nbls.py
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nbls.py
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	# -- coding: utf-8 --
	# @Author: Theo Lemaire
	# @Email: theo.lemaire@epfl.ch
	# @Date: 2016-09-29 16:16:19
	# @Last Modified by: Theo Lemaire
	# @Last Modified time: 2020-01-26 20:41:54

	import time
	from copy import deepcopy
	import logging
	import numpy as np
	import pandas as pd

	from .simulators import PWSimulator, HybridSimulator, PeriodicSimulator
	from .bls import BilayerSonophore
	from .pneuron import PointNeuron
	from .model import Model
	from .protocols import TimeProtocol, PulsedProtocol, createPulsedProtocols
	from ..utils import *
	from ..threshold import threshold
	from ..constants import *
	from ..postpro import getFixedPoints
	from .lookups import EffectiveVariablesLookup
	from ..neurons import getPointNeuron


	NEURONS_LOOKUP_DIR = os.path.abspath(os.path.split(__file__)[0] + "/../neurons/")


	class NeuronalBilayerSonophore(BilayerSonophore):
	''' This class inherits from the BilayerSonophore class and receives an PointNeuron instance
	at initialization, to define the electro-mechanical NICE model and its SONIC variant. '''

	tscale = 'ms' # relevant temporal scale of the model
	simkey = 'ASTIM' # keyword used to characterize simulations made with this model
	titration_var = 'Adrive' # name of the titration parameter

	def __init__(self, a, pneuron, Fdrive=None, embedding_depth=0.0):
	''' Constructor of the class.

	:param a: in-plane radius of the sonophore structure within the membrane (m)
	:param pneuron: point-neuron model
	:param Fdrive: frequency of acoustic perturbation (Hz)
	:param embedding_depth: depth of the embedding tissue around the membrane (m)
	'''
	# Check validity of input parameters
	if not isinstance(pneuron, PointNeuron):
	raise ValueError('Invalid neuron type: "{}" (must inherit from PointNeuron class)'
	.format(pneuron.name))
	self.pneuron = pneuron

	# Initialize BilayerSonophore parent object
	super().__init__(a, pneuron.Cm0, pneuron.Qm0, embedding_depth)

	def __repr__(self):
	s = '{}({:.1f} nm, {}'.format(self.__class__.__name__, self.a * 1e9, self.pneuron)
	if self.d > 0.:
	s += ', d={}m'.format(si_format(self.d, precision=1, space=' '))
	return s + ')'

	@classmethod
	def initFromMeta(cls, meta):
	return cls(meta['a'], getPointNeuron(meta['neuron']),
	Fdrive=meta['Fdrive'], embedding_depth=meta['d'])

	def params(self):
	return {super().params(), self.pneuron.params()}

	def getPltVars(self, wrapleft='df["', wrapright='"]'):
	return {**super().getPltVars(wrapleft, wrapright),
	**self.pneuron.getPltVars(wrapleft, wrapright)}

	def getPltScheme(self):
	return self.pneuron.getPltScheme()

	def filecode(self, *args):
	return Model.filecode(self, *args)

	@staticmethod
	def inputs():
	# Get parent input vars and supress irrelevant entries
	bls_vars = BilayerSonophore.inputs()
	pneuron_vars = PointNeuron.inputs()
	del bls_vars['Qm']
	del pneuron_vars['Astim']

	# Fill in current input vars in appropriate order
	inputvars = bls_vars
	inputvars.update(pneuron_vars)
	inputvars['fs'] = {
	'desc': 'sonophore membrane coverage fraction',
	'label': 'f_s',
	'unit': '\%',
	'factor': 1e2,
	'precision': 0
	}
	inputvars['method'] = None
	return inputvars

	def filecodes(self, Fdrive, Adrive, pp, fs, method, qss_vars):
	# Get parent codes and supress irrelevant entries
	bls_codes = super().filecodes(Fdrive, Adrive, 0.0)
	for key in ['simkey', 'Qm']:
	del bls_codes[key]
	codes = {
	'simkey': self.simkey,
	'neuron': self.pneuron.name,
	'nature': 'CW' if pp.isCW() else 'PW',
	}
	codes.update(bls_codes)
	codes.update(pp.filecodes())
	codes['fs'] = 'fs{:.0f}%'.format(fs * 1e2) if fs < 1 else None
	codes['method'] = method
	codes['qss_vars'] = qss_vars
	return codes

	@staticmethod
	def interpOnOffVariable(key, Qm, stim, lkp):
	''' Interpolate Q-dependent effective variable along ON and OFF periods of a solution.

