# AUTOGENERATED! DO NOT EDIT! File to edit: 03_modelling.ipynb (unless otherwise specified).

__all__ = ['sigmoid', 'gaussian', 'sum_of_gaussian', 'diff_of_gaussian', 'gaussian_2D', 'sum_of_2D_gaussian',
           'exponential_decay', 'sin_exponent', 'sinexp_sigm', 'fit_sigmoid', 'fit_spatial_sta', 'fit_temporal_sta',
           'fit_chirp_am', 'fit_chirp_freq_epoch', 'fit_transiency', 'repetition_quality_index',
           'onoff_transient_index']

# Cell
import math
import scipy as sp
import scipy.optimize
from scipy.interpolate import interp1d
import numpy as np
from functools import partial

# Cell
_default_sigmoid_p = (1,1,0,0)
def sigmoid(x, sigma, amp, x0, y0):
    """Sigmoid function
    params:
        - x: 1D numpy array at which to evaluate the points
        - sigma: steepness of the sigmoid
        - amp: amplitude of the sigmoid
        - x0: shift in x of the sigmoid
        - y0: shift in y of the sigmoid
    """
    return amp / (1.0 + np.exp(-sigma*(x-x0))) + y0

# Cell
_default_gaussian_p = (1,1,0,0)
def gaussian(x, sigma, amp, x0, y0):
    """Gaussian function
    params:
        - x: 1D numpy array at which to evaluate the points
        - sigma: width of the gaussian
        - amp: amplitude of the gaussian
        - x0: shift in x of the gaussian
        - y0: shift in y of the gaussian
    """
    return amp * np.exp(-(x-x0)**2 / (2*sigma**2)) + y0

# Cell
def sum_of_gaussian(t, sigma_1, amp_1, x0_1,
                       sigma_2, amp_2, x0_2, y0):
    """
    Sum of gaussian, using `gaussian`
    params:
        - t: 1D numpy array at which to evaluate the points
        - sigma_1: width of the 1st gaussian
        - amp_1: amplitude of the 1st gaussian
        - x0_1: shift in x of the 1st gaussian
        - sigma_2: width of the 2nd gaussian
        - amp_2: amplitude of the 2nd gaussian
        - x0_2: shift in x of the 2nd gaussian
        - y0: shift in y of the gaussian (shared for the two gaussians)"""
    return (gaussian(t, sigma_1, amp_1, x0_1, 0)
            + gaussian(t, sigma_2, amp_2, x0_2, 0) + y0)

def diff_of_gaussian(t, sigma_1, amp_1, x0_1,
                       sigma_2, amp_2, x0_2, y0):
    """
    Difference of gaussian, using `gaussian`
    params:
        - t: 1D numpy array at which to evaluate the points
        - sigma_1: width of the 1st gaussian
        - amp_1: amplitude of the 1st gaussian
        - x0_1: shift in x of the 1st gaussian
        - sigma_2: width of the 2nd gaussian
        - amp_2: amplitude of the 2nd gaussian
        - x0_2: shift in x of the 2nd gaussian
        - y0: shift in y of the gaussian (shared for the two gaussians)"""
    return (gaussian(t, sigma_1, amp_1, x0_1, 0)
            - gaussian(t, sigma_2, amp_2, x0_2, 0) ) + y0

# Cell
_default_2Dgaussian_p = (1,1,1,0,0,0,0)
def gaussian_2D(xz, sigma_x, sigma_z, amp, theta, x0, z0, y0):
    """Two dimensional Gaussian function
    params:
        - xz: meshgrid of x and z coordinates at which to evaluate the points
        - sigma_x: width of the gaussian
        - sigma_z: height of the gaussian
        - amp: amplitude of the gaussian
        - theta: angle of the gaussian (in radian)
        - x0: shift in x of the gaussian
        - z0: shift in z of the gaussian
        - y0: shift in y of the gaussian
    """
    (x,z) = xz
    x0, z0 = float(x0), float(z0)
    a =  (np.cos(theta)**2)/(2*sigma_x**2) + (np.sin(theta)**2)/(2*sigma_z**2)
    b = -(np.sin(2*theta)) /(4*sigma_x**2) + (np.sin(2*theta)) /(4*sigma_z**2)
    c =  (np.sin(theta)**2)/(2*sigma_x**2) + (np.cos(theta)**2)/(2*sigma_z**2)
    g = amp * np.exp( -(a*((x-x0)**2) + 2*b*(x-x0)*(z-z0) + c*((z-z0)**2))) + y0
    return g.ravel()

# Cell
def sum_of_2D_gaussian(xz, sigma_x_1, sigma_z_1, amp_1, theta_1, x0_1, z0_1,
                           sigma_x_2, sigma_z_2, amp_2, theta_2, x0_2, z0_2, y0):
    """Sum of two 2D Gaussian function. For the params, see `gaussian_2D`.
     However, both share the y0 parameter."""
    return (gaussian_2D(xz, sigma_x_1, sigma_z_1, amp_1, theta_1, x0_1, z0_1, 0)
            + gaussian_2D(xz, sigma_x_2, sigma_z_2, amp_2, theta_2, x0_2, z0_2, 0) + y0)

# Cell
def exponential_decay(x, tau, baseline, amplitude):
    """Exponential decay function
    params:
        - x: 1D numpy array at which to evaluate the points
        - tau: decay speed
        - baseline: baseline after the decay. Must be lower than the amplitude
        - amplitude: Initial amplitude before the decay
    """
    return np.exp(-x*tau)*(amplitude-baseline) + baseline

# Cell
def sin_exponent(x, amp, phi, freq, exp):
    """Sine raised to an exponent power. To the power 2, the sine is going between 0 and 1
    while its period is halfed. Subsequent round powers have the effect of narowing the peaks
    and making the rest of the sine approach zero. (graph it to see it)
    params:
        - x: 1D numpy array at which to evaluate the points
        - amp: amplitude of the sine
        - phi: phase of the sine in radian
        - freq: frequency of the sine in Hz
        - exp: Power the sine is raised to"""

    #Multiply by pi instead of 2*pi because it expect the cos to be at least squared
    #Same goes for phi that I halfed so the period is still 2*pi
    return np.power(np.sin((x*freq*np.pi)+(phi/2)), exp)*amp


# Cell
def sinexp_sigm(x, sigma, x0, y0, amp, phi, freq, exp):
    """A sine exponent weighted by a sigmoid. For parameters, see `sigmoid` and `sin_exponent`.
    However the amp and y0 (baseline) parameters are shared between the two sub-functions"""
    return sin_exponent(x, amp, phi, freq, exp) * sigmoid(x, sigma, 1, x0, 0) + y0

