1 00:00:14,940 --> 00:00:17,524 MICHALE FEE: Good morning, everybody. 2 00:00:17,524 --> 00:00:20,970 So we're going to continue today developing 3 00:00:20,970 --> 00:00:25,080 our model of a neuron. 4 00:00:25,080 --> 00:00:27,420 Again, this is called the "Equivalent Circuit Model," 5 00:00:27,420 --> 00:00:30,570 and it was developed by Alan Hodgkin and Andrew Huxley 6 00:00:30,570 --> 00:00:32,100 in the '40s and '50s. 7 00:00:34,940 --> 00:00:38,390 Let me just give a brief recap of what we've covered 8 00:00:38,390 --> 00:00:40,710 in the last couple of lectures. 9 00:00:40,710 --> 00:00:44,030 So we've been analyzing a neuron. 10 00:00:44,030 --> 00:00:47,840 We've been imagining an experiment in which we 11 00:00:47,840 --> 00:00:52,220 have a neuron in a dish filled with extracellular solution, 12 00:00:52,220 --> 00:00:54,560 which is a salt solution. 13 00:00:54,560 --> 00:00:57,380 We have an electrode in the cell that's 14 00:00:57,380 --> 00:01:00,470 measuring the voltage difference between the inside 15 00:01:00,470 --> 00:01:01,890 and the outside of the cell. 16 00:01:01,890 --> 00:01:04,879 So we have the electrode connected 17 00:01:04,879 --> 00:01:07,820 to an amplifier, a differential amplifier, and a wire 18 00:01:07,820 --> 00:01:10,400 in the bath connected to the other side of the differential 19 00:01:10,400 --> 00:01:11,330 amplifier. 20 00:01:11,330 --> 00:01:14,300 And we have a current source, which 21 00:01:14,300 --> 00:01:16,100 injects current into the cell. 22 00:01:16,100 --> 00:01:18,410 And we imagine that, as the experimenter, 23 00:01:18,410 --> 00:01:21,260 we have our hand on a knob that we 24 00:01:21,260 --> 00:01:23,510 can adjust the amount of current that's 25 00:01:23,510 --> 00:01:25,400 being injected into the cell. 26 00:01:25,400 --> 00:01:28,040 We've described the neuron basically 27 00:01:28,040 --> 00:01:31,010 as a capacitor, because it has a-- 28 00:01:31,010 --> 00:01:33,600 it's two conductors separated by an insulator. 29 00:01:33,600 --> 00:01:36,660 So we have a conductor, one conductor inside the cell, 30 00:01:36,660 --> 00:01:40,070 which is that conductive intracellular salt solution, 31 00:01:40,070 --> 00:01:42,500 and another conductor outside the cell, which 32 00:01:42,500 --> 00:01:45,480 is the conductive extracellular solution. 33 00:01:45,480 --> 00:01:47,240 And those two conductors are separated 34 00:01:47,240 --> 00:01:51,910 by an insulator, which is a phospholipid bilayer. 35 00:01:51,910 --> 00:01:53,680 We wrote down the equivalent circuit 36 00:01:53,680 --> 00:01:58,550 for this, which for this situation right here 37 00:01:58,550 --> 00:02:02,320 it has a voltage measuring device that represents 38 00:02:02,320 --> 00:02:04,990 the membrane potential as the difference 39 00:02:04,990 --> 00:02:08,389 between the intracellular voltage and the extracellular 40 00:02:08,389 --> 00:02:09,370 voltage. 41 00:02:09,370 --> 00:02:11,920 We have a current source here, and we've 42 00:02:11,920 --> 00:02:16,190 represented our neuron so far as a capacitor. 43 00:02:16,190 --> 00:02:21,790 We then introduce the idea of ion channels, or conductances, 44 00:02:21,790 --> 00:02:24,520 or pores in the membrane that allow 45 00:02:24,520 --> 00:02:27,010 ions to pass through the membrane, 46 00:02:27,010 --> 00:02:28,660 through these little pores. 47 00:02:28,660 --> 00:02:30,780 And we described the idea that there are-- 48 00:02:30,780 --> 00:02:33,280 we talked about the idea that there are many different kinds 49 00:02:33,280 --> 00:02:39,340 of ions, ion channels, that have different interesting 50 00:02:39,340 --> 00:02:40,520 properties. 51 00:02:40,520 --> 00:02:44,680 So we then extended our analysis of our neuron 52 00:02:44,680 --> 00:02:49,540 in the dish to include these ion channels. 53 00:02:49,540 --> 00:02:51,790 We began by putting-- 54 00:02:51,790 --> 00:02:55,480 by representing ion channels as a resistance 55 00:02:55,480 --> 00:02:57,700 that connects the intracellular space. 56 00:02:57,700 --> 00:03:00,340 In the extracellular space, this resistor 57 00:03:00,340 --> 00:03:03,050 is in parallel with our capacitor. 58 00:03:03,050 --> 00:03:04,750 And we represented the current going 59 00:03:04,750 --> 00:03:08,020 through that resistance, which we 60 00:03:08,020 --> 00:03:10,000 called the "leak resistance," R sub 61 00:03:10,000 --> 00:03:16,620 L. We represented the current as "leak current," I sub L. 62 00:03:16,620 --> 00:03:21,280 We noted that we were going to model this leak 63 00:03:21,280 --> 00:03:23,980 resistance using Ohm's law. 64 00:03:23,980 --> 00:03:26,950 So we wrote down that current, that 65 00:03:26,950 --> 00:03:29,380 leak current, as the membrane potential, V, 66 00:03:29,380 --> 00:03:32,440 divided by the leak resistance. 67 00:03:32,440 --> 00:03:34,180 So this is just Ohm's law. 68 00:03:34,180 --> 00:03:37,900 And we also rewrote the quantity 1 69 00:03:37,900 --> 00:03:43,520 over R, 1 over the resistance, as the leak conductance. 70 00:03:43,520 --> 00:03:46,420 So the leak current is just the leak conductance 71 00:03:46,420 --> 00:03:49,710 times the membrane potential. 72 00:03:49,710 --> 00:03:53,610 And we also introduced the idea of an I-V curve, which 73 00:03:53,610 --> 00:03:57,150 plots the amount of current flowing 74 00:03:57,150 --> 00:04:01,380 through our leak conductance here as a function 75 00:04:01,380 --> 00:04:05,130 of the membrane potential, this V. 76 00:04:05,130 --> 00:04:08,940 And for the case of a resistor, for the case where we're 77 00:04:08,940 --> 00:04:13,440 modeling our leak conductance with Ohm's law, 78 00:04:13,440 --> 00:04:15,350 the current, as a function of voltage, 79 00:04:15,350 --> 00:04:19,620 is a straight line going through the origin, whose slope is just 80 00:04:19,620 --> 00:04:22,664 the leak conductance. 81 00:04:22,664 --> 00:04:23,980 Any questions about that? 82 00:04:27,160 --> 00:04:30,040 Then we derived a simple equation 83 00:04:30,040 --> 00:04:35,470 for how the membrane potential evolves over time as a function 84 00:04:35,470 --> 00:04:37,360 of the injected current. 85 00:04:37,360 --> 00:04:40,300 And we did that using Kirchhoff's current law, which 86 00:04:40,300 --> 00:04:44,590 says that the sum of all the currents into a node 87 00:04:44,590 --> 00:04:47,560 has to equal the sum of currents out of a node, 88 00:04:47,560 --> 00:04:49,780 where a node here means a wire. 89 00:04:49,780 --> 00:04:54,010 And so the sum of these occurrences equal to 0. 90 00:04:54,010 --> 00:04:56,980 There's a minus sign here, because this current, 91 00:04:56,980 --> 00:05:01,060 the electrode injected current, is defined as positive going 92 00:05:01,060 --> 00:05:02,830 into the cell, and these currents 93 00:05:02,830 --> 00:05:06,700 are defined as positive when they go out of the cell. 94 00:05:06,700 --> 00:05:11,390 We substituted for the leak current, 95 00:05:11,390 --> 00:05:13,000 the expression from Ohm's law. 96 00:05:13,000 --> 00:05:17,230 So into this part of the equation here, this term, 97 00:05:17,230 --> 00:05:20,290 we substitute VM over R sub L. And I 98 00:05:20,290 --> 00:05:23,320 apologize for the slight inconsistency in notation here. 99 00:05:23,320 --> 00:05:25,900 Sometimes I'm using VM, and sometimes I'm 100 00:05:25,900 --> 00:05:32,560 using V. I'll try to fix that, but those-- 101 00:05:32,560 --> 00:05:37,510 VM is the same as V in what you've seen so far. 102 00:05:37,510 --> 00:05:39,420 And we also have an expression here 103 00:05:39,420 --> 00:05:42,320 for the capacitive current. 104 00:05:42,320 --> 00:05:44,760 The capacitive current through this capacitor 105 00:05:44,760 --> 00:05:46,770 is just C, dV, dt. 106 00:05:46,770 --> 00:05:51,000 So we put those two expressions, make those two substitutions, 107 00:05:51,000 --> 00:05:55,020 and we have this expression that relates the voltage in the cell 108 00:05:55,020 --> 00:05:57,420 to the injected current. 109 00:05:57,420 --> 00:05:59,940 We rewrote this a little bit by multiplying through 110 00:05:59,940 --> 00:06:05,630 by the leak resistance, and we rewrote this again making 111 00:06:05,630 --> 00:06:08,180 a couple substitutions. 112 00:06:08,180 --> 00:06:12,740 We use tau in place of RC. 113 00:06:12,740 --> 00:06:14,660 So that becomes a time constant. 114 00:06:14,660 --> 00:06:18,440 And the infinity is this expression on the right-- 115 00:06:18,440 --> 00:06:21,210 R leak times the injected current. 116 00:06:21,210 --> 00:06:23,780 So the equation that we now have, 117 00:06:23,780 --> 00:06:25,430 the differential equation that we now 118 00:06:25,430 --> 00:06:28,520 have, for the dependence of the membrane potential on 119 00:06:28,520 --> 00:06:30,410 injected current looks like this-- 120 00:06:30,410 --> 00:06:35,250 V plus tau, dV, dt, is equal to V infinity. 121 00:06:35,250 --> 00:06:37,370 When we inject current into the cell, 122 00:06:37,370 --> 00:06:41,360 we're changing the infinity. 123 00:06:41,360 --> 00:06:45,920 And then the voltage evolves, as described 124 00:06:45,920 --> 00:06:47,195 by this differential equation. 125 00:06:52,280 --> 00:06:57,380 So we're going to plot a bunch of things, the infinity 126 00:06:57,380 --> 00:07:02,910 and the voltage in the cell, as we inject a pulse of current. 127 00:07:02,910 --> 00:07:06,740 So we start out with injected current equal to 0. 128 00:07:06,740 --> 00:07:11,060 We step up to I naught, hold a constant injected current I 129 00:07:11,060 --> 00:07:15,170 naught for some period of time, and then reset the current back 130 00:07:15,170 --> 00:07:16,190 to 0. 131 00:07:16,190 --> 00:07:21,310 So what does V infinity do as a function of time, anybody? 132 00:07:24,220 --> 00:07:25,381 What does that mean? 133 00:07:25,381 --> 00:07:27,640 AUDIENCE: It does the same things. 134 00:07:27,640 --> 00:07:28,390 MICHALE FEE: Good. 135 00:07:28,390 --> 00:07:32,320 It does the same thing, but multiplied by R sub L-- 136 00:07:32,320 --> 00:07:33,400 very good. 137 00:07:33,400 --> 00:07:37,060 And what does the membrane potential do? 138 00:07:37,060 --> 00:07:39,670 So let's start the membrane potential at 0. 139 00:07:39,670 --> 00:07:42,320 So starting here, what does the membrane potential do 140 00:07:42,320 --> 00:07:44,250 [INAUDIBLE]? 141 00:07:44,250 --> 00:07:47,000 Yes? 142 00:07:47,000 --> 00:07:47,908 What's your name? 143 00:07:47,908 --> 00:07:48,366 AUDIENCE: I'm Kate. 144 00:07:48,366 --> 00:07:49,116 MICHALE FEE: Kate. 145 00:07:52,490 --> 00:07:54,200 Yes, stays at 0. 