{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solving for $d$\n",
    "\n",
    "- Minimize the average error over all $x$, i.e.,\n",
    "\n",
    "$ E = \\frac{1}{2}\\int_{-1}^1 (x-\\hat{x})^2 \\; dx $\n",
    "\n",
    "- Substitute for $\\hat{x}$:\n",
    "\n",
    "$ \n",
    "\\begin{align}\n",
    "E = \\frac{1}{2}\\int_{-1}^1 \\left(x-\\sum_i^N a_i d_i \\right)^2 \\; dx \n",
    "\\end{align}\n",
    "$\n",
    "\n",
    "- Take the derivative with respect to $d_i$:\n",
    "\n",
    "$\n",
    "\\begin{align}\n",
    "{{\\partial E} \\over {\\partial d_i}} &= {1 \\over 2} \\int_{-1}^1 2 \\left[ x-\\sum_j a_j d_j \\right] (-a_i) \\; dx \\\\\n",
    "{{\\partial E} \\over {\\partial d_i}} &= - \\int_{-1}^1 a_i x \\; dx + \\int_{-1}^1 \\sum_j a_j d_j a_i \\; dx \n",
    "\\end{align}\n",
    "$\n",
    "\n",
    "- At the minimum (i.e. smallest error), $ {{\\partial E} \\over {\\partial d_i}} = 0$\n",
    "\n",
    "$\n",
    "\\begin{align}\n",
    "\\int_{-1}^1 a_i x \\; dx &= \\int_{-1}^1 \\sum_j(a_j d_j a_i) \\; dx \\\\\n",
    "\\int_{-1}^1 a_i x \\; dx &= \\sum_j \\left(\\int_{-1}^1 a_i a_j \\; dx\\right)d_j \n",
    "\\end{align}\n",
    "$\n",
    "\n",
    "- That's a system of $N$ equations and $N$ unknowns\n",
    "- In fact, we can rewrite this in matrix form\n",
    "    \n",
    "$ \\Upsilon = \\Gamma d $\n",
    "\n",
    "where"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n",
      "\r",
      "Simulation finished in 0:00:01.                                                 \n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING: pylab import has clobbered these variables: ['piecewise']\n",
      "`%matplotlib` prevents importing * from pylab and numpy\n"
     ]
    },
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109e5a1d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "__author__ = 'Chow'\n",
    "%pylab inline\n",
    "import nengo\n",
    "from nengo.utils.functions import piecewise\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "model = nengo.Network()\n",
    "\n",
    "with model:\n",
    "    #stim = nengo.Node(lambda t: 5*np.sin(5*t))\n",
    "    stim = nengo.Node(piecewise({0.04:1, 1:0}))\n",
    "    ensA = nengo.Ensemble(200, dimensions=1)\n",
    "\n",
    "    def feedback(x):\n",
    "        return x\n",
    "\n",
    "    def input_func(u):\n",
    "        return 0.05 * u\n",
    "\n",
    "    nengo.Connection(stim, ensA, function=input_func, synapse=0.005)\n",
    "    nengo.Connection(ensA, ensA, function=feedback, synapse=0.05)\n",
    "\n",
    "    stim_p = nengo.Probe(stim)\n",
    "    ensA_p = nengo.Probe(ensA, synapse=.01)\n",
    "\n",
    "# model.config[nengo.Ensemble].neuron_type = nengo.LIFRate()\n",
    "\n",
    "sim = nengo.Simulator(model)\n",
    "sim.run(1.5)\n",
    "\n",
    "plt.plot(sim.trange(), sim.data[ensA_p], label=\"$\\hat{x}$\")\n",
    "plt.plot(sim.trange(), sim.data[stim_p], label=\"$x$\")\n",
    "plt.legend()\n",
    "# plt.ylim(-.2,1.5);\n",
    "\n",
    "\n",
    "plt.show()\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.5"
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 "nbformat_minor": 0
}