├── .gitignore ├── LICENSE ├── README.md └── code ├── MATLAB ├── PoissonProcessesTutorial.m ├── SignalAndNoiseTutorial.m └── SignalAndNoiseTutorial[Conflict].m └── Python ├── PoissonProcessesTutorial.ipynb ├── ProbabilityDistributionsTutorial.ipynb └── ReceptiveFields.ipynb /.gitignore: -------------------------------------------------------------------------------- 1 | .DS_Store 2 | 3 | # Byte-compiled / optimized / DLL files 4 | __pycache__/ 5 | *.py[cod] 6 | *$py.class 7 | 8 | # C extensions 9 | *.so 10 | 11 | # Distribution / packaging 12 | .Python 13 | build/ 14 | develop-eggs/ 15 | dist/ 16 | downloads/ 17 | eggs/ 18 | .eggs/ 19 | lib/ 20 | lib64/ 21 | parts/ 22 | sdist/ 23 | var/ 24 | wheels/ 25 | *.egg-info/ 26 | .installed.cfg 27 | *.egg 28 | MANIFEST 29 | 30 | # PyInstaller 31 | # Usually these files are written by a python script from a template 32 | # before PyInstaller builds the exe, so as to inject date/other infos into it. 33 | *.manifest 34 | *.spec 35 | 36 | # Installer logs 37 | pip-log.txt 38 | pip-delete-this-directory.txt 39 | 40 | # Unit test / coverage reports 41 | htmlcov/ 42 | .tox/ 43 | .coverage 44 | .coverage.* 45 | .cache 46 | nosetests.xml 47 | coverage.xml 48 | *.cover 49 | .hypothesis/ 50 | .pytest_cache/ 51 | 52 | # Translations 53 | *.mo 54 | *.pot 55 | 56 | # Django stuff: 57 | *.log 58 | local_settings.py 59 | db.sqlite3 60 | 61 | # Flask stuff: 62 | instance/ 63 | .webassets-cache 64 | 65 | # Scrapy stuff: 66 | .scrapy 67 | 68 | # Sphinx documentation 69 | docs/_build/ 70 | 71 | # PyBuilder 72 | target/ 73 | 74 | # Jupyter Notebook 75 | .ipynb_checkpoints 76 | 77 | # pyenv 78 | .python-version 79 | 80 | # celery beat schedule file 81 | celerybeat-schedule 82 | 83 | # SageMath parsed files 84 | *.sage.py 85 | 86 | # Environments 87 | .env 88 | .venv 89 | env/ 90 | venv/ 91 | ENV/ 92 | env.bak/ 93 | venv.bak/ 94 | 95 | # Spyder project settings 96 | .spyderproject 97 | .spyproject 98 | 99 | # Rope project settings 100 | .ropeproject 101 | 102 | # mkdocs documentation 103 | /site 104 | 105 | # mypy 106 | .mypy_cache/ 107 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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It is safest 630 | to attach them to the start of each source file to most effectively 631 | state the exclusion of warranty; and each file should have at least 632 | the "copyright" line and a pointer to where the full notice is found. 633 | 634 | 635 | Copyright (C) 636 | 637 | This program is free software: you can redistribute it and/or modify 638 | it under the terms of the GNU General Public License as published by 639 | the Free Software Foundation, either version 3 of the License, or 640 | (at your option) any later version. 641 | 642 | This program is distributed in the hope that it will be useful, 643 | but WITHOUT ANY WARRANTY; without even the implied warranty of 644 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 645 | GNU General Public License for more details. 646 | 647 | You should have received a copy of the GNU General Public License 648 | along with this program. If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # NeuroscienceTeachingCode 2 | This is a repository containing MATLAB and Python code used in teaching graduate students in neuroscience. 3 | 4 | The directory structure is organized according to language in the ./code subdirectories. Python code is contained within Jupyter notebooks, which were originally written to run on Google's Collaboratory framework. -------------------------------------------------------------------------------- /code/MATLAB/PoissonProcessesTutorial.m: -------------------------------------------------------------------------------- 1 | % PoissonProcessesTutorial.m 2 | % by Mike Manookin 11/2019 3 | 4 | 5 | % Before we get into Poisson distributions and the processes they describe, 6 | % we need to first introduce Gaussian distributions. The reason is that 7 | % Poisson distributions are just a special type of Gaussian distribution. 