# Cell
def fit_sigmoid(nonlin, t=None):
    """Fit a sigmoid to a 1D input. Used to fit the nonlinearities of the cells responses.
    Also, the data must be increasing (from left to right).
    params:
        - nonlin: 1D data to fit
        - t     : timepoints of the data to fit. Same lenght as nonlin

    return:
        - A dictionary containing the parameters of the fit for a `sigmoid`
        - A quality index of the fit (explained variance of the model)"""

    assert np.mean(nonlin[:20])<np.mean(nonlin[-20:]), "Nonlinearity must be increasing from left to right"
    if t is None:
        t = np.arange(len(nonlin))
    bounds = ([0.000001, 0.000001, -np.inf, 0],
              [np.inf  , np.inf,    np.inf, np.max(nonlin)])
    try:
        fit, _ = sp.optimize.curve_fit(sigmoid, t, nonlin, maxfev=1000000, bounds=bounds)
        fit  = {"sigma":fit[0],"amp":fit[1],"x0":fit[2],"y0":fit[3]}
    except RuntimeError:
        fit = {"sigma":1,"amp":0,"x0":0,"y0":0}
    model = sigmoid(t, **fit)
    quality_index =  1 - (np.var(nonlin-model)/np.var(nonlin))
    return fit, quality_index

def fit_spatial_sta(sta):
    """Fit a sum of 2D gaussian to a 2D matrix. Used to fit the STA obtained from checkerboard like stimulus.
    params:
        - sta: 2D data to fit

    return:
        - A dictionary containing the parameters of the fit for a `sum_of_2D_gaussian`
        - A quality index of the fit (explained variance of the model)"""
    shape_y, shape_x = sta.shape
    eps = 0.000001
    argmax = np.unravel_index(sta.argmax(), sta.shape)
    argmin = np.unravel_index(sta.argmin(), sta.shape)
    #                   sigma_x    sigma_z   amp theta      x0        z0      y0
    bounds_center   = ((0,         0,         0,  0,         0,       0,       0),
                       (shape_x*4, shape_y*4, 2,  2*math.pi, shape_x, shape_y, eps))
    if sta[argmax] < abs(sta[argmin]):
        argmax, argmin  = argmin, argmax
        bounds_center   = ((0,         0,        -2,  0,         0,       0,       0),
                           (shape_x*4, shape_y*4, 0,  2*math.pi, shape_x, shape_y, eps))

    x    = np.arange(shape_x)
    z    = np.arange(shape_y)
    x, z = np.meshgrid(x, z)
    #                   sigma_x sigma_z  amp        theta      x0        z0       y0
    init_fit_center   = (  1,     1,    sta[argmax],  0,   argmax[1],  argmax[0], 0)
    data_tofit          = sta.reshape(-1)
    if np.isnan(sp.sum(sta)): #We check that the sta exists, otherwise return default zero model
        fit  = {"sigma_x_1": 1, "sigma_z_1":1, "amp_1":0, "theta_1":0, "x0_1":0, "z0_1":0,
                "sigma_x_2": 1, "sigma_z_2":1, "amp_2":0, "theta_2":0, "x0_2":0, "z0_2":0, "y0":0}
    else:
        try:
            fit_c, _   = sp.optimize.curve_fit(gaussian_2D, (x,z), data_tofit, p0=init_fit_center, bounds=bounds_center)
            tmp_to_fit =  data_tofit - gaussian_2D((x,z), *fit_c)
            #                   sigma_x    sigma_z   amp theta        x0            z0           y0
            bounds_surround = ((0,         0,        -2,  0,         fit_c[4],     fit_c[5],     0),
                               (shape_x*4, shape_y*4, 0,  2*math.pi, fit_c[4]+eps, fit_c[5]+eps, eps))
            init_fit_surround = (  2,      2,         0,  0,         fit_c[4],     fit_c[5],     0)
            fit_s, _   = sp.optimize.curve_fit(gaussian_2D, (x,z), tmp_to_fit, p0=init_fit_surround, bounds=bounds_surround)

            init_fit_sum = (*fit_c[:-1],*fit_s[:-1],0)
            #Bounding only the position, that center fit should have found
            bounds_sum   = ((-np.inf,-np.inf,-np.inf,-np.inf,fit_c[4],fit_c[5],
                             -np.inf,-np.inf,-np.inf,-np.inf,fit_c[4],fit_c[5],-np.inf),
                            (np.inf, np.inf, np.inf, np.inf, fit_s[4]+eps,fit_s[5]+eps,
                             np.inf,np.inf,np.inf,np.inf,fit_s[4]+eps,fit_s[5]+eps,np.inf))
            fit_sum, _ = sp.optimize.curve_fit(sum_of_2D_gaussian, (x,z), data_tofit, p0=init_fit_sum, bounds=bounds_sum)
            fit  = {"sigma_x_1": fit_sum[0], "sigma_z_1":fit_sum[1], "amp_1":fit_sum[2],
                    "theta_1":fit_sum[3], "x0_1":fit_sum[4], "z0_1":fit_sum[5],
                    "sigma_x_2": fit_sum[6], "sigma_z_2":fit_sum[7], "amp_2":fit_sum[8],
                    "theta_2":fit_sum[9], "x0_2":fit_sum[10], "z0_2":fit_sum[11], "y0":fit_sum[12]}

        except RuntimeError: #If a model can't be fitted, we get a runtimeError
            fit  = {"sigma_x_1": 1, "sigma_z_1":1, "amp_1":0, "theta_1":0, "x0_1":0, "z0_1":0,
                    "sigma_x_2": 1, "sigma_z_2":1, "amp_2":0, "theta_2":0, "x0_2":0, "z0_2":0, "y0":0}
    model = sum_of_2D_gaussian((x,z), **fit)
    quality_index = 1 - (np.var(data_tofit-model)/np.var(data_tofit))
    return fit, quality_index


# Cell
def fit_temporal_sta(sta, frame_rate=60):
    """Fit a sum of gaussian to a 1D input. Used to fit the STA obtained from fullfield flicker like stimulus.
    params:
        - sta: 1D data to fit

    return:
        - A dictionary containing the parameters of the fit for a `diff_of_gaussian`. The parameter with _1 suffix
        correspond to the closest gaussian to the zero timepoint.
        - A quality index of the fit (explained variance of the model)"""

    # Cutting out the tail of the tSTA to increase the fitting error of the "interesting" part
    cell_abs = np.abs(sta)
    startSTA   = len(sta)-np.argmax(cell_abs>(cell_abs.max()*.4))
    startSTA  += np.maximum(5,(startSTA*.5)).astype(int)
    fullsta = sta.copy()
    sta = sta[-startSTA:]

    t = np.linspace((1-len(sta))/frame_rate, 0, len(sta))

