1 00:00:13,390 --> 00:00:16,840 MICHALE FEE: So today, we're going to start a new topic. 2 00:00:16,840 --> 00:00:21,740 We're going to be talking about the propagation of signals 3 00:00:21,740 --> 00:00:24,610 in dendrites and axons. 4 00:00:24,610 --> 00:00:27,790 So the model that we've considered so far 5 00:00:27,790 --> 00:00:29,430 is just a soma. 6 00:00:29,430 --> 00:00:36,280 We basically had a kind of a spherical shell of insulator 7 00:00:36,280 --> 00:00:39,160 that we've been modeling that has different kinds of ion 8 00:00:39,160 --> 00:00:42,280 channels in it that allow the cell to do things like generate 9 00:00:42,280 --> 00:00:44,630 an action potential. 10 00:00:44,630 --> 00:00:46,660 So the reason that we've been doing 11 00:00:46,660 --> 00:00:50,590 that is because in most vertebrate neurons, 12 00:00:50,590 --> 00:00:54,580 the soma is the sight in the neuron at which the decision 13 00:00:54,580 --> 00:00:56,470 to make an action potential is made. 14 00:00:56,470 --> 00:01:00,880 So all kinds of inputs come in, and then the soma 15 00:01:00,880 --> 00:01:03,790 integrates those inputs, accumulates charge, 16 00:01:03,790 --> 00:01:07,660 reaches some spiking threshold, and then generates 17 00:01:07,660 --> 00:01:10,080 an action potential. 18 00:01:10,080 --> 00:01:11,870 And so that's where the decision is about 19 00:01:11,870 --> 00:01:14,250 whether a neuron is going to spike or not. 20 00:01:14,250 --> 00:01:17,930 Now, in real neurons, relatively few of the inputs 21 00:01:17,930 --> 00:01:19,490 actually come onto the soma. 22 00:01:19,490 --> 00:01:23,000 Most of the synaptic inputs, most of the inputs 23 00:01:23,000 --> 00:01:27,350 arrive onto the dendrites, which are these branching cylinders 24 00:01:27,350 --> 00:01:29,000 of cell membrane. 25 00:01:29,000 --> 00:01:30,830 And most of the synapses actually 26 00:01:30,830 --> 00:01:34,240 form onto the dendrite at some distance from the soma. 27 00:01:34,240 --> 00:01:36,770 There are synapses that form onto the soma. 28 00:01:36,770 --> 00:01:38,810 But the vast majority of synapses 29 00:01:38,810 --> 00:01:41,180 form onto these dendrites. 30 00:01:41,180 --> 00:01:43,220 And sometimes, those synapses can 31 00:01:43,220 --> 00:01:45,590 be as far away as 1 or 2 millimeters for very 32 00:01:45,590 --> 00:01:49,350 large neurons in cortex. 33 00:01:49,350 --> 00:01:54,960 So there's a population of neurons in deep layer V 34 00:01:54,960 --> 00:01:57,300 some of you may have heard about that have dendrites 35 00:01:57,300 --> 00:01:59,700 that reach all the way up into layer I. 36 00:01:59,700 --> 00:02:02,280 And those cells can be-- 37 00:02:02,280 --> 00:02:06,660 those dendrites can be as long as a couple of millimeters. 38 00:02:06,660 --> 00:02:10,259 So we really have to think about what this means, 39 00:02:10,259 --> 00:02:15,960 how signals get from out here in the dendrite down to the soma. 40 00:02:15,960 --> 00:02:20,890 And that's what we're going to talk about today. 41 00:02:20,890 --> 00:02:23,310 So the most important thing that we're going to do 42 00:02:23,310 --> 00:02:27,180 is to simplify this-- by the way, 43 00:02:27,180 --> 00:02:29,575 anybody know what kind of cell this is? 44 00:02:29,575 --> 00:02:30,465 AUDIENCE: [INAUDIBLE] 45 00:02:30,465 --> 00:02:31,215 MICHALE FEE: Good. 46 00:02:31,215 --> 00:02:32,920 It's a Purkinje cell. 47 00:02:32,920 --> 00:02:35,590 And it was one of the-- 48 00:02:35,590 --> 00:02:39,190 this is one of the cells that Ramon Cajal drew back 49 00:02:39,190 --> 00:02:42,310 in the late 1800s. 50 00:02:42,310 --> 00:02:44,950 So the most important thing we're going to do 51 00:02:44,950 --> 00:02:50,170 is to simplify this very complex dendritic arborization. 52 00:02:50,170 --> 00:02:52,630 And we're going to basically think 53 00:02:52,630 --> 00:02:54,830 of it as a single cylinder. 54 00:02:54,830 --> 00:03:00,070 Now mathematically, there are reasons why this is not 55 00:03:00,070 --> 00:03:03,550 actually unreasonable. 56 00:03:03,550 --> 00:03:05,650 You can write down-- if you analyze 57 00:03:05,650 --> 00:03:07,600 the structure of dendritic trees, 58 00:03:07,600 --> 00:03:10,120 there is something about the way the ratio 59 00:03:10,120 --> 00:03:13,300 of the diameters of the different dendrites 60 00:03:13,300 --> 00:03:15,430 as they converge to form thicker branches 61 00:03:15,430 --> 00:03:17,290 as you get closer and closer to closer 62 00:03:17,290 --> 00:03:20,500 to the soma that mathematically makes 63 00:03:20,500 --> 00:03:26,800 this not a bad approximation for an extended dendritic arbor 64 00:03:26,800 --> 00:03:28,430 like this. 65 00:03:28,430 --> 00:03:30,850 So we're going to think about the problem of having 66 00:03:30,850 --> 00:03:35,170 a synapse out here on this cylindrical approximation 67 00:03:35,170 --> 00:03:35,980 to a dendrite. 68 00:03:35,980 --> 00:03:38,500 And we're going to imagine that we're 69 00:03:38,500 --> 00:03:41,140 measuring the voltage down here at the soma 70 00:03:41,140 --> 00:03:44,260 or at different positions along the dendrite. 71 00:03:44,260 --> 00:03:49,630 And we're going to ask, how does synaptic input out here 72 00:03:49,630 --> 00:03:54,025 on this cylinder affect the membrane potential 73 00:03:54,025 --> 00:03:56,200 in the dendrite and down here at the soma? 74 00:03:59,280 --> 00:04:04,700 And the basic conceptual picture that you should have 75 00:04:04,700 --> 00:04:08,770 is that those signals propagate some distance down the soma 76 00:04:08,770 --> 00:04:11,070 but gradually leak out. 77 00:04:11,070 --> 00:04:14,790 And there's a very simple kind of intuitive picture, which 78 00:04:14,790 --> 00:04:20,190 is that the dendrite you can think of this as a leaky 79 00:04:20,190 --> 00:04:21,540 pipe or a leaky hose. 80 00:04:21,540 --> 00:04:26,070 So imagine you took a piece of garden hose 81 00:04:26,070 --> 00:04:30,660 and you poked holes in the side of it 82 00:04:30,660 --> 00:04:35,310 or so that they're kind of close together. 83 00:04:35,310 --> 00:04:37,810 And when you hook this up to the water faucet, 84 00:04:37,810 --> 00:04:40,710 you turn the water on, that some of that water 85 00:04:40,710 --> 00:04:42,160 flows down the hose. 86 00:04:42,160 --> 00:04:46,410 But some of it also leaks out through the holes 87 00:04:46,410 --> 00:04:47,710 that you drilled. 88 00:04:47,710 --> 00:04:50,580 And you can see that eventually the water is all 89 00:04:50,580 --> 00:04:52,170 going to leak out through the sides, 90 00:04:52,170 --> 00:04:54,630 and it's not going to go all the way down to the other end 91 00:04:54,630 --> 00:04:56,820 to get to your hydrangeas or whatever 92 00:04:56,820 --> 00:04:58,830 it is that you're watering. 93 00:04:58,830 --> 00:05:01,620 And so you can see that that signal isn't 94 00:05:01,620 --> 00:05:06,900 going to get very far if the holes you drilled 95 00:05:06,900 --> 00:05:09,710 are big enough. 96 00:05:09,710 --> 00:05:12,170 And the general kind of analogy here 97 00:05:12,170 --> 00:05:15,920 is that current is like the flow of the water. 98 00:05:15,920 --> 00:05:18,800 Electrical current here is like water current flowing down 99 00:05:18,800 --> 00:05:19,730 the pipe. 100 00:05:19,730 --> 00:05:21,630 And voltage is like pressure. 101 00:05:21,630 --> 00:05:26,270 So the higher the pressure here, the higher the current flow 102 00:05:26,270 --> 00:05:26,870 you'll get. 103 00:05:29,640 --> 00:05:32,370 And we're going to develop an electrical circuit 104 00:05:32,370 --> 00:05:36,030 model for a dendrite like this that's 105 00:05:36,030 --> 00:05:41,460 going to look like a set of resistors going down 106 00:05:41,460 --> 00:05:44,130 the axis of the dendrite and a set of resistors 107 00:05:44,130 --> 00:05:47,050 that go across the membrane. 108 00:05:47,050 --> 00:05:49,980 And you can see that each little piece of membrane 109 00:05:49,980 --> 00:05:51,810 here, a little piece of the dendrite 110 00:05:51,810 --> 00:05:56,430 is going to look like a resistor divider, where you have 111 00:05:56,430 --> 00:06:00,060 a resistor along the axial direction 112 00:06:00,060 --> 00:06:02,160 and a resistance across the membrane. 113 00:06:02,160 --> 00:06:05,880 And as you make a longer and longer piece of dendrite, 114 00:06:05,880 --> 00:06:08,310 you're going to get additional voltage dividers. 115 00:06:08,310 --> 00:06:11,730 Each voltage divider divides the voltage 116 00:06:11,730 --> 00:06:14,520 by some constant factor. 117 00:06:14,520 --> 00:06:16,290 And as you stack those things up, 118 00:06:16,290 --> 00:06:20,610 the voltage drops by some constant factor per unit length 119 00:06:20,610 --> 00:06:21,450 of the dendrite. 120 00:06:21,450 --> 00:06:22,980 And so you can see-- anybody want 121 00:06:22,980 --> 00:06:26,340 to just take a guess of what kind of functional form 122 00:06:26,340 --> 00:06:28,230 that would give you if you divide 123 00:06:28,230 --> 00:06:30,450 the voltage by some constant factor 124 00:06:30,450 --> 00:06:33,013 each unit length of the dendrite? 125 00:06:33,013 --> 00:06:33,930 AUDIENCE: Exponential. 126 00:06:33,930 --> 00:06:34,972 MICHALE FEE: Exponential. 127 00:06:34,972 --> 00:06:36,640 That's right. 128 00:06:36,640 --> 00:06:40,270 And that's where this exponential falloff comes from. 129 00:06:43,950 --> 00:06:48,110 So today, we're going to do the following things. 130 00:06:48,110 --> 00:06:50,810 And we're going to basically draw 131 00:06:50,810 --> 00:06:54,020 a circuit diagram, an electrical equivalent 132 00:06:54,020 --> 00:06:56,030 circuit of a piece of dendrite. 133 00:06:56,030 --> 00:06:59,510 And I would like you to be able to make that drawing if you're 134 00:06:59,510 --> 00:07:01,040 asked to. 135 00:07:01,040 --> 00:07:03,650 We're going to be able to plot the voltage 136 00:07:03,650 --> 00:07:06,140 in a piece of dendrite as a function of distance 137 00:07:06,140 --> 00:07:10,080 for the case of a dendrite that has leaky walls 138 00:07:10,080 --> 00:07:14,540 and for the case of a dendrite that has non-leaky walls. 139 00:07:14,540 --> 00:07:17,720 And we're going to describe the concept of a length 140 00:07:17,720 --> 00:07:20,450 constant, which I'll tell you right now is just 141 00:07:20,450 --> 00:07:27,490 the 1 over the distance at which the voltage falls by 1 over e 142 00:07:27,490 --> 00:07:28,490 as a function of length. 143 00:07:28,490 --> 00:07:31,580 So it's some length over which the voltage falls 144 00:07:31,580 --> 00:07:35,600 by some amount 1 over e. 145 00:07:35,600 --> 00:07:38,250 We're going to go over how that length constant depends 146 00:07:38,250 --> 00:07:42,270 on the radius of the dendrite. 147 00:07:42,270 --> 00:07:45,180 It's a function of the size. 148 00:07:45,180 --> 00:07:47,370 And also, we're going to describe the concept 149 00:07:47,370 --> 00:07:49,290 of an electrotonic length. 