8 | % This is the equation for a Gaussian distribution: 9 | 10 | % f(x) = 1/sqrt(2*pi*sigma^2) * exp(-(x - mu).^2/(2*sigma^2)) 11 | % mu is the mean of the distribution 12 | 13 | % sigma is the standard deviation of the distribution 14 | 15 | % It's alright if equations aren't your thing. The thing to remember is 16 | % that you have some distribution of your data and that there is a 17 | % mean/average (mu) and another parameter describing how spread out the 18 | % distribution is---the standard deviation (sigma). The following code 19 | % allows us to play with those two parameters and see how it affects the 20 | % Gaussian distribution. Go ahead and change 'mu' and 'sigma' and then run 21 | % the code to see how the distribution changes. 22 | 23 | % A Poisson distribution is a special type of Gaussian distribution in which the 24 | % mean and variance are equal. Many physical and biological phenomena are well- 25 | % described by Poisson distributions. 26 | 27 | % Let's make our own Gaussian function. We can define the function as a 28 | % separate piece of code or define an 'anonymous function' in the current 29 | % code. I'll do the latter to show you how it works. (It can come in really 30 | % handy.) 31 | gaussfun = @(p,x)(1/sqrt(2*pi*p(2)^2) * exp(-(x - p(1)).^2/(2*p(2)^2))); 32 | 33 | % We can now pass x-axis values (x) and parameters for the Gaussian (p) to 34 | % the anonymous function and get our output. 35 | % Note: 36 | % p(1) = mean 37 | % p(2) = standard deviation 38 | 39 | % First, let's try to unpack the Poisson distribution a bit by making a generic 40 | % Gaussian distribution with a mean (mu) and standard deviation (sigma). Choose 41 | % whatever numbers you like. 42 | mu = 2.0; 43 | sigma = 2.6; 44 | 45 | % Define the x-axis. 46 | x = -10:0.1:10; 47 | 48 | % Pass the parameters to our anonymous function: 49 | y = gaussfun([mu,sigma],x); 50 | 51 | figure(100); clf; 52 | hold on 53 | plot(x,y); 54 | hold off; 55 | xlabel('Gaussian distribution'); 56 | 57 | %% 58 | % Instead of using a function, we can also generate a bunch of random 59 | % numbers that have a Gaussian distribution using MATLAB's built-in 'randn' 60 | % function. If you want more information about any function type 'help 61 | % functionName' or 'doc functionName' into the command window (e.g., 62 | % >> doc randn 63 | 64 | % Generate 10^5 random numbers. 65 | y2 = sigma * randn(1,1e5) + mu; 66 | 67 | % Let's plot the histogram of the numbers we generated and then plot the 68 | % function as well. 69 | 70 | figure(100); clf; 71 | hold on 72 | histogram(y2, 50, 'Normalization', 'probability'); axis tight; 73 | plot(x, y/max(y)*max(ylim), 'b', 'LineWidth', 2); % Need to scale to the same maximum value as the histogram 74 | hold off; 75 | xlabel('Gaussian distribution'); ylabel('probability'); 76 | 77 | %% 78 | % Go ahead and play around with the parameters for the Gaussian distribution. 79 | % What you'll notice as you change the parameters, is that the values along 80 | % the x-axis change, but the general, bell-curve, shape stays the same. 81 | 82 | % A Poisson distribution is just a special case in which the mean and 83 | % variance are equal. The standard deviation ( sigma ) is the square root 84 | % of the variance and, since the mean and variance are equal, it is also 85 | % the square root of the mean ( mu ). 86 | 87 | % Define the mean/variance. 88 | mu = 3.0; 89 | 90 | % The standard deviation (sigma) is just the square root of this value. 91 | sigma = sqrt(mu); 92 | 93 | % Let's do the same thing now for our Poisson distribution. 