def computePPlaneObjects(gen, xvar, yvar, state=None, tol=1e-8,
x_scale=1, max_step=1, subdomain=None, seed_pts=None,
only_var=None, do_fps=True, num_x_points=100, num_y_points=100,
strict_domains=True, with_jac=True, fast_vars=None, verbose=0,
as_array=True):
"""
THIS VERSION IS HARDWIRED FOR USE WITH HODGKIN-HUXLEY MODELS.
Compute basic phase plane objects (fixed points and nullclines)
given certain parameters (system params will come from generator):
gen Generator object from which to build nullcline from
xvar Variable to be treated as independent
yvar Variable to be treated as dependent
fast_vars Any fast variables set to their adiabatic equilibrium values
state Point or dictionary of state values
tol Tolerance scalar
max_step Maximum step in arc-length that nullcline computation
will use
subdomain Optional sub-domain of calculation (default to whole
variable domains given by Generator) given as dictionary
with [xlo, xhi] values for each x string key.
seed_pts Seed points (list of dictionaries or Points)
for any known nullcline points on either/both nullclines.
only_var Select one variable to compute nullclines for only.
do_fps Boolean switch (default True) to control computation of
fixed points.
num_x_points number of mesh points for x nullcline (default 100)
num_y_points number of mesh points for y nullcline (default 100)
strict_domains Boolean switch (default True) to control strict domain
cropping of nullclines computed.
with_jac Toggle to create Jacobian for use in fixed point and nullcline
calculations (default True)
verbose 0 = No text output; No graphic output
1 = Text output only
2 = Graphic output from this function only
No grahphic output from functions called within
3+ = Pull output from subsequently lower-level functions
called within
as_array Optional Boolean to return nullclines as array data rather
than fully interpolated spline nullcline objects (default True)
"""
## ------------------------------------------------------------
## JACOBIAN
## ------------------------------------------------------------
state = dict(state) # ensure updatable!
yvarFS = gen._FScompatibleNames(yvar)
xvarFS = gen._FScompatibleNames(xvar)
stateFS = filteredDict(gen._FScompatibleNames(state), gen.funcspec.vars)
varPars = filteredDict(stateFS, [xvarFS, yvarFS], neg=True)
# TEMP
#with_jac = False
#do_fps = False
if with_jac:
jacFS, new_fnspecsFS = prepJacobian(gen.funcspec._initargs['varspecs'], [xvarFS, yvarFS],
gen.funcspec._initargs['fnspecs'])
scopeFS = gen.pars.copy()
scopeFS.update(varPars)
scopeFS.update(new_fnspecsFS)
jac_fnFS = expr2fun(jacFS, ensure_args=['t'], **scopeFS)
jac = gen._FScompatibleNamesInv(jacFS)
scope = gen._FScompatibleNamesInv(gen.pars)
scope.update(filteredDict(state, [xvar, yvar], neg=True))
new_fnspecs = {}
# don't convert the new_fnspecs keys to . notation
keymap = Symbolic.symbolMapClass(dict(zip(gen._FScompatibleNamesInv(new_fnspecsFS.keys()), new_fnspecsFS.keys())))
for k, v in new_fnspecsFS.items():
new_fnspecs[k] = Symbolic.mapNames(keymap, gen._FScompatibleNamesInv(v))
scope.update(new_fnspecs)
jac_fn = expr2fun(jac, ensure_args=['t'], **scope)
else:
jac_fn = jac_fnFS = None
