{ "cells": [ { "cell_type": "markdown", "id": "header", "metadata": {}, "source": [ "# Hodgkin-Huxley Neuron Model\n", "\n", "This tutorial demonstrates how to implement biologically realistic neuron models using BrainState. The Hodgkin-Huxley (HH) model is one of the most important models in computational neuroscience, describing action potential generation in neurons through ion channel dynamics.\n", "\n", "## Learning Objectives\n", "\n", "By the end of this tutorial, you will:\n", "- Understand the Hodgkin-Huxley neuron model\n", "- Use BrainUnit for physical units and dimensional analysis\n", "- Implement biophysically detailed neuron dynamics\n", "- Simulate and visualize neuron spiking activity\n", "- Use BrainState's `Dynamics` class for continuous-time models\n", "\n", "## The Hodgkin-Huxley Model\n", "\n", "The HH model describes the electrical activity of neurons through:\n", "\n", "1. **Membrane Voltage** (`V`): The electrical potential across the cell membrane\n", "2. **Ion Channels**:\n", " - **Sodium (Na⁺)** channels with activation (`m`) and inactivation (`h`) gates\n", " - **Potassium (K⁺)** channels with activation (`n`) gates\n", " - **Leak** channels\n", "\n", "**Governing Equations**:\n", "```\n", "C dV/dt = -I_Na - I_K - I_leak + I_ext\n", "I_Na = gNa * m³ * h * (V - ENa)\n", "I_K = gK * n⁴ * (V - EK)\n", "I_leak = gL * (V - EL)\n", "```\n", "\n", "Each gate variable (m, h, n) follows:\n", "```\n", "dx/dt = α(V) * (1 - x) - β(V) * x\n", "```" ] }, { "cell_type": "markdown", "id": "setup", "metadata": {}, "source": [ "## Setup and Imports" ] }, { "cell_type": "code", "execution_count": 1, "id": "imports", "metadata": { "ExecuteTime": { "end_time": "2025-10-11T10:09:30.939767Z", "start_time": "2025-10-11T10:09:26.977055Z" }, "execution": { "iopub.execute_input": "2026-05-30T17:37:53.442754Z", "iopub.status.busy": "2026-05-30T17:37:53.442480Z", "iopub.status.idle": "2026-05-30T17:37:58.351704Z", "shell.execute_reply": "2026-05-30T17:37:58.350987Z" } }, "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 brainunit as u\n", "import jax.numpy as jnp\n", "import matplotlib.pyplot as plt\n", "import brainpy\n", "import brainstate\n", "\n", "# Set random seed\n", "brainstate.random.seed(42)" ] }, { "cell_type": "markdown", "id": "model_section", "metadata": {}, "source": [ "## Implementing the HH Model\n", "\n", "We'll use BrainState's `nn.Dynamics` class for continuous-time dynamics and BrainUnit for physical units:" ] }, { "cell_type": "code", "execution_count": 2, "id": "hh_model", "metadata": { "ExecuteTime": { "end_time": "2025-10-11T10:09:30.973882Z", "start_time": "2025-10-11T10:09:30.961839Z" }, "execution": { "iopub.execute_input": "2026-05-30T17:37:58.354616Z", "iopub.status.busy": "2026-05-30T17:37:58.354043Z", "iopub.status.idle": "2026-05-30T17:37:58.372858Z", "shell.execute_reply": "2026-05-30T17:37:58.371645Z" } }, "outputs": [], "source": [ "class HH(brainstate.nn.Dynamics):\n", " \"\"\"Hodgkin-Huxley neuron model.\n", " \n", " A biophysically detailed model of action potential generation.\n", " \"\"\"\n", " \n", " def __init__(\n", " self,\n", " in_size,\n", " ENa=50. * u.mV, # Sodium reversal potential\n", " gNa=120. * u.mS / u.cm ** 2, # Sodium conductance\n", " EK=-77. * u.mV, # Potassium reversal potential \n", " gK=36. * u.mS / u.cm ** 2, # Potassium conductance\n", " EL=-54.387 * u.mV, # Leak reversal potential\n", " gL=0.03 * u.mS / u.cm ** 2, # Leak conductance\n", " V_th=20. * u.mV, # Spike threshold\n", " C=1.0 * u.uF / u.cm ** 2 # Membrane capacitance\n", " ):\n", " super().