poisson_process.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.stats import poisson\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "RANDOM_SEED = 123"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Homogeneous Poisson Process"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Firing rate:\n",
    "\n",
    "$$r = \\frac{n}{T}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1000\n"
     ]
    }
   ],
   "source": [
    "r = 100 # firing rate, Hz\n",
    "T = 10 # time, seconds\n",
    "n = r*T # amount of spikes\n",
    "print(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "data_poisson =  poisson.rvs(n, size=1000, random_state=RANDOM_SEED)  \n",
    "    \n",
    "plt.hist(data_poisson, bins=50, edgeColor='black')    \n",
    "plt.xlabel('n')    \n",
    "plt.ylabel('Count')    \n",
    "plt.title('Poisson Distribution')    \n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1000.235"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data_poisson.mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1011.8257749999999"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data_poisson.var()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(RANDOM_SEED)\n",
    "\n",
    "r = 100 # firing rate, Hz\n",
    "T = 1 # time, seconds\n",
    "n = r*T # amount of spikes\n",
    "dt = 1e-3\n",
    "\n",
    "x = np.random.random(size=int(T/dt))\n",
    "spikes = (x <= r*1e-3).astype(int)\n",
    "time = np.linspace(0, T, int(T/dt))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 1440x216 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(20,3))\n",
    "xcoords = time[spikes == 1]\n",
    "for xc in xcoords:\n",
    "    plt.axvline(xc, color='black')\n",
    "plt.yticks([0,1])\n",
    "plt.xlim([0,T])\n",
    "plt.xlabel('Time, s')\n",
    "plt.title('Homogeneous Poisson Process')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "98"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spikes.sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "isi = np.diff(time[spikes == 1])\n",
    "\n",
    "plt.hist(isi, bins=15, edgeColor='black')\n",
    "plt.title('Distribution of ISI')\n",
    "plt.xlabel('ISI')\n",
    "plt.ylabel('Count')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}