## ------------------------------------------------------------
## FIXED POINTS
## ------------------------------------------------------------
if verbose >= 1:
print("Computing fixed points")
if subdomain is not None:
stateFS.update(filteredDict(gen._FScompatibleNames(subdomain),
(xvarFS, yvarFS)))
if do_fps:
# create Intervals to test inclusion in subdomain
int_x = Interval(xvar, float, subdomain[xvar])
int_y = Interval(yvar, float, subdomain[yvar])
# temporarily overwriting state with xdomains from gen,
# for fussy fsolve in find_fixedpoints
for v in (xvarFS, yvarFS):
stateFS[v] = gen.xdomain[v]
fp_coords = find_fixedpoints(gen, n=10, jac=jac_fnFS, subdomain=stateFS,
eps=tol)
fps = []
for fp in fp_coords:
if fp[xvar] in int_x and fp[yvar] in int_y:
fps.append(fixedpoint_2D(gen, Point(fp), coords=[xvar, yvar],
jac=jac_fn, description='', eps=1e-5))
# reset stateFS to use subdomain
if subdomain is not None:
stateFS.update(filteredDict(gen._FScompatibleNames(subdomain),
(xvarFS, yvarFS)))
else:
fp_coords = None
fps = []
## ------------------------------------------------------------
## NULLCLINES
## ------------------------------------------------------------
gen.set(ics=varPars)
if verbose >= 1:
print("Computing nullclines")
assert 'V' in (xvar, yvar)
if 'V' == xvar:
#Vs = linspace(gen.xdomain['V'][0], gen.xdomain['V'][1], num_x_points)
Vs = linspace(stateFS['V'][0], stateFS['V'][1], num_x_points)
other = yvar
otherFS = yvarFS
os = linspace(stateFS[yvarFS][0], stateFS[yvarFS][1], num_y_points)
n = num_y_points
else:
Vs = linspace(stateFS['V'][0], stateFS['V'][1], num_y_points)
other = xvar
otherFS = xvarFS
os = linspace(stateFS[xvarFS][0], stateFS[xvarFS][1], num_x_points)
n = num_x_points
# yes, this gets recalc'd even if only_var is 'V' but this is a cheap calc compared to
# nullcline object creation
ofn = getattr(gen.auxfns, other.split('.')[0]+'_dssrt_fn_'+other.split('.')[1]+'inf')
oinfs = [ofn(V) for V in Vs]
fn_args = gen.funcspec._initargs['fnspecs']['dssrt_fn_Vinf'][0]
# Dictionary to map Na_m into m, etc.
model_vars = gen.query('vars')
varnamemap = Symbolic.symbolMapClass(dict([(v,v.split('.')[-1]) \
for v in model_vars]))
pt = filteredDict(Symbolic.mapNames(varnamemap,
filteredDict(state, model_vars)),
fn_args)
vinfs = np.zeros((n,),float)
oarg = varnamemap[other]
if other not in model_vars and fast_vars is not None and other in fast_vars:
oarg = oarg.split('.')[-1]
os = oinfs
if fast_vars is not None and 'Na.m' in fast_vars:
pt['m'] = gen.auxfns.Na_dssrt_fn_minf(state['V'])
for i, o in enumerate(os):
pt[oarg] = o
vinfs[i] = gen.auxfns.dssrt_fn_Vinf(**pt)
if 'V' == xvar:
nulls_x = array([vinfs, os]).T
nulls_y = array([Vs, oinfs]).T
else:
nulls_y = array([os, vinfs]).T
nulls_x = array([oinfs, Vs]).T
if as_array:
nullcX = nulls_x
nullcY = nulls_y
else:
if only_var is None:
nullcX = nullcline(xvar, yvar, nulls_x, x_relative_scale=x_scale)
nullcY = nullcline(xvar, yvar, nulls_y, x_relative_scale=x_scale)
elif only_var == xvar:
nullcX = nullcline(xvar, yvar, nulls_x, x_relative_scale=x_scale)
nullcY = None
elif only_var == yvar:
nullcX = None
nullcY = nullcline(xvar, yvar, nulls_y, x_relative_scale=x_scale)
else:
raise ValueError("Invalid variable name for only_var: %s" % only_var)