94 | y = gaussfun([mu,sigma],x); 95 | y2 = sigma * randn(1,1e5) + mu; 96 | 97 | % Let's plot the histogram of the numbers we generated and then plot the 98 | % function as well. 99 | 100 | figure(100); clf; 101 | hold on 102 | histogram(y2, 50, 'Normalization', 'probability'); axis tight; 103 | plot(x, y/max(y)*max(ylim), 'b', 'LineWidth', 2); % Need to scale to the same maximum value as the histogram 104 | hold off; 105 | xlabel('Gaussian distribution'); ylabel('probability'); 106 | 107 | % 108 | % All that's happened is that we've constrained the Gaussian distribution 109 | % to have only one free parameter instead of two. Wasn't that easy?! 110 | 111 | % So, why did I make you learn this then? Well, it turns out that there are 112 | % many physical processes that are well described with Poisson statistics. 113 | % Here are a few examples that are relevant to you and me: 114 | % 115 | % 1) In physics, the frequency of photon arrivals. 116 | % 2) Neural spike counts (not exactly due to the refractory period)... 117 | % 3) Frequency of vesicle release at synapses 118 | 119 | % Let's look at concrete example. Photon arrivals at a detector obey 120 | % Poisson statistics so let's simulate a distribution of photon counts 121 | % based on a mean values. Now, if we're talking about photon counts in, 122 | % say, a 100 ms window, we aren't going to observe any counts that are less 123 | % than zero, so we can clip the negative values out. 124 | 125 | mu = 5.0; 126 | sigma = sqrt(mu); 127 | 128 | y2 = sigma * randn(1,1e5) + mu; 129 | 130 | % Photon counts are discrete (integers), so let's round. 131 | y2 = round(y2); 132 | 133 | % We can't have negative photon counts, so set them to zero. 134 | y2(y2 < 0) = 0; 135 | 136 | % Let's get the underlying function too. 137 | x = min(y2) : 0.1 : max(y2); 138 | y = gaussfun([mu,sigma],x); 139 | 140 | figure(100); clf; 141 | hold on 142 | histogram(y2, max(y2)+1, 'Normalization', 'probability'); axis tight; 143 | plot(x(x>=0), y(x>=0)/max(y)*max(ylim), 'b', 'LineWidth', 2); % Need to scale to the same maximum value as the histogram 144 | hold off; 145 | xlabel('Gaussian distribution'); ylabel('probability'); 146 | 147 | % That's all there is to it. We can see the probability (y-axis) that we 148 | % observe x number of photons (x-axis). This type of visualization can be 149 | % very helpful as we try to understand neural systems, etc. There are many 150 | % examples of this, but for the time being I would like you to become 151 | % comfortable with this concept. So, please play around with the code to 152 | % familiarize yourself with how things change as we shift the parameters 153 | % around. 154 | 155 | % Gaussian functions will keep coming up, not just in understanding 156 | % neuroscience concepts, but also in your data analysis. For example, many 157 | % of the tools that simplify statistical calculations (such as t-tests) 158 | % assume that your data conforms to a Gaussian distribution. If you're not 159 | % certain that this is the case, then you need to use other methods. 160 | 161 | %% Cumulative distribution functions 162 | % So far in this tutorial, we have explored Gaussian/Poisson probability 163 | % density functions (PDF). Now, we want to turn our attention to the 164 | % cumulative distribution functions (CDF) for these distributions. 165 | 166 | % So what is a cumulative distribution function? 167 | % 168 | % To answer that question, let's back up and give an example of a Gaussian 169 | % distribution. The classic example, of course, is the distribution of 170 | % IQ's. The mean IQ is, by definition, 100 and the standard deviation is 171 | % 15. Let's make a little IQ PDF/distribution. 172 | mu = 100; 173 | sigma = 15; 174 | 175 | x = 0:200; 176 | y = gaussfun([mu,sigma],x); 177 | y = y/sum(y); % Area under the curve must equal 1. 178 | 179 | figure(100); clf; 180 | hold on 181 | plot(x, y, 'b', 'LineWidth', 2); % Need to scale to the same maximum value as the histogram 182 | hold off; 183 | xlabel('distribution of IQ scores'); ylabel('probability'); 184 | 185 | %% 186 | % Alright, let's now look at the probability of a person having an IQ of 187 | % exactly 120. 