Judge spiking state with a piecewise quadratic function [1] [2] [3] [4] [5].

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:

$\begin{split}g(x) = \begin{cases} 0, & x < -\frac{1}{\alpha} \\ -\frac{1}{2}\alpha^2|x|x + \alpha x + \frac{1}{2}, & |x| \leq \frac{1}{\alpha} \\ 1, & x > \frac{1}{\alpha} \\ \end{cases}\end{split}$

Backward function:

$\begin{split}g'(x) = \begin{cases} 0, & |x| > \frac{1}{\alpha} \\ -\alpha^2|x|+\alpha, & |x| \leq \frac{1}{\alpha} \end{cases}\end{split}$
>>> 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

• origin (bool) – Whether to compute the original function as the feedfoward output.

Returns:

out – The spiking state.

Return type:

jax.Array

References