1 00:00:12,896 --> 00:00:15,860 MICHALE FEE: OK, good morning, everyone. 2 00:00:15,860 --> 00:00:20,810 OK, so today we are going to continue 3 00:00:20,810 --> 00:00:23,600 the process of building our equivalent circuit 4 00:00:23,600 --> 00:00:26,060 model of a neuron. 5 00:00:26,060 --> 00:00:31,040 This model was actually developed in the late '40s 6 00:00:31,040 --> 00:00:33,350 and early '50s by Alan Hodgkin and Andrew 7 00:00:33,350 --> 00:00:38,180 Huxley, who started working on the problem of understanding 8 00:00:38,180 --> 00:00:40,680 how neurons make action potentials. 9 00:00:40,680 --> 00:00:44,310 And so they studied the squid giant axon, 10 00:00:44,310 --> 00:00:46,340 which is actually a very cool preparation, 11 00:00:46,340 --> 00:00:50,520 because that axon is actually about a millimeter across, 12 00:00:50,520 --> 00:00:54,300 and so you can stick wires inside of it. 13 00:00:54,300 --> 00:00:57,620 And they did a bunch of very cool experiments 14 00:00:57,620 --> 00:01:01,368 to figure out how these different ionic conductances 15 00:01:01,368 --> 00:01:03,410 and how these different components of the circuit 16 00:01:03,410 --> 00:01:08,360 work together to make an action potential. 17 00:01:08,360 --> 00:01:12,300 So that's what we're going to continue doing, 18 00:01:12,300 --> 00:01:15,260 we're going to essentially continue 19 00:01:15,260 --> 00:01:18,050 describing and motivating the different components 20 00:01:18,050 --> 00:01:19,260 of this circuit. 21 00:01:19,260 --> 00:01:23,990 So today, we're going to get through the process 22 00:01:23,990 --> 00:01:28,280 of introducing a voltage-measuring device, 23 00:01:28,280 --> 00:01:31,610 a current source, a capacitor, a conductance, 24 00:01:31,610 --> 00:01:36,930 and we're going to start introducing a battery, OK? 25 00:01:36,930 --> 00:01:37,430 OK. 26 00:01:37,430 --> 00:01:42,260 So here's what we want to accomplish today. 27 00:01:42,260 --> 00:01:46,070 So we want to understand how kind of at the simplest level 28 00:01:46,070 --> 00:01:50,780 how neurons respond to injected currents, 29 00:01:50,780 --> 00:01:52,340 we want to understand how membrane 30 00:01:52,340 --> 00:01:56,120 capacitance and membrane resistance allows neurons 31 00:01:56,120 --> 00:01:58,670 to integrate their inputs over time, 32 00:01:58,670 --> 00:02:02,870 and to filter their inputs or smooth their inputs over time-- 33 00:02:02,870 --> 00:02:06,620 and that particular model is called a resistor capacitor 34 00:02:06,620 --> 00:02:11,500 model or an RC model of a neuron. 35 00:02:11,500 --> 00:02:14,140 We're going to go through how to derive the differential 36 00:02:14,140 --> 00:02:17,680 equations that describe that model-- it's actually quite 37 00:02:17,680 --> 00:02:20,260 simple, but some of you may not have 38 00:02:20,260 --> 00:02:22,780 been through that before, so I want to go through it 39 00:02:22,780 --> 00:02:25,030 step by step so we can really understand 40 00:02:25,030 --> 00:02:27,300 where that comes from. 41 00:02:27,300 --> 00:02:31,935 And we're going to learn to basically look at a current-- 42 00:02:31,935 --> 00:02:33,780 a pattern of current injection, and we 43 00:02:33,780 --> 00:02:36,780 should be able to intuitively see 44 00:02:36,780 --> 00:02:41,860 how the voltage of that neuron responds. 45 00:02:41,860 --> 00:02:44,950 And we're going to start working on where 46 00:02:44,950 --> 00:02:49,280 the batteries of a neuron actually come from, OK? 47 00:02:49,280 --> 00:02:49,900 OK. 48 00:02:49,900 --> 00:02:51,490 So-- all right. 49 00:02:51,490 --> 00:02:54,760 So-- all right. 50 00:02:54,760 --> 00:02:59,345 So we're going to basically talk about the following sort 51 00:02:59,345 --> 00:03:01,340 of thought experiment, OK? 52 00:03:01,340 --> 00:03:02,990 The following conceptual idea. 53 00:03:02,990 --> 00:03:05,300 We're going to take a neuron and we're 54 00:03:05,300 --> 00:03:08,090 going to put it in a bath of sailing, OK? 55 00:03:08,090 --> 00:03:10,850 A saltwater solution that represents the extracellular 56 00:03:10,850 --> 00:03:14,300 solution that neurons-- 57 00:03:14,300 --> 00:03:16,770 extracellular solution in the brain. 58 00:03:16,770 --> 00:03:19,640 And we're going to put an electrode into that neuron 59 00:03:19,640 --> 00:03:21,710 so that we can inject current, and we're 60 00:03:21,710 --> 00:03:24,120 going to put another electrode into the neuron 61 00:03:24,120 --> 00:03:25,760 so that we can measure the voltage, 62 00:03:25,760 --> 00:03:29,540 and we're going to study how this neuron responds-- 63 00:03:29,540 --> 00:03:32,950 how the voltage of the neuron responds to current injections, 64 00:03:32,950 --> 00:03:34,850 OK? 65 00:03:34,850 --> 00:03:38,120 Now why is it that we want to actually do that? 66 00:03:38,120 --> 00:03:43,510 Why is that an interesting or important experiment to do? 67 00:03:43,510 --> 00:03:45,880 Anybody have any idea why we would 68 00:03:45,880 --> 00:03:48,997 want to actually measure voltage and current 69 00:03:48,997 --> 00:03:50,080 for a neuron in the brain? 70 00:03:54,730 --> 00:03:55,230 Yes? 71 00:03:55,230 --> 00:03:58,050 AUDIENCE: Be able to use the mathematical model [INAUDIBLE] 72 00:03:58,050 --> 00:03:58,990 provided for us? 73 00:03:58,990 --> 00:04:02,130 MICHALE FEE: OK, but it's more than just 74 00:04:02,130 --> 00:04:05,760 so that we can describe it mathematically, right? 75 00:04:05,760 --> 00:04:07,410 It's because these things-- 76 00:04:07,410 --> 00:04:09,120 something about voltage and current 77 00:04:09,120 --> 00:04:13,030 are actually relevant to how a neuron functions. 78 00:04:13,030 --> 00:04:13,530 Yes? 79 00:04:13,530 --> 00:04:16,140 AUDIENCE: Like the resistance inside? 80 00:04:16,140 --> 00:04:17,130 MICHALE FEE: Yeah. 81 00:04:17,130 --> 00:04:19,230 So that's an important quantity, but we're 82 00:04:19,230 --> 00:04:21,120 looking for something more fundamental, 83 00:04:21,120 --> 00:04:23,250 like why is it actually important 84 00:04:23,250 --> 00:04:27,300 that we understand how voltage changes when a neuron has 85 00:04:27,300 --> 00:04:28,990 current injected into it? 86 00:04:28,990 --> 00:04:29,490 Habiba? 87 00:04:29,490 --> 00:04:31,880 AUDIENCE: Equals [INAUDIBLE] different like ion channels, 88 00:04:31,880 --> 00:04:35,855 a different set of voltages [INAUDIBLE].. 89 00:04:35,855 --> 00:04:36,730 MICHALE FEE: Exactly. 90 00:04:36,730 --> 00:04:40,220 So ion channels are sensitive to voltage, 91 00:04:40,220 --> 00:04:43,900 and the way they function depends very critically 92 00:04:43,900 --> 00:04:44,830 on voltage. 93 00:04:44,830 --> 00:04:48,550 So many-- if not most-- ion channels 94 00:04:48,550 --> 00:04:54,550 are voltage sensitive and are controlled by voltage, OK? 95 00:04:54,550 --> 00:04:56,070 And that's exactly why. 96 00:04:56,070 --> 00:05:00,280 So nearly every aspect of what neurons do in the brain 97 00:05:00,280 --> 00:05:05,440 as you're walking around looking at things and doing things 98 00:05:05,440 --> 00:05:07,330 is controlled by voltage, and that 99 00:05:07,330 --> 00:05:11,340 goes through the voltage sensitivity of buying channels, 100 00:05:11,340 --> 00:05:12,070 OK? 101 00:05:12,070 --> 00:05:15,700 But what is it that changes the voltage in a neuron? 102 00:05:22,030 --> 00:05:22,530 Yes? 103 00:05:22,530 --> 00:05:24,620 AUDIENCE: The action potential. 104 00:05:24,620 --> 00:05:27,790 MICHALE FEE: That's on the output side. 105 00:05:27,790 --> 00:05:28,290 Yes? 106 00:05:28,290 --> 00:05:30,490 AUDIENCE: Is it ion concentration? 107 00:05:30,490 --> 00:05:31,500 MICHALE FEE: Good. 108 00:05:31,500 --> 00:05:33,300 That's correct. 109 00:05:33,300 --> 00:05:37,080 I'm looking for something a little bit different. 110 00:05:37,080 --> 00:05:37,580 Habiba? 111 00:05:37,580 --> 00:05:39,570 AUDIENCE: Do you have pumps or [INAUDIBLE].. 112 00:05:39,570 --> 00:05:42,288 MICHALE FEE: Yeah, those are all good answers. 113 00:05:42,288 --> 00:05:43,580 Not quite what I'm looking for. 114 00:05:43,580 --> 00:05:46,320 AUDIENCE: [INAUDIBLE] 115 00:05:46,320 --> 00:05:47,880 MICHALE FEE: Yes. 116 00:05:47,880 --> 00:05:52,050 So the answer is that the voltage of a neuron 117 00:05:52,050 --> 00:05:55,270 changes because-- 118 00:05:55,270 --> 00:05:59,260 the reason current is important is because the reason voltage 119 00:05:59,260 --> 00:06:03,280 changes in a neuron is because other cells are injecting 120 00:06:03,280 --> 00:06:05,590 current into our neuron. 121 00:06:05,590 --> 00:06:10,390 Sensory inputs are injecting current into our neuron, OK? 122 00:06:10,390 --> 00:06:14,050 Everything that a neuron receives, all the information 123 00:06:14,050 --> 00:06:17,380 that a neuron receives from other neurons in the network 124 00:06:17,380 --> 00:06:20,740 and from the outside world comes from currents being 125 00:06:20,740 --> 00:06:24,190 injected into that neuron, OK? 126 00:06:24,190 --> 00:06:26,200 And so it's really important that we understand 127 00:06:26,200 --> 00:06:31,810 how the neuron transforms that current input from other cells 128 00:06:31,810 --> 00:06:36,790 and from the sensory periphery into voltage changes that 129 00:06:36,790 --> 00:06:39,670 then change the behavior of ion channels. 130 00:06:39,670 --> 00:06:40,720 Is that clear? 131 00:06:40,720 --> 00:06:45,310 That link of current inputs to voltage output 132 00:06:45,310 --> 00:06:47,170 is really crucial, and that's why we're 133 00:06:47,170 --> 00:06:49,090 doing this experiment, OK? 134 00:06:52,390 --> 00:06:55,880 OK, and that's this point right here. 135 00:06:55,880 --> 00:06:57,380 OK, so one of the first things we're 136 00:06:57,380 --> 00:07:00,320 going to see when we go through this analysis 137 00:07:00,320 --> 00:07:03,170 is that neurons can perform analog integration. 138 00:07:03,170 --> 00:07:07,270 They can perform numerical integration over time, OK? 139 00:07:07,270 --> 00:07:09,130 That's pretty cool. 140 00:07:09,130 --> 00:07:14,470 Voltage is that integral over time of the injected current. 141 00:07:14,470 --> 00:07:16,030 To first order. 142 00:07:16,030 --> 00:07:18,252 It's the simplest behavior of a neuron. 143 00:07:18,252 --> 00:07:19,960 So if you measure the voltage of a neuron 144 00:07:19,960 --> 00:07:23,410 and you turn on current and you turn the current on, 145 00:07:23,410 --> 00:07:26,620 the voltage of a neuron will ramp up, 146 00:07:26,620 --> 00:07:29,320 integrating that input over time, OK? 147 00:07:29,320 --> 00:07:30,490 Pretty cool. 148 00:07:30,490 --> 00:07:32,590 So we're going to see how that happens, 149 00:07:32,590 --> 00:07:36,430 why that happens biophysically. 150 00:07:36,430 --> 00:07:37,000 OK. 151 00:07:37,000 --> 00:07:40,000 So let's come back to our neuron in the dish. 152 00:07:40,000 --> 00:07:42,700 Let me just explain a little bit how you would actually 153 00:07:42,700 --> 00:07:43,640 do this experiment. 154 00:07:43,640 --> 00:07:46,790 So these electrodes are little pieces of glass tubing. 155 00:07:46,790 --> 00:07:49,060 So you take a fine glass tube about a millimeter 156 00:07:49,060 --> 00:07:51,970 across you heat it up in the middle over a flame, 157 00:07:51,970 --> 00:07:54,610 and you pull it apart when it melts in the middle, 158 00:07:54,610 --> 00:07:57,340 and it makes a very sharp point. 159 00:07:57,340 --> 00:08:02,140 You break off the fine little thread of glass that's left, 160 00:08:02,140 --> 00:08:05,440 and you have a tube that narrows down to a very sharp point, 161 00:08:05,440 --> 00:08:08,375 but it's still a tube, and you can literally just-- 162 00:08:08,375 --> 00:08:10,750 there are some cells, like in the old days people studied 163 00:08:10,750 --> 00:08:14,203 large neurons and in snails where the cells are 164 00:08:14,203 --> 00:08:16,120 a millimeter across, you can take an electrode 165 00:08:16,120 --> 00:08:18,640 and literally just by hand poke it into the cell. 166 00:08:18,640 --> 00:08:22,870 And then you fill that electrode with a salt solution, 167 00:08:22,870 --> 00:08:25,510 and then you put a wire in the back of that electrode, 168 00:08:25,510 --> 00:08:27,440 and you hook it up to an amplifier. 169 00:08:27,440 --> 00:08:30,170 Now we want to measure the voltage in the cell. 170 00:08:30,170 --> 00:08:33,789 Remember, voltage is always voltage difference. 171 00:08:33,789 --> 00:08:35,380 We're always measuring the difference 172 00:08:35,380 --> 00:08:38,840 between the voltage in one place and the voltage somewhere else. 173 00:08:38,840 --> 00:08:41,450 So this amplifier has two inputs. 174 00:08:41,450 --> 00:08:43,630 It's called a differential amplifier, 175 00:08:43,630 --> 00:08:45,820 and we're going to hook the electrode that's 176 00:08:45,820 --> 00:08:47,680 in the cell to the plus terminal, 177 00:08:47,680 --> 00:08:49,620 we're going to put a wire in the bath, 178 00:08:49,620 --> 00:08:51,700 hook it to the minus terminal, and this amplifier 179 00:08:51,700 --> 00:08:54,310 is measuring the difference between the voltage 180 00:08:54,310 --> 00:08:57,490 inside the cell and the voltage outside the cell, OK? 181 00:08:57,490 --> 00:08:59,690 Any questions about that? 182 00:08:59,690 --> 00:09:02,470 So we're going to take the other half of that piece of glass 183 00:09:02,470 --> 00:09:05,145 that we pulled, fill it with salt solution, 184 00:09:05,145 --> 00:09:07,270 stick it in the cell, and we're going to hook it up 185 00:09:07,270 --> 00:09:09,100 to a current source. 186 00:09:09,100 --> 00:09:13,270 Now our current source is basically just a battery, OK? 187 00:09:13,270 --> 00:09:15,440 But it's got some fancy electronics 188 00:09:15,440 --> 00:09:20,170 such that the current that flows is equal to whatever value you 189 00:09:20,170 --> 00:09:25,020 set, OK? 190 00:09:25,020 --> 00:09:26,975 And of course, remember, that voltage 191 00:09:26,975 --> 00:09:31,790 is in units of volts and current is charge per second. 