                #sigma_1, amp_1,   x0_1,                sigma_2, amp_2,  x0_2,
    bounds   = ((0,       0,      -len(sta)/frame_rate, 0,       0,      -len(sta)/frame_rate),
                (.1,      np.inf, 1/frame_rate,        .1,       np.inf, 1/frame_rate))

    diff_of_gauss = partial(diff_of_gaussian, y0=0)

    if np.isnan(sp.sum(sta)): #We check that the sta exists, otherwise return default zero model
        fit  = {"sigma_1":1,"amp_1":0,"x0_1":0,
                "sigma_2":1,"amp_2":0,"x0_2":0,"y0":0}
    else:
        try:
            fit, _ = sp.optimize.curve_fit(diff_of_gauss, t, sta, bounds=bounds, max_nfev=100000000000)

            fit  = {"sigma_1":fit[0],"amp_1":fit[1],"x0_1":fit[2],
                    "sigma_2":fit[3],"amp_2":fit[4],"x0_2":fit[5],"y0":0}
        except RuntimeError: #If a model can't be fitted, we get a runtimeError
            fit  = {"sigma_1":1,"amp_1":0,"x0_1":0,
                    "sigma_2":1,"amp_2":0,"x0_2":0,"y0":0}

    t = np.linspace((1-len(fullsta))/frame_rate, 0, len(fullsta))

    model = diff_of_gaussian(t, **fit)
    quality_index = 1 - (np.var(fullsta-model)/np.var(fullsta))
    return fit, quality_index


# Cell
def fit_chirp_am(cell_mean, start=420, stop=960, freq=1.5, frame_rate=60):
    """Fit a sinexp_sigm to the mean response of a cell to chirp_am stimulus.
    params:
        - cell_mean: Average response of the cell to the chirp_am stimulus
        - start:     Index in the cell mean where to start the fitting
        - stop:      Index in the cell mean where to stop the fitting
        - freq:      Frequency of the amplitude modulation of the stimulus in Hz

    return :
        - A dictionary containing the parameters of the fit for a `sinexp_sigm`
        - A quality index of the fit (explained variance of the model)"""

    to_fit = cell_mean[start:stop]
    t = np.linspace(0, len(to_fit)/frame_rate, len(to_fit), endpoint=False)

    #If suppressed by contrast cell, the firing should be higher in first part of chirp
    fit_positive = np.mean(to_fit[:len(to_fit)//3]) < np.mean(to_fit[len(to_fit)*2//3:])

    tmp_diff = np.inf
    best_fit = (1, 0, 0, 0, 0, freq, 0)
    phi      = np.pi
    for exp in np.exp2(range(1,10)): #Fitting the data with different sin exponents, to narrow the fit
        try:
            sinexp_sigm_part = partial(sinexp_sigm, freq=freq, exp=exp)
            if fit_positive:
                fit, _ = sp.optimize.curve_fit(sinexp_sigm_part, t, to_fit,
                                               bounds=[(  0,    0          ,        0       ,      0        , 0),
                                                        (50, len(to_fit)/60,  np.max(to_fit), np.max(to_fit), 2*np.pi)],
                                               p0 =      (1, len(to_fit)/120,       0       , np.max(to_fit), phi),
                                              maxfev=100000)
            else:
                fit, _ = sp.optimize.curve_fit(sinexp_sigm_part, t, to_fit,
                                               bounds=[(  0,    0          ,        0       ,-np.max(to_fit), 0),
                                                        (50, len(to_fit)/60,  np.max(to_fit),     0         , 2*np.pi)],
                                               p0 =     (1, len(to_fit)/120,  np.max(to_fit) ,-np.max(to_fit),phi),
                                              maxfev=100000)

            mse = np.sum(np.square(sinexp_sigm_part(t, *fit) - to_fit))
            phi = fit[4]
            if mse < tmp_diff:
                best_fit = (*fit, freq, exp)
                tmp_diff = mse
        except:
            continue
    best_fit = dict((k, v) for v, k in zip(best_fit, ["sigma","x0","y0","amp","phi","freq","exp"]))

    model  = sinexp_sigm(t, **best_fit)
    to_fit = cell_mean[start:stop]
    quality_index = 1 - (np.var(to_fit-model)/np.var(to_fit))
    return best_fit, quality_index

def fit_chirp_freq_epoch(cell_mean, start=360, freqs=[1.875,3.75,7.5,15,30], durations=[2,2,2,1,1], frame_rate=60):
    """Fit multiple sinexp_sigm to the mean response of a cell to chirp_freq_epoch stimulus.
    Each epoch is fitted independantly by a `sin_exponent`
    params:
        - cell_mean: Average response of the cell to the chirp_am stimulus
        - start:     Index in the cell mean where to start the fitting
        - freqs:     Frequencies of each epoch in Hz
        - durations: Duration of each epoch in seconds
        - freq:      Frequency of the amplitude modulation of the stimulus

    return :
        - List of dictionary containing the parameters of each epoch fit for a `sin_exponent`
        - List of quality index of each epoch fit (explained variance of the model)"""
    best_fit_l = []
    qualityidx_l = []

    cursor = start #Start of the freqs

    #We fit each freqency epoch with a cos function
    for freq, dur in zip(freqs, durations):
        n_rep, n_fr = int(dur*freq), int(frame_rate/freq)
        edgecut = 10
        cursor += edgecut
        len_fit = n_rep*n_fr - edgecut

        #Upsampling of the data to prevent low sampling effects with high freq sine
        upsamp = 10
        to_fit = cell_mean[cursor:cursor+len_fit+1]
        t = np.linspace(0, len_fit/frame_rate, len_fit*upsamp, endpoint=False)
        to_fit = interp1d(np.linspace(0,len(to_fit)/frame_rate, len(to_fit), endpoint=False), to_fit)(t)

        best_fit = None
        tmp_diff = np.inf
        for exp in np.exp2(range(1,10)): #Fitting the data with different cos exponents, to narrow the fit
            try:
                sin_exponent_part = partial(sin_exponent, freq=freq, exp=exp)
                fit, cov = sp.optimize.curve_fit(sin_exponent_part, t, to_fit, bounds=[(0,          0),
                                                                                       (max(to_fit),np.pi*2)])
                mse = np.sum(np.square(sin_exponent_part(t, *fit) - to_fit))
                if mse < tmp_diff:
                    best_cov = cov
                    best_fit = (*fit, freq, exp)
                    tmp_diff = mse
            except:
                continue
        if best_fit is not None:
            best_fit = dict((k, v) for v, k in zip(best_fit, ["amp","phi","freq","exp"]))
            best_fit_l.append(best_fit)
            model = sin_exponent(t, **best_fit)
            qualityidx_l.append(1 - (np.var(to_fit-model)/np.var(to_fit)))
        else:
            best_fit_l.append({"amp":0,"phi":0,"freq":freq,"exp":0})
            qualityidx_l.append(0)

