1 00:00:14,297 --> 00:00:15,150 MICHALE FEE: OK. 2 00:00:15,150 --> 00:00:18,770 So let's go ahead and get started. 3 00:00:18,770 --> 00:00:22,700 So what is neural computation? 4 00:00:22,700 --> 00:00:27,740 So neuroscience used to be a very descriptive field 5 00:00:27,740 --> 00:00:31,070 where you would describe the different kinds of neurons. 6 00:00:31,070 --> 00:00:36,800 Who here has seen the famous pictures-- the old pictures 7 00:00:36,800 --> 00:00:40,970 of the golgi-stained neurons, all those different types 8 00:00:40,970 --> 00:00:46,610 of neurons, describing what things look like in the brain 9 00:00:46,610 --> 00:00:49,760 and what parts of the brain are important for what 10 00:00:49,760 --> 00:00:53,540 kinds of behavior based on lesion studies. 11 00:00:53,540 --> 00:00:55,520 It used to be extremely descriptive. 12 00:00:55,520 --> 00:00:57,710 But things are changing in neuroscience, 13 00:00:57,710 --> 00:01:00,540 and have changed dramatically over the past few decades. 14 00:01:00,540 --> 00:01:06,060 Really, neuroscience now is about understanding the brain, 15 00:01:06,060 --> 00:01:08,660 how the brain works, how the brain produces behavior. 16 00:01:08,660 --> 00:01:12,500 And really trying to develop engineering-level descriptions 17 00:01:12,500 --> 00:01:16,820 of brain systems and brain circuits and neurons and ion 18 00:01:16,820 --> 00:01:19,660 channels and all the components of neurons that make the brain 19 00:01:19,660 --> 00:01:20,160 work. 20 00:01:20,160 --> 00:01:26,210 And so, for example, the level of description 21 00:01:26,210 --> 00:01:30,590 that my lab works at and that I'm most excited about 22 00:01:30,590 --> 00:01:34,250 is understanding how neural circuits-- how neurons 23 00:01:34,250 --> 00:01:37,100 are put together to make neural circuits that 24 00:01:37,100 --> 00:01:42,290 implement behaviors, or to produce 25 00:01:42,290 --> 00:01:43,800 let's say object recognition. 26 00:01:43,800 --> 00:01:45,770 So this is a figure from Jim DiCarlo, 27 00:01:45,770 --> 00:01:48,080 who is our department head. 28 00:01:48,080 --> 00:01:50,360 Basically a circuit-level description 29 00:01:50,360 --> 00:01:52,790 of how the brain goes from a visual stimulus 30 00:01:52,790 --> 00:01:58,960 to a recognition of what that object is in the stimulus. 31 00:01:58,960 --> 00:02:01,370 Now at the same time that there's 32 00:02:01,370 --> 00:02:05,040 been a big push toward understanding or generating 33 00:02:05,040 --> 00:02:08,090 an engineering level descriptions of brains 34 00:02:08,090 --> 00:02:10,970 and circuits and components, neurons, 35 00:02:10,970 --> 00:02:16,010 there's also been tremendous advances in the technologies 36 00:02:16,010 --> 00:02:18,740 that we can use to record neurons. 37 00:02:18,740 --> 00:02:24,590 So there are now imaging systems and microscopes 38 00:02:24,590 --> 00:02:28,020 that can image thousands of neurons simultaneously. 39 00:02:28,020 --> 00:02:35,360 This is an example of a movie recorded in an awake baby mouse 40 00:02:35,360 --> 00:02:38,510 that's basically dreaming. 41 00:02:38,510 --> 00:02:41,310 And let me just show you what this looks like. 42 00:02:41,310 --> 00:02:45,940 So this is a mouse that has a fluorescent protein that's 43 00:02:45,940 --> 00:02:48,340 sensitive to neural activity. 44 00:02:48,340 --> 00:02:51,460 And so when neurons in a part of the brain become active 45 00:02:51,460 --> 00:02:53,690 they become fluorescent and light up. 46 00:02:53,690 --> 00:02:58,450 And so here's a top surface of the mouse's brain. 47 00:02:58,450 --> 00:03:03,070 And you can see this spontaneous activity flickering around 48 00:03:03,070 --> 00:03:06,010 as this mouse is just dreaming and thinking 49 00:03:06,010 --> 00:03:10,050 about whatever it's thinking about. 50 00:03:10,050 --> 00:03:15,540 So one of the key challenges is to take images 51 00:03:15,540 --> 00:03:18,480 like this that represent the activity of thousands 52 00:03:18,480 --> 00:03:20,490 of neurons or millions of neurons 53 00:03:20,490 --> 00:03:24,990 and figure out how to relate that to the circuit models that 54 00:03:24,990 --> 00:03:26,080 are being developed. 55 00:03:26,080 --> 00:03:29,020 So here's another example. 56 00:03:29,020 --> 00:03:32,220 So there are these new probes where these are basically 57 00:03:32,220 --> 00:03:34,620 silicon probes that have thousands 58 00:03:34,620 --> 00:03:38,250 of little sensors on them and a computer here 59 00:03:38,250 --> 00:03:40,600 that basically reads out the pattern of activity. 60 00:03:40,600 --> 00:03:42,430 These are called neuropixels. 61 00:03:42,430 --> 00:03:44,580 So those are basically electrodes 62 00:03:44,580 --> 00:03:47,640 that can, again, record from thousands of neurons 63 00:03:47,640 --> 00:03:48,630 simultaneously. 64 00:03:48,630 --> 00:03:51,030 And they're quite long and can record 65 00:03:51,030 --> 00:03:55,260 throughout the whole brain, essentially, all at once. 66 00:03:55,260 --> 00:04:00,090 So the key now is you have these very high dimensional data set. 67 00:04:00,090 --> 00:04:04,020 How do you relate that to the circuit models 68 00:04:04,020 --> 00:04:05,140 that you're developing? 69 00:04:05,140 --> 00:04:08,370 And so one of the key challenges in neuroscience 70 00:04:08,370 --> 00:04:12,660 is to take very large data sets that 71 00:04:12,660 --> 00:04:15,120 look like this that just look like a mess 72 00:04:15,120 --> 00:04:18,870 and figure out what's going on underneath of there. 73 00:04:18,870 --> 00:04:22,110 It turns out that people are discovering that while you 74 00:04:22,110 --> 00:04:25,270 might be recording from tens of thousands of neurons 75 00:04:25,270 --> 00:04:26,910 and it looks really messy that there's 76 00:04:26,910 --> 00:04:31,140 some underlying very simple structure underneath of there. 77 00:04:31,140 --> 00:04:32,790 But you can't see it when you just 78 00:04:32,790 --> 00:04:39,030 look at big collections of neurons like this. 79 00:04:39,030 --> 00:04:42,240 So the challenge here is really to figure out 80 00:04:42,240 --> 00:04:45,340 how to not only just make those models, 81 00:04:45,340 --> 00:04:48,035 but test them by taking data and relating 82 00:04:48,035 --> 00:04:50,160 the patterns of activity that you see in these very 83 00:04:50,160 --> 00:04:54,810 high dimensional data sets, do dimensionality reduction-- 84 00:04:54,810 --> 00:04:58,470 compress that data down into a simple representation-- 85 00:04:58,470 --> 00:05:01,980 and then relate it to those models that you developed. 86 00:05:05,040 --> 00:05:07,500 One of the things we're going to try to do in this class 87 00:05:07,500 --> 00:05:13,500 is to apply these techniques of making models of neurons 88 00:05:13,500 --> 00:05:17,100 and circuits together with mathematical tools 89 00:05:17,100 --> 00:05:21,180 for analyzing data in the context of looking 90 00:05:21,180 --> 00:05:23,730 at animal behaviors. 91 00:05:23,730 --> 00:05:31,740 So for example, in my lab we study how songbirds sing, 92 00:05:31,740 --> 00:05:34,230 how they learn to produce their vocalizations. 93 00:05:34,230 --> 00:05:36,270 Songbirds learn by imitating their parents. 94 00:05:36,270 --> 00:05:38,108 They listen to their parents. 95 00:05:38,108 --> 00:05:43,088 [BIRDS SINGING] 96 00:05:49,300 --> 00:05:50,640 Here, hold on. 97 00:05:50,640 --> 00:05:54,422 I'm going to skip ahead. 98 00:05:54,422 --> 00:05:55,901 How do I do that? 99 00:05:55,901 --> 00:06:00,338 [BIRDS SINGING] 100 00:06:02,310 --> 00:06:06,254 [INAUDIBLE] bring up the-- 101 00:06:06,254 --> 00:06:08,719 I was hoping I'd be able to skip ahead. 102 00:06:12,960 --> 00:06:17,610 So this is just a setup showing how 103 00:06:17,610 --> 00:06:21,090 we can record from neurons in birds while they're singing 104 00:06:21,090 --> 00:06:22,920 and figure out how those circuits work 105 00:06:22,920 --> 00:06:24,510 to produce the song. 106 00:06:24,510 --> 00:06:27,010 This is a little micro-drive that we built. 107 00:06:27,010 --> 00:06:30,120 It's motorized so that we can move these electrodes 108 00:06:30,120 --> 00:06:34,110 around independently in the brain and record from neurons 109 00:06:34,110 --> 00:06:36,840 without the animal knowing that we're moving the electrodes 110 00:06:36,840 --> 00:06:39,870 around and looking for neurons. 111 00:06:39,870 --> 00:06:41,110 So songbirds are really cool. 112 00:06:41,110 --> 00:06:42,420 They listen to their parents. 113 00:06:42,420 --> 00:06:44,760 They store a memory of what their parents sing. 114 00:06:44,760 --> 00:06:46,470 And then they begin babbling. 115 00:06:46,470 --> 00:06:49,500 And they practice over and over again until they can learn 116 00:06:49,500 --> 00:06:50,920 a good copy of their song. 117 00:06:50,920 --> 00:06:53,160 So here's a bird that's singing with the micro-drive 118 00:06:53,160 --> 00:06:54,930 on its head. 119 00:06:54,930 --> 00:06:57,075 And you can hear the neuron in the background. 120 00:06:57,075 --> 00:06:57,700 [STATIC SOUNDS] 121 00:06:57,700 --> 00:06:59,700 Sorry, it's not over the loudspeaker here. 122 00:06:59,700 --> 00:07:03,690 But can everyone hear that? 123 00:07:03,690 --> 00:07:06,485 So we can record from neurons while the bird is singing. 124 00:07:06,485 --> 00:07:07,110 [BIRDS SINGING] 125 00:07:07,110 --> 00:07:10,050 Look at the activity in this network 126 00:07:10,050 --> 00:07:12,960 and try to figure out how that network actually 127 00:07:12,960 --> 00:07:14,880 works to produce the song. 128 00:07:14,880 --> 00:07:17,740 And also we can record in very young birds 129 00:07:17,740 --> 00:07:22,050 and figure out how the song is actually learned. 130 00:07:22,050 --> 00:07:26,850 And there's an example of a neuron generating 131 00:07:26,850 --> 00:07:31,920 action potentials, which is the basic unit of communication 132 00:07:31,920 --> 00:07:32,550 in the brain. 133 00:07:32,550 --> 00:07:37,540 [BIRDS SINGING] 134 00:07:37,540 --> 00:07:40,440 And we try to build circuit models 135 00:07:40,440 --> 00:07:44,248 and figure out how that thing actually works 136 00:07:44,248 --> 00:07:45,540 to produce and learn this song. 137 00:07:52,220 --> 00:07:53,930 So these computational approaches 138 00:07:53,930 --> 00:07:55,490 that I'm talking about are not just 139 00:07:55,490 --> 00:08:00,120 important for dissecting brain circuits related to behavior. 140 00:08:00,120 --> 00:08:03,320 The same kinds of approaches, the same kind 141 00:08:03,320 --> 00:08:07,640 of dimensionality reduction techniques we're going to learn 142 00:08:07,640 --> 00:08:12,160 are also useful in molecular genetic studies, 143 00:08:12,160 --> 00:08:15,395 like taking transcriptional profiling 144 00:08:15,395 --> 00:08:18,020 and doing clustering and looking at the different patterns that 145 00:08:18,020 --> 00:08:18,520 are there. 146 00:08:18,520 --> 00:08:21,230 It's also useful for molecular studies. 147 00:08:21,230 --> 00:08:27,050 Also, these ideas are very powerful in studying cognition. 148 00:08:27,050 --> 00:08:29,480 So if you look at the work that Josh Tenenbaum does 149 00:08:29,480 --> 00:08:34,400 and Josh McDermott, who developed mathematical models 150 00:08:34,400 --> 00:08:39,020 of how our minds work, how we learn to think about things, 151 00:08:39,020 --> 00:08:42,027 those are also very model-based and very quantitative. 152 00:08:42,027 --> 00:08:44,360 So the kinds of tools we're going to learn in this class 153 00:08:44,360 --> 00:08:46,610 are very broadly applicable. 154 00:08:46,610 --> 00:08:50,040 They're also increasingly important in medicine. 