felice-models/docs-site/docs/neuron_models/wererabbit/index.md

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