SharedTransmisión sináptica.ipynbOpen in CoCalc


import scipy as sc
import pylab as pl
from scipy.integrate import odeint
#from scipy import stats
#import scipy.linalg as lin
import matplotlib.pylab as gr


%matplotlib inline

from matplotlib import pyplot as plt
import numpy as np
from sympy import DiracDelta as delta
from sympy import Heaviside as step
from numpy import diff

diff(step(2),2)
--------------------------------------------------------------------------- AxisError Traceback (most recent call last) <ipython-input-2-89371ddce002> in <module>() ----> 1 diff(step(2),2) /usr/local/lib/python3.5/dist-packages/numpy/lib/function_base.py in diff(a, n, axis) 1933 a = asanyarray(a) 1934 nd = a.ndim -> 1935 axis = normalize_axis_index(axis, nd) 1936 1937 slice1 = [slice(None)] * nd AxisError: axis -1 is out of bounds for array of dimension 0
#Definición de los parámetros del sistema 

Use=0.067
Ase=250
# Y las escalas de tiempo 
tau_inac=3
tau_rec=800


# definiendo los spikes presináapticos.
def spikes(t, N=10, rate=1 ):
    spks = []
    for n in range(N):
        spkt = t[np.random.rand(len(t)) < rate/100.] # determina la lista de tiempo de los spikes
        spks = spkt
    return spks



t = sc.arange(0.0,30.0, 0.1)
spikes(t)

array([ 15.4])
#def Delta(t):    
#    spikes(t)==t
#definición del sistema 



def sistema(U,t):
    Re,Ef,In =U
    dRe= In/tau_rec-Use*Re
    dEf= -Ef/tau_inac+Use*Re
    dIn= 1-Re-Ef
    return dRe,dEf,dIn


U = odeint(sistema,[0.0,0.0,0.0],t)
Re = U[:,0]
Ef = U[:,1]
In = U[:,2]

pl.figure()
pl.title('transmision sináptica')
pl.plot(t, Ef, 'k')
pl.ylabel('E ')
pl.xlabel('t (seg)')
pl.show()


#el modelo con los spikes.

def sistema(U,t):
    Re,Ef,In =U
    dRe= In/tau_rec-Use*Re
    dEf= -Ef/tau_inac+Use*Re*sc.int32(delta(t-4))
    dIn= 1-Re-Ef
    return dRe,dEf,dIn


U = odeint(sistema,[0.1,0.0,0.1],t,hmax=0.1,tcrit=4)
Re = U[:,0]
Ef = U[:,1]
In = U[:,2]