146 00:07:54,200 --> 00:07:57,350 AUDIENCE: [INAUDIBLE] 147 00:07:57,350 --> 00:07:59,700 MICHALE FEE: Good. 148 00:07:59,700 --> 00:08:00,690 How do we say-- 149 00:08:00,690 --> 00:08:01,680 how do we say it? 150 00:08:01,680 --> 00:08:04,080 Exponentially toward V infinity, very good. 151 00:08:04,080 --> 00:08:07,560 So it relaxes exponentially toward infinity, which is here. 152 00:08:07,560 --> 00:08:08,520 And then what happens? 153 00:08:08,520 --> 00:08:11,842 AUDIENCE: And then it [INAUDIBLE].. 154 00:08:11,842 --> 00:08:12,800 MICHALE FEE: Very good. 155 00:08:12,800 --> 00:08:17,150 And it does-- it relaxes with a time constant of RC. 156 00:08:17,150 --> 00:08:18,860 And what that means is that the time 157 00:08:18,860 --> 00:08:23,720 it takes for it to relax to 1 over e of its original distance 158 00:08:23,720 --> 00:08:27,230 from V infinity is given by tau. 159 00:08:29,770 --> 00:08:34,750 Very good-- and you can solve this differential equation 160 00:08:34,750 --> 00:08:38,590 that was on the previous slide for periods 161 00:08:38,590 --> 00:08:41,020 where the injected current is constant, 162 00:08:41,020 --> 00:08:42,980 where V infinity is constant. 163 00:08:42,980 --> 00:08:46,300 And what you can see is that the voltage difference 164 00:08:46,300 --> 00:08:53,310 from the current V infinity relaxes exponentially to 0. 165 00:08:53,310 --> 00:08:57,580 It starts out at the initial voltage minus V infinity, 166 00:08:57,580 --> 00:09:00,790 and you multiply that difference times an exponential that 167 00:09:00,790 --> 00:09:02,140 decays to 0. 168 00:09:02,140 --> 00:09:04,600 And so the voltage difference between the voltage and V 169 00:09:04,600 --> 00:09:05,590 infinity decreases. 170 00:09:09,570 --> 00:09:12,580 So now we then introduce the idea 171 00:09:12,580 --> 00:09:14,890 that neurons have batteries. 172 00:09:14,890 --> 00:09:16,330 Where do the batteries of a neuron 173 00:09:16,330 --> 00:09:18,310 come from, anybody remember? 174 00:09:18,310 --> 00:09:19,615 Stacey, I remember you. 175 00:09:19,615 --> 00:09:21,490 Where do the batteries of a neuron come from? 176 00:09:26,924 --> 00:09:27,630 Good. 177 00:09:27,630 --> 00:09:30,690 That's what the battery produces. 178 00:09:30,690 --> 00:09:32,770 The battery produces a voltage difference. 179 00:09:32,770 --> 00:09:36,710 But biophysically, what causes that voltage difference? 180 00:09:36,710 --> 00:09:39,480 AUDIENCE: [INAUDIBLE] 181 00:09:39,480 --> 00:09:41,630 MICHALE FEE: Good, ion selective pores. 182 00:09:41,630 --> 00:09:43,400 Good, that's one component, and there's 183 00:09:43,400 --> 00:09:45,470 another important component. 184 00:09:45,470 --> 00:09:47,080 Somebody else want to answer? 185 00:09:47,080 --> 00:09:48,400 AUDIENCE: Ion concentration. 186 00:09:48,400 --> 00:09:49,608 MICHALE FEE: Good, excellent. 187 00:09:49,608 --> 00:09:52,570 So we have two components-- ion concentration gradients 188 00:09:52,570 --> 00:09:55,090 and ions selective pores. 189 00:09:55,090 --> 00:10:00,340 So basically, if you have a high concentration of potassium 190 00:10:00,340 --> 00:10:05,110 inside the cell, when you open up a potassium selective ion 191 00:10:05,110 --> 00:10:09,550 channel, the potassium ions diffuse out of the cell, 192 00:10:09,550 --> 00:10:13,000 leaving an excess of negative charges 193 00:10:13,000 --> 00:10:16,570 inside the cell that causes the voltage inside the cell 194 00:10:16,570 --> 00:10:21,500 to go down until there's a sufficient voltage 195 00:10:21,500 --> 00:10:25,370 gradient that the drift of potassium ions back 196 00:10:25,370 --> 00:10:28,040 into the cell driven by the voltage gradient 197 00:10:28,040 --> 00:10:31,280 is equal to the diffusion rate of ions 198 00:10:31,280 --> 00:10:38,750 out of the cell produced by the concentration gradient. 199 00:10:38,750 --> 00:10:41,730 At equilibrium, at the equilibrium potential, 200 00:10:41,730 --> 00:10:44,490 there's a particular voltage at which 201 00:10:44,490 --> 00:10:47,150 there is no net current into-- 202 00:10:47,150 --> 00:10:50,390 potassium current into or out of the cell. 203 00:10:50,390 --> 00:10:53,510 And that's what we mean by equilibrium potential, 204 00:10:53,510 --> 00:10:56,450 otherwise known as the "Nernst potential." 205 00:10:56,450 --> 00:10:59,090 Last time we derived the Nernst potential 206 00:10:59,090 --> 00:11:00,305 using the Boltzmann equation. 207 00:11:02,880 --> 00:11:05,270 The Boltzmann equation gives us the ratio 208 00:11:05,270 --> 00:11:08,330 of the probability of finding an ion inside or outside 209 00:11:08,330 --> 00:11:13,010 of the cell, and that's equal to e to the minus energy 210 00:11:13,010 --> 00:11:15,920 difference of an ion inside and outside of the cell divided 211 00:11:15,920 --> 00:11:17,030 by KT. 212 00:11:17,030 --> 00:11:20,150 The energy difference is given by the charge 213 00:11:20,150 --> 00:11:22,880 times the voltage difference, the charge of the ion 214 00:11:22,880 --> 00:11:26,060 that we're considering times the voltage difference. 215 00:11:26,060 --> 00:11:29,510 So we took the log of both sides, solved for V in minus V 216 00:11:29,510 --> 00:11:32,360 out, and we found that the voltage difference 217 00:11:32,360 --> 00:11:33,230 at equilibrium. 218 00:11:33,230 --> 00:11:35,360 This is at thermal equilibrium. 219 00:11:35,360 --> 00:11:37,370 That's what the Boltzmann equation tells us-- 220 00:11:37,370 --> 00:11:38,900 at thermal equilibrium. 221 00:11:38,900 --> 00:11:41,600 What's the ratio of probabilities? 222 00:11:41,600 --> 00:11:44,720 At thermal equilibrium, the voltage difference 223 00:11:44,720 --> 00:11:47,990 is 25,000 millivolts times the log, the natural log, 224 00:11:47,990 --> 00:11:53,510 of the ratio of concentrations, and that's 225 00:11:53,510 --> 00:11:59,420 E sub K. If you plug-in the potassium concentrations here, 226 00:11:59,420 --> 00:12:02,630 that gives you the equilibrium potential for potassium 227 00:12:02,630 --> 00:12:05,032 or the Nernst potential for potassium. 228 00:12:08,550 --> 00:12:12,710 So real neurons here on Earth are 229 00:12:12,710 --> 00:12:15,560 kind of like caesium neurons on that-- 230 00:12:15,560 --> 00:12:19,070 cesium ions on that alien planet. 231 00:12:19,070 --> 00:12:21,980 The potassium concentration inside of a cell 232 00:12:21,980 --> 00:12:24,050 is around 400 millimolar, and I think 233 00:12:24,050 --> 00:12:30,060 this is for squid giant axon, which is a little bit alien. 234 00:12:30,060 --> 00:12:30,560 Yes? 235 00:12:30,560 --> 00:12:30,860 Go ahead. 236 00:12:30,860 --> 00:12:32,760 AUDIENCE: From the previous [INAUDIBLE].. 237 00:12:37,510 --> 00:12:39,110 MICHALE FEE: Yep. 238 00:12:39,110 --> 00:12:42,140 So if you think about the probability 239 00:12:42,140 --> 00:12:47,660 of finding a potassium ion inside of a cell 240 00:12:47,660 --> 00:12:50,340 and finding a potassium ion outside of the cell, 241 00:12:50,340 --> 00:12:52,700 the ratio of probabilities is going 242 00:12:52,700 --> 00:12:55,635 to be proportional to the ratio of concentrations. 243 00:12:55,635 --> 00:12:56,510 Does that make sense? 244 00:13:01,430 --> 00:13:05,400 So given these concentrations, a high concentration 245 00:13:05,400 --> 00:13:08,640 of positive potassium ions inside the cell, 246 00:13:08,640 --> 00:13:11,190 a low concentration outside the cell, 247 00:13:11,190 --> 00:13:14,250 we can take the ratio of those concentrations, 248 00:13:14,250 --> 00:13:15,750 take the natural log of that. 249 00:13:15,750 --> 00:13:20,190 That's about 3 log units. 250 00:13:20,190 --> 00:13:25,380 The sign is negative, because this ratio is smaller than 1. 251 00:13:25,380 --> 00:13:28,710 So the equilibrium potential for potassium 252 00:13:28,710 --> 00:13:35,730 is about 25 millivolts, KT over Q for a monovalent iron 253 00:13:35,730 --> 00:13:38,130 is 25 millivolts. 254 00:13:38,130 --> 00:13:43,290 So the Nernst potential is 25 millivolts times minus 3. 255 00:13:43,290 --> 00:13:47,105 So the equilibrium potential is about minus 75. 256 00:13:52,120 --> 00:13:57,560 So let's now look at a different ion, sodium. 257 00:13:57,560 --> 00:14:00,850 So sodium has a very low concentration inside 258 00:14:00,850 --> 00:14:03,610 of a cell compared to outside. 259 00:14:03,610 --> 00:14:08,520 That ratio is about a factor of 10. 260 00:14:08,520 --> 00:14:14,410 The natural log of 10 is about 2 log units, right? 261 00:14:14,410 --> 00:14:17,050 So what's the equilibrium potential 262 00:14:17,050 --> 00:14:21,340 for sodium, anybody has a guess? 263 00:14:21,340 --> 00:14:22,720 [LAUGHS] 264 00:14:22,720 --> 00:14:25,490 AUDIENCE: [INAUDIBLE] 265 00:14:25,490 --> 00:14:26,440 MICHALE FEE: Yeah. 266 00:14:26,440 --> 00:14:27,400 Plus or minus? 267 00:14:27,400 --> 00:14:28,460 AUDIENCE: Oh, plus. 268 00:14:28,460 --> 00:14:29,210 MICHALE FEE: Good. 269 00:14:29,210 --> 00:14:33,110 So it's about 2 log units times 25 millivolts, right? 270 00:14:33,110 --> 00:14:37,220 So it's plus 50 or so. 271 00:14:37,220 --> 00:14:38,288 Good. 272 00:14:38,288 --> 00:14:39,080 How about chloride? 273 00:14:39,080 --> 00:14:40,038 So this is interesting. 274 00:14:40,038 --> 00:14:42,850 Chloride is a negative ion. 275 00:14:42,850 --> 00:14:45,490 It has a high concentration outside the cell 276 00:14:45,490 --> 00:14:47,750 and a low concentration inside the cell. 277 00:14:47,750 --> 00:14:51,430 So when we open up a chloride channel, what 278 00:14:51,430 --> 00:14:53,470 happens to the chloride ions? 279 00:14:53,470 --> 00:14:56,560 Which way do they go, into the cell or out of the cell? 280 00:14:59,370 --> 00:15:01,000 Good. 281 00:15:01,000 --> 00:15:01,840 They're negative. 282 00:15:01,840 --> 00:15:07,710 So what is that going to do to the voltage inside the cell? 283 00:15:07,710 --> 00:15:09,820 We're negative, good. 284 00:15:09,820 --> 00:15:14,920 And the ratio of concentrations here is about 10. 285 00:15:14,920 --> 00:15:17,020 So log of 10. 286 00:15:17,020 --> 00:15:18,130 It's 2-ish. 287 00:15:18,130 --> 00:15:22,260 So what's that Nernst potential going to be? 288 00:15:22,260 --> 00:15:24,780 About minus 50, exactly. 289 00:15:24,780 --> 00:15:28,500 If you plug-in the actual numbers, it's minus 60. 290 00:15:28,500 --> 00:15:29,530 Great. 291 00:15:29,530 --> 00:15:30,030 Good. 292 00:15:30,030 --> 00:15:32,410 Here's an interesting one-- calcium. 293 00:15:32,410 --> 00:15:37,650 Calcium is kept at an extremely low concentration 294 00:15:37,650 --> 00:15:41,190 inside of cells, because it's used as a signaling molecule. 295 00:15:41,190 --> 00:15:43,590 When calcium comes into a cell, it actually 296 00:15:43,590 --> 00:15:46,650 does things important. 