192 00:09:31,790 --> 00:09:35,150 Charge is coulombs, so coulombs per second, 193 00:09:35,150 --> 00:09:39,460 and that's equal to the unit of current, which is amperes. 194 00:09:39,460 --> 00:09:40,060 All right. 195 00:09:40,060 --> 00:09:44,770 Now let's take a closer look at our little spherical neuron, 196 00:09:44,770 --> 00:09:45,970 our little neuron. 197 00:09:45,970 --> 00:09:47,990 We've chopped all the dendrites and axons off, 198 00:09:47,990 --> 00:09:50,860 so it's just a little sphere, and you can basically 199 00:09:50,860 --> 00:09:56,470 model a neuron just like any other cell as a spherical shell 200 00:09:56,470 --> 00:09:59,740 of insulating material, OK? 201 00:09:59,740 --> 00:10:02,050 In this case, a lipid bilayer. 202 00:10:02,050 --> 00:10:05,050 This is a phospholipid bilayer. 203 00:10:05,050 --> 00:10:08,770 Phospholipids are just little fat molecules that have a polar 204 00:10:08,770 --> 00:10:11,530 head on one side-- that means they're soluble in water 205 00:10:11,530 --> 00:10:14,620 on this side, and they have a non-polar tail, 206 00:10:14,620 --> 00:10:16,780 so they don't like to be in contact with water, 207 00:10:16,780 --> 00:10:20,410 and the two polar tails go end to end-- 208 00:10:20,410 --> 00:10:22,900 sorry, the non-polar tails go end to end, 209 00:10:22,900 --> 00:10:26,320 the polar heads face out into the water. 210 00:10:26,320 --> 00:10:27,480 Does that makes sense? 211 00:10:27,480 --> 00:10:30,330 And they are very closely packed together so 212 00:10:30,330 --> 00:10:32,880 that ions can't pass through that membrane, 213 00:10:32,880 --> 00:10:35,310 so it's insulating. 214 00:10:35,310 --> 00:10:40,290 It's very thin, it's only about 23 angstroms across, OK? 215 00:10:40,290 --> 00:10:45,100 An angstrom is about the size of a hydrogen atom. 216 00:10:45,100 --> 00:10:47,030 They're very thin, OK? 217 00:10:50,030 --> 00:10:51,870 OK, we have saline inside. 218 00:10:51,870 --> 00:10:53,720 What is saline? 219 00:10:53,720 --> 00:10:55,505 What is it in our model? 220 00:10:59,220 --> 00:11:00,882 Remember on Tuesday what-- 221 00:11:00,882 --> 00:11:01,590 AUDIENCE: A wire. 222 00:11:01,590 --> 00:11:02,340 MICHALE FEE: Good. 223 00:11:02,340 --> 00:11:07,090 It's a wire and we have saline outside, which is also a wire. 224 00:11:07,090 --> 00:11:11,160 So we have two wires separated by an insulator, what is that? 225 00:11:16,100 --> 00:11:19,530 That's a capacitor, because it's two conductors separated 226 00:11:19,530 --> 00:11:21,360 by an insulator, OK? 227 00:11:21,360 --> 00:11:27,445 So an electrical component that behaves like a capacitor-- 228 00:11:27,445 --> 00:11:29,070 like if you were to build one of those, 229 00:11:29,070 --> 00:11:32,340 you would take like a piece of aluminum foil, 230 00:11:32,340 --> 00:11:35,220 put a piece of paper on it, put another piece of aluminum foil 231 00:11:35,220 --> 00:11:38,070 next to it, and attach wires to that, 232 00:11:38,070 --> 00:11:41,580 and you would squeeze the stack of aluminum foil paper 233 00:11:41,580 --> 00:11:45,450 and aluminum foil together, and that becomes a capacitor, OK? 234 00:11:45,450 --> 00:11:49,760 And it has a symbol that looks like this electrically. 235 00:11:49,760 --> 00:11:55,360 So this is now our equivalent circuit of this model neuron, 236 00:11:55,360 --> 00:11:55,970 OK? 237 00:11:55,970 --> 00:11:57,240 It's very simple. 238 00:11:57,240 --> 00:11:59,810 It's a capacitor with one wire here 239 00:11:59,810 --> 00:12:03,970 that represents the inside of the cell, another wire that 240 00:12:03,970 --> 00:12:08,160 represents the outside of the cell. 241 00:12:08,160 --> 00:12:10,680 We have a current source that connects 242 00:12:10,680 --> 00:12:13,710 the outside of the cell to the inside of the cell. 243 00:12:13,710 --> 00:12:15,180 When we turn on the current source, 244 00:12:15,180 --> 00:12:19,260 it takes charges from inside of here 245 00:12:19,260 --> 00:12:22,710 and sticks them through the electrode 246 00:12:22,710 --> 00:12:24,990 and pumps them into the cell. 247 00:12:24,990 --> 00:12:27,900 Does that makes sense? 248 00:12:27,900 --> 00:12:32,960 This is our-- this is sort of a simplified symbol 249 00:12:32,960 --> 00:12:35,690 for a voltage-measuring device. 250 00:12:35,690 --> 00:12:38,630 The voltage difference between the inside of the cell 251 00:12:38,630 --> 00:12:41,660 and the outside of the cell is what we're measuring here, 252 00:12:41,660 --> 00:12:44,930 and that difference is called the membrane potential. 253 00:12:44,930 --> 00:12:48,560 It's the voltage difference between the inside 254 00:12:48,560 --> 00:12:51,470 and the outside of the membrane, all right? 255 00:12:51,470 --> 00:12:52,970 Any questions about that? 256 00:12:52,970 --> 00:12:53,670 Yes? 257 00:12:53,670 --> 00:12:55,970 AUDIENCE: There's a narrow resistance for [INAUDIBLE].. 258 00:12:55,970 --> 00:12:56,727 MICHALE FEE: What's that? 259 00:12:56,727 --> 00:12:57,920 AUDIENCE: The resistance for-- 260 00:12:57,920 --> 00:12:58,610 MICHALE FEE: Yes, but we're going 261 00:12:58,610 --> 00:13:00,000 to do it one piece at a time. 262 00:13:00,000 --> 00:13:01,960 So we're going to start with a capacitor. 263 00:13:01,960 --> 00:13:04,340 The resistor will come in a few slides. 264 00:13:07,750 --> 00:13:09,098 Yes? 265 00:13:09,098 --> 00:13:10,535 AUDIENCE: So the [INAUDIBLE]. 266 00:13:17,455 --> 00:13:18,330 MICHALE FEE: Exactly. 267 00:13:18,330 --> 00:13:20,790 So we've simplified our neurons so that it's 268 00:13:20,790 --> 00:13:24,300 just an insulating shell, OK? 269 00:13:24,300 --> 00:13:27,090 No ion channels, no current anywhere else. 270 00:13:27,090 --> 00:13:31,020 If we want to inject current into this simple model neuron, 271 00:13:31,020 --> 00:13:34,700 we have to inject it through this electrode here, OK? 272 00:13:34,700 --> 00:13:39,730 So we're just going down to the very simplest case, 273 00:13:39,730 --> 00:13:42,780 because this is already kind of interesting enough 274 00:13:42,780 --> 00:13:44,190 to understand just by itself. 275 00:13:44,190 --> 00:13:44,996 Yes? 276 00:13:44,996 --> 00:13:47,170 AUDIENCE: So if the cell's acting as a capacitor, 277 00:13:47,170 --> 00:13:50,975 is their energy stored in their myelin? 278 00:13:50,975 --> 00:13:52,350 MICHALE FEE: The energy is stored 279 00:13:52,350 --> 00:13:54,270 in the electric field that crosses the bilayer, 280 00:13:54,270 --> 00:13:55,645 and I'll get to that in a second. 281 00:13:59,480 --> 00:14:00,530 Any other questions? 282 00:14:00,530 --> 00:14:02,868 OK, great questions. 283 00:14:02,868 --> 00:14:04,910 All right, so what happens when we inject current 284 00:14:04,910 --> 00:14:05,690 into our neuron? 285 00:14:05,690 --> 00:14:09,740 As I said, the current source is pulling charges 286 00:14:09,740 --> 00:14:13,936 from the outside and pumping them into the inside, 287 00:14:13,936 --> 00:14:16,672 all right? 288 00:14:16,672 --> 00:14:23,170 So what happens when goes on? 289 00:14:23,170 --> 00:14:25,830 So what we're doing is we are injecting current-- let's 290 00:14:25,830 --> 00:14:27,580 say this is our capacitor. 291 00:14:27,580 --> 00:14:32,930 There are charges, there are ions on the inside 292 00:14:32,930 --> 00:14:36,380 that are just up against the inside of the cell membrane. 293 00:14:36,380 --> 00:14:40,250 There are charges on the outside, OK? 294 00:14:40,250 --> 00:14:43,560 And when we inject a charge from the outside to the inside-- 295 00:14:43,560 --> 00:14:45,710 let's put one of those charges right here. 296 00:14:45,710 --> 00:14:48,460 And we're going to push it into this cell, when you inject 297 00:14:48,460 --> 00:14:52,100 a charge, you get an excessive charge 298 00:14:52,100 --> 00:14:56,060 on the inside of the cell membrane, OK? 299 00:14:56,060 --> 00:14:57,260 And what does that do? 300 00:14:57,260 --> 00:15:02,120 You now have more positive charges inside than outside, 301 00:15:02,120 --> 00:15:04,500 like-charges repel-- 302 00:15:04,500 --> 00:15:06,770 so it pushes one of those charges 303 00:15:06,770 --> 00:15:10,453 away from the outside of the membrane. 304 00:15:10,453 --> 00:15:11,370 Does that makes sense? 305 00:15:13,880 --> 00:15:15,400 OK, that's kind of interesting. 306 00:15:15,400 --> 00:15:23,650 We took a charge, we pushed it in, and a charge comes out. 307 00:15:23,650 --> 00:15:24,670 Right? 308 00:15:24,670 --> 00:15:26,290 We have a current flowing. 309 00:15:26,290 --> 00:15:29,360 We have charges coming in and charges leaving. 310 00:15:29,360 --> 00:15:32,130 We have a current flowing through an insulator. 311 00:15:32,130 --> 00:15:34,510 How is that possible? 312 00:15:34,510 --> 00:15:36,750 It's a capacitive current, OK? 313 00:15:36,750 --> 00:15:40,650 No charges are actually passing through the insulator, 314 00:15:40,650 --> 00:15:44,070 but it looks like you have a current flowing. 315 00:15:44,070 --> 00:15:46,470 That's called a capacitive current. 316 00:15:46,470 --> 00:15:51,320 And we represent that in our diagram by a current 317 00:15:51,320 --> 00:15:55,640 I sub C, capacitive current that flows through the capacitor. 318 00:15:55,640 --> 00:15:56,940 Pretty cool, right? 319 00:15:56,940 --> 00:15:59,150 You have a current flowing through an insulator. 320 00:15:59,150 --> 00:16:02,180 That's what a capacitor is. 321 00:16:02,180 --> 00:16:03,470 OK. 322 00:16:03,470 --> 00:16:06,290 Now notice that you have a charge imbalance. 323 00:16:06,290 --> 00:16:09,110 You have three positive charges here and only 324 00:16:09,110 --> 00:16:11,460 one positive charge here. 325 00:16:11,460 --> 00:16:14,330 So there is an excess of two positive charges on the inside. 326 00:16:14,330 --> 00:16:17,330 That's because we added a positive charge to the inside 327 00:16:17,330 --> 00:16:19,640 and took away a positive charge from the outside, 328 00:16:19,640 --> 00:16:23,680 so that leaves a charge imbalance of 2, OK? 329 00:16:23,680 --> 00:16:27,410 What do you get between positive charge and negative charge 330 00:16:27,410 --> 00:16:29,450 if you hold them next to each other? 331 00:16:29,450 --> 00:16:32,060 What is there in between? 332 00:16:32,060 --> 00:16:33,482 AUDIENCE: It's attraction. 333 00:16:33,482 --> 00:16:34,940 MICHALE FEE: Good, it's attraction, 334 00:16:34,940 --> 00:16:37,010 but what is it that causes that attraction? 335 00:16:37,010 --> 00:16:40,880 Remember yesterday, we talked about a something on a charge 336 00:16:40,880 --> 00:16:42,178 produces a force, what is it? 337 00:16:42,178 --> 00:16:43,220 AUDIENCE: Electric field. 338 00:16:43,220 --> 00:16:43,970 MICHALE FEE: Good. 339 00:16:43,970 --> 00:16:47,570 So there's an electric field between the positive-- 340 00:16:47,570 --> 00:16:51,230 the excess positive charges here and the excess negative charges 341 00:16:51,230 --> 00:16:52,790 here, OK? 342 00:16:52,790 --> 00:16:55,790 That's an electric field, all right? 343 00:16:55,790 --> 00:16:58,140 And that electric field stores energy. 344 00:16:58,140 --> 00:17:00,680 How do you know there's energy in this system, though? 345 00:17:00,680 --> 00:17:02,870 What could you do to demonstrate that there is 346 00:17:02,870 --> 00:17:06,630 energy stored in that system? 347 00:17:06,630 --> 00:17:07,460 Any ideas? 348 00:17:07,460 --> 00:17:10,410 You have two plates, two metal plates, let's say, 349 00:17:10,410 --> 00:17:12,170 in the metal version of this. 350 00:17:12,170 --> 00:17:14,150 Separated by an insulator. 351 00:17:14,150 --> 00:17:18,280 What would happen if you pulled away the insulator? 352 00:17:18,280 --> 00:17:20,770 Those two things would do that again, but louder-- 353 00:17:20,770 --> 00:17:21,640 boom. 354 00:17:21,640 --> 00:17:25,599 What does that take to make that sound? 355 00:17:25,599 --> 00:17:27,310 Energy, OK? 356 00:17:30,040 --> 00:17:33,380 So there's energy stored in that electric field. 357 00:17:33,380 --> 00:17:35,830 So there's a charge imbalance, there's an electric field. 358 00:17:35,830 --> 00:17:40,928 What does an electric field over some distance correspond to? 359 00:17:40,928 --> 00:17:42,220 AUDIENCE: A voltage difference. 360 00:17:42,220 --> 00:17:45,224 MICHALE FEE: A voltage difference, OK? 361 00:17:48,460 --> 00:17:51,650 Now, there's a charge imbalance and a voltage difference, 362 00:17:51,650 --> 00:17:54,060 and they're proportional to each other. 363 00:17:54,060 --> 00:17:56,510 So there's a proportionality constant 364 00:17:56,510 --> 00:17:59,830 that's called the capacitance, all right? 365 00:17:59,830 --> 00:18:04,990 If you can put a lot of charge and have a small voltage 366 00:18:04,990 --> 00:18:07,240 difference, that's a big capacitor. 367 00:18:07,240 --> 00:18:09,740 Now you can get a big capacitor just by having a big area. 368 00:18:09,740 --> 00:18:13,120 You can see, you can have a lot of charges with a small voltage 369 00:18:13,120 --> 00:18:18,610 difference if you have big plates on your capacitor, OK? 370 00:18:18,610 --> 00:18:21,760 So the capacitance is actually proportional to the area 371 00:18:21,760 --> 00:18:23,260 of the plates, and it's inversely 372 00:18:23,260 --> 00:18:26,060 proportional to the distance between them. 373 00:18:26,060 --> 00:18:29,260 It's a very thin membrane, which means 374 00:18:29,260 --> 00:18:33,700 you can get a lot of capacitance in a tiny area, OK? 375 00:18:33,700 --> 00:18:36,930 That's pretty cool. 376 00:18:36,930 --> 00:18:39,910 All right, any questions? 377 00:18:39,910 --> 00:18:44,080 So charge is coulombs, and there are 6 times 10 to the charges 378 00:18:44,080 --> 00:18:47,020 in a coulomb, the elemental charges, electron 379 00:18:47,020 --> 00:18:51,908 or monovalent ion charges. 380 00:18:51,908 --> 00:18:53,950 Voltage is in units of volts, and the capacitance 381 00:18:53,950 --> 00:18:55,270 is in units of farads. 382 00:18:58,720 --> 00:19:01,020 Any questions? 383 00:19:01,020 --> 00:19:02,640 All right. 