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Diff for: ‎.ipynb_checkpoints/HH_functions-checkpoint.py

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import numpy as np
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import math
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def alphaM(V):
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return (2.5-0.1*(V+65)) / (np.exp(2.5-0.1*(V+65)) -1)
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def betaM(V):
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return 4*np.exp(-(V+65)/18)
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def alphaH(V):
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return 0.07*np.exp(-(V+65)/20)
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def betaH(V):
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return 1/(np.exp(3.0-0.1*(V+65))+1)
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def alphaN(V):
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return (0.1-0.01*(V+65)) / (np.exp(1-0.1*(V+65)) -1)
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def betaN(V):
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return 0.125*np.exp(-(V+65)/80)
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def HH(I0,T0):
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dt = 0.01;
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T = math.ceil(T0/dt) # [ms]
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gNa0 = 120 # [mS/cm^2]
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ENa = 115; # [mV]
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gK0 = 36; # [mS/cm^2]
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EK = -12; # [mV]
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gL0 = 0.3; # [mS/cm^2]
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EL = 10.6; # [mV]
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t = np.arange(0,T)*dt
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V = np.zeros([T,1])
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m = np.zeros([T,1])
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h = np.zeros([T,1])
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n = np.zeros([T,1])
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V[0]=-70.0
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m[0]=0.05
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h[0]=0.54
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n[0]=0.34
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for i in range(0,T-1):
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V[i+1] = V[i] + dt*(gNa0*m[i]**3*h[i]*(ENa-(V[i]+65)) + gK0*n[i]**4*(EK-(V[i]+65)) + gL0*(EL-(V[i]+65)) + I0);
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m[i+1] = m[i] + dt*(alphaM(V[i])*(1-m[i]) - betaM(V[i])*m[i]);
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h[i+1] = h[i] + dt*(alphaH(V[i])*(1-h[i]) - betaH(V[i])*h[i]);
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n[i+1] = n[i] + dt*(alphaN(V[i])*(1-n[i]) - betaN(V[i])*n[i]);
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return V,m,h,n,t

Diff for: ‎01.ipynb

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"ylabel('Voltage [$\\mu$ V]') # Wrap latex characters in $..$\n",
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"show()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<a id=\"donate\"></a>\n",
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"## Donate\n",
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"If you enjoy Case-Studies-Python, and would like to share your enjoyment with us, sponsor our coffee consuption <a href=\"https://www.paypal.com/donate/?hosted_button_id=DL8P5ZGS9962U\">here</a>."
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]
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}
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],
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"metadata": {

Diff for: ‎01.md

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@@ -952,3 +952,7 @@ xlabel('Time [s]')
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ylabel('Voltage [$\mu$ V]') # Wrap latex characters in $..$
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show()
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```
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<a id="donate"></a>
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## Donate
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If you enjoy Case-Studies-Python, and would like to share your enjoyment with us, sponsor our coffee consuption <a href="https://www.paypal.com/donate/?hosted_button_id=DL8P5ZGS9962U">here</a>.

Diff for: ‎02.ipynb

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@@ -37,7 +37,8 @@
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" * [Comparing ERPs](#comparing-erps)\n",
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" * [Confidence Intervals for the ERP (Method 2)](#cis-m2)\n",
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" * [A Bootstrap Test to Compare ERPs](#bootstrap)\n",
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"* [Summary](#summary)"
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"* [Summary](#summary)\n",
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"* [Donate](#donate)"
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]
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},
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{
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"source": [
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"[Return to top](#introduction)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<a id=\"donate\"></a>\n",
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"## Donate\n",
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"If you enjoy Case-Studies-Python, and would like to share your enjoyment with us, sponsor our coffee consuption <a href=\"https://www.paypal.com/donate/?hosted_button_id=DL8P5ZGS9962U\">here</a>."
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]
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}
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],
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"metadata": {

Diff for: ‎02.md

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@@ -40,6 +40,7 @@ _**Synopsis**_
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* [Confidence Intervals for the ERP (Method 2)](#cis-m2)
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* [A Bootstrap Test to Compare ERPs](#bootstrap)
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* [Summary](#summary)
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* [Donate](#donate)
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+++
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+++
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[Return to top](#introduction)
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+++
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<a id="donate"></a>
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## Donate
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If you enjoy Case-Studies-Python, and would like to share your enjoyment with us, sponsor our coffee consuption <a href="https://www.paypal.com/donate/?hosted_button_id=DL8P5ZGS9962U">here</a>.