	:param key: lookup variable key
	:param Qm: charge density solution vector
	:param stim: stimulation state solution vector
	:param lkp: dictionary of lookups for ON and OFF states
	:return: interpolated effective variable vector
	'''
	x = np.zeros(stim.size)
	x[stim == 0] = lkp['OFF'].interpVar1D(Qm[stim == 0], key)
	x[stim == 1] = lkp['ON'].interpVar1D(Qm[stim == 1], key)
	return x

	@staticmethod
	def spatialAverage(fs, x, x0):
	''' fs-modulated spatial averaging. '''
	return fs * x + (1 - fs) * x0

	@timer
	def computeEffVars(self, Fdrive, Adrive, Qm, fs):
	''' Compute "effective" coefficients of the HH system for a specific
	combination of stimulus frequency, stimulus amplitude and charge density.

	A short mechanical simulation is run while imposing the specific charge density,
	until periodic stabilization. The HH coefficients are then averaged over the last
	acoustic cycle to yield "effective" coefficients.

	:param Fdrive: acoustic drive frequency (Hz)
	:param Adrive: acoustic drive amplitude (Pa)
	:param Qm: imposed charge density (C/m2)
	:param fs: list of sonophore membrane coverage fractions
	:return: list with computation time and a list of dictionaries of effective variables
	'''
	# Run simulation and retrieve deflection and gas content vectors from last cycle
	data = BilayerSonophore.simUntilConvergence(self, Fdrive, Adrive, Qm)
	Z_last = data.loc[-NPC_DENSE:, 'Z'].values # m
	Cm_last = self.v_capacitance(Z_last) # F/m2

	# For each coverage fraction
	effvars = []
	for x in fs:
	# Compute membrane capacitance and membrane potential vectors
	Cm = self.spatialAverage(x, Cm_last, self.Cm0) # F/m2
	Vm = Qm / Cm * 1e3 # mV

	# Compute average cycle value for membrane potential and rate constants
	effvars.append({{'V': np.mean(Vm)}, self.pneuron.getEffRates(Vm)})

	# Log process
	log = '{}: lookups @ {}Hz, {}Pa, {:.2f} nC/cm2'.format(
	self, si_format([Fdrive, Adrive], precision=1, space=' '), Qm 1e5)
	if len(fs) > 1:
	log += ', fs = {:.0f} - {:.0f}%'.format(fs.min() * 1e2, fs.max() * 1e2)
	logger.info(log)

	# Return effective coefficients
	return effvars

	def getLookupFileName(self, a=None, Fdrive=None, Adrive=None, fs=False):
	fname = '{}_lookups'.format(self.pneuron.name)
	if a is not None:
	fname += '_{:.0f}nm'.format(a * 1e9)
	if Fdrive is not None:
	fname += '_{:.0f}kHz'.format(Fdrive * 1e-3)
	if Adrive is not None:
	fname += '_{:.0f}kPa'.format(Adrive * 1e-3)
	if fs is True:
	fname += '_fs'
	return '{}.pkl'.format(fname)

	def getLookupFilePath(self, args, *kwargs):
	return os.path.join(NEURONS_LOOKUP_DIR, self.getLookupFileName(args, *kwargs))

	def getLookup(self, args, *kwargs):
	keep_tcomp = kwargs.pop('keep_tcomp', False)
	lookup_path = self.getLookupFilePath(args, *kwargs)
	lkp = EffectiveVariablesLookup.fromPickle(lookup_path)
	if not keep_tcomp:
	del lkp.tables['tcomp']
	return lkp

	def getLookup2D(self, Fdrive, fs):
	if fs < 1:
	lkp2d = self.getLookup(a=self.a, Fdrive=Fdrive, fs=True).project('fs', fs)
	else:
	lkp2d = self.getLookup().projectN({'a': self.a, 'f': Fdrive}).squeeze()
	return lkp2d

	def fullDerivatives(self, t, y, Fdrive, Adrive, fs):
	''' Compute the full system derivatives.