        cursor += len_fit
    return best_fit_l, qualityidx_l #, best_cov_l

# Cell
def fit_transiency(pref_response, frame_rate=60):
    """Fit an exponential decay to a 1D input. Used to fit the response of cells to their preferred
    sustained stimulus.
    params:
        - pref_response: 1D data to fit

    return:
        - A dictionary containing the parameters of the fit for a `exponential_decay`
        - A quality index of the fit (explained variance of the model)"""
    peak_position = np.argmax(pref_response)
    pref_response = pref_response[peak_position:]

    t = np.linspace(0, len(pref_response)/frame_rate, len(pref_response), endpoint=False)
    bounds = ([0.000001, 0],
              [np.inf  , pref_response[0]])
    if np.all(pref_response==0):
        fit = {"tau":1,"baseline":0,"amplitude":0}
    else:
        try:
            decay_part = partial(exponential_decay, amplitude=pref_response[0])
            fit, _ = sp.optimize.curve_fit(decay_part, t, pref_response, maxfev=1000000, bounds=bounds)
            fit  = {"tau":fit[0],"baseline":fit[1],"amplitude":pref_response[0]}
        except RuntimeError:
            fit = {"tau":1,"baseline":0,"amplitude":0}
    model = exponential_decay(t, **fit)
    quality_index =  1 - (np.var(pref_response-model)/np.var(pref_response))

    return fit, quality_index, peak_position

# Cell
def repetition_quality_index(cell_response):
    """Return a quality index of cell response to a repeated stimulus.
    params:
        - cell_response: response of a cell of shape (n_rep, time)"""
    tmp1 = np.var(np.mean(cell_response, axis=0)) #VAR(MEAN(C)repeat)time
    tmp2 = np.mean(np.var(cell_response, axis=0)) #MEAN(VAR(C)time)repeat
    quality = tmp1/tmp2
    return quality

def onoff_transient_index(cell_response, start_on=120, stop_on=240, start_off=240, stop_off=360):
    """Return both on-off and transient indexes of cell response. Transiency index is not always
    ideal to describe the cells response. Consider fitting an exponential decay with `fit_transiency`.
    params:
        - cell_response: response of a cell of shape (time,...)
        - start_on:  starting index of ON stimulation
        - stop_on:   stop index of ON stimulation
        - start_off: starting index of OFF stimulation
        - stop_off:  stop index of OFF stimulation
    return:
        - ON-OFF index
        - Transiency index"""
    #ON-OFF index
    on_response   = cell_response[start_on:stop_on]
    on_sum        = np.sum(on_response)
    off_response  = cell_response[start_off:stop_off]
    off_sum       = np.sum(off_response)
    onoff_indexes = (on_sum-off_sum)/(on_sum+off_sum)

    #Transient index
    if onoff_indexes>0: #Main ON response
        transient_sum   = np.sum(cell_response[start_on:start_on+30])
        sustained_sum   = np.sum(cell_response[start_on:stop_on])
    else:
        transient_sum   = np.sum(cell_response[start_off:start_off+30])
        sustained_sum   = np.sum(cell_response[start_off:stop_off])
    transient_index = transient_sum/sustained_sum

    return onoff_indexes, transient_index
# AUTOGENERATED! DO NOT EDIT! File to edit: 03_modelling.ipynb (unless otherwise specified).

__all__ = ['sigmoid', 'gaussian', 'sum_of_gaussian', 'diff_of_gaussian', 'gaussian_2D', 'sum_of_2D_gaussian',
           'exponential_decay', 'sin_exponent', 'sinexp_sigm', 'fit_sigmoid', 'fit_spatial_sta', 'fit_temporal_sta',
           'fit_chirp_am', 'fit_chirp_freq_epoch', 'fit_transiency', 'repetition_quality_index',
           'onoff_transient_index']

# Cell
import math
import scipy as sp
import scipy.optimize
from scipy.interpolate import interp1d
import numpy as np
from functools import partial

# Cell
_default_sigmoid_p = (1,1,0,0)
def sigmoid(x, sigma, amp, x0, y0):
    """Sigmoid function
    params:
        - x: 1D numpy array at which to evaluate the points
        - sigma: steepness of the sigmoid
        - amp: amplitude of the sigmoid
        - x0: shift in x of the sigmoid
        - y0: shift in y of the sigmoid
    """
    return amp / (1.0 + np.exp(-sigma*(x-x0))) + y0

# Cell
_default_gaussian_p = (1,1,0,0)
def gaussian(x, sigma, amp, x0, y0):
    """Gaussian function
    params:
        - x: 1D numpy array at which to evaluate the points
        - sigma: width of the gaussian
        - amp: amplitude of the gaussian
        - x0: shift in x of the gaussian
        - y0: shift in y of the gaussian
    """
    return amp * np.exp(-(x-x0)**2 / (2*sigma**2)) + y0

# Cell
def sum_of_gaussian(t, sigma_1, amp_1, x0_1,
                       sigma_2, amp_2, x0_2, y0):
    """
    Sum of gaussian, using `gaussian`
    params:
        - t: 1D numpy array at which to evaluate the points
        - sigma_1: width of the 1st gaussian
        - amp_1: amplitude of the 1st gaussian
        - x0_1: shift in x of the 1st gaussian
        - sigma_2: width of the 2nd gaussian
        - amp_2: amplitude of the 2nd gaussian
        - x0_2: shift in x of the 2nd gaussian
        - y0: shift in y of the gaussian (shared for the two gaussians)"""
    return (gaussian(t, sigma_1, amp_1, x0_1, 0)
            + gaussian(t, sigma_2, amp_2, x0_2, 0) + y0)

def diff_of_gaussian(t, sigma_1, amp_1, x0_1,
                       sigma_2, amp_2, x0_2, y0):
    """
    Difference of gaussian, using `gaussian`
    params:
        - t: 1D numpy array at which to evaluate the points
        - sigma_1: width of the 1st gaussian
        - amp_1: amplitude of the 1st gaussian
        - x0_1: shift in x of the 1st gaussian
        - sigma_2: width of the 2nd gaussian
        - amp_2: amplitude of the 2nd gaussian
        - x0_2: shift in x of the 2nd gaussian
        - y0: shift in y of the gaussian (shared for the two gaussians)"""
    return (gaussian(t, sigma_1, amp_1, x0_1, 0)
            - gaussian(t, sigma_2, amp_2, x0_2, 0) ) + y0