384 00:19:02,640 --> 00:19:07,340 So we have our relation between voltage difference and charge 385 00:19:07,340 --> 00:19:08,770 difference. 386 00:19:08,770 --> 00:19:11,000 And what we're going to do is we're 387 00:19:11,000 --> 00:19:14,480 going to calculate this capacitive current. 388 00:19:14,480 --> 00:19:16,760 How do you think we would calculate 389 00:19:16,760 --> 00:19:17,780 the capacitive current? 390 00:19:21,310 --> 00:19:24,650 Well, the capacitive current is just 391 00:19:24,650 --> 00:19:29,900 the rate at which the charge imbalance is changing, right? 392 00:19:29,900 --> 00:19:33,160 Current is just charge per unit of time. 393 00:19:33,160 --> 00:19:33,660 OK? 394 00:19:33,660 --> 00:19:37,340 So we're going to calculate the capacitive current as the time 395 00:19:37,340 --> 00:19:38,900 rate of change of the charge-- 396 00:19:38,900 --> 00:19:41,570 and I've dropped the deltas here. 397 00:19:41,570 --> 00:19:46,480 So capacitive current is dQ/dt, all right? 398 00:19:46,480 --> 00:19:50,725 And remember that Q is just CV, so the capacitive current 399 00:19:50,725 --> 00:19:54,490 is just C dV/dt, and the Vm here represents 400 00:19:54,490 --> 00:19:57,130 the membrane potential, OK? 401 00:19:57,130 --> 00:20:00,760 So the capacitive current through a membrane 402 00:20:00,760 --> 00:20:04,120 is just the capacitance times the time rate of change 403 00:20:04,120 --> 00:20:07,870 of the membrane potential. 404 00:20:07,870 --> 00:20:10,038 Any questions? 405 00:20:10,038 --> 00:20:10,970 OK. 406 00:20:10,970 --> 00:20:12,930 Pretty straightforward. 407 00:20:12,930 --> 00:20:15,440 Now what we're going to do now is 408 00:20:15,440 --> 00:20:19,610 we're going to relate the injected charge 409 00:20:19,610 --> 00:20:22,730 to the-- sorry, the injected current to the capacitor. 410 00:20:22,730 --> 00:20:25,680 And what does Kirchhoff's current law tell us? 411 00:20:25,680 --> 00:20:28,970 It tells us that the amount of current going into this wire 412 00:20:28,970 --> 00:20:32,830 has to be equal to the amount of current leaving that wire. 413 00:20:32,830 --> 00:20:33,330 OK? 414 00:20:33,330 --> 00:20:35,720 So we can write that down as follows. 415 00:20:35,720 --> 00:20:39,320 The difference in sign here is because the electrode current 416 00:20:39,320 --> 00:20:42,860 is defined as positive inward, the capacitive current 417 00:20:42,860 --> 00:20:46,968 is defined as positive outward. 418 00:20:46,968 --> 00:20:47,468 OK? 419 00:20:54,950 --> 00:20:57,950 So you can see that we just calculated 420 00:20:57,950 --> 00:21:00,110 the capacitive current, it's C dV/dt, 421 00:21:00,110 --> 00:21:02,180 so we can just plug that in here, 422 00:21:02,180 --> 00:21:04,370 and now we see this very simple relation 423 00:21:04,370 --> 00:21:08,250 between the injected current and the voltage. 424 00:21:14,750 --> 00:21:16,960 And again, the current has unit of amperes, 425 00:21:16,960 --> 00:21:18,280 which is coulombs per second. 426 00:21:20,800 --> 00:21:23,110 OK, so we have that. 427 00:21:23,110 --> 00:21:24,940 Now we have a differential equation 428 00:21:24,940 --> 00:21:27,400 that describes the relation between current and voltage, 429 00:21:27,400 --> 00:21:32,590 we can just integrate it to get the solution. 430 00:21:32,590 --> 00:21:34,930 So that membrane potential will just 431 00:21:34,930 --> 00:21:39,130 be some initial membrane potential at time 0 432 00:21:39,130 --> 00:21:45,190 plus 1 over C integral over time of the injected current. 433 00:21:50,890 --> 00:21:52,300 Just integrate both sides. 434 00:21:52,300 --> 00:21:55,360 You get V here, you get integral of I there, 435 00:21:55,360 --> 00:22:04,208 and divide both sides by C. Any questions? 436 00:22:04,208 --> 00:22:08,570 It's either really confusing or really obvious. 437 00:22:08,570 --> 00:22:09,070 Yeah? 438 00:22:09,070 --> 00:22:10,730 Everybody OK? 439 00:22:10,730 --> 00:22:12,010 All right, good. 440 00:22:12,010 --> 00:22:12,910 Now what is this? 441 00:22:12,910 --> 00:22:17,870 What is the integral of current over time? 442 00:22:17,870 --> 00:22:19,530 AUDIENCE: It's charge [INAUDIBLE] 443 00:22:19,530 --> 00:22:20,280 MICHALE FEE: Good. 444 00:22:20,280 --> 00:22:23,280 It's the amount of charge you injected between time 445 00:22:23,280 --> 00:22:25,680 0 and time t, right? 446 00:22:25,680 --> 00:22:30,750 And what is the amount of charge you inject-- if I tell you 447 00:22:30,750 --> 00:22:35,190 that I injected an amount of charge delta Q, how much did 448 00:22:35,190 --> 00:22:36,130 I change the voltage? 449 00:22:43,380 --> 00:22:45,960 Delta V is delta Q over C, and that's exactly. 450 00:22:45,960 --> 00:22:48,870 So the voltage is just the starting voltage 451 00:22:48,870 --> 00:22:51,740 plus delta voltage. 452 00:22:51,740 --> 00:22:54,120 Does that makes sense? 453 00:22:54,120 --> 00:22:57,000 This integral here is just-- 454 00:22:57,000 --> 00:22:59,430 that part is just the amount of charge 455 00:22:59,430 --> 00:23:03,090 you injected divided by C gives you the change in the voltage. 456 00:23:03,090 --> 00:23:08,340 So the voltage is just the starting voltage plus delta V, 457 00:23:08,340 --> 00:23:09,550 OK? 458 00:23:09,550 --> 00:23:10,050 Yes? 459 00:23:10,050 --> 00:23:11,888 AUDIENCE: [INAUDIBLE] 460 00:23:11,888 --> 00:23:12,930 MICHALE FEE: What's that? 461 00:23:12,930 --> 00:23:13,913 AUDIENCE: [INAUDIBLE] 462 00:23:13,913 --> 00:23:14,580 MICHALE FEE: Oh. 463 00:23:14,580 --> 00:23:20,340 Because this equation here just came from here. 464 00:23:20,340 --> 00:23:21,850 That was our charge-- 465 00:23:21,850 --> 00:23:26,130 our relation between charge balance and voltage difference. 466 00:23:26,130 --> 00:23:26,630 Yeah? 467 00:23:30,410 --> 00:23:32,820 If it's not clear, just please ask, thank you. 468 00:23:35,990 --> 00:23:37,000 OK. 469 00:23:37,000 --> 00:23:37,730 There we go. 470 00:23:37,730 --> 00:23:45,040 So what is the interval constant? 471 00:23:45,040 --> 00:23:47,980 It's just some constant times time, right? 472 00:23:47,980 --> 00:23:51,250 So our voltage is just some initial voltage 473 00:23:51,250 --> 00:23:55,760 plus the injected current over C times time. 474 00:23:55,760 --> 00:23:58,000 And so you can see where this comes from, right? 475 00:23:58,000 --> 00:24:00,340 When you turn the current on, the voltage 476 00:24:00,340 --> 00:24:04,150 just increases linearly over time 477 00:24:04,150 --> 00:24:08,590 with a slope that's given by the current divided 478 00:24:08,590 --> 00:24:10,156 by the capacitance. 479 00:24:10,156 --> 00:24:10,656 OK? 480 00:24:18,782 --> 00:24:22,820 All right, any questions? 481 00:24:22,820 --> 00:24:24,310 You guys are being very quiet. 482 00:24:24,310 --> 00:24:27,310 This is the point where I start feeling nervous, 483 00:24:27,310 --> 00:24:28,310 I went too fast. 484 00:24:28,310 --> 00:24:29,110 Yes? 485 00:24:29,110 --> 00:24:31,360 AUDIENCE: You continually draw the curve for a while-- 486 00:24:31,360 --> 00:24:32,068 MICHALE FEE: Yep. 487 00:24:32,068 --> 00:24:34,080 AUDIENCE: [INAUDIBLE] 488 00:24:34,080 --> 00:24:35,830 MICHALE FEE: Yes, it does. 489 00:24:35,830 --> 00:24:42,920 And it breaks at about a volt or so. 490 00:24:42,920 --> 00:24:45,470 Because the electric field gets so strong, 491 00:24:45,470 --> 00:24:48,560 it literally just rips the atoms apart 492 00:24:48,560 --> 00:24:50,355 in the molecules of the lipid bilayer. 493 00:24:53,950 --> 00:24:54,450 Yes? 494 00:24:54,450 --> 00:24:55,828 AUDIENCE: Why do you [INAUDIBLE] 495 00:24:55,828 --> 00:24:57,370 MICHALE FEE: Sorry, I shouldn't say-- 496 00:24:57,370 --> 00:25:01,740 it doesn't rip the atoms apart, it rips the molecules apart. 497 00:25:01,740 --> 00:25:04,650 You need much higher electric fields to do that. 498 00:25:04,650 --> 00:25:05,460 Yes? 499 00:25:05,460 --> 00:25:10,728 AUDIENCE: [INAUDIBLE] 500 00:25:10,728 --> 00:25:11,395 MICHALE FEE: Oh. 501 00:25:11,395 --> 00:25:17,440 Because we're integrating from time 0 to time t. 502 00:25:17,440 --> 00:25:21,460 We want to know the voltage at time t, OK? 503 00:25:21,460 --> 00:25:24,100 We're starting at 0, we're integrating the current 504 00:25:24,100 --> 00:25:29,660 from time 0 to time t, which is where we're wanting 505 00:25:29,660 --> 00:25:30,990 to know the voltage, right? 506 00:25:30,990 --> 00:25:34,010 So you have to integrate the current from 0 to t. 507 00:25:34,010 --> 00:25:39,340 We can't use t in here. t is the endpoint. 508 00:25:39,340 --> 00:25:39,980 Yeah. 509 00:25:39,980 --> 00:25:40,978 Does that makes sense? 510 00:25:40,978 --> 00:25:42,020 Good question, thank you. 511 00:25:50,180 --> 00:25:51,590 OK, everybody all right? 512 00:25:51,590 --> 00:25:55,770 I'm going to stand here until I hear one more question. 513 00:25:55,770 --> 00:25:56,270 Yes? 514 00:25:56,270 --> 00:26:00,483 AUDIENCE: [INAUDIBLE] 515 00:26:00,483 --> 00:26:01,400 MICHALE FEE: Oh, here. 516 00:26:01,400 --> 00:26:05,840 Because it's a current of value I0. 517 00:26:05,840 --> 00:26:07,790 Great question. 518 00:26:07,790 --> 00:26:08,290 Yes? 519 00:26:08,290 --> 00:26:11,510 AUDIENCE: So to maintain this constant current [INAUDIBLE] 520 00:26:11,510 --> 00:26:15,270 the amount of [INAUDIBLE] you're pumping it and-- 521 00:26:15,270 --> 00:26:16,410 MICHALE FEE: Yes. 522 00:26:16,410 --> 00:26:17,040 That's right. 523 00:26:17,040 --> 00:26:19,920 This-- OK, I should have maybe been more clear. 524 00:26:19,920 --> 00:26:24,150 This current source has a knob on it that I get to set. 525 00:26:24,150 --> 00:26:26,670 I get to like-- 526 00:26:26,670 --> 00:26:28,140 there would be an app now, and I'd 527 00:26:28,140 --> 00:26:30,360 pull out my current source app and I 528 00:26:30,360 --> 00:26:31,938 type in 10 milliamps, boom. 529 00:26:31,938 --> 00:26:34,230 And this thing-- because there's a Bluetooth connection 530 00:26:34,230 --> 00:26:37,410 to this thing, and it like sets this thing to 10 milliamps, 531 00:26:37,410 --> 00:26:39,960 and it just keeps pumping 10 milliamps until you tell 532 00:26:39,960 --> 00:26:44,120 it to do something else, OK? 533 00:26:44,120 --> 00:26:44,620 Yes? 534 00:26:44,620 --> 00:26:47,690 AUDIENCE: [INAUDIBLE] make the right direction instead of it 535 00:26:47,690 --> 00:26:50,307 being constant current state that usually-- 536 00:26:50,307 --> 00:26:51,140 MICHALE FEE: Oh, OK. 537 00:26:51,140 --> 00:26:54,000 AUDIENCE: --that would create some weird kind of-- 538 00:26:54,000 --> 00:26:55,150 MICHALE FEE: Sure. 539 00:26:55,150 --> 00:26:55,650 Yeah. 540 00:26:55,650 --> 00:26:56,880 What would that be, actually? 541 00:26:56,880 --> 00:26:59,685 If you put in a linear ramp in current? 542 00:26:59,685 --> 00:27:02,400 It would be a parabolic voltage profile. 543 00:27:02,400 --> 00:27:03,600 Yeah, very good. 544 00:27:03,600 --> 00:27:04,830 That's exactly it. 545 00:27:04,830 --> 00:27:09,630 This current profile that you-- this voltage profile 546 00:27:09,630 --> 00:27:13,660 is literally just the numerical integral of this function. 547 00:27:13,660 --> 00:27:15,330 So all you have to do is look at this 548 00:27:15,330 --> 00:27:17,490 and integrate it in your head, and you can 549 00:27:17,490 --> 00:27:20,670 see what the voltage does, OK? 550 00:27:20,670 --> 00:27:21,420 Great. 551 00:27:21,420 --> 00:27:22,980 That's exactly right. 552 00:27:22,980 --> 00:27:24,280 So let's do another example. 553 00:27:24,280 --> 00:27:26,280 Let's put in a current pulse. 554 00:27:26,280 --> 00:27:29,160 So we start at zero current, we step it up to I0, 555 00:27:29,160 --> 00:27:32,580 we hold it there for tau, then we turn the current off. 556 00:27:32,580 --> 00:27:34,680 So let's start our neuron right here. 557 00:27:34,680 --> 00:27:36,462 What's going to happen? 558 00:27:36,462 --> 00:27:37,170 What's your name? 559 00:27:37,170 --> 00:27:39,420 I'm going to ask you to do this problem. 560 00:27:39,420 --> 00:27:40,210 What-- yeah. 561 00:27:40,210 --> 00:27:40,878 AUDIENCE: Sammy. 562 00:27:40,878 --> 00:27:41,670 MICHALE FEE: Sammy. 563 00:27:41,670 --> 00:27:44,448 What's this voltage going to do? 564 00:27:44,448 --> 00:27:45,930 AUDIENCE: So I'd say constant. 565 00:27:45,930 --> 00:27:46,680 MICHALE FEE: Good. 566 00:27:46,680 --> 00:27:48,183 Because it's zero current. 567 00:27:48,183 --> 00:27:49,350 Then what's going to happen? 568 00:27:49,350 --> 00:27:50,610 AUDIENCE: And it's going to [INAUDIBLE] 569 00:27:50,610 --> 00:27:51,360 MICHALE FEE: Good. 570 00:27:51,360 --> 00:27:53,130 AUDIENCE: [INAUDIBLE] 571 00:27:53,130 --> 00:27:54,294 MICHALE FEE: Good. 572 00:27:54,294 --> 00:27:57,655 AUDIENCE: And then the [INAUDIBLE] go back to constant 573 00:27:57,655 --> 00:27:58,620 at that point. 574 00:27:58,620 --> 00:27:59,700 MICHALE FEE: Awesome. 575 00:27:59,700 --> 00:28:00,370 That's it. 576 00:28:00,370 --> 00:28:03,240 It's that simple. 577 00:28:03,240 --> 00:28:05,755 OK? 578 00:28:05,755 --> 00:28:08,550 Good. 579 00:28:08,550 --> 00:28:09,120 All right. 580 00:28:09,120 --> 00:28:10,740 Now somebody brought up resistors. 581 00:28:10,740 --> 00:28:13,020 Who brought up ion conductances? 582 00:28:13,020 --> 00:28:14,700 Somebody mentioned that. 583 00:28:17,330 --> 00:28:19,340 So that's the next thing we're going to add. 584 00:28:19,340 --> 00:28:23,580 This is sort of the zero order model of a neuron. 585 00:28:23,580 --> 00:28:28,040 It has the simplest view and it's often not so bad, OK? 586 00:28:28,040 --> 00:28:30,000 For short periods of time. 587 00:28:30,000 --> 00:28:33,680 But neurons actually have ion channels, all right? 588 00:28:33,680 --> 00:28:36,540 Allow current to flow through the membrane. 589 00:28:36,540 --> 00:28:38,300 So we're going to start today by analyzing 590 00:28:38,300 --> 00:28:40,880 the case of the simplest kind of ion channel 591 00:28:40,880 --> 00:28:43,610 which is the kind of ion channel you get when you take a needle 592 00:28:43,610 --> 00:28:47,670 and you poke a hole in the membrane, OK? 593 00:28:47,670 --> 00:28:52,030 It's called the leak or a hole, all right? 