jmxpearson/spiking_power.ipynb

## spiking_power.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Spiking power analysis\n",
    "\n",
    "This notebook illustrates a simple simulation-based spiking analysis\n",
    "\n",
    "We will assume:\n",
    "\n",
    "- a baseline firing period prior to each trial $\\lambda_0$\n",
    "- a baseline period of $T_0$ seconds\n",
    "- a stimulus presentation period on each trial $\\alpha \\lambda_0$\n",
    "- a stimulus period of $T$ seconds\n",
    "- a firing rate response that is more variable than Poisson; this is modeled as a trial-to-trial Gaussian noise on the log firing rate: $\\log \\lambda_i = \\log (\\alpha\\lambda) + \\epsilon$\n",
    "- a paired t-test on firing rates to determine significance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy.stats as stats\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "np.random.seed(12345)  # for reproducibility"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, let's go through it carefully and plot along the way:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# define some numbers\n",
    "baseline = 20.  # firing rate in Hz\n",
    "Tbaseline = 1.\n",
    "Tstim = 1.\n",
    "effect = 2  # fractional change from baseline\n",
    "trial_noise_std = 0.0  # trial-to-trial noise as a percentage\n",
    "Ntrials = 500  # number of trials for each stimulus\n",
    "Nexpt = 1000  # number of synthetic experiments to conduct\n",
    "alpha = 0.05  # nominal false positive rate"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Make some fake firing rates:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "baseline_rates = np.exp(np.log(baseline) * (1 + trial_noise_std * np.random.randn(Ntrials)))\n",
    "stim_rates = np.exp(np.log(effect * baseline) * (1 + trial_noise_std * np.random.randn(Ntrials)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e9b8c88>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(baseline_rates, label='baseline');\n",
    "plt.hist(stim_rates, label='stim');\n",
    "plt.legend();\n",
    "plt.title('Single trial firing rates');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and use these to generate fake spike counts:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "baseline_counts = stats.poisson.rvs(baseline_rates * Tbaseline)\n",
    "stim_counts = stats.poisson.rvs(stim_rates * Tstim)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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97Lvvvhx11FEAnHzyyUybNg2A6dOnc8IJJ3DVVVcxYMAARo0axe9+9zumT5/OsGHDqK+v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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e9c7390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(baseline_counts, label='baseline');\n",
    "plt.hist(stim_counts, label='stim');\n",
    "plt.legend();\n",
    "plt.title('Single trial spike counts');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculate firing rates from data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "obs_baseline_rates = baseline_counts / Tbaseline\n",
    "obs_stim_rates = stim_counts / Tstim"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-62.191816367 3.41747897325e-237\n",
      "True\n"
     ]
    }
   ],
   "source": [
    "tstat, pval = stats.ttest_rel(obs_baseline_rates, obs_stim_rates)\n",
    "print(tstat, pval)\n",
    "print(pval < alpha)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Now put it in a function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def run_expt(baseline, effect, Tbaseline, Tstim, trial_noise_std, Ntrials):\n",
    "    # run a single synthetic experiment, return p-value for paired t-test\n",
    "    # of baseline vs stim spike rate\n",
    "    \n",
    "    # random rates\n",
    "    baseline_rates = np.exp(np.log(baseline) * (1 + trial_noise_std * np.random.randn(Ntrials)))\n",
    "    stim_rates = np.exp(np.log(effect * baseline) * (1 + trial_noise_std * np.random.randn(Ntrials)))\n",
    "    \n",
    "    # random counts\n",
    "    baseline_counts = stats.poisson.rvs(baseline_rates * Tbaseline)\n",
    "    stim_counts = stats.poisson.rvs(stim_rates * Tstim)\n",
    "    \n",
    "    # observed rates\n",
    "    obs_baseline_rates = baseline_counts / Tbaseline\n",
    "    obs_stim_rates = stim_counts / Tstim\n",
    "    \n",
    "    # test\n",
    "    return stats.ttest_rel(obs_baseline_rates, obs_stim_rates)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def calc_power(alpha, Nexpt, exptfun, *args):\n",
    "    # calculate power for an experiment defined by false positive rate alpha and repeated Nexpt times\n",
    "    # exptfun returns an object with a pvalue\n",
    "    # args are passed on to exptfun\n",
    "    \n",
    "    pvals = np.array([exptfun(*args).pvalue for _ in range(Nexpt)])\n",
    "    \n",
    "    return np.mean(pvals < alpha)\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Now calculate power"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.68400000000000005"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alpha = 0.05\n",
    "Nexpt = 1000\n",
    "baseline = 20.  # firing rate in Hz\n",
    "Tbaseline = 1.\n",
    "Tstim = 1.\n",
    "effect = 1.2  # fractional change from baseline\n",
    "trial_noise_std = 0.10  # trial-to-trial noise as a percentage\n",
    "Ntrials = 50  # number of trials for each stimulus\n",
    "\n",
    "calc_power(alpha, Nexpt, run_expt, baseline, effect, Tbaseline, Tstim, trial_noise_std, Ntrials)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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