So here is Wilfred Rall, my own grand eh, eh great mentor. And Will Rall developed what we call the Cable Theory for Dendrites. So I want to explain to you a little bit about the cable theory and, and the idea is really of the cable theory. The purpose of Rall's Cable Theory for Dendrite is to understand mathematically the impact of remote dendritic synapses, the input as you know, synapses impinges, impinge as a whole on the dendritic tree. So the sum of them are remote from the output region which is in the soma/axon. So Rhode tried to develop a theory, a mathematical theory, explaining how remote dendritic synapse affect the output, the spike generation in the axle. So Rall, actually, in '64, wrote this paper, he wrote the original paper, it was from '59, but in '64, he wrote this paper, showing the contrast, the dramatical contrast, between the schematic neuron, the McCulloch and Pitts neuron, which is essentially a point neuron, summing on this point. Summing on this point, all the synaptic input. This is the McCulloch and Pitts neuron. So this is a schematic neuron, and this is a real neuron. You already saw these purkinje cells, reconstructed by Romonica Hall, and other cells. So this is the real neuron, the distributed neuron, the histological and anatomical neuron and, and so Rall felt of course, that there must be some kind of a dissonance between this simplification, which could be important for certain things, but maybe certain things are missed if you don't into account. If you don't take into consideration the full extent of the dendritic tree, with all the synapses that impinge on this dendritic tree. And, and the first intuition for Rall came from the following basic fact: Suppose you have a neuron, so this is the cell body of a neuron. These are the dendrites emerging from the cell body. And suppose you inject current into the cell. Or suppose that current comes from synapses that are located remotely. So current flows into the cell. With a pen and pencil you can see, and a paper, you can sh-, you can very see that most of the current, most of the membrane current, does not flow through the membrane of the soma. But because you have so many, so many, dendrites and path for current to flow out, most of the current that arrives to the soma, or that is injected directly to the soma, most of the current is not flowing through the membrane of the soma but rather outside, away from the soma. This means that you cannot think about the soma as an isopotential when it is located in a real neuron where so many dendrites pop out from the soma. So you cannot think about the neuron as the isopotential point neuron. It's not correct to think like this. Maybe it's the first approximation. But atomically, physiologically, it's incorrect. That's what Rall was trying to show us when he drew this schematic, but realistic way of thinking about current flow from soma out. Most of the input current flows into the dendrites, and not into the soma membrane. Okay, so what does it mean? It means the dendrites are not isopotential electrical devices. It must mean that voltage attenuates from the sym-, s-, synapse to the soma, we'll discuss it in a second. Because the dendrites and the soma and the whole system is not isopotential, there are some location with one potential and another location with another potential. So it's non-iso-potential. It's a distributed electrical system. Rall already understood that because you have a distributed system, there must be delay. It will take time from the synapse to reach the soma. So the sima, synapse may be active there. It will take some time if we have a delay to see the effect of a synapse at the cell body. And eventually, the location of the synapse, whether it's near the cell body, or whether it's away from the cell body must make some effect. Must have some effect on the overall integration of inputs coming to the cell. So to show it schematically, if you have a cell, a dendrite. Rall look at the denditric tree as composed of set of connected cylinders. So this distal part from the soma is probably this, this, this cylinder, this other region, is this cylindrical membrane, so this is a distributed system. A distributed system whereby synapses may originate here or here or here, not at the cell body. And he was trying to understand, what does it mean in terms of electrical flow of current. What does it mean to have a distributed system like this represented as a set of cylindrical membranes connected to each other to form this particular geometry, with this particular diameter and particular length, diameter, length, it has geometry. So, let's look very briefly at Rall's cable theory ideas, very schematically. So, suppose you have a cylinder. This cylinder is composed of membrane, wrapped with the membrane, and there is, inside of the cylinder, inside of the dendrite, some axial resistivity. Okay, so inside the dendrite, inside the cytoplasm, of the cell, inside, there are ingredients also inside that behave like a resistance, axial resistance. Okay? And suppose that I am activating the synapse. This is a red synapse activated at this location here. And, already you know that when a synapse is active there are opening of channels, ion channels. And maybe ion channels in this case will flow inside, from outside inside, like this arrow shows. So this is the origin as we discussed before, the origin of currents flowing from the outside, into the, into the cell. Cross-membrane. From out, in or in, out, depending on the type