297 00:15:46,650 --> 00:15:48,660 So the cell buffers calcium. 298 00:15:48,660 --> 00:15:51,930 It sequesters calcium into the endoplasmic reticulum 299 00:15:51,930 --> 00:15:55,410 and keeps the concentration in the cytoplasm very low. 300 00:15:55,410 --> 00:15:59,060 The concentration outside the cell is about 2. 301 00:15:59,060 --> 00:16:03,510 The ratio is a pretty big number. 302 00:16:03,510 --> 00:16:06,810 So if you-- 303 00:16:06,810 --> 00:16:07,310 OK. 304 00:16:07,310 --> 00:16:12,440 Why is the coefficient here, KT over Q, 305 00:16:12,440 --> 00:16:15,600 why is that 12 millivolts here instead of 25? 306 00:16:21,050 --> 00:16:21,550 Excellent. 307 00:16:21,550 --> 00:16:25,720 So it's KT over 2 times the electron charge. 308 00:16:25,720 --> 00:16:27,250 So that's 12 millivolts, very good. 309 00:16:27,250 --> 00:16:31,160 And so the equilibrium potential for calcium is very positive. 310 00:16:33,890 --> 00:16:34,670 Great. 311 00:16:34,670 --> 00:16:36,630 Any questions about that? 312 00:16:36,630 --> 00:16:42,510 So these are the most important ions 313 00:16:42,510 --> 00:16:45,840 that we have to think about in terms of ionic conductances 314 00:16:45,840 --> 00:16:49,670 across the membrane. 315 00:16:49,670 --> 00:16:53,060 Any questions? 316 00:16:53,060 --> 00:16:55,460 Good. 317 00:16:55,460 --> 00:16:59,030 So let's go back to our neuron in the dish, 318 00:16:59,030 --> 00:17:02,750 and we're going to consider a neuron that has a potassium 319 00:17:02,750 --> 00:17:08,010 conductance and a high potassium concentration inside the cell. 320 00:17:08,010 --> 00:17:10,819 Remember, we can write down the magnitude 321 00:17:10,819 --> 00:17:13,849 of that conductance as G sub K. And now 322 00:17:13,849 --> 00:17:15,359 we're going to do an experiment. 323 00:17:15,359 --> 00:17:18,140 We're going to measure the voltage 324 00:17:18,140 --> 00:17:20,869 in our cell, the steady state voltage in our cell, 325 00:17:20,869 --> 00:17:23,810 as a function of the amount of current that 326 00:17:23,810 --> 00:17:27,650 is either being injected or passing through the membrane-- 327 00:17:27,650 --> 00:17:28,580 its steady state. 328 00:17:28,580 --> 00:17:30,510 Those two things are the same. 329 00:17:30,510 --> 00:17:35,400 So we're going to plot the potassium current 330 00:17:35,400 --> 00:17:37,260 through the membrane as a function 331 00:17:37,260 --> 00:17:40,710 of the voltage of the cell. 332 00:17:40,710 --> 00:17:45,190 So let's say that we inject 0 current. 333 00:17:45,190 --> 00:17:49,384 What's the steady state voltage in the cell going to be, 334 00:17:49,384 --> 00:17:50,338 anybody? 335 00:17:58,940 --> 00:18:00,080 We just went through this. 336 00:18:05,410 --> 00:18:07,300 Uh-oh, I have to-- 337 00:18:07,300 --> 00:18:07,910 oh, OK. 338 00:18:07,910 --> 00:18:09,150 I have a volunteer back here. 339 00:18:12,460 --> 00:18:15,430 Why 25 millivolts? 340 00:18:15,430 --> 00:18:17,530 And we will assume the channel is open. 341 00:18:20,370 --> 00:18:21,360 So what is the-- 342 00:18:21,360 --> 00:18:23,861 so your answer is that the-- 343 00:18:23,861 --> 00:18:28,310 if you inject 0 current, the voltage 344 00:18:28,310 --> 00:18:30,610 is going to be the Nernst potential for potassium. 345 00:18:30,610 --> 00:18:36,100 And that is correct, but what is the actual number? 346 00:18:36,100 --> 00:18:36,600 Good. 347 00:18:36,600 --> 00:18:40,020 It's going to be around negative 75 millivolts. 348 00:18:40,020 --> 00:18:44,580 So if we inject 0 current, we know a voltage of our neuron 349 00:18:44,580 --> 00:18:49,960 is going to be around EK or minus 75 millivolts. 350 00:18:49,960 --> 00:18:54,860 Now, let's start injecting current into the cell 351 00:18:54,860 --> 00:18:59,150 until the voltage gets to 0. 352 00:18:59,150 --> 00:19:01,760 Now, are we going to be injecting 353 00:19:01,760 --> 00:19:04,190 positive current into this cell or negative current? 354 00:19:04,190 --> 00:19:07,490 What do we have to do, inject positive or negative current 355 00:19:07,490 --> 00:19:09,650 to make the voltage get up to 0? 356 00:19:12,190 --> 00:19:15,070 Positive, right, because the voltage inside is negative. 357 00:19:15,070 --> 00:19:17,230 So we need to inject positive charges. 358 00:19:17,230 --> 00:19:19,930 And how much current do we need to inject? 359 00:19:19,930 --> 00:19:22,990 If the conductance is g, how much current 360 00:19:22,990 --> 00:19:25,562 do we need to inject? 361 00:19:25,562 --> 00:19:26,062 What? 362 00:19:30,050 --> 00:19:34,750 V is the voltage inside of our cell, so you're getting there. 363 00:19:34,750 --> 00:19:35,760 You're really close. 364 00:19:35,760 --> 00:19:36,510 What's the answer? 365 00:19:40,250 --> 00:19:43,610 A times G. So we're going to inject the current 366 00:19:43,610 --> 00:19:46,880 into our cell until the voltage gets to 0. 367 00:19:46,880 --> 00:19:48,890 And if we inject different amounts of current 368 00:19:48,890 --> 00:19:51,920 into the cell and measure what the voltage is, 369 00:19:51,920 --> 00:19:53,810 we get kind of a straight line. 370 00:19:58,340 --> 00:20:00,940 So for a potassium conductance, if you 371 00:20:00,940 --> 00:20:05,870 hold the voltage positive above the equilibrium potential, 372 00:20:05,870 --> 00:20:09,940 potassium ions are going to flow out through the membrane. 373 00:20:09,940 --> 00:20:11,350 Is that clear? 374 00:20:11,350 --> 00:20:16,820 Let's hold the voltage at 0. 375 00:20:16,820 --> 00:20:18,580 Which way are our ions flowing? 376 00:20:18,580 --> 00:20:22,260 So the ions are going to be flowing out. 377 00:20:22,260 --> 00:20:25,990 To hold the voltage at EK, what happens 378 00:20:25,990 --> 00:20:28,765 to the potassium current? 379 00:20:28,765 --> 00:20:32,060 It goes to 0, because you're now at the Nernst potential. 380 00:20:32,060 --> 00:20:39,900 And if you hold the voltage below, the inside of the cell 381 00:20:39,900 --> 00:20:43,150 is now negative relative to the Nernst potential, 382 00:20:43,150 --> 00:20:44,960 and current's going to flow in. 383 00:20:44,960 --> 00:20:46,540 So notice that the current actually 384 00:20:46,540 --> 00:20:51,080 reverses sine around EK. 385 00:20:51,080 --> 00:20:54,640 And so sometimes this voltage is called the "reversal 386 00:20:54,640 --> 00:20:55,840 potential." 387 00:20:55,840 --> 00:20:58,540 So you'll hear me sometimes refer to the reversal 388 00:20:58,540 --> 00:21:01,990 potential, and that's just the same as the Nernst potential 389 00:21:01,990 --> 00:21:03,231 or the equilibrium potential. 390 00:21:07,000 --> 00:21:09,700 So the equation for something that 391 00:21:09,700 --> 00:21:13,240 looks like this, a straight line that's offset from 0, 392 00:21:13,240 --> 00:21:14,140 is just this. 393 00:21:14,140 --> 00:21:17,680 So this expression right here is going 394 00:21:17,680 --> 00:21:20,710 to be our basic model for how we describe 395 00:21:20,710 --> 00:21:23,740 the current through an ion channel as a function 396 00:21:23,740 --> 00:21:25,360 of the voltage of the membrane. 397 00:21:25,360 --> 00:21:27,610 So current, the potassium current, 398 00:21:27,610 --> 00:21:30,370 is just the potassium conductance 399 00:21:30,370 --> 00:21:33,790 times the difference of the membrane potential 400 00:21:33,790 --> 00:21:36,760 from the reversal potential. 401 00:21:36,760 --> 00:21:41,680 So IK equals GK times V minus EK. 402 00:21:41,680 --> 00:21:45,940 We also have a circuit, a little simple equivalent circuit, 403 00:21:45,940 --> 00:21:50,960 that describes this relation, and this is what it looks like. 404 00:21:50,960 --> 00:21:54,610 We have a-- what we're going to do 405 00:21:54,610 --> 00:22:05,740 to include the effects of this ion-specific conductance 406 00:22:05,740 --> 00:22:08,140 in the presence of a concentration gradient 407 00:22:08,140 --> 00:22:12,010 is to take our conductor, our resistor here, our conductance, 408 00:22:12,010 --> 00:22:14,830 and put it in series with a battery. 409 00:22:14,830 --> 00:22:16,720 So that's the basic circuit element 410 00:22:16,720 --> 00:22:20,320 that describes this kind of I-V relation. 411 00:22:20,320 --> 00:22:21,920 Why is that? 412 00:22:21,920 --> 00:22:24,135 So let's break this down a little bit. 413 00:22:24,135 --> 00:22:25,510 Basically, what we're going to do 414 00:22:25,510 --> 00:22:29,750 is we're going to equate this membrane potential difference 415 00:22:29,750 --> 00:22:32,040 between the inside and the outside of the cell, 416 00:22:32,040 --> 00:22:36,650 this potential difference, to the sum of the voltage drops 417 00:22:36,650 --> 00:22:38,070 across these two elements. 418 00:22:38,070 --> 00:22:42,930 Is it 1.5 volts, the same as the battery in here? 419 00:22:42,930 --> 00:22:43,640 No. 420 00:22:43,640 --> 00:22:44,140 What is it? 421 00:22:46,760 --> 00:22:47,635 AUDIENCE: [INAUDIBLE] 422 00:22:47,635 --> 00:22:48,343 MICHALE FEE: Yes. 423 00:22:48,343 --> 00:22:50,640 That's the battery we've been talking about, exactly. 424 00:22:50,640 --> 00:22:55,020 So the voltage drop across this battery is EK. 425 00:22:55,020 --> 00:22:59,130 What's the voltage drop across this resistor, this potassium 426 00:22:59,130 --> 00:23:01,920 conductance? 427 00:23:01,920 --> 00:23:03,930 What does it depend on? 428 00:23:03,930 --> 00:23:05,490 What if the current is 0? 429 00:23:05,490 --> 00:23:10,500 What's the voltage drop across a resistor whose current is 0? 430 00:23:10,500 --> 00:23:11,070 0. 431 00:23:11,070 --> 00:23:13,720 Ohm's law, right? 432 00:23:13,720 --> 00:23:15,960 What is it at arbitrary current? 433 00:23:15,960 --> 00:23:19,380 What's the voltage drop across a resistor 434 00:23:19,380 --> 00:23:21,124 as a function of current? 435 00:23:21,124 --> 00:23:22,007 AUDIENCE: [INAUDIBLE] 436 00:23:22,007 --> 00:23:22,757 MICHALE FEE: Good. 437 00:23:22,757 --> 00:23:25,600 Can somebody tell me what it is in terms of the quantities 438 00:23:25,600 --> 00:23:26,170 that we have? 439 00:23:28,950 --> 00:23:31,080 We're looking for a delta V across here. 440 00:23:34,400 --> 00:23:37,310 So VM, the membrane potential, has 441 00:23:37,310 --> 00:23:40,880 to just equal the sum of those two voltage drops, right? 442 00:23:40,880 --> 00:23:44,360 So VM equals EK plus IK over GK. 443 00:23:44,360 --> 00:23:46,505 And now let's just solve this for IK. 444 00:23:49,500 --> 00:23:52,440 So that's why this little circuit element describes 445 00:23:52,440 --> 00:23:55,640 our potassium conductance. 446 00:23:55,640 --> 00:23:57,956 That's the equation that goes with it. 447 00:23:57,956 --> 00:23:59,227 Any questions? 