{ "cells": [ { "cell_type": "markdown", "id": "2912659fbdc55848", "metadata": {}, "source": [ "# Simulation and Analysis of Excitatory-Inhibitory Neuronal Networks (E-I Networks)\n", "\n", "This example demonstrates the implementation of a classical excitatory-inhibitory (E-I) neuronal network using the `braincell` framework. By constructing an E-I network composed of Hodgkin-Huxley model neurons, you will learn how to implement hierarchical modeling from single neurons to networks in `braincell`, and analyze the spike dynamics of the network.\n", "\n", "## Preparation\n", "First, ensure that the necessary libraries (`braincell`, `brainstate`, `brainunit`, `matplotlib`) are installed, and import the required modules:\n" ] }, { "cell_type": "code", "execution_count": 1, "id": "9d0cab5ba9e88f66", "metadata": { "ExecuteTime": { "end_time": "2025-10-13T09:24:18.860294Z", "start_time": "2025-10-13T09:24:18.857911Z" }, "execution": { "iopub.execute_input": "2026-05-25T10:03:15.738474Z", "iopub.status.busy": "2026-05-25T10:03:15.738303Z", "iopub.status.idle": "2026-05-25T10:03:19.260935Z", "shell.execute_reply": "2026-05-25T10:03:19.259928Z" } }, "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 brainunit as u\n", "import matplotlib.pyplot as plt\n", "import brainstate\n", "import braincell" ] }, { "cell_type": "markdown", "id": "7c40b3a6e85f45ae", "metadata": {}, "source": [ "## Code Explanation\n", "\n", "### Parameter Definition\n", "\n", "First, define the key biophysical parameters, which determine the electrophysiological properties of the neurons and the network size:\n" ] }, { "cell_type": "code", "execution_count": 2, "id": "c22585ffe4f19bdd", "metadata": { "ExecuteTime": { "end_time": "2025-10-13T09:24:18.963848Z", "start_time": "2025-10-13T09:24:18.960475Z" }, "execution": { "iopub.execute_input": "2026-05-25T10:03:19.263567Z", "iopub.status.busy": "2026-05-25T10:03:19.263145Z", "iopub.status.idle": "2026-05-25T10:03:19.267983Z", "shell.execute_reply": "2026-05-25T10:03:19.266919Z" } }, "outputs": [], "source": [ "# Neuron action potential firing threshold\n", "V_th = -20. * u.mV\n", "\n", "# Neuron membrane area\n", "area = 20000 * u.um **2\n", "area = area.in_unit(u.cm** 2)\n", "\n", "# Membrane capacitance\n", "Cm = (1 * u.uF * u.cm ** -2) * area # Total capacitance = specific capacitance × area" ] }, { "cell_type": "markdown", "id": "12e040fbf99cff1b", "metadata": {}, "source": [ "### Define HH single neuron model\n", "\n", "Use `SingleCompartment` to construct a single neuron based on the HH model, including sodium channel `INa`, potassium channel `IK`, and leak current `IL`. These channels jointly determine the firing properties of the neuron:\n" ] }, { "cell_type": "code", "execution_count": 3, "id": "c31fb6de04535794", "metadata": { "ExecuteTime": { "end_time": "2025-10-13T09:24:19.081543Z", "start_time": "2025-10-13T09:24:19.077656Z" }, "execution": { "iopub.execute_input": "2026-05-25T10:03:19.270259Z", "iopub.status.busy": "2026-05-25T10:03:19.270070Z", "iopub.status.idle": "2026-05-25T10:03:19.275371Z", "shell.execute_reply": "2026-05-25T10:03:19.274150Z" } }, "outputs": [], "source": [ "class HH(braincell.SingleCompartment):\n", " def __init__(self, in_size):\n", " # Initialize single-compartment neuron\n", " super().__init__(in_size, C=Cm, solver='ind_exp_euler')\n", "\n", " # Sodium channel (INa)\n", " self.na = braincell.ion.SodiumFixed(in_size, E=50. * u.mV)\n", " self.na.add(\n", " # Maximum conductance\n", " INa=braincell.channel.Na_TM1991(in_size, g_max=(100. * u.mS * u.cm **-2) * area, V_sh=-63. * u.mV)\n", " )\n", "\n", " # Potassium channel (IK)\n", " self.k = braincell.ion.PotassiumFixed(in_size, E=-90 * u.mV)\n", " self.k.add(\n", " # Maximum conductance\n", " IK=braincell.channel.K_TM1991(in_size, g_max=(30. * u.mS * u.cm** -2) * area, V_sh=-63. * u.mV)\n", " )\n", "\n", " # Leak current (IL)\n", " self.IL = braincell.channel.IL(\n", " in_size,\n", " E=-60. * u.mV,\n", " g_max=(5. * u.nS * u.cm **-2) * area # Maximum conductance\n", " )\n" ] }, { "cell_type": "markdown", "id": "a70a4069f943e1b", "metadata": {}, "source": [ "### Define E-I Network: Connections Between Excitatory and Inhibitory Neurons\n", "\n", "Construct a network composed of excitatory (E) and inhibitory (I) neurons to simulate the commonly observed E-I balance mechanism in cortical networks:\n" ] }, { "cell_type": "code", "execution_count": 4, "id": "e72dfafb35da107c", "metadata": { "ExecuteTime": { "end_time": "2025-10-13T09:24:19.192953Z", "start_time": "2025-10-13T09:24:19.189095Z" }, "execution": { "iopub.execute_input": "2026-05-25T10:03:19.278003Z", "iopub.status.busy": "2026-05-25T10:03:19.277734Z", "iopub.status.idle": "2026-05-25T10:03:19.283539Z", "shell.execute_reply": "2026-05-25T10:03:19.282553Z" } }, "outputs": [], "source": [ "class EINet(brainstate.nn.Module):\n", " def __init__(self):\n", " super().