braincore.surrogate.piecewise_quadratic

braincore.surrogate.piecewise_quadratic(x, alpha=1.0)

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 braincore as bc
>>> import matplotlib.pyplot as plt
>>> xs = jax.numpy.linspace(-3, 3, 1000)
>>> for alpha in [0.5, 1., 2., 4.]:
>>>   grads = bc.vector_grad(bc.surrogate.piecewise_quadratic)(xs, alpha)
>>>   plt.plot(xs, grads, label=r'$\alpha$=' + str(alpha))
>>> plt.legend()
>>> plt.show()

(Source code)

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

[1]

Esser S K, Merolla P A, Arthur J V, et al. Convolutional networks for fast, energy-efficient neuromorphic computing[J]. Proceedings of the national academy of sciences, 2016, 113(41): 11441-11446.

[2]

Wu Y, Deng L, Li G, et al. Spatio-temporal backpropagation for training high-performance spiking neural networks[J]. Frontiers in neuroscience, 2018, 12: 331.

[3]

Bellec G, Salaj D, Subramoney A, et al. Long short-term memory and learning-to-learn in networks of spiking neurons[C]//Proceedings of the 32nd International Conference on Neural Information Processing Systems. 2018: 795-805.

[4]

Neftci E O, Mostafa H, Zenke F. Surrogate gradient learning in spiking neural networks: Bringing the power of gradient-based optimization to spiking neural networks[J]. IEEE Signal Processing Magazine, 2019, 36(6): 51-63.

[5]

Panda P, Aketi S A, Roy K. Toward scalable, efficient, and accurate deep spiking neural networks with backward residual connections, stochastic softmax, and hybridization[J]. Frontiers in Neuroscience, 2020, 14.