minor typos fixes (NeuromatchAcademy#1064)

* minor typos fixes * Process tutorial notebooks --------- Co-authored-by: GitHub Action <[email protected]>
Yuzuki-Ishikawa · Jul 20, 2023 · 6c083a1 · 6c083a1
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diff --git a/tutorials/W2D3_BiologicalNeuronModels/W2D3_Tutorial2.ipynb b/tutorials/W2D3_BiologicalNeuronModels/W2D3_Tutorial2.ipynb
@@ -546,11 +546,17 @@
     "\n",
     "\\begin{align}\n",
     "r_{ij} &= \\frac{cov(I_i, I_j)}{\\sqrt{var(I_i)} \\sqrt{var(I_j)}}\\\\\n",
-    "cov(I_i, I_j) &= \\sum_{k=1}^L (I_i^k -\\bar{I}_i)(I_j^k -\\bar{I}_j) \\\\\n",
+    "cov(I_i, I_j) &= \\sum_{k=1}^L (I_i^k -\\bar{I_i})(I_j^k -\\bar{I_j}) \\\\\n",
     "var(I_i) &= \\sum_{k=1}^L (I_i^k -\\bar{I}_i)^2\n",
     "\\end{align}\n",
     "\n",
-    "where $\\bar{I}_i$ is the sample mean, k is the time bin, and L is the length of $I$.  This means that $I_i^k$ is current i at time $k\\cdot dt$. Note that the equations above are not accurate for sample covariances and variances as they should be additionally divided by L-1 - we have dropped this term because it cancels out in the sample correlation coefficient formula.\n",
+    "where $\\bar{I_i}$ is the sample mean, $k$ is the time bin, and $L$ is the length of $I$. This means that $I_i^k$ is current $i$ at time $k\\cdot dt$.\n",
+    "\n",
+    "<br>\n",
+    "\n",
+    "**Important note:** The equations above are not accurate for sample covariances and variances as they should be additionally divided by $L-1$. We have dropped this term because it cancels out in the sample correlation coefficient formula.\n",
+    "\n",
+    "<br>\n",
     "\n",
     "The _sample correlation coefficient_ may also be referred to as the _sample Pearson correlation coefficient_. Here, is a beautiful paper that explains multiple ways to calculate and understand correlations [Rodgers and Nicewander 1988](https://www.stat.berkeley.edu/~rabbee/correlation.pdf).\n",
     "\n",
@@ -598,6 +604,7 @@
     "\n",
     "  return rij\n",
     "\n",
+    "\n",
     "example_plot_myCC()"
    ]
   },
@@ -1246,18 +1253,19 @@
    },
    "source": [
     "## Section 3.2: What is the rationale behind varying $\\mu$ and $\\sigma$?\n",
+    "\n",
     "The mean and the variance of the synaptic current depends on the spike rate of a Poisson process. We can use something called [Campbell's theorem](https://en.wikipedia.org/wiki/Campbell%27s_theorem_(probability)) to estimate the mean and the variance of the synaptic current:\n",
     "\n",
     "\\begin{align}\n",
-    "\\mu_{\\rm syn} = \\lambda J \\int P(t) \\\\\n",
+    "\\mu_{\\rm syn} = \\lambda J \\int P(t) dt \\\\\n",
     "\\sigma_{\\rm syn} = \\lambda J \\int P(t)^2 dt\n",
     "\\end{align}\n",
     "\n",
     "where $\\lambda$ is the firing rate of the Poisson input, $J$ the amplitude of the postsynaptic current and $P(t)$ is the shape of the postsynaptic current as a function of time.\n",
     "\n",
     "Therefore, when we varied $\\mu$ and/or $\\sigma$ of the GWN, we mimicked a change in the input firing rate. Note that, if we change the firing rate, both $\\mu$ and $\\sigma$ will change simultaneously, not independently.\n",
     "\n",
-    "Here, since we observe an effect of $\\mu$ and $\\sigma$ on correlation transfer, this implies that the input rate has an impact on the correlation transfer function.\n"
+    "Here, since we observe an effect of $\\mu$ and $\\sigma$ on correlation transfer, this implies that the input rate has an impact on the correlation transfer function."