	:param t: specific instant in time (s)
	:param y: vector of state variables
	:param Fdrive: acoustic drive frequency (Hz)
	:param Adrive: acoustic drive amplitude (Pa)
	:param fs: sonophore membrane coevrage fraction (-)
	:return: vector of derivatives
	'''
	dydt_mech = BilayerSonophore.derivatives(
	self, t, y[:3], Fdrive, Adrive, y[3])
	dydt_elec = self.pneuron.derivatives(
	t, y[3:], Cm=self.spatialAverage(fs, self.capacitance(y[1]), self.Cm0))
	return dydt_mech + dydt_elec

	def effDerivatives(self, t, y, lkp1d, qss_vars):
	''' Compute the derivatives of the n-ODE effective system variables,
	based on 1-dimensional linear interpolation of "effective" coefficients
	that summarize the system's behaviour over an acoustic cycle.

	:param t: specific instant in time (s)
	:param y: vector of HH system variables at time t
	:param lkp: dictionary of 1D data points of "effective" coefficients
	over the charge domain, for specific frequency and amplitude values.
	:param qss_vars: list of QSS variables
	:return: vector of effective system derivatives at time t
	'''
	# Unpack values and interpolate lookup at current charge density
	Qm, *states = y
	lkp0d = lkp1d.interpolate1D(Qm)

	# Compute states dictionary from differential and QSS variables
	states_dict = {}
	i = 0
	for k in self.pneuron.statesNames():
	if k in qss_vars:
	states_dict[k] = self.pneuron.quasiSteadyStates()[k](lkp0d)
	else:
	states_dict[k] = states[i]
	i += 1

	# Compute charge density derivative
	dQmdt = - self.pneuron.iNet(lkp0d['V'], states_dict) * 1e-3

	# Compute states derivative vector only for differential variable
	dstates = []
	for k in self.pneuron.statesNames():
	if k not in qss_vars:
	dstates.append(self.pneuron.derEffStates()[k](lkp0d, states_dict))

	return [dQmdt, *dstates]

	def __simFull(self, Fdrive, Adrive, pp, fs):
	# Determine time step
	dt = 1 / (NPC_DENSE * Fdrive)

	# Compute initial non-zero deflection
	Z = self.computeInitialDeflection(Adrive, Fdrive, self.Qm0, dt)

	# Set initial conditions
	ss0 = self.pneuron.getSteadyStates(self.pneuron.Vm0)
	y0 = np.concatenate(([0., 0., self.ng0, self.Qm0], ss0))
	y1 = np.concatenate(([0., Z, self.ng0, self.Qm0], ss0))

	# Initialize simulator and compute solution
	logger.debug('Computing detailed solution')
	simulator = PWSimulator(
	lambda t, y: self.fullDerivatives(t, y, Fdrive, Adrive, fs),
	lambda t, y: self.fullDerivatives(t, y, 0., 0., fs))
	t, y, stim = simulator(
	y1, dt, pp,
	target_dt=CLASSIC_TARGET_DT,
	print_progress=logger.getEffectiveLevel() <= logging.INFO,
	monitor_func=None)
	# monitor_func=lambda t, y: f't = {t * 1e3:.5f} ms, Qm = {y[3] * 1e5:.2f} nC/cm2')

	# Prepend initial conditions (prior to stimulation)
	t, y, stim = simulator.prependSolution(t, y, stim, y0=y0)

	# Store output in dataframe and return
	data = pd.DataFrame({
	't': t,
	'stimstate': stim,
	'Z': y[:, 1],
	'ng': y[:, 2],
	'Qm': y[:, 3]
	})
	data['Vm'] = data['Qm'].values / self.spatialAverage(
	fs, self.v_capacitance(data['Z'].values), self.Cm0) * 1e3 # mV
	for i in range(len(self.pneuron.states)):
	data[self.pneuron.statesNames()[i]] = y[:, i + 4]
	return data

	def __simHybrid(self, Fdrive, Adrive, pp, fs):
	# Determine time steps
	dt_dense, dt_sparse = [1. / (n * Fdrive) for n in [NPC_DENSE, NPC_SPARSE]]