594 00:28:52,030 --> 00:28:54,680 And we're going to analyze what our neuron does 595 00:28:54,680 --> 00:28:55,410 when you do that. 596 00:28:55,410 --> 00:29:00,490 So what we're going to find is that the ion channel-- 597 00:29:00,490 --> 00:29:04,400 a leak conductance, it can be represented in our model 598 00:29:04,400 --> 00:29:09,070 simply by a resistor, OK? 599 00:29:09,070 --> 00:29:12,220 And we're going to have our capacitive current, membrane 600 00:29:12,220 --> 00:29:16,300 capacitive current, and a membrane ionic current that's 601 00:29:16,300 --> 00:29:20,700 due to ions flowing through ion channels in the membrane, OK? 602 00:29:24,410 --> 00:29:26,070 And that current will be-- 603 00:29:26,070 --> 00:29:29,130 we're going to call that our leak resistance, 604 00:29:29,130 --> 00:29:35,730 and that current will be our leak current, OK? 605 00:29:35,730 --> 00:29:38,490 So now, Kirchhoff's current law tells us what? 606 00:29:38,490 --> 00:29:42,060 That the leak current plus the capacitive current 607 00:29:42,060 --> 00:29:45,728 has to equal the injected current, all right? 608 00:29:50,710 --> 00:29:53,860 We know the capacitive current is just C dV/dt, 609 00:29:53,860 --> 00:29:56,140 so we just plug that in, and now we have I 610 00:29:56,140 --> 00:30:00,160 leak plus C dV/dt equals the injected electrode current. 611 00:30:02,790 --> 00:30:05,880 That is called membrane ionic current, that 612 00:30:05,880 --> 00:30:08,500 is called membrane capacitive current, 613 00:30:08,500 --> 00:30:11,230 and that is our electrode current. 614 00:30:11,230 --> 00:30:15,730 There's a sign convention in neuroscience, 615 00:30:15,730 --> 00:30:18,600 which is that membrane ionic currents that 616 00:30:18,600 --> 00:30:20,850 are outward from the inside of the cell 617 00:30:20,850 --> 00:30:25,410 to the outside of the cell membrane are positive in sign. 618 00:30:25,410 --> 00:30:29,720 Positive charges leaving the cell have a positive sign. 619 00:30:29,720 --> 00:30:33,010 It's just convention, it could have been the other way. 620 00:30:33,010 --> 00:30:37,130 But you have to choose something, so that's what-- 621 00:30:37,130 --> 00:30:39,860 I think it was Hodgkin and Huxley, actually, 622 00:30:39,860 --> 00:30:41,990 who decided that. 623 00:30:41,990 --> 00:30:44,690 Inward currents, positive charges entering the cell 624 00:30:44,690 --> 00:30:47,150 through the membrane going this way from extracellular 625 00:30:47,150 --> 00:30:51,533 to intracellular, are defined as negative. 626 00:30:51,533 --> 00:30:52,950 Electrode currents, the other way. 627 00:30:52,950 --> 00:30:55,410 Electrode current-- inward current is-- 628 00:30:55,410 --> 00:30:57,482 into the cell is positive. 629 00:31:00,860 --> 00:31:03,440 OK, so we're going to poke a hole in our membrane 630 00:31:03,440 --> 00:31:08,860 and we're going to model that ion channel using Ohm's law. 631 00:31:08,860 --> 00:31:12,160 So the leak current is just membrane potential divided 632 00:31:12,160 --> 00:31:15,220 by resistance, leak resistance. 633 00:31:19,950 --> 00:31:22,850 So what do we get if we plug this into our-- 634 00:31:22,850 --> 00:31:25,050 I leak plus I capacitance? 635 00:31:25,050 --> 00:31:27,570 Capacitive equals injected current. 636 00:31:27,570 --> 00:31:32,640 We get V membrane potential, Vm over RL plus C dV/dt 637 00:31:32,640 --> 00:31:36,295 equals injected current, OK? 638 00:31:39,070 --> 00:31:45,460 We multiply both sides by resistance, we get V plus RC-- 639 00:31:45,460 --> 00:31:48,320 that why it's called an RC model-- 640 00:31:48,320 --> 00:31:52,260 dV/dt equals leak resistance times 641 00:31:52,260 --> 00:31:56,360 the injected current, all right? 642 00:31:56,360 --> 00:31:58,740 Now that's looking a little complicated, 643 00:31:58,740 --> 00:32:01,940 but we're going to simplify things now. 644 00:32:01,940 --> 00:32:05,970 Tau-- I'm sorry, RC is resistance times capacitance, 645 00:32:05,970 --> 00:32:09,270 and it turns out, that has units of time. 646 00:32:09,270 --> 00:32:13,110 And so we're going to call that tau-- 647 00:32:13,110 --> 00:32:14,767 got a little ahead of myself. 648 00:32:14,767 --> 00:32:16,350 That's called tau, we're going to make 649 00:32:16,350 --> 00:32:18,920 that substitution in a minute. 650 00:32:18,920 --> 00:32:21,170 But first we're going to calculate the steady state 651 00:32:21,170 --> 00:32:24,050 solution to that little equation, OK? 652 00:32:24,050 --> 00:32:26,537 So bear with me, hang on, it's all going 653 00:32:26,537 --> 00:32:27,620 to make sense in a minute. 654 00:32:30,947 --> 00:32:33,030 Does anyone know how to calculate the steady state 655 00:32:33,030 --> 00:32:35,650 solution of a differential equation? 656 00:32:35,650 --> 00:32:36,150 Yes? 657 00:32:36,150 --> 00:32:37,290 AUDIENCE: [INAUDIBLE] 658 00:32:37,290 --> 00:32:38,040 MICHALE FEE: Good. 659 00:32:38,040 --> 00:32:38,610 What's your name? 660 00:32:38,610 --> 00:32:39,360 AUDIENCE: Rebecca. 661 00:32:39,360 --> 00:32:40,910 MICHALE FEE: Rebecca. 662 00:32:40,910 --> 00:32:42,870 So you said the derivative-- 663 00:32:42,870 --> 00:32:44,270 so let's do that. 664 00:32:44,270 --> 00:32:45,850 Set dV/dt equal to 0. 665 00:32:48,550 --> 00:32:50,640 And what do you find? 666 00:32:50,640 --> 00:32:52,490 Sorry, we flashed the answer up there. 667 00:32:52,490 --> 00:32:56,126 What do you get if you said dV/dt equals 0? 668 00:32:56,126 --> 00:32:57,860 AUDIENCE: [INAUDIBLE] 669 00:32:57,860 --> 00:32:58,790 MICHALE FEE: Good. 670 00:32:58,790 --> 00:33:01,690 So you inject some current, we're 671 00:33:01,690 --> 00:33:04,530 going to hold the current constant, let's say, OK? 672 00:33:04,530 --> 00:33:08,720 We've put some current in and we hold it constant. 673 00:33:08,720 --> 00:33:11,210 The voltage will change, and eventually things 674 00:33:11,210 --> 00:33:15,630 will settle down and the dV/dt will go to 0. 675 00:33:15,630 --> 00:33:18,180 At that point, we know the voltage. 676 00:33:18,180 --> 00:33:21,840 It's just RL times Ie. 677 00:33:21,840 --> 00:33:25,020 What is the voltage equals resistance times current? 678 00:33:25,020 --> 00:33:26,107 What is that? 679 00:33:26,107 --> 00:33:26,940 AUDIENCE: Ohm's law. 680 00:33:26,940 --> 00:33:29,190 MICHALE FEE: Ohm's law. 681 00:33:29,190 --> 00:33:33,790 It's just when we inject current, a bunch of stuff 682 00:33:33,790 --> 00:33:37,720 happens, and when the dust settles, 683 00:33:37,720 --> 00:33:41,580 the voltage difference is just the injected current 684 00:33:41,580 --> 00:33:45,090 divided by the resistance. 685 00:33:45,090 --> 00:33:46,420 Does that makes sense? 686 00:33:46,420 --> 00:33:46,920 Yes? 687 00:33:46,920 --> 00:33:52,567 AUDIENCE: Where is [INAUDIBLE] 688 00:33:52,567 --> 00:33:54,650 MICHALE FEE: Well, right now we just took a needle 689 00:33:54,650 --> 00:33:57,380 and poked a hole in our cell. 690 00:33:57,380 --> 00:33:58,450 So-- 691 00:33:58,450 --> 00:33:59,700 AUDIENCE: How big the hole is? 692 00:33:59,700 --> 00:34:01,490 MICHALE FEE: Yep, exactly. 693 00:34:01,490 --> 00:34:03,530 It's how big the hole is. 694 00:34:03,530 --> 00:34:07,950 Now cells-- real cells do have leak channels. 695 00:34:07,950 --> 00:34:11,630 They're actually ion channels that 696 00:34:11,630 --> 00:34:14,590 leak kind of any ion that's in there. 697 00:34:14,590 --> 00:34:18,298 It's not very common. 698 00:34:18,298 --> 00:34:19,590 And you'll see why in a minute. 699 00:34:22,870 --> 00:34:26,139 Actually, that's not quite true. 700 00:34:26,139 --> 00:34:31,080 There are ion channels that look essentially like leaks, 701 00:34:31,080 --> 00:34:35,389 and it turns out that the ion channels 702 00:34:35,389 --> 00:34:39,590 that many neurotransmitter receptors like 703 00:34:39,590 --> 00:34:43,969 glutamate and acetylcholine, the ion channels actually 704 00:34:43,969 --> 00:34:45,350 look like little leaks. 705 00:34:45,350 --> 00:34:50,210 They pass multiple ions that makes them look like leaks. 706 00:34:55,690 --> 00:34:56,510 OK. 707 00:34:56,510 --> 00:35:02,210 So in steady state, the membrane potential goes to RL times Ie, 708 00:35:02,210 --> 00:35:06,680 and we call that voltage something special, V infinity, 709 00:35:06,680 --> 00:35:10,940 because it's the voltage that the system reaches at time 710 00:35:10,940 --> 00:35:14,760 equals infinity, OK? 711 00:35:14,760 --> 00:35:15,850 Any questions about that? 712 00:35:18,570 --> 00:35:19,070 OK. 713 00:35:19,070 --> 00:35:24,650 So we're going to just rewrite this equation as Vm plus tau 714 00:35:24,650 --> 00:35:26,670 dV/dt equals V infinity. 715 00:35:26,670 --> 00:35:30,170 And that equation we're going to see over and over 716 00:35:30,170 --> 00:35:34,050 and over again in this class in many different contexts, all 717 00:35:34,050 --> 00:35:34,550 right? 718 00:35:34,550 --> 00:35:36,830 It's a first order linear differential equation 719 00:35:36,830 --> 00:35:42,700 and it's very powerful, so I want you to get used to it. 720 00:35:42,700 --> 00:35:46,100 So what does this mean? 721 00:35:46,100 --> 00:35:48,410 So let's rewrite this equation a little bit. 722 00:35:48,410 --> 00:35:52,060 Let's move this term to the other side 723 00:35:52,060 --> 00:35:55,060 and divide both sides by tau, and here's what you get. 724 00:35:55,060 --> 00:35:59,950 dV/dt equals minus 1 over tau V minus-- 725 00:35:59,950 --> 00:36:03,835 times V minus V infinity. 726 00:36:03,835 --> 00:36:06,630 OK? 727 00:36:06,630 --> 00:36:10,520 Now let's take a look at what the derivative dV/dt looks 728 00:36:10,520 --> 00:36:11,800 like as a function of voltage. 729 00:36:11,800 --> 00:36:12,300 Yes? 730 00:36:12,300 --> 00:36:13,883 AUDIENCE: So why couldn't we have said 731 00:36:13,883 --> 00:36:16,260 we didn't mean to [INAUDIBLE]. 732 00:36:20,220 --> 00:36:25,554 MICHALE FEE: V infinity is defined as RL times Ie. 733 00:36:25,554 --> 00:36:27,350 AUDIENCE: Oh, so that's a [INAUDIBLE].. 734 00:36:27,350 --> 00:36:28,760 MICHALE FEE: It's a definition. 735 00:36:28,760 --> 00:36:31,900 Sorry, I should have put like three lines there 736 00:36:31,900 --> 00:36:33,610 to indicate that it's the definition. 737 00:36:33,610 --> 00:36:34,490 AUDIENCE: OK. 738 00:36:34,490 --> 00:36:35,240 MICHALE FEE: Yeah. 739 00:36:35,240 --> 00:36:37,690 Sorry, that's a very important question. 740 00:36:37,690 --> 00:36:38,676 What's your name? 741 00:36:38,676 --> 00:36:39,570 AUDIENCE: Rishi. 742 00:36:39,570 --> 00:36:40,110 MICHALE FEE: Rishi. 743 00:36:40,110 --> 00:36:40,610 OK. 744 00:36:40,610 --> 00:36:43,375 I'm going to make an attempt to remember names. 745 00:36:46,090 --> 00:36:48,120 So is everyone clear about that? 746 00:36:48,120 --> 00:36:52,420 V infinity is defined as the resistance 747 00:36:52,420 --> 00:36:54,080 times this injected current. 748 00:36:54,080 --> 00:36:58,780 So the injected current, when we inject current into our neuron, 749 00:36:58,780 --> 00:37:02,150 you're changing V infinity, OK? 750 00:37:02,150 --> 00:37:04,103 You're controlling it. 751 00:37:04,103 --> 00:37:05,020 Does that makes sense? 752 00:37:08,570 --> 00:37:09,190 OK. 753 00:37:09,190 --> 00:37:12,038 So let's look at how the derivative changes 754 00:37:12,038 --> 00:37:13,830 as a function of voltage, it's very simple. 755 00:37:13,830 --> 00:37:14,530 Bear with me. 756 00:37:14,530 --> 00:37:17,020 All of this is going to crystallize 757 00:37:17,020 --> 00:37:21,220 in your mind in one beautiful construct very shortly. 758 00:37:21,220 --> 00:37:22,063 Yes? 759 00:37:22,063 --> 00:37:25,444 AUDIENCE: [INAUDIBLE] 760 00:37:25,444 --> 00:37:27,560 MICHALE FEE: Yep. 761 00:37:27,560 --> 00:37:33,280 Resistance times capacitance has units time, OK? 762 00:37:33,280 --> 00:37:34,750 And so we call it tau. 763 00:37:34,750 --> 00:37:37,720 Tau is just a constant, OK? 764 00:37:37,720 --> 00:37:40,610 So hang on, bear with me. 765 00:37:40,610 --> 00:37:44,550 The derivative is a function of voltage. 766 00:37:44,550 --> 00:37:49,850 And at V equals V infinity, the derivative is 0, right? 767 00:37:53,420 --> 00:37:55,530 That's the definition of the infinity, 768 00:37:55,530 --> 00:37:59,576 it's the voltage at which the derivative is 0. 769 00:37:59,576 --> 00:38:02,390 Yeah? 770 00:38:02,390 --> 00:38:09,490 And the voltage is less than V infinity, the derivative 771 00:38:09,490 --> 00:38:13,270 is positive, right? 772 00:38:13,270 --> 00:38:15,670 When the voltage is below the infinity, 773 00:38:15,670 --> 00:38:17,920 the derivative of voltage is positive. 774 00:38:17,920 --> 00:38:20,820 So what is the voltage doing? 775 00:38:20,820 --> 00:38:22,990 It's approaching V infinity. 776 00:38:22,990 --> 00:38:26,090 If the voltage is above V infinity, 777 00:38:26,090 --> 00:38:30,590 voltage greater than V infinity, the derivative is negative. 778 00:38:30,590 --> 00:38:32,955 So voltage does what? 779 00:38:32,955 --> 00:38:33,830 AUDIENCE: [INAUDIBLE] 780 00:38:33,830 --> 00:38:34,870 MICHALE FEE: Yes, but it's-- 781 00:38:34,870 --> 00:38:35,787 AUDIENCE: Approaches-- 782 00:38:35,787 --> 00:38:37,860 MICHALE FEE: It approaches V infinity. 783 00:38:37,860 --> 00:38:39,860 So no matter where voltage is, it's 784 00:38:39,860 --> 00:38:41,300 always approaching V infinity. 785 00:38:41,300 --> 00:38:43,910 If it's below V infinity, the slope is positive, 786 00:38:43,910 --> 00:38:45,740 and it approaches V infinity. 787 00:38:45,740 --> 00:38:50,000 If it's above V infinity, the slope is negative, 788 00:38:50,000 --> 00:38:53,240 and it approaches V infinity from above. 789 00:38:53,240 --> 00:38:54,710 Pretty cool, right? 790 00:38:54,710 --> 00:38:59,690 So V is always just relaxing toward V infinity. 791 00:38:59,690 --> 00:39:00,980 And how does it get there? 792 00:39:00,980 --> 00:39:02,630 Does it go linearly? 