Diff for: ‎03.ipynb

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@@ -43,6 +43,7 @@
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" 5. [Decibel scaling](#decibel-scaling)\n",
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" 6. [The spectrogram](#the-spectrogram)\n",
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"* [Summary](#summary)\n",
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"* [Donate](#donate)\n",
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"---\n",
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"* [Supplements](#supplements)\n",
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" 1. [Biased versus unbiased autocovariance](#supplement-acv)\n",
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"[Return to top](#top)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<a id=\"donate\"></a>\n",
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"## Donate\n",
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"If you enjoy Case-Studies-Python, and would like to share your enjoyment with us, sponsor our coffee consuption <a href=\"https://www.paypal.com/donate/?hosted_button_id=DL8P5ZGS9962U\">here</a>."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},

Diff for: ‎03.md

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@@ -45,6 +45,7 @@ _**Synopsis**_
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5. [Decibel scaling](#decibel-scaling)
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6. [The spectrogram](#the-spectrogram)
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* [Summary](#summary)
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* [Donate](#donate)
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---
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* [Supplements](#supplements)
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1. [Biased versus unbiased autocovariance](#supplement-acv)
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+++
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<a id="donate"></a>
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## Donate
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If you enjoy Case-Studies-Python, and would like to share your enjoyment with us, sponsor our coffee consuption <a href="https://www.paypal.com/donate/?hosted_button_id=DL8P5ZGS9962U">here</a>.
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+++
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---
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+++

Diff for: ‎04.ipynb

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" 4. [Beyond the Hanning Taper—the Multitaper Method](#multitaper)\n",
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" 5. [Confidence Intervals of the Spectrum](#ci)\n",
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"* [Summary](#summary)\n",
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"* [Donate](#donate)\n",
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"* [Appendix: Multiplication and Convolution in Different Domains](#appendix)"
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]
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},
@@ -1298,6 +1299,15 @@
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"[Return to top](#top)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<a id=\"donate\"></a>\n",
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"## Donate\n",
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"If you enjoy Case-Studies-Python, and would like to share your enjoyment with us, sponsor our coffee consuption <a href=\"https://www.paypal.com/donate/?hosted_button_id=DL8P5ZGS9962U\">here</a>."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},

Diff for: ‎04.md

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@@ -39,6 +39,7 @@ _**Synopsis**_
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4. [Beyond the Hanning Taper—the Multitaper Method](#multitaper)
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5. [Confidence Intervals of the Spectrum](#ci)
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* [Summary](#summary)
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* [Donate](#donate)
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* [Appendix: Multiplication and Convolution in Different Domains](#appendix)
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+++
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+++
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<a id="donate"></a>
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## Donate
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If you enjoy Case-Studies-Python, and would like to share your enjoyment with us, sponsor our coffee consuption <a href="https://www.paypal.com/donate/?hosted_button_id=DL8P5ZGS9962U">here</a>.
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+++
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<a id="appendix"></a>
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## Appendix: Multiplication and Convolution in Different Domains
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Diff for: ‎05.ipynb

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" * [Visualizing the Phase Difference across Trials](#Visualizing_the_Phase_Difference)\n",
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" * [Single-Trial Coherence](#single_trial_coherence)\n",
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"* [Relation between Statistical Modeling and Coherence](#Relation_between_Statistical_Modeling_and_Coherence)\n",
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"* [Summary](#summary)"
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"* [Summary](#summary)\n",
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"* [Donate](#summary)"
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]
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},
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{
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"\n",
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"- <a href=\"http://numerical.recipes/\" rel=\"external\">Numerical recipes</a>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<a id=\"donate\"></a>\n",
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"## Donate\n",
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"If you enjoy Case-Studies-Python, and would like to share your enjoyment with us, sponsor our coffee consuption <a href=\"https://www.paypal.com/donate/?hosted_button_id=DL8P5ZGS9962U\">here</a>."
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]
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}
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],
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"metadata": {