	# Compute initial non-zero deflection
	Z = self.computeInitialDeflection(Adrive, Fdrive, self.Qm0, dt_dense)

	# Set initial conditions
	ss0 = self.pneuron.getSteadyStates(self.pneuron.Vm0)
	y0 = np.concatenate(([0., 0., self.ng0, self.Qm0], ss0))
	y1 = np.concatenate(([0., Z, self.ng0, self.Qm0], ss0))

	# Initialize simulator and compute solution
	is_dense_var = np.array([True] * 3 + [False] * (len(self.pneuron.states) + 1))
	logger.debug('Computing hybrid solution')
	simulator = HybridSimulator(
	lambda t, y: self.fullDerivatives(t, y, Fdrive, Adrive, fs),
	lambda t, y: self.fullDerivatives(t, y, 0., 0., fs),
	lambda t, y, Cm: self.pneuron.derivatives(
	t, y, Cm=self.spatialAverage(fs, Cm, self.Cm0)),
	lambda yref: self.capacitance(yref[1]),
	is_dense_var,
	ivars_to_check=[1, 2])
	t, y, stim = simulator(y1, dt_dense, dt_sparse, 1. / Fdrive, pp)

	# Prepend initial conditions (prior to stimulation)
	t, y, stim = simulator.prependSolution(t, y, stim, y0=y0)

	# Store output in dataframe and return
	data = pd.DataFrame({
	't': t,
	'stimstate': stim,
	'Z': y[:, 1],
	'ng': y[:, 2],
	'Qm': y[:, 3]
	})
	data['Vm'] = data['Qm'].values / self.spatialAverage(
	fs, self.v_capacitance(data['Z'].values), self.Cm0) * 1e3 # mV
	for i in range(len(self.pneuron.states)):
	data[self.pneuron.statesNames()[i]] = y[:, i + 4]
	return data

	def __simSonic(self, Fdrive, Adrive, pp, fs, qss_vars=None, pavg=False):
	# Load appropriate 2D lookups
	lkp2d = self.getLookup2D(Fdrive, fs)

	# Interpolate 2D lookups at zero and US amplitude
	logger.debug('Interpolating lookups at A = %.2f kPa and A = 0', Adrive * 1e-3)
	lkps1d = {'ON': lkp2d.project('A', Adrive), 'OFF': lkp2d.project('A', 0.)}

	# Adapt lookups and pulsing protocol if pulse-average mode is selected
	if pavg:
	lkps1d['ON'] = lkps1d['ON'] * pp.DC + lkps1d['OFF'] * (1 - pp.DC)
	tstim = (int(pp.tstim * pp.PRF) - 1 + pp.DC) / pp.PRF
	toffset = pp.tstim + pp.toffset - tstim
	tp = TimeProtocol(tstim, toffset)

	# # Determine QSS and differential variables
	if qss_vars is None:
	qss_vars = []
	diff_vars = [item for item in self.pneuron.statesNames() if item not in qss_vars]

	# Create 1D lookup of QSS variables with reference charge vector
	QSS_1D_lkp = {
	key: EffectiveVariablesLookup(
	lkps1d['ON'].refs,
	{k: self.pneuron.quasiSteadyStates()[k](val) for k in qss_vars})
	for key, val in lkps1d.items()}

	# Set initial conditions
	sstates = [self.pneuron.steadyStates()[k](self.pneuron.Vm0) for k in diff_vars]
	y0 = np.array([self.Qm0, *sstates])

	# Initialize simulator and compute solution
	logger.debug('Computing effective solution')
	simulator = PWSimulator(
	lambda t, y: self.effDerivatives(t, y, lkps1d['ON'], qss_vars),
	lambda t, y: self.effDerivatives(t, y, lkps1d['OFF'], qss_vars))
	t, y, stim = simulator(y0, self.pneuron.chooseTimeStep(), pp)

	# Prepend initial conditions (prior to stimulation)
	t, y, stim = simulator.prependSolution(t, y, stim)

	# Store output vectors in dataframe: time, stim state, charge, potential
	# and other differential variables
	data = pd.DataFrame({
	't': t,
	'stimstate': stim,
	'Qm': y[:, 0]
	})
	data['Vm'] = self.interpOnOffVariable('V', data['Qm'].values, stim, lkps1d)
	for key in ['Z', 'ng']:
	data[key] = np.full(t.size, np.nan)
	for i, k in enumerate(diff_vars):
	data[k] = y[:, i + 1]