150 00:07:49,290 --> 00:07:50,820 And then finally, we're going to go 151 00:07:50,820 --> 00:07:53,880 to some sort of extreme simplifications, 152 00:07:53,880 --> 00:07:57,090 even beyond taking that very complex dendrite, 153 00:07:57,090 --> 00:07:59,010 simplifying it as a cylinder. 154 00:07:59,010 --> 00:08:01,320 We're going to go to an even simpler case 155 00:08:01,320 --> 00:08:05,130 where we can just treat the cell as a soma connected 156 00:08:05,130 --> 00:08:07,620 to a resistor connected by a resistor 157 00:08:07,620 --> 00:08:10,350 to a separate compartment. 158 00:08:10,350 --> 00:08:12,900 And that's sort of the most extreme simplification 159 00:08:12,900 --> 00:08:13,800 of a dendrite. 160 00:08:13,800 --> 00:08:17,520 But, in fact, it's an extremely powerful one 161 00:08:17,520 --> 00:08:20,100 from which you can get a lot of intuition 162 00:08:20,100 --> 00:08:23,910 about how signals are integrated in dendrites. 163 00:08:26,510 --> 00:08:30,470 So we're going to analyze a piece of dendrite 164 00:08:30,470 --> 00:08:33,980 using a technique called finite element analysis. 165 00:08:33,980 --> 00:08:35,480 We're going to imagine-- we're going 166 00:08:35,480 --> 00:08:38,270 to approximate our piece of dendrite as a cylinder 167 00:08:38,270 --> 00:08:42,770 of constant radius a, an axial dimension that we're 168 00:08:42,770 --> 00:08:45,230 going to label x. 169 00:08:45,230 --> 00:08:48,590 We're going to break up this cylinder into little slices. 170 00:08:48,590 --> 00:08:50,720 So imagine we just took a little knife, 171 00:08:50,720 --> 00:08:55,220 and we cut little slices of this dendrite. 172 00:08:55,220 --> 00:08:58,020 And they're going to be very small slices. 173 00:08:58,020 --> 00:09:00,290 And we're going to model each one of those slices 174 00:09:00,290 --> 00:09:02,153 with a separate little circuit. 175 00:09:02,153 --> 00:09:04,070 And then we're going to connect them together. 176 00:09:04,070 --> 00:09:05,778 And we're going to let the length of that 177 00:09:05,778 --> 00:09:07,010 slice be delta x. 178 00:09:07,010 --> 00:09:10,640 And then eventually, we're going to let delta x go to 0. 179 00:09:10,640 --> 00:09:12,950 We're going to get some differential equations that 180 00:09:12,950 --> 00:09:15,740 describe that relationship between the voltage 181 00:09:15,740 --> 00:09:18,510 and the current in this piece of dendrite. 182 00:09:21,250 --> 00:09:27,290 So let's start with a model for the inside of this cylinder. 183 00:09:27,290 --> 00:09:31,540 So remember, in a cell, we had the inside 184 00:09:31,540 --> 00:09:34,720 of the cell modeled by a wire. 185 00:09:34,720 --> 00:09:38,650 In a dendrite, we can't just use a wire. 186 00:09:38,650 --> 00:09:40,960 And the reason is that current is going to flow along 187 00:09:40,960 --> 00:09:42,663 the inside of the dendrite. 188 00:09:42,663 --> 00:09:44,080 It's going to flow, and it's going 189 00:09:44,080 --> 00:09:45,850 to experience voltage drops. 190 00:09:45,850 --> 00:09:50,020 So we have to actually model the resistance 191 00:09:50,020 --> 00:09:51,550 of the inside of the dendrite. 192 00:09:51,550 --> 00:09:56,290 And we're going to model it like the resistance 193 00:09:56,290 --> 00:10:00,060 between each one of those slices with a resistor value little r. 194 00:10:02,650 --> 00:10:07,390 We're going to model the outside of the axon or the dendrite 195 00:10:07,390 --> 00:10:08,140 as a wire. 196 00:10:08,140 --> 00:10:11,170 And the reason we're going to put resistors inside and just 197 00:10:11,170 --> 00:10:15,700 a wire outside is because the resistance-- remember the axon 198 00:10:15,700 --> 00:10:17,260 or dendrite is very small. 199 00:10:17,260 --> 00:10:20,110 In the brain, dendrites might be about 2 microns across. 200 00:10:20,110 --> 00:10:23,410 So the current is constrained to a very small space. 201 00:10:23,410 --> 00:10:25,840 When currents then flow outside, they're 202 00:10:25,840 --> 00:10:30,280 flowing in a much larger volume, and so the effective resistance 203 00:10:30,280 --> 00:10:31,760 is much smaller. 204 00:10:31,760 --> 00:10:34,270 And we're going to essentially ignore that resistance 205 00:10:34,270 --> 00:10:36,760 and treat the outside as just a wire. 206 00:10:39,560 --> 00:10:43,790 Now we have to model the membrane. 207 00:10:43,790 --> 00:10:46,730 Anybody want to take a guess how we're 208 00:10:46,730 --> 00:10:48,125 going to model the membrane? 209 00:10:50,635 --> 00:10:51,990 AUDIENCE: [INAUDIBLE] 210 00:10:51,990 --> 00:10:53,500 MICHALE FEE: What's that? 211 00:10:53,500 --> 00:10:56,470 I heard two correct answers. 212 00:10:56,470 --> 00:10:57,830 What did you say Jasmine? 213 00:10:57,830 --> 00:10:58,830 AUDIENCE: The capacitor. 214 00:10:58,830 --> 00:10:59,490 MICHALE FEE: Capacitor. 215 00:10:59,490 --> 00:11:00,160 And? 216 00:11:00,160 --> 00:11:03,750 AUDIENCE: [INAUDIBLE] 217 00:11:03,750 --> 00:11:04,720 MICHALE FEE: Excellent. 218 00:11:04,720 --> 00:11:05,220 Whoops. 219 00:11:05,220 --> 00:11:06,435 I wasn't quite there. 220 00:11:06,435 --> 00:11:07,940 Let's put that up. 221 00:11:07,940 --> 00:11:08,440 Good. 222 00:11:08,440 --> 00:11:10,200 So we're going to have a capacitance. 223 00:11:10,200 --> 00:11:12,360 We're going to imagine that this membrane might 224 00:11:12,360 --> 00:11:16,390 have an ion selective ion channel with some conductance G 225 00:11:16,390 --> 00:11:21,760 sub l and an equilibrium or reversal potential E sub l. 226 00:11:21,760 --> 00:11:25,470 Now coming back to these terms here, we're going to model. 227 00:11:25,470 --> 00:11:27,660 We're going to write down the voltage 228 00:11:27,660 --> 00:11:31,110 in each one of our little slices of the dendrite. 229 00:11:31,110 --> 00:11:33,270 So let's do that. 230 00:11:33,270 --> 00:11:37,740 Let's just pick one of them as V, the voltage, 231 00:11:37,740 --> 00:11:41,732 at position, x, and time t. 232 00:11:41,732 --> 00:11:44,800 The voltage in the next slides over 233 00:11:44,800 --> 00:11:50,710 is going to be V at x plus delta x of t. 234 00:11:50,710 --> 00:11:54,760 And the voltage in this slice over here is V of x minus delta 235 00:11:54,760 --> 00:11:57,400 x and t. 236 00:11:57,400 --> 00:12:01,450 So now, we can also write down the current that goes axially 237 00:12:01,450 --> 00:12:03,760 through that piece of-- 238 00:12:03,760 --> 00:12:05,140 that slice of our dendrite. 239 00:12:05,140 --> 00:12:09,070 We're going to call that I of x and t. 240 00:12:09,070 --> 00:12:12,420 And we can write down also the current in every other time-- 241 00:12:12,420 --> 00:12:14,950 in every other slice of the dendrite, I 242 00:12:14,950 --> 00:12:16,930 of x minus delta x and t. 243 00:12:21,090 --> 00:12:25,700 And we're going to model this piece of membrane 244 00:12:25,700 --> 00:12:30,290 in each one of those slices as well. 245 00:12:30,290 --> 00:12:31,400 Any questions about that? 246 00:12:31,400 --> 00:12:32,690 That's the basic setup. 247 00:12:32,690 --> 00:12:39,400 That's the basic finite element model of a dendrite. 248 00:12:39,400 --> 00:12:40,450 No questions? 249 00:12:43,660 --> 00:12:49,590 Now we also have to model the current through the membrane. 250 00:12:49,590 --> 00:12:54,300 That's going to be I sub m, m for membrane. 251 00:12:54,300 --> 00:12:56,730 And it's going to be a current per unit 252 00:12:56,730 --> 00:12:59,490 length of the dendrite. 253 00:12:59,490 --> 00:13:01,640 We're going to imagine that there's current 254 00:13:01,640 --> 00:13:05,180 flowing from the inside to the outside through the membrane. 255 00:13:05,180 --> 00:13:08,180 And there's going to be some current per unit 256 00:13:08,180 --> 00:13:09,285 length of the dendrite. 257 00:13:11,790 --> 00:13:15,630 And we can also imagine that we have current being injected, 258 00:13:15,630 --> 00:13:18,660 let's say, through a synapse or through an electrode 259 00:13:18,660 --> 00:13:22,620 that we can also model as coming in at any position x. 260 00:13:22,620 --> 00:13:28,125 And this is, again, current per unit length times delta x. 261 00:13:28,125 --> 00:13:29,000 Does that make sense? 262 00:13:36,910 --> 00:13:39,880 So the first thing we're going to do 263 00:13:39,880 --> 00:13:48,685 is we're going to write down the relation between V 264 00:13:48,685 --> 00:13:53,260 in each node and the current going through that node. 265 00:13:53,260 --> 00:13:54,710 So let's do that. 266 00:13:54,710 --> 00:13:56,440 We're going to use Ohm's law. 267 00:13:56,440 --> 00:14:00,130 So the voltage difference between here and here 268 00:14:00,130 --> 00:14:05,235 is just going to be the current times that resistance. 269 00:14:05,235 --> 00:14:06,110 Does that make sense? 270 00:14:06,110 --> 00:14:07,910 We're just going to use Ohm's law-- 271 00:14:07,910 --> 00:14:09,990 very simple. 272 00:14:09,990 --> 00:14:19,870 So V of x and t minus V of x plus delta x, t 273 00:14:19,870 --> 00:14:23,830 is just equal to little r times that current. 274 00:14:26,880 --> 00:14:28,560 And now we're going to rewrite this. 275 00:14:28,560 --> 00:14:34,090 Let's divide both sides of this equation by delta x. 276 00:14:34,090 --> 00:14:38,530 So you see 1 over delta x V of x minus V of x plus delta 277 00:14:38,530 --> 00:14:43,300 x is equal to little r over delta x times the current. 278 00:14:43,300 --> 00:14:48,540 And can anyone tell me what that thing is as delta x goes to 0? 279 00:14:48,540 --> 00:14:49,440 AUDIENCE: [INAUDIBLE] 280 00:14:49,440 --> 00:14:50,190 MICHALE FEE: Good. 281 00:14:50,190 --> 00:14:53,430 It's the derivative of-- it's the spatial derivative 282 00:14:53,430 --> 00:14:56,480 of the voltage. 283 00:14:56,480 --> 00:14:58,910 That's just the definition of derivative 284 00:14:58,910 --> 00:15:00,650 when delta x goes to 0. 285 00:15:00,650 --> 00:15:03,020 So let's write that out. 286 00:15:03,020 --> 00:15:05,610 Notice that it's the negative of the derivative 287 00:15:05,610 --> 00:15:08,420 because the derivative would have V 288 00:15:08,420 --> 00:15:10,490 of x plus delta x minus V of x. 289 00:15:10,490 --> 00:15:12,730 So it's a negative of the derivative. 290 00:15:12,730 --> 00:15:18,660 So negative dv/dx is equal to some resistance 291 00:15:18,660 --> 00:15:19,750 times the current. 292 00:15:19,750 --> 00:15:23,010 And notice that this capital R sub a 293 00:15:23,010 --> 00:15:26,430 is called the axial resistance per unit length. 294 00:15:26,430 --> 00:15:33,060 It's this resistance per unit length of the dendrite. 295 00:15:33,060 --> 00:15:38,760 Now notice that if you pass current down that dendrite, 296 00:15:38,760 --> 00:15:42,540 the voltage drop is going to keep increasing. 297 00:15:42,540 --> 00:15:45,060 The resistance is going to keep increasing the longer 298 00:15:45,060 --> 00:15:46,710 that piece of dendrite is. 299 00:15:46,710 --> 00:15:48,750 So you can think about resistance 300 00:15:48,750 --> 00:15:50,970 in a piece of dendrite more appropriately 301 00:15:50,970 --> 00:15:55,620 as resistance per unit length. 