# Cell
_default_2Dgaussian_p = (1,1,1,0,0,0,0)
def gaussian_2D(xz, sigma_x, sigma_z, amp, theta, x0, z0, y0):
    """Two dimensional Gaussian function
    params:
        - xz: meshgrid of x and z coordinates at which to evaluate the points
        - sigma_x: width of the gaussian
        - sigma_z: height of the gaussian
        - amp: amplitude of the gaussian
        - theta: angle of the gaussian (in radian)
        - x0: shift in x of the gaussian
        - z0: shift in z of the gaussian
        - y0: shift in y of the gaussian
    """
    (x,z) = xz
    x0, z0 = float(x0), float(z0)
    a =  (np.cos(theta)**2)/(2*sigma_x**2) + (np.sin(theta)**2)/(2*sigma_z**2)
    b = -(np.sin(2*theta)) /(4*sigma_x**2) + (np.sin(2*theta)) /(4*sigma_z**2)
    c =  (np.sin(theta)**2)/(2*sigma_x**2) + (np.cos(theta)**2)/(2*sigma_z**2)
    g = amp * np.exp( -(a*((x-x0)**2) + 2*b*(x-x0)*(z-z0) + c*((z-z0)**2))) + y0
    return g.ravel()

# Cell
def sum_of_2D_gaussian(xz, sigma_x_1, sigma_z_1, amp_1, theta_1, x0_1, z0_1,
                           sigma_x_2, sigma_z_2, amp_2, theta_2, x0_2, z0_2, y0):
    """Sum of two 2D Gaussian function. For the params, see `gaussian_2D`.
     However, both share the y0 parameter."""
    return (gaussian_2D(xz, sigma_x_1, sigma_z_1, amp_1, theta_1, x0_1, z0_1, 0)
            + gaussian_2D(xz, sigma_x_2, sigma_z_2, amp_2, theta_2, x0_2, z0_2, 0) + y0)

# Cell
def exponential_decay(x, tau, baseline, amplitude):
    """Exponential decay function
    params:
        - x: 1D numpy array at which to evaluate the points
        - tau: decay speed
        - baseline: baseline after the decay. Must be lower than the amplitude
        - amplitude: Initial amplitude before the decay
    """
    return np.exp(-x*tau)*(amplitude-baseline) + baseline

# Cell
def sin_exponent(x, amp, phi, freq, exp):
    """Sine raised to an exponent power. To the power 2, the sine is going between 0 and 1
    while its period is halfed. Subsequent round powers have the effect of narowing the peaks
    and making the rest of the sine approach zero. (graph it to see it)
    params:
        - x: 1D numpy array at which to evaluate the points
        - amp: amplitude of the sine
        - phi: phase of the sine in radian
        - freq: frequency of the sine in Hz
        - exp: Power the sine is raised to"""

    #Multiply by pi instead of 2*pi because it expect the cos to be at least squared
    #Same goes for phi that I halfed so the period is still 2*pi
    return np.power(np.sin((x*freq*np.pi)+(phi/2)), exp)*amp


# Cell
def sinexp_sigm(x, sigma, x0, y0, amp, phi, freq, exp):
    """A sine exponent weighted by a sigmoid. For parameters, see `sigmoid` and `sin_exponent`.
    However the amp and y0 (baseline) parameters are shared between the two sub-functions"""
    return sin_exponent(x, amp, phi, freq, exp) * sigmoid(x, sigma, 1, x0, 0) + y0

# Cell
def fit_sigmoid(nonlin, t=None):
    """Fit a sigmoid to a 1D input. Used to fit the nonlinearities of the cells responses.
    Also, the data must be increasing (from left to right).
    params:
        - nonlin: 1D data to fit
        - t     : timepoints of the data to fit. Same lenght as nonlin

    return:
        - A dictionary containing the parameters of the fit for a `sigmoid`
        - A quality index of the fit (explained variance of the model)"""

    assert np.mean(nonlin[:20])<np.mean(nonlin[-20:]), "Nonlinearity must be increasing from left to right"
    if t is None:
        t = np.arange(len(nonlin))
    bounds = ([0.000001, 0.000001, -np.inf, 0],
              [np.inf  , np.inf,    np.inf, np.max(nonlin)])
    try:
        fit, _ = sp.optimize.curve_fit(sigmoid, t, nonlin, maxfev=1000000, bounds=bounds)
        fit  = {"sigma":fit[0],"amp":fit[1],"x0":fit[2],"y0":fit[3]}
    except RuntimeError:
        fit = {"sigma":1,"amp":0,"x0":0,"y0":0}
    model = sigmoid(t, **fit)
    quality_index =  1 - (np.var(nonlin-model)/np.var(nonlin))
    return fit, quality_index

def fit_spatial_sta(sta):
    """Fit a sum of 2D gaussian to a 2D matrix. Used to fit the STA obtained from checkerboard like stimulus.
    params:
        - sta: 2D data to fit

    return:
        - A dictionary containing the parameters of the fit for a `sum_of_2D_gaussian`
        - A quality index of the fit (explained variance of the model)"""
    shape_y, shape_x = sta.shape
    eps = 0.000001
    argmax = np.unravel_index(sta.argmax(), sta.shape)
    argmin = np.unravel_index(sta.argmin(), sta.shape)
    #                   sigma_x    sigma_z   amp theta      x0        z0      y0
    bounds_center   = ((0,         0,         0,  0,         0,       0,       0),
                       (shape_x*4, shape_y*4, 2,  2*math.pi, shape_x, shape_y, eps))
    if sta[argmax] < abs(sta[argmin]):
        argmax, argmin  = argmin, argmax
        bounds_center   = ((0,         0,        -2,  0,         0,       0,       0),
                           (shape_x*4, shape_y*4, 0,  2*math.pi, shape_x, shape_y, eps))

    x    = np.arange(shape_x)
    z    = np.arange(shape_y)
    x, z = np.meshgrid(x, z)
    #                   sigma_x sigma_z  amp        theta      x0        z0       y0
    init_fit_center   = (  1,     1,    sta[argmax],  0,   argmax[1],  argmax[0], 0)
    data_tofit          = sta.reshape(-1)
    if np.isnan(sp.sum(sta)): #We check that the sta exists, otherwise return default zero model
        fit  = {"sigma_x_1": 1, "sigma_z_1":1, "amp_1":0, "theta_1":0, "x0_1":0, "z0_1":0,
                "sigma_x_2": 1, "sigma_z_2":1, "amp_2":0, "theta_2":0, "x0_2":0, "z0_2":0, "y0":0}
    else:
        try:
            fit_c, _   = sp.optimize.curve_fit(gaussian_2D, (x,z), data_tofit, p0=init_fit_center, bounds=bounds_center)
            tmp_to_fit =  data_tofit - gaussian_2D((x,z), *fit_c)
            #                   sigma_x    sigma_z   amp theta        x0            z0           y0
            bounds_surround = ((0,         0,        -2,  0,         fit_c[4],     fit_c[5],     0),
                               (shape_x*4, shape_y*4, 0,  2*math.pi, fit_c[4]+eps, fit_c[5]+eps, eps))
            init_fit_surround = (  2,      2,         0,  0,         fit_c[4],     fit_c[5],     0)
            fit_s, _   = sp.optimize.curve_fit(gaussian_2D, (x,z), tmp_to_fit, p0=init_fit_surround, bounds=bounds_surround)

