WereRabbit
The wererabbit neuron model is a two coupled oscillator that follows a predator- prey dynamic with a switching in the diagonal of the phaseplane. When the z in equation 1c represents the “moon phase”, when ever it cross that threshold, the rabbit (prey) becomes the predator.
Circuit equation
\[
\begin{align}
C\frac{du}{dt} &= z I_{bias} - I_{n0} e^{\kappa v / U_t} [z + 26e^{-2} (0.5 - u) z] - I_a \\
C\frac{dv}{dt} &= -z I_{bias} + I_{n0} e^{\kappa u / U_t} [z + 26e^{-2} (0.5 - v) z] - I_a \\
z &= tanh(\rho (u-v))\\
I_a &= \sigma I_{bias} \\
\end{align}
\]
| Parameter | Symbol | Definition | Value |
|---|---|---|---|
| Capacitance | C | Circuit capacitance | \(0.1\,pF\) |
| Bias current | \(I_{bias}\) | DC bias current for the fixpoint location | \(100\,pA\) |
| Leakage current | \(I_{n0}\) | Transistor leakage current | \(0.129\,pA\) |
| Subthreshold slope | \(\kappa\) | Transistor subthreshold slope factor | \(0.39\) |
| Thermal voltage | \(U_t\) | Thermal voltage at room temperature | \(25\,mV\) |
| Bias scale | \(\sigma\) | Scaling factor for the distance between fixpoints | \(0.6\) |
| Steepness | \(\rho\) | Tanh steepness for the moonphase | \(5\)s |
Abstraction
To simplify the analysis of the model for simulation purposes, we can introduce a dimensionless time variable \(\tau=tI_{bias}/C\), transforming the derivate of the equations in \(\frac{d}{dt}=\frac{I_{bias}}{C}\frac{d}{d\tau}\). Substituting this time transformation on equation~\ref{eq:wererabbit:circ}
\[
\begin{equation}
C\frac{I_{bias}}{C}\frac{du}{d\tau} = z I_{bias} - I_{n0} e^{\kappa v / U_t} [z + 26e^{-2} (0.5 - u) z] - \sigma I_{bias}
\end{equation}
\]
And dividing by \(I_{bias}\) on both sides:
\[
\begin{equation}
\frac{du}{d\tau} = z - \frac{I_{n0}}{I_{bias}} e^{\kappa v / U_t} [z + 26e^{-2} (0.5 - u) z] - \sigma
\end{equation}
\]
Obtaining the following set of equations:
\[
\begin{align}
z &= tanh(\kappa (u-v)) \\
\frac{du}{dt} &= z - z \alpha e^{\beta v} [1 + \gamma (0.5 - u)] - \sigma \\
\frac{dv}{dt} &= -z - z \alpha e^{\beta u} [1 + \gamma (0.5 - v)] - \sigma
\end{align}
\]
| Parameter | Definition | Value |
|---|---|---|
| \(\tau\) | \(tI_{bias}/C\) | -- |
| \(\alpha\) | \(I_{n0}/I_{bias}\) | \(0.0129\) |
| \(\beta\) | \(\kappa/U_t\) | 15.6 |
| \(\gamma\) | -- | \(26e^{-2}\) |
| \(\rho\) | Tanh steepness for the moonphase | 5 |
| \(\sigma\) | Scaling factor for the distance between fixpoints | 0.6 |
Examples
See the following interactive notebook for a practical example:
- Basic Usage Example - Introduction to the WereRabbit model