188 | p120 = y(x == 120) 189 | 190 | % For our example, that turns out to be about 1%... It turns out that the 191 | % 120 threshold is pretty important... In Malcolm Gladwell's book 192 | % 'Outliers' he cites studies indicating that up to 120, IQ is a reliable 193 | % predictor of a person's success, but above that threshold, it no longer 194 | % correlates with success. The argument is that the IQ test captures 195 | % something about intelligence, but there are aspects of intellect that it 196 | % does not assess adequately. I digress, but it's a fun read. 197 | 198 | % OK, so what if we wanted to know what percentage of the population have 199 | % an IQ less than 120? How would we calculate that? Well, knowing the value 200 | % at 120, isn't particularly informative... This is where the cumulative 201 | % distribution function becomes useful. 202 | 203 | % To find the probability of someone having an IQ less than 120, we need to 204 | % take the definite integral from [0,119]. This calculates the area under 205 | % the distribution (i.e., cumulative distribution function) from 0 to 119. 206 | cumPr120 = sum(y(x <= 120)) 207 | 208 | % That gives us a value of 0.90, meaning that 90% of the population has an 209 | % IQ less than 120. We could do the same thing and get the probability that 210 | % a random person on the street has an IQ less than or equal to any value. 211 | % One way to calculate this would be in a 'for' loop: 212 | 213 | cumPr = zeros(size(x)); % Good practice to pre-allocate memory... 214 | for k = 1 : length(x) % This is looping through each value in x... 215 | cumPr(k) = sum(y(x <= x(k))); 216 | end 217 | 218 | % Let's look at the result. 219 | figure(100); clf; 220 | hold on 221 | plot(x, cumPr, 'b', 'LineWidth', 2); 222 | 223 | % Let's highlight the value at 120 for fun. 224 | plot(120, cumPr(x == 120), 'ko'); 225 | plot(120*ones(1,2), [0 cumPr(x == 120)],'k--'); 226 | plot([0 120], cumPr(x == 120)*ones(1,2),'k--'); 227 | 228 | hold off; 229 | xlabel('IQ score'); ylabel('cumulative probability'); 230 | 231 | % I've highlighted the value at 120 to illustrate what we discussed above. 232 | % This is the cumulative distribution function (CDF) for the Gaussian 233 | % distribution that we looked at earlier. That's all there is to it!! 234 | 235 | %% 236 | % It turns out that there are several ways to calculate the CDF. The for 237 | % loop example above is one way, and I would like to show you two other 238 | % ways of doing the calculation. Both ways are easier than the for loop 239 | % example: 240 | 241 | % 1) 'cumsum': Matlab has a built-in function called 'cumsum' that does 242 | % exactly what we did in the for loop, but using this function just means 243 | % fewer lines of code that we have to write. Let's do it! 244 | cumPrCumSum = cumsum(y); 245 | 246 | % Let's plot the output from our for loop above and the new way to make 247 | % sure they align. 248 | 249 | figure(100); clf; 250 | hold on 251 | plot(x, cumPr, 'b', 'LineWidth', 2); 252 | plot(x, cumPrCumSum, 'r', 'LineWidth', 2); 253 | hold off; 254 | xlabel('IQ score'); ylabel('cumulative probability'); 255 | 256 | % Yep. We only see one curve, because the two curves are identical. Here's 257 | % the second way. 258 | 259 | % 2) 'normcdf': This function generates a Gaussian (also called 'Normal') 260 | % cumulative distribution function based on mean and standard deviation. 261 | % Check it out. 262 | cumPrFun = normcdf(x, mu, sigma); 263 | 264 | % Let's check this as well. 265 | figure(100); clf; 266 | hold on 267 | plot(x, cumPr, 'b', 'LineWidth', 2); 268 | plot(x, cumPrFun, 'r', 'LineWidth', 2); 269 | hold off; 270 | xlabel('IQ score'); ylabel('cumulative probability'); 271 | 272 | % No big deal, right?! You're just taking the integral under the 273 | % distribution. Note: if you're dealing with a Poisson distribution, the 274 | % function is 'poisscdf'. 