302 00:15:55,620 --> 00:15:58,530 So there's Ohm's law-- 303 00:15:58,530 --> 00:16:03,060 minus dv/dx equals axial resistance per unit length 304 00:16:03,060 --> 00:16:03,810 times the current. 305 00:16:08,160 --> 00:16:09,298 Any questions? 306 00:16:14,020 --> 00:16:16,370 And notice that according to this, 307 00:16:16,370 --> 00:16:19,910 current flow to the right, positive I 308 00:16:19,910 --> 00:16:21,650 is defined as current to the right here 309 00:16:21,650 --> 00:16:24,650 produces a negative gradient in the voltage. 310 00:16:24,650 --> 00:16:28,280 So the voltage is high on this side and low on that side. 311 00:16:28,280 --> 00:16:30,028 So the slope is negative. 312 00:16:33,900 --> 00:16:36,360 So now let's take this, and let's 313 00:16:36,360 --> 00:16:40,080 analyze this for some simple cases where 314 00:16:40,080 --> 00:16:42,540 we have no membrane current. 315 00:16:42,540 --> 00:16:44,880 So we're going to just ignore those. 316 00:16:44,880 --> 00:16:48,380 And we're just going to include these axial resistances. 317 00:16:48,380 --> 00:16:51,000 And we're going to analyze what this equation tells us 318 00:16:51,000 --> 00:16:53,572 about the voltage inside of the dendrite. 319 00:16:53,572 --> 00:16:56,160 Does that makes sense? 320 00:16:56,160 --> 00:16:57,330 So let's do that. 321 00:16:57,330 --> 00:17:03,290 So if we take that equation, we can write down the current 322 00:17:03,290 --> 00:17:05,390 at, let's say, these two different nodes-- 323 00:17:05,390 --> 00:17:08,450 I of x minus delta x and I of x. 324 00:17:08,450 --> 00:17:12,560 And because there are no membrane currents, 325 00:17:12,560 --> 00:17:14,180 you can see that those two currents 326 00:17:14,180 --> 00:17:16,369 have to be equal to each other. 327 00:17:16,369 --> 00:17:19,670 Kirchoff's Current Law says that the current into this node 328 00:17:19,670 --> 00:17:22,010 has to equal the current out of that node. 329 00:17:22,010 --> 00:17:24,619 So if there are no membrane currents, 330 00:17:24,619 --> 00:17:26,420 there's nothing leaking out here, 331 00:17:26,420 --> 00:17:30,910 then those two currents have to equal each other. 332 00:17:30,910 --> 00:17:32,350 And we can call that I0. 333 00:17:35,990 --> 00:17:40,670 So now, dv/dx is minus axial resistance times I0. 334 00:17:40,670 --> 00:17:43,820 And what does that tell us about how the voltage changes 335 00:17:43,820 --> 00:17:48,080 in a piece of dendrite if there's no membrane current, 336 00:17:48,080 --> 00:17:50,612 if there's no leaky membrane? 337 00:17:50,612 --> 00:17:52,070 There's no leakage in the membrane. 338 00:17:55,620 --> 00:17:58,730 If dv/dx is a constant, what does it tell us? 339 00:18:06,005 --> 00:18:07,510 AUDIENCE: [INAUDIBLE]. 340 00:18:07,510 --> 00:18:08,260 MICHALE FEE: Yeah. 341 00:18:08,260 --> 00:18:09,520 But decreases how? 342 00:18:09,520 --> 00:18:10,780 What functional form? 343 00:18:10,780 --> 00:18:11,200 AUDIENCE: [INAUDIBLE] 344 00:18:11,200 --> 00:18:11,720 MICHALE FEE: Good. 345 00:18:11,720 --> 00:18:12,553 It changes linearly. 346 00:18:15,220 --> 00:18:17,410 So if there are no membrane conductances, 347 00:18:17,410 --> 00:18:19,585 then the membrane potential changes linearly. 348 00:18:23,760 --> 00:18:26,660 So you can see that the voltage as a function of position-- 349 00:18:26,660 --> 00:18:28,700 sorry I forgot to label that voltage-- 350 00:18:28,700 --> 00:18:31,790 just changes linearly from some initial voltage 351 00:18:31,790 --> 00:18:35,960 to some final voltage over some length l. 352 00:18:35,960 --> 00:18:41,710 We're considering a case of a piece of dendrite of length l. 353 00:18:41,710 --> 00:18:42,476 Yes? 354 00:18:42,476 --> 00:18:47,592 AUDIENCE: [INAUDIBLE] 355 00:18:47,592 --> 00:18:48,300 MICHALE FEE: Yep. 356 00:18:48,300 --> 00:18:50,460 So I just rewrote this equation. 357 00:18:50,460 --> 00:18:53,880 Sorry, I just rewrote this equation moving the minus sign 358 00:18:53,880 --> 00:18:56,210 to that side. 359 00:18:56,210 --> 00:18:56,710 Yep. 360 00:19:01,130 --> 00:19:02,710 Good. 361 00:19:02,710 --> 00:19:07,810 Now you can see that the delta V that the voltage difference 362 00:19:07,810 --> 00:19:09,490 from the left side to the right side 363 00:19:09,490 --> 00:19:14,810 is just the total resistance times the current-- just Ohm's 364 00:19:14,810 --> 00:19:17,060 law again. 365 00:19:17,060 --> 00:19:20,030 And the total resistance is the axial resistance 366 00:19:20,030 --> 00:19:22,880 per unit length times the length. 367 00:19:22,880 --> 00:19:24,530 Really simple. 368 00:19:24,530 --> 00:19:27,130 Voltage changes linearly. 369 00:19:27,130 --> 00:19:30,640 If you don't have any membrane conductances, 370 00:19:30,640 --> 00:19:33,400 and you can just write down the relation 371 00:19:33,400 --> 00:19:35,500 between the voltage difference on the two 372 00:19:35,500 --> 00:19:37,060 sides and the current. 373 00:19:42,290 --> 00:19:44,630 So, in general, let's think a little bit more 374 00:19:44,630 --> 00:19:49,940 about this problem of being able to what you need to write down 375 00:19:49,940 --> 00:19:52,820 the solution to this equation. 376 00:19:52,820 --> 00:19:54,840 It's a very simple equation. 377 00:19:54,840 --> 00:19:56,420 If you integrate this over x, you 378 00:19:56,420 --> 00:19:58,940 can see that the voltage as a function of position 379 00:19:58,940 --> 00:20:03,320 is some initial voltage minus a resistance 380 00:20:03,320 --> 00:20:06,822 times the current times x. 381 00:20:06,822 --> 00:20:08,930 And that, again, just looks like this. 382 00:20:08,930 --> 00:20:11,540 That's where that solution came from. 383 00:20:11,540 --> 00:20:15,400 It's just integrating this over x. 384 00:20:15,400 --> 00:20:18,040 And you can see that in order to write down 385 00:20:18,040 --> 00:20:20,890 the solution to this equation, we need a couple of things. 386 00:20:20,890 --> 00:20:24,520 We need to either know the voltages at the beginning 387 00:20:24,520 --> 00:20:26,380 and end, or we need to know the current. 388 00:20:26,380 --> 00:20:30,040 We need to know some combination of those three things. 389 00:20:30,040 --> 00:20:32,670 So let's write down the voltage here. 390 00:20:32,670 --> 00:20:33,790 Let's call it V0. 391 00:20:33,790 --> 00:20:38,730 Let's write down the voltage there, V sub l, 392 00:20:38,730 --> 00:20:40,110 and plug those in. 393 00:20:40,110 --> 00:20:42,560 And you can see that-- 394 00:20:42,560 --> 00:20:43,540 there's V0. 395 00:20:43,540 --> 00:20:44,800 There's V sub l. 396 00:20:44,800 --> 00:20:47,600 You can see that if you know any two of those quantities-- 397 00:20:47,600 --> 00:20:50,860 V0, V sub l, or Io-- 398 00:20:50,860 --> 00:20:52,450 you can calculate the third. 399 00:20:52,450 --> 00:20:56,200 So if you know V0 and Vl, you can calculate the current. 400 00:20:56,200 --> 00:21:00,010 If you know V0 and the current, you can calculate V sub l. 401 00:21:00,010 --> 00:21:06,100 That is the concept of boundary conditions. 402 00:21:06,100 --> 00:21:10,750 You can write down the voltages or the currents 403 00:21:10,750 --> 00:21:13,570 at some positions on the dendrite 404 00:21:13,570 --> 00:21:18,370 and figure out the total solution 405 00:21:18,370 --> 00:21:21,220 to the voltage [AUDIO OUT] of position. 406 00:21:21,220 --> 00:21:23,140 Does that make sense? 407 00:21:23,140 --> 00:21:26,170 If you don't know some of those quantities, 408 00:21:26,170 --> 00:21:30,220 you can't write down the solution to the equation. 409 00:21:30,220 --> 00:21:32,530 It's just the simple idea that when 410 00:21:32,530 --> 00:21:34,420 you integrate a differential equation, 411 00:21:34,420 --> 00:21:37,000 you need to have an initial condition in order 412 00:21:37,000 --> 00:21:39,810 to actually solve the equation. 413 00:21:43,918 --> 00:21:44,960 Any questions about that? 414 00:21:47,830 --> 00:21:50,830 So let's think about a couple of different kinds of boundary 415 00:21:50,830 --> 00:21:54,200 conditions that you might encounter. 416 00:21:54,200 --> 00:21:57,670 So this boundary condition right here-- so let's 417 00:21:57,670 --> 00:22:01,750 say that we inject a x amount of current I0 418 00:22:01,750 --> 00:22:04,540 into a piece of dendrite. 419 00:22:04,540 --> 00:22:06,430 And we take that piece of dendrite 420 00:22:06,430 --> 00:22:08,110 and we inject current on one end, 421 00:22:08,110 --> 00:22:12,520 and we cut the other end so that it's open. 422 00:22:12,520 --> 00:22:15,860 What does that produce at the other end? 423 00:22:15,860 --> 00:22:18,923 So we have a wire that describes the inside of the dendrite. 424 00:22:18,923 --> 00:22:21,340 We have a wire that describes the outside of the dendrite. 425 00:22:21,340 --> 00:22:25,720 And if you cut the end of the dendrite off so that they're-- 426 00:22:25,720 --> 00:22:28,610 it's leaky-- so it's an open end-- 427 00:22:28,610 --> 00:22:31,734 what does that look like electrically? 428 00:22:31,734 --> 00:22:34,350 Like what's the word for-- 429 00:22:34,350 --> 00:22:36,770 like those two wires are touching each other. 430 00:22:36,770 --> 00:22:38,510 What's that called? 431 00:22:38,510 --> 00:22:40,650 They're shorts. 432 00:22:40,650 --> 00:22:42,310 If you cut the end of a dendrite off, 433 00:22:42,310 --> 00:22:43,740 you've created a short circuit. 434 00:22:43,740 --> 00:22:48,700 The inside is connected to the outside. 435 00:22:48,700 --> 00:22:53,170 So that's called an open end boundary condition. 436 00:22:53,170 --> 00:22:58,657 And what can you say about the voltage at this end? 437 00:22:58,657 --> 00:23:00,240 If the outside is [AUDIO OUT] what can 438 00:23:00,240 --> 00:23:05,465 you say about the voltage inside the dendrite at that end? 439 00:23:05,465 --> 00:23:06,340 AUDIENCE: [INAUDIBLE] 440 00:23:06,340 --> 00:23:08,080 MICHALE FEE: It's 0. 441 00:23:08,080 --> 00:23:08,860 Good. 442 00:23:08,860 --> 00:23:11,620 So we have injected current. 443 00:23:11,620 --> 00:23:14,650 We have V0, the voltage at this end. 444 00:23:14,650 --> 00:23:17,050 And we know, if we have an open end, 445 00:23:17,050 --> 00:23:19,780 that the voltage here is 0. 446 00:23:19,780 --> 00:23:21,340 Now we can write down. 447 00:23:21,340 --> 00:23:24,310 We know that the initial voltage is V0. 448 00:23:24,310 --> 00:23:28,180 The voltage at position L is 0. 449 00:23:28,180 --> 00:23:31,640 And now you can-- 450 00:23:31,640 --> 00:23:35,410 you know that the current here is equal to the current there, 451 00:23:35,410 --> 00:23:40,990 and you can write down the equation and solve V0. 452 00:23:40,990 --> 00:23:46,960 So V0 is just the resistance, the total resistance 453 00:23:46,960 --> 00:23:49,420 of the dendrite times the injected current. 454 00:23:53,700 --> 00:23:58,550 And that Rin is known as input impedance. 455 00:23:58,550 --> 00:24:00,690 It's just the resistance of the dendrite. 456 00:24:03,320 --> 00:24:05,930 It tells you how much voltage change 457 00:24:05,930 --> 00:24:10,740 you will get if you inject a given amount of current. 