448 00:24:02,636 --> 00:24:03,610 No? 449 00:24:03,610 --> 00:24:05,170 All right, let's push on. 450 00:24:05,170 --> 00:24:08,450 This quantity right here, by the way, V minus EK, 451 00:24:08,450 --> 00:24:10,660 is called the driving potential. 452 00:24:10,660 --> 00:24:15,642 If V is equal to EK, then the driving potential is 0, 453 00:24:15,642 --> 00:24:16,600 and there's no current. 454 00:24:22,300 --> 00:24:25,880 The current through the channel is proportional to the driving 455 00:24:25,880 --> 00:24:26,480 potential. 456 00:24:31,910 --> 00:24:35,130 So let's just-- so there is our new circuit 457 00:24:35,130 --> 00:24:40,530 diagram for our cell that's a capacitor whose membrane has 458 00:24:40,530 --> 00:24:42,600 little leaks in it, where there's 459 00:24:42,600 --> 00:24:50,430 a ion-specific permeability of the pore and ion concentration 460 00:24:50,430 --> 00:24:53,190 gradient to produce a battery. 461 00:24:53,190 --> 00:24:54,820 There's our new circuit element. 462 00:24:54,820 --> 00:24:56,820 You can see that we're getting awfully close now 463 00:24:56,820 --> 00:25:01,755 to the whole equivalent circuit model that Hodgkin 464 00:25:01,755 --> 00:25:05,160 and Huxley wrote down. 465 00:25:05,160 --> 00:25:10,080 So let's just flesh out this equation here. 466 00:25:10,080 --> 00:25:12,240 What is I sub K? 467 00:25:12,240 --> 00:25:19,730 I sub K is just times V minus EK, very good. 468 00:25:19,730 --> 00:25:24,110 Remember that the resistance, RK is just 1 over GK, 469 00:25:24,110 --> 00:25:28,100 and tau, again, is RK times C. So let's 470 00:25:28,100 --> 00:25:30,180 massage this a little bit more. 471 00:25:30,180 --> 00:25:36,020 We're going to write this down as V plus dV, 472 00:25:36,020 --> 00:25:40,955 dt equals EK plus RKIE. 473 00:25:40,955 --> 00:25:43,140 So what is that? 474 00:25:43,140 --> 00:25:44,780 Any guess what that is? 475 00:25:48,182 --> 00:25:49,160 AUDIENCE: [INAUDIBLE] 476 00:25:49,160 --> 00:25:50,118 MICHALE FEE: Very good. 477 00:25:54,380 --> 00:25:56,280 Can everyone see why that's V infinity? 478 00:25:56,280 --> 00:26:00,210 Because if you set dV, dt equal to 0, 479 00:26:00,210 --> 00:26:03,825 V is equal to IK plus RKIE. 480 00:26:07,000 --> 00:26:11,590 If you inject a constant current at steady state, 481 00:26:11,590 --> 00:26:15,490 dV, dt equals 0, you can see that the injected current 482 00:26:15,490 --> 00:26:18,820 is just equal to the potassium current 483 00:26:18,820 --> 00:26:21,490 leaking out through the membrane. 484 00:26:21,490 --> 00:26:23,090 That all makes great sense. 485 00:26:23,090 --> 00:26:24,640 So that's V infinity. 486 00:26:24,640 --> 00:26:26,710 The voltage is just-- 487 00:26:26,710 --> 00:26:29,580 the differential equation is V plus tau dV, 488 00:26:29,580 --> 00:26:31,090 dt equals V infinity. 489 00:26:31,090 --> 00:26:34,100 It's exactly the same equation we had before. 490 00:26:34,100 --> 00:26:37,710 So you know that at every moment V is going to be doing what? 491 00:26:41,880 --> 00:26:48,690 Everybody together-- relaxing toward V infinity, right? 492 00:26:48,690 --> 00:26:51,710 Where that's our new V infinity. 493 00:26:51,710 --> 00:26:54,890 So we're going to do the same experiment again. 494 00:26:54,890 --> 00:26:57,030 Here is our pulse of injected current. 495 00:26:57,030 --> 00:27:00,870 V infinity is now starting at EK, 496 00:27:00,870 --> 00:27:05,460 jumps up to EK plus RKI naught and then goes back down. 497 00:27:05,460 --> 00:27:10,446 And the voltage of the cell is relaxing toward V infinity. 498 00:27:16,770 --> 00:27:22,840 So adding that just shifted this voltage trace down from 0. 499 00:27:22,840 --> 00:27:25,300 Remember, before this was sitting at 0, 500 00:27:25,300 --> 00:27:28,130 and now it's sitting at minus 75 millivolts. 501 00:27:34,770 --> 00:27:35,985 Any questions? 502 00:27:42,340 --> 00:27:53,486 So we're going to come back in a minute 503 00:27:53,486 --> 00:28:01,290 and finish fleshing out this model where 504 00:28:01,290 --> 00:28:04,890 we make those conductances is dependent on time 505 00:28:04,890 --> 00:28:10,090 as a way of describing the spikes that a neuron produces. 506 00:28:10,090 --> 00:28:11,770 But before we do that, we're going 507 00:28:11,770 --> 00:28:15,790 to talk about a much simpler model of how 508 00:28:15,790 --> 00:28:18,810 a spiking neuron behaves. 509 00:28:18,810 --> 00:28:22,680 Basically, what we're going to do is we're going to 510 00:28:22,680 --> 00:28:28,050 instead of write down a detailed biophysical model of action 511 00:28:28,050 --> 00:28:30,660 potentials, which is where we're heading, 512 00:28:30,660 --> 00:28:34,710 we're just going to ask some simpler questions about how 513 00:28:34,710 --> 00:28:37,120 a spiking neuron behaves. 514 00:28:37,120 --> 00:28:42,030 So notice that action potentials are really important. 515 00:28:42,030 --> 00:28:44,430 They're the way neurons communicate with other neurons. 516 00:28:44,430 --> 00:28:47,910 They send a signal down their axon, release neurotransmitter 517 00:28:47,910 --> 00:28:49,410 on other neurons. 518 00:28:49,410 --> 00:28:53,052 But most of the time a neuron is not spiking. 519 00:28:53,052 --> 00:28:53,760 What is it doing? 520 00:29:02,530 --> 00:29:04,230 Well, could be. 521 00:29:06,820 --> 00:29:08,060 Other thoughts? 522 00:29:08,060 --> 00:29:09,955 Certainly sometimes it could be resting. 523 00:29:12,850 --> 00:29:13,660 Right. 524 00:29:13,660 --> 00:29:15,310 Well, and how does it do that? 525 00:29:20,612 --> 00:29:21,576 [LAUGHS] 526 00:29:22,550 --> 00:29:24,262 It's part of it. 527 00:29:24,262 --> 00:29:25,640 Anybody? 528 00:29:25,640 --> 00:29:28,360 Yes? 529 00:29:28,360 --> 00:29:32,590 Integrating its inputs, and it integrates those inputs, 530 00:29:32,590 --> 00:29:35,550 and eventually it spikes. 531 00:29:35,550 --> 00:29:38,490 So we're going to develop a model 532 00:29:38,490 --> 00:29:42,900 or look at a model now called, not surprisingly, "integrate 533 00:29:42,900 --> 00:29:47,160 and fire," that captures exactly that idea. 534 00:29:47,160 --> 00:29:51,720 That a neuron spends most of its time integrating its inputs, 535 00:29:51,720 --> 00:29:53,550 making a decision about when to spike, 536 00:29:53,550 --> 00:29:56,130 and then spiking, and then starting over again. 537 00:29:59,340 --> 00:30:01,890 So the other crucial piece of this 538 00:30:01,890 --> 00:30:04,750 is that for most types of neurons 539 00:30:04,750 --> 00:30:07,690 the spikes are really all the same. 540 00:30:07,690 --> 00:30:11,280 The details of the spike wave form 541 00:30:11,280 --> 00:30:14,520 don't carry extra information beyond the fact 542 00:30:14,520 --> 00:30:15,820 that there's a spike. 543 00:30:15,820 --> 00:30:19,320 We're going to treat our spikes as delta functions, just 544 00:30:19,320 --> 00:30:23,370 discrete events at a single time, 545 00:30:23,370 --> 00:30:27,210 and the spikes are going to occur 546 00:30:27,210 --> 00:30:30,150 when the voltage in the neuron reaches 547 00:30:30,150 --> 00:30:34,350 a particular membrane potential, called the "spike threshold." 548 00:30:34,350 --> 00:30:38,480 Now, that's a reasonable approximation for many neurons. 549 00:30:38,480 --> 00:30:41,120 It's not absolutely the case, but many neurons 550 00:30:41,120 --> 00:30:46,160 spike when the neuron reaches a particular voltage threshold. 551 00:30:46,160 --> 00:30:51,515 And that's captured in this model called the integrate 552 00:30:51,515 --> 00:30:52,250 and fire neuron. 553 00:30:52,250 --> 00:30:53,708 And what we're going to do is we're 554 00:30:53,708 --> 00:30:55,880 going to take our Hodgkin-Huxley model, 555 00:30:55,880 --> 00:31:01,310 and we're going to replace these sodium and potassium 556 00:31:01,310 --> 00:31:04,110 conductances that actually generate the spike. 557 00:31:04,110 --> 00:31:06,260 We're going to replace that with a very simplified 558 00:31:06,260 --> 00:31:11,160 model of a spike generator, and it's going to look like this. 559 00:31:11,160 --> 00:31:16,880 So basically, the idea is that the cell gets input 560 00:31:16,880 --> 00:31:22,380 from either an electrode or from a synaptic input, 561 00:31:22,380 --> 00:31:25,500 and it integrates that input until it reaches a voltage, 562 00:31:25,500 --> 00:31:28,870 called V threshold. 563 00:31:28,870 --> 00:31:31,800 And once it reaches a threshold, we simply 564 00:31:31,800 --> 00:31:37,110 reset the voltage back down to a lower value, called "V reset." 565 00:31:37,110 --> 00:31:39,750 And then we just-- 566 00:31:39,750 --> 00:31:42,840 if we want to, we can just draw a line at that time, 567 00:31:42,840 --> 00:31:43,860 and that's the spike. 568 00:31:47,960 --> 00:31:49,840 But what really happens to the voltage here 569 00:31:49,840 --> 00:31:52,690 is that once the voltage hits V threshold, 570 00:31:52,690 --> 00:31:57,190 it gets reset back down to this [INAUDIBLE].. 571 00:31:57,190 --> 00:31:58,386 Any questions? 572 00:32:02,200 --> 00:32:04,855 Now, that kind of behavior is not-- 573 00:32:07,490 --> 00:32:09,300 it's fairly common in neurons. 574 00:32:09,300 --> 00:32:14,360 So this is an example of, does the voltage in a cell in motor 575 00:32:14,360 --> 00:32:15,980 cortex of the songbird-- and this 576 00:32:15,980 --> 00:32:20,420 is from a paper from Rich Mooney's lab back in 1992. 577 00:32:20,420 --> 00:32:22,530 And you can see that if you inject current, 578 00:32:22,530 --> 00:32:25,580 the voltage in this cell ramps up until it hits a threshold 579 00:32:25,580 --> 00:32:27,050 voltage. 580 00:32:27,050 --> 00:32:29,720 It makes a spike, which is very narrow in time, but then 581 00:32:29,720 --> 00:32:34,740 the voltage resets down to a lower voltage. 582 00:32:34,740 --> 00:32:36,790 And that process just repeats over and over. 583 00:32:41,110 --> 00:32:43,050 So now what we're going to do is we're 584 00:32:43,050 --> 00:32:48,210 going to calculate the rate at which a neuron fires 585 00:32:48,210 --> 00:32:51,150 as a function of how much current gets 586 00:32:51,150 --> 00:32:56,490 injected into the cell, how much input a cell gets. 587 00:32:56,490 --> 00:32:58,320 So we're going to start by considering 588 00:32:58,320 --> 00:33:01,800 the case where the cell has no leaks in its membrane, 589 00:33:01,800 --> 00:33:03,720 no conductances. 590 00:33:03,720 --> 00:33:07,120 So let's plot voltage as a function of time, 591 00:33:07,120 --> 00:33:09,420 so this is voltage as a function of time, 592 00:33:09,420 --> 00:33:12,990 when we inject a step of current into the cell. 593 00:33:12,990 --> 00:33:14,500 So what's going to happen? 