275 | 276 | %% Fitting a cumulative distribution function 277 | 278 | % One thing you'll be doing a lot of in your research is trying to model 279 | % your data. Imagine that you have some data that you want to present. The 280 | % first thing to do is show the data. Let's make some pretend data of some 281 | % IQ scores: 282 | dataHist = sigma * randn(1,1e3) + mu; 283 | [data, binEdges] = histcounts(dataHist, 100,'Normalization', 'cdf'); 284 | x = (binEdges(1:end-1) + binEdges(2:end))/2; % Get the center of the bins. 285 | 286 | % Alright, we've created a cumulative distribution of IQ scores. Now, let's 287 | % fit the data. I'll show you two different functions that I typically use 288 | % for nonlinear fitting... 289 | 290 | % Another anonymous function. 291 | fitfun = @(p,x)(normcdf(x,p(1),p(2))); 292 | 293 | % 1) 'nlinfit' 294 | params = nlinfit(x,data,fitfun,[100 15]); 295 | 296 | % Let's take a look at our fitted parameters: 297 | params 298 | 299 | % Hopefully, they're pretty close to the mu and sigma values we used to 300 | % generate our pretend data. 301 | 302 | % 2) 'lsqcurvefit' 303 | params2 = lsqcurvefit(fitfun,[100 15],x,data); 304 | params2 305 | 306 | % Let's see how good our fits were. 307 | figure(100); clf; 308 | hold on 309 | plot(x, data, 'b.'); 310 | plot(x, fitfun(params,x), 'r', 'LineWidth', 1); 311 | hold off; 312 | xlabel('IQ score'); ylabel('cumulative probability'); 313 | 314 | % There! We fit the data. That's all there is to it. The two fitting 315 | % function each have their own advantages and I'm happy to discuss those, 316 | % but for now, I just wanted you to see the process in action. 317 | 318 | -------------------------------------------------------------------------------- /code/MATLAB/SignalAndNoiseTutorial.m: -------------------------------------------------------------------------------- 1 | % SignalAndNoiseTutorial.m 2 | % by Mike Manookin 11/2019 3 | 4 | x = 0 : 10; 5 | 6 | p1 = poisspdf(x, 1); 7 | 8 | 9 | p2 = poisspdf(x, 2); 10 | 11 | x = 0:2; 12 | y = [0.81 0.17 0.02 0.64 0.28 0.08]; 13 | 14 | % Let's define anonymous functions. 15 | 16 | % This would be for the single-photon condition. 17 | pfun1 = @(m,k)([(m.^k * exp(-m))./factorial(k),((2*m).^k * exp(-2*m))./factorial(k)]); 18 | 19 | % This would be for the coincident model: 20 | pfun2 = @(m,k)([(m.^k * exp(-m))./factorial(k)+(m.^(k-1) * exp(-m))./factorial(k-1),((2*m).^k * exp(-2*m))./factorial(k)+((2*m).^(k-1) * exp(-2*m))./factorial(k-1)]); 21 | 22 | m = 0.21; 23 | 24 | % Single-photon model. 25 | p1 = 0; 26 | 27 | 28 | 29 | % Let's generate data from Equation 10 from Baylor et al 1979: 30 | m = 0.53; 31 | a = 1.2; 32 | sigma0 = 0.15; 33 | sigma1 = 0.3; 34 | x = -1:0.1:3; 35 | 36 | p = zeros(1, length(x)); 37 | for j = 1 : length(x) 38 | r = x(j); 39 | for k = 0:3 40 | p(j) = p(j) + (exp(-m)*m^k)/factorial(k) .* 1./(2*pi*(sigma0^2+k*sigma1^2)).^0.5 .* exp(-(r - k*a).^2 / (2*(sigma0^2 + k*sigma1^2))); 41 | end 42 | end 43 | 44 | % Divide by the number of k's you've iterated over.. 45 | p = p / 4; 46 | 47 | figure(1); clf; 48 | plot(x,p) 49 | 50 | -------------------------------------------------------------------------------- /code/MATLAB/SignalAndNoiseTutorial[Conflict].m: -------------------------------------------------------------------------------- 1 | % SignalAndNoiseTutorial.m 2 | % by Mike Manookin 11/2019 3 | 4 | x = 0:2; 5 | y = [0.81 0.17 0.02 0.64 0.28 0.08]; 6 | 7 | % Let's define anonymous functions. 8 | 9 | % This would be for the single-photon condition. 10 | pfun1 = @(m,k)([(m.^k * exp(-m))./factorial(k),((2*m).^k * exp(-2*m))./factorial(k)]); 11 | 12 | % This would be for the coincident model: 13 | pfun2 = @(m,k)([(m.^k * exp(-m))./factorial(k)+(m.^(k-1) * exp(-m))./factorial(k-1),((2*m).^k * exp(-2*m))./factorial(k)+((2*m).^(k-1) * exp(-2*m))./factorial(k-1)]); 14 | 15 | m = 0.21; 16 | 17 | % Single-photon model. 