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# 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