793 00:39:02,630 --> 00:39:04,310 Well, you can see that the slope-- 794 00:39:04,310 --> 00:39:06,050 the rate at which it approaches V 795 00:39:06,050 --> 00:39:10,630 infinity is proportional to how far away it is from V infinity. 796 00:39:10,630 --> 00:39:13,970 And so it doesn't just go vroom, boom, crash into V infinity, 797 00:39:13,970 --> 00:39:22,360 it kind of slowly approaches V infinity, OK? 798 00:39:22,360 --> 00:39:25,890 Anybody know what that function is called? 799 00:39:25,890 --> 00:39:26,890 AUDIENCE: Exponential. 800 00:39:26,890 --> 00:39:30,430 MICHALE FEE: It's an exponential, good. 801 00:39:30,430 --> 00:39:36,920 And it approaches with a timescale of tau. 802 00:39:36,920 --> 00:39:40,430 So if tau is small, it approaches quickly. 803 00:39:40,430 --> 00:39:42,600 If tau is long, it approaches slow. 804 00:39:42,600 --> 00:39:47,970 You can see that if tau is big, the derivatives are small. 805 00:39:47,970 --> 00:39:52,004 If tau is small, the derivatives are bigger, all right? 806 00:40:00,280 --> 00:40:01,735 Any questions about that? 807 00:40:01,735 --> 00:40:02,980 Yes? 808 00:40:02,980 --> 00:40:05,211 AUDIENCE: So a tau usually is a-- 809 00:40:05,211 --> 00:40:06,586 MICHALE FEE: Sorry, say it again? 810 00:40:06,586 --> 00:40:08,290 AUDIENCE: --times equals to tau-- 811 00:40:08,290 --> 00:40:08,998 MICHALE FEE: Yes. 812 00:40:08,998 --> 00:40:12,090 AUDIENCE: --is V infinity plus 1 over e. 813 00:40:12,090 --> 00:40:15,646 MICHALE FEE: At time tau, the-- 814 00:40:15,646 --> 00:40:17,020 AUDIENCE: Times V0. 815 00:40:17,020 --> 00:40:18,520 MICHALE FEE: At time tau-- 816 00:40:18,520 --> 00:40:22,000 at time 0, the difference between V infinity-- sorry, V 817 00:40:22,000 --> 00:40:25,330 and V infinity is V0 minus V infinity. 818 00:40:25,330 --> 00:40:27,820 At time tau, that initial difference 819 00:40:27,820 --> 00:40:29,320 drops by about a third. 820 00:40:29,320 --> 00:40:30,010 AUDIENCE: OK. 821 00:40:30,010 --> 00:40:34,220 MICHALE FEE: About 1 over e, which I think is 2.7-something, 822 00:40:34,220 --> 00:40:35,050 OK? 823 00:40:35,050 --> 00:40:37,600 So in 1 tau, this voltage difference 824 00:40:37,600 --> 00:40:40,250 falls by about a-- drops by about a third. 825 00:40:40,250 --> 00:40:43,020 And in another tau, it drops by another third, 826 00:40:43,020 --> 00:40:46,970 it keeps going, OK? 827 00:40:46,970 --> 00:40:47,470 All right. 828 00:40:47,470 --> 00:40:51,076 So let's just write down the general solution. 829 00:40:51,076 --> 00:40:53,350 The general solution for the case 830 00:40:53,350 --> 00:41:02,700 where you have constant current, that voltage difference 831 00:41:02,700 --> 00:41:05,370 from the voltage at time t to V infinity 832 00:41:05,370 --> 00:41:08,140 is just equal to the initial voltage difference times e 833 00:41:08,140 --> 00:41:11,775 to the minus t over tau, OK? 834 00:41:14,550 --> 00:41:19,950 So if t is equal to tau, then this is e to the minus 1, 835 00:41:19,950 --> 00:41:23,730 so the voltage difference will be 1/3 of the original voltage 836 00:41:23,730 --> 00:41:24,630 difference. 837 00:41:24,630 --> 00:41:27,290 Is that clear? 838 00:41:27,290 --> 00:41:27,790 OK. 839 00:41:27,790 --> 00:41:31,930 So now let's see what this looks like in our neuron. 840 00:41:31,930 --> 00:41:34,320 We have our neuron, we have a current pulse, 841 00:41:34,320 --> 00:41:37,260 we have zero current, we turn the current 842 00:41:37,260 --> 00:41:40,970 on to I0 at this time, we hold the current constant, 843 00:41:40,970 --> 00:41:45,660 and we turn current off at this time right here, OK? 844 00:41:45,660 --> 00:41:48,480 So what does the voltage do? 845 00:41:48,480 --> 00:41:50,580 Let's go step by step. 846 00:41:50,580 --> 00:41:52,470 The first thing is that voltage-- 847 00:41:52,470 --> 00:41:55,995 sorry-- the current controls what in that equation? 848 00:41:58,902 --> 00:42:03,070 The current controls V infinity. 849 00:42:03,070 --> 00:42:05,100 So we can plot V infinity immediately, 850 00:42:05,100 --> 00:42:07,440 because V infinity is just the resistance 851 00:42:07,440 --> 00:42:08,800 times the injected current. 852 00:42:08,800 --> 00:42:10,220 So what does V infinity look like? 853 00:42:13,230 --> 00:42:16,908 It's constant here, and then what happens to V infinity? 854 00:42:16,908 --> 00:42:19,540 AUDIENCE: Increases. 855 00:42:19,540 --> 00:42:21,948 MICHALE FEE: It increases, that's correct, but-- 856 00:42:21,948 --> 00:42:23,490 AUDIENCE: It'll just be [INAUDIBLE].. 857 00:42:23,490 --> 00:42:24,240 MICHALE FEE: Good. 858 00:42:24,240 --> 00:42:26,320 It will just be the resistance times the current. 859 00:42:26,320 --> 00:42:30,886 Resistance is a constant, so V infinity will just go up here, 860 00:42:30,886 --> 00:42:31,386 right? 861 00:42:31,386 --> 00:42:31,886 Good. 862 00:42:31,886 --> 00:42:39,300 It'll go up, and then it stays constant at R times I. 863 00:42:39,300 --> 00:42:42,050 And then at this point, the current 864 00:42:42,050 --> 00:42:47,420 goes back to 0, so V infinity is resistance time 0, 865 00:42:47,420 --> 00:42:51,300 so V infinity drops back to 0. 866 00:42:51,300 --> 00:42:52,720 Does that makes sense? 867 00:42:52,720 --> 00:42:57,160 That's V infinity, that's not the voltage of the cell. 868 00:42:57,160 --> 00:43:00,880 That's the steady state voltage of the cell. 869 00:43:00,880 --> 00:43:04,770 So now what does voltage of the cell actually do? 870 00:43:07,470 --> 00:43:11,338 So let's start our voltage here at 0. 871 00:43:11,338 --> 00:43:11,880 What happens? 872 00:43:15,720 --> 00:43:16,740 Good. 873 00:43:16,740 --> 00:43:18,470 What is it doing? 874 00:43:18,470 --> 00:43:19,970 AUDIENCE: Approaching from infinity. 875 00:43:19,970 --> 00:43:20,750 MICHALE FEE: Good. 876 00:43:20,750 --> 00:43:21,960 It approaches. 877 00:43:21,960 --> 00:43:27,650 So V at every point is relaxing toward V 878 00:43:27,650 --> 00:43:31,290 infinity exponentially, right? 879 00:43:31,290 --> 00:43:32,370 And that looks like-- 880 00:43:32,370 --> 00:43:39,680 at some time constant, and that looks like this, all right? 881 00:43:39,680 --> 00:43:42,185 Now what happens here? 882 00:43:42,185 --> 00:43:43,160 AUDIENCE: [INAUDIBLE] 883 00:43:43,160 --> 00:43:43,910 MICHALE FEE: Good. 884 00:43:43,910 --> 00:43:47,180 Because V infinity suddenly change to 0, 885 00:43:47,180 --> 00:43:50,570 and so V relaxes toward V infinity exponentially 886 00:43:50,570 --> 00:43:52,130 with some time constant. 887 00:43:57,366 --> 00:43:57,866 OK? 888 00:44:01,220 --> 00:44:02,397 Any questions? 889 00:44:07,870 --> 00:44:08,370 OK. 890 00:44:12,080 --> 00:44:18,690 Now this-- that is our RC model neuron, OK? 891 00:44:18,690 --> 00:44:20,320 Resistance times capacitance-- they 892 00:44:20,320 --> 00:44:24,380 got a resistor and a capacitor, but the solutions 893 00:44:24,380 --> 00:44:27,740 are just exponential decays toward some steady state 894 00:44:27,740 --> 00:44:30,490 solution. 895 00:44:30,490 --> 00:44:35,110 Now it turns out that an RC system, 896 00:44:35,110 --> 00:44:40,650 a first order linear system acts like a filter, OK? 897 00:44:40,650 --> 00:44:45,510 So remember, our neuron that just has a capacitor 898 00:44:45,510 --> 00:44:48,660 is an integrator, it integrates over time. 899 00:44:48,660 --> 00:44:51,960 When you add a resistor, this thing-- 900 00:44:51,960 --> 00:44:54,210 it's kind of integrating here, but then it 901 00:44:54,210 --> 00:44:57,630 gets tired and stops integrating, OK? 902 00:44:57,630 --> 00:45:01,860 It relaxes to some steady state. 903 00:45:01,860 --> 00:45:05,750 So this actually looks like a filter. 904 00:45:05,750 --> 00:45:09,820 It takes time to respond to something. 905 00:45:09,820 --> 00:45:15,550 So that system responds well to things that are changing slowly 906 00:45:15,550 --> 00:45:19,150 in time, and it responds very weakly to things that 907 00:45:19,150 --> 00:45:21,410 are changing rapidly in time. 908 00:45:21,410 --> 00:45:22,570 So here's an example-- 909 00:45:22,570 --> 00:45:26,350 I'll put together this demonstration. 910 00:45:26,350 --> 00:45:28,280 In red is the injected current. 911 00:45:28,280 --> 00:45:32,050 So if you have long pulses of injected current, 912 00:45:32,050 --> 00:45:35,290 the time constant of this garners about 10 milliseconds-- 913 00:45:35,290 --> 00:45:36,930 I think this is probably a-- 914 00:45:36,930 --> 00:45:39,750 what is that, a 50 or 100-millisecond pulse? 915 00:45:39,750 --> 00:45:40,250 80-- 916 00:45:40,250 --> 00:45:41,125 AUDIENCE: Nanofarads. 917 00:45:41,125 --> 00:45:42,225 MICHALE FEE: Yeah. 918 00:45:42,225 --> 00:45:44,390 You can see that in blue is the voltage, 919 00:45:44,390 --> 00:45:47,670 it relaxes toward V infinity. 920 00:45:47,670 --> 00:45:50,000 And then the current goes off, it relaxes back. 921 00:45:50,000 --> 00:45:53,510 And you can see that the voltage is responding very well 922 00:45:53,510 --> 00:45:55,220 to the current injection. 923 00:45:55,220 --> 00:45:58,420 But now let's make really short pulses 924 00:45:58,420 --> 00:46:00,760 that are much shorter than tau. 925 00:46:00,760 --> 00:46:04,630 You can see that the voltage starts 926 00:46:04,630 --> 00:46:07,027 relaxing toward V infinity, but it doesn't get very far, 927 00:46:07,027 --> 00:46:08,860 and all of a sudden the current's turned off 928 00:46:08,860 --> 00:46:10,540 and it relaxes back. 929 00:46:10,540 --> 00:46:13,450 And so you can plot the peak voltage response 930 00:46:13,450 --> 00:46:16,220 as a function of the width of these pulses, 931 00:46:16,220 --> 00:46:20,650 and you can see that for long pulses, it responds very well, 932 00:46:20,650 --> 00:46:22,520 but for short pulses-- 933 00:46:22,520 --> 00:46:24,310 really responds at all. 934 00:46:24,310 --> 00:46:28,180 And that's called a low pass filter, OK? 935 00:46:28,180 --> 00:46:31,840 It responds well to slowly-changing things, 936 00:46:31,840 --> 00:46:35,210 but barely responds to rapidly-changing things. 937 00:46:35,210 --> 00:46:40,651 So it's passing low frequencies, low pass filter, OK? 938 00:46:43,340 --> 00:46:44,070 All right. 939 00:46:44,070 --> 00:46:44,760 Any questions? 940 00:46:44,760 --> 00:46:46,540 That was a lot of stuff all at once. 941 00:46:46,540 --> 00:46:47,040 Yes? 942 00:46:47,040 --> 00:46:49,900 AUDIENCE: I'm just curious, like on what order is 943 00:46:49,900 --> 00:46:53,030 the capacitance for nanofarads? 944 00:46:53,030 --> 00:46:54,100 MICHALE FEE: OK. 945 00:46:54,100 --> 00:46:57,760 It's 10 nanofarads per-- 946 00:46:57,760 --> 00:46:59,990 1 microfarads per square centimeter. 947 00:46:59,990 --> 00:47:03,953 10 nanofarads per square millimeter, OK? 948 00:47:03,953 --> 00:47:05,870 We're going to get to that in a second, that's 949 00:47:05,870 --> 00:47:07,110 a great question. 950 00:47:07,110 --> 00:47:09,950 We're going to get to what the actual numbers look like 951 00:47:09,950 --> 00:47:13,190 for real neurons, OK? 952 00:47:13,190 --> 00:47:14,640 I think you had a question. 953 00:47:14,640 --> 00:47:17,095 AUDIENCE: [INAUDIBLE] 954 00:47:17,095 --> 00:47:18,470 MICHALE FEE: Sorry, say it again? 955 00:47:18,470 --> 00:47:19,870 AUDIENCE: Past [INAUDIBLE]? 956 00:47:19,870 --> 00:47:22,795 Is that [INAUDIBLE] reacts to [INAUDIBLE]?? 957 00:47:22,795 --> 00:47:24,670 MICHALE FEE: What happens is it reacts to it, 958 00:47:24,670 --> 00:47:27,640 but because it's changing kind of linearly 959 00:47:27,640 --> 00:47:30,220 at these short times, it doesn't get very far. 960 00:47:30,220 --> 00:47:32,060 If the current stays on for a long time, 961 00:47:32,060 --> 00:47:35,290 you can see it has exactly the same profile here, 962 00:47:35,290 --> 00:47:37,802 but it just has time to reach V infinity. 963 00:47:37,802 --> 00:47:39,760 Here, it doesn't have time to reach V infinity, 964 00:47:39,760 --> 00:47:46,590 it just gets a little bit away from 0 and then it decays back. 965 00:47:46,590 --> 00:47:47,350 Yes? 966 00:47:47,350 --> 00:47:50,162 AUDIENCE: Are there other sorts of filters for non-responses 967 00:47:50,162 --> 00:47:52,930 like [INAUDIBLE]? 968 00:47:52,930 --> 00:47:55,690 MICHALE FEE: You can build different kinds of filters 969 00:47:55,690 --> 00:47:58,990 from circuits of neurons, but neurons themselves 970 00:47:58,990 --> 00:48:03,160 tend to be high pass filters. 971 00:48:03,160 --> 00:48:05,360 You can-- sorry, low pass filters. 972 00:48:05,360 --> 00:48:10,840 You can make-- so you can put in different kinds of ion channels 973 00:48:10,840 --> 00:48:14,080 that change the properties of neurons, 974 00:48:14,080 --> 00:48:16,180 but sort of to first order, you should think 975 00:48:16,180 --> 00:48:18,040 of neurons as low pass filters. 976 00:48:21,976 --> 00:48:22,960 Yeah? 977 00:48:22,960 --> 00:48:31,282 AUDIENCE: Can you show the [INAUDIBLE] 978 00:48:31,282 --> 00:48:32,740 MICHALE FEE: I wrote it like this-- 979 00:48:32,740 --> 00:48:35,980 you can write it as V equals V infinity plus this other stuff, 980 00:48:35,980 --> 00:48:38,620 but I wrote it like this because what you should really be 981 00:48:38,620 --> 00:48:42,610 seeing here is that the voltage difference from-- 982 00:48:42,610 --> 00:48:48,230 time between the voltage and V infinity decays exponentially. 983 00:48:48,230 --> 00:48:51,370 So the distance you are from V infinity decays 984 00:48:51,370 --> 00:48:53,974 exponentially, OK? 985 00:48:56,720 --> 00:48:58,380 It makes it more obvious that you're 986 00:48:58,380 --> 00:49:02,915 decaying toward V infinity, OK? 987 00:49:06,800 --> 00:49:07,950 Yes? 988 00:49:07,950 --> 00:49:10,492 AUDIENCE: If you were to arrange like the physical properties 989 00:49:10,492 --> 00:49:15,870 of the neuron itself, you [INAUDIBLE] 990 00:49:15,870 --> 00:49:17,400 MICHALE FEE: Not-- not really. 991 00:49:17,400 --> 00:49:18,420 Not simply. 