	# Interpolate QSS variables along charge vector and store them in dataframe
	for k in qss_vars:
	data[k] = self.interpOnOffVariable(k, data['Qm'].values, stim, QSS_1D_lkp)

	return data

	def intMethods(self):
	''' Listing of model integration methods. '''
	return {
	'full': self.__simFull,
	'hybrid': self.__simHybrid,
	'sonic': self.__simSonic
	}

	@classmethod
	@Model.checkOutputDir
	def simQueue(cls, freqs, amps, durations, offsets, PRFs, DCs, fs, methods, qss_vars, **kwargs):
	''' Create a serialized 2D array of all parameter combinations for a series of individual
	parameter sweeps, while avoiding repetition of CW protocols for a given PRF sweep.

	:param freqs: list (or 1D-array) of US frequencies
	:param amps: list (or 1D-array) of acoustic amplitudes
	:param durations: list (or 1D-array) of stimulus durations
	:param offsets: list (or 1D-array) of stimulus offsets (paired with durations array)
	:param PRFs: list (or 1D-array) of pulse-repetition frequencies
	:param DCs: list (or 1D-array) of duty cycle values
	:param fs: sonophore membrane coverage fractions (-)
	:params methods: integration methods
	:param qss_vars: QSS variables
	:return: list of parameters (list) for each simulation
	'''
	if ('full' in methods or 'hybrid' in methods) and kwargs['outputdir'] is None:
	logger.warning('Running cumbersome simulation(s) without file saving')

	if amps is None:
	amps = [None]
	ppqueue = createPulsedProtocols(durations, offsets, PRFs, DCs)
	queue = []
	for f in freqs:
	for A in amps:
	for item in ppqueue:
	for cov in fs:
	for method in methods:
	queue.append([f, A, item, cov, method, qss_vars])
	return queue

	def checkInputs(self, Fdrive, Adrive, pp, fs, method, qss_vars):
	for k, v in {'Fdrive': Fdrive, 'Adrive': Adrive, 'fs': fs}.items():
	if not isinstance(v, float):
	raise TypeError(f'Invalid {k} parameter (must be float typed)')
	if not isinstance(pp, PulsedProtocol):
	raise TypeError('Invalid pulsed protocol (must be "PulsedProtocol" instance)')
	if Fdrive <= 0:
	raise ValueError('Invalid US driving frequency: {} kHz (must be strictly positive)'
	.format(Fdrive * 1e-3))
	if Adrive < 0:
	raise ValueError('Invalid US pressure amplitude: {} kPa (must be positive or null)'
	.format(Adrive * 1e-3))
	if qss_vars is not None:
	if not isIterable(qss_vars) or not isinstance(qss_vars[0], str):
	raise ValueError('Invalid QSS variables: must be None or an iterable of strings')
	sn = self.pneuron.statesNames()
	for item in qss_vars:
	if item not in sn:
	raise ValueError(f'Invalid QSS variable: {item} (must be in {sn}')
	if method not in list(self.intMethods().keys()):
	raise ValueError(f'Invalid integration method: "{method}"')

	@Model.logNSpikes
	@Model.checkTitrate
	@Model.addMeta
	@Model.logDesc
	@Model.checkSimParams
	def simulate(self, Fdrive, Adrive, pp, fs=1., method='sonic', qss_vars=None):
	''' Simulate the electro-mechanical model for a specific set of US stimulation parameters,
	and return output data in a dataframe.