__init__()\n", " # Network scale\n", " self.n_exc = 3200\n", " self.n_inh = 800\n", " self.num = self.n_exc + self.n_inh # Total number of neurons: 4000\n", "\n", " # Initialize neuron population\n", " self.N = HH(self.num)\n", "\n", " # Excitatory synaptic projection\n", " self.E = brainpy.state.AlignPostProj(\n", " # Connection rule\n", " comm=brainstate.nn.EventFixedProb(\n", " self.n_exc, self.num, conn_num=0.02, # Connection probability\n", " conn_weight=6. * u.nS # Synaptic weight\n", " ),\n", " # Synaptic dynamics\n", " syn=brainpy.state.Expon(self.num, tau=5. * u.ms),\n", " # Postsynaptic effect\n", " out=brainpy.state.COBA(E=0. * u.mV),\n", " post=self.N # Projection target is neuron population N\n", " )\n", "\n", " # Inhibitory synaptic projection\n", " self.I = brainpy.state.AlignPostProj(\n", " # Connection rule\n", " comm=brainstate.nn.EventFixedProb(\n", " self.n_inh, self.num, conn_num=0.02,\n", " conn_weight=67. * u.nS\n", " ),\n", " # Synaptic dynamics\n", " syn=brainpy.state.Expon(self.num, tau=10. * u.ms),\n", " # Postsynaptic effect\n", " out=brainpy.state.COBA(E=-80. * u.mV),\n", " post=self.N # Projection target is neuron population N\n", " )\n", "\n", " def update(self, t):\n", " # Define network update rules over time\n", " with brainstate.environ.context(t=t):\n", " # Get spike signals at current time\n", " spk = self.N.spike.value\n", "\n", " # Spikes from excitatory neurons drive excitatory synaptic projection\n", " self.E(spk[:self.n_exc])\n", "\n", " # Spikes from inhibitory neurons drive inhibitory synaptic projection\n", " self.I(spk[self.n_exc:])\n", "\n", " # Neurons update their states after receiving synaptic inputs, return new spike signals\n", " spk = self.N(0. * u.nA)\n", "\n", " return spk\n" ] }, { "cell_type": "markdown", "id": "fa97af7731e5f5e8", "metadata": {}, "source": [ "### Run network simulation\n", "\n", "Initialize the network and run the simulation, recording the spike activity of neurons at each time point:\n" ] }, { "cell_type": "code", "execution_count": 5, "id": "655509eeebf82b38", "metadata": { "ExecuteTime": { "end_time": "2025-10-13T09:24:19.848845Z", "start_time": "2025-10-13T09:24:19.207206Z" }, "execution": { "iopub.execute_input": "2026-05-25T10:03:19.286392Z", "iopub.status.busy": "2026-05-25T10:03:19.285980Z", "iopub.status.idle": "2026-05-25T10:03:23.810213Z", "shell.execute_reply": "2026-05-25T10:03:23.809344Z" } }, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "b9114a26acf946c9ad8ede8e7688c2ef", "version_major": 2, "version_minor": 0 }, "text/plain": [ " 0%| | 0/1000 [00:00" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Extract spike times and neuron indices\n", "t_indices, n_indices = u.math.where(spikes)\n", "\n", "# Plot raster plot\n", "plt.scatter(times[t_indices] / u.ms, n_indices, s=1)\n", "plt.xlabel('Time (ms)') # X-axis: time\n", "plt.ylabel('Neuron index') # Y-axis: neuron index\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "55ef3973da8a763e", "metadata": {}, "source": [ "## Results Interpretation\n", "\n", "After running the code, you will see a raster plot, as shown below:\n", "\n", "![raster plot](../_static/cobahh2007.png)\n", "\n", "Each point represents a spike fired by a neuron at a specific time. Typical dynamics of an E-I network have the following characteristics:\n", "- Asynchronous firing: Neurons fire at dispersed times without obvious synchronous rhythms.\n", "- Sparse activity: Most neurons fire only a few times within 100 ms.\n", "- No burst synchronization: Rapid feedback from inhibitory synapses prevents large-scale synchronous bursting.\n", "\n", "These features indicate that the E-I network maintains a stable and physiologically realistic activity pattern through dynamic balance between excitatory and inhibitory neurons.\n", "\n", "## Extended Exercises\n", "\n", "- Adjust inhibitory synaptic weights and observe whether the network exhibits excessive synchronization or bursting activity.\n", "- Increase the proportion of excitatory neurons to analyze the impact of E-I imbalance on network dynamics.\n", "- Extend the simulation duration to see if the network maintains stable asynchronous activity.\n", "\n", "Through these extensions, you can gain deeper insights into the critical role of E-I balance in maintaining neural network function, and the flexibility of `braincell` in modeling complex networks.\n" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "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": "3.13.11" }, "widgets": { "application/vnd.jupyter.widget-state+json": { "state": { "117cc2e23eda4f65b6248e97bf9504a2": { "model_module": "@jupyter-widgets/base", "model_module_version": "2.0.0", "model_name": "LayoutModel", "state": { "_model_module": "@jupyter-widgets/base", "_model_module_version": "2.0.0", "_model_name": "LayoutModel", "_view_count": null, "_view_module": "@jupyter-widgets/base", "_view_module_version": "2.0.0", "_view_name": "LayoutView", "align_content": null, "align_items": null, "align_self": null, 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