         [-d|--dump-lut] [--APstatistics] [-v|--validate]
         [-sDOUBLE|--save-time=DOUBLE] [-fSTRING|--save-file=STRING]
         [-rSTRING|--restore=STRING] [-FSTRING|--save-ini-file=STRING]
         [-SDOUBLE|--save-ini-time=DOUBLE] [-RSTRING|--read-ini-file=STRING]
  or : LIMPET [--list-imps] [--plugin-outputs] [--imp-info]
  or : LIMPET [--stim-times=STRING]
  or : LIMPET [--numstim=INT] [-iDOUBLE|--stim-start=DOUBLE]
  or : LIMPET --restitute=STRING [--res-file=STRING] [--res-trace]
  or : LIMPET --clamp-ini=DOUBLE --clamp-file=STRING

  -h, --help                   Print help and exit
      --detailed-help          Print help, including all details and hidden
                                 options, and exit
      --full-help              Print help, including hidden options, and exit
  -V, --version                Print version and exit

 Mode: regstim
  regular periodic stimuli
      --numstim=INT            number of stimuli  (default=`1')
  -i, --stim-start=DOUBLE      start stimulation  (default=`1.')
  -b, --bcl=DOUBLE             basic cycle length  (default=`1000.0')

 Mode: neqstim
  irregularly timed stimuli
      --stim-times=STRING      comma separated list of stim times  (default=`')

 Mode: restitute
  restitution curves
      --restitute=STRING       restitution experiment  (possible
                                 values="S1S2", "dyn", "S1S2f")
      --res-file=STRING        definition file for restitution parameters
      --res-trace              output ionic model trace  (default=off)

 Mode: info
  output IMP information
      --list-imps              list all available IMPs  (default=off)
      --plugin-outputs         list the outputs of available plugins
      --imp-info               print tunable parameters and state variables for
                                 particular IMPs  (default=off)

 Mode: vclamp
  Constant Voltage clamp
      --clamp-ini=DOUBLE       outside of clamp pulse, clamp Vm to this value
      --clamp-file=STRING      clamp voltage to a signal read from a file

 Group: stimtype
  stimulus type
  -c, --stim-curr=FLOAT        stimulation current  (default=`60.0')
      --stim-volt=FLOAT        stimulation voltage
      --stim-file=STRING       use signal from a file for current stim

Simulation control:
  -a, --duration=DOUBLE        duration of simulation  (default=`1000.')
  -D, --dt=DOUBLE              time step  (default=`.01')
  -n, --num=INT                number of cells  (default=`1')
      --ext-vm-update          update Vm externally  (default=off)
      --rseed=INT              random number seed  (default=`1')

  -T, --stim-dur=DOUBLE        duration of stimulus pulse  (default=`1.')
  -A, --stim-assign            stimulus current assignment  (default=off)
      --stim-species=STRING    concentrations that should be affected by
                                 stimuli, e.g. 'K_i:Cl_i'  (default=`K_i')
      --stim-ratios=STRING     proportions of stimlus current carried by each
                                 species, e.g. '0.7:0.3'  (default=`1.0')
      --resistance=DOUBLE      coupling resistance [kOhm]  (default=`100.0')

Ionic Models:
  -I, --imp=STRING             IMP to use  (default=`MBRDR')
  -p, --imp-par=STRING         params to modify IMP  (default=`')
  -P, --plug-in=STRING         plugins to use, separate with ':'  (default=`')
  -m, --plug-par=STRING        params to modify plug-ins, separate params with
                                 ',' and plugin params with ':'  (default=`')
      --load-module=STRING     load a module for use with bench (implies --imp)

Voltage clamp pulse:
  -l, --clamp=DOUBLE           clamp Vm to this value  (default=`0.0')
  -L, --clamp-dur=DOUBLE       duration of Vm clamp pulse  (default=`0.0')
      --clamp-start=DOUBLE     start time of Vm clamp pulse  (default=`10.0')

      --clamp-SVs=STRING       colon separated list of state variable to clamp
      --SV-clamp-files=STRING  : separated list of files from which to read
                                 state variables  (default=`')
      --SV-I-trigger           apply SV clamps at each current stim
      --AP-clamp-file=STRING   action potential trace applied at each stimulus