594 00:33:14,500 --> 00:33:17,220 So the cell starts at some voltage, 595 00:33:17,220 --> 00:33:19,130 and the current turns on. 596 00:33:19,130 --> 00:33:22,430 What's going to happen? 597 00:33:22,430 --> 00:33:25,224 Some fresh [INAUDIBLE]. 598 00:33:30,950 --> 00:33:31,450 Yes? 599 00:33:31,450 --> 00:33:36,882 AUDIENCE: [INAUDIBLE] 600 00:33:36,882 --> 00:33:37,840 MICHALE FEE: Excellent. 601 00:33:37,840 --> 00:33:38,840 Like that? 602 00:33:38,840 --> 00:33:39,432 AUDIENCE: Yes. 603 00:33:39,432 --> 00:33:40,390 MICHALE FEE: Very good. 604 00:33:40,390 --> 00:33:41,557 Then what's going to happen? 605 00:33:44,818 --> 00:33:45,318 Yeah. 606 00:33:50,280 --> 00:33:51,750 Excellent-- like that. 607 00:33:51,750 --> 00:33:55,505 And then what's going to happen? 608 00:33:55,505 --> 00:33:59,000 AUDIENCE: [INAUDIBLE] 609 00:33:59,000 --> 00:34:02,090 MICHALE FEE: So we inject a constant current into our cell, 610 00:34:02,090 --> 00:34:04,940 and now our cell is going to generate 611 00:34:04,940 --> 00:34:10,880 spikes, action potentials, at regular intervals. 612 00:34:10,880 --> 00:34:12,840 And the interval between those spikes 613 00:34:12,840 --> 00:34:17,310 is going to be controlled by how long it takes that capacitor 614 00:34:17,310 --> 00:34:22,389 to charge up from the reset voltage to the threshold. 615 00:34:22,389 --> 00:34:24,719 So we can actually just very simply calculate 616 00:34:24,719 --> 00:34:27,420 the firing rate of this neuron as a function 617 00:34:27,420 --> 00:34:32,040 of how much current we inject into the cell. 618 00:34:32,040 --> 00:34:35,823 So the firing rate is just 1 over that interval 619 00:34:35,823 --> 00:34:36,615 between the spikes. 620 00:34:40,360 --> 00:34:43,360 So now what do we do? 621 00:34:43,360 --> 00:34:46,110 Anybody want to guess what the next step is? 622 00:34:55,820 --> 00:34:57,610 How do we figure out how long it takes? 623 00:34:57,610 --> 00:34:59,570 We know this distance. 624 00:34:59,570 --> 00:35:02,260 This is kind of like some third grade word problem here. 625 00:35:02,260 --> 00:35:04,230 Come on, somebody help me out. 626 00:35:04,230 --> 00:35:05,350 You cross a river. 627 00:35:05,350 --> 00:35:10,030 You're paddling one second. 628 00:35:10,030 --> 00:35:11,770 The river's 10 meters across. 629 00:35:18,300 --> 00:35:20,345 Anybody, anybody? 630 00:35:28,680 --> 00:35:33,900 How about right there, white shirt, what do you think? 631 00:35:33,900 --> 00:35:34,400 Yes. 632 00:35:34,400 --> 00:35:35,394 [LAUGHS] 633 00:35:38,380 --> 00:35:40,460 Well, we're trying to calculate the time 634 00:35:40,460 --> 00:35:43,640 it takes for the voltage to go from here to here. 635 00:35:56,830 --> 00:35:57,850 What would it depend on? 636 00:35:57,850 --> 00:35:58,350 Yes? 637 00:36:02,890 --> 00:36:03,390 Yeah. 638 00:36:03,390 --> 00:36:04,432 Well, what are you doing? 639 00:36:09,650 --> 00:36:11,150 Yeah, but what are you-- 640 00:36:11,150 --> 00:36:12,260 you're calculating what? 641 00:36:12,260 --> 00:36:13,523 AUDIENCE: Delta t. 642 00:36:13,523 --> 00:36:15,440 MICHALE FEE: Yeah, you're calculating delta t. 643 00:36:15,440 --> 00:36:17,310 And you're using what about this line? 644 00:36:20,200 --> 00:36:23,660 Slope, exactly. 645 00:36:23,660 --> 00:36:25,630 So if we knew the slope of this line, 646 00:36:25,630 --> 00:36:29,200 we could calculate how long it takes to go from here to here. 647 00:36:31,960 --> 00:36:32,990 So let's do that. 648 00:36:32,990 --> 00:36:35,980 So we have this equation, C, dV, dt 649 00:36:35,980 --> 00:36:38,650 equals I sub E. What's the slope of this line? 650 00:36:44,446 --> 00:36:48,380 Yeah, it's just dV, dt. 651 00:36:48,380 --> 00:36:49,210 Good. 652 00:36:49,210 --> 00:36:51,480 So it's dV, dt. 653 00:36:51,480 --> 00:36:53,820 That's delta V over delta t. 654 00:36:53,820 --> 00:36:56,760 Delta V is just this voltage difference, right? 655 00:36:56,760 --> 00:36:59,970 And delta t is what we're trying to calculate. 656 00:36:59,970 --> 00:37:01,370 So we just solve-- 657 00:37:01,370 --> 00:37:03,650 we just plug this into here, and we just 658 00:37:03,650 --> 00:37:07,470 solve for 1 over delta t. 659 00:37:07,470 --> 00:37:11,777 And that's just C delta V times the injected current. 660 00:37:11,777 --> 00:37:13,610 That's probably what you were saying, right? 661 00:37:18,518 --> 00:37:19,560 Any questions about that? 662 00:37:24,290 --> 00:37:25,880 So let's look at what that looks like. 663 00:37:25,880 --> 00:37:30,470 The firing rate is proportional to the projected current. 664 00:37:30,470 --> 00:37:31,760 It goes through 0. 665 00:37:31,760 --> 00:37:34,880 At 0 injected current, the voltage is constant 666 00:37:34,880 --> 00:37:37,220 and the neuron will never fire. 667 00:37:37,220 --> 00:37:39,260 But if we inject a tiny bit of current, 668 00:37:39,260 --> 00:37:42,440 the voltage will slowly ramp up, and it will eventually 669 00:37:42,440 --> 00:37:45,010 hit the threshold and then reset. 670 00:37:45,010 --> 00:37:47,990 So the firing rate is proportional to the injected 671 00:37:47,990 --> 00:37:52,220 current slope 1 over C delta V. If the capacitor 672 00:37:52,220 --> 00:37:54,980 is much bigger, what happens to the firing rate? 673 00:37:58,350 --> 00:38:01,020 It slows down, because as you inject current, 674 00:38:01,020 --> 00:38:04,350 the voltage increases more slowly because the capacitance 675 00:38:04,350 --> 00:38:04,860 is bigger. 676 00:38:09,430 --> 00:38:14,770 Now, let's add our leak conductance back in. 677 00:38:19,090 --> 00:38:21,460 So this leak conductance, we're going 678 00:38:21,460 --> 00:38:26,282 to think of it as a potassium conductance. 679 00:38:26,282 --> 00:38:28,730 So I'm going to call it "G leak," because I'm 680 00:38:28,730 --> 00:38:32,870 going to use something like that later in Hodgkin-Huxley model, 681 00:38:32,870 --> 00:38:34,460 but you should think about it just 682 00:38:34,460 --> 00:38:36,290 as a potassium conductance. 683 00:38:39,230 --> 00:38:40,380 So what's going to happen? 684 00:38:40,380 --> 00:38:43,370 Let's go to our plot of voltage as a function of time 685 00:38:43,370 --> 00:38:45,260 during our injected current. 686 00:38:45,260 --> 00:38:53,920 Let's say that we start at V reset, our voltage at V reset. 687 00:38:59,510 --> 00:39:02,440 Or actually, let's start it E leak. 688 00:39:02,440 --> 00:39:06,950 Now what happens, anybody? 689 00:39:06,950 --> 00:39:07,575 Yes? 690 00:39:07,575 --> 00:39:11,460 AUDIENCE: It relaxes [INAUDIBLE].. 691 00:39:11,460 --> 00:39:12,690 MICHALE FEE: Good. 692 00:39:12,690 --> 00:39:17,250 The voltage starts here, because there's 0 injected current. 693 00:39:17,250 --> 00:39:20,740 As soon as you inject current, V infinity jumps up to here, 694 00:39:20,740 --> 00:39:22,760 let's say. 695 00:39:22,760 --> 00:39:24,890 The voltage relaxes exponentially 696 00:39:24,890 --> 00:39:30,110 toward V infinity until it hits the threshold-- very good. 697 00:39:30,110 --> 00:39:30,920 Then what happens? 698 00:39:30,920 --> 00:39:31,730 Somebody else. 699 00:39:34,586 --> 00:39:37,450 Yeah? 700 00:39:37,450 --> 00:39:39,121 How about you in the gray shirt. 701 00:39:39,121 --> 00:39:41,608 AUDIENCE: It drops. 702 00:39:41,608 --> 00:39:42,400 MICHALE FEE: Great. 703 00:39:42,400 --> 00:39:44,670 So if you know the answer, raise your hand. 704 00:39:44,670 --> 00:39:47,260 We'll go much faster. 705 00:39:47,260 --> 00:39:52,630 So at this point, it will jump back down to where? 706 00:39:52,630 --> 00:39:53,350 Good. 707 00:39:53,350 --> 00:39:55,081 And then what. 708 00:39:55,081 --> 00:39:56,160 Anybody else? 709 00:39:56,160 --> 00:39:58,620 If you know the answer, raise your hand. 710 00:39:58,620 --> 00:40:00,570 Let's just do an exercise. 711 00:40:00,570 --> 00:40:03,090 Raise your hand, everybody who knows-- 712 00:40:03,090 --> 00:40:03,645 up high. 713 00:40:03,645 --> 00:40:05,490 Raise your hand up high. 714 00:40:05,490 --> 00:40:09,860 Now, one of you say it. 715 00:40:09,860 --> 00:40:10,804 [LAUGHTER] 716 00:40:11,750 --> 00:40:13,215 Everybody, say it. 717 00:40:13,215 --> 00:40:14,860 AUDIENCE: It's relaxes. 718 00:40:14,860 --> 00:40:17,600 MICHALE FEE: Great, music to my ears. 719 00:40:17,600 --> 00:40:21,770 It relaxes exponentially back V infinity 720 00:40:21,770 --> 00:40:26,950 until it hits threshold and it jumps back. 721 00:40:26,950 --> 00:40:27,860 It keeps doing it. 722 00:40:27,860 --> 00:40:29,360 Now, what happens here? 723 00:40:29,360 --> 00:40:30,650 The current turns off. 724 00:40:30,650 --> 00:40:31,430 What happens? 725 00:40:31,430 --> 00:40:33,500 Who knows the answer to that question? 726 00:40:33,500 --> 00:40:35,350 Raise your hands up high. 727 00:40:35,350 --> 00:40:36,140 Shout it out. 728 00:40:36,140 --> 00:40:37,432 AUDIENCE: It goes back to the-- 729 00:40:37,432 --> 00:40:38,330 AUDIENCE: It relaxes. 730 00:40:38,330 --> 00:40:39,288 MICHALE FEE: Excellent. 731 00:40:39,288 --> 00:40:40,780 It relaxes back to? 732 00:40:40,780 --> 00:40:45,370 AUDIENCE: [INAUDIBLE] 733 00:40:45,370 --> 00:40:50,865 MICHALE FEE: Why is it a leak, not V reset? 734 00:40:54,250 --> 00:40:57,260 When there's no current injected, that's V infinity. 735 00:40:57,260 --> 00:41:01,240 So that's why it relaxes back to E leak. 736 00:41:01,240 --> 00:41:02,990 Any questions? 737 00:41:02,990 --> 00:41:03,490 No? 738 00:41:03,490 --> 00:41:04,670 It's pretty simple, right? 739 00:41:07,660 --> 00:41:09,250 So it's pretty simple. 740 00:41:09,250 --> 00:41:11,650 And we can actually derive the expression now 741 00:41:11,650 --> 00:41:14,230 for this delta t, and therefore, we 742 00:41:14,230 --> 00:41:16,180 can drive the equation for the firing rate 743 00:41:16,180 --> 00:41:18,550 as a function of injected current. 744 00:41:18,550 --> 00:41:21,340 It's a little more complicated than the other one. 745 00:41:21,340 --> 00:41:26,740 I'm going to go through it in a little bit less detail, 746 00:41:26,740 --> 00:41:28,540 because the answer turns out to actually 747 00:41:28,540 --> 00:41:30,220 be pretty simple [AUDIO OUT]. 748 00:41:30,220 --> 00:41:33,160 So we're going to calculate the firing rate, which is just 1 749 00:41:33,160 --> 00:41:35,380 over this delta t here. 750 00:41:35,380 --> 00:41:37,750 Before we actually get into the math, 751 00:41:37,750 --> 00:41:42,140 what happens if v infinity is down here? 752 00:41:42,140 --> 00:41:45,680 Who knows the answer to that question? 753 00:41:45,680 --> 00:41:48,000 Raise your hands. 754 00:41:48,000 --> 00:41:50,460 Shout it out. 755 00:41:50,460 --> 00:41:52,340 It won't spike, right? 