992 00:49:18,420 --> 00:49:22,050 You can put in certain ion channels 993 00:49:22,050 --> 00:49:29,950 that could make a neuron less responsive at low frequencies, 994 00:49:29,950 --> 00:49:30,450 OK? 995 00:49:30,450 --> 00:49:34,620 So you can make them kind of responsive to some middle range 996 00:49:34,620 --> 00:49:37,265 of frequencies that won't respond 997 00:49:37,265 --> 00:49:38,640 to very high frequencies and they 998 00:49:38,640 --> 00:49:43,040 won't respond much to very low frequencies, 999 00:49:43,040 --> 00:49:45,080 but for the most part, again, just 1000 00:49:45,080 --> 00:49:47,330 you should-- at this point, let's just think of them 1001 00:49:47,330 --> 00:49:50,210 as low pass filters. 1002 00:49:50,210 --> 00:49:52,940 We're going to start adding fancy stuff 1003 00:49:52,940 --> 00:49:58,110 to our neuron that's going to make it more complicated. 1004 00:49:58,110 --> 00:50:01,070 So don't get too hung up on this. 1005 00:50:01,070 --> 00:50:03,994 All right, let me just make this point. 1006 00:50:03,994 --> 00:50:07,630 This one right here, V equals tau-- 1007 00:50:07,630 --> 00:50:11,660 V plus tau dV/dt equals V infinity appears everywhere, 1008 00:50:11,660 --> 00:50:17,920 it's ubiquitous in physics, chemistry, biology, we'll 1009 00:50:17,920 --> 00:50:20,380 be using it in multiple different contexts 1010 00:50:20,380 --> 00:50:24,790 in different parts of the class, and in-- 1011 00:50:24,790 --> 00:50:26,920 computation, OK? 1012 00:50:26,920 --> 00:50:32,770 And even slightly more complicated versions of this, 1013 00:50:32,770 --> 00:50:37,500 like the Michaelis-Menten equations in chemistry, 1014 00:50:37,500 --> 00:50:39,880 you can kind of understand them in simple terms. 1015 00:50:39,880 --> 00:50:43,290 If you like have a handle on this, 1016 00:50:43,290 --> 00:50:45,270 other slightly more complicated things 1017 00:50:45,270 --> 00:50:49,360 become much more intuitive, OK? 1018 00:50:49,360 --> 00:50:53,260 All right, so try to really make sure 1019 00:50:53,260 --> 00:50:57,250 that you understand this equation 1020 00:50:57,250 --> 00:51:02,250 and how we derived it, OK? 1021 00:51:02,250 --> 00:51:05,040 OK, let's talk about the origin of this-- 1022 00:51:05,040 --> 00:51:06,510 the timescale of a neuron. 1023 00:51:06,510 --> 00:51:08,370 So the tau of a neuron-- 1024 00:51:08,370 --> 00:51:11,460 of most neurons is about 100 milliseconds to-- 1025 00:51:11,460 --> 00:51:13,650 sorry, 10 milliseconds to 100 milliseconds, 1026 00:51:13,650 --> 00:51:15,250 kind of in that range. 1027 00:51:15,250 --> 00:51:18,600 And it comes from the values of resistance and capacitance 1028 00:51:18,600 --> 00:51:19,990 of a neuron. 1029 00:51:19,990 --> 00:51:24,660 So a resistor-- the resistance of a neuron-- 1030 00:51:24,660 --> 00:51:29,770 range of 100 million ohms, OK? 1031 00:51:29,770 --> 00:51:33,510 And the capacitance is about 10 to the minus 10 ohms 1032 00:51:33,510 --> 00:51:37,640 or about 100 picofarads. 1033 00:51:37,640 --> 00:51:39,780 And you multiply those two things together 1034 00:51:39,780 --> 00:51:44,910 and you get a time constant of about 10 milliseconds, OK? 1035 00:51:44,910 --> 00:51:49,510 So what that means is if you inject current into a neuron, 1036 00:51:49,510 --> 00:51:52,420 it takes about 10 to 100 milliseconds for it 1037 00:51:52,420 --> 00:51:58,010 to fully respond to that step of current. 1038 00:51:58,010 --> 00:52:00,260 The voltage will jump up and relax 1039 00:52:00,260 --> 00:52:05,510 to the new V infinity in about 10 to 100 milliseconds, OK? 1040 00:52:05,510 --> 00:52:07,190 So let's take a little bit closer 1041 00:52:07,190 --> 00:52:11,000 look at the resistance-- what this resistance in capacitance 1042 00:52:11,000 --> 00:52:14,420 looks like in a neuron. 1043 00:52:14,420 --> 00:52:18,700 So we've described the relation between leak current 1044 00:52:18,700 --> 00:52:24,850 and voltage as current equals voltage over resistance, 1045 00:52:24,850 --> 00:52:29,810 but rather than using resistance to think about currents flowing 1046 00:52:29,810 --> 00:52:33,290 through a membrane, it's much more useful, usually, 1047 00:52:33,290 --> 00:52:35,730 to think about something called conductance, 1048 00:52:35,730 --> 00:52:39,550 and conductance is just 1 over resistance, OK? 1049 00:52:39,550 --> 00:52:43,790 So conductance, G-- and we use the simple G for conductance-- 1050 00:52:43,790 --> 00:52:45,620 it's equal to 1 over resistance. 1051 00:52:45,620 --> 00:52:47,960 So now we can write Ohm's law as I 1052 00:52:47,960 --> 00:52:55,820 equals G times V. Resistance has units of ohms, G-- 1053 00:52:55,820 --> 00:52:58,730 conductance has units of inverse ohms or siemens 1054 00:52:58,730 --> 00:53:02,930 is the SI unit for conductance. 1055 00:53:02,930 --> 00:53:06,340 So if we have conductances, if we 1056 00:53:06,340 --> 00:53:10,400 have two-- let's say two ion channels in the membrane, 1057 00:53:10,400 --> 00:53:11,830 they operate in parallel. 1058 00:53:11,830 --> 00:53:15,125 Current flows through them separately, right? 1059 00:53:15,125 --> 00:53:17,500 They're not in series, like it flows through one and then 1060 00:53:17,500 --> 00:53:19,070 flows through the other, right? 1061 00:53:19,070 --> 00:53:22,150 They are in parallel-- the current can flow 1062 00:53:22,150 --> 00:53:26,210 through both like this up in parallel. 1063 00:53:26,210 --> 00:53:30,160 And we can write down the current using Kirchhoff's law, 1064 00:53:30,160 --> 00:53:32,200 the total current is just the sum 1065 00:53:32,200 --> 00:53:36,130 of the current through those two separate conductances, right? 1066 00:53:36,130 --> 00:53:38,140 Now the total current is-- 1067 00:53:38,140 --> 00:53:40,840 we can just expand this in terms of the inductance 1068 00:53:40,840 --> 00:53:41,750 of each one of those. 1069 00:53:41,750 --> 00:53:47,560 So the total current is G1 times the voltage difference 1070 00:53:47,560 --> 00:53:50,680 plus G2 times the voltage difference 1071 00:53:50,680 --> 00:53:53,610 for this conductance. 1072 00:53:53,610 --> 00:53:55,490 So the total current is just-- 1073 00:53:55,490 --> 00:53:59,720 you factor out the V, the total current is just G1 plus G2, 1074 00:53:59,720 --> 00:54:05,910 so we can write down the total conductance as just G1 plus G2. 1075 00:54:05,910 --> 00:54:07,290 Does that makes sense? 1076 00:54:07,290 --> 00:54:10,230 So conductances in parallel add together. 1077 00:54:13,563 --> 00:54:15,230 So if we have a piece of membrane that's 1078 00:54:15,230 --> 00:54:18,860 got some ion channels in it-- or holes, 1079 00:54:18,860 --> 00:54:21,080 and we add another piece of membrane that kind of has 1080 00:54:21,080 --> 00:54:23,440 the same density of ion channels, 1081 00:54:23,440 --> 00:54:26,080 you have twice the holes, twice the current, 1082 00:54:26,080 --> 00:54:27,500 and twice the conductance. 1083 00:54:33,230 --> 00:54:36,400 So we can write the current as conductance times membrane 1084 00:54:36,400 --> 00:54:39,010 potential, but we can rewrite that conductance 1085 00:54:39,010 --> 00:54:43,680 as the area times that conductance per unit area, 1086 00:54:43,680 --> 00:54:47,200 and that's called specific membrane conductance-- 1087 00:54:47,200 --> 00:54:49,540 in this case, it's a leak, so we call it specific leak 1088 00:54:49,540 --> 00:54:53,980 conductance, and it has units of conductance per area. 1089 00:54:57,700 --> 00:55:03,070 We multiply that by the area and we get that total conductance. 1090 00:55:03,070 --> 00:55:04,330 Any questions about this? 1091 00:55:09,077 --> 00:55:09,577 No? 1092 00:55:12,920 --> 00:55:17,190 So you can see that we can now plot 1093 00:55:17,190 --> 00:55:21,940 the current through the membrane as a function of voltage. 1094 00:55:21,940 --> 00:55:25,080 This is called a IV current plotted 1095 00:55:25,080 --> 00:55:26,170 as a function of voltage. 1096 00:55:26,170 --> 00:55:28,877 You can see that the current is linear 1097 00:55:28,877 --> 00:55:30,210 as a function of voltage, right? 1098 00:55:30,210 --> 00:55:33,230 That's just Ohm's law. 1099 00:55:33,230 --> 00:55:36,180 And you can see that for a low conductance, G is small, 1100 00:55:36,180 --> 00:55:38,380 so the slope is small. 1101 00:55:38,380 --> 00:55:40,760 For a high conductance, you get a lot 1102 00:55:40,760 --> 00:55:43,460 of current for a little bit of voltage, and so the-- 1103 00:55:43,460 --> 00:55:45,070 deeper, OK? 1104 00:55:45,070 --> 00:55:48,580 So if you plot current versus voltage, you get a curve, 1105 00:55:48,580 --> 00:55:54,414 and the slope of that curve is just equal to the conductance, 1106 00:55:54,414 --> 00:55:56,800 all right? 1107 00:55:56,800 --> 00:55:57,340 OK. 1108 00:55:57,340 --> 00:56:00,200 Now let's look at capacitance. 1109 00:56:00,200 --> 00:56:04,600 The total current through these two capacitors in parallel 1110 00:56:04,600 --> 00:56:07,140 is just the current through one capacitor plus the current 1111 00:56:07,140 --> 00:56:08,160 through the other. 1112 00:56:08,160 --> 00:56:10,800 We can write the current through each capacitor separately. 1113 00:56:10,800 --> 00:56:16,080 I total equals C1 dV/dt plus C2 dV/dt. 1114 00:56:16,080 --> 00:56:19,050 Factor out the dV/dt and you get that the total current is just 1115 00:56:19,050 --> 00:56:21,320 C1 plus C2 dV/dt. 1116 00:56:21,320 --> 00:56:25,410 So the total capacitance is the sum of the capacitances. 1117 00:56:25,410 --> 00:56:27,570 So if you have a patch of membrane, 1118 00:56:27,570 --> 00:56:31,090 you measure the capacitance, if you put another one next to it, 1119 00:56:31,090 --> 00:56:33,820 you'll get the sum of those two capacitances. 1120 00:56:33,820 --> 00:56:36,675 So the capacitance also scales with area. 1121 00:56:39,270 --> 00:56:42,590 So we can write down the total membrane capacitance 1122 00:56:42,590 --> 00:56:48,500 as capacitance per unit area times the area of the cell, 1123 00:56:48,500 --> 00:56:50,380 right? 1124 00:56:50,380 --> 00:56:53,650 And the area of a cell is like the-- if it's a sphere, 1125 00:56:53,650 --> 00:56:59,020 it's 4 pi r squared where is the radius, OK? 1126 00:56:59,020 --> 00:56:59,520 All right. 1127 00:56:59,520 --> 00:57:02,760 And here's the-- this C sub m is called 1128 00:57:02,760 --> 00:57:05,580 this specific membrane capacitance, 1129 00:57:05,580 --> 00:57:11,580 and it's 10 nanofarads per square millimeter, all right? 1130 00:57:11,580 --> 00:57:14,860 OK, now, I want to show you something really cool. 1131 00:57:14,860 --> 00:57:18,450 We have a cell that has a membrane with some-- 1132 00:57:18,450 --> 00:57:20,400 this cell has some time constant-- remember, 1133 00:57:20,400 --> 00:57:22,390 10 milliseconds. 1134 00:57:22,390 --> 00:57:26,280 Now you might think, oh, the capacitance 1135 00:57:26,280 --> 00:57:29,160 of the cell depends on how big it is, right? 1136 00:57:29,160 --> 00:57:31,140 And so the time constant will change 1137 00:57:31,140 --> 00:57:34,650 depending on how big the cell is, you might think. 1138 00:57:34,650 --> 00:57:36,240 But let's actually calculate the time 1139 00:57:36,240 --> 00:57:41,230 constant from this capacitance and this conductance, OK? 1140 00:57:41,230 --> 00:57:42,310 So here we go. 1141 00:57:42,310 --> 00:57:48,730 Time constant C. R is just 1 over the conductance, right? 1142 00:57:48,730 --> 00:57:50,500 So the time constant is capacitance 1143 00:57:50,500 --> 00:57:52,990 divided by conductance-- total capacitance 1144 00:57:52,990 --> 00:57:55,900 divided by total conductance. 1145 00:57:55,900 --> 00:57:58,860 But you can rewrite this capacitance as capacitance 1146 00:57:58,860 --> 00:58:01,140 per unit area times area, you can 1147 00:58:01,140 --> 00:58:04,950 rewrite that conductance as conductance per unit 1148 00:58:04,950 --> 00:58:10,630 area times the area of the cell, and the areas cancel. 1149 00:58:10,630 --> 00:58:14,590 And so the time constant is just that capacitance 1150 00:58:14,590 --> 00:58:17,770 per unit area of the membrane divided by the conductance 1151 00:58:17,770 --> 00:58:19,880 per unit area of the membrane. 1152 00:58:19,880 --> 00:58:23,470 And what that means is that the time constant of a cell-- 1153 00:58:23,470 --> 00:58:25,380 nothing to do with the cell. 1154 00:58:25,380 --> 00:58:28,870 The time constant is the membrane time constant, 1155 00:58:28,870 --> 00:58:31,165 and it's a property only of the membrane. 1156 00:58:34,110 --> 00:58:35,270 That's pretty cool, right? 1157 00:58:43,126 --> 00:58:44,580 Any questions about that? 1158 00:58:53,600 --> 00:58:56,420 Now in a more complicated neuron where 1159 00:58:56,420 --> 00:58:59,480 you have a soma and dendrites and axons, 1160 00:58:59,480 --> 00:59:05,310 different parts of the cell can have different conductance 1161 00:59:05,310 --> 00:59:07,560 per unit area-- like more ion channels 1162 00:59:07,560 --> 00:59:11,010 out here on the dendrite and maybe fewer on the soma. 1163 00:59:11,010 --> 00:59:14,700 And so one part of a cell can have a different membrane time 1164 00:59:14,700 --> 00:59:18,870 constant than some other part of the cell, OK? 1165 00:59:18,870 --> 00:59:20,880 But again, it's a property of the membrane. 1166 00:59:27,478 --> 00:59:28,520 Any questions about that? 1167 00:59:28,520 --> 00:59:29,294 Yes? 1168 00:59:29,294 --> 00:59:32,305 AUDIENCE: So in that case, like different time constants, 1169 00:59:32,305 --> 00:59:35,150 do you have to like consider flow between different areas? 1170 00:59:35,150 --> 00:59:36,580 MICHALE FEE: You sure do. 1171 00:59:36,580 --> 00:59:37,490 Absolutely. 1172 00:59:37,490 --> 00:59:41,660 That's one of the interesting things about-- 1173 00:59:41,660 --> 00:59:43,442 when cells have multiple-- 1174 00:59:43,442 --> 00:59:44,900 they have different properties, you 1175 00:59:44,900 --> 00:59:46,940 have current flowing between them, 1176 00:59:46,940 --> 00:59:50,390 but you have to understand this kind of basic stuff 1177 00:59:50,390 --> 00:59:53,300 before you even get anywhere close to understanding a more 1178 00:59:53,300 --> 00:59:55,240 complicated neuron, right? 