18 | p1 = 0; 19 | 20 | 21 | -------------------------------------------------------------------------------- /code/Python/ProbabilityDistributionsTutorial.ipynb: -------------------------------------------------------------------------------- 1 | {"nbformat":4,"nbformat_minor":0,"metadata":{"colab":{"name":"ProbabilityDistributionsTutorial.ipynb","provenance":[],"collapsed_sections":[]},"kernelspec":{"name":"python3","display_name":"Python 3"}},"cells":[{"cell_type":"markdown","metadata":{"id":"_O2wjKwgPJsP"},"source":["# **Probability distributions**\n","\n","Today we're going to focus on three general probability distributions and two special types of those distributions: \n","\n","1. **Gaussian distributions**\n","2. **Poisson distribution**\n","3. **Uniform distributions**\n","4. **Binomal distributions**\n"," * Special case: **Bernoulli distribution**\n","\n","\n","## **Normal (Gaussian) distributions**\n","The Gaussian distribution is arguably the most useful probability distributions out there. The reason is that many random processes are described well by a Gaussian distribution when the number of samples is large (for more on why, look up the central limit theorem).\n","\n","When data have a Gaussian distribution, there are some extra mathematical tricks that we can use to analyze those data. One of the reasons for this is that (if we have sampled the data sufficiently) we can describe the data with two variables: the mean ($\\mu$) and the variance ($\\sigma^2$).\n","\n","This is the equation for a Gaussian distribution:\n","\n","$f(x) = \\frac{1}{\\sigma \\sqrt{2 \\pi}}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}$\n","\n","$\\mu$ is the mean of the distribution\n","\n","$\\sigma$ is the standard deviation of the distribution (square root of the variance)\n","\n","It's alright if equations aren't your thing. The thing to remember is that you have some distribution of your data and that there is a mean/average ($\\mu$) and another parameter describing how spread out the distribution is---the standard deviation ($\\sigma$). The following code allows us to play with those two parameters and see how it affects the Gaussian distribution. Go ahead and change 'mu' and 'sigma' and then push the \"play\" button to see how the distribution changes."]},{"cell_type":"code","metadata":{"colab":{"base_uri":"https://localhost:8080/","height":296},"id":"t9TqkEblIR8y","executionInfo":{"status":"ok","timestamp":1630553613765,"user_tz":420,"elapsed":1364,"user":{"displayName":"Mike Manookin","photoUrl":"https://lh3.googleusercontent.com/a-/AOh14Ggqm5Gu3fF6zZeMUl4BlfJw_UYWrxC4O6YnC9Zsbg=s64","userId":"13097636622245819913"}},"outputId":"f62e0405-17b9-46d4-a2b2-2186840e7b91"},"source":["# Numpy module\n","import numpy as np\n","\n","# Seaborn for visualizing distributions\n","import seaborn as sns\n","\n","numSamples = 10000;\n","numBins = 25;\n","\n","# The distribution mean\n","mu = 1.0\n","\n","# The standard deviation (sigma).\n","sigma = 0.3\n","\n","# 'randn' just generates a bunch of random numbers that have a Gaussian distribution\n","y = sigma * np.random.randn(numSamples) + mu\n","\n","\n","ax = sns.histplot(data=y, stat=\"probability\", bins=numBins, kde=True, element='poly')\n","ax.set(xlabel='Gaussian distribution')"],"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["[Text(0.5, 0, 'Gaussian distribution')]"]},"metadata":{},"execution_count":1},{"output_type":"display_data","data":{"image/png":"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\n","text/plain":["
"]},"metadata":{"needs_background":"light"}}]},{"cell_type":"markdown","metadata":{"id":"mTzeQHlO186Q"},"source":["Go ahead and play around with the parameters for the Gaussian distribution. What you'll notice as you change the parameters, is that the values along the x-axis change, but the general, bell-curve, shape stays the same.\n","\n","Gaussian functions will keep coming up, not just in understanding neuroscience concepts, but also in your data analysis. For example, many of the tools that simplify statistical calculations (such as t-tests) assume that your data conforms to a Gaussian distribution. If you're not certain that this is the case, then you need to use other methods.