458 00:24:10,740 --> 00:24:11,240 All right. 459 00:24:11,240 --> 00:24:12,680 Any questions about that? 460 00:24:12,680 --> 00:24:14,690 Let's consider another case. 461 00:24:14,690 --> 00:24:18,680 Rather than having an open end, let's leave [? the end of ?] 462 00:24:18,680 --> 00:24:23,660 the dendrite closed so that it's sealed closed. 463 00:24:23,660 --> 00:24:27,440 So we're going to consider a piece of dendrite 464 00:24:27,440 --> 00:24:29,720 that, one end, we're injecting current in, 465 00:24:29,720 --> 00:24:31,842 and the other end is closed. 466 00:24:31,842 --> 00:24:33,800 So what do you think that's going to look like? 467 00:24:33,800 --> 00:24:36,470 It's called a closed end. 468 00:24:36,470 --> 00:24:39,170 What does that look like here? 469 00:24:39,170 --> 00:24:40,490 It's an open circuit. 470 00:24:40,490 --> 00:24:44,610 Those two wires are not connected to each other. 471 00:24:44,610 --> 00:24:47,910 There's no resistance between them. 472 00:24:47,910 --> 00:24:52,190 Let's say we define the voltage here as V0. 473 00:24:52,190 --> 00:24:53,690 What can you say-- well, what you 474 00:24:53,690 --> 00:24:56,750 can say about the current there is that the current is 0, 475 00:24:56,750 --> 00:24:58,670 because it's an open circuit. 476 00:24:58,670 --> 00:25:01,020 There's no current flowing. 477 00:25:01,020 --> 00:25:05,572 And so the current flowing through this at this end is 0. 478 00:25:05,572 --> 00:25:07,370 Does that make sense? 479 00:25:07,370 --> 00:25:11,090 So what can you say about the current everywhere? 480 00:25:11,090 --> 00:25:11,590 AUDIENCE: 0. 481 00:25:11,590 --> 00:25:12,560 MICHALE FEE: It's 0. 482 00:25:12,560 --> 00:25:16,170 And what can you say about the voltage everywhere? 483 00:25:16,170 --> 00:25:16,710 It's V0. 484 00:25:19,600 --> 00:25:20,170 Exactly. 485 00:25:20,170 --> 00:25:22,690 So the voltage everywhere becomes V0. 486 00:25:25,570 --> 00:25:26,940 And the input impedance? 487 00:25:26,940 --> 00:25:29,970 Anybody want to guess what the input impedance is? 488 00:25:29,970 --> 00:25:34,440 How much-- what's the ratio of the voltage at this end 489 00:25:34,440 --> 00:25:35,790 and the current at this end? 490 00:25:40,029 --> 00:25:41,525 AUDIENCE: Infinite? 491 00:25:41,525 --> 00:25:42,650 MICHALE FEE: It's infinite. 492 00:25:42,650 --> 00:25:43,192 That's right. 493 00:25:46,220 --> 00:25:50,700 So we're just trying to build some intuition about how 494 00:25:50,700 --> 00:25:53,040 voltage looks [AUDIO OUT] of distance 495 00:25:53,040 --> 00:25:57,130 for one special case, which is a piece of dendrite 496 00:25:57,130 --> 00:26:03,530 of some finite length for which you have no membrane currents. 497 00:26:03,530 --> 00:26:10,660 And you can see that the voltage profile you get is linear, 498 00:26:10,660 --> 00:26:14,260 and the slope of it depends on the boundary conditions, 499 00:26:14,260 --> 00:26:16,690 depends on whether the piece of dendrite 500 00:26:16,690 --> 00:26:19,165 has a sealed end, whether it's open. 501 00:26:26,760 --> 00:26:27,260 All right. 502 00:26:27,260 --> 00:26:29,360 So now we're going to come back to the case 503 00:26:29,360 --> 00:26:32,810 where we have membrane currents, and we're 504 00:26:32,810 --> 00:26:36,140 going to derive the general solution 505 00:26:36,140 --> 00:26:43,550 to the voltage in a piece of dendrite for the case 506 00:26:43,550 --> 00:26:46,070 where we have membrane capacitance and membrane 507 00:26:46,070 --> 00:26:48,450 currents. 508 00:26:48,450 --> 00:26:48,950 All right. 509 00:26:48,950 --> 00:26:52,970 And I don't expect you to be able to reproduce this, 510 00:26:52,970 --> 00:26:55,340 but we're going to derive what's called the cable 511 00:26:55,340 --> 00:26:58,640 equation, which is the general mathematical description, 512 00:26:58,640 --> 00:27:03,080 the most general mathematical description for the voltage 513 00:27:03,080 --> 00:27:08,210 in a cylindrical tube, of which-- 514 00:27:08,210 --> 00:27:09,990 that's what dendrites look like. 515 00:27:09,990 --> 00:27:13,490 So we're going to write down that differential equation, 516 00:27:13,490 --> 00:27:16,520 and I want you to just see what it looks like 517 00:27:16,520 --> 00:27:18,573 and where it comes from, but I don't expect 518 00:27:18,573 --> 00:27:19,740 you to be able to derive it. 519 00:27:23,890 --> 00:27:24,970 All right. 520 00:27:24,970 --> 00:27:28,990 So let's come back to this simple model that we started. 521 00:27:28,990 --> 00:27:34,300 We're going to put our model for the membrane back in. 522 00:27:34,300 --> 00:27:40,060 Remember, that's a capacitor and a conductance in parallel. 523 00:27:40,060 --> 00:27:45,680 We're going to-- we can write down the membrane current, 524 00:27:45,680 --> 00:27:48,230 and we're going to have an injected current per unit 525 00:27:48,230 --> 00:27:49,260 length. 526 00:27:49,260 --> 00:27:51,020 So Kirchoff's current law tells us 527 00:27:51,020 --> 00:27:56,690 the sum of all of those currents into each node has to be 0. 528 00:27:56,690 --> 00:27:58,795 So let's just write down-- 529 00:27:58,795 --> 00:28:00,170 let's just write down an equation 530 00:28:00,170 --> 00:28:02,840 that sums together all of those and sets them to 0. 531 00:28:02,840 --> 00:28:07,400 So the membrane current leaking out minus that 532 00:28:07,400 --> 00:28:08,930 injected current coming in. 533 00:28:08,930 --> 00:28:12,380 They have positive signs because one is defined as positive 534 00:28:12,380 --> 00:28:17,150 going into the dendrite, and the other one 535 00:28:17,150 --> 00:28:20,540 is defined as positive going out. 536 00:28:20,540 --> 00:28:22,580 So those two, the membrane currents, 537 00:28:22,580 --> 00:28:26,810 plus the current going out this way minus the current coming 538 00:28:26,810 --> 00:28:28,910 in that way is 0. 539 00:28:32,440 --> 00:28:34,660 So we're going to do the same trick we did last time. 540 00:28:34,660 --> 00:28:37,700 We're going to divide by delta x. 541 00:28:37,700 --> 00:28:42,350 So, again, membrane current per unit length times the length 542 00:28:42,350 --> 00:28:44,630 of this finite element. 543 00:28:44,630 --> 00:28:46,010 We're going to divide by delta x. 544 00:28:46,010 --> 00:28:50,630 So this thing right here, i membrane minus i 545 00:28:50,630 --> 00:28:55,760 electrode, I guess, equals minus 1 over delta x I of x minus I 546 00:28:55,760 --> 00:28:58,250 of x minus delta x. 547 00:28:58,250 --> 00:29:00,480 So what is this? 548 00:29:00,480 --> 00:29:03,130 You've seen something like that before. 549 00:29:03,130 --> 00:29:05,310 It's just a derivative. 550 00:29:05,310 --> 00:29:10,130 First derivative of I with respect to position. 551 00:29:10,130 --> 00:29:13,460 So now what you see is that the membrane current 552 00:29:13,460 --> 00:29:17,090 minus the injected current is just the first derivative of I. 553 00:29:17,090 --> 00:29:19,010 So hang in there. 554 00:29:19,010 --> 00:29:21,530 We're going to substitute that with something that 555 00:29:21,530 --> 00:29:23,660 depends on voltage. 556 00:29:23,660 --> 00:29:25,140 So how do we do that? 557 00:29:25,140 --> 00:29:27,170 We're going to take Ohm's law. 558 00:29:27,170 --> 00:29:28,460 There's Ohm's law. 559 00:29:28,460 --> 00:29:31,610 Let's take the derivative of that with respect to position. 560 00:29:31,610 --> 00:29:34,930 So now we get the second derivative of voltage 561 00:29:34,930 --> 00:29:37,620 with respect to position is just equal to minus 562 00:29:37,620 --> 00:29:40,750 Ra times the first derivative of current. 563 00:29:40,750 --> 00:29:47,310 And you can see we can just take this and substitute it there. 564 00:29:47,310 --> 00:29:51,270 So here's what we get, that the second derivative of voltage 565 00:29:51,270 --> 00:29:57,480 with respect to position is just equal to the membrane 566 00:29:57,480 --> 00:30:01,440 or injected current coming into the dendrite at any position. 567 00:30:01,440 --> 00:30:07,130 So the curvature of the voltage, how curved it is, 568 00:30:07,130 --> 00:30:10,865 just depends on what's coming in through the membrane. 569 00:30:14,650 --> 00:30:17,110 Remember, in the case where we had no membrane current 570 00:30:17,110 --> 00:30:20,530 and no injected current, the curvature was 0, 571 00:30:20,530 --> 00:30:24,280 d2V dx squared is 0, which, if the curvature is 0, 572 00:30:24,280 --> 00:30:26,423 then what do you have? 573 00:30:26,423 --> 00:30:27,090 A straight line. 574 00:30:33,380 --> 00:30:37,820 Now, we're going to plug in the right equation 575 00:30:37,820 --> 00:30:40,250 for our membrane current. 576 00:30:40,250 --> 00:30:41,906 What is that? 577 00:30:41,906 --> 00:30:43,050 That we know. 578 00:30:43,050 --> 00:30:45,660 It's just a sum of two terms. 579 00:30:45,660 --> 00:30:46,180 What is it? 580 00:30:46,180 --> 00:30:47,580 It's the sum of-- 581 00:30:47,580 --> 00:30:53,220 remember, this is going to be the same as our soma model. 582 00:30:53,220 --> 00:30:53,940 What was that? 583 00:30:53,940 --> 00:30:55,590 We had two terms. 584 00:30:55,590 --> 00:30:58,330 What were they? 585 00:30:58,330 --> 00:31:00,690 The current through the membrane in the model, 586 00:31:00,690 --> 00:31:02,850 in the Hodgkin-Huxley model is? 587 00:31:06,024 --> 00:31:07,176 What's that? 588 00:31:07,176 --> 00:31:08,590 AUDIENCE: [INAUDIBLE]. 589 00:31:08,590 --> 00:31:09,340 MICHALE FEE: Good. 590 00:31:09,340 --> 00:31:13,270 It's a capacitive current and a membrane ionic current. 591 00:31:13,270 --> 00:31:15,530 So let's just plug that in. 592 00:31:15,530 --> 00:31:19,500 We're just going to substitute into here the current 593 00:31:19,500 --> 00:31:21,060 through the capacitor and the current 594 00:31:21,060 --> 00:31:23,490 through this conductance. 595 00:31:23,490 --> 00:31:30,515 That's just C dV dt G times V minus EL. 596 00:31:30,515 --> 00:31:35,690 It's a capacitive part and a resistive part. 597 00:31:35,690 --> 00:31:38,080 Now, the capacitance is a little funny. 598 00:31:38,080 --> 00:31:43,270 It's capacitance per unit length times the length of the element 599 00:31:43,270 --> 00:31:43,840 plus-- 600 00:31:43,840 --> 00:31:46,540 and the [AUDIO OUT] is conductance per unit length 601 00:31:46,540 --> 00:31:48,355 times the length of our finite element. 602 00:31:53,070 --> 00:31:56,250 Capacitance per unit length and ionic conductance 603 00:31:56,250 --> 00:31:58,000 per unit length. 604 00:31:58,000 --> 00:32:00,730 And we're going to plug that into there. 605 00:32:00,730 --> 00:32:05,210 We're first going to notice that this E leak is just an offset, 606 00:32:05,210 --> 00:32:06,370 so we can just ignore it. 607 00:32:06,370 --> 00:32:08,260 We can just set it to 0. 608 00:32:08,260 --> 00:32:11,770 We can always add it back later if we want. 609 00:32:11,770 --> 00:32:14,650 We divide both sides by the membrane conductance per unit 610 00:32:14,650 --> 00:32:17,510 length to get this equation. 