1179 00:59:58,670 --> 01:00:00,630 OK. 1180 01:00:00,630 --> 01:00:04,555 So we're going to add a new component to our model. 1181 01:00:07,370 --> 01:00:12,080 It's a battery, and it's going to solve one really 1182 01:00:12,080 --> 01:00:15,050 fatal problem with this model. 1183 01:00:15,050 --> 01:00:17,990 What's the problem with this model? 1184 01:00:17,990 --> 01:00:18,790 Can anyone see-- 1185 01:00:18,790 --> 01:00:20,570 I'm kind of showing it right here. 1186 01:00:25,370 --> 01:00:30,280 What happens to this neuron if I turn the current off? 1187 01:00:30,280 --> 01:00:31,330 It goes back to zero. 1188 01:00:36,240 --> 01:00:38,790 And in order to get the voltage to go different from zero, 1189 01:00:38,790 --> 01:00:43,460 I have to inject current through my current source. 1190 01:00:43,460 --> 01:00:46,850 Without me, the experimenter with an electrode in it 1191 01:00:46,850 --> 01:00:50,120 injecting current, this neuron literally just sits at zero 1192 01:00:50,120 --> 01:00:53,210 and stays there, all right? 1193 01:00:53,210 --> 01:00:56,170 It's actually a good model of a dead neuron, OK? 1194 01:00:59,170 --> 01:01:04,345 So in order to change that, we need to add a battery here, OK? 1195 01:01:08,150 --> 01:01:11,270 And that battery is going to power this thing up, 1196 01:01:11,270 --> 01:01:13,710 so now it can change its own voltage. 1197 01:01:16,710 --> 01:01:19,230 And then things start getting really interesting. 1198 01:01:19,230 --> 01:01:21,870 So how can these batteries allow a neuron 1199 01:01:21,870 --> 01:01:23,950 to change its own voltage? 1200 01:01:23,950 --> 01:01:28,500 Well, the way a neuron controls its own voltage is it has ion 1201 01:01:28,500 --> 01:01:30,720 channels-- conductances-- 1202 01:01:30,720 --> 01:01:33,720 that have little knobs on them that the cell 1203 01:01:33,720 --> 01:01:39,140 can control called voltage. 1204 01:01:39,140 --> 01:01:42,320 So these conductances are voltage-dependent, 1205 01:01:42,320 --> 01:01:45,200 and now the cell connects these batteries 1206 01:01:45,200 --> 01:01:51,170 to its inside wire at different times in different ways. 1207 01:01:51,170 --> 01:01:53,600 So let's say we want to make an action potential. 1208 01:01:53,600 --> 01:01:56,380 So we have a battery that's got a positive voltage, 1209 01:01:56,380 --> 01:01:58,990 we have a battery that's got a negative voltage, 1210 01:01:58,990 --> 01:02:03,760 and we make an action potential by connecting the back 1211 01:02:03,760 --> 01:02:08,080 to the inside of the cell by turning on this conductance, 1212 01:02:08,080 --> 01:02:10,930 and then we're going to connect the battery 1213 01:02:10,930 --> 01:02:13,120 with a negative voltage to our cell, 1214 01:02:13,120 --> 01:02:15,520 and we're going to do that one after the other. 1215 01:02:15,520 --> 01:02:17,140 So watch this. 1216 01:02:19,720 --> 01:02:21,880 We're going to connect the positive battery 1217 01:02:21,880 --> 01:02:23,885 and the voltage is going to go up; 1218 01:02:23,885 --> 01:02:25,760 we're going to turn off the positive battery, 1219 01:02:25,760 --> 01:02:29,050 connect the negative battery, the voltage goes down; 1220 01:02:29,050 --> 01:02:32,070 and then we're going to turn off both batteries, 1221 01:02:32,070 --> 01:02:36,610 and the voltage just relaxes, OK? 1222 01:02:36,610 --> 01:02:37,800 Cool, right? 1223 01:02:37,800 --> 01:02:42,180 So now the neuron can control its own voltage. 1224 01:02:42,180 --> 01:02:46,914 But before we do that, we need to put batteries in our neuron, 1225 01:02:46,914 --> 01:02:48,950 OK? 1226 01:02:48,950 --> 01:02:49,450 All right. 1227 01:02:49,450 --> 01:02:52,656 So anybody know what the-- yes? 1228 01:02:52,656 --> 01:03:00,508 AUDIENCE: [INAUDIBLE] What does that [INAUDIBLE] 1229 01:03:00,508 --> 01:03:02,425 MICHALE FEE: Good, we're going to get to that. 1230 01:03:02,425 --> 01:03:05,000 That was-- exact next question. 1231 01:03:05,000 --> 01:03:08,546 What is it that makes a battery in a neuron? 1232 01:03:08,546 --> 01:03:09,480 Yeah? 1233 01:03:09,480 --> 01:03:12,444 AUDIENCE: Well I mean, like, you have like something, right? 1234 01:03:12,444 --> 01:03:13,230 Even in like-- 1235 01:03:13,230 --> 01:03:13,520 MICHALE FEE: Good. 1236 01:03:13,520 --> 01:03:14,680 AUDIENCE: --concentration gradient-- 1237 01:03:14,680 --> 01:03:15,430 MICHALE FEE: Good. 1238 01:03:15,430 --> 01:03:17,470 AUDIENCE: --so that give us a [INAUDIBLE] 1239 01:03:17,470 --> 01:03:18,220 MICHALE FEE: Here. 1240 01:03:18,220 --> 01:03:20,710 You give the rest of the lecture. 1241 01:03:20,710 --> 01:03:22,600 That's exactly right. 1242 01:03:22,600 --> 01:03:23,140 OK? 1243 01:03:23,140 --> 01:03:25,760 Well concentration gradients. 1244 01:03:25,760 --> 01:03:28,270 There's one more thing we need. 1245 01:03:28,270 --> 01:03:31,330 Concentration gradients by themselves don't do it. 1246 01:03:31,330 --> 01:03:36,200 We need ion channels that are permeable only to certain ions, 1247 01:03:36,200 --> 01:03:36,700 OK? 1248 01:03:36,700 --> 01:03:38,283 And that's what we're going to do now. 1249 01:03:40,840 --> 01:03:42,870 So you need concentration gradients 1250 01:03:42,870 --> 01:03:45,360 and ion-selective permeability, OK? 1251 01:03:45,360 --> 01:03:48,370 So we're going to go through that. 1252 01:03:48,370 --> 01:03:51,870 So let's take a beaker, fill it with water, 1253 01:03:51,870 --> 01:03:54,570 have a membrane, dividing it into two. 1254 01:03:54,570 --> 01:03:56,670 We're going to have an electrode on one side, 1255 01:03:56,670 --> 01:03:58,830 we measure the voltage difference-- sorry, 1256 01:03:58,830 --> 01:04:00,270 we have an electron on both sides, 1257 01:04:00,270 --> 01:04:02,310 we hook it up to our differential amplifier, 1258 01:04:02,310 --> 01:04:05,100 and measure the voltage difference on the two sides, 1259 01:04:05,100 --> 01:04:06,390 OK? 1260 01:04:06,390 --> 01:04:09,745 Then we're going to put-- 1261 01:04:13,050 --> 01:04:14,580 we're going to take a-- 1262 01:04:14,580 --> 01:04:18,820 we're going to buy some potassium chloride from sigma, 1263 01:04:18,820 --> 01:04:20,400 we're going to take a spoonful of it 1264 01:04:20,400 --> 01:04:23,460 and dump it into this side of the beaker. 1265 01:04:23,460 --> 01:04:27,630 Stir it up, and now you're going to have lots of potassium ions 1266 01:04:27,630 --> 01:04:29,980 and chloride ions on this side of the beaker, right? 1267 01:04:32,850 --> 01:04:35,550 Now we're going to take a needle and poke 1268 01:04:35,550 --> 01:04:37,660 a hole in that membrane. 1269 01:04:37,660 --> 01:04:41,460 That becomes a leak, a leak channel. 1270 01:04:41,460 --> 01:04:44,940 It's a non-specific-- a non-selective pore 1271 01:04:44,940 --> 01:04:49,107 that passes any iron, OK? 1272 01:04:49,107 --> 01:04:50,190 So what's going to happen? 1273 01:04:54,375 --> 01:04:56,710 AUDIENCE: [INAUDIBLE] 1274 01:04:56,710 --> 01:04:57,490 MICHALE FEE: Good. 1275 01:04:57,490 --> 01:04:59,697 The ions are going to diffuse. 1276 01:04:59,697 --> 01:05:00,530 From where to where? 1277 01:05:00,530 --> 01:05:01,670 Somebody else. 1278 01:05:01,670 --> 01:05:03,470 AUDIENCE: To the lower concentration. 1279 01:05:03,470 --> 01:05:05,387 MICHALE FEE: To the lower concentration, good. 1280 01:05:05,387 --> 01:05:08,230 So some of those ions are going to diffuse from here to here. 1281 01:05:08,230 --> 01:05:12,132 And we can plot the potassium-- let's focus on potassium now. 1282 01:05:12,132 --> 01:05:14,090 We're going to plot-- we can plot the potassium 1283 01:05:14,090 --> 01:05:16,710 concentration on this side over time 1284 01:05:16,710 --> 01:05:18,620 and on this side over time. 1285 01:05:18,620 --> 01:05:21,530 By the way, this side of the beaker 1286 01:05:21,530 --> 01:05:23,900 is going to represent the inside of the neuron, which 1287 01:05:23,900 --> 01:05:25,910 has lots of potassium, and this is 1288 01:05:25,910 --> 01:05:28,250 going to represent the outside of our neuron, which 1289 01:05:28,250 --> 01:05:31,340 has very little potassium, OK? 1290 01:05:31,340 --> 01:05:35,280 So if we plot that potassium concentration on this side, 1291 01:05:35,280 --> 01:05:37,160 it's going to increase over time. 1292 01:05:37,160 --> 01:05:38,480 It's going to take a long-- 1293 01:05:38,480 --> 01:05:40,610 and the concentration here will decrease 1294 01:05:40,610 --> 01:05:44,240 and eventually they'll meet in the middle somewhere. 1295 01:05:44,240 --> 01:05:47,245 They'll become equal. 1296 01:05:47,245 --> 01:05:49,370 And that's going to take a really long time, right? 1297 01:05:49,370 --> 01:05:51,950 Because it takes a long time for this half 1298 01:05:51,950 --> 01:05:53,810 of this spoonful of potassium chloride 1299 01:05:53,810 --> 01:05:56,196 to diffuse to the other side. 1300 01:05:56,196 --> 01:05:59,070 Yeah? 1301 01:05:59,070 --> 01:06:01,800 OK. 1302 01:06:01,800 --> 01:06:06,590 Now let's get a different kind of needle, 1303 01:06:06,590 --> 01:06:08,060 a very special needle. 1304 01:06:08,060 --> 01:06:11,210 It's really small and poke a hole 1305 01:06:11,210 --> 01:06:13,850 in the membrane that is only big enough 1306 01:06:13,850 --> 01:06:16,760 to pass potassium ions but not chloride ions. 1307 01:06:19,680 --> 01:06:20,930 So what's going to happen now? 1308 01:06:26,490 --> 01:06:26,990 Yeah? 1309 01:06:26,990 --> 01:06:30,920 Somebody-- yes? 1310 01:06:30,920 --> 01:06:33,280 AUDIENCE: Half of it flows to the other side-- 1311 01:06:33,280 --> 01:06:34,030 MICHALE FEE: Good. 1312 01:06:34,030 --> 01:06:37,640 AUDIENCE: [INAUDIBLE] 1313 01:06:37,640 --> 01:06:39,080 MICHALE FEE: Good. 1314 01:06:39,080 --> 01:06:45,140 So some potassium ions are going to diffuse through this pore 1315 01:06:45,140 --> 01:06:48,460 and go to the other side. 1316 01:06:48,460 --> 01:06:50,770 And what's going to happen is if we plot the potassium 1317 01:06:50,770 --> 01:06:53,950 concentration here, it will decrease-- sorry-- 1318 01:06:53,950 --> 01:06:56,380 over here it will increase a little bit, 1319 01:06:56,380 --> 01:06:58,420 and the potassium concentration on this side 1320 01:06:58,420 --> 01:07:01,870 will decrease a little bit, but then it will stop changing, 1321 01:07:01,870 --> 01:07:05,690 and it will never come to equilibrium. 1322 01:07:05,690 --> 01:07:08,650 It will also take a very short time for that equilibrium 1323 01:07:08,650 --> 01:07:10,510 to happen, because just a few potassium 1324 01:07:10,510 --> 01:07:14,880 ions need to go to the other side before it stops. 1325 01:07:14,880 --> 01:07:18,360 So why does the concentration stop changing here? 1326 01:07:18,360 --> 01:07:21,390 Well, it's because the potassium current 1327 01:07:21,390 --> 01:07:24,320 from this side to this side goes to zero, 1328 01:07:24,320 --> 01:07:25,695 and it goes to zero very quickly. 1329 01:07:31,070 --> 01:07:31,880 So why is that? 1330 01:07:34,860 --> 01:07:41,390 Well, one hint to the answer to that question 1331 01:07:41,390 --> 01:07:44,300 comes if we look at the voltage difference between the two 1332 01:07:44,300 --> 01:07:44,800 sides. 1333 01:07:44,800 --> 01:07:48,020 So what you see is that the voltage difference started 1334 01:07:48,020 --> 01:07:54,250 at 0, and when we poked that hole, all of a sudden 1335 01:07:54,250 --> 01:07:56,920 there was a rapidly-developing voltage difference 1336 01:07:56,920 --> 01:08:00,470 across the two sides, OK? 1337 01:08:04,210 --> 01:08:08,350 Why does that voltage go negative? 1338 01:08:08,350 --> 01:08:10,800 Anybody? 1339 01:08:10,800 --> 01:08:13,940 What happened when these positive charges started 1340 01:08:13,940 --> 01:08:18,115 diffusing from this side to this side? 1341 01:08:18,115 --> 01:08:19,615 AUDIENCE: Didn't you say it can bond 1342 01:08:19,615 --> 01:08:24,479 to the positive and negative charge [INAUDIBLE] 1343 01:08:24,479 --> 01:08:26,939 MICHALE FEE: Basically this is like a capacitor, right? 1344 01:08:26,939 --> 01:08:30,930 And some charges diffused from here to here, 1345 01:08:30,930 --> 01:08:33,779 some positive charges diffused from here to here, 1346 01:08:33,779 --> 01:08:37,649 that charges up this side, and so the voltage is positive. 1347 01:08:37,649 --> 01:08:40,200 We put positive charge-- more positive charges here, 1348 01:08:40,200 --> 01:08:43,915 the voltage here goes up, OK? 1349 01:08:47,310 --> 01:08:54,529 So if the voltage here is higher than the voltage here-- 1350 01:08:54,529 --> 01:08:58,130 we're plotting V1 minus V2, the voltage here 1351 01:08:58,130 --> 01:09:02,420 is lower than the voltage here, and so this 1352 01:09:02,420 --> 01:09:05,789 is going negative, OK? 1353 01:09:09,250 --> 01:09:13,090 And that voltage difference, that negative voltage here, 1354 01:09:13,090 --> 01:09:16,880 positive voltage here, what does that do? 1355 01:09:16,880 --> 01:09:19,850 It repels, it makes-- the positive side 1356 01:09:19,850 --> 01:09:23,100 starts repelling which ions? 1357 01:09:23,100 --> 01:09:25,609 It's repelling positive ions. 1358 01:09:25,609 --> 01:09:29,840 So it keeps more potassium ions from diffusing 1359 01:09:29,840 --> 01:09:32,663 through the hole. 1360 01:09:32,663 --> 01:09:33,580 Does that makes sense? 1361 01:09:43,830 --> 01:09:45,240 Blank stares. 1362 01:09:45,240 --> 01:09:46,810 Are we OK? 1363 01:09:46,810 --> 01:09:48,710 OK, good. 1364 01:09:48,710 --> 01:09:53,569 And it continues to drop until it reaches a constant voltage, 1365 01:09:53,569 --> 01:09:56,045 and that's called the equilibrium potential, OK? 1366 01:09:56,045 --> 01:09:59,120 The voltage changes until it comes to equilibrium, 1367 01:09:59,120 --> 01:10:01,760 and that voltage difference is called the equilibrium 1368 01:10:01,760 --> 01:10:04,160 potential. 