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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 This entire theory is built on the idea that everything is normalized as input into the brain. i.e. all values are between 0 and 1. This is necessary because the learning rule has an adaptive learning rate that is $\sigma^4$. If everything is normalized, the probability of $\sigma^2$ being greater than 1 is very low
Step1:  I can assume $q(x)$ has two forms
$$q(x) = \frac{1}{\sqrt{2 \pi \sigma^2}}exp{-\frac{(x-\mu)^2}{2\sigma^2}}$$
or 
$$q(x) = exp{-\frac{(x-\mu)^2}{\sigma^2}}$$
When I assume the second form and remove the extra $\sigma$ term from the learning equations it no longer converges smoothly. However, if I add an 'astrocyte' to normalize all of them periodically by averaging over the output it works again. Perhaps astrocytes 'normalizing' the neurons is the biological mechanism for keeping the output roughly normal.
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Python Code:
p = GMM([1.0], np.array([[0.5,0.05]]))
num_samples = 1000
beg = 0.0
end = 1.0
t = np.linspace(beg,end,num_samples)
num_neurons = len(p.pis)
colors = [np.random.rand(num_neurons,) for i in range(num_neurons)]
p_y = p(t)
p_max = p_y.max()
np.random.seed(110)
num_neurons = 1
neuron = Neuron((1,1),[[0.6]], bias=0.0006, decay=0.00005)
samples, labels = p.sample(10000)
ys = []
lbls = []
colors = [np.random.rand(3,) for i in range(num_neurons)]
def f(i=0):
    x = np.array(samples[i])
    l = labels[i]
    y = np.array(neuron(x.reshape(1,1,1)))
    
    ys.append(y)
    c = 'b' if l else 'g'
    lbls.append(c)
    fig, ax = plt.subplots(figsize=(15,5))
    ax.plot(t, p_y/p_max, c='r', lw=3, label='$p(x)$')
    ax.plot([x,x],[0,p_max],label="$x\sim p(x)$", lw=4)
   
    y = [neuron(t.reshape(num_samples,1,1),update=0)]
   
    for j,yi in enumerate(y):
        yj_max = y[j].max()
        ax.plot(t, y[j]/yj_max, c=colors[j], lw=3, label="$q(x)$")
    
    ax.set_ylim(0.,1.5)
    ax.set_xlim(beg,end)
    plt.savefig('for_colloquium/fig%03i.png'%(i))
    plt.show()
    
interactive_plot = interactive(f, i=(0, 9999))
output = interactive_plot.children[-1]
output.layout.height = '450px'
interactive_plot
[n.weights for n in list(network.neurons.items())[0][1]]
[np.sqrt(n.bias) for n in list(network.neurons.items())[0][1]]
[n.pi for n in list(network.neurons.items())[0][1]]
Explanation: This entire theory is built on the idea that everything is normalized as input into the brain. i.e. all values are between 0 and 1. This is necessary because the learning rule has an adaptive learning rate that is $\sigma^4$. If everything is normalized, the probability of $\sigma^2$ being greater than 1 is very low
End of explanation
def s(x):
    return (1/(1+np.exp(-10*(x-0.25))))
x = np.linspace(0,1,100)
plt.plot(x,s(x))
plt.show()
Explanation: I can assume $q(x)$ has two forms
$$q(x) = \frac{1}{\sqrt{2 \pi \sigma^2}}exp{-\frac{(x-\mu)^2}{2\sigma^2}}$$
or 
$$q(x) = exp{-\frac{(x-\mu)^2}{\sigma^2}}$$
When I assume the second form and remove the extra $\sigma$ term from the learning equations it no longer converges smoothly. However, if I add an 'astrocyte' to normalize all of them periodically by averaging over the output it works again. Perhaps astrocytes 'normalizing' the neurons is the biological mechanism for keeping the output roughly normal.
End of explanation
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