611 00:32:17,510 --> 00:32:20,440 And that's called the cable equation. 612 00:32:20,440 --> 00:32:24,310 It's got a term with the second derivative of voltage 613 00:32:24,310 --> 00:32:26,740 with respect to position, and it's 614 00:32:26,740 --> 00:32:29,920 got a term that's the first derivative of voltage 615 00:32:29,920 --> 00:32:32,250 with respect to time. 616 00:32:32,250 --> 00:32:34,160 That's because of the capacitor. 617 00:32:36,720 --> 00:32:43,080 And then it's got a term that just depends on [AUDIO OUT].. 618 00:32:43,080 --> 00:32:46,530 Now, that's the most general equation. 619 00:32:46,530 --> 00:32:50,240 It describes how the voltage changes in a dendrite 620 00:32:50,240 --> 00:32:52,230 if you inject a pulse of current, 621 00:32:52,230 --> 00:32:56,125 how that current will propagate down the dendrite or down 622 00:32:56,125 --> 00:32:56,625 an axon. 623 00:33:00,840 --> 00:33:04,960 We're going to take a simplifying case. 624 00:33:04,960 --> 00:33:08,940 Next, we're going to study the case just of the steady state 625 00:33:08,940 --> 00:33:11,760 solution to this. 626 00:33:11,760 --> 00:33:15,770 But I want you to see this and to see 627 00:33:15,770 --> 00:33:22,710 how it was derived just using finite element analysis, 628 00:33:22,710 --> 00:33:28,080 deriving Ohm's law in a one-dimensional continuous 629 00:33:28,080 --> 00:33:29,100 medium. 630 00:33:29,100 --> 00:33:33,150 And by plugging in the equation for the membrane that 631 00:33:33,150 --> 00:33:36,500 includes the capacitive and resistive parts, 632 00:33:36,500 --> 00:33:39,440 you can derive this full equation 633 00:33:39,440 --> 00:33:44,230 for how the voltage changes in a piece of dendrite. 634 00:33:44,230 --> 00:33:46,720 Now, there are a couple of interesting constants here 635 00:33:46,720 --> 00:33:48,160 that are important-- 636 00:33:48,160 --> 00:33:49,960 lambda and tau. 637 00:33:49,960 --> 00:33:52,520 So lambda has units of length. 638 00:33:52,520 --> 00:33:54,370 Notice that all of the denominators 639 00:33:54,370 --> 00:33:56,750 here have units of voltage. 640 00:33:56,750 --> 00:33:59,530 So this is voltage per distance squared. 641 00:33:59,530 --> 00:34:01,030 So in order to have the right units, 642 00:34:01,030 --> 00:34:02,770 you have to multiply by something 643 00:34:02,770 --> 00:34:05,990 that's distance squared. 644 00:34:05,990 --> 00:34:07,970 This is voltage per unit time, so you 645 00:34:07,970 --> 00:34:12,190 have to multiply by something that has units of time. 646 00:34:12,190 --> 00:34:14,969 So that is the length constant right there, 647 00:34:14,969 --> 00:34:16,830 and that is a time constant. 648 00:34:16,830 --> 00:34:20,489 And the length constant is defined as 1 649 00:34:20,489 --> 00:34:22,739 over membrane conductance. 650 00:34:22,739 --> 00:34:27,010 That's the conductance of the membrane, through the membrane, 651 00:34:27,010 --> 00:34:32,429 and this is the axial resistance down the dendrite. 652 00:34:32,429 --> 00:34:36,699 So this is conductance per unit length, 653 00:34:36,699 --> 00:34:39,940 and this is resistance per unit length. 654 00:34:39,940 --> 00:34:42,219 And when you multiply those things together, 655 00:34:42,219 --> 00:34:47,045 you get two per unit length down in the denominator. 656 00:34:47,045 --> 00:34:48,670 So when you put those in the numerator, 657 00:34:48,670 --> 00:34:51,060 you get length squared. 658 00:34:51,060 --> 00:34:52,480 And then you take the square root, 659 00:34:52,480 --> 00:34:54,870 and that gives you units of length. 660 00:34:54,870 --> 00:34:57,960 The time constant is just the capacitance 661 00:34:57,960 --> 00:35:00,900 per unit length divided by the conductance per unit length. 662 00:35:00,900 --> 00:35:03,060 And that is the membrane time constant, 663 00:35:03,060 --> 00:35:05,490 and that's exactly the same as the membrane time 664 00:35:05,490 --> 00:35:09,780 constant that we had for our cell. 665 00:35:09,780 --> 00:35:13,260 It's a property of the membrane, not the geometry. 666 00:35:13,260 --> 00:35:15,250 So any questions about that? 667 00:35:15,250 --> 00:35:17,400 It was-- it's a lot. 668 00:35:17,400 --> 00:35:20,610 I just wanted you to see it. 669 00:35:20,610 --> 00:35:21,333 Yes, [INAUDIBLE]. 670 00:35:21,333 --> 00:35:25,715 AUDIENCE: Like, two slides ago [INAUDIBLE] 671 00:35:25,715 --> 00:35:26,840 MICHALE FEE: This one, or-- 672 00:35:26,840 --> 00:35:29,412 AUDIENCE: One more slide [INAUDIBLE].. 673 00:35:29,412 --> 00:35:30,370 MICHALE FEE: Yes, here. 674 00:35:30,370 --> 00:35:33,268 AUDIENCE: So when you plug that in for the derivative of V, 675 00:35:33,268 --> 00:35:37,293 were we not assuming that there was no membrane [INAUDIBLE]?? 676 00:35:37,293 --> 00:35:37,960 MICHALE FEE: No. 677 00:35:37,960 --> 00:35:41,170 That equation is still correct. 678 00:35:41,170 --> 00:35:42,220 AUDIENCE: OK. 679 00:35:42,220 --> 00:35:45,550 MICHALE FEE: It's-- voltage is the derivative with respect 680 00:35:45,550 --> 00:35:49,400 to position as a function of the axial current. 681 00:35:49,400 --> 00:35:50,080 AUDIENCE: OK. 682 00:35:50,080 --> 00:35:50,747 MICHALE FEE: OK? 683 00:36:00,700 --> 00:36:04,300 Remember, going back up to here, notice 684 00:36:04,300 --> 00:36:08,650 that when we derive this equation right here, 685 00:36:08,650 --> 00:36:11,560 we didn't even have to include these membrane. 686 00:36:11,560 --> 00:36:12,797 They don't change anything. 687 00:36:12,797 --> 00:36:13,630 It's just Ohm's law. 688 00:36:13,630 --> 00:36:16,300 It's the voltage here minus the voltage 689 00:36:16,300 --> 00:36:19,630 there has to equal the current flowing through that resistor. 690 00:36:19,630 --> 00:36:22,300 Doesn't matter what other currents-- whether current 691 00:36:22,300 --> 00:36:23,980 is flowing in other directions here. 692 00:36:23,980 --> 00:36:24,800 AUDIENCE: OK. 693 00:36:24,800 --> 00:36:27,410 MICHALE FEE: Does that make sense? 694 00:36:27,410 --> 00:36:29,700 The current through that resistor 695 00:36:29,700 --> 00:36:32,610 is just given by the voltage difference 696 00:36:32,610 --> 00:36:34,520 on either side of it. 697 00:36:34,520 --> 00:36:35,240 That's Ohm's law. 698 00:36:44,410 --> 00:36:46,430 So now we're going to take a simple example. 699 00:36:46,430 --> 00:36:48,010 We're going to solve that equation 700 00:36:48,010 --> 00:36:51,550 for the case of steady state. 701 00:36:51,550 --> 00:36:53,890 How are we going to take the steady state? 702 00:36:53,890 --> 00:36:57,610 How are we going to find the steady state 703 00:36:57,610 --> 00:36:59,350 version of this equation? 704 00:36:59,350 --> 00:37:01,210 Any idea? 705 00:37:01,210 --> 00:37:02,560 AUDIENCE: [INAUDIBLE]. 706 00:37:02,560 --> 00:37:03,310 MICHALE FEE: Good. 707 00:37:03,310 --> 00:37:05,320 We just set dV dt to 0, and we're 708 00:37:05,320 --> 00:37:09,990 left with this equals that. 709 00:37:09,990 --> 00:37:13,770 So we're going to take a piece of our cable, 710 00:37:13,770 --> 00:37:16,500 and we're going to imagine that we 711 00:37:16,500 --> 00:37:18,750 take a piece of dendrite that's infinitely long 712 00:37:18,750 --> 00:37:21,060 in either direction. 713 00:37:21,060 --> 00:37:23,840 And somewhere here in the middle of it, we're going to inject-- 714 00:37:23,840 --> 00:37:25,590 we're going to put an electrode, and we're 715 00:37:25,590 --> 00:37:29,755 going to inject current at one position. 716 00:37:32,880 --> 00:37:36,760 So it's injecting current at position 0. 717 00:37:36,760 --> 00:37:40,110 How many of you have heard of a delta function, a Dirac delta 718 00:37:40,110 --> 00:37:40,610 function? 719 00:37:40,610 --> 00:37:41,110 OK. 720 00:37:41,110 --> 00:37:44,220 So we're going to define the current 721 00:37:44,220 --> 00:37:47,400 as a function of position as just a current times 722 00:37:47,400 --> 00:37:50,160 a Dirac delta function of x, that just says 723 00:37:50,160 --> 00:37:52,800 that all the current is going in at position 0, 724 00:37:52,800 --> 00:37:58,020 and no current is going in anywhere else. 725 00:37:58,020 --> 00:38:00,440 So the Dirac delta function is just-- 726 00:38:00,440 --> 00:38:05,270 it's a peaky thing that is very narrow and very tall, such 727 00:38:05,270 --> 00:38:07,190 that when you integrate over it, you get a 1. 728 00:38:12,750 --> 00:38:16,480 So we're going to go to the steady state solution. 729 00:38:16,480 --> 00:38:18,540 And now let's write down that. 730 00:38:18,540 --> 00:38:22,310 So there's the steady state cable equation. 731 00:38:22,310 --> 00:38:25,230 And we're going to inject current at a single point. 732 00:38:25,230 --> 00:38:26,817 So that's what it looks like. 733 00:38:26,817 --> 00:38:28,400 Does anyone know the solution to this? 734 00:38:28,400 --> 00:38:31,640 Notice, what this says is we have a function. 735 00:38:31,640 --> 00:38:35,320 It's equal to the second derivative of that function. 736 00:38:35,320 --> 00:38:36,470 Anybody know? 737 00:38:36,470 --> 00:38:42,780 There's only one function that does this. 738 00:38:42,780 --> 00:38:44,790 It's an exponential. 739 00:38:44,790 --> 00:38:45,930 That's right. 740 00:38:45,930 --> 00:38:51,090 So the solution to this equation is an exponential. 741 00:38:51,090 --> 00:38:57,740 V of position is V0, some voltage in the middle, 742 00:38:57,740 --> 00:39:00,020 e to the minus x over lambda. 743 00:39:00,020 --> 00:39:01,580 Why do I have an absolute value? 744 00:39:01,580 --> 00:39:03,650 What is the voltage going to look like if I 745 00:39:03,650 --> 00:39:06,290 inject current right here? 746 00:39:06,290 --> 00:39:07,850 You're going to have current flowing. 747 00:39:07,850 --> 00:39:09,183 Where's the current going to go? 748 00:39:11,900 --> 00:39:14,680 If I inject current into the middle of a piece of dendrite, 749 00:39:14,680 --> 00:39:17,290 is it all going to go this way? 750 00:39:17,290 --> 00:39:17,790 No. 751 00:39:17,790 --> 00:39:19,260 What's it going to do? 752 00:39:19,260 --> 00:39:20,385 It's going to go both ways. 753 00:39:23,328 --> 00:39:25,620 And the current-- the voltage is going to be high here, 754 00:39:25,620 --> 00:39:30,560 and it's going to fall as you go in both directions. 755 00:39:30,560 --> 00:39:32,580 That's why we have an absolute value here. 756 00:39:32,580 --> 00:39:35,700 So the voltage is going to start at some V0 that 757 00:39:35,700 --> 00:39:37,870 depends on how much current we're injecting, 758 00:39:37,870 --> 00:39:44,010 and it's going to drop exponentially on both sides. 759 00:39:44,010 --> 00:39:48,930 And notice what's right here. 760 00:39:48,930 --> 00:39:55,970 The lambda tells us the 1 over e point, 761 00:39:55,970 --> 00:39:59,150 how far away the 1 over e point of the voltage is. 762 00:39:59,150 --> 00:40:00,890 What that means is that the voltage is 763 00:40:00,890 --> 00:40:06,960 going to fall to 1 over e of V0 at a distance lambda 764 00:40:06,960 --> 00:40:11,815 from the side at which the current is injected. 765 00:40:11,815 --> 00:40:12,840 Does that make sense? 