Introducing Learning & Saturation Effects into the Prior Voting History Neuron

Word Problem

We want to have a Neuron on which Power is accrued through the count of SCF past participations. The total vote power should accrue slowly initially, accelerate (e.g. increase the marginal power per newly participated round) until it hits a inflection point, on which it deaccelerates until it hits a asymptotic level, which can either be a slowly-growing slope or a constant.

Sources: [1], [2], [3]

Requirements

• R1:
  $n\ge 0$
• R2:
  $P(n=0)=0$
• R3:
  $P(n=1)=P_0$
• R4:
  $P(n\to \mathrm{\infty })=P^{+}\text{ or }\mathrm{\infty }$
• R5:
  $\dot{P}(n=0)=0$
• R6:
  $\dot{P}(n\to \mathrm{\infty })=0$
• R7: R3:
  $\ddot{P}(n^{\ast })=0$
• R8:
  $\ddot{P}(n<n^{\ast })>0$
• R9:
  $\ddot{P}(n>n^{\ast })<0$
• R10:
  $P(n=n^{\ast })=P_{\ast }$

Hypothesis

https://en.wikipedia.org/wiki/Generalised_logistic_function

$${\displaystyle Y(t)=A+\frac{K-A}{(C+Q{e}^{-Bt}{)}^{1/\nu }}}$$

• R2 can be interpreted as
  $A:=0$
• R3 and R10 is a constraint that can be used for finding
  $C$,
  $Q$ or
  $\nu$.
• R4 can be implemented as
  $K:=P^{+}$ if
  $C=1$ or
  $K:=A+\frac{K-A}{C^{\frac{1}{\nu }}}$
• R6 is implemented by the adoption of GLF
• R5, R7, R8 and R9 can be implemented through
  $t:=n-n^{\ast }$
  • By considering R2, we can also have that
    $Y(n-n^{\ast })\to 0$, which implies
    $(C+Q{e}^{-B(n-n^{\ast })}{)}^{\frac{1}{\nu }}\to \mathrm{\infty }$
  • Alternatively, we can consider a threshold on which
    $Y(\overline{t})<Y^{-}⟹Y(\overline{t}):=0$

PoC

https://colab.research.google.com/drive/1vHfveIP8PBLqg6_1efNGWqkdD8P2JOfq?usp=sharing