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Initial commit
Co-authored-by: Aradhana Dube <a.dube@rug.nl> Co-authored-by: Renzo I. Barraza Altamirano <r.i.barraza.altamirano@rug.nl> Co-authored-by: Paolo Gibertini <p.gibertini@rug.nl> Co-authored-by: Luca D. Fehlings <l.d.fehlings@rug.nl>
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# WereRabbit
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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.
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## Circuit equation
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$$
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\begin{align}
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C\frac{du}{dt} &= z I_{bias} - I_{n0} e^{\kappa v / U_t} [z + 26e^{-2} (0.5 - u) z] - I_a \\
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C\frac{dv}{dt} &= -z I_{bias} + I_{n0} e^{\kappa u / U_t} [z + 26e^{-2} (0.5 - v) z] - I_a \\
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z &= tanh(\rho (u-v))\\
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I_a &= \sigma I_{bias} \\
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\end{align}
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$$
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| **Parameter** | **Symbol** | **Definition** | **Value** |
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|-----------|--------|------------|-------|
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| Capacitance | C | Circuit capacitance | $0.1\,pF$ |
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| Bias current | $I_{bias}$ | DC bias current for the fixpoint location | $100\,pA$
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Leakage current | $I_{n0}$ | Transistor leakage current | $0.129\,pA$
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Subthreshold slope | $\kappa$ | Transistor subthreshold slope factor | $0.39$
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Thermal voltage | $U_t$ | Thermal voltage at room temperature | $25\,mV$
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Bias scale | $\sigma$ | Scaling factor for the distance between fixpoints | $0.6$
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Steepness | $\rho$ | Tanh steepness for the moonphase | $5$s
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## Abstraction
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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}
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$$
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\begin{equation}
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C\frac{I_{bias}}{C}\frac{du}{d\tau} = z I_{bias} - I_{n0} e^{\kappa v / U_t} [z + 26e^{-2} (0.5 - u) z] - \sigma I_{bias}
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\end{equation}
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$$
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And dividing by $I_{bias}$ on both sides:
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$$
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\begin{equation}
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\frac{du}{d\tau} = z - \frac{I_{n0}}{I_{bias}} e^{\kappa v / U_t} [z + 26e^{-2} (0.5 - u) z] - \sigma
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\end{equation}
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$$
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Obtaining the following set of equations:
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$$
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\begin{align}
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z &= tanh(\kappa (u-v)) \\
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\frac{du}{dt} &= z - z \alpha e^{\beta v} [1 + \gamma (0.5 - u)] - \sigma \\
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\frac{dv}{dt} &= -z - z \alpha e^{\beta u} [1 + \gamma (0.5 - v)] - \sigma
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\end{align}
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$$
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| **Parameter** | **Definition** | **Value** |
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|---------------|----------------|-----------|
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| $\tau$ | $tI_{bias}/C$ | -- |
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| $\alpha$ | $I_{n0}/I_{bias}$ | $0.0129$ |
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| $\beta$ | $\kappa/U_t$ | 15.6 |
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| $\gamma$ | -- | $26e^{-2}$ |
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| $\rho$ | Tanh steepness for the moonphase | 5 |
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| $\sigma$ | Scaling factor for the distance between fixpoints | 0.6 |
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## Examples
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See the following interactive notebook for a practical example:
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- [Basic Usage Example](wererabbit.ipynb) - Introduction to the WereRabbit model
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@@ -0,0 +1,189 @@
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 10,
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"id": "cbf60fc3",
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"metadata": {},
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"outputs": [],
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"source": [
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"import diffrax as dfx\n",
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"import jax\n",
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"import jax.numpy as jnp\n",
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"import jax.random as jrand\n",
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"import matplotlib as mpl\n",
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"import matplotlib.pyplot as plt\n",
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"\n",
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"from felice.neuron_models import WereRabbit\n",
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"\n",
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"jax.config.update(\"jax_enable_x64\", True)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"id": "5f28e42f",
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"metadata": {},
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"outputs": [],
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"source": [
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"key = jrand.key(0)\n",
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"max_time = 40\n",
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"\n",
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"model = WereRabbit(dtype=jnp.float64)\n",
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"\n",
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"\n",
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"def state_at_t(comp_times):\n",
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" sol = dfx.diffeqsolve(\n",
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" terms=dfx.ODETerm(model.dynamics),\n",
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" solver=dfx.Tsit5(),\n",
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" t0=0.0,\n",
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" t1=max_time,\n",
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" dt0=1e-3,\n",
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" y0=model.init_state(1)\n",