\n","\n","\n","\n","## **Poisson distributions**\n","\n","A Poisson distribution is a very important type of distribution in neuroscience. This type of distribution is discrete, meaning that every data entry is an integer value such as 0, 1, 2... (not 1.3439034).\n","\n","Why is the Poisson distribution so useful? Well, it turns out that there are many physical processes that are well described with Poisson statistics. Here are a few examples that are relevant to you and me:\n","\n","1) In physics, the frequency of photon arrivals.\n","\n","2) Neural spike counts (not exactly due to the refractory period)...\n","\n","3) Frequency of vesicle release at synapses\n","\n","One of the key properties of the Poisson distribution is how many variables are required to describe them. You'll recall from our discussion of Gaussian distributions that we need two numbers to describe them: 1) the mean ($\\mu$) tells you where the distribution is centered and 2) the variance ($\\sigma^2$) tells you how spread out the distribution is. Well, a Poisson distribution is even easier, you just need one number to describe it because the mean and variance are equal.\n","\n","The standard deviation ($\\sigma$) is the square root of the variance and, since the mean and variance are equal, it is also the square root of the mean ($\\mu$).\n","\n","$\\sigma^2 = \\mu$\n","\n","$\\sigma = \\sqrt{\\mu}$\n","\n","\n","Let's look at concrete example. Photon arrivals at a detector obey Poisson statistics so let's simulate a distribution of photon counts based on a mean values."]},{"cell_type":"code","metadata":{"id":"ytTYtvzNamwT","colab":{"base_uri":"https://localhost:8080/","height":296},"executionInfo":{"status":"ok","timestamp":1630536801243,"user_tz":420,"elapsed":967,"user":{"displayName":"Mike Manookin","photoUrl":"https://lh3.googleusercontent.com/a-/AOh14Ggqm5Gu3fF6zZeMUl4BlfJw_UYWrxC4O6YnC9Zsbg=s64","userId":"13097636622245819913"}},"outputId":"40ec2c38-9bc1-43e8-fc4c-9c3335295dd3"},"source":["# Numpy module\n","import numpy as np\n","\n","import math\n","\n","# Seaborn for visualizing distributions\n","import seaborn as sns\n","\n","numSamples = 10000;\n","\n","# Define the mean/variance of the distribution\n","mu = 3.0\n","\n","# This is a way of generating a Poisson distribution.\n","y = np.random.poisson(mu, size=numSamples)\n","\n","ax = sns.histplot(data=y, stat=\"probability\", discrete=True)\n","ax.set(xlabel='photon count per bin')\n"],"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["[Text(0.5, 0, 'photon count per bin')]"]},"metadata":{},"execution_count":56},{"output_type":"display_data","data":{"image/png":"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\n","text/plain":["
"]},"metadata":{"needs_background":"light"}}]},{"cell_type":"markdown","metadata":{"id":"d8tmgh8Gsa5v"},"source":["## **Uniform distributions**\n","\n","Data can also be distributed uniformly: a case in which all of the possible outcomes are equally likely. If we think of this in the context of the photon count example above, having a count of {0,1,2,...} would occur with equal probability. \n","\n","Let's look at some code for generating a visualizing data with a uniform distribution.\n","\n"]},{"cell_type":"code","metadata":{"colab":{"base_uri":"https://localhost:8080/","height":296},"id":"VctlcwJxNBJ1","executionInfo":{"status":"ok","timestamp":1630536390831,"user_tz":420,"elapsed":428,"user":{"displayName":"Mike Manookin","photoUrl":"https://lh3.googleusercontent.com/a-/AOh14Ggqm5Gu3fF6zZeMUl4BlfJw_UYWrxC4O6YnC9Zsbg=s64","userId":"13097636622245819913"}},"outputId":"c58248b5-8130-469d-db0d-00ea888b4307"},"source":["# Numpy module\n","import numpy as np\n","\n","# Seaborn for visualizing distributions\n","import seaborn as sns\n","\n","numSamples = 10000;\n","numBins = 25;\n","\n","y = np.random.rand(numSamples)\n","\n","\n","ax = sns.histplot(data=y, stat=\"probability\", bins=numBins, kde=False, element='poly')\n","ax.set(xlabel='Uniform distribution')"],"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["[Text(0.5, 0, 'Uniform distribution')]"]},"metadata":{},"execution_count":52},{"output_type":"display_data","data":{"image/png":"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\n","text/plain":["