    ]
   },
   {
@@ -1276,9 +1284,9 @@
     "\n",
     "References:\n",
     "\n",
-    "- De La Rocha, Jaime, et al. \"Correlation between neural spike trains increases with firing rate.\" Nature (2007). doi: [10.1038/nature06028](https://doi.org/10.1038/nature06028)\n",
+    "- de la Rocha J, Doiron B, Shea-Brown E, Josić K, Reyes A (2007). Correlation between neural spike trains increases with firing rate. Nature 448:802-806. doi: [10.1038/nature06028](https://doi.org/10.1038/nature06028)\n",
     "\n",
-    "- Bujan AF, Aertsen A, Kumar A. Role of input correlations in shaping the variability and noise correlations of evoked activity in the neocortex. Journal of Neuroscience. 2015 Jun 3;35(22):8611-25. doi: [10.1523/JNEUROSCI.4536-14.2015](https://doi.org/10.1523/JNEUROSCI.4536-14.2015)"
+    "- Bujan AF, Aertsen A, Kumar A (2015). Role of input correlations in shaping the variability and noise correlations of evoked activity in the neocortex, Journal of Neuroscience 35(22):8611-25. doi: [10.1523/JNEUROSCI.4536-14.2015](https://doi.org/10.1523/JNEUROSCI.4536-14.2015)"
    ]
   },
   {

diff --git a/tutorials/W2D3_BiologicalNeuronModels/instructor/W2D3_Tutorial2.ipynb b/tutorials/W2D3_BiologicalNeuronModels/instructor/W2D3_Tutorial2.ipynb
@@ -546,11 +546,17 @@
     "\n",
     "\\begin{align}\n",
     "r_{ij} &= \\frac{cov(I_i, I_j)}{\\sqrt{var(I_i)} \\sqrt{var(I_j)}}\\\\\n",
-    "cov(I_i, I_j) &= \\sum_{k=1}^L (I_i^k -\\bar{I}_i)(I_j^k -\\bar{I}_j) \\\\\n",
+    "cov(I_i, I_j) &= \\sum_{k=1}^L (I_i^k -\\bar{I_i})(I_j^k -\\bar{I_j}) \\\\\n",
     "var(I_i) &= \\sum_{k=1}^L (I_i^k -\\bar{I}_i)^2\n",
     "\\end{align}\n",
     "\n",
-    "where $\\bar{I}_i$ is the sample mean, k is the time bin, and L is the length of $I$.  This means that $I_i^k$ is current i at time $k\\cdot dt$. Note that the equations above are not accurate for sample covariances and variances as they should be additionally divided by L-1 - we have dropped this term because it cancels out in the sample correlation coefficient formula.\n",
+    "where $\\bar{I_i}$ is the sample mean, $k$ is the time bin, and $L$ is the length of $I$. This means that $I_i^k$ is current $i$ at time $k\\cdot dt$.\n",
+    "\n",
+    "<br>\n",
+    "\n",
+    "**Important note:** The equations above are not accurate for sample covariances and variances as they should be additionally divided by $L-1$. We have dropped this term because it cancels out in the sample correlation coefficient formula.\n",
+    "\n",
+    "<br>\n",
     "\n",
     "The _sample correlation coefficient_ may also be referred to as the _sample Pearson correlation coefficient_. Here, is a beautiful paper that explains multiple ways to calculate and understand correlations [Rodgers and Nicewander 1988](https://www.stat.berkeley.edu/~rabbee/correlation.pdf).\n",
     "\n",
@@ -598,6 +604,7 @@
     "\n",
     "  return rij\n",
     "\n",
+    "\n",
     "example_plot_myCC()\n",
     "\n",
     "```"
@@ -1250,18 +1257,19 @@
    },
    "source": [
     "## Section 3.2: What is the rationale behind varying $\\mu$ and $\\sigma$?\n",
+    "\n",
     "The mean and the variance of the synaptic current depends on the spike rate of a Poisson process. We can use something called [Campbell's theorem](https://en.wikipedia.org/wiki/Campbell%27s_theorem_(probability)) to estimate the mean and the variance of the synaptic current:\n",