of synapse. So you inject the current into this location, and then this current starts to flow. It can either flow to the right, or it can flow to the left. And then some of this current leaks out. So you can see that this axial current, some of it leaks out through the membrane because you know that this membrane behaves like an RC circuit. And in this RC circuit, current can escape. Through the resistance. Or can charge the membrane capacitors. So you lose some of the current here. Some is left, flows to the right. Some is lost. Some continues to axially flow and lost and so on. So you can see, that just thinking about a cable, cylindrical cable, or what Rall called the core conductor, it's a cable, it's a cylinder with a resistance inside, means that you lose current. And because you lose current, the voltage here through the membrane, the membrane voltage here and the membrane voltage here, and so on, will attenuate. There will be less and less voltage because you have less and less current charging the membrane as you go away from the synapse. This is the origin of cable theory. Okay so you have membrane current that is lost this is the blue current, that is lost through the membrane, and you have the axial current, flowing, axially. And eventually this will mean that if you inject here a synapse or a current, you will go locally. You will get the maximum voltage here, and this local voltage will attenuate along your structure as here. It will attenuate along the structure, and then it will encounter bifurcation maybe in the dendrite. It will continue to attenuate. And, and the question is, how do you describe mathematically a cable, a passive cable. So the membrane is passive RC membrane passive membrane. How do you describe the attenuation of voltage in a bifurcating dendritic tree, passive dendritic tree. So I'm not going to go into the details of the mathematics here. It's a whole lesson about ca, or several lessons about cable theory. But the idea is really mathematically Intuitively is the following. You have an axial current that is proportional to the derivative of voltage with distance. Here, dV dx. So this is the axial current, coming in from left. This current, the axial current, is lost. Some of it is lost through the membrane. And we know how to describe membrane current because membrane current in ways we know, is composed of capacitative current and resistive current. So the ask, axial current is either becoming membrane current or continuing to the right. Yes? As we saw before. This means that, that, that losing an axial current, the change in axial current, at each point is actually the second derivative. This is just the axial current, and I want the change in axial current, so you have to, once again, take the derivative of the axiom current and this will give you dV squared to dX squared, because you derived it again, you [UNKNOWN] it again. And so if you want to eventually think about the cable equation, it will look something like that. Okay, you will say that the change in axial current, the change in axial current, which is proportional to the second derivative of voltage with distance, is equal, basically to the membrane current, c dV dt plus V divided by r. Okay, so this is the membrane current, this is the change in axial current and they are equal. And so the sum of the change in axial current plus the membrane current must be zero, unless you inject current from the outside. This is the foundation for Rall's cable theory. It's a passive cable theory, because all the parameter, the membrane resistivity, and the capacitance are passive, are static. Okay, so this is a partial dif, linear partial differential equation. Because this is linear, and this is linear. And it's partial differential equation because you have x and you have t. This means that the voltage changes at each, each location. The voltage changes with distance and with time. Okay, and this is the dimensionless cable equation, and we don't need to go into the details, but this linear partial differential equation can be solved analytically. If you have the initial conditions for solving this equation, V0 at some location, or I0 when you inject I to some location, and if you also have the boundary condition. Is it an infinite cylinder? Is it a closed, short cylinder with some boundary at the end? So you need two additional information, the initial condition and the boundary condition in order to solve this equation. And that's exactly what Rall did. So we solve this differential equation, the cable equation, for different boundary conditions. I just want to show you one example here just to let you some intuition about what does it mean to solve the cable equation for particular boundary conditions. Let's say that you're initital condition is at some location of the cylinder. It's a cylinder. Is some one voltage at some location is one. At location zero, the initial condition is normalized to one. We are now looking at the steady-state solution, which is easier. Steady-state means that there is no change in time, so you take this out, and you only look at the change in space, in distance, from the location, from the initiation of some voltage at this location. And you can see that the voltage indeed attenuates with distance. From the site of injection, a way and, and the, and the slope and the shape of attenuation depends on the property of the cable. For example, in this case, the membrane cable is infinite long, infinitely long. In this case, for example, the