"""
Modeling synaptic responses to a random train of inputs -- see, eg,
http://www.jneurosci.org/cgi/content/full/21/15/5781
"""
import os
import numpy
import scipy.optimize
from matplotlib.mlab import load
import matplotlib.cbook as cbook
from pylab import figure, show
class Model(object):
"""
Helper class for parametric modeling of spike train data
"""
def __init__(self, t, s, model):
"""
Initialize with the times of the impulses as array t, the
amplitude of the spikes as array s, and a model function which
has signature
m = model(t, pars)
where pars is a parameter vector that the model will be
evaluated over and m is the model's prediction on times t for
those parameter. m is an array that has the same length as t
and s.
"""
self.t = t
self.s = s
self.model = model
def rms(self, pars):
"""
call model with signature m = model(t, pars) and return the
root-mean-square RMS error between actual spikes s and model m
"""
m = self.model(self.t, pars)
d = self.s-m
err = numpy.sqrt(numpy.mean(d*d))
#print pars, err
return err
def fit(self, guess, bounds=None):
"""
Use scipy.optimize to get the best fit of the model to the
data. See scipy.optimize.fmin_l_bfgs_b
"""
bestpars, err, infod = scipy.optimize.fmin_l_bfgs_b(
self.rms, guess, approx_grad=True, bounds=bounds)
#print bestpars, err, infod
return bestpars
def plot_data(self, ax, pars=None):
"""
plot the data in axes ax. If pars is not None, overlay the
model estimated at those parameters
"""
ax.vlines(self.t, 0, self.s)
if pars is not None:
m = self.model(self.t, pars)
ax.plot(self.t, m, 'o')
return
def plot_isi(self, ax):
"""
make a plot of the interspike interval histogram (the
interspike interval is the difference between adjacent spike
times. ax is an axes instance to make the plot in
"""
return ax.hist(numpy.diff(self.t), bins=100)
def one_exponential(t, (a, alpha)):
"""
a one factor exponential model. t is a vector of times of the
delta impulses. The underlying factor is given by the convolution
of the response function $a exp(-alpha t)$ with the delta inputs,
and by the sift theorem we reduce the convolution to
factor[0] = a
factor[k] = factor[k-1] * exp(-alpha t) + a
and the predicted amplitude
m[k] = 1 + factor[k]
"""
m = a*numpy.ones((len(t),), dtype=numpy.float_)
for k, tk in enumerate(t):
if k==0:
m[k] = a
else:
m[k] = m[k-1]*numpy.exp(-alpha*(t[k]-t[k-1])) + a
return m + 1.
# data files are located in 'data' and have filenames
# synapse_times.dat and synapse_data.dat, each of which contain a
# single column of ASCII floating point numbers. load each into an
# array t and s, where t is the array of times and s an equal length
# array of synapse amplitudes
t = load(os.path.join('data', 'synapse_times.dat'))
s = load(os.path.join('data', 'synapse_data.dat'))
guess = (-0.05, 0.3)
# create the model and do the best fit
model = Model(t, s, one_exponential)
bounds = [(None, 0.), (0., None)]
bestpars = model.fit(guess, bounds=bounds)
# plot the interspike-interval histogram, the actual data, and the best model fit
fig = figure()
ax1 = fig.add_subplot(311)
model.plot_isi(ax1)
ax2 = fig.add_subplot(312)
model.plot_data(ax2, bestpars)
### OPTIONAL BELOW THIS POINT
# for the nexponential model, provide ordered pairs of a, alpha for
# the amplitude and rate constant of an exponential factor. Eg, if
# pars is length(6), this is a 3 factor model with parameters (a1,
# alpha1, a2, alpha2, a3, alpha3). If a is negative the factor is
# depression, and if a is positive to factor is excitatory
def nexponential(t, pars):
"""
a N-factor linear exponential model
pars is an even length sequence of a, alpha pairs where a is the
amplitude and alpha is the rate constant of an exponential factor
if n is the number of factors, the impulse response function is
r(t) = a1 exp(-alpha1 t) + a2 exp(-alpha2 t) + ... an exp(-alphan t)
and the spike amplitude is given by
m(t) = 1 + s(t) * r(t) where * denotes convolution.
Since the inputs s(t) = sum_k delta(t-tk) where delta is the Dirac
delta function, we can write the output
m_k = 1 + sum_k r(t-tk) where tk is the time of the k-th spike <= t
pars is an (a1, alpha1, a2, alpha2, ..., an, alphan) tuple
"""
N,remainder = divmod(len(pars),2)
assert(remainder==0)
m = numpy.ones((len(t), ), dtype=numpy.float_) # the model amplitudes
factors = numpy.zeros((N,), dtype=numpy.float_) # the N exponential factors
exp = numpy.exp # local attribute lookup faster
as = pars[::2] # the N amplitudes
alphas = pars[1::2] # the N rate constants
tlast = t[0]
for k, tk in enumerate(t):
factors = factors*exp(-alphas*(tk-tlast)) + as
m[k] += factors.sum()
tlast = tk
return m
def autobounds(pars):
"""
if pars is an nexponential parameter vector, return bounds
assuming the sign of the amplitude guess and positive rate
contants. return value is a len(pars) list of (pmin, pmax) bounds
"""
pos = 0., None
neg = None, 0.
bounds = []
for a, alpha in cbook.pieces(pars):
if a<0: abounds = neg
else: abounds = pos
bounds.append(abounds) # amplitude bounds
bounds.append(pos) # alpha bounds strictly pos
return bounds
guess = (-0.05, 0.3, -0.1, 1.0)
#guess = (-0.05, 0.3, -0.1, 1.0, 0.3, 25.)
# an nexponential model with auto-bounds
model = Model(t, s, nexponential)
bestpars = model.fit(guess, bounds=autobounds(guess))
ax3 = fig.add_subplot(313)
model.plot_data(ax3, bestpars)
show()