return {'nullcX': nullcX, 'nullcY': nullcY, 'fps': fps}
# n=3 uses three starting points in the domain to find nullcline parts, to an
# accuracy of eps=1e-8, and a maximum step for the solver of 0.1 units.
# The fixed points found is also provided to help locate the nullclines.
if all_plots:
nulls_x, nulls_y = pp.find_nullclines(ode_sys, 'phi', 'nu', n=3,
eps=1e-6, max_step=0.1, fps=fp_coords)
# plot the nullclines
if all_plots:
plt.plot(nulls_x[:,0], nulls_x[:,1], 'b')
plt.plot(nulls_y[:,0], nulls_y[:,1], 'g')
# plot the fixed points
fps = []
for fp_coord in fp_coords:
fps.append( pp.fixedpoint_2D(ode_sys, dst.Point(fp_coord)) )
saddle = fps[1]
plotter.set_active_layer('fp_data')
plot_PP_fp(saddle, 'fp_data', do_evecs=True, markersize=7)
gui.buildPlotter2D((8,8), with_times=False)
# magBound change ensures quicker determination of divergence during
# manifold computations. max_pts must be larger when we are further
# away from the fixed point.
ode_sys.set(algparams={'magBound': 10000})
def plot_manifold(man, which, style='k.-'):
for sgn in (-1, 1):
# let solution settle
transient = fhn.compute('trans')
fhn.set(ics=transient(10),
tdata=[0,20])
# More of your code here
# Your code here for the frequency plot
1/0 # comment this to apply phase plane picture to whatever
# are the current parameters of FHN model
## Optional code
fp_coord = pp.find_fixedpoints(fhn, n=25, eps=1e-6)[0]
fp = pp.fixedpoint_2D(fhn, Point(fp_coord), eps=1e-6)
nulls_x, nulls_y = pp.find_nullclines(fhn, 'x', 'y', n=3, eps=1e-6,
max_step=0.1, fps=[fp_coord])
plt.figure(3)
pp.plot_PP_fps(fp)
plt.plot(nulls_x[:,0], nulls_x[:,1], 'b')
plt.plot(nulls_y[:,0], nulls_y[:,1], 'g')
plt.show()
}
DSargs.ics = icdict
vdp = Dopri_ODEsystem(DSargs) # Vode_ODEsystem
vdp_e = Euler_ODEsystem(DSargs)
traj = vdp.compute('v')
pts = traj.sample()
traj_e = vdp_e.compute('e')
pts_e = traj_e.sample()
plt.plot(pts['x'], pts['y'], 'k.-', linewidth=2)
plt.plot(pts_e['x'], pts_e['y'], 'r.-', linewidth=2)
fp_coord = pp.find_fixedpoints(vdp, n=4, eps=1e-8)[0]
fp = pp.fixedpoint_2D(vdp, Point(fp_coord), eps=1e-8)
nulls_x, nulls_y = pp.find_nullclines(vdp,
'x',
'y',
n=3,
eps=1e-8,
max_step=0.1,
fps=[fp_coord])
pp.plot_PP_fps(fp)
plt.plot(nulls_x[:, 0], nulls_x[:, 1], 'b')
plt.plot(nulls_y[:, 0], nulls_y[:, 1], 'g')
plt.show()
PC = ContClass(vdp)