155 00:08:50,040 --> 00:08:54,890 So at some point we're going to take a little bit of a detour 156 00:08:54,890 --> 00:08:58,910 to look at a particular disease that's caused 157 00:08:58,910 --> 00:09:02,042 by a defect in an ion channel. 158 00:09:02,042 --> 00:09:03,500 And it turns out you can understand 159 00:09:03,500 --> 00:09:06,590 exactly how that defect in that ion channel 160 00:09:06,590 --> 00:09:11,260 relates to the phenotype of the disease. 161 00:09:11,260 --> 00:09:15,260 And you can do that by creating a mathematical model of how 162 00:09:15,260 --> 00:09:18,440 a neuron behaves when it has an ion channel that 163 00:09:18,440 --> 00:09:20,210 has this defect in it. 164 00:09:20,210 --> 00:09:20,960 So it's very cool. 165 00:09:20,960 --> 00:09:22,790 And once you model it, you can really 166 00:09:22,790 --> 00:09:24,125 understand why that happens. 167 00:09:27,840 --> 00:09:29,500 So here are some of the course goals. 168 00:09:29,500 --> 00:09:31,110 So we're going to start by working 169 00:09:31,110 --> 00:09:33,750 on basic biophysics of neurons and networks 170 00:09:33,750 --> 00:09:38,280 and other principles underlying brain and cognitive functions. 171 00:09:38,280 --> 00:09:40,530 We're going to develop mathematical techniques 172 00:09:40,530 --> 00:09:46,020 to analyze those models and to analyze the behavioral data 173 00:09:46,020 --> 00:09:48,060 and neural data that you would take 174 00:09:48,060 --> 00:09:50,580 to study those brain circuits. 175 00:09:50,580 --> 00:09:54,720 And along the way, we're going to become proficient at using 176 00:09:54,720 --> 00:09:58,170 MATLAB to do these things. 177 00:09:58,170 --> 00:10:01,670 So how many of you have experience with MATLAB? 178 00:10:01,670 --> 00:10:02,960 OK, great. 179 00:10:02,960 --> 00:10:05,510 And not? 180 00:10:05,510 --> 00:10:09,050 So anybody who doesn't have experience with MATLAB, 181 00:10:09,050 --> 00:10:12,830 we're going to really make an effort to bring you up 182 00:10:12,830 --> 00:10:14,720 to speed very quickly. 183 00:10:14,720 --> 00:10:18,440 Daniel has actually just created a very nice MATLAB cheat 184 00:10:18,440 --> 00:10:20,870 sheet that's just amazing. 185 00:10:20,870 --> 00:10:26,110 So there will be lots of help with programming. 186 00:10:26,110 --> 00:10:27,840 So let me just mention some of the topics 187 00:10:27,840 --> 00:10:29,010 that we'll be covering. 188 00:10:29,010 --> 00:10:32,370 So we'll be talking about equivalent circuit 189 00:10:32,370 --> 00:10:35,715 model of neurons. 190 00:10:35,715 --> 00:10:38,320 So let me just explain how this is broken down. 191 00:10:38,320 --> 00:10:40,560 So these are topics that we'll be covering. 192 00:10:40,560 --> 00:10:43,080 And these are the mathematical tools 193 00:10:43,080 --> 00:10:45,150 that go along with those topics that we'll 194 00:10:45,150 --> 00:10:47,790 be learning about in parallel. 195 00:10:47,790 --> 00:10:50,220 So we'll be studying neuronal biophysics. 196 00:10:50,220 --> 00:10:52,620 And we'll be doing some differential equations 197 00:10:52,620 --> 00:10:55,230 along the way for that, just first-order linear differential 198 00:10:55,230 --> 00:10:58,020 equations, nothing to be scared of. 199 00:10:58,020 --> 00:10:59,910 We'll talk about neuronal responses 200 00:10:59,910 --> 00:11:01,830 to stimuli and tuning curves. 201 00:11:01,830 --> 00:11:03,330 And along the way, we'll be learning 202 00:11:03,330 --> 00:11:06,000 about spike sorting and peristimulus, time histograms, 203 00:11:06,000 --> 00:11:09,330 and ways of analyzing firing patterns. 204 00:11:09,330 --> 00:11:13,060 We talked about neural coding and receptive fields. 205 00:11:13,060 --> 00:11:15,270 And we'll learn about correlation and convolution 206 00:11:15,270 --> 00:11:17,270 for that topic. 207 00:11:17,270 --> 00:11:19,617 We'll talk about feed forward networks and perceptrons. 208 00:11:19,617 --> 00:11:21,200 And then we're going to start bringing 209 00:11:21,200 --> 00:11:25,890 a lot of linear algebra, which is really fun. 210 00:11:25,890 --> 00:11:27,320 It's really powerful. 211 00:11:27,320 --> 00:11:30,590 And that linear algebra sets the stage 212 00:11:30,590 --> 00:11:33,170 for then doing dimensionality, reduction 213 00:11:33,170 --> 00:11:36,890 on data, and principal component analysis, and singular value 214 00:11:36,890 --> 00:11:39,580 decomposition, and other things. 215 00:11:39,580 --> 00:11:44,590 We'll then take an additional extension of neural networks 216 00:11:44,590 --> 00:11:45,940 from feed forward networks. 217 00:11:45,940 --> 00:11:48,130 We'll figure out how to make them talk back 218 00:11:48,130 --> 00:11:50,590 to themselves so they can start doing things 219 00:11:50,590 --> 00:11:54,790 like remember things and make decisions. 220 00:11:54,790 --> 00:11:59,170 And that involves more linear algebra, eigenvalues. 221 00:11:59,170 --> 00:12:01,340 And then I'm not sure we're going 222 00:12:01,340 --> 00:12:06,070 to get time to sensory integration and Bayes' rule. 223 00:12:06,070 --> 00:12:07,720 So by the end of the class, there 224 00:12:07,720 --> 00:12:11,180 are some important skills that you'll have. 225 00:12:11,180 --> 00:12:16,180 You'll be able to think about a neuron very clearly 226 00:12:16,180 --> 00:12:19,120 and how its components work together 227 00:12:19,120 --> 00:12:23,220 to give that neuron its properties. 228 00:12:23,220 --> 00:12:27,060 And how neurons themselves can connect together 229 00:12:27,060 --> 00:12:29,580 to give a neural circuit its properties. 230 00:12:29,580 --> 00:12:33,030 You'll be able to write MATLAB programs that 231 00:12:33,030 --> 00:12:34,770 simulate those models. 232 00:12:34,770 --> 00:12:38,760 You'll be able to analyze data using MATLAB. 233 00:12:38,760 --> 00:12:41,850 You'll be able to visualize high dimensional data sets. 234 00:12:41,850 --> 00:12:44,970 And one of my goals in this class 235 00:12:44,970 --> 00:12:47,010 is that you guys should be able to go 236 00:12:47,010 --> 00:12:51,390 into any lab in the department and do 237 00:12:51,390 --> 00:12:53,910 cool things that even the graduate students may not 238 00:12:53,910 --> 00:12:55,840 know how to do. 239 00:12:55,840 --> 00:12:58,200 And so you can do really great stuff as a UROP. 240 00:13:06,070 --> 00:13:10,360 So one of the most important things about this class 241 00:13:10,360 --> 00:13:12,130 is problem sets because that's where 242 00:13:12,130 --> 00:13:15,610 you're going to get the hands-on experience to do that data 243 00:13:15,610 --> 00:13:21,280 analysis and write programs and analyze the data. 244 00:13:21,280 --> 00:13:22,240 Please install that. 245 00:13:22,240 --> 00:13:27,290 It's really important, if you don't already have that. 246 00:13:27,290 --> 00:13:30,010 We use live scripts for problems set submissions. 247 00:13:30,010 --> 00:13:33,510 And Daniel made some nice examples on Stellar. 248 00:13:33,510 --> 00:13:36,830 And of course the guidelines for Pset submissions 249 00:13:36,830 --> 00:13:39,810 are also on Stellar. 250 00:13:39,810 --> 00:13:40,888 OK, that's it. 251 00:13:40,888 --> 00:13:41,930 Any questions about that? 252 00:13:46,020 --> 00:13:46,850 No? 253 00:13:46,850 --> 00:13:48,050 All right, good. 254 00:13:48,050 --> 00:13:51,430 So let's go ahead and get started then 255 00:13:51,430 --> 00:13:52,375 with the first topic. 256 00:14:00,340 --> 00:14:01,210 OK. 257 00:14:01,210 --> 00:14:04,750 So the first thing we're going to do 258 00:14:04,750 --> 00:14:10,150 is we're going to build a model of a neuron. 259 00:14:10,150 --> 00:14:12,550 This model is very particular. 260 00:14:12,550 --> 00:14:18,890 It uses electrical components to describe the neuron. 261 00:14:18,890 --> 00:14:22,630 Now that may not be surprising since a neuron is basically 262 00:14:22,630 --> 00:14:24,220 an electrical device. 263 00:14:24,220 --> 00:14:27,370 It has components that are sensitive to voltages, 264 00:14:27,370 --> 00:14:30,400 that generate currents, that control currents. 265 00:14:30,400 --> 00:14:33,820 And so we're going to build our model using electrical circuit 266 00:14:33,820 --> 00:14:34,580 components. 267 00:14:34,580 --> 00:14:37,030 And one of the nice things about doing that 268 00:14:37,030 --> 00:14:40,370 is that every electrical circuit component, 269 00:14:40,370 --> 00:14:42,640 like a resistor or a capacitor, has 270 00:14:42,640 --> 00:14:47,950 a very well-defined mathematical relation between the current 271 00:14:47,950 --> 00:14:50,260 and the voltage, the current that 272 00:14:50,260 --> 00:14:52,870 flows through that device and the voltage 273 00:14:52,870 --> 00:14:54,670 across the terminals of that device. 274 00:14:54,670 --> 00:14:58,300 So you can write down very precisely, mathematically, 275 00:14:58,300 --> 00:15:02,040 what each of those components does. 276 00:15:02,040 --> 00:15:07,050 So then you can then take all those components 277 00:15:07,050 --> 00:15:12,270 and construct a set of equations or in general a set 278 00:15:12,270 --> 00:15:16,890 of differential equations that allows you to basically evolve 279 00:15:16,890 --> 00:15:19,950 that circuit over time and plot, let's say, 280 00:15:19,950 --> 00:15:23,530 the voltage on the inside of the cell as a function of time. 281 00:15:23,530 --> 00:15:27,390 And you can see that that model neuron can actually 282 00:15:27,390 --> 00:15:33,410 very precisely replicate many of the properties of neurons. 283 00:15:33,410 --> 00:15:37,010 Now neurons are actually really complicated. 284 00:15:37,010 --> 00:15:40,850 And this is the real reason why we need to write down a model. 285 00:15:40,850 --> 00:15:42,740 So there are many different kinds of neurons. 286 00:15:42,740 --> 00:15:45,500 Each type of neuron has a different pattern 287 00:15:45,500 --> 00:15:46,800 of genes that are expressed. 288 00:15:46,800 --> 00:15:51,530 So this is a cluster diagram of neuron type based 289 00:15:51,530 --> 00:15:55,760 on a transcriptional profiling of the RNA 290 00:15:55,760 --> 00:15:59,000 that I think it was about 13,000 neurons that were extracted 291 00:15:59,000 --> 00:16:00,830 from a part of the brain. 292 00:16:00,830 --> 00:16:02,420 You do a transcriptional profiling. 293 00:16:02,420 --> 00:16:04,880 It gives you a map of all the different genes 294 00:16:04,880 --> 00:16:06,908 are expressed in each neuron. 295 00:16:06,908 --> 00:16:08,450 And then you can cluster them and you 296 00:16:08,450 --> 00:16:11,480 can see that this particular part of the brain, 297 00:16:11,480 --> 00:16:15,200 which is in the hypothalamus, expresses all 298 00:16:15,200 --> 00:16:17,672 of these different cell types. 299 00:16:17,672 --> 00:16:19,130 Now what are those different genes? 300 00:16:19,130 --> 00:16:21,170 Many of those different genes are actually 301 00:16:21,170 --> 00:16:22,430 different ion channels. 302 00:16:22,430 --> 00:16:26,090 And there are hundreds of different kinds of ion channels 303 00:16:26,090 --> 00:16:29,300 that control the flow of current across the membrane 304 00:16:29,300 --> 00:16:30,600 of the neuron. 305 00:16:30,600 --> 00:16:34,610 So this is just a diagram showing different potassium ion 306 00:16:34,610 --> 00:16:37,160 channels, different calcium ion channels. 307 00:16:37,160 --> 00:16:41,505 You can see they have families and different subtypes. 308 00:16:41,505 --> 00:16:43,130 And all of those different ion channels 309 00:16:43,130 --> 00:16:45,110 have different timescales on which 310 00:16:45,110 --> 00:16:48,230 the current varies as a function of voltage change. 311 00:16:48,230 --> 00:16:50,090 They have different voltage ranges 312 00:16:50,090 --> 00:16:51,770 that they're sensitive to. 313 00:16:51,770 --> 00:16:53,420 They have different inactivation. 