756 00:41:52,340 --> 00:41:52,840 Good. 757 00:41:55,830 --> 00:41:58,590 So V infinity actually has to be above V threshold 758 00:41:58,590 --> 00:42:00,660 in order for that neuron to spike. 759 00:42:00,660 --> 00:42:03,210 Does that makes sense? 760 00:42:03,210 --> 00:42:05,010 So something is different already, 761 00:42:05,010 --> 00:42:09,180 because before we found in the current 762 00:42:09,180 --> 00:42:12,270 we injected the cell would eventually spike. 763 00:42:12,270 --> 00:42:15,130 But now what we see is when we have a leak, 764 00:42:15,130 --> 00:42:18,100 the V infinity actually has to be above V threshold. 765 00:42:18,100 --> 00:42:20,310 And so that means there's going to be some threshold 766 00:42:20,310 --> 00:42:22,978 current below which the neuron won't spike. 767 00:42:22,978 --> 00:42:23,770 That's pretty cool. 768 00:42:23,770 --> 00:42:25,810 We can see that right away. 769 00:42:25,810 --> 00:42:30,130 So I just wanted to find one quantity, 770 00:42:30,130 --> 00:42:34,690 called the "rheobase," and that is that current at which 771 00:42:34,690 --> 00:42:36,620 the neuron begins to spike. 772 00:42:36,620 --> 00:42:41,140 So if we start our cell at V reset, 773 00:42:41,140 --> 00:42:43,150 because let's say it just spiked, 774 00:42:43,150 --> 00:42:45,430 so it relaxes exponentially toward V infinity, 775 00:42:45,430 --> 00:42:49,810 but let's say that V infinity is right at threshold. 776 00:42:49,810 --> 00:42:52,930 So you can see that the time to reach the threshold 777 00:42:52,930 --> 00:42:54,370 is actually very long. 778 00:42:54,370 --> 00:42:56,680 If V infinity is equal to V threshold, 779 00:42:56,680 --> 00:42:59,320 it will never actually reach it. 780 00:42:59,320 --> 00:43:02,300 Even when V infinity is equal to V threshold, 781 00:43:02,300 --> 00:43:04,450 the firing rate is 0. 782 00:43:04,450 --> 00:43:06,670 Because if you inject just a tiny bit more current, 783 00:43:06,670 --> 00:43:07,930 now it'll begin to spike. 784 00:43:10,850 --> 00:43:13,110 It can calculate the injected current 785 00:43:13,110 --> 00:43:14,430 required to reach threshold. 786 00:43:14,430 --> 00:43:15,900 That's called the rheobase. 787 00:43:15,900 --> 00:43:20,022 We just set V infinity equal to V threshold, right? 788 00:43:20,022 --> 00:43:22,460 And we use our equation for-- 789 00:43:22,460 --> 00:43:26,960 that's V infinity, and we just set it equal to threshold. 790 00:43:26,960 --> 00:43:29,140 Now we just derive the injected current 791 00:43:29,140 --> 00:43:30,880 we just solved for I sub E. That's 792 00:43:30,880 --> 00:43:35,650 the injected current required to make V infinity 793 00:43:35,650 --> 00:43:37,510 reach V threshold, and you can see 794 00:43:37,510 --> 00:43:42,340 that it's just G leak times V threshold minus E leak. 795 00:43:45,840 --> 00:43:49,075 And we call that I threshold. 796 00:43:49,075 --> 00:43:50,200 So here's the way it looks. 797 00:43:50,200 --> 00:43:56,010 The firing rate of this neuron is 0 for low currents. 798 00:43:56,010 --> 00:43:59,520 As you inject more current, V infinity increases, 799 00:43:59,520 --> 00:44:02,770 but the cell still can't reach threshold 800 00:44:02,770 --> 00:44:06,190 until you inject an amount of current, 801 00:44:06,190 --> 00:44:08,680 such that V infinity reaches V threshold, 802 00:44:08,680 --> 00:44:10,120 and then it begins to spike. 803 00:44:10,120 --> 00:44:15,620 And the firing rate increases rapidly, and then going up. 804 00:44:15,620 --> 00:44:17,360 Does that makes sense? 805 00:44:17,360 --> 00:44:20,720 So many neurons have that property 806 00:44:20,720 --> 00:44:25,850 of having a threshold current above which you have to-- 807 00:44:25,850 --> 00:44:27,560 below which the cell won't spike. 808 00:44:31,730 --> 00:44:36,310 So now let's actually derive the equation 809 00:44:36,310 --> 00:44:42,910 for this firing rate as a function of injected current. 810 00:44:42,910 --> 00:44:45,468 So here's how we're going to do that. 811 00:44:45,468 --> 00:44:47,010 The cell just spiked, and we're going 812 00:44:47,010 --> 00:44:50,160 to calculate the amount of time before the cell spikes again. 813 00:44:50,160 --> 00:44:53,040 So we're going to start the voltage at V reset. 814 00:44:53,040 --> 00:44:56,940 We know that at some injected current above threshold 815 00:44:56,940 --> 00:44:59,175 the cell relaxes exponentially to V infinity. 816 00:44:59,175 --> 00:45:01,050 And we're just going to calculate how long it 817 00:45:01,050 --> 00:45:03,460 takes equal to threshold. 818 00:45:03,460 --> 00:45:05,770 So you know that that's an exponential, right? 819 00:45:05,770 --> 00:45:08,350 In fact, we wrote down the solution to that exponential 820 00:45:08,350 --> 00:45:10,470 a bunch of times. 821 00:45:10,470 --> 00:45:12,800 The difference from V to V infinity 822 00:45:12,800 --> 00:45:14,900 decreases exponentially. 823 00:45:14,900 --> 00:45:16,970 But we know a bunch of these values, right? 824 00:45:16,970 --> 00:45:22,250 We know that we're calculating these voltages when tau equals, 825 00:45:22,250 --> 00:45:27,100 sorry, when t equals delta t. 826 00:45:27,100 --> 00:45:29,870 And we know the initial voltage, that's V reset. 827 00:45:29,870 --> 00:45:34,430 And we know the voltage at time delta t, 828 00:45:34,430 --> 00:45:36,120 and that's just equal to V threshold. 829 00:45:36,120 --> 00:45:39,960 So we can just substitute those quantities into this equation. 830 00:45:39,960 --> 00:45:44,750 So now we have V [AUDIO OUT] minus V infinity equals V reset 831 00:45:44,750 --> 00:45:48,070 minus V infinity times E to the minus delta t over tau. 832 00:45:48,070 --> 00:45:49,610 Does that make sense? 833 00:45:49,610 --> 00:45:52,700 Everyone see what I just did? 834 00:45:52,700 --> 00:45:56,470 We're just calculating this time that it takes the neuron 835 00:45:56,470 --> 00:46:02,170 to relax exponentially from V reset to V threshold. 836 00:46:02,170 --> 00:46:03,910 We know all these quantities, so we just 837 00:46:03,910 --> 00:46:05,290 stick them into this equation. 838 00:46:05,290 --> 00:46:06,370 And what do we solve for? 839 00:46:09,310 --> 00:46:12,040 Delta t, good. 840 00:46:12,040 --> 00:46:15,130 So we take the natural log of-- well, 841 00:46:15,130 --> 00:46:17,950 we divide through by V reset minus V 842 00:46:17,950 --> 00:46:22,120 infinity, take the natural log, and we solve for delta t. 843 00:46:22,120 --> 00:46:26,950 So delta t equals minus tau, natural log, V infinity minus V 844 00:46:26,950 --> 00:46:30,430 threshold over V infinity minus V reset. 845 00:46:30,430 --> 00:46:32,770 It's kind of messy, right? 846 00:46:32,770 --> 00:46:34,330 I don't know. 847 00:46:34,330 --> 00:46:39,150 The shape of that doesn't really leap to my mind. 848 00:46:39,150 --> 00:46:44,800 So what we're going to do is actually just 849 00:46:44,800 --> 00:46:48,040 simplify this expression in a limit, 850 00:46:48,040 --> 00:46:50,860 in a limit that the injected current is large. 851 00:46:55,140 --> 00:46:58,180 So what happens here when the injected current is large? 852 00:46:58,180 --> 00:46:59,260 What gets big? 853 00:46:59,260 --> 00:47:00,850 V infinity gets big, right? 854 00:47:00,850 --> 00:47:02,290 When you inject a lot of current, 855 00:47:02,290 --> 00:47:04,532 V infinity is very high. 856 00:47:04,532 --> 00:47:06,615 Does that make sense? 857 00:47:06,615 --> 00:47:09,180 And when V infinity is really big, 858 00:47:09,180 --> 00:47:11,400 this expression approaches what? 859 00:47:14,760 --> 00:47:16,030 1. 860 00:47:16,030 --> 00:47:17,820 And what is the log of 1? 861 00:47:17,820 --> 00:47:18,600 AUDIENCE: 0. 862 00:47:18,600 --> 00:47:19,440 MICHALE FEE: 0. 863 00:47:19,440 --> 00:47:23,820 So this expression, this expression approaches 1, 864 00:47:23,820 --> 00:47:28,170 and this expression, the log of that, approaches 0. 865 00:47:28,170 --> 00:47:32,430 So we can do is do a linear approximation of this turn 866 00:47:32,430 --> 00:47:33,530 right here. 867 00:47:33,530 --> 00:47:35,640 So here's what we're going to do. 868 00:47:35,640 --> 00:47:37,800 We're going to work in the limit that V infinity 869 00:47:37,800 --> 00:47:40,800 is much bigger than V reset or V threshold. 870 00:47:40,800 --> 00:47:45,600 We're going to use the approximation that log 871 00:47:45,600 --> 00:47:49,440 of 1 plus alpha is just alpha. 872 00:47:49,440 --> 00:47:54,490 So as this approaches 1, you can write it as 1 plus alpha. 873 00:47:54,490 --> 00:47:57,570 That whole thing there approximates to alpha. 874 00:47:57,570 --> 00:48:01,080 And when you do that and you solve for the firing rate, what 875 00:48:01,080 --> 00:48:03,360 you find is that the firing rate is just 1 876 00:48:03,360 --> 00:48:07,170 over C delta V times the injected current 877 00:48:07,170 --> 00:48:09,820 minus the threshold current. 878 00:48:09,820 --> 00:48:12,340 Threshold current is just the rheobase 879 00:48:12,340 --> 00:48:15,030 that we calculated before. 880 00:48:15,030 --> 00:48:16,420 Well, what does that look like? 881 00:48:16,420 --> 00:48:19,040 Well, what is the firing rate, first of all? 882 00:48:19,040 --> 00:48:21,840 I kind of simplified this a bit. 883 00:48:21,840 --> 00:48:27,980 What is the firing rate when the current is below I threshold? 884 00:48:27,980 --> 00:48:28,850 0. 885 00:48:28,850 --> 00:48:31,550 And then if it's above I threshold-- so this expression 886 00:48:31,550 --> 00:48:33,890 that I wrote right here is true only 887 00:48:33,890 --> 00:48:36,980 if the injected current greater than I threshold. 888 00:48:36,980 --> 00:48:38,430 And it's zero below that. 889 00:48:38,430 --> 00:48:41,186 So now, what does this look like? 890 00:48:41,186 --> 00:48:44,290 The firing rate is 0 until you hit threshold. 891 00:48:44,290 --> 00:48:45,870 And then what? 892 00:48:45,870 --> 00:48:46,708 It increases-- 893 00:48:46,708 --> 00:48:47,500 AUDIENCE: Linearly. 894 00:48:47,500 --> 00:48:50,750 MICHALE FEE: --linearly. 895 00:48:50,750 --> 00:48:55,210 So this equation is linearly in the injected current. 896 00:48:55,210 --> 00:48:58,900 The slope is actually exactly the same as it was for the case 897 00:48:58,900 --> 00:49:02,520 where there was no leak. 898 00:49:02,520 --> 00:49:04,530 So the fine rate is 0. 899 00:49:04,530 --> 00:49:10,180 Once you hit the threshold, the firing rate of the neuron 900 00:49:10,180 --> 00:49:16,330 increases approximately linearly for large currents. 901 00:49:16,330 --> 00:49:20,470 And if you look down here, what you see is that the current 902 00:49:20,470 --> 00:49:21,400 actually-- 903 00:49:21,400 --> 00:49:23,830 the actual solution is that the firing rate 904 00:49:23,830 --> 00:49:28,290 jumps up at threshold and then tracks 905 00:49:28,290 --> 00:49:30,390 along the linear approximation. 