	:param Fdrive: acoustic drive frequency (Hz)
	:param Adrive: acoustic drive amplitude (Pa)
	:param pp: pulse protocol object
	:param fs: sonophore membrane coverage fraction (-)
	:param method: selected integration method
	:return: output dataframe
	'''
	# Call appropriate simulation function and return
	simfunc = self.intMethods()[method]
	simargs = [Fdrive, Adrive, pp, fs]
	if method == 'sonic':
	simargs.append(qss_vars)
	return simfunc(*simargs)

	def meta(self, Fdrive, Adrive, pp, fs, method, qss_vars):
	return {
	'simkey': self.simkey,
	'neuron': self.pneuron.name,
	'a': self.a,
	'd': self.d,
	'Fdrive': Fdrive,
	'Adrive': Adrive,
	'pp': pp,
	'fs': fs,
	'method': method,
	'qss_vars': qss_vars
	}

	def desc(self, meta):
	s = '{}: {} simulation @ f = {}Hz, A = {}Pa, {}'.format(
	self, meta['method'], *si_format([meta['Fdrive'], meta['Adrive']], 2), meta['pp'].pprint())
	if meta['fs'] < 1.0:
	s += f', fs = {(meta["fs"] * 1e2):.2f}%'
	if 'qss_vars' in meta and meta['qss_vars'] is not None:
	s += f" - QSS ({', '.join(meta['qss_vars'])})"
	return s

	@staticmethod
	def getNSpikes(data):
	return PointNeuron.getNSpikes(data)

	@logCache(os.path.join(os.path.split(__file__)[0], 'astim_titrations.log'))
	def titrate(self, Fdrive, pp, fs=1., method='sonic', qss_vars=None, xfunc=None, Arange=None):
	''' Use a binary search to determine the threshold amplitude needed to obtain
	neural excitation for a given frequency and pulsed protocol.

	:param Fdrive: US frequency (Hz)
	:param pp: pulse protocol object
	:param fs: sonophore membrane coverage fraction (-)
	:param method: integration method
	:param xfunc: function determining whether condition is reached from simulation output
	:param Arange: search interval for Adrive, iteratively refined
	:return: determined threshold amplitude (Pa)
	'''
	# Default output function
	if xfunc is None:
	xfunc = self.pneuron.titrationFunc

	# Default amplitude interval
	if Arange is None:
	Arange = [0., self.getLookup().refs['A'].max()]

	return threshold(
	lambda x: xfunc(self.simulate(Fdrive, x, pp, fs=fs, method=method, qss_vars=qss_vars)[0]),
	Arange, x0=ASTIM_AMP_INITIAL, eps_thr=ASTIM_ABS_CONV_THR, rel_eps_thr=1e0, precheck=True)

	def getQuasiSteadyStates(self, Fdrive, amps=None, charges=None, DC=1.0, squeeze_output=False):
	''' Compute the quasi-steady state values of the neuron's gating variables
	for a combination of US amplitudes, charge densities,
	at a specific US frequency and duty cycle.

	:param Fdrive: US frequency (Hz)
	:param amps: US amplitudes (Pa)
	:param charges: membrane charge densities (C/m2)
	:param DC: duty cycle
	:return: 4-tuple with reference values of US amplitude and charge density,
	as well as interpolated Vmeff and QSS gating variables
	'''
	# Get DC-averaged lookups interpolated at the appropriate amplitudes and charges
	lkp = self.getLookup().projectDC(amps=amps, DC=DC).projectN({'a': self.a, 'f': Fdrive})
	if charges is not None:
	lkp = lkp.project('Q', charges)

	# Specify dimensions with A as the first axis
	A_axis = lkp.getAxisIndex('A')
	lkp.move('A', 0)
	nA = lkp.dims()[0]

	# Compute QSS states using these lookups
	QSS = EffectiveVariablesLookup(
	lkp.refs,
	{k: v(lkp) for k, v in self.pneuron.quasiSteadyStates().items()})

	# Compress outputs if needed
	if squeeze_output:
	QSS = QSS.squeeze()
	lkp = lkp.squeeze()

	return lkp, QSS

	def iNetQSS(self, Qm, Fdrive, Adrive, DC):
	''' Compute quasi-steady state net membrane current for a given combination
	of US parameters and a given membrane charge density.

	:param Qm: membrane charge density (C/m2)
	:param Fdrive: US frequency (Hz)
	:param Adrive: US amplitude (Pa)
	:param DC: duty cycle (-)
	:return: net membrane current (mA/m2)
	'''
	lkp, QSS = self.getQuasiSteadyStates(
	Fdrive, amps=Adrive, charges=Qm, DC=DC, squeeze_output=True)
	return self.pneuron.iNet(lkp['V'], QSS) # mA/m2


	def fixedPointsQSS(self, Fdrive, Adrive, DC, lkp, dQdt):
	''' Compute QSS fixed points along the charge dimension for a given combination
	of US parameters, and determine their stability.