766 00:40:12,840 --> 00:40:16,980 That is the steady state space constant. 767 00:40:16,980 --> 00:40:19,632 It has units of length. 768 00:40:19,632 --> 00:40:24,650 It's how far away do you have to go so that the voltage falls 769 00:40:24,650 --> 00:40:26,390 to 1 over-- 770 00:40:26,390 --> 00:40:33,405 falls to 1 over 2.7 of the initial voltage. 771 00:40:42,156 --> 00:40:44,330 Any questions? 772 00:40:44,330 --> 00:40:46,640 It's pretty simple. 773 00:40:46,640 --> 00:40:53,110 We took an unusually complicated route to get there, 774 00:40:53,110 --> 00:40:56,650 but that's the-- the nice thing about that is you've seen 775 00:40:56,650 --> 00:41:00,180 the most general solution to how a cable-- 776 00:41:00,180 --> 00:41:03,070 a dendrite will behave when you inject current into it. 777 00:41:07,000 --> 00:41:10,780 So now we can calculate the current 778 00:41:10,780 --> 00:41:12,070 as a function of position. 779 00:41:12,070 --> 00:41:14,540 Any idea how to do that? 780 00:41:14,540 --> 00:41:17,620 What-- if you know voltage, what do you 781 00:41:17,620 --> 00:41:20,470 use to calculate current? 782 00:41:20,470 --> 00:41:21,093 Which law? 783 00:41:21,093 --> 00:41:21,760 AUDIENCE: Ohm's. 784 00:41:21,760 --> 00:41:23,410 MICHALE FEE: Ohm's law. 785 00:41:23,410 --> 00:41:26,020 Anybody remember what Ohm's law looks like here? 786 00:41:29,796 --> 00:41:33,100 AUDIENCE: [INAUDIBLE]. 787 00:41:33,100 --> 00:41:34,020 MICHALE FEE: Yes. 788 00:41:34,020 --> 00:41:36,050 And we have to do something else. 789 00:41:36,050 --> 00:41:40,080 The-- remember, the current is what? 790 00:41:40,080 --> 00:41:43,290 Ohm's law in a continuous medium, the current 791 00:41:43,290 --> 00:41:46,380 is just going to be what of the voltage, 792 00:41:46,380 --> 00:41:48,810 the blank of the voltage? 793 00:41:48,810 --> 00:41:49,800 AUDIENCE: Derivative? 794 00:41:49,800 --> 00:41:51,190 MICHALE FEE: The derivative of the voltage. 795 00:41:51,190 --> 00:41:53,565 So we're just going to take this and take the derivative. 796 00:41:53,565 --> 00:41:54,390 That's it. 797 00:41:54,390 --> 00:41:57,810 dV dx is just equal to minus R times I. 798 00:41:57,810 --> 00:42:00,360 So the current is proportional to the derivative of this. 799 00:42:00,360 --> 00:42:02,190 What's the derivative of an exponential? 800 00:42:04,950 --> 00:42:07,715 Just another exponential. 801 00:42:07,715 --> 00:42:08,340 So there we go. 802 00:42:08,340 --> 00:42:10,090 The current, and then there's some-- 803 00:42:10,090 --> 00:42:12,002 you have to bring the lambda down 804 00:42:12,002 --> 00:42:13,210 when you take the derivative. 805 00:42:13,210 --> 00:42:17,520 So the current is now just minus 1 over the axial resistance per 806 00:42:17,520 --> 00:42:21,700 unit length times minus V0 over lambda-- 807 00:42:21,700 --> 00:42:25,220 lambda comes down when you take the derivative-- 808 00:42:25,220 --> 00:42:28,760 times e to the minus x over lambda. 809 00:42:28,760 --> 00:42:32,190 Notice, the current is to the right on this side, 810 00:42:32,190 --> 00:42:34,310 so the current is positive it's flowing 811 00:42:34,310 --> 00:42:37,490 to the left on that side, so the current is negative. 812 00:42:37,490 --> 00:42:40,190 So to do this properly, you'd have-- this is 813 00:42:40,190 --> 00:42:42,320 the solution on the right side. 814 00:42:42,320 --> 00:42:44,930 You'd have to write another version of this for the current 815 00:42:44,930 --> 00:42:47,013 on the left side, but I haven't put that in there. 816 00:42:49,720 --> 00:42:55,150 And, again, the current starts out at I0, 817 00:42:55,150 --> 00:43:01,430 and drops exponentially, and it falls to 1 over e 818 00:43:01,430 --> 00:43:03,560 at a distance lambda. 819 00:43:03,560 --> 00:43:04,190 Why is that? 820 00:43:04,190 --> 00:43:06,890 Because the current is leaking out 821 00:43:06,890 --> 00:43:11,530 through the holes in our garden hose. 822 00:43:11,530 --> 00:43:16,330 So as you go further down, less and less of the current 823 00:43:16,330 --> 00:43:18,780 is still going down the dendrite. 824 00:43:21,520 --> 00:43:24,530 I don't expect you to be able to derive this, 825 00:43:24,530 --> 00:43:27,820 but, again, just know where it comes from. 826 00:43:27,820 --> 00:43:28,810 Comes from Ohm's law. 827 00:43:33,680 --> 00:43:38,840 So I want to show you one really cool thing about the space 828 00:43:38,840 --> 00:43:39,770 constant. 829 00:43:39,770 --> 00:43:42,530 It has a really important dependence 830 00:43:42,530 --> 00:43:46,333 on the size of the dendrite. 831 00:43:46,333 --> 00:43:47,750 And we're going to learn something 832 00:43:47,750 --> 00:43:53,060 really interesting about why the brain has action potentials. 833 00:43:53,060 --> 00:43:57,050 So let's take a closer look at the space constant, 834 00:43:57,050 --> 00:43:58,670 and how you calculate it, and how 835 00:43:58,670 --> 00:44:01,950 it depends on the size, this diameter, 836 00:44:01,950 --> 00:44:04,450 this radius of the dendrite. 837 00:44:04,450 --> 00:44:08,330 So we're going to take a little cylinder of dendrite 838 00:44:08,330 --> 00:44:10,580 of radius a length little l. 839 00:44:13,120 --> 00:44:16,550 G sub m is the membrane conductance per unit length. 840 00:44:16,550 --> 00:44:21,590 Let's just derive what that would look like. 841 00:44:21,590 --> 00:44:27,450 The total membrane conductance of this little cylinder 842 00:44:27,450 --> 00:44:33,510 of dendrite, little cylinder of cell membrane, 843 00:44:33,510 --> 00:44:42,100 is just the surface area of that cylinder times the conductance 844 00:44:42,100 --> 00:44:43,420 per unit area. 845 00:44:43,420 --> 00:44:45,340 Remember, this is the same idea that we've 846 00:44:45,340 --> 00:44:47,050 talked about when we were talking 847 00:44:47,050 --> 00:44:50,530 about the area of our soma. 848 00:44:50,530 --> 00:44:53,020 We have a conductance per unit area 849 00:44:53,020 --> 00:44:55,810 that just depends on the number of ion channels 850 00:44:55,810 --> 00:44:59,560 and how open they are on that piece of membrane. 851 00:44:59,560 --> 00:45:01,720 So the total conductance is just going 852 00:45:01,720 --> 00:45:06,410 to be the conductance per unit area times the area. 853 00:45:06,410 --> 00:45:09,710 And the area of that cylinder is 2 pi a-- 854 00:45:09,710 --> 00:45:11,270 that gives us the circumference-- 855 00:45:11,270 --> 00:45:13,520 times the length, 2 pi al. 856 00:45:20,120 --> 00:45:22,070 And the conductance per unit length 857 00:45:22,070 --> 00:45:27,200 is just that total conductance divided by the length. 858 00:45:27,200 --> 00:45:33,190 So it's 2 pi a times g sub l, the conductance per unit area. 859 00:45:33,190 --> 00:45:36,410 So that's membrane conductance per unit length. 860 00:45:40,150 --> 00:45:42,760 The axial resistance per unit length 861 00:45:42,760 --> 00:45:45,580 along this piece, this little cylinder of dendrite, 862 00:45:45,580 --> 00:45:47,780 we can calculate in a similar way. 863 00:45:47,780 --> 00:45:52,750 The total axial resistance along that dendrite is-- 864 00:45:52,750 --> 00:45:56,080 can be calculated using this equation that we developed 865 00:45:56,080 --> 00:45:59,950 on the very first day, the resistance of a wire 866 00:45:59,950 --> 00:46:03,790 in the brain, the resistance of a chunk of extracellular 867 00:46:03,790 --> 00:46:04,990 or intracellular solution. 868 00:46:07,660 --> 00:46:10,900 The resistance is just the resistivity 869 00:46:10,900 --> 00:46:14,390 times the length divided by the area. 870 00:46:14,390 --> 00:46:20,000 The longer-- for a given medium of some resistivity, 871 00:46:20,000 --> 00:46:23,690 the longer you have to run your current through, 872 00:46:23,690 --> 00:46:26,120 the bigger the resistance is going to be. 873 00:46:26,120 --> 00:46:29,300 And the bigger the area, the lower the resistance 874 00:46:29,300 --> 00:46:31,340 is going to be. 875 00:46:31,340 --> 00:46:34,690 So that total resistance is-- it has units of ohm-millimeters. 876 00:46:34,690 --> 00:46:38,150 So it's the resistivity times l divided by A. 877 00:46:38,150 --> 00:46:42,410 In intracellular space, that's around 2,000 ohm-millimeters. 878 00:46:42,410 --> 00:46:47,310 And the cross-sectional area is just pi times a squared. 879 00:46:47,310 --> 00:46:49,440 So now we can calculate the axial resistance 880 00:46:49,440 --> 00:46:50,160 per unit length. 881 00:46:50,160 --> 00:46:53,100 That's the total resistance divided by l. 882 00:46:53,100 --> 00:46:55,690 So that's just resistivity divided by A, 883 00:46:55,690 --> 00:46:59,700 which is resistivity divided by pi radius squared, just 884 00:46:59,700 --> 00:47:02,160 the cross-sectional area, and that has 885 00:47:02,160 --> 00:47:04,170 units of ohms per millimeter. 886 00:47:04,170 --> 00:47:07,260 So now we can calculate the steady state space constant. 887 00:47:07,260 --> 00:47:11,100 Conductance per unit length and axial resistance 888 00:47:11,100 --> 00:47:14,250 for unit length-- the space constant is just 1 889 00:47:14,250 --> 00:47:17,550 over the product of those two, square root. 890 00:47:17,550 --> 00:47:21,020 We're just going to notice that that's 891 00:47:21,020 --> 00:47:25,130 siemens per millimeter, ohms per millimeter, inverse ohms. 892 00:47:25,130 --> 00:47:28,310 So those cancel, and you're left with millimeter squared, 893 00:47:28,310 --> 00:47:30,090 square root, which is just millimeters. 894 00:47:30,090 --> 00:47:33,700 So, again, that has the right units, units of length. 895 00:47:33,700 --> 00:47:36,520 But now let's plug these two things 896 00:47:36,520 --> 00:47:38,505 into this equation for the space constant 897 00:47:38,505 --> 00:47:43,682 and calculate how it depends on a. 898 00:47:43,682 --> 00:47:44,390 So let's do that. 899 00:47:47,243 --> 00:47:48,910 Actually, the first thing I wanted to do 900 00:47:48,910 --> 00:47:51,040 is just show you what a typical lambda is 901 00:47:51,040 --> 00:47:53,050 for a piece of dendrite. 902 00:47:53,050 --> 00:47:54,550 So let's do that. 903 00:47:54,550 --> 00:48:00,430 Conductance per area is around 5 times 10 to the minus 7, 904 00:48:00,430 --> 00:48:01,030 typically. 905 00:48:01,030 --> 00:48:02,860 So the conductance per unit length 906 00:48:02,860 --> 00:48:06,070 of a dendrite, 6 nanosiemens per millimeter. 907 00:48:06,070 --> 00:48:07,990 You don't have to remember that. 908 00:48:07,990 --> 00:48:10,570 We're just calculating the length constant. 909 00:48:10,570 --> 00:48:13,990 Axial resistance is-- plugging in the numbers 910 00:48:13,990 --> 00:48:19,735 for a piece of dendrite that's about 2 microns in radius, 911 00:48:19,735 --> 00:48:21,850 the axial resistance per unit length 912 00:48:21,850 --> 00:48:26,420 is about 60 megaohms per millimeter. 913 00:48:26,420 --> 00:48:30,470 And so when you plug those two things to calculate lambda, 914 00:48:30,470 --> 00:48:34,250 you find that lambda for a typical piece of dendrite 915 00:48:34,250 --> 00:48:35,920 is about a millimeter. 