# brainpy.math.surrogate.erf#

brainpy.math.surrogate.erf(x, alpha=1.0)[source]#

Judge spiking state with an erf function [1] [2] [3].

If origin=False, computes the forward function:

$\begin{split}g(x) = \begin{cases} 1, & x \geq 0 \\ 0, & x < 0 \\ \end{cases}\end{split}$

If origin=True, computes the original function:

$\begin{split}\begin{split} g(x) &= \frac{1}{2}(1-\text{erf}(-\alpha x)) \\ &= \frac{1}{2} \text{erfc}(-\alpha x) \\ &= \frac{1}{\sqrt{\pi}}\int_{-\infty}^{\alpha x}e^{-t^2}dt \end{split}\end{split}$

Backward function:

$g'(x) = \frac{\alpha}{\sqrt{\pi}}e^{-\alpha^2x^2}$
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import matplotlib.pyplot as plt
>>> bp.visualize.get_figure(1, 1, 4, 6)
>>> xs = bm.linspace(-3, 3, 1000)
>>> for alpha in [0.5, 1., 2., 4.]:
>>>   plt.plot(xs, grads, label=r'$\alpha$=' + str(alpha))
>>> plt.legend()
>>> plt.show()

Parameters:
• x (jax.Array, Array) – The input data.

• alpha (float) – Parameter to control smoothness of gradient

Returns:

out – The spiking state.

Return type:

jax.Array

References