dmeliza/learning-models.ipynb

## learning-models.ipynb
{
 "metadata": {
  "name": "",
  "signature": "sha256:a905b039fcac5aa3f055d6512113f7060d9baee4f585082816e9c5f2f2d1f571"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import matplotlib.pyplot as plt\n",
      "import numpy as np"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 47
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Learning Models\n",
      "\n",
      "Classical learning models predict what happens to the strength (V) of the association between CS and US on each trial. Let's build a learning model that can predict the following phenomena:\n",
      "\n",
      "- learning curve (defined by rate and asymptote)\n",
      "- extinction\n",
      "- blocking\n",
      "- overshadowing\n",
      "- inhibitory conditioning\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## The simplest learning rule\n",
      "\n",
      "$V^{(n)} - V^{(n-1)} = \\Delta V = \\alpha \\lambda$\n",
      "\n",
      "- $\\alpha$ - the strength or *associability* of the CS\n",
      "- $\\lambda$ - the strength of the US\n",
      "\n",
      "To see how this learning rule works, let's apply it iteratively:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "V_0 = 0\n",
      "alpha = 0.2\n",
      "lmbda = 100\n",
      "\n",
      "print (alpha * lmbda)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "20.0\n"
       ]
      }
     ],
     "prompt_number": 48
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "V = V_0\n",
      "for trial in range(0, 20):\n",
      "    plt.plot(trial, V, 'ok', hold=1)\n",
      "    V = V + alpha * lmbda\n",
      "    \n",
      "plt.xlabel(\"Trial\")\n",
      "plt.ylabel(\"Associative Strength (V)\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 49,
       "text": [
        "<matplotlib.text.Text at 0x10df8e0b8>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10de987f0>"
       ]
      }
     ],
     "prompt_number": 49
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Adding an asymptote\n",
      "\n",
      "One major problem with the simple model is that it predicts a linear growth in the CR, whereas what we observe is an asymptote. In other words, learning rate *decreases* with training. Here's a second try:\n",
      "\n",
      "$\\Delta V = \\alpha (\\lambda - V)$\n",
      "\n",
      "And let's see how that looks:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "V = V_0\n",
      "for trial in range(0, 20):\n",
      "    plt.plot(trial, V, 'ok', hold=1)\n",
      "    V = V + alpha * (lmbda - V)\n",
      "    \n",
      "plt.xlabel(\"Trial\")\n",
      "plt.ylabel(\"Associative Strength (V)\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's play around with $\\alpha$ and $\\lambda$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "V = V_0\n",
      "for trial in range(0, 20):\n",
      "    plt.plot(trial, V, 'ok', hold=1)\n",
      "    V = V + alpha * (lmbda - V)\n",
      "\n",
      "alpha = 0.4    \n",
      "V = V_0\n",
      "for trial in range(0, 20):\n",
      "    plt.plot(trial, V, 'or', hold=1)\n",
      "    V = V + alpha * (lmbda - V)\n",
      "    \n",
      "plt.xlabel(\"Trial\")\n",
      "plt.ylabel(\"Associative Strength (V)\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Rescorla-Wagner\n",
      "\n",
      "To explain blocking, overshadowing, and inhibitory conditioning, we need to expand the model to deal with compound stimuli. One simple way of doing this is to assume that each component of a compound stimulus contributes independently to the CR:\n",
      "\n",
      "$V = \\Sigma_i V_i$\n",