314 00:16:53,420 --> 00:16:56,610 So many ion channels, when you turn them on, they stay on. 315 00:16:56,610 --> 00:16:58,700 But other ion channels, they turn on 316 00:16:58,700 --> 00:17:01,640 and then they slowly decay away. 317 00:17:01,640 --> 00:17:03,170 The current slowly decays away. 318 00:17:03,170 --> 00:17:04,609 And that's called inactivation. 319 00:17:04,609 --> 00:17:06,109 And all these different ion channels 320 00:17:06,109 --> 00:17:08,960 have different combinations of those properties. 321 00:17:08,960 --> 00:17:12,290 And it's really hard to predict when 322 00:17:12,290 --> 00:17:15,319 you think about how this neuron will 323 00:17:15,319 --> 00:17:18,450 behave with a different kind of ion channel here. 324 00:17:18,450 --> 00:17:22,579 It's super hard to just look at the properties of an ion 325 00:17:22,579 --> 00:17:25,332 channel and just see how that's going to work in a neuron 326 00:17:25,332 --> 00:17:27,290 because you have all these different parts that 327 00:17:27,290 --> 00:17:29,285 are working together. 328 00:17:29,285 --> 00:17:31,790 And so it's really important to be able to write down 329 00:17:31,790 --> 00:17:32,990 a mathematical model. 330 00:17:32,990 --> 00:17:35,930 If you have a neuron that has a different kind of ion channel, 331 00:17:35,930 --> 00:17:40,400 you can actually predict how the neuron's going to behave. 332 00:17:40,400 --> 00:17:43,970 Now that's just the ion channel components. 333 00:17:43,970 --> 00:17:47,010 Neurons also have complex morphologies. 334 00:17:50,090 --> 00:17:52,220 This is a Purkinje cell in the cerebellum. 335 00:17:52,220 --> 00:17:56,240 They have these very densely elaborated dendrites. 336 00:17:56,240 --> 00:17:58,160 Other neurons have very long dendrites 337 00:17:58,160 --> 00:17:59,990 with just a few branches. 338 00:17:59,990 --> 00:18:02,960 Other neurons have very short stubby dendrites. 339 00:18:02,960 --> 00:18:07,670 And each of those different morphological patterns 340 00:18:07,670 --> 00:18:12,440 also affects how a neuron responds to its inputs, 341 00:18:12,440 --> 00:18:15,020 because now a neuron can have inputs 342 00:18:15,020 --> 00:18:17,450 out here at the end of the dendrite or up close 343 00:18:17,450 --> 00:18:19,340 to the soma. 344 00:18:19,340 --> 00:18:22,640 And all of those, the spatial structure, 345 00:18:22,640 --> 00:18:25,700 also affects how a neuron responds. 346 00:18:25,700 --> 00:18:28,310 And those produce very different firing patterns. 347 00:18:28,310 --> 00:18:31,760 So some neurons, if you put in a constant current, 348 00:18:31,760 --> 00:18:34,130 they just fire regularly up. 349 00:18:34,130 --> 00:18:37,070 So it turns out we can really understand 350 00:18:37,070 --> 00:18:41,360 why all these different things happen 351 00:18:41,360 --> 00:18:44,030 if we build a model like this. 352 00:18:47,265 --> 00:18:48,640 So let me just point out a couple 353 00:18:48,640 --> 00:18:50,560 of other interesting things about this model. 354 00:18:50,560 --> 00:18:52,390 Different parts of this circuit actually 355 00:18:52,390 --> 00:18:54,610 do cool different things. 356 00:18:54,610 --> 00:18:58,300 So neurons have not just one power supply. 357 00:18:58,300 --> 00:19:00,767 They've got multiple power supplies to power 358 00:19:00,767 --> 00:19:02,350 up different parts of the circuit that 359 00:19:02,350 --> 00:19:05,290 do different things. 360 00:19:05,290 --> 00:19:08,380 Neurons have capacitances that allow 361 00:19:08,380 --> 00:19:14,050 a neuron to accumulate over time, act as an integrator. 362 00:19:14,050 --> 00:19:17,470 If you combine a capacitor with a resistor, 363 00:19:17,470 --> 00:19:20,200 that circuit now looks like a filter. 364 00:19:20,200 --> 00:19:24,430 It smooths its past inputs over time. 365 00:19:24,430 --> 00:19:27,430 And these two components here, this sodium current 366 00:19:27,430 --> 00:19:30,790 and this potassium current, make a spike generator 367 00:19:30,790 --> 00:19:32,650 that generates an action potential that 368 00:19:32,650 --> 00:19:34,930 then talks to other neurons. 369 00:19:34,930 --> 00:19:36,610 And you put that whole thing together, 370 00:19:36,610 --> 00:19:38,620 and that thing can act like an oscillator. 371 00:19:38,620 --> 00:19:40,900 It can act like a coincidence detector. 372 00:19:40,900 --> 00:19:44,800 It can do all kinds of different cool things. 373 00:19:44,800 --> 00:19:48,170 And all that stuff is understandable 374 00:19:48,170 --> 00:19:52,490 if you just write down a simple model like this. 375 00:19:52,490 --> 00:19:53,533 Any questions? 376 00:19:56,970 --> 00:19:59,360 So what we're going to do is we're 377 00:19:59,360 --> 00:20:04,610 going to just start describing this network. 378 00:20:04,610 --> 00:20:07,880 We're going to build it up one piece at a time. 379 00:20:07,880 --> 00:20:13,260 And we're going to start with a capacitance. 380 00:20:13,260 --> 00:20:14,990 But before we get to the capacitor, 381 00:20:14,990 --> 00:20:16,820 we need to do one more thing. 382 00:20:16,820 --> 00:20:18,620 We need to do one thing first, which 383 00:20:18,620 --> 00:20:22,760 is figure out what the wires are in the brain. 384 00:20:22,760 --> 00:20:26,220 For an electrical circuit, you need to have wires. 385 00:20:26,220 --> 00:20:30,700 So what are the wires in the brain? 386 00:20:30,700 --> 00:20:33,110 What do wires do in a circuit? 387 00:20:33,110 --> 00:20:35,890 They carry current. 388 00:20:35,890 --> 00:20:42,060 So what are the wires in a neuron? 389 00:20:42,060 --> 00:20:43,388 AUDIENCE: Axons? 390 00:20:43,388 --> 00:20:44,430 MICHALE FEE: What's that? 391 00:20:44,430 --> 00:20:45,142 AUDIENCE: Axons? 392 00:20:45,142 --> 00:20:46,170 MICHALE FEE: Axons. 393 00:20:46,170 --> 00:20:48,580 So axons carry information. 394 00:20:48,580 --> 00:20:52,110 They carry a spike that travels down the axon 395 00:20:52,110 --> 00:20:53,430 and goes to other neurons. 396 00:20:53,430 --> 00:20:57,940 But there is even a simpler answer than that. 397 00:20:57,940 --> 00:20:58,440 Yes? 398 00:20:58,440 --> 00:20:59,691 AUDIENCE: Ion channels? 399 00:20:59,691 --> 00:21:03,600 MICHALE FEE: Ion channels are these resistors here. 400 00:21:03,600 --> 00:21:08,084 But what is it that connects all those components to each other? 401 00:21:08,084 --> 00:21:09,834 AUDIENCE: Intracellular and extracellular. 402 00:21:09,834 --> 00:21:10,930 MICHALE FEE: Excellent. 403 00:21:10,930 --> 00:21:14,680 It's the intracellular and extracellular solution. 404 00:21:14,680 --> 00:21:16,810 And so what we're going to do today 405 00:21:16,810 --> 00:21:20,530 is to understand how the intracellular and extracellular 406 00:21:20,530 --> 00:21:25,420 solution acts as a wire in our neuron. 407 00:21:25,420 --> 00:21:31,150 And it's not quite as simple as a piece of metal. 408 00:21:31,150 --> 00:21:32,800 It's a bit more complicated. 409 00:21:32,800 --> 00:21:34,570 There are different ways you can get 410 00:21:34,570 --> 00:21:39,315 current flow in intracellular and extracellular solution. 411 00:21:39,315 --> 00:21:40,690 So we're going to go through that 412 00:21:40,690 --> 00:21:43,630 and we're going to analyze that in some detail. 413 00:21:47,410 --> 00:21:53,200 So in the brain, the wires are the intracellular 414 00:21:53,200 --> 00:21:55,480 and extracellular salt solutions. 415 00:21:55,480 --> 00:21:58,240 And you get current flow that results 416 00:21:58,240 --> 00:22:03,730 from the movement of ions in that aqueous solution. 417 00:22:03,730 --> 00:22:07,420 So the solution consists of ions. 418 00:22:07,420 --> 00:22:10,690 Like in the extracellular, it's mostly 419 00:22:10,690 --> 00:22:15,080 sodium ions and chloride ions that are dissolved in water. 420 00:22:15,080 --> 00:22:16,630 Water is a polar solvent. 421 00:22:16,630 --> 00:22:19,970 That means that the negative parts, the oxygen that's 422 00:22:19,970 --> 00:22:21,520 slightly negatively charged. 423 00:22:21,520 --> 00:22:25,810 Oxygen is attracted toward positive ions. 424 00:22:25,810 --> 00:22:28,540 And the intracellular and extracellular space 425 00:22:28,540 --> 00:22:31,120 are filled with salt solution at a concentration 426 00:22:31,120 --> 00:22:33,091 of about 100 millimolar. 427 00:22:36,390 --> 00:22:38,430 And that corresponds to having one 428 00:22:38,430 --> 00:22:45,310 of these ions about every 25 angstroms apart. 429 00:22:45,310 --> 00:22:49,200 So at those concentrations, there 430 00:22:49,200 --> 00:22:52,580 are a lot of ions floating around. 431 00:22:52,580 --> 00:22:57,000 And those ions can move under different conditions 432 00:22:57,000 --> 00:22:59,540 to produce currents. 433 00:22:59,540 --> 00:23:03,620 So currents flow in the brain through two 434 00:23:03,620 --> 00:23:05,450 primary different mechanisms. 435 00:23:05,450 --> 00:23:11,510 Diffusion, which is caused by variations 436 00:23:11,510 --> 00:23:14,150 in the concentration. 437 00:23:14,150 --> 00:23:18,000 And drifts of particles in an electric field. 438 00:23:18,000 --> 00:23:19,530 So when you put an electric field, 439 00:23:19,530 --> 00:23:22,490 so if you take a beaker filled with salt solution, 440 00:23:22,490 --> 00:23:25,490 you put two metal electrodes in it, 441 00:23:25,490 --> 00:23:27,170 you produce an electric field that 442 00:23:27,170 --> 00:23:30,380 causes these ions to drift in-- 443 00:23:30,380 --> 00:23:32,533 and that's another source of current 444 00:23:32,533 --> 00:23:33,950 that we're going to look at today. 445 00:23:37,330 --> 00:23:41,030 So here are our learning objectives for today. 446 00:23:41,030 --> 00:23:44,350 We're going to understand how the timescales of diffusion 447 00:23:44,350 --> 00:23:46,420 relate to the length scales. 448 00:23:46,420 --> 00:23:48,040 That's a really interesting story. 449 00:23:48,040 --> 00:23:50,710 That's very important. 450 00:23:50,710 --> 00:23:53,700 We're going to understand how concentration gradients lead 451 00:23:53,700 --> 00:23:54,630 to currents. 452 00:23:54,630 --> 00:23:57,680 That's known as Fick's First Law. 453 00:23:57,680 --> 00:24:01,700 And we're going to understand how charges drift 454 00:24:01,700 --> 00:24:05,120 in an electric field in a way that leads to current, 455 00:24:05,120 --> 00:24:07,850 and the mathematical relation that 456 00:24:07,850 --> 00:24:09,760 describes voltage differences. 457 00:24:09,760 --> 00:24:12,660 And this is called Ohm's Law in the brain. 458 00:24:12,660 --> 00:24:15,388 And we're going to learn about the concept of resistivity. 459 00:24:19,970 --> 00:24:21,860 So the first thing we need to talk about, 460 00:24:21,860 --> 00:24:25,380 if we're going to talk about diffusion, is thermal energy. 461 00:24:25,380 --> 00:24:28,880 So every particle in the world is 462 00:24:28,880 --> 00:24:35,350 being jostled by other particles that are crashing into it. 463 00:24:35,350 --> 00:24:38,740 And at thermal equilibrium, every degree of freedom, 464 00:24:38,740 --> 00:24:42,160 every way that a particle can move, either forward 465 00:24:42,160 --> 00:24:46,300 and backward, left and right, up and down, or rotations, 466 00:24:46,300 --> 00:24:48,980 this way or this way, or whichever way, 467 00:24:48,980 --> 00:24:54,610 I didn't show yet, come to equilibrium 468 00:24:54,610 --> 00:24:58,870 at a particular energy that's proportional to temperature. 469 00:24:58,870 --> 00:25:01,570 In other words, if a particle is moving 470 00:25:01,570 --> 00:25:03,640 in this direction in equilibrium, 471 00:25:03,640 --> 00:25:08,740 it will have a kinetic energy in that direction that's 472 00:25:08,740 --> 00:25:10,160 proportional to the temperature. 