Quick examination of the Reduced Wong Wang model

rww-solutions.ipynb

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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import sys\n",
      "# replace with path to your copy\n",
      "sys.path.append('/home/duke/dev/tvb/library')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 81
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from tvb.simulator.models import ReducedWongWang"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Create model with default parameters"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "rww = ReducedWongWang()\n",
      "rww.state_variable_range"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 83,
       "text": [
        "{'S': array([ 0.,  1.])}"
       ]
      }
     ],
     "prompt_number": 83
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Compute derivative with range of state variable values spanning reasonable range [0, 1] & null coupling"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# pretend \"network\" of 1000 nodes, varying state\n",
      "state = r_[0:1:1000j].reshape((1, -1))\n",
      "\n",
      "# same shape array, but all zero\n",
      "coupling = state*0.0\n",
      "\n",
      "# compute derivative\n",
      "deriv = rww.dfun(state, coupling)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 84
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Visualize 1D phase flow with stable & unstable fixed points"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# compute fixed points in 1D\n",
      "offset = diff(state[0,:2])/2\n",
      "stable = state[0,   c_[deriv[0,:-1]>0, deriv[0,1:]<0].all(axis=1)] + offset\n",
      "unstable = state[0, c_[deriv[0,:-1]<0, deriv[0,1:]>0].all(axis=1)] + offset\n",
      "\n",
      "plot(state[0], deriv[0], 'k')\n",
      "plot(stable, zeros(stable.shape), 'ko')\n",
      "plot(unstable, zeros(unstable.shape), 'kx')\n",
      "xlabel('s(t)')\n",
      "ylabel('d/dt s(t)')\n",
      "grid(True);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Wn332GXPnzuW9994jOjqawYMHEx0drQkEEhDUOpMCc54zH+q8mYuwsDD+/Oc/849//IMj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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x633bf90>"
       ]
      }
     ],
     "prompt_number": 85
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Wherever the derivative is negative, $s(t)$ will decrease; where positive, $s(t)$ increases. Therefore, we can label points where $s(t)=0$ as fixed points, such the $s(t)$ doesn't change. \n",
      "\n",
      "If $d s(t)/dt$ is positive to the left, we know the point is attractive, stable, and vice versa if negative. \n",
      "\n",
      "To verify this, we can simulate (from two different initial conditions) for a long time to obtain the steady state solution:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "T, S = rww.stationary_trajectory(initial_conditions=array([[0.01, 0.99]]), n_skip=100, n_step=int(5e4), dt=0.1)\n",
      "plot(T, S[:, 0, 0, :], 'k')\n",
      "xlabel('time (s)')\n",
      "ylabel('s(t)')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 80,
       "text": [
        "<matplotlib.text.Text at 0x634ee10>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x4cfcd10>"
       ]
      }
     ],
     "prompt_number": 80
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Of course, the value of the fixed point computed above should be close to the last value of the simulation:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(S[-1, 0, 0, 0] - stable[0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 79,
       "text": [
        "0.00042085555921959894"
       ]
      }
     ],
     "prompt_number": 79
    }
   ],
   "metadata": {}
  }
 ]
}