  -N, --strain=DOUBLE          amount of strain to apply  (default=`0')
  -t, --strain-time=DOUBLE     time to strain  (default=`200')
  -y, --strain-dur=FLOAT       duration of strain (default=tonic) [ms]
      --strain-rate=DOUBLE     time to apply/remove strain  (default=`2')

  -o, --dt-out=DOUBLE          temporal output granularity  (default=`1.0')
  -O, --fout[=STRING]          output to files  (default=`BENCH_REG')
      --no-trace               do not output trace  (default=off)
      --trace-no=INT           number for trace file name  (default=`0')
  -B, --bin                    write binary files  (default=off)
  -u, --imp-sv-dump=STRING     sv dump list  (default=`')
  -g, --plug-sv-dump=STRING    : separated sv dump list for plug-ins, lists
                                 separated by semicolon  (default=`')
  -d, --dump-lut               dump lookup tables  (default=off)
      --APstatistics           compute AP statistics  (default=off)
  -v, --validate               output all SVs  (default=off)

State saving/restoring:
  -s, --save-time=DOUBLE       time at which to save state  (default=`0')
  -f, --save-file=STRING       file in which to save state  (default=`')
  -r, --restore=STRING         restore saved state file  (default=`')
  -F, --save-ini-file=STRING   text file in which to save state of a single
                                 cell  (default=`')
  -S, --save-ini-time=DOUBLE   time at which to save single cell state
  -R, --read-ini-file=STRING   text file from which to read state of a single
                                 cell  (default=`')
      --SV-init=STRING         colon separated list of comma separated

Available models

The --list-imps argument outputs a list of all available ionic models and plugins.

bench --list-imps

Ionic models:

Show model state variables and exposed parameters

The --imp-info argument lists all state variables of a model and all exposed parameters which can be modified. For instance, to interrogate this for the tenTusscher 2 model run:

bench --imp TT2 --imp-info

                                 big_cnt        0
                                   fmode        0
                                      Ko        5.4
                                     Cao        2
                                     Nao        140
                                      Vc        0.016404
                                     Vsr        0.001094
                                    Bufc        0.2
                                   Kbufc        0.001
                                   Bufsr        10
                                  Kbufsr        0.3
                                  Vmaxup        0.006375
                                     Kup        0.00025
                                       R        8314.47
                                       F        96485.3
                                       T        310
                                   RTONF        26.7138
                             CAPACITANCE        0.185
                                     Gkr        0.153
                                    pKNa        0.03
                                     Gks        0.392
                                     GK1        5.405
                                     Gto        0.294
                                     GNa        14.838
                                    GbNa        0.00029
                                     KmK        1
                                    KmNa        40
                                    knak        2.724
                                    GCaL        3.98e-05
                                    GbCa        0.000592
                                   knaca        1000
                                   KmNai        87.5
                                    KmCa        1.38
                                    ksat        0.1
                                       n        0.35
                                    GpCa        0.1238
                                    KpCa        0.0005
                                     GpK        0.0146
                               scl_tau_f        1
                                   flags        ENDO|EPI|MCELL
        State variables:

The --plugin-outputs argument provides the same information as --imp-info, but lists in addition all the outputs of plugins, that is, global quantities modified by a given plugins.

        converted_EP     Iion
        converted_EP_RS  Iion
        converted_IA     Iion
        EP               Iion
        EP_RS            Iion
        ExcConNHS        Tension
        I_ACh            Iion
        IA               Iion
        IFun             Iion
        I_KATP           K_e Iion
        INa_Bond_Markov  Iion
        I_NaH            Na_e
        Muscon           Tension
        NPStress         Tension
        FxStress         Tension
        TanhStress       Tension
        Rice             Tension
        LandStress       Tension
        NHSstress        Tension
        Lumens           delLambda Tension
        SAC_KS           Iion
        YHuSAC           Iion

Define a stimulus

The arguments --stim-curr and --stim-volt select the magnitude of a current or voltage stimulus pulse in \mu A/cm^2 and mV, respectively. The duration of a stimulus is set through the stim-dur argument. Alternatively, a trace file can be provided through --stim-file to impose a stimulus of arbitrary shape.