            init_fit_sum = (*fit_c[:-1],*fit_s[:-1],0)
            #Bounding only the position, that center fit should have found
            bounds_sum   = ((-np.inf,-np.inf,-np.inf,-np.inf,fit_c[4],fit_c[5],
                             -np.inf,-np.inf,-np.inf,-np.inf,fit_c[4],fit_c[5],-np.inf),
                            (np.inf, np.inf, np.inf, np.inf, fit_s[4]+eps,fit_s[5]+eps,
                             np.inf,np.inf,np.inf,np.inf,fit_s[4]+eps,fit_s[5]+eps,np.inf))
            fit_sum, _ = sp.optimize.curve_fit(sum_of_2D_gaussian, (x,z), data_tofit, p0=init_fit_sum, bounds=bounds_sum)
            fit  = {"sigma_x_1": fit_sum[0], "sigma_z_1":fit_sum[1], "amp_1":fit_sum[2],
                    "theta_1":fit_sum[3], "x0_1":fit_sum[4], "z0_1":fit_sum[5],
                    "sigma_x_2": fit_sum[6], "sigma_z_2":fit_sum[7], "amp_2":fit_sum[8],
                    "theta_2":fit_sum[9], "x0_2":fit_sum[10], "z0_2":fit_sum[11], "y0":fit_sum[12]}

        except RuntimeError: #If a model can't be fitted, we get a runtimeError
            fit  = {"sigma_x_1": 1, "sigma_z_1":1, "amp_1":0, "theta_1":0, "x0_1":0, "z0_1":0,
                    "sigma_x_2": 1, "sigma_z_2":1, "amp_2":0, "theta_2":0, "x0_2":0, "z0_2":0, "y0":0}
    model = sum_of_2D_gaussian((x,z), **fit)
    quality_index = 1 - (np.var(data_tofit-model)/np.var(data_tofit))
    return fit, quality_index


# Cell
def fit_temporal_sta(sta, frame_rate=60):
    """Fit a sum of gaussian to a 1D input. Used to fit the STA obtained from fullfield flicker like stimulus.
    params:
        - sta: 1D data to fit

    return:
        - A dictionary containing the parameters of the fit for a `diff_of_gaussian`. The parameter with _1 suffix
        correspond to the closest gaussian to the zero timepoint.
        - A quality index of the fit (explained variance of the model)"""

    # Cutting out the tail of the tSTA to increase the fitting error of the "interesting" part
    cell_abs = np.abs(sta)
    startSTA   = len(sta)-np.argmax(cell_abs>(cell_abs.max()*.4))
    startSTA  += np.maximum(5,(startSTA*.5)).astype(int)
    fullsta = sta.copy()
    sta = sta[-startSTA:]

    t = np.linspace((1-len(sta))/frame_rate, 0, len(sta))

                #sigma_1, amp_1,   x0_1,                sigma_2, amp_2,  x0_2,
    bounds   = ((0,       0,      -len(sta)/frame_rate, 0,       0,      -len(sta)/frame_rate),
                (.1,      np.inf, 1/frame_rate,        .1,       np.inf, 1/frame_rate))

    diff_of_gauss = partial(diff_of_gaussian, y0=0)

    if np.isnan(sp.sum(sta)): #We check that the sta exists, otherwise return default zero model
        fit  = {"sigma_1":1,"amp_1":0,"x0_1":0,
                "sigma_2":1,"amp_2":0,"x0_2":0,"y0":0}
    else:
        try:
            fit, _ = sp.optimize.curve_fit(diff_of_gauss, t, sta, bounds=bounds, max_nfev=100000000000)

            fit  = {"sigma_1":fit[0],"amp_1":fit[1],"x0_1":fit[2],
                    "sigma_2":fit[3],"amp_2":fit[4],"x0_2":fit[5],"y0":0}
        except RuntimeError: #If a model can't be fitted, we get a runtimeError
            fit  = {"sigma_1":1,"amp_1":0,"x0_1":0,
                    "sigma_2":1,"amp_2":0,"x0_2":0,"y0":0}

    t = np.linspace((1-len(fullsta))/frame_rate, 0, len(fullsta))

    model = diff_of_gaussian(t, **fit)
    quality_index = 1 - (np.var(fullsta-model)/np.var(fullsta))
    return fit, quality_index


# Cell
def fit_chirp_am(cell_mean, start=420, stop=960, freq=1.5, frame_rate=60):
    """Fit a sinexp_sigm to the mean response of a cell to chirp_am stimulus.
    params:
        - cell_mean: Average response of the cell to the chirp_am stimulus
        - start:     Index in the cell mean where to start the fitting
        - stop:      Index in the cell mean where to stop the fitting
        - freq:      Frequency of the amplitude modulation of the stimulus in Hz

    return :
        - A dictionary containing the parameters of the fit for a `sinexp_sigm`
        - A quality index of the fit (explained variance of the model)"""

    to_fit = cell_mean[start:stop]
    t = np.linspace(0, len(to_fit)/frame_rate, len(to_fit), endpoint=False)

    #If suppressed by contrast cell, the firing should be higher in first part of chirp
    fit_positive = np.mean(to_fit[:len(to_fit)//3]) < np.mean(to_fit[len(to_fit)*2//3:])

    tmp_diff = np.inf
    best_fit = (1, 0, 0, 0, 0, freq, 0)
    phi      = np.pi
    for exp in np.exp2(range(1,10)): #Fitting the data with different sin exponents, to narrow the fit
        try:
            sinexp_sigm_part = partial(sinexp_sigm, freq=freq, exp=exp)
            if fit_positive:
                fit, _ = sp.optimize.curve_fit(sinexp_sigm_part, t, to_fit,
                                               bounds=[(  0,    0          ,        0       ,      0        , 0),
                                                        (50, len(to_fit)/60,  np.max(to_fit), np.max(to_fit), 2*np.pi)],
                                               p0 =      (1, len(to_fit)/120,       0       , np.max(to_fit), phi),
                                              maxfev=100000)
            else:
                fit, _ = sp.optimize.curve_fit(sinexp_sigm_part, t, to_fit,
                                               bounds=[(  0,    0          ,        0       ,-np.max(to_fit), 0),
                                                        (50, len(to_fit)/60,  np.max(to_fit),     0         , 2*np.pi)],
                                               p0 =     (1, len(to_fit)/120,  np.max(to_fit) ,-np.max(to_fit),phi),
                                              maxfev=100000)