916 00:48:40,630 --> 00:48:45,490 So that's a number that I would hope that you would remember. 917 00:48:45,490 --> 00:48:47,870 That's a typical space constant. 918 00:48:47,870 --> 00:48:56,180 So if you inject a signal into a piece of dendrite, it's gone-- 919 00:48:56,180 --> 00:49:01,160 it's mostly gone or about 2/3 gone in a millimeter. 920 00:49:03,730 --> 00:49:07,870 And that's how you can have dendrites 921 00:49:07,870 --> 00:49:11,110 that are up in the range of close to a millimeter, 922 00:49:11,110 --> 00:49:13,300 and they still are able to conduct 923 00:49:13,300 --> 00:49:17,980 a signal from synaptic inputs out onto the dendrite down 924 00:49:17,980 --> 00:49:18,820 to the soma. 925 00:49:23,800 --> 00:49:27,420 So a millimeter is a typical length scale 926 00:49:27,420 --> 00:49:30,018 for how far signals propagate. 927 00:49:32,990 --> 00:49:35,180 So now let's plug in those-- 928 00:49:35,180 --> 00:49:40,370 the expressions that we derived for conductance per unit length 929 00:49:40,370 --> 00:49:44,060 and axial resistance for unit length of a 930 00:49:44,060 --> 00:49:47,100 into this equation for the space constant. 931 00:49:47,100 --> 00:49:49,820 And what you find is that the space constant 932 00:49:49,820 --> 00:49:52,700 is a divided by 2 times the resistivity 933 00:49:52,700 --> 00:49:55,520 times the membrane conductance per unit 934 00:49:55,520 --> 00:50:00,420 area, per area to the 1/2. 935 00:50:00,420 --> 00:50:02,540 It goes as the square root. 936 00:50:02,540 --> 00:50:07,340 The space constant, the length, goes as the square root 937 00:50:07,340 --> 00:50:10,390 of the radius. 938 00:50:10,390 --> 00:50:13,160 And notice that the space constant 939 00:50:13,160 --> 00:50:19,520 gets bigger as you increase the size of the dendrite. 940 00:50:19,520 --> 00:50:22,580 As you make a dendrite bigger, what happens 941 00:50:22,580 --> 00:50:28,100 is the resistance down the middle gets smaller. 942 00:50:28,100 --> 00:50:31,460 And so the current can go further down the dendrite 943 00:50:31,460 --> 00:50:33,506 before it leaks out. 944 00:50:33,506 --> 00:50:36,130 Does that make sense? 945 00:50:36,130 --> 00:50:38,850 But the resistance [AUDIO OUT] is 946 00:50:38,850 --> 00:50:43,140 dropping as the square of the area, 947 00:50:43,140 --> 00:50:48,680 but the surface area is only increasing linearly. 948 00:50:48,680 --> 00:50:53,220 And so the resistance down the middle 949 00:50:53,220 --> 00:50:54,420 is dropping as the square. 950 00:50:57,732 --> 00:51:03,050 The conductance out the side is growing more slowly. 951 00:51:03,050 --> 00:51:05,970 And so the signal can propagate further 952 00:51:05,970 --> 00:51:09,630 the bigger the dendrite is. 953 00:51:09,630 --> 00:51:16,230 So that's why-- it's very closely related to why 954 00:51:16,230 --> 00:51:19,070 the squid giant axon is big. 955 00:51:19,070 --> 00:51:21,110 Because the current has more access 956 00:51:21,110 --> 00:51:28,180 to propagate down the axon the bigger the cylinder is. 957 00:51:28,180 --> 00:51:30,220 But there are limits to this. 958 00:51:30,220 --> 00:51:33,430 So you know that, in our brains, neurons 959 00:51:33,430 --> 00:51:36,760 need to be able to send signals from one side of our head 960 00:51:36,760 --> 00:51:42,540 to the other side of our head, which is about how big? 961 00:51:42,540 --> 00:51:43,840 How far is that? 962 00:51:48,390 --> 00:51:53,115 Not in Homer, but in [AUDIO OUT] seen the cartoon with little-- 963 00:51:53,115 --> 00:51:53,740 OK, never mind. 964 00:51:56,780 --> 00:51:59,400 How big across is the brain? 965 00:51:59,400 --> 00:52:02,720 How many millimeters, about? 966 00:52:02,720 --> 00:52:03,240 Yes. 967 00:52:03,240 --> 00:52:09,030 Order of magnitude, let's call it 100. 968 00:52:09,030 --> 00:52:14,740 So a piece of dendrite 2 microns across 969 00:52:14,740 --> 00:52:19,690 has a length constant of a millimeter. 970 00:52:19,690 --> 00:52:27,320 How-- what diameter dendrite would we need if we needed 971 00:52:27,320 --> 00:52:35,610 to send a signal across the brain passively through a piece 972 00:52:35,610 --> 00:52:36,650 of-- 973 00:52:36,650 --> 00:52:40,660 a cylindrical piece of dendrite like this? 974 00:52:45,640 --> 00:52:48,660 So lambda scales with radius. 975 00:52:48,660 --> 00:52:56,130 2 microns diameter, radius, gives you 1 millimeter. 976 00:52:56,130 --> 00:53:02,100 Now you want to go to-- you want to go 100 times further. 977 00:53:02,100 --> 00:53:06,773 How-- by what factor larger does the radius have to be? 978 00:53:06,773 --> 00:53:07,720 AUDIENCE: [INAUDIBLE]. 979 00:53:07,720 --> 00:53:09,416 MICHALE FEE: 10,000. 980 00:53:09,416 --> 00:53:10,630 Good. 981 00:53:10,630 --> 00:53:18,190 And so how big does our 2-micron radius piece of dendrite 982 00:53:18,190 --> 00:53:23,950 have to be to send a signal 100 millimeters? 983 00:53:23,950 --> 00:53:28,130 10,000 times 2 microns, what is that? 984 00:53:28,130 --> 00:53:28,983 Anybody? 985 00:53:32,847 --> 00:53:34,790 AUDIENCE: [INAUDIBLE]. 986 00:53:34,790 --> 00:53:36,290 MICHALE FEE: 2 centimeters. 987 00:53:40,430 --> 00:53:45,050 So if you want to make a piece of dendrite that 988 00:53:45,050 --> 00:53:47,030 sends a signal from one side of your brain 989 00:53:47,030 --> 00:53:50,056 to the other 100 millimeters away, 990 00:53:50,056 --> 00:53:51,675 you need 2 centimeters across. 991 00:53:56,230 --> 00:53:58,800 Actually, that's the radius. 992 00:53:58,800 --> 00:54:02,240 It needs to be 4 centimeters. across. 993 00:54:02,240 --> 00:54:05,620 Doesn't work, does it? 994 00:54:05,620 --> 00:54:09,160 So you can make things-- you can make signals propagate further 995 00:54:09,160 --> 00:54:11,590 by making dendrites bigger, but it only 996 00:54:11,590 --> 00:54:13,400 goes as the square root. 997 00:54:13,400 --> 00:54:15,910 It's like diffusion. 998 00:54:15,910 --> 00:54:21,258 it's only-- it increases very slowly. 999 00:54:21,258 --> 00:54:23,550 So in order to get a signal from one side of your brain 1000 00:54:23,550 --> 00:54:25,500 to the other with the same kind of membrane, 1001 00:54:25,500 --> 00:54:30,070 your dendrite would have to be 4 centimeters in diameter. 1002 00:54:33,380 --> 00:54:36,260 So that's why the brain doesn't use 1003 00:54:36,260 --> 00:54:39,830 passive propagation of signals to get from one place 1004 00:54:39,830 --> 00:54:41,330 to the other. 1005 00:54:41,330 --> 00:54:47,420 It uses action potentials that actively propagate down axons. 1006 00:54:53,118 --> 00:54:53,910 Pretty cool, right? 1007 00:55:00,930 --> 00:55:02,130 All right. 1008 00:55:02,130 --> 00:55:08,070 So I want to just introduce you to the concept 1009 00:55:08,070 --> 00:55:10,680 of electrotonic length. 1010 00:55:10,680 --> 00:55:12,790 And the idea is very simple. 1011 00:55:12,790 --> 00:55:16,530 If we have a piece of dendrite that has some physical length 1012 00:55:16,530 --> 00:55:21,370 l, you can see that that length l 1013 00:55:21,370 --> 00:55:31,560 might be very good at conducting signals to the soma if what? 1014 00:55:31,560 --> 00:55:34,800 If-- what aspects of that dendrite 1015 00:55:34,800 --> 00:55:38,810 would make it very good at conducting signals to the soma? 1016 00:55:38,810 --> 00:55:39,970 AUDIENCE: [INAUDIBLE]. 1017 00:55:39,970 --> 00:55:41,470 MICHALE FEE: So it's big. 1018 00:55:41,470 --> 00:55:42,430 Or what else? 1019 00:55:42,430 --> 00:55:44,120 AUDIENCE: Short. 1020 00:55:44,120 --> 00:55:46,330 MICHALE FEE: It's got a fixed physical length l, 1021 00:55:46,330 --> 00:55:48,736 so let's think of something else. 1022 00:55:48,736 --> 00:55:50,550 AUDIENCE: [INAUDIBLE]. 1023 00:55:50,550 --> 00:55:52,250 MICHALE FEE: Less leaky. 1024 00:55:52,250 --> 00:55:53,190 Right. 1025 00:55:53,190 --> 00:55:53,790 OK. 1026 00:55:53,790 --> 00:55:56,940 So depending on the properties of that dendrite, 1027 00:55:56,940 --> 00:55:59,700 that piece of dendrite of physical length l 1028 00:55:59,700 --> 00:56:04,080 might be very good at sending signals to the soma, 1029 00:56:04,080 --> 00:56:08,970 or it might be very bad if it's really thin, really leaky. 1030 00:56:08,970 --> 00:56:13,210 So we have to compare the physical length 1031 00:56:13,210 --> 00:56:15,790 to the space constant. 1032 00:56:15,790 --> 00:56:19,300 So in this case, there's very little decay. 1033 00:56:19,300 --> 00:56:24,940 The signal is able to propagate from the site of the synapse 1034 00:56:24,940 --> 00:56:26,410 to the soma. 1035 00:56:26,410 --> 00:56:29,260 In this case, a slightly smaller piece of dendrite 1036 00:56:29,260 --> 00:56:31,240 might have a shorter lambda, and so there 1037 00:56:31,240 --> 00:56:34,430 would be more decay by the time you get to the soma. 1038 00:56:34,430 --> 00:56:38,500 And in this case, the lambda is really short, 1039 00:56:38,500 --> 00:56:40,540 and so the signal really decays away 1040 00:56:40,540 --> 00:56:44,280 before you get to the soma. 1041 00:56:44,280 --> 00:56:47,240 So people often refer to a quantity as the-- 1042 00:56:47,240 --> 00:56:50,060 referred to as the electrotonic length, 1043 00:56:50,060 --> 00:56:54,440 which is simply the ratio of the physical length to the space 1044 00:56:54,440 --> 00:56:56,030 constant. 1045 00:56:56,030 --> 00:56:58,710 So you can see that in this case, 1046 00:56:58,710 --> 00:57:02,810 the physical length is about the same as lambda, 1047 00:57:02,810 --> 00:57:06,340 and so the electrotonic length is 1. 1048 00:57:06,340 --> 00:57:08,470 In this case, the physical length 1049 00:57:08,470 --> 00:57:12,430 is twice as long as lambda, and so the electrotonic length 1050 00:57:12,430 --> 00:57:14,390 of that piece of dendrite is 2. 1051 00:57:14,390 --> 00:57:17,600 And in this case, it's 4. 1052 00:57:17,600 --> 00:57:20,970 And you can see that the amount of signal it gets from this end 1053 00:57:20,970 --> 00:57:24,060 to that end will go like what? 1054 00:57:24,060 --> 00:57:28,908 Will depend how on the electrotonic length? 1055 00:57:28,908 --> 00:57:34,240 It will depend something like e to the minus L. 1056 00:57:34,240 --> 00:57:39,550 So a piece of dendrite that has a low electrotonic length 1057 00:57:39,550 --> 00:57:42,550 means that the synapse out here at the other end of it 1058 00:57:42,550 --> 00:57:46,000 is effectively very close to the soma. 1059 00:57:46,000 --> 00:57:49,690 It's very effective at transmitting that signal. 1060 00:57:49,690 --> 00:57:52,150 If the electrotonic length is large, 1061 00:57:52,150 --> 00:57:56,680 it's telling you that some input out here at the end of it 1062 00:57:56,680 --> 00:57:58,120 is very far away. 1063 00:57:58,120 --> 00:58:01,090 The signal can't propagate to the soma. 1064 00:58:01,090 --> 00:58:03,460 And the amount of signal that gets to the soma 1065 00:58:03,460 --> 00:58:07,870 goes as e to the minus L, e to the minus [AUDIO OUT].. 