To initiate an action potential in the TT2 model by applying a current stimulus of 60 \mu A/cm^2 and 2 ms duration run

bench --stim-curr 60 --stim-dur 2

or, to stimulate with an arbitrary stimulus current profile run

bench --stim-file mypulse.trc

where the stimulus definition file mypulse.trc must comply with the following format:

t0  y0
t1  y1
t2  y2
tN  yN

Define a stimulus protocol

Two types of protocols are supported, a train of regular periodic stimuli or a train of irregularly time stimuli. In the regular case the total number of stimuli to be delivered is prescribed by --numstim and the interval between stimuli by the basic cycle length argument --bcl. The stimulus protocol is initiated at a given time prescribed by --stim-start.

bench --imp TT2 --numstim 50 --stim-start 10. --bcl 350

An irregular stimulus sequence is specified through the --stim-times argument.

bench --imp TT2 --stim-times 10,360,700,920

The response to this stimulus definition is illustrated here. When using the --stim-times options the number of stimuli depends solely on the times provided, using the --num-stim causes a conflict.


Fig. 35 Irregular basic cycle length due to stimulation using --stim-times.

Read the default command line output

Depending on the ionic model and the plugins used global variables which may diffuse at the tissue level are written to the console. The output will look similar to

bench --imp TT2

Outputting the following quantities at each time:
   Time              Vm            Iion

  0.000 -8.62000000e+01 +0.00000000e+00
  1.000 -8.61618521e+01 -3.46181728e-02
  2.000 +4.10176938e+01 -6.35646133e+01
  3.000 +3.51969845e+01 +1.22923870e+01
  4.000 +2.62015535e+01 +5.84205961e+00
  5.000 +2.29077661e+01 +1.52032351e+00

To quickly visualize AP shape or ionic current redirect the output to a file and visualize with your preferred tool.

Basic stimulation control

The overall duration of simulation is set by --duration in [ms] and the integration time step is chosen by --dt in [ms]. The time stepping method cannot be chosen at the command line as ODE integration in LIMPET is hardcoded, but the source code for any ionic model can be regenerated with a different integration scheme (see for details). To simulate 10 action potentials at a basic cycle length of 500 ms the overall duration of the simulation should be picked as 10 \times 500 ms = 5000 ms.

bench --duration 5000 --bcl 500 --num-stim 10 --stim-start 0

Simple AP visualization

To quickly visualize the shape of an action potential or the ionic current redirect the output to a file

bench --duration 400 --dt-out 0.1 --fout=AP

Time t, transmembrane voltage V_{\mathrm m} and ionic current, I_{\mathrm {ion}} are written to the file AP.txt. Use your favourite plotting tool such as, for instance, gnuplot.


# plot Vm
plot 'AP.txt' using 1:2 with lines

# plot Iion
plot 'AP.txt' using 1:3 with lines

Fig. 36 Simple AP visualization with gnuplot

Performance benchmarking

For the sake of performance benchmarking the number of cell solved can be increased using --num. Performance metrics of the ODE solver are reported in any simulation run at the end of the output. To run a short benchmark simulating 10 ms of activity in 10000 cells use:

mpiexec -n 16 bench --num 100000 --duration 10

This outputs a basic performance statistics as shown below:

setup time          0.001208 s
initialization time 0.036891 s
main loop time      2.801997 s
total ode time      2.788086 s

mn/avg/mx loop time 2.064943 2.799198 5.839109 ms
mn/avg/mx ODE time  2.063036 2.785301 5.825996 ms
real time factor    0.003587

Model modification

Model behavior can be modified by providing parameter modification string through --imp-par. Modifications strings are of the following form:


where param is the name of the parameter one wants to modify such as, for instance, the peak conductivity of the sodium channel, GNa. Not all parameters of ionic models are exposed for modification, but a list of those which are can be retrieved for each model using imp-info. As an example, let’s assume we are interested in specializing a generic human ventricular myocyte based on the ten Tusscher-Panfilov model to match the cellular dynamics of myocytes as they are found in the halo around an infarct, the peri-infarct zone (PZ). In the PZ various current are downregulated. Patch-clamp studies of cells harvested from the peri-infract zone have reported a 62% reduction in peak sodium current, 69% reduction in L-type calcium current and a reduction of 70% and 80% in potassium currents I_{\mathrm {Kr}} and I_{\mathrm K}, respectively. In the model these modifications are implemented by altering the peak conductivities of the respective currents as follows:

# all modification strings yield the same results
bench --imp TT2 --imp-par="GNa-62%,GCaL-69%,Gkr-70%,GK1-80%" --numstim 10 --bcl 500 --duration 5000

# or
bench --imp TT2 --imp-par="GNa*0.38,GCaL*0.31,Gkr*0.3,GK1*0.2" --numstim 10 --bcl 500 --duration 5000

To make sure that the provided parameter modifications are used as expected look at the log messages printed to the console. You should see messages confirming the use of the specified parameter modifications:

Ionic model: TT2
        GNa                  modifier: -62%            value: 5.63844
        GCaL                 modifier: -69%            value: 1.2338e-05
        Gkr                  modifier: -70%            value: 0.0459
        GK1                  modifier: -80%            value: 1.081

To observe the effect of the modification relative to the baseline model simulate a train of action potentials with both baseline and modified model. Run

# run baseline model
bench --imp TT2 --numstim 10 --bcl 500 --duration 5000 --fout=TT2_baseline

# run modified PZ model
bench --imp TT2 --imp-par="GNa-62%,GCaL-69%,Gkr-70%,GK1-80%" --numstim 10 --bcl 500 --duration 5000 --fout=TT2_PZ

Fig. 37 illustrates the effect of the parameter modifications upon AP features.


Fig. 37 Comparison of TT2 APs with baseline (red trace) and modified PZ (blue trace) settings. As a result of the modifications, the PZ APs feature a longer duration, decreased upstroke velocity and decreased peak amplitude compared to baseline APs.

Model augmentation

Published models of cellular dynamics are not complete and comprehensive representations of the electrophysiological behavior of a myocyte. Rather, the represent a subset of features which were deemed relevant for a given purpose. This is true for any published model, none of them is able to represent myocyte behavior under an arbitrary range of experimental circumstances. For the sake of verification it is crucial to be able to run models under the exact same conditions under which published results were obtained. However, after verification it is often necessary to augment a model with additional features which remained unaccounted for in the original publication. The --plug-in mechanism is designed for such applications where additional currents or physical effects such as the generation of active tension are considered without recompiling or directly modifying the model as published. The underlying concept is illustrated in Fig. 38. Typical augmentation is used for incorporating additional currents which play a role when driving the transmembrane potential V_{\mathrm m} beyond the physiological range during the application of a strong depolarizing/hyperpolarizing stimuli such as an electroporation current, EP, an electroporation current with pore resealing, EP_RS, or a hypothetical time-independent outward rectifying potassium current IA, or to account for the effect of adrenergic stimulation by incorporating an acetylcholine-dependent potassium current I_ACh


Fig. 38 Design concept underlying the implmented augmentation mechanism in LIMPET.

For instance, to augment a model of cellular dynamics to be used for studying the effect of defibrillation shocks, one would activate the plugins EP_RS and IA. This is achieved as follows:

bench --imp TT2 --plug-in "EP_RS:IA"

A frequent use of plugins is the augmentation with active stress models. For instance, to specialize a human ventricular myocyte model for use in a ventricular electro-mechanical modeling study one would enable an active stress model such as LandHumanStress:

bench --imp TT2 --plug-in "LandHumanStress"

As shown above in the Section on usage of –imp-par, the default behavior of plugins can also be modified in the same way as for the parent cell model. For instance, to reduce the peak force generated by 25 % one would use a plug-par string as follows:

bench --imp TT2 --plug-in "LandHumanStress" --plug-par="Tref-25%"

When using more than one plugin, parameter modification strings for the various plugins are separated by a :. For instance, if we further augment with the ATP-sensitive potassium current I_KATP and modify the f_{\mathrm ATP}

bench --imp TT2 --plug-in "I_KATP:LandHumanStress" --imp-par "GNa*0.8" --plug-par "f_ATP-10%:Tref-10%,TRPN_n+25%"

Again, a message is printed to the console confirming that the requested parameter modifications have been performed:

Ionic model: TT2
        GNa                  modifier: *0.8            value: 11.8704
Plug-in: I_KATP
        f_ATP                modifier: -10%            value: 0.00099
Plug-in: LandHumanStress
        Tref                 modifier: -10%            value: 108
        TRPN_n               modifier: +25%            value: 2.5

Limit cycle

Action potentials and the trajectory of state variables depend on the basic cycle length by which myocytes are activated. Typically, a train of pacing pulses of a certain length are required until the electrical activity in a myocyte arrives at a limit cycle, that is, the transmembrane voltage V_{\mathrm m}(t) and all state variables in the state vector \boldsymbol{\eta}(t) traverse the exact trajectory in the state space after each perturbation with the same stimulus. Mathematically, this boils down to

  \boldsymbol{\eta}(t+T) = \boldsymbol{\eta}(t) \hspace{1cm} \forall t > t_{\mathrm lc} \nonumber

where T is the basic cycle length, bcl, and t_{\mathrm lc} is an instant in time beyond which we deem the time evolution of the state variables to follow a limit cycle. In general, more recent ionic models, particularly those which allow intracellular ionic concentrations to vary, are unlikely to arrive at a stable limit cycle. Instead of demanding the exact same trajectories are traversed in each cycle one could use various norms to demand that the beat-to-beat deviations in the state space remain within a certain limit, \delta. For instance, we could demand

  \frac{1}{T} \int_{t}^{t+T} \frac{|| \boldsymbol{\eta}(t+T) - \boldsymbol{\eta}(t) ||}{|| \boldsymbol{\eta}(t) ||} \, dt < \delta
  \hspace{1cm} \forall t > t_{\mathrm lc} \nonumber

which would the relative difference between two cycles is, on average, less than a desired relative error \delta. Under this norm it is still possible that relative differences exceed

While using norms to detect the instant t_{\mathrm lc} beyond which the state trajectories traverse a stable limit cycle is certainly feasible, one has to keep in mind that various myocyte models will not arrive at a stable limit cycle in the strict mathematical sense and that myocytes in vivo never operate in a regime that would comply with such a rigorous constraint. A more pragmatic approach is to determine t_{\mathrm lc} by visual inspection. This is easily achieved with limpetGUI by pacing a cell for a sufficiently larger number of pacing cycles and simply observe the envelope of all state time traces. For the purpose of illustration an example is shown in Fig. 39 and Fig. 40.


Fig. 39 Visualization of a limit cycle pacing experiment using limpetGUI. After 20 seconds pacing at a bcl of 500 ms most state variables settle in at a limit cycle with the exception of intracellular concentration of sodium and potassium ions.


Fig. 40 Zoom on state traces over the last three pacing cycles of the same pacing experiments shown above in Fig. 39.

Initial state

Models of cellular dynamics are virtually never used with the values given for the initial state vector. Rather, cells are based to a given limit cycle to approximate the experimental conditions one is interested in. The state of the cell can be frozen at any given instant in time and this state can be reused then in subsequent simulations as initial state vector. Typically, the state of a cell is frozen in its limit cycle right at the end of the last diastolic interval. The standard procedure based on bench would proceed along the following steps:

  • Parameterize and augment a cell model corresponding to the experimental conditions of interest and for the intended application

  • Run bench then for a sufficiently large number of pacing pulses numstim until a reasonable approximation of a stable limit cycle is achieved for the given cycle length bcl. Confirm the arrival at a limit cycle at least visually using limpetGUI.

  • Store the initial state vector at the very end of the pacing protocol, that is, for numstim stimuli and a given bcl determine the end time of the pacing protocol as

    tsave = tstart + bcl*numstim

  • Save the state vector at the determined time tsave into a state vector file.