1369 01:10:04,160 --> 01:10:10,660 And that voltage difference is a battery, OK? 1370 01:10:10,660 --> 01:10:11,500 And I'm going to-- 1371 01:10:11,500 --> 01:10:13,930 we're going to explain a little bit-- 1372 01:10:13,930 --> 01:10:16,690 I think it's in the next lecture, the one that's 1373 01:10:16,690 --> 01:10:20,860 on tape that explains how you actually can justify 1374 01:10:20,860 --> 01:10:22,810 representing that as a battery. 1375 01:10:22,810 --> 01:10:25,210 But right now, I'm going to show you 1376 01:10:25,210 --> 01:10:28,810 how to calculate what that voltage difference actually 1377 01:10:28,810 --> 01:10:30,910 is-- what's the size of the battery, how 1378 01:10:30,910 --> 01:10:34,400 big is the battery, OK? 1379 01:10:34,400 --> 01:10:37,070 So you can see that when ions-- 1380 01:10:37,070 --> 01:10:39,950 positive ions defuse to this side, this voltage goes up, 1381 01:10:39,950 --> 01:10:44,210 you have a voltage gradient that corresponds to a field pointing 1382 01:10:44,210 --> 01:10:46,460 in this direction. 1383 01:10:46,460 --> 01:10:50,160 And that electric field pushes against the ions-- remember, 1384 01:10:50,160 --> 01:10:52,560 we talked about drift in an electric field, 1385 01:10:52,560 --> 01:10:55,130 so when those ions are trying to diffuse across, 1386 01:10:55,130 --> 01:10:59,000 that electric field is literally dragging them back 1387 01:10:59,000 --> 01:11:04,160 to this side, right? 1388 01:11:04,160 --> 01:11:08,480 So we have a current flowing this way from diffusion, 1389 01:11:08,480 --> 01:11:11,990 and we have a current flowing that way from being 1390 01:11:11,990 --> 01:11:13,410 dragged in the electric field. 1391 01:11:13,410 --> 01:11:19,600 And so we can calculate that voltage difference because 1392 01:11:19,600 --> 01:11:22,900 at equilibrium, the current flowing this way from diffusion 1393 01:11:22,900 --> 01:11:28,850 has to equal the current flowing that way from drift 1394 01:11:28,850 --> 01:11:30,260 in an electric field, OK? 1395 01:11:30,260 --> 01:11:34,040 So one way to calculate this is we're 1396 01:11:34,040 --> 01:11:37,490 going to calculate the current due to drift, 1397 01:11:37,490 --> 01:11:39,600 current due to diffusion, add those up, 1398 01:11:39,600 --> 01:11:42,470 and that's equal to the total, and at equilibrium, 1399 01:11:42,470 --> 01:11:46,330 that has to equal to 0, right? 1400 01:11:46,330 --> 01:11:50,440 So what I'm showing you now is just 1401 01:11:50,440 --> 01:11:52,030 sort of the framework for how you 1402 01:11:52,030 --> 01:11:58,870 would calculate this using this drift and diffusion equation. 1403 01:11:58,870 --> 01:12:02,200 So we have Ohm's law that tells us 1404 01:12:02,200 --> 01:12:05,230 how the voltage difference produces 1405 01:12:05,230 --> 01:12:08,920 a current due to drift, and we have Fick's law that tells us 1406 01:12:08,920 --> 01:12:12,520 how much current is due to diffusion, 1407 01:12:12,520 --> 01:12:18,280 and you can set the sum of those two things to equal zero, 1408 01:12:18,280 --> 01:12:19,588 all right? 1409 01:12:19,588 --> 01:12:21,130 And so this is the way it would look. 1410 01:12:21,130 --> 01:12:25,090 I don't expect you to follow anything on this slide 1411 01:12:25,090 --> 01:12:29,140 just except to see that it can be done in this way, OK? 1412 01:12:29,140 --> 01:12:34,240 So you don't even have to write this down, OK? 1413 01:12:34,240 --> 01:12:36,700 So the drift current is proportion-- it's 1414 01:12:36,700 --> 01:12:39,910 some constant times voltage. 1415 01:12:39,910 --> 01:12:44,350 Fisk's law is some constant times that concentration 1416 01:12:44,350 --> 01:12:46,135 gradient, remember that. 1417 01:12:46,135 --> 01:12:50,748 And we just set those two things equal and solve for delta V, 1418 01:12:50,748 --> 01:12:51,790 and that's what you find. 1419 01:12:51,790 --> 01:12:54,670 What you find is the delta V is just some constant times 1420 01:12:54,670 --> 01:12:58,330 the log of the difference in concentrations 1421 01:12:58,330 --> 01:13:02,994 on the inside and outside, OK? 1422 01:13:02,994 --> 01:13:06,220 Now it turns out, there's a way of calculating this that's 1423 01:13:06,220 --> 01:13:08,980 much simpler and more elegant, and I'm 1424 01:13:08,980 --> 01:13:13,130 going to show you that calculation, OK? 1425 01:13:13,130 --> 01:13:13,639 Yes? 1426 01:13:13,639 --> 01:13:15,597 AUDIENCE: And so like the concentrations inside 1427 01:13:15,597 --> 01:13:18,382 and outside, like at the [INAUDIBLE] beginning of the-- 1428 01:13:18,382 --> 01:13:19,840 MICHALE FEE: Yes, at the beginning. 1429 01:13:19,840 --> 01:13:23,080 And the answer is that the concentrations don't change 1430 01:13:23,080 --> 01:13:25,720 very much through this process, so you can even 1431 01:13:25,720 --> 01:13:28,870 ignore that change, OK? 1432 01:13:28,870 --> 01:13:31,160 All right, everybody got this? 1433 01:13:31,160 --> 01:13:31,660 All right. 1434 01:13:31,660 --> 01:13:33,520 So here's how we're going to calculate 1435 01:13:33,520 --> 01:13:35,110 an alternative way of calculating 1436 01:13:35,110 --> 01:13:39,660 this voltage difference, just really beautiful. 1437 01:13:39,660 --> 01:13:41,420 We're going to use the Boltzmann equation. 1438 01:13:41,420 --> 01:13:45,290 The Boltzmann equation tells the probability 1439 01:13:45,290 --> 01:13:48,350 of a particle being in two states 1440 01:13:48,350 --> 01:13:51,280 as a function of the energy difference between those two 1441 01:13:51,280 --> 01:13:51,780 states. 1442 01:13:51,780 --> 01:13:54,560 And one of those states is going to correspond to a particle 1443 01:13:54,560 --> 01:13:58,700 being on the left side of the beaker or inside of our cell, 1444 01:13:58,700 --> 01:14:01,772 and that particle being outside of our cell 1445 01:14:01,772 --> 01:14:03,230 or on the other side of the beaker. 1446 01:14:03,230 --> 01:14:05,780 And those two, left side and right side, 1447 01:14:05,780 --> 01:14:08,600 have different energies, OK? 1448 01:14:08,600 --> 01:14:11,990 So our system has two states, a high-energy state 1449 01:14:11,990 --> 01:14:13,460 and a low-energy state. 1450 01:14:13,460 --> 01:14:15,518 Boltzmann equation just says the probability 1451 01:14:15,518 --> 01:14:17,810 of being in state 1 divided by the probability of being 1452 01:14:17,810 --> 01:14:22,050 in state 2 is just e to the minus energy difference divided 1453 01:14:22,050 --> 01:14:22,550 by kT. 1454 01:14:25,210 --> 01:14:27,820 k is the Boltzmann constant, T is temperature-- 1455 01:14:27,820 --> 01:14:29,710 this is the same kT that we talked about 1456 01:14:29,710 --> 01:14:30,550 in the last lecture. 1457 01:14:33,450 --> 01:14:36,500 Now, you can see if the temperature is very low, 1458 01:14:36,500 --> 01:14:37,700 all those particles are-- 1459 01:14:37,700 --> 01:14:39,860 if the temperature is 0, they're not being jostled, 1460 01:14:39,860 --> 01:14:42,350 they're just sitting there quietly. 1461 01:14:42,350 --> 01:14:44,720 They can't move, they can't get into state 1, 1462 01:14:44,720 --> 01:14:48,020 they just sit in state 2, OK? 1463 01:14:48,020 --> 01:14:49,970 So the probability of being in state 1 divided 1464 01:14:49,970 --> 01:14:54,250 by the probability of being in state 2 is 0, OK? 1465 01:14:54,250 --> 01:14:57,650 If kT is 0, this is a very big number, 1466 01:14:57,650 --> 01:14:59,530 e to the minus big number is 0. 1467 01:15:02,040 --> 01:15:03,810 Now let's say that we heat things up 1468 01:15:03,810 --> 01:15:07,290 a bit so that now kT gets big. 1469 01:15:07,290 --> 01:15:12,690 So the Katie actually gets approximately the same size 1470 01:15:12,690 --> 01:15:16,650 as the energy difference between our two states. 1471 01:15:16,650 --> 01:15:18,200 So now some of those particles can 1472 01:15:18,200 --> 01:15:21,450 get jostled over into state 1. 1473 01:15:25,210 --> 01:15:27,130 And we can write down-- 1474 01:15:27,130 --> 01:15:28,690 we can just calculate-- 1475 01:15:28,690 --> 01:15:31,180 you can see now that the probability of being in state 1 1476 01:15:31,180 --> 01:15:32,020 is-- 1477 01:15:32,020 --> 01:15:33,970 divided by the probability of being in state 2 1478 01:15:33,970 --> 01:15:35,890 is bigger than 0 now. 1479 01:15:35,890 --> 01:15:37,720 You can actually just calculate it. 1480 01:15:37,720 --> 01:15:41,770 If the energy difference is just twice kT, then 1481 01:15:41,770 --> 01:15:45,880 energy difference, 2kT divided by kT, 1482 01:15:45,880 --> 01:15:50,308 that ratio of probabilities is just e to the minus 2, OK? 1483 01:15:53,550 --> 01:15:57,240 If the energy difference is bigger, then you can see, 1484 01:15:57,240 --> 01:15:59,100 that probability ratio is smaller. 1485 01:15:59,100 --> 01:16:00,630 The probability of being a state 1 1486 01:16:00,630 --> 01:16:04,015 goes to 0 if you increase that energy. 1487 01:16:04,015 --> 01:16:05,640 If you make that energy difference very 1488 01:16:05,640 --> 01:16:08,580 small at temperature kT, you'd see 1489 01:16:08,580 --> 01:16:13,150 that if the energy difference is about the same as kT, 1490 01:16:13,150 --> 01:16:14,830 then the probability of being at state 1 1491 01:16:14,830 --> 01:16:16,660 is equal to the probability in state 2. 1492 01:16:16,660 --> 01:16:19,450 Those particles just jostle back and forth 1493 01:16:19,450 --> 01:16:21,880 between the two states. 1494 01:16:21,880 --> 01:16:24,410 OK, so we're ready to do this. 1495 01:16:24,410 --> 01:16:26,810 Ratio of probabilities is e to the minus energy 1496 01:16:26,810 --> 01:16:27,870 difference over KT. 1497 01:16:27,870 --> 01:16:30,950 What's the energy of a particle here versus here? 1498 01:16:30,950 --> 01:16:33,200 Well, that's a charged particle, the energy difference 1499 01:16:33,200 --> 01:16:36,050 is just given by the voltage difference. 1500 01:16:36,050 --> 01:16:38,900 So energy is charged times voltage where 1501 01:16:38,900 --> 01:16:41,036 Q is the charge of an ion. 1502 01:16:41,036 --> 01:16:43,710 The ratio of probabilities is e to the-- 1503 01:16:43,710 --> 01:16:46,035 Q times voltage difference over kT. 1504 01:16:50,330 --> 01:16:52,400 Take the log of both sides, solve 1505 01:16:52,400 --> 01:16:55,310 for the voltage difference, V in minus V out 1506 01:16:55,310 --> 01:17:00,440 equals minus kT over Q times the log of the probability ratio, 1507 01:17:00,440 --> 01:17:05,600 but the probability is just this concentration. 1508 01:17:05,600 --> 01:17:09,350 And so we can write this as delta V-- 1509 01:17:09,350 --> 01:17:12,910 the voltage difference is equal to kT 1510 01:17:12,910 --> 01:17:18,340 over Q, which is 25 millivolts, times the log of the ratio 1511 01:17:18,340 --> 01:17:21,460 of potassium concentration outside to potassium 1512 01:17:21,460 --> 01:17:24,000 concentration inside. 1513 01:17:24,000 --> 01:17:26,160 And that's exactly the same equation 1514 01:17:26,160 --> 01:17:30,660 that we get if you do that much more complicated derivation 1515 01:17:30,660 --> 01:17:35,070 based on Fick's law and Ohm's law, balancing Fick's 1516 01:17:35,070 --> 01:17:38,790 law and Ohm's law, OK? 1517 01:17:38,790 --> 01:17:40,920 And this is equilibrium potential here, 1518 01:17:40,920 --> 01:17:44,120 not electric field. 1519 01:17:44,120 --> 01:17:46,610 OK, so let's take a look at potassium concentrations 1520 01:17:46,610 --> 01:17:48,700 in a real cell. 1521 01:17:48,700 --> 01:17:50,470 This is actually from squid giant axon. 1522 01:17:50,470 --> 01:17:52,450 400 millimolar inside. 1523 01:17:52,450 --> 01:17:56,200 So 20 millimolar outside, so there's a lot of potassium 1524 01:17:56,200 --> 01:17:59,590 inside of a cell, not very much outside. 1525 01:17:59,590 --> 01:18:01,360 Plug those into our equation. 1526 01:18:01,360 --> 01:18:04,090 kT over Q is 25,000 millimolar-- temperature 1527 01:18:04,090 --> 01:18:06,980 at room temperature. 1528 01:18:06,980 --> 01:18:10,310 The log of that concentration ratio is minus 3, 1529 01:18:10,310 --> 01:18:12,430 so ek is minus 75 millivolts. 1530 01:18:12,430 --> 01:18:14,410 That means if we start with a lot of potassium 1531 01:18:14,410 --> 01:18:20,110 inside of our cell, open up a potassium-selective channel, 1532 01:18:20,110 --> 01:18:21,280 what happens? 1533 01:18:21,280 --> 01:18:26,010 Potassium diffuses out through that channel 1534 01:18:26,010 --> 01:18:30,460 and the voltage goes to minus 75 millivolts. 1535 01:18:30,460 --> 01:18:31,810 How do you know it's negative? 1536 01:18:31,810 --> 01:18:34,810 Like, I always can't remember whether this is concentration 1537 01:18:34,810 --> 01:18:37,820 inside or outside, outside over inside, I don't know. 1538 01:18:37,820 --> 01:18:39,965 But the point is, you don't actually have to know, 1539 01:18:39,965 --> 01:18:41,590 because you can just look at it and see 1540 01:18:41,590 --> 01:18:44,080 the answer what sign it is. 1541 01:18:44,080 --> 01:18:45,930 If you have positive ions-- 1542 01:18:45,930 --> 01:18:50,800 a high concentration of positive ions inside, they diffuse out, 1543 01:18:50,800 --> 01:18:54,970 the voltage inside of the cell when positive ions leave 1544 01:18:54,970 --> 01:18:57,630 is going to do what? 1545 01:18:57,630 --> 01:19:02,040 It's going to go down, so that's why it's minus, OK? 1546 01:19:02,040 --> 01:19:06,500 So-- the battery and in those video 1547 01:19:06,500 --> 01:19:09,320 modules that I recorded for you, it's 1548 01:19:09,320 --> 01:19:12,260 going to explain how you actually incorporate that 1549 01:19:12,260 --> 01:19:16,040 into a battery in our circuit model. 1550 01:19:16,040 --> 01:19:17,880 And so we've done all of these things, 1551 01:19:17,880 --> 01:19:20,270 we've looked at how membrane capacitance and resistance 1552 01:19:20,270 --> 01:19:23,120 allows neurons to integrate over time, 1553 01:19:23,120 --> 01:19:25,740 we've learned how to write down the differential equations, 1554 01:19:25,740 --> 01:19:28,520 we're now able to just look at a current input 1555 01:19:28,520 --> 01:19:30,530 and figure out the voltage change, 1556 01:19:30,530 --> 01:19:33,320 and we now understand where the batteries in a neuron 1557 01:19:33,320 --> 01:19:35,110 come from.