1066 00:58:07,870 --> 00:58:10,030 So if I told you that a signal is 1067 00:58:10,030 --> 00:58:11,410 at the end of a piece of dendrite 1068 00:58:11,410 --> 00:58:14,950 that has electrotonic length 2, how much of that signal 1069 00:58:14,950 --> 00:58:16,050 arrives at the soma. 1070 00:58:16,050 --> 00:58:24,270 The answer is e to the minus 2, about 10%, whatever that is. 1071 00:58:30,150 --> 00:58:32,520 So I want to tell you a little bit more 1072 00:58:32,520 --> 00:58:40,240 about the way people model complex dendrites in-- 1073 00:58:40,240 --> 00:58:41,420 sort of in real life. 1074 00:58:41,420 --> 00:58:46,210 So most of the time, we're not integrating or solving 1075 00:58:46,210 --> 00:58:47,180 the cable equation. 1076 00:58:47,180 --> 00:58:49,360 The cable equation is really most 1077 00:58:49,360 --> 00:58:52,630 powerful in terms of giving intuition 1078 00:58:52,630 --> 00:58:55,320 about how cables respond. 1079 00:58:55,320 --> 00:58:57,310 So you can write down exact solutions 1080 00:58:57,310 --> 00:59:00,250 to things like pulses of current input 1081 00:59:00,250 --> 00:59:03,850 at some position, how the voltage propagates down 1082 00:59:03,850 --> 00:59:09,810 the dendrite, the functional form of the voltage 1083 00:59:09,810 --> 00:59:11,260 as a function of distance. 1084 00:59:11,260 --> 00:59:14,530 But when you actually want to sort of model a neuron, 1085 00:59:14,530 --> 00:59:17,230 you're not usually integrating the cable equation. 1086 00:59:17,230 --> 00:59:20,140 And so people do different approximations 1087 00:59:20,140 --> 00:59:23,590 to a very complex dendritic structure like this. 1088 00:59:23,590 --> 00:59:26,440 And one common way that that's done 1089 00:59:26,440 --> 00:59:28,695 is called multi-compartment model. 1090 00:59:28,695 --> 00:59:30,070 So, basically, what you can do is 1091 00:59:30,070 --> 00:59:35,860 you can model the soma with this capacitor-resistor combination. 1092 00:59:35,860 --> 00:59:38,710 And then you can model the connection 1093 00:59:38,710 --> 00:59:42,430 to another part of the dendrite through a resistor 1094 00:59:42,430 --> 00:59:45,880 to another sort of finite element slice, 1095 00:59:45,880 --> 00:59:48,550 but we're gonna let the slices go to 0 length. 1096 00:59:48,550 --> 00:59:52,240 We're just going to model them as, like, chunks of dendrites, 1097 00:59:52,240 --> 00:59:56,860 that are going to be modeled by a compartment like this. 1098 00:59:56,860 --> 00:59:59,290 And then that can branch to connect 1099 00:59:59,290 --> 01:00:01,360 to other parts of the dendrite, and that 1100 01:00:01,360 --> 01:00:04,760 can branch to connect to other parts of the model 1101 01:00:04,760 --> 01:00:06,550 that model other pieces [AUDIO OUT] 1102 01:00:06,550 --> 01:00:08,920 So you can basically take something like this 1103 01:00:08,920 --> 01:00:11,800 and make it arbitrarily complicated 1104 01:00:11,800 --> 01:00:16,780 and arbitrarily close to a representation 1105 01:00:16,780 --> 01:00:20,010 of the physical structure of a real dendrite. 1106 01:00:20,010 --> 01:00:22,400 And so there are labs that do this, 1107 01:00:22,400 --> 01:00:28,150 that take a picture of a neuron like this 1108 01:00:28,150 --> 01:00:30,220 and break it up into little chunks, 1109 01:00:30,220 --> 01:00:32,230 and model each one of those little chunks, 1110 01:00:32,230 --> 01:00:35,650 and model the branching structure of the real dendrite. 1111 01:00:35,650 --> 01:00:40,060 And you can put in real ionic conductances of different types 1112 01:00:40,060 --> 01:00:41,530 out here in this model. 1113 01:00:41,530 --> 01:00:45,650 And you get a gazillion differential equations. 1114 01:00:45,650 --> 01:00:47,860 And you can [AUDIO OUT] those differential equations 1115 01:00:47,860 --> 01:00:53,180 and actually compute, sort of predict 1116 01:00:53,180 --> 01:00:56,150 the behavior of a complex piece of dendrite like this. 1117 01:00:59,080 --> 01:01:03,710 Now, that's not my favorite way of doing modeling. 1118 01:01:03,710 --> 01:01:07,250 Any idea why that would be-- 1119 01:01:07,250 --> 01:01:12,910 why there could be a better way of modeling a complex dendrite? 1120 01:01:12,910 --> 01:01:16,310 I mean, what's the-- one of the problems 1121 01:01:16,310 --> 01:01:19,640 here is that, in a sense, your model 1122 01:01:19,640 --> 01:01:24,130 gets to be as complicated as the real thing. 1123 01:01:24,130 --> 01:01:28,870 So it would be-- it's a great way to simulate some behavior, 1124 01:01:28,870 --> 01:01:32,140 but it's not a great way of getting an intuition about how 1125 01:01:32,140 --> 01:01:33,205 something works. 1126 01:01:36,210 --> 01:01:41,880 So people take simplified versions of this, 1127 01:01:41,880 --> 01:01:44,490 and they can take this very complex model 1128 01:01:44,490 --> 01:01:48,100 and simplify it even more by doing something like this. 1129 01:01:48,100 --> 01:01:51,050 So you take a soma and a dendrite. 1130 01:01:51,050 --> 01:01:53,390 You can basically just break off the dendrite 1131 01:01:53,390 --> 01:01:56,048 into a separate piece and connect it to the soma 1132 01:01:56,048 --> 01:01:56,840 through a resistor. 1133 01:01:59,720 --> 01:02:02,390 Now, we can simplify this even more 1134 01:02:02,390 --> 01:02:07,470 by just turning it into another little module, 1135 01:02:07,470 --> 01:02:11,490 a little compartment, that's kind of like the soma. 1136 01:02:11,490 --> 01:02:14,280 It just has a capacitor, and a membrane resistance, 1137 01:02:14,280 --> 01:02:16,810 and whatever ion channels in it you want. 1138 01:02:16,810 --> 01:02:19,290 And it's a dendritic compartment that's 1139 01:02:19,290 --> 01:02:22,530 connected to the somatic compartment through a resistor. 1140 01:02:22,530 --> 01:02:26,040 And if you write that down, it just looks like this. 1141 01:02:26,040 --> 01:02:28,080 So you have a somatic compartment 1142 01:02:28,080 --> 01:02:31,140 that has a somatic membrane capacitance, 1143 01:02:31,140 --> 01:02:35,085 somatic membrane conductances, a somatic voltage. 1144 01:02:35,085 --> 01:02:37,710 You have a dendritic compartment that has all the same things-- 1145 01:02:37,710 --> 01:02:40,893 dendritic membrane capacitance, conductances, and voltage, 1146 01:02:40,893 --> 01:02:43,185 and they're just connected through a coupling resistor. 1147 01:02:46,480 --> 01:02:49,990 It turns out that that very simple model 1148 01:02:49,990 --> 01:02:54,430 can explain a lot of complicated things about neurons. 1149 01:02:54,430 --> 01:02:57,190 So there are some really beautiful studies 1150 01:02:57,190 --> 01:02:59,950 showing that this kind of model can really 1151 01:02:59,950 --> 01:03:05,800 explain very diverse kinds of electrophysiological 1152 01:03:05,800 --> 01:03:07,090 [AUDIO OUT] neurons. 1153 01:03:07,090 --> 01:03:16,570 So you can take, for example, a simple model of a layer 2/3 1154 01:03:16,570 --> 01:03:22,540 pyramidal cell that has a simple, compact dendrite. 1155 01:03:22,540 --> 01:03:24,500 And you can write down a model like this 1156 01:03:24,500 --> 01:03:27,470 where you have different conduct [AUDIO OUT] dendrite. 1157 01:03:27,470 --> 01:03:31,850 You have Hodgkin-Huxley conductances in the soma. 1158 01:03:31,850 --> 01:03:34,202 You connect them through this resistor. 1159 01:03:34,202 --> 01:03:35,660 And now, basically, what you can do 1160 01:03:35,660 --> 01:03:37,910 is you can model that spiking behavior. 1161 01:03:37,910 --> 01:03:42,180 And what you find is that if you have the same conductances 1162 01:03:42,180 --> 01:03:45,060 in the dendrite and in the soma but you simply 1163 01:03:45,060 --> 01:03:50,922 increase the area, the total area of this compartment, 1164 01:03:50,922 --> 01:03:54,820 just increase the total capacitance and conductances, 1165 01:03:54,820 --> 01:03:56,280 that you can see that-- 1166 01:03:56,280 --> 01:03:59,050 and that would model a layer 5 neuron that 1167 01:03:59,050 --> 01:04:01,000 has one of these very large dendrites-- 1168 01:04:01,000 --> 01:04:04,780 you can see that the spiking behavior of that neuron 1169 01:04:04,780 --> 01:04:06,490 just totally changes. 1170 01:04:06,490 --> 01:04:09,400 And that's exactly what the spiking behavior of layer 5 1171 01:04:09,400 --> 01:04:11,920 neurons looks like. 1172 01:04:11,920 --> 01:04:18,610 And so you could imagine building a very complicated 1173 01:04:18,610 --> 01:04:21,570 thousand-compartment model to simulate this, 1174 01:04:21,570 --> 01:04:23,760 but you wouldn't really understand much more 1175 01:04:23,760 --> 01:04:25,530 about why it behaves that way. 1176 01:04:25,530 --> 01:04:27,630 Whereas [AUDIO OUT] a simple two-compartment 1177 01:04:27,630 --> 01:04:30,960 model and analyze it, and really understand 1178 01:04:30,960 --> 01:04:32,940 what are the properties of a neuron 1179 01:04:32,940 --> 01:04:35,070 that give this kind of behavior as opposed 1180 01:04:35,070 --> 01:04:37,390 to some other kind of behavior. 1181 01:04:37,390 --> 01:04:39,480 It's very similar to the approach 1182 01:04:39,480 --> 01:04:45,110 that David Corey took in modeling the effect of the T 1183 01:04:45,110 --> 01:04:51,540 tubules on muscle fiber spiking in the case of sodium-- 1184 01:04:51,540 --> 01:04:55,170 failures of the sodium channel to inactivate. 1185 01:04:55,170 --> 01:04:57,030 That was also a two-compartment model. 1186 01:05:02,600 --> 01:05:04,580 So you can get a lot of intuition 1187 01:05:04,580 --> 01:05:07,490 about the properties of neurons [AUDIO OUT] 1188 01:05:07,490 --> 01:05:11,840 simple extensions of an additional compartment 1189 01:05:11,840 --> 01:05:14,820 onto the soma. 1190 01:05:14,820 --> 01:05:20,530 And, next time, on Thursday, we're 1191 01:05:20,530 --> 01:05:22,840 going to extend a model like this 1192 01:05:22,840 --> 01:05:29,410 to include a model of a [AUDIO OUT] 1193 01:05:29,410 --> 01:05:33,390 So let me just remind you of what we learned about today. 1194 01:05:33,390 --> 01:05:36,890 So you should be able to draw a circuit 1195 01:05:36,890 --> 01:05:41,270 diagram of a dendrite, just that kind of finite element picture, 1196 01:05:41,270 --> 01:05:45,500 with maybe three or four elements on it. 1197 01:05:45,500 --> 01:05:48,240 Be able to plot the voltage in a dendrite 1198 01:05:48,240 --> 01:05:50,970 as a function of distance in steady state 1199 01:05:50,970 --> 01:05:52,770 for leaky and non-leaky dendrites, 1200 01:05:52,770 --> 01:05:56,770 and understand the concept of a length constant. 1201 01:05:56,770 --> 01:05:59,500 Know how the length constant depends on dendritic radius. 1202 01:06:03,030 --> 01:06:05,915 You should understand the idea of an electrotonic length 1203 01:06:05,915 --> 01:06:09,120 and be able to say how much a signal will 1204 01:06:09,120 --> 01:06:13,260 decay for a dendrite of a given electrotonic length. 1205 01:06:13,260 --> 01:06:15,210 And be able to draw the circuit diagram 1206 01:06:15,210 --> 01:06:16,590 of a two-compartment model. 1207 01:06:16,590 --> 01:06:20,600 And we're going to spend more time on that on Thursday.