     "\n",
     "\\begin{align}\n",
-    "\\mu_{\\rm syn} = \\lambda J \\int P(t) \\\\\n",
+    "\\mu_{\\rm syn} = \\lambda J \\int P(t) dt \\\\\n",
     "\\sigma_{\\rm syn} = \\lambda J \\int P(t)^2 dt\n",
     "\\end{align}\n",
     "\n",
     "where $\\lambda$ is the firing rate of the Poisson input, $J$ the amplitude of the postsynaptic current and $P(t)$ is the shape of the postsynaptic current as a function of time.\n",
     "\n",
     "Therefore, when we varied $\\mu$ and/or $\\sigma$ of the GWN, we mimicked a change in the input firing rate. Note that, if we change the firing rate, both $\\mu$ and $\\sigma$ will change simultaneously, not independently.\n",
     "\n",
-    "Here, since we observe an effect of $\\mu$ and $\\sigma$ on correlation transfer, this implies that the input rate has an impact on the correlation transfer function.\n"
+    "Here, since we observe an effect of $\\mu$ and $\\sigma$ on correlation transfer, this implies that the input rate has an impact on the correlation transfer function."
    ]
   },
   {
@@ -1280,9 +1288,9 @@
     "\n",
     "References:\n",
     "\n",
-    "- De La Rocha, Jaime, et al. \"Correlation between neural spike trains increases with firing rate.\" Nature (2007). doi: [10.1038/nature06028](https://doi.org/10.1038/nature06028)\n",
+    "- de la Rocha J, Doiron B, Shea-Brown E, Josić K, Reyes A (2007). Correlation between neural spike trains increases with firing rate. Nature 448:802-806. doi: [10.1038/nature06028](https://doi.org/10.1038/nature06028)\n",
     "\n",
-    "- Bujan AF, Aertsen A, Kumar A. Role of input correlations in shaping the variability and noise correlations of evoked activity in the neocortex. Journal of Neuroscience. 2015 Jun 3;35(22):8611-25. doi: [10.1523/JNEUROSCI.4536-14.2015](https://doi.org/10.1523/JNEUROSCI.4536-14.2015)"
+    "- Bujan AF, Aertsen A, Kumar A (2015). Role of input correlations in shaping the variability and noise correlations of evoked activity in the neocortex, Journal of Neuroscience 35(22):8611-25. doi: [10.1523/JNEUROSCI.4536-14.2015](https://doi.org/10.1523/JNEUROSCI.4536-14.2015)"
    ]
   },
   {

diff --git a/...rials/W2D3_BiologicalNeuronModels/static/W2D3_Tutorial2_Solution_313f41e4_0.png b/...rials/W2D3_BiologicalNeuronModels/static/W2D3_Tutorial2_Solution_313f41e4_0.png
diff --git a/tutorials/W2D3_BiologicalNeuronModels/student/W2D3_Tutorial2.ipynb b/tutorials/W2D3_BiologicalNeuronModels/student/W2D3_Tutorial2.ipynb
@@ -546,11 +546,17 @@
     "\n",
     "\\begin{align}\n",
     "r_{ij} &= \\frac{cov(I_i, I_j)}{\\sqrt{var(I_i)} \\sqrt{var(I_j)}}\\\\\n",
-    "cov(I_i, I_j) &= \\sum_{k=1}^L (I_i^k -\\bar{I}_i)(I_j^k -\\bar{I}_j) \\\\\n",
+    "cov(I_i, I_j) &= \\sum_{k=1}^L (I_i^k -\\bar{I_i})(I_j^k -\\bar{I_j}) \\\\\n",
     "var(I_i) &= \\sum_{k=1}^L (I_i^k -\\bar{I}_i)^2\n",
     "\\end{align}\n",
     "\n",
-    "where $\\bar{I}_i$ is the sample mean, k is the time bin, and L is the length of $I$.  This means that $I_i^k$ is current i at time $k\\cdot dt$. Note that the equations above are not accurate for sample covariances and variances as they should be additionally divided by L-1 - we have dropped this term because it cancels out in the sample correlation coefficient formula.\n",
+    "where $\\bar{I_i}$ is the sample mean, $k$ is the time bin, and $L$ is the length of $I$. This means that $I_i^k$ is current $i$ at time $k\\cdot dt$.\n",