__init__(in_size)\n", " \n", " # Store parameters\n", " self.ENa = ENa\n", " self.EK = EK\n", " self.EL = EL\n", " self.gNa = gNa\n", " self.gK = gK\n", " self.gL = gL\n", " self.C = C\n", " self.V_th = V_th\n", " \n", " # Sodium channel activation (m)\n", " def m_alpha(self, V):\n", " return 1. / u.math.exprel(-(V / u.mV + 40) / 10)\n", " \n", " def m_beta(self, V):\n", " return 4.0 * jnp.exp(-(V / u.mV + 65) / 18)\n", " \n", " def m_inf(self, V):\n", " return self.m_alpha(V) / (self.m_alpha(V) + self.m_beta(V))\n", " \n", " def dm(self, m, t, V):\n", " return (self.m_alpha(V) * (1 - m) - self.m_beta(V) * m) / u.ms\n", " \n", " # Sodium channel inactivation (h)\n", " def h_alpha(self, V):\n", " return 0.07 * jnp.exp(-(V / u.mV + 65) / 20.)\n", " \n", " def h_beta(self, V):\n", " return 1 / (1 + jnp.exp(-(V / u.mV + 35) / 10))\n", " \n", " def h_inf(self, V):\n", " return self.h_alpha(V) / (self.h_alpha(V) + self.h_beta(V))\n", " \n", " def dh(self, h, t, V):\n", " return (self.h_alpha(V) * (1 - h) - self.h_beta(V) * h) / u.ms\n", " \n", " # Potassium channel activation (n)\n", " def n_alpha(self, V):\n", " return 0.1 / u.math.exprel(-(V / u.mV + 55) / 10)\n", " \n", " def n_beta(self, V):\n", " return 0.125 * jnp.exp(-(V / u.mV + 65) / 80)\n", " \n", " def n_inf(self, V):\n", " return self.n_alpha(V) / (self.n_alpha(V) + self.n_beta(V))\n", " \n", " def dn(self, n, t, V):\n", " return (self.n_alpha(V) * (1 - n) - self.n_beta(V) * n) / u.ms\n", " \n", " def init_state(self, batch_size=None):\n", " \"\"\"Initialize state variables at rest.\"\"\"\n", " self.V = brainstate.HiddenState(\n", " jnp.ones(self.varshape, brainstate.environ.dftype()) * -65. * u.mV\n", " )\n", " self.m = brainstate.HiddenState(self.m_inf(self.V.value))\n", " self.h = brainstate.HiddenState(self.h_inf(self.V.value))\n", " self.n = brainstate.HiddenState(self.n_inf(self.V.value))\n", " \n", " def dV(self, V, t, m, h, n, I):\n", " \"\"\"Membrane voltage dynamics.\"\"\"\n", " # Sodium current\n", " I_Na = (self.gNa * m * m * m * h) * (V - self.ENa)\n", " \n", " # Potassium current\n", " n2 = n * n\n", " I_K = (self.gK * n2 * n2) * (V - self.EK)\n", " \n", " # Leak current\n", " I_leak = self.gL * (V - self.EL)\n", " \n", " # Total current\n", " dVdt = (- I_Na - I_K - I_leak + I) / self.C\n", " return dVdt\n", " \n", " def update(self, x=0. * u.mA / u.cm ** 2):\n", " \"\"\"\n", " Update neuron state for one time step.\n", " \n", " Args:\n", " x: Input current density\n", " \n", " Returns:\n", " spike: Boolean spike indicator\n", " \"\"\"\n", " t = brainstate.environ.get('t')\n", " \n", " # Update voltage and gating variables using exponential Euler\n", " V = brainstate.nn.exp_euler_step(\n", " self.dV, self.V.value, t, \n", " self.m.value, self.h.value, self.n.value, x\n", " )\n", " m = brainstate.nn.exp_euler_step(self.dm, self.m.value, t, self.V.value)\n", " h = brainstate.nn.exp_euler_step(self.dh, self.h.value, t, self.V.value)\n", " n = brainstate.nn.exp_euler_step(self.dn, self.n.value, t, self.V.value)\n", " \n", " # Detect spike (threshold crossing)\n", " spike = jnp.logical_and(self.V.value < self.V_th, V >= self.V_th)\n", " \n", " # Update states\n", " self.V.value = V\n", " self.m.value = m\n", " self.h.value = h\n", " self.n.value = n\n", " \n", " return spike" ] }, { "cell_type": "markdown", "id": "simulation_section", "metadata": {}, "source": [ "## Simulating the HH Neuron\n", "\n", "### Create and Initialize Neuron" ] }, { "cell_type": "code", "execution_count": 