473 00:25:10,160 --> 00:25:13,270 And that temperature is in units of kelvin 474 00:25:13,270 --> 00:25:15,040 relative to absolute zero. 475 00:25:15,040 --> 00:25:19,550 And the proportionality constant is the Boltzmann constant, 476 00:25:19,550 --> 00:25:24,020 which has units of joules per kelvin. 477 00:25:24,020 --> 00:25:27,610 So when you multiply the Boltzmann constant k times 478 00:25:27,610 --> 00:25:30,910 temperature, what you find is that every degree of freedom 479 00:25:30,910 --> 00:25:33,550 will come to equilibrium at 4 times 10 480 00:25:33,550 --> 00:25:39,910 to the minus 21 joules, which is an amount of energy, 481 00:25:39,910 --> 00:25:42,310 at room temperature. 482 00:25:42,310 --> 00:25:46,120 At zero temperature, you can see that every degree of freedom 483 00:25:46,120 --> 00:25:47,320 has zero energy. 484 00:25:47,320 --> 00:25:48,550 And so nothing is moving. 485 00:25:48,550 --> 00:25:52,390 Nothing's rotating, nothing's moving any direction. 486 00:25:52,390 --> 00:25:53,680 Everything's perfectly still. 487 00:25:56,660 --> 00:26:01,670 So let's calculate how fast particles 488 00:26:01,670 --> 00:26:04,820 move at thermal equilibrium in room temperature. 489 00:26:04,820 --> 00:26:11,660 So you may remember from your first physics class 490 00:26:11,660 --> 00:26:13,550 that the kinetic energy of a particle 491 00:26:13,550 --> 00:26:17,640 is proportional to the velocity squared, 1/2 mv squared. 492 00:26:17,640 --> 00:26:20,990 So the average velocity squared of a particle 493 00:26:20,990 --> 00:26:27,610 at thermal equilibrium is just 1/2 times that much energy. 494 00:26:27,610 --> 00:26:29,090 That makes sense? 495 00:26:29,090 --> 00:26:34,430 Now we [AUDIO OUT] how fast a particle is moving-- 496 00:26:34,430 --> 00:26:38,680 for example, a sodium ion. 497 00:26:38,680 --> 00:26:41,980 So you can see that the average velocity squared is just 498 00:26:41,980 --> 00:26:43,150 kT over m. 499 00:26:43,150 --> 00:26:45,890 We just divide both sides by m. 500 00:26:45,890 --> 00:26:48,200 So the average velocity squared is kT over m. 501 00:26:48,200 --> 00:26:50,360 The mass of a sodium ion is this. 502 00:26:50,360 --> 00:26:52,490 So the average velocity squared is 503 00:26:52,490 --> 00:26:54,980 10 to the 5 meter squared per second squared. 504 00:26:54,980 --> 00:26:56,690 Just take the square root that, and you 505 00:26:56,690 --> 00:27:01,230 get the average velocity is 320 meters per second. 506 00:27:01,230 --> 00:27:05,150 So that means that the air molecules, which 507 00:27:05,150 --> 00:27:08,840 have a similar mass to sodium ion, 508 00:27:08,840 --> 00:27:11,790 are whizzing around at 300 meters per second. 509 00:27:11,790 --> 00:27:17,955 So that would cross this room in a few hundredths of a second. 510 00:27:17,955 --> 00:27:19,580 But of course, that's not what happens. 511 00:27:19,580 --> 00:27:25,550 Particles don't just go whizzing along at 300 meters per second. 512 00:27:25,550 --> 00:27:28,705 What happens to them? 513 00:27:28,705 --> 00:27:30,410 AUDIENCE: Bump into each other. 514 00:27:30,410 --> 00:27:31,618 MICHALE FEE: Into each other. 515 00:27:31,618 --> 00:27:35,520 They're all crashing into each other constantly. 516 00:27:35,520 --> 00:27:40,440 So in solution, a particle collides with a water molecule 517 00:27:40,440 --> 00:27:44,910 every about 10 to the 13 times per second. 518 00:27:44,910 --> 00:27:47,850 10 to the minus 13 seconds between collisions. 519 00:27:47,850 --> 00:27:50,750 So that means the particle is moving a little bit crashing, 520 00:27:50,750 --> 00:27:52,500 moving in a different direction, crashing, 521 00:27:52,500 --> 00:27:55,410 moving in a different direction and crashing. 522 00:27:55,410 --> 00:28:01,200 So if you follow one particle, it's just jumping around, 523 00:28:01,200 --> 00:28:02,403 it's diffusing. 524 00:28:02,403 --> 00:28:03,570 So what does that look like? 525 00:28:03,570 --> 00:28:08,260 Daniel made a little video that shows to scale. 526 00:28:08,260 --> 00:28:10,650 This is position in micron. 527 00:28:10,650 --> 00:28:13,000 And time is in real-time. 528 00:28:13,000 --> 00:28:17,280 So this video shows in real-time what the motion of a particle 529 00:28:17,280 --> 00:28:18,060 might look like. 530 00:28:22,590 --> 00:28:27,750 In each point, it's moving, colliding, and moving off 531 00:28:27,750 --> 00:28:30,050 in some random direction. 532 00:28:35,690 --> 00:28:37,840 You can actually see this. 533 00:28:37,840 --> 00:28:41,670 If you look at a very small particle-- 534 00:28:41,670 --> 00:28:45,480 who was it, Daniel, who did that experiment looking at pollen? 535 00:28:45,480 --> 00:28:46,592 It's Brownian, at Brown. 536 00:28:46,592 --> 00:28:47,175 AUDIENCE: Yup. 537 00:28:47,175 --> 00:28:49,590 MICHALE FEE: What was his first name? 538 00:28:49,590 --> 00:28:50,790 Brown. 539 00:28:50,790 --> 00:28:51,720 Brownian motion. 540 00:28:51,720 --> 00:28:53,620 Have you heard of Brownian motion? 541 00:28:53,620 --> 00:28:56,070 So somebody named Brown was looking 542 00:28:56,070 --> 00:29:02,010 at pollen particles in water and noticing that they jump around, 543 00:29:02,010 --> 00:29:04,820 just like this. 544 00:29:04,820 --> 00:29:07,790 And he hypothesized that they were being jostled around 545 00:29:07,790 --> 00:29:08,450 by the water. 546 00:29:13,520 --> 00:29:14,720 Any questions? 547 00:29:18,790 --> 00:29:22,370 So what can we say about this? 548 00:29:22,370 --> 00:29:25,840 There's something really interesting about diffusion 549 00:29:25,840 --> 00:29:29,480 that's very non-intuitive at first. 550 00:29:29,480 --> 00:29:33,250 Diffusion has some really strange aspect to it. 551 00:29:33,250 --> 00:29:36,950 That a distance that a particle can diffuse 552 00:29:36,950 --> 00:29:41,390 depends very much on the time that you allow. 553 00:29:41,390 --> 00:29:43,850 And it's not just a simple relation. 554 00:29:43,850 --> 00:29:46,200 So let's just look at this. 555 00:29:46,200 --> 00:29:48,200 So let's ask how much time does it 556 00:29:48,200 --> 00:29:51,860 take for an ion to diffuse a short distance, 557 00:29:51,860 --> 00:29:57,320 like across the soma of a neuron. 558 00:29:57,320 --> 00:30:01,150 So an ion can diffuse across the soma of a neuron 559 00:30:01,150 --> 00:30:02,560 in about a 20th of a second. 560 00:30:07,548 --> 00:30:08,840 How about down it at dendrites. 561 00:30:08,840 --> 00:30:12,960 So let's start our ion in the cell body. 562 00:30:12,960 --> 00:30:14,640 And ask, how long does it take an iron 563 00:30:14,640 --> 00:30:17,980 to reach the end of a dendrite that can be about a millimeter 564 00:30:17,980 --> 00:30:18,480 away. 565 00:30:21,610 --> 00:30:25,610 Can take about 10 minutes on average. 566 00:30:25,610 --> 00:30:27,350 That's how long it will take an iron 567 00:30:27,350 --> 00:30:29,540 to get that far away from its starting point. 568 00:30:32,640 --> 00:30:35,880 So you can see, 20th of a second here. 569 00:30:35,880 --> 00:30:41,250 And here it's like 500 seconds. 570 00:30:41,250 --> 00:30:43,050 About 10 minutes. 571 00:30:43,050 --> 00:30:45,390 How long does it take an ion, starting at the cell body, 572 00:30:45,390 --> 00:30:47,392 to diffuse all the way down-- 573 00:30:47,392 --> 00:30:49,350 so you know there are neurons in your body that 574 00:30:49,350 --> 00:30:54,110 start in your spinal cord and go all the way down to your feet. 575 00:30:54,110 --> 00:30:59,030 So motor neurons in your spinal cord can have very long axons. 576 00:30:59,030 --> 00:31:03,590 So how long does it take an ion to get from the soma 577 00:31:03,590 --> 00:31:07,400 all the way down to the end of an axon, a long axon? 578 00:31:13,220 --> 00:31:16,960 Somebody just take a guess. 579 00:31:16,960 --> 00:31:19,840 It's 20th of a second here, 10 minutes here. 580 00:31:23,110 --> 00:31:25,010 Anybody want to guess? 581 00:31:25,010 --> 00:31:27,170 An hour, yup. 582 00:31:27,170 --> 00:31:29,180 10 years. 583 00:31:29,180 --> 00:31:30,560 OK. 584 00:31:30,560 --> 00:31:31,310 Why is that? 585 00:31:31,310 --> 00:31:33,595 That's crazy, right? 586 00:31:33,595 --> 00:31:34,470 How is that possible? 587 00:31:34,470 --> 00:31:35,860 And that's an ion. 588 00:31:35,860 --> 00:31:40,320 So a cell body is making proteins and all kinds of stuff 589 00:31:40,320 --> 00:31:42,930 that have to get down to build synapses 590 00:31:42,930 --> 00:31:44,760 at the other end of that axon. 591 00:31:44,760 --> 00:31:49,350 And proteins diffuse a heck of a lot slower than ions do. 592 00:31:49,350 --> 00:31:54,480 So basically a cell body could make stuff for the axon, 593 00:31:54,480 --> 00:31:58,430 and it would never get there in your entire lifetime. 594 00:31:58,430 --> 00:32:02,730 And that's why cells have to actually make little trains. 595 00:32:02,730 --> 00:32:04,260 They literally make little trains. 596 00:32:04,260 --> 00:32:06,490 They package up stuff and put it on the train 597 00:32:06,490 --> 00:32:12,070 and it just marches down the axon until it gets to the end. 598 00:32:12,070 --> 00:32:15,250 And this is the reason why. 599 00:32:15,250 --> 00:32:18,070 So what we're going to do is I'm going to just walk you 600 00:32:18,070 --> 00:32:21,590 through a very simple derivation of why this is true 601 00:32:21,590 --> 00:32:24,790 and how to think about this. 602 00:32:24,790 --> 00:32:27,250 So here's what we're going to do. 603 00:32:27,250 --> 00:32:30,580 So normally things diffuse in three dimensions, right? 604 00:32:30,580 --> 00:32:32,870 But it's just much harder to analyze things in three 605 00:32:32,870 --> 00:32:33,370 dimensions.. 606 00:32:33,370 --> 00:32:36,430 So you can get basically the right answer 607 00:32:36,430 --> 00:32:39,520 just by analyzing how things diffuse in one dimension. 608 00:32:39,520 --> 00:32:41,950 So Daniel made this little video to show you 609 00:32:41,950 --> 00:32:43,160 what this looks like. 610 00:32:43,160 --> 00:32:47,968 This is I think 100 particles all lined up near zero. 611 00:32:47,968 --> 00:32:49,510 And we're going to turn on the video. 612 00:32:49,510 --> 00:32:52,300 We're going to let them all start diffusing at one moment. 613 00:32:52,300 --> 00:32:54,037 So you can just watch what happens to all 614 00:32:54,037 --> 00:32:55,120 these different particles. 615 00:32:58,360 --> 00:33:01,380 So you can see that some particles end up 616 00:33:01,380 --> 00:33:02,550 over here on the left. 617 00:33:02,550 --> 00:33:05,230 Other particles end up over here on the right. 618 00:33:05,230 --> 00:33:08,790 You can see that the distribution of particles 619 00:33:08,790 --> 00:33:11,490 spreads out. 620 00:33:11,490 --> 00:33:16,740 And so we're going to figure out why that is, why that happens. 621 00:33:16,740 --> 00:33:18,490 So the first thing I just want to tell you 622 00:33:18,490 --> 00:33:21,430 is that the distribution of particles, 623 00:33:21,430 --> 00:33:24,880 if they all start at zero, and they diffuse 624 00:33:24,880 --> 00:33:28,870 in 1D away from zero, the distribution that you get 625 00:33:28,870 --> 00:33:30,500 is Gaussian. 626 00:33:30,500 --> 00:33:34,720 And the basic reason is that, let's start at the center, 627 00:33:34,720 --> 00:33:37,210 and on every time step they have a probability of 1/2 628 00:33:37,210 --> 00:33:41,560 of going to the right and 1/2 of going to the left. 629 00:33:41,560 --> 00:33:45,760 And so basically there are many more combinations 630 00:33:45,760 --> 00:33:49,330 of ways a particle can do some lefts and do some rights 631 00:33:49,330 --> 00:33:51,250 and end up back where it started. 