# n=3 uses three starting points in the domain to find nullcline parts, to an
# accuracy of eps=1e-8, and a maximum step for the solver of 0.1 units.
# The fixed points found is also provided to help locate the nullclines.
nulls_x, nulls_y = pp.find_nullclines(ode_sys,
'phi',
'nu',
n=3,
eps=1e-6,
max_step=0.1,
fps=fp_coords)
# plot the fixed points
fps = []
for fp_coord in fp_coords:
fps.append(pp.fixedpoint_2D(ode_sys, dst.Point(fp_coord)))
for fp_obj in fps:
plot_PP_fps_custom(fp_obj, do_evecs=True, markersize=7, flip_coords=True)
# plot the nullclines
plt.plot(nulls_x[:, 0], nulls_x[:, 1], 'b')
plt.plot(nulls_y[:, 0], nulls_y[:, 1], 'g')
plt.axis('tight')
plt.title('Phase plane')
plt.xlabel('phi')
plt.ylabel('nu')
# you may not need to run these commands on your system
plt.draw()
plt.plot(pts['t'], pts['y'], 'r', linewidth=2)
# figure 2 is the phase plane
plt.figure(2)
# phase plane tools are in the Toolbox module
from PyDSTool.Toolbox import phaseplane as pp
# plot vector field, using a scale exponent to ensure arrows are well spaced
# and sized
pp.plot_PP_vf(vdp, 'x', 'y', scale_exp=-1)
# only one fixed point, hence [0] at end.
# n=4 uses three starting points in the domain to find any fixed points, to an
# accuracy of eps=1e-8.
fp_coord = pp.find_fixedpoints(vdp, n=4, eps=1e-8)[0]
fp = pp.fixedpoint_2D(vdp, dst.Point(fp_coord), eps=1e-8)
# n=3 uses three starting points in the domain to find nullcline parts, to an
# accuracy of eps=1e-8, and a maximum step for the solver of 0.1 units.
# The fixed point found is also provided to help locate the nullclines.
nulls_x, nulls_y = pp.find_nullclines(vdp, 'x', 'y', n=3, eps=1e-8,
max_step=0.1, fps=[fp_coord])
# plot the fixed point
pp.plot_PP_fps(fp)
# plot the nullclines
plt.plot(nulls_x[:,0], nulls_x[:,1], 'b')
plt.plot(nulls_y[:,0], nulls_y[:,1], 'g')
nulls_x, nulls_y = pp.find_nullclines(dmModel, 's1', 's2', n=3, \
eps=1e-8, max_step=0.01, fps=fp_coord)
plot(nulls_x[:,0], nulls_x[:,1], 'b')
plot(nulls_y[:,0], nulls_y[:,1], 'g')
# compute the jacobian matrix
jac, new_fnspecs = dst.prepJacobian(dmModel.funcspec._initargs['varspecs'],
['s1','s2'],dmModel.funcspec._initargs['fnspecs'])
scope = dst.copy(dmModel.pars)
scope.update(new_fnspecs)
jac_fn = dst.expr2fun(jac, ensure_args=['t'],**scope)
# add fixed points to the phase portrait
for i in range(0,len(fp_coord)):
fp = pp.fixedpoint_2D(dmModel, dst.Point(fp_coord[i]),
jac = jac_fn, eps=1e-8)
pp.plot_PP_fps(fp)
# compute and plot projectories
traj = dmModel.compute('trajectory1')
pts = traj.sample()
plot(pts['s1'], pts['s2'], 'r-o')
xlabel('s_1')
ylabel('s_2')
title('Phase plane I1=0.035 nA, I2=0.035 nA')
# savefig('pp2')
# show()
# Your code here for the frequency plot
Is = linspace(0, 2, 100)
fs = []
for I in Is:
fs.append(freq(I))
plt.figure(2)
plt.plot(Is, fs, 'k.')
plt.xlabel('I')
plt.ylabel('frequencies')
1/0 # comment this to apply phase plane picture to whatever
# are the current parameters of FHN model
## Optional code
fp_coord = pp.find_fixedpoints(fhn, n=25, eps=1e-6)[0]
fp = pp.fixedpoint_2D(fhn, Point(fp_coord), eps=1e-6)
nulls_x, nulls_y = pp.find_nullclines(fhn, 'x', 'y', n=3, eps=1e-6,
max_step=0.1, fps=[fp_coord])
plt.figure(3)
pp.plot_PP_fps(fp)
plt.plot(nulls_x[:,0], nulls_x[:,1], 'b')
plt.plot(nulls_y[:,0], nulls_y[:,1], 'g')
plt.show()