Correction: Anastassiou, G.A. Composition of Activation Functions and the Reduction to Finite Domain. Mathematics 2025, 13, 3177

There was an error in the original publication [1]. A correction has been made to 2. Basics, Paragraph 6–8:

So we have h₂ : ℝ → (−1, 1), h₁|(−1,1) : (−1, 1) → (−1, 1), and the strictly increasing function H := h₁|_(−1,1) ◦ h₂ : ℝ → (−1, 1), with the graph of H containing an arc of finite length, such that H(0) = 0, starting at (−1, h₁(h₂(−1))) and terminating at (1, h₁(h₂(1))). We call this arc also H. In particular H is negative and convex over (−1, 0], and it is positive and concave over [0, 1).

So it has compact support [−1, 1] and it is like a squashing function, see [3], Ch. 1, p. 8.

We will work from now on with |H|, which has as a graph a cusp joining the points (−1, |h₁(h₂(−1))|), (0, 0), (1, h₁(h₂(1))) and with compact support, again, [−1, 1]. The points (−1, |h₁(h₂(−1))|), (1, h₁(h₂(1))) belong to the graph of |H| and (0, 0) too.

Typically H has a steeper slope than of h₂, but it is flatter and closer to the x-axis than h₂ is, e.g. tanh(tanh x) has asymptotes ±0.76, while tanh x has asymptotes ±1, notice that tanh(1) = 0.76. Clearly H has applications in spiking neural networks.

The authors state that the scientific conclusions are unaffected. This correction was approved by the Section Editor-in-Chief. The original publication has also been updated.