	:param Fdrive: US frequency (Hz)
	:param Adrive: US amplitude (Pa)
	:param DC: duty cycle (-)
	:param lkp: lookup dictionary for effective variables along charge dimension
	:param dQdt: charge derivative profile along charge dimension
	:return: 2-tuple with values of stable and unstable fixed points
	'''
	pltvars = self.getPltVars()
	logger.debug('A = {:.2f} kPa, DC = {:.0f}%'.format(Adrive * 1e-3, DC * 1e2))

	# Extract fixed points from QSS charge variation profile
	def dfunc(Qm):
	return - self.iNetQSS(Qm, Fdrive, Adrive, DC)
	fixed_points = getFixedPoints(
	lkp.refs['Q'], dQdt, filter='both', der_func=dfunc).tolist()
	dfunc = lambda x: np.array(self.effDerivatives(_, x, lkp))

	# classified_fixed_points = {'stable': [], 'unstable': [], 'saddle': []}
	classified_fixed_points = []

	np.set_printoptions(precision=2)

	# For each fixed point
	for i, Qm in enumerate(fixed_points):

	# Re-compute QSS at fixed point
	*_, QSS = self.getQuasiSteadyStates(Fdrive, amps=Adrive, charges=Qm, DC=DC,
	squeeze_output=True)

	# Classify fixed point stability by numerically evaluating its Jacobian and
	# computing its eigenvalues
	x = np.array([Qm, *QSS.tables.values()])
	eigvals, key = classifyFixedPoint(x, dfunc)
	# classified_fixed_points[key].append(Qm)

	classified_fixed_points.append((x, eigvals, key))
	# eigenvalues.append(eigvals)
	logger.debug(f'{key} point @ Q = {(Qm * 1e5):.1f} nC/cm2')

	# eigenvalues = np.array(eigenvalues).T
	# print(eigenvalues.shape)

	return classified_fixed_points

	def isStableQSS(self, Fdrive, Adrive, DC):
	lookups, QSS = self.getQuasiSteadyStates(
	Fdrive, amps=Adrive, DCs=DC, squeeze_output=True)
	dQdt = -self.pneuron.iNet(lookups['V'], QSS.tables) # mA/m2
	classified_fixed_points = self.fixedPointsQSS(Fdrive, Adrive, DC, lookups, dQdt)
	return len(classified_fixed_points['stable']) > 0



	class DrivenNeuronalBilayerSonophore(NeuronalBilayerSonophore):

	simkey = 'DASTIM' # keyword used to characterize simulations made with this model

	def __init__(self, Idrive, args, *kwargs):
	self.Idrive = Idrive
	super().__init__(args, *kwargs)

	def __repr__(self):
	return super().__repr__()[:-1] + f', Idrive = {self.Idrive:.2f} mA/m2)'

	@classmethod
	def initFromMeta(cls, meta):
	return cls(meta['Idrive'], meta['a'], getPointNeuron(meta['neuron']),
	Fdrive=meta['Fdrive'], embedding_depth=meta['d'])

	def params(self):
	return {{'Idrive': self.Idrive}, super().params()}

	@staticmethod
	def inputs():
	inputvars = NeuronalBilayerSonophore.inputs()
	inputvars['Idrive'] = {
	'desc': 'driving current density',
	'label': 'I_{drive}',
	'unit': 'mA/m2',
	'factor': 1e0,
	'precision': 0
	}
	return inputvars

	def filecodes(self, *args):
	codes = super().filecodes(*args)
	codes['Idrive'] = f'Idrive{self.Idrive:.1f}mAm2'
	return codes

	def fullDerivatives(self, *args):
	dydt = super().fullDerivatives(*args)
	dydt[3] += self.Idrive * 1e-3
	return dydt

	def effDerivatives(self, *args):
	dQmdt, dstates = super().effDerivatives(args)
	dQmdt += self.Idrive * 1e-3
	return [dQmdt, *dstates]

	def meta(self, Fdrive, Adrive, pp, fs, method, qss_vars):
	d = super().meta(Fdrive, Adrive, pp, fs, method, qss_vars)
	d['Idrive'] = self.Idrive
	return d

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