"]},"metadata":{"needs_background":"light"}}]},{"cell_type":"markdown","metadata":{"id":"XDBJuKJUBt06"},"source":["## **Binomial distributions**\n","\n","Binomial distributions are an important type of discrete distribution (as opposed to a continuous distribution, such as a Gaussian distribution).\n","\n","$f(k; p) = \\big( \\frac{n!}{k!(n-k)!} \\big) p^k (1 - p)^{n-k}$ for k $\\in$ {0, 1, ..., n}\n","\n","Let's look at an example of how this distribution might be used. Let's say I give you a coin and ask you to flip the coin five times (that's your $n$ variable). If you know it's a fair coin, what are the chances that you end up with zero heads or one heads, etc? That's your $k$ variable! \n"]},{"cell_type":"code","metadata":{"colab":{"base_uri":"https://localhost:8080/","height":296},"id":"x2q1fAZ_Bscf","executionInfo":{"status":"ok","timestamp":1630986008348,"user_tz":420,"elapsed":882,"user":{"displayName":"Mike Manookin","photoUrl":"https://lh3.googleusercontent.com/a-/AOh14Ggqm5Gu3fF6zZeMUl4BlfJw_UYWrxC4O6YnC9Zsbg=s64","userId":"13097636622245819913"}},"outputId":"0f9a917b-218a-4924-eeb0-bcf0c7b4af3a"},"source":["# Numpy module\n","import numpy as np\n","\n","# Seaborn for visualizing distributions\n","import seaborn as sns\n","\n","numSamples = 10000;\n","\n","n = 5\n","\n","# This is the probability that a value of '1' is observed. Go ahead and play \n","# with this value and see how the distribution changes. It needs to be a value \n","# between 0-1. For a fair coin, the value should be 0.5.\n","p = 0.5\n","\n","rng = np.random.default_rng()\n","y = rng.binomial(n, p, numSamples)\n","\n","ax = sns.histplot(data=y, stat=\"probability\", discrete=True)\n","ax.set(xlabel='Binomial distribution')"],"execution_count":1,"outputs":[{"output_type":"execute_result","data":{"text/plain":["[Text(0.5, 0, 'Binomial distribution')]"]},"metadata":{},"execution_count":1},{"output_type":"display_data","data":{"image/png":"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\n","text/plain":["
"]},"metadata":{"needs_background":"light"}}]},{"cell_type":"markdown","metadata":{"id":"GQ6HV8U7aWri"},"source":["## **Bernoulli distribution**\n","\n","The Bernoulli distribution is a type of binomial distribution. Data with this distribution have values that are either 0 or 1.\n","\n","$f(k; p) = p^k (1 - p)^{1-k}$ for k $\\in$ {0, 1}\n","\n"]},{"cell_type":"code","metadata":{"id":"QboHmkDnavjQ","colab":{"base_uri":"https://localhost:8080/","height":296},"executionInfo":{"status":"ok","timestamp":1630539284255,"user_tz":420,"elapsed":382,"user":{"displayName":"Mike Manookin","photoUrl":"https://lh3.googleusercontent.com/a-/AOh14Ggqm5Gu3fF6zZeMUl4BlfJw_UYWrxC4O6YnC9Zsbg=s64","userId":"13097636622245819913"}},"outputId":"9d570ff3-b800-4926-aa29-073e01dfe245"},"source":["# Numpy module\n","import numpy as np\n","\n","# Seaborn for visualizing distributions\n","import seaborn as sns\n","\n","numSamples = 10000;\n","\n","n = 1\n","\n","# This is the probability that a value of '1' is observed. Go ahead and play \n","# with this value and see how the distribution changes. It needs to be a value \n","# between 0-1\n","p = 0.1\n","\n","rng = np.random.default_rng()\n","y = rng.binomial(n, p, numSamples)\n","\n","ax = sns.histplot(data=y, stat=\"probability\")\n","ax.set(xlabel='Bernoulli distribution')"],"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["[Text(0.5, 0, 'Bernoulli distribution')]"]},"metadata":{},"execution_count":59},{"output_type":"display_data","data":{"image/png":"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\n","text/plain":["
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