The last step of saving the state vector is triggered using the parameters --save-ini-time and --save-ini-file. If we want to save the state of a myocyte after 25 pacing pulses at a basic cycle length of 500 ms with the time of delivering the first pacing pulse at 0 ms one saves the initial state vector as follows:

# run baseline model
bench --imp TT2 --plug-in "LandHumanStress" --fout=TT2_baseline \
      --stim-start 0. --numstim 25 --bcl 500 --duration 12500 \
      --save-ini-time 12500 --save-ini-file

# run modified PZ model
bench --imp TT2 --plug-in "LandHumanStress" --imp-par="GNa-62%,GCaL-69%,Gkr-70%,GK1-80%"  --fout=TT2_PZ \
      --stim-start 0. --numstim 25 --bcl 500 --duration 12500 \
      --save-ini-time 12500 --save-ini-file

These runs generate two state vector files, and which store the state of each state variable at the instant t=12500 ms under the above given pacing protocol. For instance, the state vector under baseline conditions in the file will be similar to:

-85.2262            # Vm
1                   # Lambda
0                   # delLambda
0.308053            # Tension
-                   # K_e
-                   # Na_e
-                   # Ca_e
0.00256503          # Iion
-                   # tension_component
0.126796            # Ca_i
3.62845             # CaSR
0.000380231         # CaSS
0.907029            # R_
2.68967e-07         # O
7.62609             # Na_i
137.689             # K_i
0.00172638          # M
0.743346            # H
0.678447            # J
0.0150679           # Xr1
0.470865            # Xr2
0.0143382           # Xs
2.43132e-08         # R
0.999995            # S
3.3857e-05          # D
0.729795            # F
0.959628            # F2
0.995618            # FCaSS

0.0249847           # TRPN
0.999196            # TmBlocked
0.000641671         # XS
8.33073e-05         # XW
0                   # ZETAS
0                   # ZETAW
0                   # __Ca_i_local

Action Potential Duration (APD) Restitution Experiments

The single cell EP tool bench provides built-in capabilities for performing restitution experiments. Restitution experiments with bench are steered through the following command line arguments --restitute, --res-file and --res-trace.

             pick the protocol for performing the restitution experiment.
             Possible values are "S1S2", "dyn" or "S1S2f".

             definition file for restitution parameters.
             For a description of restitution file formats see :ref:`restitution protocol definition <restitution-protocol-def>`.
             By default no restitution protocol defintion file is provided.

             Can be on or off, toggles writing of trace files

Action potential analysis data are written in the following format:

#Beat Pre(P||*) lc  APD(n)  DI(n)   DI(n-1)  Tri   VmMax   VmMin

1     *          0  288.95  711.04   11.69   0.00  44.30  -86.08
2     *          0  286.97  713.02  711.04   0.00  45.27  -86.04
3     *          0  288.58  711.42  713.02   0.00  45.31  -86.01
4     *          0  304.53  695.00  711.42  78.29  45.37  -85.98
5     *          0  305.50  694.50  695.00  78.54  45.26  -85.95
6     *          0  305.90  694.10  694.50  78.74  45.31  -85.92
7     *          0  306.26  693.74  694.10  78.93  45.33  -85.90
8     *          0  306.59  693.40  693.74  79.11  45.20  -85.87

where the column headers refer to: #Beat - the total beat number in the protocol; Pre - whether beat was premature P or not ***; lc - whether AP traversed a limit cycle in the state space, 1, or not, 0; DI(n) - diastolic interval in ms of the current beat and DI(n-1) of the previous beat; Tri - Triangulation of the AP; VmMax and VmMin - maximum and minimum value of AP within pacing cycle.

Restitution data are written following the same format, but keeping only those beats which were marked as premature in the AP analysis file.

33    P          0  276.07  723.94  180.25  85.35  41.02  -85.58
46    P          0  272.73  727.27  170.63  85.93  40.11  -85.53
59    P          0  268.79  731.22  161.30  86.36  39.19  -85.49
72    P          0  264.53  735.47  151.97  86.84  38.27  -85.46
85    P          0  259.94  740.07  142.59  87.34  37.29  -85.43
98    P          0  255.01  744.99  133.16  87.94  36.20  -85.40
111   P          0  249.66  750.35  123.70  88.61  35.02  -85.37
124   P          0  243.87  756.14  114.14  89.45  33.68  -85.35
137   P          0  237.56  762.45  104.56  90.45  32.18  -85.32
150   P          0  230.68  769.32   94.94  91.71  30.47  -85.30

For an introduction into using bench as a tool for measuring restitution curves see the restitution experiments in the tutorials.

Code Generation

A tutorial on using the code generation tool is given here.