906 00:49:30,390 --> 00:49:33,870 So the dashed line there is the actual solution. 907 00:49:33,870 --> 00:49:37,620 The solid line is that linear approximation. 908 00:49:37,620 --> 00:49:41,310 And that right there is really a very good model 909 00:49:41,310 --> 00:49:42,285 for a lot of neurons. 910 00:49:46,170 --> 00:49:50,255 Most neurons sort of saturate a little bit. 911 00:49:50,255 --> 00:49:52,380 Their firing rate kind of flattens out a little bit 912 00:49:52,380 --> 00:49:54,060 as you go to very high firing rates. 913 00:49:54,060 --> 00:49:56,210 But that's a pretty good approximation. 914 00:49:56,210 --> 00:49:57,050 Yes? 915 00:49:57,050 --> 00:49:59,690 AUDIENCE: What's [INAUDIBLE]? 916 00:49:59,690 --> 00:50:04,040 MICHALE FEE: Delta V is the difference 917 00:50:04,040 --> 00:50:08,970 from V reset to V threshold. 918 00:50:08,970 --> 00:50:11,850 Those are just parameters of the model. 919 00:50:11,850 --> 00:50:13,280 There was another question here. 920 00:50:13,280 --> 00:50:13,535 Skyler? 921 00:50:13,535 --> 00:50:14,370 AUDIENCE: I had the same question. 922 00:50:14,370 --> 00:50:15,020 MICHALE FEE: Same question? 923 00:50:15,020 --> 00:50:15,520 OK. 924 00:50:20,310 --> 00:50:21,060 Anything else? 925 00:50:25,740 --> 00:50:27,560 That's the integrate and fire neuron. 926 00:50:27,560 --> 00:50:32,060 That's probably the most commonly used model 927 00:50:32,060 --> 00:50:35,880 of neurons in neuroscience-- 928 00:50:35,880 --> 00:50:36,480 pretty simple. 929 00:50:40,553 --> 00:50:42,720 Or I should say, that's the most commonly used model 930 00:50:42,720 --> 00:50:45,230 of spiking neurons. 931 00:50:45,230 --> 00:50:49,960 So you can actually take neurons that behave like this, 932 00:50:49,960 --> 00:50:54,070 and assemble them into complex networks, 933 00:50:54,070 --> 00:50:57,640 and study how network interactions occur 934 00:50:57,640 --> 00:50:59,080 with a spiking model. 935 00:50:59,080 --> 00:51:04,990 And this model captures most of the interesting, important 936 00:51:04,990 --> 00:51:07,150 behavior of spiking neurons. 937 00:51:10,560 --> 00:51:11,260 Question, Danny? 938 00:51:17,930 --> 00:51:18,780 Any other questions? 939 00:51:26,250 --> 00:51:30,960 So let's come back to our Hodgkin-Huxley model, 940 00:51:30,960 --> 00:51:32,340 our equivalent circuit. 941 00:51:32,340 --> 00:51:37,000 So what we just described was a model neuron 942 00:51:37,000 --> 00:51:40,750 in which we replaced these sodium and potassium 943 00:51:40,750 --> 00:51:43,150 conductances that actually produce these action 944 00:51:43,150 --> 00:51:47,775 potentials with a very simple spike generator. 945 00:51:47,775 --> 00:51:49,150 But now what we're going to do is 946 00:51:49,150 --> 00:51:50,960 we're going to come back to our model, 947 00:51:50,960 --> 00:51:55,120 and we're going to flesh out the biophysical details that 948 00:51:55,120 --> 00:51:59,650 allow these two conductances right here to produce 949 00:51:59,650 --> 00:52:01,870 action potentials. 950 00:52:01,870 --> 00:52:08,480 Now, in fact, most of the time when 951 00:52:08,480 --> 00:52:12,410 we model networks of neurons, we simplify the spike generator 952 00:52:12,410 --> 00:52:16,640 to something like an integrate and fire spike generator. 953 00:52:16,640 --> 00:52:19,730 But the framework that Hodgkin and Huxley developed 954 00:52:19,730 --> 00:52:24,890 for describing a time-dependent and voltage-dependent 955 00:52:24,890 --> 00:52:29,600 conductances is so powerful and commonly used 956 00:52:29,600 --> 00:52:34,250 to describe conductances that it's really worth understanding 957 00:52:34,250 --> 00:52:38,690 that mathematical description, that physical description, 958 00:52:38,690 --> 00:52:42,680 of ionic conductances and how they 959 00:52:42,680 --> 00:52:44,937 depend on voltage and time. 960 00:52:44,937 --> 00:52:46,520 So that's what we're going to do next. 961 00:52:49,400 --> 00:52:51,290 So the first thing we do is we notice 962 00:52:51,290 --> 00:52:56,930 that in the Hodgkin-Huxley model we have three conductances. 963 00:52:56,930 --> 00:53:00,140 We have conductance which is very 964 00:53:00,140 --> 00:53:03,890 much like the lead conductance that we just 965 00:53:03,890 --> 00:53:06,530 used in the integrate and fire model. 966 00:53:06,530 --> 00:53:11,990 It has a reversal potential of around minus 50 millivolts, 967 00:53:11,990 --> 00:53:14,150 and it's just always on. 968 00:53:14,150 --> 00:53:16,780 It's just constant conductance. 969 00:53:16,780 --> 00:53:19,330 We have these two other conductances, a sodium 970 00:53:19,330 --> 00:53:22,000 conductance and a potassium conductance, 971 00:53:22,000 --> 00:53:26,280 that are both time-dependent and voltage-dependent. 972 00:53:26,280 --> 00:53:27,750 Each one of those conductances as 973 00:53:27,750 --> 00:53:30,300 has a current associated with it, 974 00:53:30,300 --> 00:53:33,900 currents flowing through ion channels. 975 00:53:33,900 --> 00:53:36,398 The total membrane current is just the sum 976 00:53:36,398 --> 00:53:37,440 of all of those currents. 977 00:53:37,440 --> 00:53:40,780 That's just definition. 978 00:53:40,780 --> 00:53:44,010 The total ionic membrane current is just 979 00:53:44,010 --> 00:53:48,180 the sum of contributions from sodium channels, potassium 980 00:53:48,180 --> 00:53:49,770 channels, and a leak. 981 00:53:53,594 --> 00:53:56,850 So the equation for our Hodgkin-Huxley model, 982 00:53:56,850 --> 00:53:58,900 again using Kirchhoff's current law, 983 00:53:58,900 --> 00:54:02,470 the sum of all the currents into these nodes has equal 0. 984 00:54:02,470 --> 00:54:05,560 So the membrane ionic current plus the capacitive current 985 00:54:05,560 --> 00:54:08,470 equals the injected electrode current. 986 00:54:08,470 --> 00:54:13,670 Now, each one of these conductances, 987 00:54:13,670 --> 00:54:17,490 or each one of these currents, can be written down 988 00:54:17,490 --> 00:54:20,790 in the same form that we developed before 989 00:54:20,790 --> 00:54:22,890 for our potassium conductance. 990 00:54:22,890 --> 00:54:27,600 It's just a, sorry, for the potassium current. 991 00:54:27,600 --> 00:54:32,320 So the sodium current is just a sodium conductance times what? 992 00:54:32,320 --> 00:54:35,040 What is that? 993 00:54:35,040 --> 00:54:36,150 Driving potential, right? 994 00:54:36,150 --> 00:54:38,940 The driving potential for sodium. 995 00:54:38,940 --> 00:54:44,250 But the sodium conductance now is voltage- and time-dependent. 996 00:54:44,250 --> 00:54:46,560 And it's that voltage and time dependence 997 00:54:46,560 --> 00:54:53,690 that gives sodium channels the properties that they need 998 00:54:53,690 --> 00:54:55,880 to generate action potentials. 999 00:54:55,880 --> 00:54:59,750 It's just analogous to what we already did before 1000 00:54:59,750 --> 00:55:02,360 for the potassium conductance. 1001 00:55:02,360 --> 00:55:04,430 And here's the potassium current. 1002 00:55:04,430 --> 00:55:08,780 It's just GK times the driving potential for potassium, 1003 00:55:08,780 --> 00:55:13,430 and the potassium conductance is voltage- and time-dependent. 1004 00:55:13,430 --> 00:55:18,490 Again, EK is minus 75, ENA is plus 55. 1005 00:55:18,490 --> 00:55:21,640 And our leak conductance, the leaked current rather, 1006 00:55:21,640 --> 00:55:24,610 is just the leak conductance times the driving potential 1007 00:55:24,610 --> 00:55:27,770 for the leak current. 1008 00:55:27,770 --> 00:55:30,240 The difference is that, in this model 1009 00:55:30,240 --> 00:55:32,180 now, the leak conductance is just constant. 1010 00:55:32,180 --> 00:55:34,520 There's no time dependence, it's always there, 1011 00:55:34,520 --> 00:55:38,458 and its voltage independent. 1012 00:55:38,458 --> 00:55:39,500 Any questions about that? 1013 00:55:44,350 --> 00:55:46,650 So the name of the game here in understanding 1014 00:55:46,650 --> 00:55:51,090 how this thing works is to figure out 1015 00:55:51,090 --> 00:55:55,016 where this time dependence and voltage dependence comes from. 1016 00:56:02,640 --> 00:56:03,514 What's that? 1017 00:56:03,514 --> 00:56:07,390 AUDIENCE: [INAUDIBLE] 1018 00:56:07,390 --> 00:56:09,130 MICHALE FEE: They're very close. 1019 00:56:09,130 --> 00:56:13,570 The potassium equilibrium potential is always very close 1020 00:56:13,570 --> 00:56:14,770 to minus 75. 1021 00:56:14,770 --> 00:56:18,980 In some neurons, it might be as low as minus 95, 1022 00:56:18,980 --> 00:56:21,180 but it's always in that range. 1023 00:56:21,180 --> 00:56:25,710 The sodium reversal potential is always plus 50-ish. 1024 00:56:30,840 --> 00:56:35,070 Well, it's highly consistent across mammals, 1025 00:56:35,070 --> 00:56:41,630 and the numbers for squid are pretty close to that as well. 1026 00:56:41,630 --> 00:56:45,070 I think these are the numbers for squid that 1027 00:56:45,070 --> 00:56:46,705 come from Hodgkin and Huxley. 1028 00:56:49,894 --> 00:56:50,852 Questions? 1029 00:56:56,121 --> 00:57:03,148 Now, how do these things generate an action potential? 1030 00:57:03,148 --> 00:57:04,190 That's the next question. 1031 00:57:04,190 --> 00:57:06,380 Just in principle, how do you think 1032 00:57:06,380 --> 00:57:11,910 about conductances like this generating action potentials? 1033 00:57:11,910 --> 00:57:16,680 So let's say that the conductances here are 0. 1034 00:57:16,680 --> 00:57:19,080 We set those to 0. 1035 00:57:19,080 --> 00:57:22,920 Those little arrows mean variable or adjustable. 1036 00:57:22,920 --> 00:57:26,550 So we can imagine that we have our hand on the knob, 1037 00:57:26,550 --> 00:57:29,430 and let's just turn those both down to 0. 1038 00:57:29,430 --> 00:57:32,060 So what's the voltage in the cell going to be? 1039 00:57:32,060 --> 00:57:33,190 So what does the cell do? 1040 00:57:37,138 --> 00:57:38,930 Well, if this is one of those moments where 1041 00:57:38,930 --> 00:57:41,388 everybody knows the answer and they're just not saying it-- 1042 00:57:44,890 --> 00:57:45,390 anybody? 1043 00:57:45,390 --> 00:57:46,290 Skylar? 1044 00:57:46,290 --> 00:57:52,390 AUDIENCE: [INAUDIBLE] 1045 00:57:52,390 --> 00:57:53,140 MICHALE FEE: Good. 1046 00:57:53,140 --> 00:57:58,840 So roughly, it's close to that. 1047 00:57:58,840 --> 00:58:00,890 It's minus 50, in this case. 1048 00:58:00,890 --> 00:58:01,390 Good. 1049 00:58:01,390 --> 00:58:06,660 So if these conductances are 0, then the cell 1050 00:58:06,660 --> 00:58:10,140 is going to be sitting at the vault of this battery. 