Spike triggered Average in Tensorflow.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Spike Triggered Average in TensorFlow\n",
    "\n",
    "The data is an experimentally recorded set of spikes recorded from the famous H1 motion-sensitive neuron of the fly (Calliphora vicina) from the lab of Dr Robert de Ruyter van Steveninck.\n",
    "\n",
    "This is a complete rewrite of non-tensorflow code in the Coursera course [Computational Neuroscience](https://www.coursera.org/learn/computational-neuroscience/home/welcome) by University of Washington.  I am thoroughly enjoying this course!\n",
    "\n",
    "Here we use TensorFlow to find out how the neuron is reacting to the data, to see what causes the neuron to trigger."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import pickle\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import tensorflow as tf\n",
    "sess = tf.InteractiveSession()\n",
    "\n",
    "FILENAME = 'data.pickle'\n",
    "\n",
    "with open(FILENAME, 'rb') as f:\n",
    "    data = pickle.load(f)\n",
    "\n",
    "stim = tf.constant(data['stim'])\n",
    "rho = tf.constant(data['rho'])\n",
    "sampling_period = 2 # The data was sampled at 500hz = 2ms\n",
    "window_size = 150 # Let's use a 300ms / sampling_period sliding window"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now have our data loaded into tensorflow as a constant, which means that we can easily 'run' our tensorflow graph.  For example, let's examine stim and rho:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Spike-train time-series = [0 0 0 ..., 0 0 0] \n",
      "Stimulus time-series    = [-111.94824219  -81.80664062   10.21972656 ...,    9.78515625   24.11132812\n",
      "   50.25390625]\n"
     ]
    }
   ],
   "source": [
    "print(\"Spike-train time-series =\", rho.eval(), \"\\nStimulus time-series    =\", stim.eval())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "rho is an binary array where a 1 indicates a spike.  Let's turn that into an array of indices of where the value is 1, but ignoring the first window_size elements.\n",
    "\n",
    "**Note:** We can use the [] and + operations on a tensorflow variable, and it correctly adds those operations to the graph.  This is equivalent to using  the tf.slice and tf.add operations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Time indices where there is a spike:\n",
      " [[   158]\n",
      " [   160]\n",
      " [   162]\n",
      " ..., \n",
      " [599936]\n",
      " [599941]\n",
      " [599947]]\n"
     ]
    }
   ],
   "source": [
    "spike_times = tf.where(tf.not_equal(rho[window_size:-1], 0)) + window_size\n",
    "print(\"Time indices where there is a spike:\\n\", spike_times.eval())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Spike triggered averaged is: [-0.33083048 -0.29083503 -0.23076012 -0.24636984 -0.10962767] (truncated)\n"
     ]
    }
   ],
   "source": [
    "def getStimWindow(index):\n",
    "    i = tf.cast(index, tf.int32)\n",
    "    return stim[i-window_size+1:i+1]\n",
    "stim_windows = tf.map_fn(lambda x: getStimWindow(x[0]), spike_times, dtype=tf.float64)\n",
    "spike_triggered_average = tf.reduce_mean(stim_windows, 0).eval()\n",
    "print(\"Spike triggered averaged is:\", spike_triggered_average[0:5], \"(truncated)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's plot this!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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WTBo17DuDC7KSZ5oJSwTGmJRT3dLJ/vr2Id8/ECqZFqexRGCMSTkrd9cDcOaU0cM+RzKt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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f4880ad4a58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "time = (np.arange(-window_size, 0) + 1) * sampling_period\n",
    "plt.plot(time, spike_triggered_average)\n",
    "plt.xlabel('Time (ms)')\n",
    "plt.ylabel('Stimulus')\n",
    "plt.title('Spike-Triggered Average')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's...  **beautiful**!\n",
    "\n",
    "What we are looking at here, is that we've discovered that our neuron is doing a [leaky integration](https://en.wikipedia.org/wiki/Leaky_integrator) of the stimulus.  And when that integration adds up to a certain value, it triggers."
   ]
  }
 ],
 "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": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}