            mse = np.sum(np.square(sinexp_sigm_part(t, *fit) - to_fit))
            phi = fit[4]
            if mse < tmp_diff:
                best_fit = (*fit, freq, exp)
                tmp_diff = mse
        except:
            continue
    best_fit = dict((k, v) for v, k in zip(best_fit, ["sigma","x0","y0","amp","phi","freq","exp"]))

    model  = sinexp_sigm(t, **best_fit)
    to_fit = cell_mean[start:stop]
    quality_index = 1 - (np.var(to_fit-model)/np.var(to_fit))
    return best_fit, quality_index

def fit_chirp_freq_epoch(cell_mean, start=360, freqs=[1.875,3.75,7.5,15,30], durations=[2,2,2,1,1], frame_rate=60):
    """Fit multiple sinexp_sigm to the mean response of a cell to chirp_freq_epoch stimulus.
    Each epoch is fitted independantly by a `sin_exponent`
    params:
        - cell_mean: Average response of the cell to the chirp_am stimulus
        - start:     Index in the cell mean where to start the fitting
        - freqs:     Frequencies of each epoch in Hz
        - durations: Duration of each epoch in seconds
        - freq:      Frequency of the amplitude modulation of the stimulus

    return :
        - List of dictionary containing the parameters of each epoch fit for a `sin_exponent`
        - List of quality index of each epoch fit (explained variance of the model)"""
    best_fit_l = []
    qualityidx_l = []

    cursor = start #Start of the freqs

    #We fit each freqency epoch with a cos function
    for freq, dur in zip(freqs, durations):
        n_rep, n_fr = int(dur*freq), int(frame_rate/freq)
        edgecut = 10
        cursor += edgecut
        len_fit = n_rep*n_fr - edgecut

        #Upsampling of the data to prevent low sampling effects with high freq sine
        upsamp = 10
        to_fit = cell_mean[cursor:cursor+len_fit+1]
        t = np.linspace(0, len_fit/frame_rate, len_fit*upsamp, endpoint=False)
        to_fit = interp1d(np.linspace(0,len(to_fit)/frame_rate, len(to_fit), endpoint=False), to_fit)(t)

        best_fit = None
        tmp_diff = np.inf
        for exp in np.exp2(range(1,10)): #Fitting the data with different cos exponents, to narrow the fit
            try:
                sin_exponent_part = partial(sin_exponent, freq=freq, exp=exp)
                fit, cov = sp.optimize.curve_fit(sin_exponent_part, t, to_fit, bounds=[(0,          0),
                                                                                       (max(to_fit),np.pi*2)])
                mse = np.sum(np.square(sin_exponent_part(t, *fit) - to_fit))
                if mse < tmp_diff:
                    best_cov = cov
                    best_fit = (*fit, freq, exp)
                    tmp_diff = mse
            except:
                continue
        if best_fit is not None:
            best_fit = dict((k, v) for v, k in zip(best_fit, ["amp","phi","freq","exp"]))
            best_fit_l.append(best_fit)
            model = sin_exponent(t, **best_fit)
            qualityidx_l.append(1 - (np.var(to_fit-model)/np.var(to_fit)))
        else:
            best_fit_l.append({"amp":0,"phi":0,"freq":freq,"exp":0})
            qualityidx_l.append(0)

        cursor += len_fit
    return best_fit_l, qualityidx_l #, best_cov_l

# Cell
def fit_transiency(pref_response, frame_rate=60):
    """Fit an exponential decay to a 1D input. Used to fit the response of cells to their preferred
    sustained stimulus.
    params:
        - pref_response: 1D data to fit

    return:
        - A dictionary containing the parameters of the fit for a `exponential_decay`
        - A quality index of the fit (explained variance of the model)"""
    peak_position = np.argmax(pref_response)
    pref_response = pref_response[peak_position:]

    t = np.linspace(0, len(pref_response)/frame_rate, len(pref_response), endpoint=False)
    bounds = ([0.000001, 0],
              [np.inf  , pref_response[0]])
    if np.all(pref_response==0):
        fit = {"tau":1,"baseline":0,"amplitude":0}
    else:
        try:
            decay_part = partial(exponential_decay, amplitude=pref_response[0])
            fit, _ = sp.optimize.curve_fit(decay_part, t, pref_response, maxfev=1000000, bounds=bounds)
            fit  = {"tau":fit[0],"baseline":fit[1],"amplitude":pref_response[0]}
        except RuntimeError:
            fit = {"tau":1,"baseline":0,"amplitude":0}
    model = exponential_decay(t, **fit)
    quality_index =  1 - (np.var(pref_response-model)/np.var(pref_response))

    return fit, quality_index, peak_position

# Cell
def repetition_quality_index(cell_response):
    """Return a quality index of cell response to a repeated stimulus.
    params:
        - cell_response: response of a cell of shape (n_rep, time)"""
    tmp1 = np.var(np.mean(cell_response, axis=0)) #VAR(MEAN(C)repeat)time
    tmp2 = np.mean(np.var(cell_response, axis=0)) #MEAN(VAR(C)time)repeat
    quality = tmp1/tmp2
    return quality

def onoff_transient_index(cell_response, start_on=120, stop_on=240, start_off=240, stop_off=360):
    """Return both on-off and transient indexes of cell response. Transiency index is not always
    ideal to describe the cells response. Consider fitting an exponential decay with `fit_transiency`.
    params:
        - cell_response: response of a cell of shape (time,...)
        - start_on:  starting index of ON stimulation
        - stop_on:   stop index of ON stimulation
        - start_off: starting index of OFF stimulation
        - stop_off:  stop index of OFF stimulation
    return:
        - ON-OFF index
        - Transiency index"""
    #ON-OFF index
    on_response   = cell_response[start_on:stop_on]
    on_sum        = np.sum(on_response)
    off_response  = cell_response[start_off:stop_off]
    off_sum       = np.sum(off_response)
    onoff_indexes = (on_sum-off_sum)/(on_sum+off_sum)