632 00:33:51,250 --> 00:33:53,290 It's very unlikely that the particle 633 00:33:53,290 --> 00:33:57,670 will do a whole bunch of going right all in a row. 634 00:33:57,670 --> 00:34:01,780 And so that's why the density and the distribution 635 00:34:01,780 --> 00:34:03,995 is very low down here. 636 00:34:03,995 --> 00:34:05,620 And so you end up with something that's 637 00:34:05,620 --> 00:34:07,735 just a Gaussian distribution. 638 00:34:12,690 --> 00:34:15,790 So let's analyze this in a little more detail. 639 00:34:15,790 --> 00:34:19,469 So we're going to just make a very simple model of particles 640 00:34:19,469 --> 00:34:21,434 stepping to the right or to the left. 641 00:34:26,159 --> 00:34:29,100 We're going to consider a particle that 642 00:34:29,100 --> 00:34:33,790 is moving left or right at a fixed velocity vx for some time 643 00:34:33,790 --> 00:34:35,580 tau before a collision. 644 00:34:35,580 --> 00:34:37,830 And we're going to imagine that each time the particle 645 00:34:37,830 --> 00:34:41,250 collides it resets its velocity randomly, either to the left 646 00:34:41,250 --> 00:34:43,280 or to the right. 647 00:34:43,280 --> 00:34:46,489 So on every time step, half the particles 648 00:34:46,489 --> 00:34:49,500 will step right by a distance delta, 649 00:34:49,500 --> 00:34:52,699 which is the velocity times the time tau. 650 00:34:52,699 --> 00:34:54,260 And the other half of the particles 651 00:34:54,260 --> 00:34:57,500 will step left by that same distance. 652 00:34:57,500 --> 00:35:00,440 So they're going either to the left or to the right 653 00:35:00,440 --> 00:35:01,460 by a distance delta. 654 00:35:06,720 --> 00:35:10,820 So if we start with n particles and all of them 655 00:35:10,820 --> 00:35:17,660 start at position 0 at time 0, then 656 00:35:17,660 --> 00:35:23,060 we can write down the position of every particle at time step 657 00:35:23,060 --> 00:35:27,480 n, the i-th particle at time step n. 658 00:35:27,480 --> 00:35:30,270 And we're going to assume that each particle is independent, 659 00:35:30,270 --> 00:35:34,760 each doing their own thing, ignoring each other. 660 00:35:34,760 --> 00:35:38,560 So now you can see that you can write down 661 00:35:38,560 --> 00:35:42,850 the position of the particle at time step n is just 662 00:35:42,850 --> 00:35:45,490 the position of the particle at the previous time 663 00:35:45,490 --> 00:35:48,420 step, plus or minus this little delta. 664 00:35:51,270 --> 00:35:52,740 Any questions about that? 665 00:35:57,280 --> 00:36:03,250 So please, if you ever just haven't followed one step 666 00:36:03,250 --> 00:36:05,390 that I do, just let me know. 667 00:36:05,390 --> 00:36:07,030 I'm happy to explain it again. 668 00:36:07,030 --> 00:36:11,170 I often am watching somebody explaining something really 669 00:36:11,170 --> 00:36:14,440 simple, and my brain is just in some funny state 670 00:36:14,440 --> 00:36:16,660 and I just don't get it. 671 00:36:16,660 --> 00:36:21,020 So it's totally fine if you want me to explain something again. 672 00:36:21,020 --> 00:36:23,020 You don't have to be embarrassed because happens 673 00:36:23,020 --> 00:36:23,830 to me all the time. 674 00:36:26,920 --> 00:36:30,460 So now what we can do is use this expression, 675 00:36:30,460 --> 00:36:33,700 compute how that distribution evolves over time, 676 00:36:33,700 --> 00:36:39,070 how that distribution of particles, this i-th particle 677 00:36:39,070 --> 00:36:42,340 over time, time step n. 678 00:36:42,340 --> 00:36:44,650 All right, so let's calculate what the average position 679 00:36:44,650 --> 00:36:46,060 of the ensemble is. 680 00:36:46,060 --> 00:36:48,320 So these brackets mean average. 681 00:36:48,320 --> 00:36:51,490 So the bracket with an i, that I'm averaging 682 00:36:51,490 --> 00:36:55,525 this quantity over i particles. 683 00:36:55,525 --> 00:37:00,130 And so it's just the sum of positions 684 00:37:00,130 --> 00:37:02,800 for every particle, divided by the number of particles. 685 00:37:02,800 --> 00:37:04,910 That's the average position. 686 00:37:04,910 --> 00:37:08,590 So again, the position of the i-th particle at time step n 687 00:37:08,590 --> 00:37:11,570 is just the position of that particle at the previous time 688 00:37:11,570 --> 00:37:12,940 step, plus or minus delta. 689 00:37:12,940 --> 00:37:16,480 We just plug that into there, into there. 690 00:37:16,480 --> 00:37:18,320 And now we calculate the sum. 691 00:37:18,320 --> 00:37:19,640 But we have two terms. 692 00:37:19,640 --> 00:37:20,980 We have this term and that term. 693 00:37:20,980 --> 00:37:23,480 Let's break them up into two separate sums. 694 00:37:23,480 --> 00:37:27,880 So this is equal to the sum over the previous positions, 695 00:37:27,880 --> 00:37:33,310 plus the sum over how much the change was 696 00:37:33,310 --> 00:37:35,020 from one time step to the next. 697 00:37:35,020 --> 00:37:37,400 Does that makes sense? 698 00:37:37,400 --> 00:37:39,040 But what is this sum? 699 00:37:39,040 --> 00:37:41,800 We're summing over all the particles, 700 00:37:41,800 --> 00:37:44,770 how much they changed from the previous time step 701 00:37:44,770 --> 00:37:46,780 to this time step. 702 00:37:46,780 --> 00:37:49,200 Well, half of them moved to the right and half of them 703 00:37:49,200 --> 00:37:50,050 the left. 704 00:37:50,050 --> 00:37:54,260 So that sum is just zero. 705 00:37:54,260 --> 00:37:59,210 So you can see that the average position 706 00:37:59,210 --> 00:38:01,220 of the particles at this time step 707 00:38:01,220 --> 00:38:04,550 is just equal to the average position of the particles 708 00:38:04,550 --> 00:38:05,840 at the previous time step. 709 00:38:05,840 --> 00:38:08,690 And what that means is that the center of the distribution 710 00:38:08,690 --> 00:38:11,840 hasn't changed. 711 00:38:11,840 --> 00:38:15,800 If you start all the particles at zero, they diffuse around. 712 00:38:15,800 --> 00:38:18,970 The average position is still zero. 713 00:38:18,970 --> 00:38:19,650 Yes? 714 00:38:19,650 --> 00:38:22,555 AUDIENCE: [INAUDIBLE] bracket [INAUDIBLE].. 715 00:38:22,555 --> 00:38:23,500 MICHALE FEE: Yes. 716 00:38:23,500 --> 00:38:30,650 So this here is just this. 717 00:38:30,650 --> 00:38:35,480 So this bracket means I'm averaging over this quantity i. 718 00:38:35,480 --> 00:38:37,230 So you can see that's what I'm doing here. 719 00:38:37,230 --> 00:38:40,712 I'm summing over i and dividing by the number of particles. 720 00:38:40,712 --> 00:38:41,712 AUDIENCE: And what is i? 721 00:38:41,712 --> 00:38:44,270 MICHALE FEE: I is the particle number. 722 00:38:44,270 --> 00:38:48,920 So if we have 10 particles, i goes from 1 to 10. 723 00:38:48,920 --> 00:38:49,420 Thank you. 724 00:38:54,540 --> 00:38:56,970 So that's a little boring. 725 00:38:56,970 --> 00:39:00,030 But we used a trick here that we're 726 00:39:00,030 --> 00:39:02,040 going to use now to actually calculate 727 00:39:02,040 --> 00:39:05,640 the interesting thing, which is on average how far do 728 00:39:05,640 --> 00:39:09,950 the particles get from where they started. 729 00:39:09,950 --> 00:39:13,070 So what we're going to do is not calculate the average position 730 00:39:13,070 --> 00:39:15,350 of all the particles. 731 00:39:15,350 --> 00:39:19,610 We're going to calculate the average absolute value 732 00:39:19,610 --> 00:39:21,155 from where they started. 733 00:39:21,155 --> 00:39:22,800 Does that makes sense? 734 00:39:22,800 --> 00:39:27,132 We're going to ask, on average, how far did they get from where 735 00:39:27,132 --> 00:39:28,340 they started, which was zero. 736 00:39:32,260 --> 00:39:37,180 So absolute values, nobody likes. 737 00:39:37,180 --> 00:39:38,390 They're hard to deal with. 738 00:39:38,390 --> 00:39:43,580 But this is exactly the same as calculating the square root 739 00:39:43,580 --> 00:39:46,590 of the average square. 740 00:39:46,590 --> 00:39:50,942 It's the same as calculating the variance. 741 00:39:50,942 --> 00:39:52,493 Does that makes sense? 742 00:39:52,493 --> 00:39:53,910 So what we're going to do is we're 743 00:39:53,910 --> 00:39:57,380 going to calculate the variance of that distribution. 744 00:39:57,380 --> 00:39:58,890 And the square root of that variance 745 00:39:58,890 --> 00:40:02,670 is just the standard deviation, which is just how wide it is, 746 00:40:02,670 --> 00:40:07,500 which is just how far on average the particles got 747 00:40:07,500 --> 00:40:09,402 from where they started. 748 00:40:09,402 --> 00:40:11,490 Does that makes sense? 749 00:40:11,490 --> 00:40:13,930 So let's push on. 750 00:40:13,930 --> 00:40:17,412 We're going to calculate the average square distance. 751 00:40:17,412 --> 00:40:19,870 Now we're just going to take the square of that at the end. 752 00:40:19,870 --> 00:40:23,890 So the average of the position squared, we're 753 00:40:23,890 --> 00:40:26,740 going to plug this into here. 754 00:40:26,740 --> 00:40:28,190 So we're going to square it. 755 00:40:28,190 --> 00:40:31,240 So the position of the particle squared 756 00:40:31,240 --> 00:40:33,230 is just this quantity squared. 757 00:40:33,230 --> 00:40:35,450 Let's factor it out. 758 00:40:35,450 --> 00:40:38,530 So we have this term squared plus twice 759 00:40:38,530 --> 00:40:43,720 that times that here, plus that term squared. 760 00:40:43,720 --> 00:40:47,450 And we're going to now plug that average. 761 00:40:47,450 --> 00:40:50,620 So the average position squared is just the average. 762 00:40:50,620 --> 00:40:56,230 The average position squared at this time step n 763 00:40:56,230 --> 00:40:59,200 is the average position squared at the previous time 764 00:40:59,200 --> 00:41:01,940 step plus some other stuff. 765 00:41:01,940 --> 00:41:05,190 And let's take a look at what that other stuff is. 766 00:41:05,190 --> 00:41:06,400 What is this? 767 00:41:06,400 --> 00:41:10,180 This is plus or minus 2 times delta, 768 00:41:10,180 --> 00:41:15,060 which is the step it takes, the size of the step times x. 769 00:41:15,060 --> 00:41:18,520 So what is that average? 770 00:41:18,520 --> 00:41:21,660 Half of these are positive and half of these are negative. 771 00:41:21,660 --> 00:41:24,630 So the average is zero. 772 00:41:24,630 --> 00:41:27,450 And quantity is the average of delta squared. 773 00:41:27,450 --> 00:41:33,570 Well, delta squared is always positive, right? 774 00:41:33,570 --> 00:41:35,580 So what does this say? 775 00:41:35,580 --> 00:41:43,050 What this says is that the variance at this time step 776 00:41:43,050 --> 00:41:45,680 is just the variance at a previous time step 777 00:41:45,680 --> 00:41:47,410 is a constant. 778 00:41:47,410 --> 00:41:49,260 So let's analyze that. 779 00:41:49,260 --> 00:41:51,660 What this says is that at each time step, 780 00:41:51,660 --> 00:41:56,950 the variance grows by some constant. 781 00:41:56,950 --> 00:41:58,090 Delta is a distance. 782 00:41:58,090 --> 00:42:02,660 Delta squared is just the units of variance of a distribution 783 00:42:02,660 --> 00:42:03,910 that's a function of distance. 784 00:42:06,740 --> 00:42:10,010 So if the variance at time step 0 is 0, 785 00:42:10,010 --> 00:42:12,560 that means they're all lined up at the origin. 786 00:42:12,560 --> 00:42:15,660 One time step later, the variance will be delta squared. 787 00:42:15,660 --> 00:42:17,730 The next time step, it will be two delta squared. 788 00:42:17,730 --> 00:42:20,180 The next time step, dot, dot, dot. 789 00:42:20,180 --> 00:42:23,540 Up at some time step n, it will be n times delta squared. 790 00:42:23,540 --> 00:42:24,860 So you see what's happening? 791 00:42:24,860 --> 00:42:26,720 The variance of this distribution 792 00:42:26,720 --> 00:42:27,665 is growing linearly. 