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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 ABC calibration of $I_\text{Kur}$ in Nygren model to unified dataset.
Note the term $I_\text{sus}$ for sustained outward Potassium current is used throughout the notebook.
Step1:  Initial set-up
Load experiments used for unified dataset calibration
Step2:  Plot steady-state and time constant functions of original model
Step3:  Activation gate ($r$) calibration
Combine model and experiments to produce
Step4:  Set up prior ranges for each parameter in the model.
See the modelfile for further information on specific parameters. Prepending `log_' has the effect of setting the parameter in log space.
Step5:  Run ABC calibration
Step6:  Database results analysis
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Python Code:
import os, tempfile
import logging
import matplotlib as mpl
import matplotlib.pyplot as plt
import seaborn as sns
import numpy as np
from ionchannelABC import theoretical_population_size
from ionchannelABC import IonChannelDistance, EfficientMultivariateNormalTransition, IonChannelAcceptor
from ionchannelABC.experiment import setup
from ionchannelABC.visualization import plot_sim_results, plot_kde_matrix_custom
import myokit
from pyabc import Distribution, RV, History, ABCSMC
from pyabc.epsilon import MedianEpsilon
from pyabc.sampler import MulticoreEvalParallelSampler, SingleCoreSampler
from pyabc.populationstrategy import ConstantPopulationSize
Explanation: ABC calibration of $I_\text{Kur}$ in Nygren model to unified dataset.
Note the term $I_\text{sus}$ for sustained outward Potassium current is used throughout the notebook.
End of explanation
from experiments.isus_wang import (wang_act_and_kin)
from experiments.isus_courtemanche import (courtemanche_deact)
modelfile = 'models/nygren_isus.mmt'
Explanation: Initial set-up
Load experiments used for unified dataset calibration:
 - Steady-state activation [Wang1993]
 - Activation time constant [Wang1993]
 - Deactivation time constant [Courtemanche1998]
 - Steady-state inactivation [Firek1995]
 - Inactivation time constant [Nygren1998]
 - Recovery time constant [Nygren1998]
End of explanation
from ionchannelABC.visualization import plot_variables
sns.set_context('talk')
V = np.arange(-100, 40, 0.01)
nyg_par_map = {'ri': 'isus.r_inf',
            'si': 'isus.s_inf',
            'rt': 'isus.tau_r',
            'st': 'isus.tau_s'}
f, ax = plot_variables(V, nyg_par_map, modelfile, figshape=(2,2))
Explanation: Plot steady-state and time constant functions of original model
End of explanation
observations, model, summary_statistics = setup(modelfile,
                                                wang_act_and_kin,
                                                courtemanche_deact)
assert len(observations)==len(summary_statistics(model({})))
g = plot_sim_results(modelfile,
                     wang_act_and_kin,
                     courtemanche_deact)
Explanation: Activation gate ($r$) calibration
Combine model and experiments to produce:
 - observations dataframe
 - model function to run experiments and return traces
 - summary statistics function to accept traces
End of explanation
limits = {'isus.p1': (-100, 100),
          'isus.p2': (1e-7, 50),
          'log_isus.p3': (-5, 0),
          'isus.p4': (-100, 100),
          'isus.p5': (1e-7, 50),
          'log_isus.p6': (-6, -1)}
prior = Distribution(**{key: RV("uniform", a, b - a)
                        for key, (a,b) in limits.items()})
Explanation: Set up prior ranges for each parameter in the model.
See the modelfile for further information on specific parameters. Prepending `log_' has the effect of setting the parameter in log space.
End of explanation
db_path = ("sqlite:///" + os.path.join(tempfile.gettempdir(), "nygren_isus_rgate_unified.db"))
logging.basicConfig()
abc_logger = logging.getLogger('ABC')
abc_logger.setLevel(logging.DEBUG)
eps_logger = logging.getLogger('Epsilon')
eps_logger.setLevel(logging.DEBUG)
pop_size = theoretical_population_size(2, len(limits))
print("Theoretical minimum population size is {} particles".format(pop_size))
abc = ABCSMC(models=model,
             parameter_priors=prior,
             distance_function=IonChannelDistance(
                 exp_id=list(observations.exp_id),
                 variance=list(observations.variance),
                 delta=0.05),
             population_size=ConstantPopulationSize(1000),
             summary_statistics=summary_statistics,
             transitions=EfficientMultivariateNormalTransition(),
             eps=MedianEpsilon(initial_epsilon=100),
             sampler=MulticoreEvalParallelSampler(n_procs=16),
             acceptor=IonChannelAcceptor())
obs = observations.to_dict()['y']
obs = {str(k): v for k, v in obs.items()}
abc_id = abc.new(db_path, obs)
history = abc.run(minimum_epsilon=0., max_nr_populations=100, min_acceptance_rate=0.01)
Explanation: Run ABC calibration
End of explanation
history = History('sqlite:///results/nygren/isus/unified/nygren_isus_rgate_unified.db')
df, w = history.get_distribution()
df.describe()
sns.set_context('poster')
mpl.rcParams['font.size'] = 14
mpl.rcParams['legend.fontsize'] = 14
g = plot_sim_results(modelfile,
                     wang_act_and_kin,
                    courtemanche_deact,
                     df=df, w=w)
plt.tight_layout()
import pandas as pd
N = 100
nyg_par_samples = df.sample(n=N, weights=w, replace=True)
nyg_par_samples = nyg_par_samples.set_index([pd.Index(range(N))])
nyg_par_samples = nyg_par_samples.to_dict(orient='records')
sns.set_context('talk')
mpl.rcParams['font.size'] = 14
mpl.rcParams['legend.fontsize'] = 14
f, ax = plot_variables(V, nyg_par_map, 
                       'models/nygren_isus.mmt', 
                       [nyg_par_samples],
                       figshape=(2,2))
from ionchannelABC.visualization import plot_kde_matrix_custom
import myokit
import numpy as np
m,_,_ = myokit.load(modelfile)
originals = {}
for name in limits.keys():
    if name.startswith("log"):
        name_ = name[4:]
    else:
        name_ = name
    val = m.value(name_)
    if name.startswith("log"):
        val_ = np.log10(val)
    else:
        val_ = val
    originals[name] = val_
sns.set_context('paper')
g = plot_kde_matrix_custom(df, w, limits=limits, refval=originals)
plt.tight_layout()
Explanation: Database results analysis
End of explanation
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