      "\n",
      "But that the change in associative strength for stimulus $i$ is determined by the **total** $V$:\n",
      "\n",
      "$\\Delta V_i = \\alpha_i (\\lambda - V)$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "V_1 = V_2 = 0\n",
      "alpha_1 = 0.2\n",
      "alpha_2 = 0.3\n",
      "lmbda = 100\n",
      "\n",
      "print (\"Delta V_1 = %f\" % (alpha_1 * (lmbda - (V_1 + V_2))))\n",
      "print (\"Delta V_2 = %f\" % (alpha_2 * (lmbda - (V_1 + V_2))))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Blocking\n",
      "\n",
      "Recall that the experimental animals were first trained on a single CS:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "V_1 = V_2 = 0\n",
      "for trial in range(0, 20):\n",
      "    plt.plot(trial, V_1, 'or', trial, V_2, 'ob', hold=1)\n",
      "    V_1 = V_1 + alpha_1 * (lmbda - (V_1 + V_2))\n",
      "    \n",
      "plt.xlabel(\"Trial\")\n",
      "plt.ylabel(\"Associative Strength (V)\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now they're trained on a compound CS. What value is $V_1$?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print(\"V1 = %f, V2 = %f\" % (V_1, V_2))\n",
      "for trial in range(0, 20):\n",
      "    plt.plot(trial, V_1, 'or', trial, V_2, 'ob', hold=1)\n",
      "    V_1 = V_1 + alpha_1 * (lmbda - (V_1 + V_2))\n",
      "    V_2 = V_2 + alpha_2 * (lmbda - (V_1 + V_2))\n",
      "    \n",
      "plt.xlabel(\"Trial\")\n",
      "plt.ylabel(\"Associative Strength (V)\")\n",
      "\n",
      "print(\"V1 = %f, V2 = %f\" % (V_1, V_2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "V_1 = V_2 = 0\n",
      "for trial in range(0, 20):\n",
      "    plt.plot(trial, V_1, 'or', trial, V_2, 'ob', trial, V_1 + V_2, 'ok', hold=1)\n",
      "    V_1 = V_1 + alpha_1 * (lmbda - (V_1 + V_2))\n",
      "    V_2 = V_2 + alpha_2 * (lmbda - (V_1 + V_2))\n",
      "    \n",
      "plt.xlabel(\"Trial\")\n",
      "plt.ylabel(\"Associative Strength (V)\")\n",
      "\n",
      "print(\"V1 = %f, V2 = %f\" % (V_1, V_2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Overshadowing\n",
      "\n",
      "We can see overshadowing in the previous example for the control animals: the associative strength of $CS_1$ and $CS_2$ is less than the total associative strength that either one would reach if conditioned alone.\n",
      "\n",
      "What happens if $\\alpha_1 \\gg \\alpha_2$?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "alpha_1 = 0.9\n",
      "alpha_2 = 0.1\n",
      "V_1 = V_2 = 0\n",
      "for trial in range(0, 20):\n",
      "    plt.plot(trial, V_1, 'or', trial, V_2, 'ob', trial, V_1 + V_2, 'ok', hold=1)\n",
      "    V_1 = V_1 + alpha_1 * (lmbda - (V_1 + V_2))\n",
      "    V_2 = V_2 + alpha_2 * (lmbda - (V_1 + V_2))\n",
      "    \n",
      "plt.xlabel(\"Trial\")\n",
      "plt.ylabel(\"Associative Strength (V)\")\n",
      "\n",
      "print(\"V1 = %f, V2 = %f\" % (V_1, V_2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Inhibitory conditioning\n",
      "\n",
      "Finally, let's consider the case where a CS is paired with the *absence* of a US:\n",
      "\n",
      "$\\Delta V_1 = \\alpha_1 (\\lambda - (V_1 + V_2))$\n",
      "\n",
      "$\\Delta V_2 = \\alpha_2 (0 - (V_1 + V_2))$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "alpha_1 = 0.2\n",
      "alpha_2 = 0.2\n",
      "V_1 = 100 \n",
      "V_2 = 0\n",
      "for trial in range(0, 20):\n",
      "    plt.plot(trial, V_1, 'or', trial, V_2, 'ob', trial, V_1 + V_2, 'ok', hold=1)\n",
      "    V_1 = V_1 + alpha_1 * (lmbda - (V_1 + V_2))\n",
      "    V_2 = V_2 + alpha_2 * (0 - (V_1 + V_2))\n",