pl.figure()
pl.title('transmision sináptica')
pl.plot(t, Ef, 'k')
pl.ylabel('E ')
pl.xlabel('t (seg)')
pl.show()
--------------------------------------------------------------------------- TypeError Traceback (most recent call last) <ipython-input-17-8517955f461e> in <module>() 9 10 ---> 11 U = odeint(sistema,[0.1,0.0,0.1],t,hmax=0.1,tcrit=4) 12 Re = U[:,0] 13 Ef = U[:,1] /usr/local/lib/python3.4/dist-packages/scipy/integrate/odepack.py in odeint(func, y0, t, args, Dfun, col_deriv, full_output, ml, mu, rtol, atol, tcrit, h0, hmax, hmin, ixpr, mxstep, mxhnil, mxordn, mxords, printmessg) 213 output = _odepack.odeint(func, y0, t, args, Dfun, col_deriv, ml, mu, 214 full_output, rtol, atol, tcrit, h0, hmax, hmin, --> 215 ixpr, mxstep, mxhnil, mxordn, mxords) 216 if output[-1] < 0: 217 warning_msg = _msgs[output[-1]] + " Run with full_output = 1 to get quantitative information." <ipython-input-17-8517955f461e> in sistema(U, t) 4 Re,Ef,In =U 5 dRe= In/tau_rec-Use*Re ----> 6 dEf= -Ef/tau_inac+Use*Re*sc.int32(delta(t-4)) 7 dIn= 1-Re-Ef 8 return dRe,dEf,dIn /usr/local/lib/python3.4/dist-packages/sympy/core/expr.py in __int__(self) 193 # places) we need to test whether we are off by one. 194 from sympy import Dummy --> 195 r = self.round(2) 196 if not r.is_Number: 197 raise TypeError("can't convert complex to int") /usr/local/lib/python3.4/dist-packages/sympy/core/expr.py in round(self, p) 3144 if not pure_complex(xn, or_real=True): 3145 raise TypeError('Expected a number but got %s:' % -> 3146 getattr(getattr(x,'func', x), '__name__', type(x))) 3147 elif x in (S.NaN, S.Infinity, S.NegativeInfinity, S.ComplexInfinity): 3148 return x TypeError: Expected a number but got DiracDelta:
spikes1 = 1000*sc.rand(10) # tren de 10 spikes
spikes1
array([ 357.58958498, 726.28010855, 893.33723847, 752.60324619, 176.03423421, 943.84290786, 97.66847695, 414.95317082, 855.8267547 , 901.73633242])
help(odeint)
Help on function odeint in module scipy.integrate.odepack: odeint(func, y0, t, args=(), Dfun=None, col_deriv=0, full_output=0, ml=None, mu=None, rtol=None, atol=None, tcrit=None, h0=0.0, hmax=0.0, hmin=0.0, ixpr=0, mxstep=0, mxhnil=0, mxordn=12, mxords=5, printmessg=0) Integrate a system of ordinary differential equations. Solve a system of ordinary differential equations using lsoda from the FORTRAN library odepack. Solves the initial value problem for stiff or non-stiff systems of first order ode-s:: dy/dt = func(y, t0, ...) where y can be a vector. *Note*: The first two arguments of ``func(y, t0, ...)`` are in the opposite order of the arguments in the system definition function used by the `scipy.integrate.ode` class. Parameters ---------- func : callable(y, t0, ...) Computes the derivative of y at t0. y0 : array Initial condition on y (can be a vector). t : array A sequence of time points for which to solve for y. The initial value point should be the first element of this sequence. args : tuple, optional Extra arguments to pass to function. Dfun : callable(y, t0, ...) Gradient (Jacobian) of `func`. col_deriv : bool, optional True if `Dfun` defines derivatives down columns (faster), otherwise `Dfun` should define derivatives across rows. full_output : bool, optional True if to return a dictionary of optional outputs as the second output printmessg : bool, optional Whether to print the convergence message Returns ------- y : array, shape (len(t), len(y0)) Array containing the value of y for each desired time in t, with the initial value `y0` in the first row. infodict : dict, only returned if full_output == True Dictionary containing additional output information ======= ============================================================ key meaning ======= ============================================================ 'hu' vector of step sizes successfully used for each time step. 'tcur' vector with the value of t reached for each time step. (will always be at least as large as the input times). 'tolsf' vector of tolerance scale factors, greater than 1.0, computed when a request for too much accuracy was detected. 'tsw' value of t at the time of the last method switch (given for each time step) 'nst' cumulative number of time steps 'nfe' cumulative number of function evaluations for each time step 'nje' cumulative number of jacobian evaluations for each time step 'nqu' a vector of method orders for each successful step. 