793 00:42:34,900 --> 00:42:38,870 We can change from time steps to continuous time. 794 00:42:38,870 --> 00:42:45,190 So the step number is just time divided 795 00:42:45,190 --> 00:42:48,880 by tau, which is some interval in time 796 00:42:48,880 --> 00:42:51,930 like the interval between collisions. 797 00:42:51,930 --> 00:42:54,790 And so you can see that the variance is just 798 00:42:54,790 --> 00:43:01,120 growing linearly in time where the variance is just 2 times d 799 00:43:01,120 --> 00:43:06,070 times T, where d is what we call the diffusion coefficient. 800 00:43:06,070 --> 00:43:09,010 It's just length squared divided by time. 801 00:43:13,400 --> 00:43:14,030 Why is that? 802 00:43:14,030 --> 00:43:20,155 Because as time grows, the variance grows linearly. 803 00:43:23,890 --> 00:43:28,180 So if we want to take time, multiply it 804 00:43:28,180 --> 00:43:30,430 by something that gives us variance, 805 00:43:30,430 --> 00:43:35,600 it has to be variance per unit time. 806 00:43:35,600 --> 00:43:38,090 And variance, for something that's 807 00:43:38,090 --> 00:43:42,760 a distribution of position, has to have position squared. 808 00:43:42,760 --> 00:43:43,260 Yes? 809 00:43:43,260 --> 00:43:46,324 AUDIENCE: But do we like [INAUDIBLE],, like that? 810 00:43:49,150 --> 00:43:51,400 MICHALE FEE: It's built into the definition 811 00:43:51,400 --> 00:43:53,460 of the diffusion constant, OK? 812 00:44:02,440 --> 00:44:03,490 Any questions about that? 813 00:44:08,110 --> 00:44:10,380 And now here here's the answer. 814 00:44:10,380 --> 00:44:13,170 So the variance is growing linearly in time. 815 00:44:13,170 --> 00:44:17,540 What that means is that the standard deviation, 816 00:44:17,540 --> 00:44:20,450 the average distance from the starting point, 817 00:44:20,450 --> 00:44:23,140 is growing as the square root of time. 818 00:44:25,980 --> 00:44:28,050 And that's key. 819 00:44:28,050 --> 00:44:29,430 That I want you to remember. 820 00:44:34,970 --> 00:44:40,930 The distance that a particle diffuses 821 00:44:40,930 --> 00:44:43,930 from its starting point on average grows 822 00:44:43,930 --> 00:44:45,210 is the square root of time. 823 00:44:52,240 --> 00:44:56,590 So for a small molecule, a typical small molecule, 824 00:44:56,590 --> 00:45:00,490 the diffusion constant is 10 to the minus 5 centimeters squared 825 00:45:00,490 --> 00:45:02,650 per second. 826 00:45:02,650 --> 00:45:07,140 And so now we can just plug in some distances in times 827 00:45:07,140 --> 00:45:12,480 and see how long it takes this particle 828 00:45:12,480 --> 00:45:14,850 to diffuse some distance. 829 00:45:14,850 --> 00:45:15,870 So let's do that. 830 00:45:15,870 --> 00:45:18,510 Let's plug in a length of 10 microns. 831 00:45:18,510 --> 00:45:20,730 That was our soma, our cell body. 832 00:45:20,730 --> 00:45:23,430 It's 10 to the minus 3 centimeters. 833 00:45:23,430 --> 00:45:27,980 Time is that squared, length squared. 834 00:45:27,980 --> 00:45:30,490 So it's 10 to the minus 6 centimeters squared divided 835 00:45:30,490 --> 00:45:32,410 by the diffusion constant. 836 00:45:32,410 --> 00:45:35,708 2 times the diffusion constant, 2 times 10 to the minus 5 837 00:45:35,708 --> 00:45:37,000 centimeters squared per second. 838 00:45:37,000 --> 00:45:39,010 You can see centimeter squareds cancel. 839 00:45:39,010 --> 00:45:41,160 That leaves us time. 840 00:45:41,160 --> 00:45:44,450 50 milliseconds. 841 00:45:44,450 --> 00:45:47,000 Now let's put in one millimeter. 842 00:45:47,000 --> 00:45:50,240 That was the length of our dendrite. 843 00:45:50,240 --> 00:45:53,570 So that's 10 to the minus 1 centimeter. 844 00:45:53,570 --> 00:45:56,670 So we plug that into our equation for time. 845 00:45:56,670 --> 00:45:58,790 Time is just L squared-- 846 00:45:58,790 --> 00:46:02,670 I forgot to actually write that down. 847 00:46:02,670 --> 00:46:04,395 Here's the equation that I'm solving. 848 00:46:08,200 --> 00:46:10,720 So what this equation at the bottom here is saying 849 00:46:10,720 --> 00:46:18,760 is some distance is equal to the square root of 2dT. 850 00:46:18,760 --> 00:46:22,600 And I'm just saying L squared is equal to 2 dT. 851 00:46:22,600 --> 00:46:28,000 And I'm solving for T, L squared over 2d. 852 00:46:28,000 --> 00:46:29,920 That's the equation I'm solving. 853 00:46:29,920 --> 00:46:33,795 I'm giving you a length and I'm calculating how long it takes. 854 00:46:40,110 --> 00:46:42,690 So if you put in 10 to the minus 1 here, 855 00:46:42,690 --> 00:46:44,900 you get 10 to the minus 2 divided by 2 times 10 856 00:46:44,900 --> 00:46:49,910 to the minus 500 seconds, which is about 10 minutes. 857 00:46:49,910 --> 00:46:54,610 And now if you ask how long does it take to go a meter, 858 00:46:54,610 --> 00:46:56,840 that's 10 to the 2 centimeters. 859 00:46:56,840 --> 00:47:00,180 That's 10 to the 4 divided by 10 to the minus 5. 860 00:47:00,180 --> 00:47:02,930 Somebody over here figured it out right away. 861 00:47:05,820 --> 00:47:09,160 About 5 times 10 to the 8 seconds, 862 00:47:09,160 --> 00:47:10,200 which is about 10 years. 863 00:47:10,200 --> 00:47:13,320 A year is pi times 10 to the 7 seconds, by the way. 864 00:47:17,700 --> 00:47:19,020 Plus or minus a few percent. 865 00:47:25,296 --> 00:47:26,550 Any questions about that? 866 00:47:33,400 --> 00:47:35,730 Cool, right? 867 00:47:35,730 --> 00:47:39,590 So neurons and cells and biology has 868 00:47:39,590 --> 00:47:45,130 to go to extraordinary lengths to overcome this craziness 869 00:47:45,130 --> 00:47:53,820 of diffusion, which explains a lot of the structure you 870 00:47:53,820 --> 00:47:54,600 see in cells. 871 00:48:06,870 --> 00:48:08,570 So you can see that diffusion causes 872 00:48:08,570 --> 00:48:12,380 the movement of ions from places where 873 00:48:12,380 --> 00:48:15,170 they're concentrated to places where there aren't so many 874 00:48:15,170 --> 00:48:15,950 ions. 875 00:48:15,950 --> 00:48:19,580 So let's take a little bit slightly more 876 00:48:19,580 --> 00:48:21,345 detail look at that idea. 877 00:48:21,345 --> 00:48:22,970 So what I'm going to tell you about now 878 00:48:22,970 --> 00:48:24,830 is called Fick's First Law. 879 00:48:24,830 --> 00:48:26,810 And the idea is that diffusion produces 880 00:48:26,810 --> 00:48:30,530 a net flow of particles from regions of high concentration 881 00:48:30,530 --> 00:48:34,010 to regions of lower concentration. 882 00:48:34,010 --> 00:48:36,260 And the flux of particles is proportional 883 00:48:36,260 --> 00:48:38,120 to the concentration gradient. 884 00:48:38,120 --> 00:48:40,070 Now this is just really obvious, right? 885 00:48:40,070 --> 00:48:45,770 If you have a box, and on the left side of the box you have n 886 00:48:45,770 --> 00:48:46,370 particles. 887 00:48:46,370 --> 00:48:48,830 Then on the right side of the box then 888 00:48:48,830 --> 00:48:51,643 you're going to have particles diffusing from here to there. 889 00:48:51,643 --> 00:48:53,060 And you're going to have particles 890 00:48:53,060 --> 00:48:55,190 diffusing from there to there. 891 00:48:55,190 --> 00:48:57,300 But because there are more of them over here, 892 00:48:57,300 --> 00:49:00,250 they're just going to be more particles going this way 893 00:49:00,250 --> 00:49:02,756 than there are that way. 894 00:49:02,756 --> 00:49:03,910 Does that makes sense? 895 00:49:07,370 --> 00:49:10,080 Let's say each particle here might 896 00:49:10,080 --> 00:49:13,530 have a 50% chance of diffusing here or staying here 897 00:49:13,530 --> 00:49:15,600 or diffusing somewhere else. 898 00:49:15,600 --> 00:49:18,047 Particles here also equally have probability 899 00:49:18,047 --> 00:49:18,880 of going either way. 900 00:49:18,880 --> 00:49:21,280 But just because there are more of them here, 901 00:49:21,280 --> 00:49:25,710 there's going to be more particles going that way. 902 00:49:25,710 --> 00:49:27,780 You can just calculate the number 903 00:49:27,780 --> 00:49:30,690 of particles going this way minus the number of particles 904 00:49:30,690 --> 00:49:33,270 going that way. 905 00:49:33,270 --> 00:49:34,680 And that gives you the net number 906 00:49:34,680 --> 00:49:36,930 of particles going to the right. 907 00:49:36,930 --> 00:49:40,920 But what does that look like? 908 00:49:40,920 --> 00:49:47,760 You have the number here minus the number some distance away. 909 00:49:47,760 --> 00:49:51,047 And what if you were to divide that by the distance? 910 00:49:51,047 --> 00:49:52,130 What would that look like? 911 00:49:54,660 --> 00:49:55,420 Good. 912 00:49:55,420 --> 00:49:56,980 It looks like a derivative. 913 00:49:56,980 --> 00:50:00,320 So if you calculate the flux, it's 914 00:50:00,320 --> 00:50:02,390 minus the diffusion constant times 915 00:50:02,390 --> 00:50:05,750 1 over delta, the separation between these boxes. 916 00:50:05,750 --> 00:50:09,140 It's the concentration here minus the concentration there. 917 00:50:09,140 --> 00:50:11,920 And that is just a derivative. 918 00:50:11,920 --> 00:50:15,050 And that's Fick's First Law. 919 00:50:15,050 --> 00:50:18,290 I have a few slides at the end of the lecture that 920 00:50:18,290 --> 00:50:21,540 do this derivation more completely. 921 00:50:21,540 --> 00:50:25,280 So please take a look at that if you have time. 922 00:50:27,940 --> 00:50:31,660 So now this is really an important concept. 923 00:50:31,660 --> 00:50:36,430 This Fick's First Law, the fact that concentration gradients 924 00:50:36,430 --> 00:50:40,720 produce a flow of ions, of particles, 925 00:50:40,720 --> 00:50:44,840 is so fundamental to how neurons work. 926 00:50:44,840 --> 00:50:46,640 And here we're going to be building 927 00:50:46,640 --> 00:50:50,410 that up over the course of the next couple lectures. 928 00:50:50,410 --> 00:50:52,230 So imagine that you have a cell that 929 00:50:52,230 --> 00:50:56,160 has a lot of potassium ions inside and very 930 00:50:56,160 --> 00:50:58,720 few potassium ions outside. 931 00:50:58,720 --> 00:51:00,100 Now you can see that you're going 932 00:51:00,100 --> 00:51:03,035 to have potassium ions diffusing from here. 933 00:51:03,035 --> 00:51:04,410 Sorry, and I forgot to say, let's 934 00:51:04,410 --> 00:51:07,710 say that your cell has a hole in it. 935 00:51:07,710 --> 00:51:10,710 So you're going to have potassium ions diffusing 936 00:51:10,710 --> 00:51:13,830 from inside to outside through the hole. 937 00:51:13,830 --> 00:51:16,200 You also have some potassium ions out here. 938 00:51:16,200 --> 00:51:19,030 And some of those might diffuse in. 939 00:51:19,030 --> 00:51:21,360 But there are just so many more potassium ions 940 00:51:21,360 --> 00:51:24,480 inside than outside concentration-wise 941 00:51:24,480 --> 00:51:26,760 that the probability of one going out through the hole 942 00:51:26,760 --> 00:51:29,760 is just much higher than the probability of a potassium ion 943 00:51:29,760 --> 00:51:31,410 going back into the cell. 944 00:51:36,240 --> 00:51:39,510 So here I'm just zooming in on that channel, 945 00:51:39,510 --> 00:51:41,400 on that pore through the membrane. 946 00:51:41,400 --> 00:51:44,160 Lots of potassium ions here. 947 00:51:44,160 --> 00:51:47,610 On average, there's going to be a net flow of potassium 948 00:51:47,610 --> 00:51:50,610 out through that hole. 949 00:51:50,610 --> 00:51:54,172 And we can plot the concentration gradient 950 00:51:54,172 --> 00:51:54,880 through the hole. 951 00:51:54,880 --> 00:51:57,490 And you can see it's high here, it decreases, 952 00:51:57,490 --> 00:52:00,670 and it's low outside. 953 00:52:00,670 --> 00:52:05,160 And so there's a net flow that's proportional to the steepness 954 00:52:05,160 --> 00:52:09,220 of concentration profile. 