      "    \n",
      "plt.xlabel(\"Trial\")\n",
      "plt.ylabel(\"Associative Strength (V)\")\n",
      "\n",
      "print(\"V1 = %f, V2 = %f\" % (V_1, V_2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Observe that $CS_2$ now has a negative associative strength. We can explain the results of a summation test by pairing $CS_2$ with a novel $CS_3$. We can explain the retardation test by comparing how long it takes to reach asymptote relative to controls"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "V_1 = 0\n",
      "for trial in range(0, 20):\n",
      "    plt.plot(trial, V_1, 'or', trial, V_2, 'ob', hold=1)\n",
      "    V_1 = V_1 + alpha_1 * (lmbda - V_1)\n",
      "    V_2 = V_2 + alpha_2 * (lmbda - V_2)\n",
      "    \n",
      "plt.xlabel(\"Trial\")\n",
      "plt.ylabel(\"Associative Strength (V)\")\n",
      "\n",
      "print(\"V1 = %f, V2 = %f\" % (V_1, V_2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}
	{
	"metadata": {
	"name": "",
	"signature": "sha256:a905b039fcac5aa3f055d6512113f7060d9baee4f585082816e9c5f2f2d1f571"
	},
	"nbformat": 3,
	"nbformat_minor": 0,
	"worksheets": [
	{
	"cells": [
	{
	"cell_type": "code",
	"collapsed": false,
	"input": [
	"%matplotlib inline\n",
	"import matplotlib.pyplot as plt\n",
	"import numpy as np"
	],
	"language": "python",
	"metadata": {},
	"outputs": [],
	"prompt_number": 47
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"# Learning Models\n",
	"\n",
	"Classical learning models predict what happens to the strength (V) of the association between CS and US on each trial. Let's build a learning model that can predict the following phenomena:\n",
	"\n",
	"- learning curve (defined by rate and asymptote)\n",
	"- extinction\n",
	"- blocking\n",
	"- overshadowing\n",
	"- inhibitory conditioning\n"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"## The simplest learning rule\n",
	"\n",
	"$V^{(n)} - V^{(n-1)} = \\Delta V = \\alpha \\lambda$\n",
	"\n",
	"- $\\alpha$ - the strength or associability of the CS\n",
	"- $\\lambda$ - the strength of the US\n",
	"\n",
	"To see how this learning rule works, let's apply it iteratively:"
	]
	},
	{
	"cell_type": "code",
	"collapsed": false,
	"input": [
	"V_0 = 0\n",
	"alpha = 0.2\n",
	"lmbda = 100\n",
	"\n",
	"print (alpha * lmbda)"
	],
	"language": "python",
	"metadata": {},
	"outputs": [
	{
	"output_type": "stream",
	"stream": "stdout",
	"text": [
	"20.0\n"
	]
	}
	],
	"prompt_number": 48
	},
	{
	"cell_type": "code",
	"collapsed": false,
	"input": [
	"V = V_0\n",
	"for trial in range(0, 20):\n",
	" plt.plot(trial, V, 'ok', hold=1)\n",
	" V = V + alpha * lmbda\n",
	" \n",
	"plt.xlabel(\"Trial\")\n",
	"plt.ylabel(\"Associative Strength (V)\")"
	],
	"language": "python",
	"metadata": {},
	"outputs": [
	{
	"metadata": {},
	"output_type": "pyout",
	"prompt_number": 49,
	"text": [
	"<matplotlib.text.Text at 0x10df8e0b8>"
	]
	},
	{
	"metadata": {},
	"output_type": "display_data",
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	"text": [
	"<matplotlib.figure.Figure at 0x10de987f0>"
	]
	}
	],
	"prompt_number": 49
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"## Adding an asymptote\n",
	"\n",
	"One major problem with the simple model is that it predicts a linear growth in the CR, whereas what we observe is an asymptote. In other words, learning rate decreases with training. Here's a second try:\n",
	"\n",
	"$\\Delta V = \\alpha (\\lambda - V)$\n",
	"\n",
	"And let's see how that looks:"
	]
	},
	{
	"cell_type": "code",
	"collapsed": false,
	"input": [
	"V = V_0\n",
	"for trial in range(0, 20):\n",
	" plt.plot(trial, V, 'ok', hold=1)\n",