1051 00:58:10,140 --> 00:58:13,680 There's 0 current, steady state, 0 current. 1052 00:58:13,680 --> 00:58:17,170 The voltage drop across here is 0, because there is 0 current. 1053 00:58:17,170 --> 00:58:19,050 And so the inside of the cell had better 1054 00:58:19,050 --> 00:58:21,720 be sitting at that voltage of that battery, which 1055 00:58:21,720 --> 00:58:24,630 is minus 50. 1056 00:58:24,630 --> 00:58:29,540 So now, what happens if we suddenly turn on a conductance? 1057 00:58:29,540 --> 00:58:31,790 What do I mean by "turn on a conductance"? 1058 00:58:31,790 --> 00:58:34,470 We make the resistance really small, 1059 00:58:34,470 --> 00:58:37,430 or we make the conductance really big. 1060 00:58:37,430 --> 00:58:38,520 So what are we doing? 1061 00:58:38,520 --> 00:58:42,000 Let's turn that conductance on as much as possible, 1062 00:58:42,000 --> 00:58:46,010 which means we're setting that resistor to 0. 1063 00:58:46,010 --> 00:58:50,240 Actually, if this resistance is 0, 1064 00:58:50,240 --> 00:58:52,460 then the voltage drop across-- 1065 00:58:55,850 --> 00:58:59,570 the voltage inside has to just be-- 1066 00:59:02,580 --> 00:59:05,795 it has to be set to the voltage of that battery. 1067 00:59:05,795 --> 00:59:06,670 Does that make sense? 1068 00:59:12,680 --> 00:59:15,590 Well, if we have no conductance here, 1069 00:59:15,590 --> 00:59:22,760 then the voltage inside the cell will relax toward e sub l. 1070 00:59:22,760 --> 00:59:25,480 But if we now turn on this conductance, 1071 00:59:25,480 --> 00:59:29,770 we set that resistor to 0, the voltage 1072 00:59:29,770 --> 00:59:35,698 will jump up that resistor back up to some big value. 1073 00:59:35,698 --> 00:59:36,990 What's the voltage going to do? 1074 00:59:40,220 --> 00:59:42,710 Relax back to e sub l. 1075 00:59:42,710 --> 00:59:45,650 So Daniel made this nice simulation of what happens. 1076 00:59:45,650 --> 00:59:48,320 So here's what is going to happen. 1077 00:59:48,320 --> 00:59:51,470 These conductances are going to start at 0. 1078 00:59:51,470 --> 00:59:55,200 We're going to-- oh, wait. 1079 00:59:55,200 --> 00:59:57,480 These conductances are to start at 0, 1080 00:59:57,480 --> 01:00:00,720 and then we're going to make this resistor small-- 1081 01:00:00,720 --> 01:00:03,450 not 0, but small. 1082 01:00:03,450 --> 01:00:05,580 Then we're going to make this resistor big, 1083 01:00:05,580 --> 01:00:08,960 and we're going to make that resistor small. 1084 01:00:08,960 --> 01:00:11,960 So watch what happens. 1085 01:00:11,960 --> 01:00:15,260 So the conductances are going to be plotted along the bottom, 1086 01:00:15,260 --> 01:00:17,210 and the voltage of the cell is going 1087 01:00:17,210 --> 01:00:20,480 to be plotted as a function of time here. 1088 01:00:20,480 --> 01:00:24,980 And the green and red show the reversal potentials 1089 01:00:24,980 --> 01:00:27,662 for the sodium and potassium. 1090 01:00:34,560 --> 01:00:37,020 There, the sodium conductance just got turned on. 1091 01:00:37,020 --> 01:00:38,330 And what happened? 1092 01:00:38,330 --> 01:00:42,330 The voltage of the cell jumps up to [INAUDIBLE] when 1093 01:00:42,330 --> 01:00:45,690 we turn on the potassium conductance, 1094 01:00:45,690 --> 01:00:47,700 and we turn off the sodium conductance, 1095 01:00:47,700 --> 01:00:51,360 and the voltage gets dragged down to EK. 1096 01:00:51,360 --> 01:00:53,520 Then when we turn off the potassium conductance, 1097 01:00:53,520 --> 01:00:57,580 the voltage relaxes back up to [INAUDIBLE].. 1098 01:00:57,580 --> 01:01:03,590 So look, conductances are just knobs 1099 01:01:03,590 --> 01:01:05,900 that allow the cell to control its voltage-- 1100 01:01:08,680 --> 01:01:13,240 an anthropomorphic way to put it. 1101 01:01:13,240 --> 01:01:16,080 You can control the voltage of the cell 1102 01:01:16,080 --> 01:01:21,440 just by setting the values of those resistors. 1103 01:01:21,440 --> 01:01:25,970 If you make this resistor really small and that resistor big, 1104 01:01:25,970 --> 01:01:29,900 the voltage jumps up toward [INAUDIBLE].. 1105 01:01:29,900 --> 01:01:30,420 Why is that? 1106 01:01:30,420 --> 01:01:32,420 Because you're connecting the inside of the cell 1107 01:01:32,420 --> 01:01:35,310 to that battery. 1108 01:01:35,310 --> 01:01:38,670 Turn that off, and then turn on this conductance, 1109 01:01:38,670 --> 01:01:41,270 or setting that resistor to be very small, 1110 01:01:41,270 --> 01:01:43,530 now you're connecting the inside of your cell 1111 01:01:43,530 --> 01:01:47,280 to this battery, which is negative, down here. 1112 01:01:47,280 --> 01:01:50,010 So you can just control the voltage of the cell up and down 1113 01:01:50,010 --> 01:01:51,765 just by twiddling these knobs. 1114 01:01:54,370 --> 01:01:56,350 Another way to think about it is this. 1115 01:01:56,350 --> 01:02:02,810 If this resistor is big, and if you set that resistor 1116 01:02:02,810 --> 01:02:08,810 to be really small, then the voltage here dominates. 1117 01:02:14,750 --> 01:02:18,150 So over the timescale of any process that we're considering 1118 01:02:18,150 --> 01:02:21,300 here, the ionic concentrations inside the cell and outside 1119 01:02:21,300 --> 01:02:24,566 of the cell don't change. 1120 01:02:24,566 --> 01:02:27,036 Does that make sense? 1121 01:02:31,760 --> 01:02:32,260 Yes? 1122 01:02:32,260 --> 01:02:34,620 AUDIENCE: So if we set both of them? 1123 01:02:34,620 --> 01:02:36,590 MICHALE FEE: So you set both of these to 0? 1124 01:02:36,590 --> 01:02:37,257 AUDIENCE: Not 0. 1125 01:02:37,257 --> 01:02:38,810 [INAUDIBLE] 1126 01:02:38,810 --> 01:02:39,560 MICHALE FEE: Yeah. 1127 01:02:39,560 --> 01:02:40,252 AUDIENCE: [INAUDIBLE] 1128 01:02:40,252 --> 01:02:42,230 MICHALE FEE: If the conductance is really high, 1129 01:02:42,230 --> 01:02:44,390 that means setting the resistors really small. 1130 01:02:44,390 --> 01:02:45,140 AUDIENCE: Exactly. 1131 01:02:45,140 --> 01:02:46,682 MICHALE FEE: That's a great question. 1132 01:02:46,682 --> 01:02:47,580 What happens? 1133 01:02:47,580 --> 01:02:48,725 AUDIENCE: [INAUDIBLE] 1134 01:02:48,725 --> 01:02:49,600 MICHALE FEE: Exactly. 1135 01:02:49,600 --> 01:02:50,680 AUDIENCE: Isn't that like-- 1136 01:02:50,680 --> 01:02:52,097 MICHALE FEE: That's exactly right. 1137 01:02:52,097 --> 01:02:53,800 If you set both of these, if you turn 1138 01:02:53,800 --> 01:02:56,080 on both of these conductances, then the voltage 1139 01:02:56,080 --> 01:02:58,480 goes somewhere in the middle. 1140 01:02:58,480 --> 01:03:04,060 And the voltage that it goes to is actually-- 1141 01:03:04,060 --> 01:03:05,170 you have to [AUDIO OUT]. 1142 01:03:05,170 --> 01:03:09,970 The Nernst potential that we calculated 1143 01:03:09,970 --> 01:03:14,530 is calculated only for the case where you have one ion. 1144 01:03:14,530 --> 01:03:16,960 If you want to calculate the equilibrium potential when 1145 01:03:16,960 --> 01:03:20,530 you have multiple pores open, then 1146 01:03:20,530 --> 01:03:25,210 you have to use a different method of calculating. 1147 01:03:25,210 --> 01:03:27,790 You have to actually calculate the currents flowing 1148 01:03:27,790 --> 01:03:31,780 in each one of those channels, and based 1149 01:03:31,780 --> 01:03:33,880 on the permeabilities of the channels, 1150 01:03:33,880 --> 01:03:37,030 and you get a different expression, 1151 01:03:37,030 --> 01:03:42,470 called the Goldman-Hodgkin-Katz equation. 1152 01:03:42,470 --> 01:03:44,470 But the bottom line is, if you open up 1153 01:03:44,470 --> 01:03:47,783 both of those conductances, it's somewhere in the middle. 1154 01:03:47,783 --> 01:03:49,700 And if you want to get the exact right answer, 1155 01:03:49,700 --> 01:03:51,250 you have to use the GHK equation. 1156 01:03:54,510 --> 01:03:55,010 Yes? 1157 01:03:55,010 --> 01:04:01,500 AUDIENCE: [INAUDIBLE] 1158 01:04:01,500 --> 01:04:02,370 MICHALE FEE: Almost. 1159 01:04:02,370 --> 01:04:03,610 We're not quite there yet. 1160 01:04:03,610 --> 01:04:07,020 We're going to get there. 1161 01:04:07,020 --> 01:04:09,080 But you can see what happens here 1162 01:04:09,080 --> 01:04:13,440 is that these conductances are actually voltage-dependent. 1163 01:04:13,440 --> 01:04:16,220 So for example, the sodium conductance 1164 01:04:16,220 --> 01:04:18,680 turns on at higher voltages. 1165 01:04:18,680 --> 01:04:21,740 When this conductance starts getting bigger, 1166 01:04:21,740 --> 01:04:24,320 what happens to the voltage in the cell? 1167 01:04:24,320 --> 01:04:26,120 It starts going up, right? 1168 01:04:26,120 --> 01:04:28,130 But if this is voltage-dependent and it 1169 01:04:28,130 --> 01:04:30,020 turns on at a higher voltage, what 1170 01:04:30,020 --> 01:04:32,900 happens to the conductance? 1171 01:04:32,900 --> 01:04:34,700 When the voltage gets a little bit bigger, 1172 01:04:34,700 --> 01:04:38,780 what happens to the conductance, which does what to the voltage? 1173 01:04:38,780 --> 01:04:41,810 Makes it go up even faster, right? 1174 01:04:41,810 --> 01:04:43,580 And so you get this runaway process 1175 01:04:43,580 --> 01:04:48,230 where the disconductance turns on very quickly 1176 01:04:48,230 --> 01:04:49,420 and the voltage jumps up. 1177 01:04:51,950 --> 01:04:54,560 That's the essence of the action potential. 1178 01:04:54,560 --> 01:04:56,540 But the essential picture that I wanted 1179 01:04:56,540 --> 01:05:03,590 you to get from this slide is that these are just knobs. 1180 01:05:03,590 --> 01:05:05,750 And when you turn the knobs, you can 1181 01:05:05,750 --> 01:05:08,810 change the volt-- that controls the voltage in the cell. 1182 01:05:11,660 --> 01:05:14,660 Each one of these is causing the cell 1183 01:05:14,660 --> 01:05:19,310 to be dragged toward its reversal potential. 1184 01:05:19,310 --> 01:05:25,310 So if this conductance is big, this conductance is big, 1185 01:05:25,310 --> 01:05:28,470 the voltage in the cell gets dragged toward ENA. 1186 01:05:28,470 --> 01:05:30,940 If the conductance is big, the voltage in the cell 1187 01:05:30,940 --> 01:05:32,262 gets dragged toward EK. 1188 01:05:35,760 --> 01:05:45,520 And next time, we're going to go through the process. 1189 01:05:45,520 --> 01:05:47,110 We're going to describe the process 1190 01:05:47,110 --> 01:05:50,830 to drive the voltage dependence of those ion 1191 01:05:50,830 --> 01:05:53,770 channels, the sodium and potassium ion channels, 1192 01:05:53,770 --> 01:05:56,530 to derive the voltage dependence and the time dependence that 1193 01:05:56,530 --> 01:06:02,470 explains how you get this action potential in a neuron. 1194 01:06:02,470 --> 01:06:05,190 So that's next Tuesday.