955 00:52:09,220 --> 00:52:11,850 So that's true, you get a net flow, 956 00:52:11,850 --> 00:52:14,770 even if each particle is diffusing independently. 957 00:52:14,770 --> 00:52:18,590 They don't know anything about each other. 958 00:52:18,590 --> 00:52:21,695 And yet that concentration gradient produces a current. 959 00:52:26,000 --> 00:52:29,030 All concentration gradients go away. 960 00:52:29,030 --> 00:52:31,360 Why is that? 961 00:52:31,360 --> 00:52:34,330 Because calcium ions will flow from the inside of the cell 962 00:52:34,330 --> 00:52:37,810 to the outside of the cell until they're the same concentration. 963 00:52:37,810 --> 00:52:39,790 And then you'll have just as many 964 00:52:39,790 --> 00:52:43,570 flowing back inside as you have flowing outside. 965 00:52:43,570 --> 00:52:44,290 Why? 966 00:52:44,290 --> 00:52:46,810 So eventually that would happen to all of our cells. 967 00:52:46,810 --> 00:52:47,810 Why doesn't that happen? 968 00:52:52,067 --> 00:52:54,432 AUDIENCE: [INAUDIBLE] because they're alive. 969 00:52:54,432 --> 00:52:57,070 MICHALE FEE: Well, that's exactly the right answer, 970 00:52:57,070 --> 00:53:01,440 but there are a few intermediate steps. 971 00:53:01,440 --> 00:53:04,470 If you were to not be alive anymore, 972 00:53:04,470 --> 00:53:06,390 the potassium ions would just diffuse out. 973 00:53:06,390 --> 00:53:08,520 And that would be the end. 974 00:53:08,520 --> 00:53:12,360 But what happens is there are other proteins 975 00:53:12,360 --> 00:53:14,970 in the membrane that take those potassium ions from here 976 00:53:14,970 --> 00:53:17,940 and pump them back inside and maintain the concentration 977 00:53:17,940 --> 00:53:18,660 gradient. 978 00:53:18,660 --> 00:53:20,100 But that costs energy. 979 00:53:20,100 --> 00:53:22,260 Those proteins use ATP. 980 00:53:22,260 --> 00:53:24,240 And that ATP comes from eating. 981 00:53:30,120 --> 00:53:32,850 But eventually all concentration gradients go away. 982 00:53:39,640 --> 00:53:43,420 So that is how we get current flow 983 00:53:43,420 --> 00:53:45,460 from concentration gradients. 984 00:53:45,460 --> 00:53:52,590 Now the next topic has to do with the diffusion of ions 985 00:53:52,590 --> 00:53:57,340 in the presence of voltage differences, 986 00:53:57,340 --> 00:54:00,800 in the presence of voltage gradients. 987 00:54:00,800 --> 00:54:03,790 The bottom line here that I want you to know, 988 00:54:03,790 --> 00:54:07,330 that I want you to understand, is that current flow in neurons 989 00:54:07,330 --> 00:54:08,640 obeys Ohm's Law. 990 00:54:08,640 --> 00:54:09,650 Now what does that mean? 991 00:54:09,650 --> 00:54:12,290 Let's imagine that we have a resistor. 992 00:54:12,290 --> 00:54:17,380 Let's say across a membrane or in the intracellular 993 00:54:17,380 --> 00:54:21,790 or extracellular space of a neuron. 994 00:54:21,790 --> 00:54:26,230 The current flow through that resistive medium 995 00:54:26,230 --> 00:54:31,780 is proportional to the voltage difference. 996 00:54:31,780 --> 00:54:33,250 So that's Ohm's Law. 997 00:54:33,250 --> 00:54:38,080 The current is proportional to the voltage difference 998 00:54:38,080 --> 00:54:41,020 across the two terminals, the two sides of the resistor. 999 00:54:41,020 --> 00:54:44,290 And the proportionality constant is 1 over the resistance. 1000 00:54:47,960 --> 00:54:50,680 So here current has units of amperes. 1001 00:54:53,260 --> 00:54:55,750 The voltage difference is units of volts. 1002 00:54:55,750 --> 00:54:59,600 And the resistance has units of ohms. 1003 00:54:59,600 --> 00:55:00,650 Any questions about that? 1004 00:55:06,130 --> 00:55:08,240 So let's go through-- 1005 00:55:08,240 --> 00:55:10,160 let's develop this idea a little bit more 1006 00:55:10,160 --> 00:55:15,350 and understand why it is that a voltage difference produces 1007 00:55:15,350 --> 00:55:18,470 a current that's proportional to voltage. 1008 00:55:18,470 --> 00:55:20,220 So let's go back to our little [AUDIO OUT] 1009 00:55:20,220 --> 00:55:21,800 filled with salt solution. 1010 00:55:21,800 --> 00:55:24,800 There are ions in here dissolved in the water. 1011 00:55:24,800 --> 00:55:26,060 We have two metal plates. 1012 00:55:26,060 --> 00:55:28,760 We've put a battery between the two metal plates that 1013 00:55:28,760 --> 00:55:31,580 holds those two plates at some fixed voltage difference 1014 00:55:31,580 --> 00:55:39,780 delta v. And we're going to ask what happens. 1015 00:55:39,780 --> 00:55:41,050 So let's zoom in here. 1016 00:55:41,050 --> 00:55:43,070 There is one plate that's at one potential. 1017 00:55:43,070 --> 00:55:45,500 There's another plate at another potential. 1018 00:55:45,500 --> 00:55:47,420 There's some voltage difference between those 1019 00:55:47,420 --> 00:55:52,940 that's delta v. The two plates are separated by a distance L. 1020 00:55:52,940 --> 00:56:02,660 And that voltage difference produces an electric field 1021 00:56:02,660 --> 00:56:06,530 that points from the high voltage region 1022 00:56:06,530 --> 00:56:07,800 to the low voltage region. 1023 00:56:10,970 --> 00:56:14,390 So an electric field produces a force on a charge-- 1024 00:56:14,390 --> 00:56:16,670 we have lots of charges in here-- 1025 00:56:16,670 --> 00:56:22,700 that's proportional to the charge and the electric field. 1026 00:56:22,700 --> 00:56:24,260 So what is that force going to do? 1027 00:56:24,260 --> 00:56:29,600 That force is just going to drag that particle 1028 00:56:29,600 --> 00:56:31,770 through the liquid, through the water. 1029 00:56:36,840 --> 00:56:39,520 So why is it? 1030 00:56:39,520 --> 00:56:44,100 So if this were a vacuum in here and we put a charge there 1031 00:56:44,100 --> 00:56:47,190 and metal plates and we put a battery across, 1032 00:56:47,190 --> 00:56:48,540 what would that particle do? 1033 00:56:51,820 --> 00:56:53,580 It would move. 1034 00:56:53,580 --> 00:56:56,080 But what would this force do to that particle? 1035 00:56:56,080 --> 00:56:57,880 AUDIENCE: [INTERPOSING VOICES] 1036 00:56:57,880 --> 00:56:58,780 MICHALE FEE: Exactly. 1037 00:56:58,780 --> 00:57:00,910 So what would the velocity do? 1038 00:57:00,910 --> 00:57:01,780 AUDIENCE: Increase. 1039 00:57:01,780 --> 00:57:03,230 MICHALE FEE: It would just increase linearly. 1040 00:57:03,230 --> 00:57:04,920 So the particle would start moving. 1041 00:57:04,920 --> 00:57:06,930 And it would start moving slowly and it'd go-- 1042 00:57:06,930 --> 00:57:09,585 poof-- crash into the plate. 1043 00:57:09,585 --> 00:57:10,960 But that's not what happens here. 1044 00:57:10,960 --> 00:57:13,358 Why is that? 1045 00:57:13,358 --> 00:57:14,774 AUDIENCE: [INAUDIBLE] 1046 00:57:14,774 --> 00:57:17,190 MICHALE FEE: Because there's stuff in the way. 1047 00:57:17,190 --> 00:57:22,540 And so it accelerates, and it gets hit by a water molecule. 1048 00:57:22,540 --> 00:57:24,770 And it gets pushed off in some direction. 1049 00:57:24,770 --> 00:57:29,110 And then it accelerates in this direction, gets hit again. 1050 00:57:29,110 --> 00:57:32,680 But it's constantly being accelerated in one direction 1051 00:57:32,680 --> 00:57:34,855 before it collides. 1052 00:57:34,855 --> 00:57:35,980 And so here's what happens. 1053 00:57:46,580 --> 00:57:48,970 So it's diffusing around. 1054 00:57:48,970 --> 00:57:52,600 But on each step, it has a little bit of acceleration 1055 00:57:52,600 --> 00:57:54,610 in this direction, in the direction 1056 00:57:54,610 --> 00:57:55,540 of the electric field. 1057 00:58:03,370 --> 00:58:07,360 And so you can show using the same kind of analysis 1058 00:58:07,360 --> 00:58:12,940 that we used in calculating the distribution, the change 1059 00:58:12,940 --> 00:58:15,520 in mean and variance, you can show 1060 00:58:15,520 --> 00:58:20,200 that mean of a distribution of particles 1061 00:58:20,200 --> 00:58:24,000 that starts at zero shifts-- 1062 00:58:24,000 --> 00:58:27,670 of positive particles shifts in the electric field 1063 00:58:27,670 --> 00:58:31,220 linearly in time. 1064 00:58:31,220 --> 00:58:35,150 And you can just think about that as the electric field 1065 00:58:35,150 --> 00:58:38,300 reaches in, grabs that charged particle, and pulls it 1066 00:58:38,300 --> 00:58:41,780 in this direction against viscous drag. 1067 00:58:41,780 --> 00:58:46,370 So now a force produces a constant velocity, not 1068 00:58:46,370 --> 00:58:49,160 acceleration. 1069 00:58:49,160 --> 00:58:51,440 And that velocity is called the drift velocity. 1070 00:58:54,330 --> 00:59:01,190 So the force is proportional to drift velocity. 1071 00:59:01,190 --> 00:59:03,630 What is that little f there? 1072 00:59:03,630 --> 00:59:04,720 Anybody know what that is? 1073 00:59:04,720 --> 00:59:06,095 AUDIENCE: Frictional coefficient. 1074 00:59:06,095 --> 00:59:08,600 MICHALE FEE: It's the coefficient 1075 00:59:08,600 --> 00:59:10,580 of friction of that particle. 1076 00:59:10,580 --> 00:59:14,900 And Einstein cleverly noticed that the coefficient 1077 00:59:14,900 --> 00:59:20,390 of friction of a particle being dragged through a liquid 1078 00:59:20,390 --> 00:59:21,735 is related to what? 1079 00:59:21,735 --> 00:59:22,235 Any guess? 1080 00:59:26,360 --> 00:59:29,900 Diffusion coefficient of that particle. 1081 00:59:29,900 --> 00:59:30,525 Is that cool? 1082 00:59:30,525 --> 00:59:32,260 That just gives me chills. 1083 00:59:35,010 --> 00:59:38,770 The frictional coefficient is just 1084 00:59:38,770 --> 00:59:41,750 kT over the diffusion constant. 1085 00:59:41,750 --> 00:59:44,320 So if you actually just go through that same analysis 1086 00:59:44,320 --> 00:59:48,160 of calculating the mean of the distribution, what you find 1087 00:59:48,160 --> 00:59:52,450 is that v moves linearly in time. 1088 00:59:52,450 --> 00:59:54,130 But it's also very intuitive. 1089 00:59:54,130 --> 01:00:01,000 If you're in a swimming pool, you put your hand in the water, 1090 01:00:01,000 --> 01:00:05,180 and you push your hand with a constant force. 1091 01:00:05,180 --> 01:00:05,840 What happens? 1092 01:00:05,840 --> 01:00:07,590 Well, let me flip it around. 1093 01:00:07,590 --> 01:00:10,790 You move your hand through the water at a constant velocity. 1094 01:00:10,790 --> 01:00:14,620 What is the force feel like? 1095 01:00:14,620 --> 01:00:16,197 The force is constant, right? 1096 01:00:16,197 --> 01:00:17,530 So flip it the other way around. 1097 01:00:17,530 --> 01:00:19,072 If the force is constant, then you're 1098 01:00:19,072 --> 01:00:20,840 going to get a constant velocity. 1099 01:00:23,620 --> 01:00:24,120 Yes? 1100 01:00:24,120 --> 01:00:26,430 AUDIENCE: So side question, but you can also 1101 01:00:26,430 --> 01:00:28,555 look at that like a terminal velocity problem? 1102 01:00:28,555 --> 01:00:29,430 MICHALE FEE: Exactly. 1103 01:00:29,430 --> 01:00:30,660 It's exactly the same thing. 1104 01:00:34,210 --> 01:00:39,250 So the drift velocity is proportional to the force 1105 01:00:39,250 --> 01:00:43,420 by proportionality constant, 1 over the coefficient 1106 01:00:43,420 --> 01:00:46,270 of friction, which is now d over kT. 1107 01:00:46,270 --> 01:00:49,180 And what is this force proportional to? 1108 01:00:49,180 --> 01:00:50,530 Anybody remember? 1109 01:00:50,530 --> 01:00:54,695 The force was proportional to the electric field. 1110 01:00:59,050 --> 01:01:03,590 And so let's calculate the current. 1111 01:01:03,590 --> 01:01:06,280 So I'm going to argue that the current is 1112 01:01:06,280 --> 01:01:10,950 proportional to the drift velocity times the area. 1113 01:01:10,950 --> 01:01:12,770 Now why is that? 1114 01:01:12,770 --> 01:01:15,410 So if I have an electric field, it 1115 01:01:15,410 --> 01:01:18,980 makes these particles, all the particles in this area 1116 01:01:18,980 --> 01:01:24,250 here drift at a constant velocity in this direction. 