cable ends after a certain distance here. And it ends with a sealed end, the boundary condition. The boundary condition at the end is sealed. Meaning that there is no axial current at the boundary. So there is a boundary. The current cannot cross the boundary. So this is a sealed end boundary condition. DV DX, the axial, the axial current, is zero at X equal L, here. So you can see that the slope of attenuation very strongly depends on the boundary conditions at the end of the cable. For the infinite case, It attenuates exponentially with distance. For sealed-end short cylinders, like dendrites are, it will attenuate less steeply with distance. As you can see here, or here. So that's the general idea, of solving the cable equations for dendrites. Eh, taking into account the boundary conditions. And here is the most fundamental, the most important solution for the cable equations for, a branched dendritic tree model here. So this is the model, and you can see something important. That Rall and Rinzel and Rall, actually, Rall and Rinzel in '73 already solved the case for this idealized tree, and here is what they showed. Suppose you have a branched tree like this. And you inject current here, at the very distant tip of your dendritic tree, the passive dendritic tree. So you start with a certain voltage here. So this is voltage. And you see that there is a very very steep attenuation here. From your input side to the first branch point, here. Then, when the voltage or when the current reaches here, the injected current reaches here, some of it will go to the sibling branch s. And some will continue to the father branch. So very steep attenuation in the input branch. And very shallow attenuation to the sibling identical branch. As, as the input one. So these are twin branches. Then again steep to the farther branch, then to the next farther branch and then eventually it reaches the soma. So what do we see here? We see very steep attenuation at the input branch meaning that if the synapse was sitting here, activated here, locally it would generate large voltage. Because the input resistance locally is big. So locally, the injected synaptic current will enable to build up a large voltage. Especially if it's on the spine head. But from the input side, very fast, very nearby, you will get a very steep attenuation. And this is, by the way, because at the boundary here, at the boundadry of this short cable, you have a leaky end. There is an increase in diameter, so a lot of current flows into this increased diameter. But a very little current flows through the side branch because the side branch here is sealed. So, it's a very short cylinder with sealed end. That means that there is almost no attenuation, almost no attenuation. You can see that when the branch is very short, there is almost no attenuation towards the sealed end and that's exactly what happened there. So you can see the big asymmetry in attenuation of dendritic trees in dendritic trees. Although these two branches are identical, the boundary conditions are not identical for current flowing this direction into current flowing this direction. There is steeper generation here, very shallow generation here. And eventually some of this current reaches the soma. So this is a very highly non-isopotential system. Very large voltage near the, near the synapse and very low voltage, eventually at the soma. Let's say that near the synapse, it could be 20, 30, maybe 40 millivolts, millivolts. At the synapse post synaptically, local PSP will be big, but at the soma, eventually we will see a very small EPSP, maybe one millivolt. So in this case, 40 fold attenuation, maybe hundred-folds attenuation from the input site to the soma. That's one property of dendrites. The other interesting property, I'm not going to extend much about this, is the fact that if you take the same synaptic current And you inject it directly to the soma, here. Instead of injecting the current here, the synapse is now directly, the electrode is now directly at the soma. You see that you don't lose very much in terms of voltage if you compare what you would gain at the soma now with direct attenuation to the soma compared to what remains of the soma from the distal input. So locally its very big, and eventually you get of the soma less than you would get the soma if you inject directly in the soma, but the difference is not so big. So you don't lose much charge because most of the charge that you inject here eventually reaches the soma. But, locally, you have much larger voltage than at the soma. That's the important thing here. So this is one of the main results coming from, from Rall's Cable Theory. And so we can think now of the dendrite as essentially building from, because it is electrically distributed, from building from sub regions. So for example your syn, if your synapse is here, this local region will feel large voltage. So now that this is a color coded, red means that all local region feels the same voltage, and the distal region does not. If your synapse is near the cell body here, all this region will see all this soma and basal dendrites will see about the same voltage. But distally this voltage is not being seen. So each synapse has a neighborhood, a neighborhood territory or a neighborhood like a unit or sub-region that is affected by this synapse very strongly, locally. And then the distal part is less affected by this synapse, so we may start to think, and I'll show you later that you can use this notion of functional