{ "cells": [ { "cell_type": "markdown", "id": "834fe6f5", "metadata": {}, "source": [ "# How to reproduce a paper: gamma oscillation\n", "\n", "**Task.** Reproduce a published result — the interneuron-network gamma rhythm of\n", "Wang & Buzsáki (1996) — to show how a custom biophysical model is expressed in\n", "`brainpy.state`.\n", "\n", "**Audience.** Simulation. Comfortable with custom `Neuron`/`Synapse` subclasses.\n", "\n", "Reproducing a paper usually means writing the model's exact equations. This guide\n", "implements the Wang–Buzsáki single-compartment Hodgkin–Huxley interneuron and an\n", "all-to-all GABAergic synapse, drives the network with a constant current, and\n", "recovers the population gamma oscillation. It demonstrates: subclassing\n", "`brainpy.state.Neuron` / `Synapse`, integrating ODEs with `exp_euler_step`, and a\n", "`CurrentProj` with all-to-all connectivity.\n", "\n", "Reference: Wang, X-J. & Buzsáki, G. (1996), *Gamma oscillation by synaptic\n", "inhibition in a hippocampal interneuronal network model*, J. Neurosci. 16(20)." ] }, { "cell_type": "code", "execution_count": 1, "id": "4bd09e78", "metadata": { "execution": { "iopub.execute_input": "2026-06-17T09:08:13.200424Z", "iopub.status.busy": "2026-06-17T09:08:13.200211Z", "iopub.status.idle": "2026-06-17T09:08:17.388219Z", "shell.execute_reply": "2026-06-17T09:08:17.387154Z" } }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "An NVIDIA GPU may be present on this machine, but a CUDA-enabled jaxlib is not installed. Falling back to cpu.\n" ] } ], "source": [ "import brainpy\n", "import brainstate\n", "import braintools\n", "import brainunit as u\n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "markdown", "id": "f971cc27", "metadata": {}, "source": [ "## The Wang–Buzsáki interneuron\n", "\n", "A single-compartment model with fast sodium, delayed-rectifier potassium, and a\n", "leak current. We integrate the gating variables and the membrane potential with\n", "the exponential Euler step." ] }, { "cell_type": "code", "execution_count": 2, "id": "b9e829bc", "metadata": { "execution": { "iopub.execute_input": "2026-06-17T09:08:17.390521Z", "iopub.status.busy": "2026-06-17T09:08:17.389958Z", "iopub.status.idle": "2026-06-17T09:08:17.397177Z", "shell.execute_reply": "2026-06-17T09:08:17.396426Z" } }, "outputs": [], "source": [ "class WBNeuron(brainpy.state.Neuron):\n", " def __init__(self, in_size, ENa=55. * u.mV, EK=-90. * u.mV, EL=-65. * u.mV,\n", " C=1.0 * u.uF, gNa=35. * u.msiemens, gK=9. * u.msiemens,\n", " gL=0.1 * u.msiemens, V_th=20. * u.mV, phi=5.0):\n", " super().__init__(in_size)\n", " self.ENa, self.EK, self.EL, self.C = ENa, EK, EL, C\n", " self.gNa, self.gK, self.gL = gNa, gK, gL\n", " self.V_th, self.phi = V_th, phi\n", "\n", " def init_state(self, *args, **kwargs):\n", " self.V = brainstate.HiddenState(\n", " -70. * u.mV + brainstate.random.randn(*self.varshape) * 20. * u.mV)\n", " self.h = brainstate.HiddenState(\n", " braintools.init.param(braintools.init.Constant(0.6), self.varshape))\n", " self.n = brainstate.HiddenState(\n", " braintools.init.param(braintools.init.Constant(0.3), self.varshape))\n", " self.spike = brainstate.HiddenState(\n", " braintools.init.param(lambda s: u.math.zeros(s, dtype=bool), self.varshape))\n", "\n", " def dh(self, h, t, V):\n", " alpha = 0.07 * u.math.exp(-(V / u.mV + 58) / 20)\n", " beta = 1 / (u.math.exp(-0.1 * (V / u.mV + 28)) + 1)\n", " return self.phi * (alpha * (1 - h) - beta * h) / u.ms\n", "\n", " def dn(self, n, t, V):\n", " alpha = -0.01 * (V / u.mV + 34) / (u.math.exp(-0.1 * (V / u.mV + 34)) - 1)\n", " beta = 0.125 * u.math.exp(-(V / u.mV + 44) / 80)\n", " return self.phi * (alpha * (1 - n) - beta * n) / u.ms\n", "\n", " def dV(self, V, t, h, n, Iext):\n", " m_alpha = -0.1 * (V / u.mV + 35) / (u.math.exp(-0.1 * (V / u.mV + 35)) - 1)\n", " m_beta = 4 * u.math.exp(-(V / u.mV + 60) / 18)\n", " m = m_alpha / (m_alpha + m_beta)\n", " INa = self.gNa * m ** 3 * h * (V - self.ENa)\n", " IK = self.gK * n ** 4 * (V - self.EK)\n", " IL = self.gL * (V - self.EL)\n", " return (-INa - IK - IL + self.sum_current_inputs(Iext, V)) / self.C\n", "\n", " def update(self, x=0. * u.uA):\n", " t = brainstate.environ.get('t')\n", " V = brainstate.nn.exp_euler_step(self.dV, self.V.value, t, self.h.value, self.n.value, x)\n", " h = brainstate.nn.exp_euler_step(self.dh, self.h.value, t, V)\n", " n = brainstate.nn.exp_euler_step(self.dn, self.n.value, t, V)\n", " self.spike.value = u.math.logical_and(self.V.value < self.V_th, V >= self.V_th)\n", " self.V.value, self.h.value, self.n.value = V, h, n\n", " return self.V.value" ] }, { "cell_type": "markdown", "id": "c13303d6", "metadata": {}, "source": [ "## The GABAergic synapse\n", "\n", "A first-order kinetic synapse gated by a sigmoid of the presynaptic voltage." ] }, { "cell_type": "code", "execution_count": 3, "id": "c46b6d9b", "metadata": { "execution": { "iopub.execute_input": "2026-06-17T09:08:17.399002Z", "iopub.status.busy": "2026-06-17T09:08:17.398833Z", "iopub.status.idle": "2026-06-17T09:08:17.403557Z", "shell.execute_reply": "2026-06-17T09:08:17.402700Z" } }, "outputs": [], "source": [ "class GABASyn(brainpy.state.Synapse):\n", " def __init__(self, in_size, alpha=12. / u.ms, beta=0.1 / u.ms):\n", " super().__init__(in_size=in_size)\n", " self.alpha, self.beta = alpha, beta\n", "\n", " def init_state(self, *args, **kwargs):\n", " self.g = brainstate.HiddenState(\n", " braintools.init.param(braintools.init.ZeroInit(), self.varshape))\n", "\n", " def update(self, pre_V):\n", " f_v = lambda v: 1 / (1 + u.math.exp(-v / u.mV / 2))\n", " ds = lambda s: self.alpha * f_v(pre_V) * (1 - s) - self.beta * s\n", " self.g.value = brainstate.nn.exp_euler_step(ds, self.g.value)\n", " return self.g.value" ] }, { "cell_type": "markdown", "id": "876ec8f5", "metadata": {}, "source": [ "## Wire the network\n", "\n", "The synapse's conductance `g` is prefetched into a `CurrentProj` with all-to-all\n", "(no self) inhibitory connectivity and a `COBA` output at the chloride reversal\n", "potential." ] }, { "cell_type": "code", "execution_count": 4, "id": "a7e81b8f", "metadata": { "execution": { "iopub.execute_input": "2026-06-17T09:08:17.405294Z", "iopub.status.busy": "2026-06-17T09:08:17.405139Z", "iopub.status.idle": "2026-06-17T09:08:17.409582Z", "shell.execute_reply": "2026-06-17T09:08:17.408736Z" } }, "outputs": [], "source": [ "class GammaNet(brainstate.nn.Module):\n", " def __init__(self, num=100):\n", " super().__init__()\n", " self.neu = WBNeuron(num)\n", " self.syn = GABASyn(num)\n", " self.proj = brainpy.state.CurrentProj(\n", " self.syn.prefetch('g'),\n", " comm=brainstate.nn.AllToAll(\n", " self.neu.varshape, self.neu.varshape,\n", " include_self=False, w_init=0.1 * u.msiemens / num),\n", " out=brainpy.state.COBA(E=-75. * u.mV),\n", " post=self.neu)\n", "\n", " def update(self, t, drive):\n", " with brainstate.environ.context(t=t):\n", " self.proj()\n", " self.syn(self.neu(drive))\n", " return self.neu.spike.value, self.neu.V.value[:5]" ] }, { "cell_type": "markdown", "id": "38d8ef92", "metadata": {}, "source": [ "## Simulate and visualize\n", "\n", "Drive every neuron with a 1 µA current for 500 ms. The raster reveals the\n", "synchronous gamma rhythm produced by mutual inhibition." ] }, { "cell_type": "code", "execution_count": 5, "id": "bca607ea", "metadata": { "execution": { "iopub.execute_input": "2026-06-17T09:08:17.411530Z", "iopub.status.busy": "2026-06-17T09:08:17.411251Z", "iopub.status.idle": "2026-06-17T09:08:21.757306Z", "shell.execute_reply": "2026-06-17T09:08:21.756477Z" } }, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "8e3bd002a4d0495f8f44bf954c5ea55a", "version_major": 2, "version_minor": 0 }, "text/plain": [ " 0%| | 0/50000 [00:00" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "net = GammaNet()\n", "brainstate.nn.init_all_states(net)\n", "\n", "with brainstate.environ.context(dt=0.01 * u.ms):\n", " times = u.math.arange(0. * u.ms, 500. * u.ms, brainstate.environ.get_dt())\n", " spikes, vs = brainstate.transform.for_loop(\n", " lambda t: net.update(t, 1.0 * u.uA), times,\n", " pbar=brainstate.transform.ProgressBar(10))\n", "\n", "fig, gs = braintools.visualize.get_figure(1, 2, 4.0, 5.0)\n", "fig.add_subplot(gs[0, 0])\n", "plt.plot(times.to_decimal(u.ms), vs.to_decimal(u.mV))\n", "plt.xlabel('Time (ms)'); plt.ylabel('V (mV)'); plt.title('Sample membrane potentials')\n", "fig.add_subplot(gs[0, 1])\n", "ti, ni = u.math.where(spikes)\n", "plt.plot(times[ti].to_decimal(u.ms), ni, 'k.', markersize=1)\n", "plt.xlabel('Time (ms)'); plt.ylabel('Neuron index'); plt.title('Population raster (gamma)')\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "b245e75d", "metadata": {}, "source": [ "## See also\n", "\n", "- {doc}`/concepts/model-anatomy` — the `Neuron`/`Synapse` base classes you\n", " subclassed here.\n", "- {doc}`../tutorials/03-ei-balanced-network` — a network from built-in models.\n", "- {doc}`/examples/brainpy-gallery` — more reproductions and complete scripts." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", 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