+    "\n",
+    "<br>\n",
+    "\n",
+    "**Important note:** The equations above are not accurate for sample covariances and variances as they should be additionally divided by $L-1$. We have dropped this term because it cancels out in the sample correlation coefficient formula.\n",
+    "\n",
+    "<br>\n",
     "\n",
     "The _sample correlation coefficient_ may also be referred to as the _sample Pearson correlation coefficient_. Here, is a beautiful paper that explains multiple ways to calculate and understand correlations [Rodgers and Nicewander 1988](https://www.stat.berkeley.edu/~rabbee/correlation.pdf).\n",
     "\n",
@@ -598,6 +604,7 @@
     "\n",
     "  return rij\n",
     "\n",
+    "\n",
     "example_plot_myCC()"
    ]
   },
@@ -1157,18 +1164,19 @@
    },
    "source": [
     "## Section 3.2: What is the rationale behind varying $\\mu$ and $\\sigma$?\n",
+    "\n",
     "The mean and the variance of the synaptic current depends on the spike rate of a Poisson process. We can use something called [Campbell's theorem](https://en.wikipedia.org/wiki/Campbell%27s_theorem_(probability)) to estimate the mean and the variance of the synaptic current:\n",
     "\n",
     "\\begin{align}\n",
-    "\\mu_{\\rm syn} = \\lambda J \\int P(t) \\\\\n",
+    "\\mu_{\\rm syn} = \\lambda J \\int P(t) dt \\\\\n",
     "\\sigma_{\\rm syn} = \\lambda J \\int P(t)^2 dt\n",
     "\\end{align}\n",
     "\n",
     "where $\\lambda$ is the firing rate of the Poisson input, $J$ the amplitude of the postsynaptic current and $P(t)$ is the shape of the postsynaptic current as a function of time.\n",
     "\n",
     "Therefore, when we varied $\\mu$ and/or $\\sigma$ of the GWN, we mimicked a change in the input firing rate. Note that, if we change the firing rate, both $\\mu$ and $\\sigma$ will change simultaneously, not independently.\n",
     "\n",
-    "Here, since we observe an effect of $\\mu$ and $\\sigma$ on correlation transfer, this implies that the input rate has an impact on the correlation transfer function.\n"
+    "Here, since we observe an effect of $\\mu$ and $\\sigma$ on correlation transfer, this implies that the input rate has an impact on the correlation transfer function."
    ]
   },
   {
@@ -1187,9 +1195,9 @@
     "\n",
     "References:\n",
     "\n",
-    "- De La Rocha, Jaime, et al. \"Correlation between neural spike trains increases with firing rate.\" Nature (2007). doi: [10.1038/nature06028](https://doi.org/10.1038/nature06028)\n",
+    "- de la Rocha J, Doiron B, Shea-Brown E, Josić K, Reyes A (2007). Correlation between neural spike trains increases with firing rate. Nature 448:802-806. doi: [10.1038/nature06028](https://doi.org/10.1038/nature06028)\n",
     "\n",
-    "- Bujan AF, Aertsen A, Kumar A. Role of input correlations in shaping the variability and noise correlations of evoked activity in the neocortex. Journal of Neuroscience. 2015 Jun 3;35(22):8611-25. doi: [10.1523/JNEUROSCI.4536-14.2015](https://doi.org/10.1523/JNEUROSCI.4536-14.2015)"
+    "- Bujan AF, Aertsen A, Kumar A (2015). Role of input correlations in shaping the variability and noise correlations of evoked activity in the neocortex, Journal of Neuroscience 35(22):8611-25. doi: [10.1523/JNEUROSCI.4536-14.2015](https://doi.org/10.1523/JNEUROSCI.4536-14.2015)"
    ]
   },
   {