3, "id": "create_neuron", "metadata": { "ExecuteTime": { "end_time": "2025-10-11T10:09:31.405628Z", "start_time": "2025-10-11T10:09:30.984238Z" }, "execution": { "iopub.execute_input": "2026-05-30T17:37:58.375023Z", "iopub.status.busy": "2026-05-30T17:37:58.374861Z", "iopub.status.idle": "2026-05-30T17:38:02.208126Z", "shell.execute_reply": "2026-05-30T17:38:02.207116Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Created (10,) HH neurons\n", "Initial membrane potential: -65. mV\n" ] } ], "source": [ "# Create a population of 10 HH neurons\n", "hh = HH(10)\n", "\n", "# Initialize states\n", "brainstate.nn.init_all_states(hh)\n", "\n", "# Set simulation parameters\n", "dt = 0.01 * u.ms\n", "brainstate.environ.set(dt=dt)\n", "\n", "print(f\"Created {hh.varshape} HH neurons\")\n", "print(f\"Initial membrane potential: {hh.V.value[0]}\")" ] }, { "cell_type": "markdown", "id": "run_section", "metadata": {}, "source": [ "### Define Simulation Function" ] }, { "cell_type": "code", "execution_count": 4, "id": "run_function", "metadata": { "ExecuteTime": { "end_time": "2025-10-11T10:09:31.412660Z", "start_time": "2025-10-11T10:09:31.409641Z" }, "execution": { "iopub.execute_input": "2026-05-30T17:38:02.210416Z", "iopub.status.busy": "2026-05-30T17:38:02.210055Z", "iopub.status.idle": "2026-05-30T17:38:02.213690Z", "shell.execute_reply": "2026-05-30T17:38:02.212919Z" } }, "outputs": [], "source": [ "def run(t, inp):\n", " \"\"\"Run neuron for one time step.\n", " \n", " Args:\n", " t: Current time\n", " inp: Input current\n", " \n", " Returns:\n", " V: Membrane voltage\n", " \"\"\"\n", " with brainstate.environ.context(t=t, dt=dt):\n", " hh(inp)\n", " return hh.V.value" ] }, { "cell_type": "markdown", "id": "simulate_section", "metadata": {}, "source": [ "### Run Simulation with Random Input" ] }, { "cell_type": "code", "execution_count": 5, "id": "simulate", "metadata": { "ExecuteTime": { "end_time": "2025-10-11T10:09:31.835315Z", "start_time": "2025-10-11T10:09:31.433111Z" }, "execution": { "iopub.execute_input": "2026-05-30T17:38:02.215384Z", "iopub.status.busy": "2026-05-30T17:38:02.215245Z", "iopub.status.idle": "2026-05-30T17:38:02.798535Z", "shell.execute_reply": "2026-05-30T17:38:02.797505Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Running simulation for 100. ms...\n" ] }, { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "360b9bdb538f4197bd4c3626cad8387a", "version_major": 2, "version_minor": 0 }, "text/plain": [ " 0%| | 0/10000 [00:00" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Convert to milliseconds and millivolts for plotting\n", "times_ms = times.to_decimal(u.ms)\n", "vs_mv = vs.to_decimal(u.mV)\n", "\n", "# Plot voltage traces for first 3 neurons\n", "fig, axes = plt.subplots(3, 1, figsize=(12, 8), sharex=True)\n", "\n", "for i in range(3):\n", " axes[i].plot(times_ms, vs_mv[:, i], linewidth=1.5)\n", " axes[i].set_ylabel(f'V{i} (mV)', fontsize=11)\n", " axes[i].grid(True, alpha=0.3)\n", " axes[i].axhline(y=20, color='r', linestyle='--', alpha=0.5, label='Threshold')\n", " if i == 0:\n", " axes[i].legend(fontsize=9)\n", "\n", "axes[2].set_xlabel('Time (ms)', fontsize=11)\n", "plt.suptitle('Hodgkin-Huxley Neuron Activity', fontsize=14, fontweight='bold')\n", "plt.tight_layout()\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "analysis_section", "metadata": {}, "source": [ "### Analyze Spiking Statistics" ] }, { "cell_type": "code", "execution_count": 