	" V = V + alpha * (lmbda - V)\n",
	" \n",
	"plt.xlabel(\"Trial\")\n",
	"plt.ylabel(\"Associative Strength (V)\")"
	],
	"language": "python",
	"metadata": {},
	"outputs": []
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"Let's play around with $\\alpha$ and $\\lambda$:"
	]
	},
	{
	"cell_type": "code",
	"collapsed": false,
	"input": [
	"V = V_0\n",
	"for trial in range(0, 20):\n",
	" plt.plot(trial, V, 'ok', hold=1)\n",
	" V = V + alpha * (lmbda - V)\n",
	"\n",
	"alpha = 0.4 \n",
	"V = V_0\n",
	"for trial in range(0, 20):\n",
	" plt.plot(trial, V, 'or', hold=1)\n",
	" V = V + alpha * (lmbda - V)\n",
	" \n",
	"plt.xlabel(\"Trial\")\n",
	"plt.ylabel(\"Associative Strength (V)\")"
	],
	"language": "python",
	"metadata": {},
	"outputs": []
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"## Rescorla-Wagner\n",
	"\n",
	"To explain blocking, overshadowing, and inhibitory conditioning, we need to expand the model to deal with compound stimuli. One simple way of doing this is to assume that each component of a compound stimulus contributes independently to the CR:\n",
	"\n",
	"$V = \\Sigma_i V_i$\n",
	"\n",
	"But that the change in associative strength for stimulus $i$ is determined by the total $V$:\n",
	"\n",
	"$\\Delta V_i = \\alpha_i (\\lambda - V)$"
	]
	},
	{
	"cell_type": "code",
	"collapsed": false,
	"input": [
	"V_1 = V_2 = 0\n",
	"alpha_1 = 0.2\n",
	"alpha_2 = 0.3\n",
	"lmbda = 100\n",
	"\n",
	"print (\"Delta V_1 = %f\" % (alpha_1 * (lmbda - (V_1 + V_2))))\n",
	"print (\"Delta V_2 = %f\" % (alpha_2 * (lmbda - (V_1 + V_2))))"
	],
	"language": "python",
	"metadata": {},
	"outputs": []
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"### Blocking\n",
	"\n",
	"Recall that the experimental animals were first trained on a single CS:"
	]
	},
	{
	"cell_type": "code",
	"collapsed": false,
	"input": [
	"V_1 = V_2 = 0\n",
	"for trial in range(0, 20):\n",
	" plt.plot(trial, V_1, 'or', trial, V_2, 'ob', hold=1)\n",
	" V_1 = V_1 + alpha_1 * (lmbda - (V_1 + V_2))\n",
	" \n",
	"plt.xlabel(\"Trial\")\n",
	"plt.ylabel(\"Associative Strength (V)\")"
	],
	"language": "python",
	"metadata": {},
	"outputs": []
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"Now they're trained on a compound CS. What value is $V_1$?"
	]
	},
	{
	"cell_type": "code",
	"collapsed": false,
	"input": [
	"print(\"V1 = %f, V2 = %f\" % (V_1, V_2))\n",
	"for trial in range(0, 20):\n",
	" plt.plot(trial, V_1, 'or', trial, V_2, 'ob', hold=1)\n",
	" V_1 = V_1 + alpha_1 * (lmbda - (V_1 + V_2))\n",
	" V_2 = V_2 + alpha_2 * (lmbda - (V_1 + V_2))\n",
	" \n",
	"plt.xlabel(\"Trial\")\n",
	"plt.ylabel(\"Associative Strength (V)\")\n",
	"\n",
	"print(\"V1 = %f, V2 = %f\" % (V_1, V_2))"
	],
	"language": "python",
	"metadata": {},
	"outputs": []
	},
	{
	"cell_type": "code",
	"collapsed": false,
	"input": [
	"V_1 = V_2 = 0\n",
	"for trial in range(0, 20):\n",
	" plt.plot(trial, V_1, 'or', trial, V_2, 'ob', trial, V_1 + V_2, 'ok', hold=1)\n",
	" V_1 = V_1 + alpha_1 * (lmbda - (V_1 + V_2))\n",
	" V_2 = V_2 + alpha_2 * (lmbda - (V_1 + V_2))\n",
	" \n",
	"plt.xlabel(\"Trial\")\n",
	"plt.ylabel(\"Associative Strength (V)\")\n",
	"\n",
	"print(\"V1 = %f, V2 = %f\" % (V_1, V_2))"
	],
	"language": "python",
	"metadata": {},
	"outputs": []
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"### Overshadowing\n",
	"\n",
	"We can see overshadowing in the previous example for the control animals: the associative strength of $CS_1$ and $CS_2$ is less than the total associative strength that either one would reach if conditioned alone.\n",
	"\n",
	"What happens if $\\alpha_1 \\gg \\alpha_2$?"