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" + jrand.uniform(key, shape=(1, 2), minval=0.1, maxval=0.5),\n",
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" saveat=dfx.SaveAt(ts=comp_times),\n",
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" max_steps=100000,\n",
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" )\n",
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"\n",
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" return sol.ts, sol.ys"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"id": "28924cfa",
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"metadata": {},
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"outputs": [],
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"source": [
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"comp_times = jnp.linspace(0.0, max_time, 2000)\n",
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"_, state = state_at_t(comp_times)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 13,
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"id": "9d4d0829",
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"metadata": {},
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"outputs": [],
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"source": [
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"def compute_nullclines(snn, u_range, v_range, resolution=200):\n",
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" \"\"\"\n",
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" Compute nullclines\n",
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" du/dt = 0 (u-nullcline)\n",
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" dv/dt = 0 (v-nullcline)\n",
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" \"\"\"\n",
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" u_vals = jnp.linspace(u_range[0], u_range[1], resolution)\n",
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" v_vals = jnp.linspace(v_range[0], v_range[1], resolution)\n",
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" U, V = jnp.meshgrid(u_vals, v_vals)\n",
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"\n",
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" UV = jnp.stack(\n",
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" [U.reshape(-1), V.reshape(-1), jnp.ones((resolution * resolution,))], axis=1\n",
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" )\n",
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" dS = snn.vector_field(UV)\n",
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" dU = dS[:, 0].reshape(U.shape)\n",
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" dV = dS[:, 1].reshape(V.shape)\n",
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"\n",
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" return U, V, dU, dV\n",
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"\n",
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"\n",
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"def plot_vf(ax, snn, u_range, v_range):\n",
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" import numpy as np\n",
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"\n",
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" u_sparse = jnp.linspace(u_range[0], u_range[1], 20)\n",
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" v_sparse = jnp.linspace(v_range[0], v_range[1], 20)\n",
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"\n",
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" Us, Vs = jnp.meshgrid(u_sparse, v_sparse)\n",
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"\n",
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" U, V, dU, dV = compute_nullclines(snn, u_range, v_range, 200)\n",
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"\n",
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" UVs = jnp.stack([Us.reshape(-1), Vs.reshape(-1), jnp.ones((20 * 20,))], axis=1)\n",
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" dS = snn.vector_field(UVs)\n",
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" dUs = dS[:, 0].reshape(Us.shape)\n",
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" dVs = dS[:, 1].reshape(Vs.shape)\n",
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"\n",
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" # Normalize for visualization\n",
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" magnitude = np.sqrt(dUs**2 + dVs**2)\n",
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" magnitude[magnitude == 0] = 1\n",
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" dUs_norm = dUs / magnitude\n",
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" dVs_norm = dVs / magnitude\n",
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"\n",
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" # Nullclines\n",
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" ax.contour(U, V, dU, levels=[0], colors=\"blue\", linewidths=1, linestyles=\"-\")\n",
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" ax.contour(U, V, dV, levels=[0], colors=\"red\", linewidths=1, linestyles=\"-\")\n",
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"\n",
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" # Vector field\n",
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" ax.quiver(Us, Vs, dUs_norm, dVs_norm, magnitude, cmap=\"viridis\", alpha=0.6)\n",
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"\n",
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" ax.set_xlabel(\"u (Prey)\")\n",
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" ax.set_ylabel(\"v (Predator)\")\n",
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" ax.set_title(\"Wererabbit: Phase Portrait\")\n",
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" ax.legend([\"u-nullcline (du/dt=0)\", \"v-nullcline (dv/dt=0)\"], loc=\"upper right\")\n",
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" ax.set_xlim(u_range)\n",
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" ax.set_ylim(v_range)\n",
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" ax.axhline(y=0, color=\"gray\", linestyle=\"--\", alpha=0.3)\n",
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" ax.axvline(x=0, color=\"gray\", linestyle=\"--\", alpha=0.3)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 14,
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"id": "b1e166ef",
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"metadata": {},
|
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"outputs": [
|
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{
|
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"data": {
|
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"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 1380x520 with 2 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"with mpl.style.context(\"boilerplot.ieeetran\"):\n",
|
||||
" fig, ax = plt.subplots(1, 2, figsize=(6.9, 2.6), dpi=200)\n",
|
||||
" ax[0].plot(comp_times, state[:, 0, 0], label=\"x1\")\n",
|
||||
" ax[0].plot(comp_times, state[:, 0, 1], label=\"x2\")\n",
|
||||
" ax[0].legend([\"x1\", \"x2\"])\n",
|
||||
"\n",
|
||||
" plot_vf(ax[1], model, [-0.2, 0.5], [-0.2, 0.5])\n",
|
||||
" ax[1].plot(state[:, 0, 0], state[:, 0, 1])\n",
|
||||
" ax[1].plot(state[0, 0, 0], state[0, 0, 1], \".\", label=\"start\")\n",
|
||||
" ax[1].plot(state[-1, 0, 0], state[-1, 0, 1], \".\", label=\"end\")\n",
|
||||
" ax[1].legend()\n",
|
||||
" plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "f87852ec",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "felice",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.13.5"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
Reference in new issue
Block a user