7, "id": "spike_stats", "metadata": { "ExecuteTime": { "end_time": "2025-10-11T10:09:32.465674Z", "start_time": "2025-10-11T10:09:32.300998Z" }, "execution": { "iopub.execute_input": "2026-05-30T17:38:03.095259Z", "iopub.status.busy": "2026-05-30T17:38:03.094949Z", "iopub.status.idle": "2026-05-30T17:38:03.283235Z", "shell.execute_reply": "2026-05-30T17:38:03.282085Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Spike counts per neuron:\n", " Neuron 0: 6 spikes\n", " Neuron 1: 6 spikes\n", " Neuron 2: 6 spikes\n", " Neuron 3: 6 spikes\n", " Neuron 4: 6 spikes\n", "\n", "Average firing rate: 60.00 Hz\n" ] } ], "source": [ "# Detect spikes (threshold crossings)\n", "import numpy as np\n", "\n", "threshold = 20.0 # mV\n", "spike_times = []\n", "spike_counts = []\n", "\n", "for i in range(hh.varshape[0]):\n", " # Find threshold crossings\n", " above_threshold = vs_mv[:, i] > threshold\n", " spike_indices = np.where(np.diff(above_threshold.astype(int)) > 0)[0]\n", " \n", " spike_times.append(times_ms[spike_indices])\n", " spike_counts.append(len(spike_indices))\n", "\n", "print(\"Spike counts per neuron:\")\n", "for i, count in enumerate(spike_counts[:5]):\n", " print(f\" Neuron {i}: {count} spikes\")\n", "\n", "print(f\"\\nAverage firing rate: {np.mean(spike_counts) / (duration.to_decimal(u.ms) / 1000):.2f} Hz\")" ] }, { "cell_type": "markdown", "id": "raster_section", "metadata": {}, "source": [ "### Raster Plot" ] }, { "cell_type": "code", "execution_count": 8, "id": "raster_plot", "metadata": { "ExecuteTime": { "end_time": "2025-10-11T10:09:32.621283Z", "start_time": "2025-10-11T10:09:32.480680Z" }, "execution": { "iopub.execute_input": "2026-05-30T17:38:03.286109Z", "iopub.status.busy": "2026-05-30T17:38:03.285805Z", "iopub.status.idle": "2026-05-30T17:38:03.416759Z", "shell.execute_reply": "2026-05-30T17:38:03.415795Z" } }, "outputs": [ { "data": { "image/png": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.figure(figsize=(12, 6))\n", "\n", "for i, spikes in enumerate(spike_times[:10]):\n", " plt.scatter(spikes, [i] * len(spikes), marker='|', s=100, color='black')\n", "\n", "plt.xlabel('Time (ms)', fontsize=12)\n", "plt.ylabel('Neuron Index', fontsize=12)\n", "plt.title('Spike Raster Plot', fontsize=14, fontweight='bold')\n", "plt.ylim(-0.5, 9.5)\n", "plt.grid(True, alpha=0.3, axis='x')\n", "plt.tight_layout()\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "next_steps", "metadata": {}, "source": [ "## Next Steps\n", "\n", "- **Network Models**: Build networks of neurons with synapses\n", "- **Learning Rules**: Implement STDP and other plasticity mechanisms\n", "- **Brain Regions**: Model cortical columns, hippocampus, etc.\n", "- **Spiking Networks**: Training spiking neural networks\n", "\n", "## References\n", "\n", "- [Hodgkin & Huxley (1952) Original Paper](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1392413/)\n", "- [Neuronal Dynamics Book](https://neuronaldynamics.epfl.ch/)\n", "- [BrainState Documentation](https://brainx.chaobrain.com/brainstate/)\n", "- [BrainUnit Documentation](https://brainx.chaobrain.com/brainunit/)" ] } ], "metadata": { "kernelspec": { "display_name": "Ecosystem-py", "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": { "360b9bdb538f4197bd4c3626cad8387a": { "model_module": "@jupyter-widgets/controls", "model_module_version": "2.0.0", "model_name": "HBoxModel", "state": { "_dom_classes": [], "_model_module": "@jupyter-widgets/controls", "_model_module_version": "2.0.0", "_model_name": "HBoxModel", "_view_count": null, 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