Diff for: ‎05.md

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@@ -49,6 +49,7 @@ _**Synopsis**_
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* [Single-Trial Coherence](#single_trial_coherence)
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* [Relation between Statistical Modeling and Coherence](#Relation_between_Statistical_Modeling_and_Coherence)
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* [Summary](#summary)
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* [Donate](#summary)
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- <a href="https://www.elsevier.com/books/spectral-analysis-and-time-series-two-volume-set/priestley/978-0-08-057055-6" rel="external">Priestly, 1982</a>
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- <a href="http://numerical.recipes/" rel="external">Numerical recipes</a>
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+++
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<a id="donate"></a>
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## Donate
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If you enjoy Case-Studies-Python, and would like to share your enjoyment with us, sponsor our coffee consuption <a href="https://www.paypal.com/donate/?hosted_button_id=DL8P5ZGS9962U">here</a>.

Diff for: ‎06.ipynb

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" 4. [More Sophisticated Filtering](#advanced-filters)\n",
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" 5. [What’s Phase Got to Do with It?](#phase)\n",
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" 6. [Analysis of the Filtered EEG Data](#analysis)\n",
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"* [Summary](#summary)"
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"* [Summary](#summary)\n",
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"* [Donate](#donate)"
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]
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},
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{
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"cell_type": "markdown",
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"metadata": {},
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"outputs": [],
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"source": []
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"source": [
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"<a id=\"donate\"></a>\n",
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"## Donate\n",
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"If you enjoy Case-Studies-Python, and would like to share your enjoyment with us, sponsor our coffee consuption <a href=\"https://www.paypal.com/donate/?hosted_button_id=DL8P5ZGS9962U\">here</a>."
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]
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}
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],
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"metadata": {

Diff for: ‎06.md

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5. [What’s Phase Got to Do with It?](#phase)
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6. [Analysis of the Filtered EEG Data](#analysis)
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* [Summary](#summary)
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* [Donate](#donate)
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[Back to Top](#top)
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```{code-cell} ipython3
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+++
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```
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<a id="donate"></a>
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## Donate
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If you enjoy Case-Studies-Python, and would like to share your enjoyment with us, sponsor our coffee consuption <a href="https://www.paypal.com/donate/?hosted_button_id=DL8P5ZGS9962U">here</a>.

Diff for: ‎07.ipynb

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" 3. [Determine if the phase and amplitude are related](#step3)\n",
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" * [The phase-amplitude plot](#m1) \n",
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"* [Summary](#summary)\n",
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"* [Donate](#donate)\n",
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"* [Appendix](#appendix)"
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"In developing these approaches, we utilized expertise and procedures developed in other modules. In particular, we relied on the notions of frequency and power, amplitude and phase, filtering, and resampling. Such a multifaceted approach is typical in analysis of neural data, where we leverage the skills developed in analyzing other datasets."
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<a id=\"donate\"></a>\n",
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"## Donate\n",
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"If you enjoy Case-Studies-Python, and would like to share your enjoyment with us, sponsor our coffee consuption <a href=\"https://www.paypal.com/donate/?hosted_button_id=DL8P5ZGS9962U\">here</a>."
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]
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},
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{
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"metadata": {},
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"\n",
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"This time domain representation of the HIlbert Transform is equivalent to the frequency domain representation. However, the time domain representation is much less intuitive. Compare the previous equation to the statement, \"*The Hilbert Transform is a 90-degree phase shift in the frequency domain.*\" The latter, we propose, is much more intuitive."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {

Diff for: ‎07.md

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3. [Determine if the phase and amplitude are related](#step3)
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* [The phase-amplitude plot](#m1)
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* [Summary](#summary)
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* [Donate](#donate)
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* [Appendix](#appendix)
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<a id="donate"></a>
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## Donate
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If you enjoy Case-Studies-Python, and would like to share your enjoyment with us, sponsor our coffee consuption <a href="https://www.paypal.com/donate/?hosted_button_id=DL8P5ZGS9962U">here</a>.
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## Appendix: Hilbert Transform in the Time Domain
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$$
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This time domain representation of the HIlbert Transform is equivalent to the frequency domain representation. However, the time domain representation is much less intuitive. Compare the previous equation to the statement, "*The Hilbert Transform is a 90-degree phase shift in the frequency domain.*" The latter, we propose, is much more intuitive.
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```{code-cell} ipython3
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```

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