	]
	},
	{
	"cell_type": "code",
	"collapsed": false,
	"input": [
	"alpha_1 = 0.9\n",
	"alpha_2 = 0.1\n",
	"V_1 = V_2 = 0\n",
	"for trial in range(0, 20):\n",
	" plt.plot(trial, V_1, 'or', trial, V_2, 'ob', trial, V_1 + V_2, 'ok', hold=1)\n",
	" V_1 = V_1 + alpha_1 * (lmbda - (V_1 + V_2))\n",
	" V_2 = V_2 + alpha_2 * (lmbda - (V_1 + V_2))\n",
	" \n",
	"plt.xlabel(\"Trial\")\n",
	"plt.ylabel(\"Associative Strength (V)\")\n",
	"\n",
	"print(\"V1 = %f, V2 = %f\" % (V_1, V_2))"
	],
	"language": "python",
	"metadata": {},
	"outputs": []
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"### Inhibitory conditioning\n",
	"\n",
	"Finally, let's consider the case where a CS is paired with the absence of a US:\n",
	"\n",
	"$\\Delta V_1 = \\alpha_1 (\\lambda - (V_1 + V_2))$\n",
	"\n",
	"$\\Delta V_2 = \\alpha_2 (0 - (V_1 + V_2))$"
	]
	},
	{
	"cell_type": "code",
	"collapsed": false,
	"input": [
	"alpha_1 = 0.2\n",
	"alpha_2 = 0.2\n",
	"V_1 = 100 \n",
	"V_2 = 0\n",
	"for trial in range(0, 20):\n",
	" plt.plot(trial, V_1, 'or', trial, V_2, 'ob', trial, V_1 + V_2, 'ok', hold=1)\n",
	" V_1 = V_1 + alpha_1 * (lmbda - (V_1 + V_2))\n",
	" V_2 = V_2 + alpha_2 * (0 - (V_1 + V_2))\n",
	" \n",
	"plt.xlabel(\"Trial\")\n",
	"plt.ylabel(\"Associative Strength (V)\")\n",
	"\n",
	"print(\"V1 = %f, V2 = %f\" % (V_1, V_2))"
	],
	"language": "python",
	"metadata": {},
	"outputs": []
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"Observe that $CS_2$ now has a negative associative strength. We can explain the results of a summation test by pairing $CS_2$ with a novel $CS_3$. We can explain the retardation test by comparing how long it takes to reach asymptote relative to controls"
	]
	},
	{
	"cell_type": "code",
	"collapsed": false,
	"input": [
	"V_1 = 0\n",
	"for trial in range(0, 20):\n",
	" plt.plot(trial, V_1, 'or', trial, V_2, 'ob', hold=1)\n",
	" V_1 = V_1 + alpha_1 * (lmbda - V_1)\n",
	" V_2 = V_2 + alpha_2 * (lmbda - V_2)\n",
	" \n",
	"plt.xlabel(\"Trial\")\n",
	"plt.ylabel(\"Associative Strength (V)\")\n",
	"\n",
	"print(\"V1 = %f, V2 = %f\" % (V_1, V_2))"
	],
	"language": "python",
	"metadata": {},
	"outputs": []
	},
	{
	"cell_type": "code",
	"collapsed": false,
	"input": [],
	"language": "python",
	"metadata": {},
	"outputs": []
	},
	{
	"cell_type": "code",
	"collapsed": false,
	"input": [],
	"language": "python",
	"metadata": {},
	"outputs": []
	}
	],
	"metadata": {}
	}
	]
	}