    #Transient index
    if onoff_indexes>0: #Main ON response
        transient_sum   = np.sum(cell_response[start_on:start_on+30])
        sustained_sum   = np.sum(cell_response[start_on:stop_on])
    else:
        transient_sum   = np.sum(cell_response[start_off:start_off+30])
        sustained_sum   = np.sum(cell_response[start_off:stop_off])
    transient_index = transient_sum/sustained_sum

    return onoff_indexes, transient_index
# AUTOGENERATED! DO NOT EDIT! File to edit: 03_modelling.ipynb (unless otherwise specified). # Cell # Cell Sigmoid function     params:         - x: 1D numpy array at which to evaluate the points         - sigma: steepness of the sigmoid         - amp: amplitude of the sigmoid         - x0: shift in x of the sigmoid         - y0: shift in y of the sigmoid # Cell Gaussian function     params:         - x: 1D numpy array at which to evaluate the points         - sigma: width of the gaussian         - amp: amplitude of the gaussian         - x0: shift in x of the gaussian         - y0: shift in y of the gaussian # Cell Sum of gaussian, using `gaussian`     params:         - t: 1D numpy array at which to evaluate the points         - sigma_1: width of the 1st gaussian         - amp_1: amplitude of the 1st gaussian         - x0_1: shift in x of the 1st gaussian         - sigma_2: width of the 2nd gaussian         - amp_2: amplitude of the 2nd gaussian         - x0_2: shift in x of the 2nd gaussian         - y0: shift in y of the gaussian (shared for the two gaussians) Difference of gaussian, using `gaussian`     params:         - t: 1D numpy array at which to evaluate the points         - sigma_1: width of the 1st gaussian         - amp_1: amplitude of the 1st gaussian         - x0_1: shift in x of the 1st gaussian         - sigma_2: width of the 2nd gaussian         - amp_2: amplitude of the 2nd gaussian         - x0_2: shift in x of the 2nd gaussian         - y0: shift in y of the gaussian (shared for the two gaussians) # Cell Two dimensional Gaussian function     params:         - xz: meshgrid of x and z coordinates at which to evaluate the points         - sigma_x: width of the gaussian         - sigma_z: height of the gaussian         - amp: amplitude of the gaussian         - theta: angle of the gaussian (in radian)         - x0: shift in x of the gaussian         - z0: shift in z of the gaussian         - y0: shift in y of the gaussian # Cell Sum of two 2D Gaussian function. For the params, see `gaussian_2D`.      However, both share the y0 parameter. # Cell Exponential decay function     params:         - x: 1D numpy array at which to evaluate the points         - tau: decay speed         - baseline: baseline after the decay. Must be lower than the amplitude         - amplitude: Initial amplitude before the decay # Cell Sine raised to an exponent power. To the power 2, the sine is going between 0 and 1     while its period is halfed. Subsequent round powers have the effect of narowing the peaks     and making the rest of the sine approach zero. (graph it to see it)     params:         - x: 1D numpy array at which to evaluate the points         - amp: amplitude of the sine         - phi: phase of the sine in radian         - freq: frequency of the sine in Hz         - exp: Power the sine is raised to #Multiply by pi instead of 2*pi because it expect the cos to be at least squared #Same goes for phi that I halfed so the period is still 2*pi # Cell A sine exponent weighted by a sigmoid. For parameters, see `sigmoid` and `sin_exponent`.     However the amp and y0 (baseline) parameters are shared between the two sub-functions # Cell Fit a sigmoid to a 1D input. Used to fit the nonlinearities of the cells responses.     Also, the data must be increasing (from left to right).     params:         - nonlin: 1D data to fit         - t     : timepoints of the data to fit. Same lenght as nonlin      return:         - A dictionary containing the parameters of the fit for a `sigmoid`         - A quality index of the fit (explained variance of the model) Fit a sum of 2D gaussian to a 2D matrix. Used to fit the STA obtained from checkerboard like stimulus.     params:         - sta: 2D data to fit      return:         - A dictionary containing the parameters of the fit for a `sum_of_2D_gaussian`         - A quality index of the fit (explained variance of the model) #                   sigma_x    sigma_z   amp theta      x0        z0      y0 #                   sigma_x sigma_z  amp        theta      x0        z0       y0 #We check that the sta exists, otherwise return default zero model #                   sigma_x    sigma_z   amp theta        x0            z0           y0 #Bounding only the position, that center fit should have found #If a model can't be fitted, we get a runtimeError # Cell Fit a sum of gaussian to a 1D input. Used to fit the STA obtained from fullfield flicker like stimulus.     params:         - sta: 1D data to fit      return:         - A dictionary containing the parameters of the fit for a `diff_of_gaussian`. The parameter with _1 suffix         correspond to the closest gaussian to the zero timepoint.         - A quality index of the fit (explained variance of the model) # Cutting out the tail of the tSTA to increase the fitting error of the "interesting" part #sigma_1, amp_1,   x0_1,                sigma_2, amp_2,  x0_2, #We check that the sta exists, otherwise return default zero model #If a model can't be fitted, we get a runtimeError # Cell Fit a sinexp_sigm to the mean response of a cell to chirp_am stimulus.     params:         - cell_mean: Average response of the cell to the chirp_am stimulus         - start:     Index in the cell mean where to start the fitting         - stop:      Index in the cell mean where to stop the fitting         - freq:      Frequency of the amplitude modulation of the stimulus in Hz      return :         - A dictionary containing the parameters of the fit for a `sinexp_sigm`         - A quality index of the fit (explained variance of the model) #If suppressed by contrast cell, the firing should be higher in first part of chirp #Fitting the data with different sin exponents, to narrow the fit Fit multiple sinexp_sigm to the mean response of a cell to chirp_freq_epoch stimulus.     Each epoch is fitted independantly by a `sin_exponent`     params:         - cell_mean: Average response of the cell to the chirp_am stimulus         - start:     Index in the cell mean where to start the fitting         - freqs:     Frequencies of each epoch in Hz         - durations: Duration of each epoch in seconds         - freq:      Frequency of the amplitude modulation of the stimulus      return :         - List of dictionary containing the parameters of each epoch fit for a `sin_exponent`         - List of quality index of each epoch fit (explained variance of the model) #Start of the freqs #We fit each freqency epoch with a cos function #Upsampling of the data to prevent low sampling effects with high freq sine #Fitting the data with different cos exponents, to narrow the fit #, best_cov_l # Cell Fit an exponential decay to a 1D input. Used to fit the response of cells to their preferred     sustained stimulus.     params:         - pref_response: 1D data to fit      return:         - A dictionary containing the parameters of the fit for a `exponential_decay`         - A quality index of the fit (explained variance of the model) # Cell Return a quality index of cell response to a repeated stimulus.     params:         - cell_response: response of a cell of shape (n_rep, time) #VAR(MEAN(C)repeat)time #MEAN(VAR(C)time)repeat Return both on-off and transient indexes of cell response. Transiency index is not always     ideal to describe the cells response. Consider fitting an exponential decay with `fit_transiency`.     params:         - cell_response: response of a cell of shape (time,...)         - start_on:  starting index of ON stimulation         - stop_on:   stop index of ON stimulation         - start_off: starting index of OFF stimulation         - stop_off:  stop index of OFF stimulation     return:         - ON-OFF index         - Transiency index #ON-OFF index #Transient index #Main ON response