'imxer' index of the component of largest magnitude in the weighted local error vector (e / ewt) on an error return, -1 otherwise. 'lenrw' the length of the double work array required. 'leniw' the length of integer work array required. 'mused' a vector of method indicators for each successful time step: 1: adams (nonstiff), 2: bdf (stiff) ======= ============================================================ Other Parameters ---------------- ml, mu : int, optional If either of these are not None or non-negative, then the Jacobian is assumed to be banded. These give the number of lower and upper non-zero diagonals in this banded matrix. For the banded case, `Dfun` should return a matrix whose rows contain the non-zero bands (starting with the lowest diagonal). Thus, the return matrix `jac` from `Dfun` should have shape ``(ml + mu + 1, len(y0))`` when ``ml >=0`` or ``mu >=0``. The data in `jac` must be stored such that ``jac[i - j + mu, j]`` holds the derivative of the `i`th equation with respect to the `j`th state variable. If `col_deriv` is True, the transpose of this `jac` must be returned. rtol, atol : float, optional The input parameters `rtol` and `atol` determine the error control performed by the solver. The solver will control the vector, e, of estimated local errors in y, according to an inequality of the form ``max-norm of (e / ewt) <= 1``, where ewt is a vector of positive error weights computed as ``ewt = rtol * abs(y) + atol``. rtol and atol can be either vectors the same length as y or scalars. Defaults to 1.49012e-8. tcrit : ndarray, optional Vector of critical points (e.g. singularities) where integration care should be taken. h0 : float, (0: solver-determined), optional The step size to be attempted on the first step. hmax : float, (0: solver-determined), optional The maximum absolute step size allowed. hmin : float, (0: solver-determined), optional The minimum absolute step size allowed. ixpr : bool, optional Whether to generate extra printing at method switches. mxstep : int, (0: solver-determined), optional Maximum number of (internally defined) steps allowed for each integration point in t. mxhnil : int, (0: solver-determined), optional Maximum number of messages printed. mxordn : int, (0: solver-determined), optional Maximum order to be allowed for the non-stiff (Adams) method. mxords : int, (0: solver-determined), optional Maximum order to be allowed for the stiff (BDF) method. See Also -------- ode : a more object-oriented integrator based on VODE. quad : for finding the area under a curve. Examples -------- The second order differential equation for the angle `theta` of a pendulum acted on by gravity with friction can be written:: theta''(t) + b*theta'(t) + c*sin(theta(t)) = 0 where `b` and `c` are positive constants, and a prime (') denotes a derivative. To solve this equation with `odeint`, we must first convert it to a system of first order equations. By defining the angular velocity ``omega(t) = theta'(t)``, we obtain the system:: theta'(t) = omega(t) omega'(t) = -b*omega(t) - c*sin(theta(t)) Let `y` be the vector [`theta`, `omega`]. We implement this system in python as: >>> def pend(y, t, b, c): ... theta, omega = y ... dydt = [omega, -b*omega - c*np.sin(theta)] ... return dydt ... We assume the constants are `b` = 0.25 and `c` = 5.0: >>> b = 0.25 >>> c = 5.0 For initial conditions, we assume the pendulum is nearly vertical with `theta(0)` = `pi` - 0.1, and it initially at rest, so `omega(0)` = 0. Then the vector of initial conditions is >>> y0 = [np.pi - 0.1, 0.0] We generate a solution 101 evenly spaced samples in the interval 0 <= `t` <= 10. So our array of times is: >>> t = np.linspace(0, 10, 101) Call `odeint` to generate the solution. To pass the parameters `b` and `c` to `pend`, we give them to `odeint` using the `args` argument. >>> from scipy.integrate import odeint >>> sol = odeint(pend, y0, t, args=(b, c)) The solution is an array with shape (101, 2). The first column is `theta(t)`, and the second is `omega(t)`. The following code plots both components. >>> import matplotlib.pyplot as plt >>> plt.plot(t, sol[:, 0], 'b', label='theta(t)') >>> plt.plot(t, sol[:, 1], 'g', label='omega(t)') >>> plt.legend(loc='best') >>> plt.xlabel('t') >>> plt.grid() >>> plt.show()
print unit_step, type(unit_step)
print 'at -3 :', unit_step(-3)
print 'at 4 :', unit_step(4)
File "<ipython-input-1-b2784db7da8b>", line 1 print unit_step, type(unit_step) ^ SyntaxError: invalid syntax