1117 01:01:24,250 --> 01:01:25,900 So there is a certain amount of current 1118 01:01:25,900 --> 01:01:29,706 that's flowing in this area right here. 1119 01:01:29,706 --> 01:01:31,700 Does that makes sense? 1120 01:01:31,700 --> 01:01:35,690 Now if my electrodes are big and I also have electric field 1121 01:01:35,690 --> 01:01:38,360 up here, then that electric field 1122 01:01:38,360 --> 01:01:42,067 is causing current to flow up here too. 1123 01:01:42,067 --> 01:01:43,650 And if there's electric field up here, 1124 01:01:43,650 --> 01:01:45,780 then there will be current flowing up here too. 1125 01:01:45,780 --> 01:01:48,570 And so you can see that the amount of current that's 1126 01:01:48,570 --> 01:01:51,720 flowing between the electrodes is proportional to the drift 1127 01:01:51,720 --> 01:01:57,120 velocity and the cross-sectional area between the two 1128 01:01:57,120 --> 01:01:58,520 electrodes. 1129 01:01:58,520 --> 01:02:00,130 Yes? 1130 01:02:00,130 --> 01:02:01,370 So that's really important. 1131 01:02:05,480 --> 01:02:10,410 Now we figured out that the drift velocity 1132 01:02:10,410 --> 01:02:13,500 is proportional to the electric field. 1133 01:02:13,500 --> 01:02:16,440 So the current is proportional to the electric field 1134 01:02:16,440 --> 01:02:17,910 times the area. 1135 01:02:17,910 --> 01:02:21,030 And the electric field is just the voltage difference 1136 01:02:21,030 --> 01:02:25,890 divided by the spacing between the electrodes. 1137 01:02:25,890 --> 01:02:27,870 And so the current is proportional to voltage 1138 01:02:27,870 --> 01:02:30,750 times area divided by length. 1139 01:02:30,750 --> 01:02:32,090 So we have a proportionality. 1140 01:02:32,090 --> 01:02:34,490 Current is proportional to voltage 1141 01:02:34,490 --> 01:02:36,420 times area divided by length. 1142 01:02:36,420 --> 01:02:39,980 And now let's plug in what that proportionality constant is. 1143 01:02:47,520 --> 01:02:49,350 This is now like Ohm's Law, right? 1144 01:02:49,350 --> 01:02:51,630 We're saying the current is proportional to voltage 1145 01:02:51,630 --> 01:02:53,170 difference. 1146 01:02:53,170 --> 01:02:55,600 The thing that the proportionality constant 1147 01:02:55,600 --> 01:02:57,750 here is something called resistivity. 1148 01:02:57,750 --> 01:02:59,860 Otherwise known as conductivity. 1149 01:02:59,860 --> 01:03:02,583 But we're going to use resistivity. 1150 01:03:05,690 --> 01:03:07,490 So this is just Ohm's Law. 1151 01:03:07,490 --> 01:03:13,230 It says current is proportional to voltage difference. 1152 01:03:13,230 --> 01:03:15,210 Let's rewrite that a little bit so that it 1153 01:03:15,210 --> 01:03:16,680 looks more like Ohm's Law. 1154 01:03:16,680 --> 01:03:20,010 Current is proportional to voltage difference. 1155 01:03:20,010 --> 01:03:23,580 And that thing, that thingy right there, 1156 01:03:23,580 --> 01:03:25,770 should have units of what? 1157 01:03:25,770 --> 01:03:27,510 1 over ohms. 1158 01:03:27,510 --> 01:03:28,820 Right? 1159 01:03:28,820 --> 01:03:30,980 So that is 1 over resistance. 1160 01:03:30,980 --> 01:03:34,670 Let's just write down what the resistance is. 1161 01:03:34,670 --> 01:03:39,660 Resistance is just resistivity times length divided by area. 1162 01:03:39,660 --> 01:03:42,290 So let's just stop and take a breath 1163 01:03:42,290 --> 01:03:45,230 and think about why this makes sense. 1164 01:03:48,100 --> 01:03:51,100 Resistance is how much resistance there 1165 01:03:51,100 --> 01:03:53,630 is to flow at a given voltage, right? 1166 01:03:53,630 --> 01:03:58,130 So what happens if we make ours really small? 1167 01:03:58,130 --> 01:04:01,130 What happens to the resistance? 1168 01:04:01,130 --> 01:04:03,130 AUDIENCE: [INAUDIBLE] really big. 1169 01:04:03,130 --> 01:04:05,960 MICHALE FEE: The resistance gets big. 1170 01:04:05,960 --> 01:04:10,070 The amount of current gets small because there's less area 1171 01:04:10,070 --> 01:04:13,272 that the electric field is in. 1172 01:04:13,272 --> 01:04:14,480 And so the current goes down. 1173 01:04:14,480 --> 01:04:15,855 That means the resistance is big. 1174 01:04:15,855 --> 01:04:17,820 If we make our plates really big, 1175 01:04:17,820 --> 01:04:20,790 the resistance gets smaller. 1176 01:04:20,790 --> 01:04:24,270 What happens if we pull our plates further apart? 1177 01:04:24,270 --> 01:04:26,120 What happens to the resistance? 1178 01:04:26,120 --> 01:04:27,800 AUDIENCE: [INAUDIBLE] further apart. 1179 01:04:27,800 --> 01:04:28,730 MICHALE FEE: Good. 1180 01:04:28,730 --> 01:04:30,950 If the plates are further apart, L is bigger, 1181 01:04:30,950 --> 01:04:32,030 and resistance is bigger. 1182 01:04:32,030 --> 01:04:34,580 But conceptually, what's going on? 1183 01:04:34,580 --> 01:04:37,000 Physically, what's going? 1184 01:04:37,000 --> 01:04:39,010 The plates are further apart, so what happens? 1185 01:04:39,010 --> 01:04:40,110 AUDIENCE: [INAUDIBLE] 1186 01:04:40,110 --> 01:04:41,500 MICHALE FEE: Right. 1187 01:04:41,500 --> 01:04:43,450 The voltage difference is the same, 1188 01:04:43,450 --> 01:04:46,430 but the distance is bigger. 1189 01:04:46,430 --> 01:04:50,620 And so the electric field, which is voltage per distance, 1190 01:04:50,620 --> 01:04:51,730 is smaller. 1191 01:04:51,730 --> 01:04:58,120 And that smaller electric field produces a drift velocity. 1192 01:04:58,120 --> 01:05:02,600 And that's why the resistance goes up. 1193 01:05:02,600 --> 01:05:05,250 Cool, right? 1194 01:05:05,250 --> 01:05:06,690 OK. 1195 01:05:06,690 --> 01:05:12,560 Now, let's talk for a minute about resistivity. 1196 01:05:12,560 --> 01:05:18,140 So resistivity in the brain is really, really lousy. 1197 01:05:18,140 --> 01:05:23,270 The wires of the brain are just awful. 1198 01:05:23,270 --> 01:05:26,110 So if you look at the resistivity for copper, which 1199 01:05:26,110 --> 01:05:30,700 is which is the wire that's used in electronics, 1200 01:05:30,700 --> 01:05:36,940 the resistivity is 1.6 microohms times centimeters. 1201 01:05:36,940 --> 01:05:40,780 What that means is if I took a block of copper, a centimeter 1202 01:05:40,780 --> 01:05:44,080 on a side, and I put electrodes on the side of it, 1203 01:05:44,080 --> 01:05:48,730 and I measured the resistance, it would be 1.6 microohms. 1204 01:05:51,705 --> 01:05:59,460 That means I could run an amp, that thing with 1.6 microvolts. 1205 01:06:05,550 --> 01:06:09,810 Now the resistivity of the brain is 60 ohms centimeters. 1206 01:06:09,810 --> 01:06:14,190 That means a centimeter of block of saline solution, 1207 01:06:14,190 --> 01:06:18,770 intracellular or extracellular solution, 1208 01:06:18,770 --> 01:06:24,890 has a resistance of 60 ohms instead of 1.6 microohms. 1209 01:06:24,890 --> 01:06:27,395 It's more than a million times worse. 1210 01:06:30,030 --> 01:06:35,380 And what that means is that when you try to send current 1211 01:06:35,380 --> 01:06:38,990 through brain, you try to send some current, 1212 01:06:38,990 --> 01:06:41,580 the voltage just drops. 1213 01:06:41,580 --> 01:06:45,360 You need huge voltage drops to produce tiny currents. 1214 01:06:51,640 --> 01:06:55,910 That's why the brain has invented things-- 1215 01:06:55,910 --> 01:07:00,710 axons-- because the wires are so bad that you 1216 01:07:00,710 --> 01:07:02,900 can't send a signal from one part of the brain 1217 01:07:02,900 --> 01:07:05,360 to another part of the brain through the wire. 1218 01:07:05,360 --> 01:07:08,270 You have to invent this special gimmick called an action 1219 01:07:08,270 --> 01:07:14,240 potential to send a signal more than a few microns away. 1220 01:07:17,910 --> 01:07:20,580 It's pretty cool, right? 1221 01:07:20,580 --> 01:07:24,810 That's why it's so interesting to understand the basic physics 1222 01:07:24,810 --> 01:07:28,260 of something, the basic mechanisms by which something 1223 01:07:28,260 --> 01:07:31,230 works because most of what you see 1224 01:07:31,230 --> 01:07:36,170 is a hack to compensate for weird physics, right? 1225 01:07:36,170 --> 01:07:36,670 Yes? 1226 01:07:36,670 --> 01:07:38,594 AUDIENCE: Does this [INAUDIBLE]? 1227 01:07:44,366 --> 01:07:50,570 MICHALE FEE: This high resistivity-- 1228 01:07:50,570 --> 01:07:54,290 you're asking what causes that high resistivity. 1229 01:07:54,290 --> 01:07:59,660 It basically has to do with things like the mean-free path 1230 01:07:59,660 --> 01:08:01,190 of the particle. 1231 01:08:01,190 --> 01:08:06,050 So in a metal, particles can go further effectively 1232 01:08:06,050 --> 01:08:06,980 before they collide. 1233 01:08:09,620 --> 01:08:11,810 So the resistivity is lower. 1234 01:08:11,810 --> 01:08:15,007 AUDIENCE: Is that slope [INAUDIBLE]?? 1235 01:08:15,007 --> 01:08:17,340 MICHALE FEE: It's a little bit different inside the cell 1236 01:08:17,340 --> 01:08:19,410 because there's more gunk inside of a cell 1237 01:08:19,410 --> 01:08:20,979 than there is outside of a cell. 1238 01:08:20,979 --> 01:08:23,310 And so the resistivity is a little bit worse. 1239 01:08:23,310 --> 01:08:26,380 It's 2,000 ohms centimeters, or 1,000 or 2,000 1240 01:08:26,380 --> 01:08:28,770 inside the cell and more like 60 outside. 1241 01:08:28,770 --> 01:08:30,900 AUDIENCE: [INAUDIBLE] 1242 01:08:30,900 --> 01:08:32,830 MICHALE FEE: Yes once you're outside the cell, 1243 01:08:32,830 --> 01:08:34,956 it's basically the same everywhere. 1244 01:08:39,630 --> 01:08:40,130 OK? 1245 01:08:44,250 --> 01:08:45,060 So that's it. 1246 01:08:45,060 --> 01:08:48,029 So here's what we learned about today. 1247 01:08:48,029 --> 01:08:53,890 We understood the relation between the timescale 1248 01:08:53,890 --> 01:08:55,529 of diffusion and length scales. 1249 01:08:55,529 --> 01:09:00,840 And we learned that the distance that a particle can diffuse 1250 01:09:00,840 --> 01:09:03,029 grows only as the square root of time. 1251 01:09:06,069 --> 01:09:09,970 We understood how concentration gradients lead to currents. 1252 01:09:09,970 --> 01:09:13,420 And we talked about Fick's First Law 1253 01:09:13,420 --> 01:09:16,870 that says that concentration differences lead 1254 01:09:16,870 --> 01:09:18,250 to particle flux. 1255 01:09:18,250 --> 01:09:21,729 The flux is proportional to the gradient or the derivative 1256 01:09:21,729 --> 01:09:23,560 of the concentration. 1257 01:09:23,560 --> 01:09:27,850 And we also talked about how the drift of charged particles 1258 01:09:27,850 --> 01:09:31,630 in an electric field leads to currents, 1259 01:09:31,630 --> 01:09:37,000 and how the voltage current relation obeys Ohm's Law. 1260 01:09:37,000 --> 01:09:40,090 And we also talked about the concept of resistivity 1261 01:09:40,090 --> 01:09:43,689 and how the resistivity in the brain is really high 1262 01:09:43,689 --> 01:09:45,700 and makes the wires in the brain really bad. 1263 01:09:48,649 --> 01:09:51,470 So that's all I have. 1264 01:09:51,470 --> 01:09:54,360 I will take any questions. 1265 01:09:54,360 --> 01:09:54,900 Yes, Daniel? 1266 01:09:54,900 --> 01:09:56,900 AUDIENCE: I just wanted to introduce David. 1267 01:09:56,900 --> 01:09:57,950 MICHALE FEE: OK. 1268 01:09:57,950 --> 01:09:58,970 Our other TA is here. 1269 01:10:01,690 --> 01:10:02,915 Any questions? 1270 01:10:05,410 --> 01:10:05,910 Great. 1271 01:10:05,910 --> 01:10:08,160 So we will see you-- 1272 01:10:08,160 --> 01:10:09,570 when is the first [AUDIO OUT]? 1273 01:10:09,570 --> 01:10:09,810 Is that-- 1274 01:10:09,810 --> 01:10:10,602 AUDIENCE: Tomorrow. 1275 01:10:10,602 --> 01:10:12,320 MICHALE FEE: Tomorrow. 1276 01:10:12,320 --> 01:10:15,320 So I will see you Thursday.