Arctan

class braintools.surrogate.Arctan(alpha=1.0)

Judge spiking state with an arctan function.

This class implements a surrogate gradient using the arctangent function.

The forward function:

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

The original function:

\[g(x) = \frac{1}{\pi} \arctan(\frac{\pi}{2}\alpha x) + \frac{1}{2}\]

Backward function:

\[g'(x) = \frac{\alpha}{2(1 + (\frac{\pi}{2}\alpha x)^2)}\]

Parameters:

alpha (float, optional) – Parameter controlling the steepness of the surrogate gradient. Higher values make the transition sharper. Default is 1.0.

See also

arctan

Function version of this class.

Examples

>>> import brainstate
>>> import jax.numpy as jnp
>>>
>>> # Create an arctangent surrogate
>>> arctan_fn = braintools.surrogate.Arctan(alpha=2.0)
>>>
>>> # Apply to membrane potentials
>>> x = jnp.array([-1.0, 0.0, 1.0])
>>> spikes = arctan_fn(x)
>>> print(spikes)  # Binary spike output

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

(Source code)

Notes

The arctangent function provides smooth gradients with polynomial decay, offering a balance between the fast decay of exponential functions and the slow decay of linear functions.

surrogate_fun(x)

The surrogate function.

surrogate_grad(x)

The gradient function of the surrogate function.