sub-units, regional, regional sub-units is doing specific computations. You can use this distributed, this distributed electrical system, whereby something here is not so much felt by something there, to subdivide the dendrites into functional subunits. I want just to highlight, just to full completion, I want just to complete this issue about the analytics and mathematics of cable theory, showing you that you can also solve the full equation, the full equation. You can also solve both in space and in time. And eventually you'll see something like this. If you inject the current here, at some time t, the distribution of voltage with space at this time will look like this. For a symmetrical infinite cable, with time if you look at the next time step, you see that everything [UNKNOWN] goes down, and the voltage starts to distribute distally, and with time it becomes more and more isopotential, with time. If you wait enough time, everything goes down. And because current leaks out. And eventually, you go back to resting, which is a iso-potential system at rest. But let me show you a very important thing that we are going to use very, very soon. A very important thing is the fact that, as the role already predicted by understanding the cable equation without even solving it. That, it, it, that the shape of the EPSP, if you inject current at some location, X equals 0, and you look at distance from where you inject the current, you see that the voltage, at distance from your input, has a shape that is changing with distance. As you go away from our injection point, the [UNKNOWN] attenuates, as we discussed. It takes more time, it is delayed, toward the distal point, and it becomes broader. Your EPSP becomes broader as you go away from the synapse. So this is a thinner EPSP, a broader EPSP, a broader EPSP as you go away from the synapse. And Rall, very beautifully, made use of this property. So here I show you a result, a theoretical result coming from the cable theory. It was extremely, extremely useful for experimentalists. So, this is the power, as I mentioned before, of a good theory that gives you a prediction. The prediction is, the distal synapses. Distal synapses are broader and delayed compared to proximal synapses. Here is an example. So here is a cable. Compartment number one is the soma. This is away from the soma. That this is a simplification of this complicated dendritic tree. So the soma is linked to compartment number one, and the distal dendrites, now, are all collapsed to compartment number ten, so this is now a simplification of the complicated dendritic tree. Okay. And suppose you record at the soma. I record always here, but I inject my synapse once directly to the soma here, here, here and here, and I record always at the soma. And now I normalize the synapses here, just to show you the shape. Of the EPSP once when record, when injected into the soma, or to compartment number one here, once when injected to compartment number four here, once when injected to compartment number eight always recorded at the soma. And when I normalize it, of course this one is attenuated. The distal one is attenuated, if you inject the same current. But, if I normalize all the peaks at the soma just to compare the shape, you can see how the distant synapse, the one that comes from compartment eight to the soma the distant synapse is delayed, is delayed, and is much broader. [SOUND]. Then the synapse that is directly sitting on the soma. It is not delayed. It is immediately at the soma. And it's less broad. It is not as wide, not as broad as the EPSP from distal. So this is broader and this EPSP is briefer. So Rall said, okay, now you record an EPSP at the cell body. You activate a certain axon and the synapse is active. And you want to know where the synapse comes from, where it is coming from. You can plot this curve. Here you can plot the time to peak. The time to peak from here to here or from here to here the time to peak of EPSP at the soma. And here you can plot the half width, the width at half amplitude here or the width of half amplitude here on this axis and you should get a curve like this. From the distal synapse, the time to peak is delayed. It takes time to reach the peak. It takes time to reach the peak, and also the half-width is big. If your synapse is near the cell body, the time to peak is brief, and also you're half-width is small. So this is the fantastic success of a theory because people started to use the shape of the EPSP from cell body as if there is a huge tree. You see to the cell body, record only here, only here. And suddenly, you can guess, you can predict. Where is the origin of the synapse on the dendrite? Is the synapse near you? Or is the synapse distal from you when you see to the soma. This is a big success of, of a theory. It is still a theory. But eventually, later on, a group from Australia, Steve Redmond and others, succeeded to really now both record from the cell body and also reconstruct the location of a synapse. And so they knew where the synapse is. And they compare the location to the prediction of the shape of the EPSP. They found a beautiful match between the theory and the experiments. So today we can see that the cell body record an EPSP and guess very well where the synapse sits on the dendrites. So this is really a very compact, fast summary of the cable equations. Or the implication of cable theory, of the dendritic cable theory of Rall, to try to understand the implication of the location of the synapse as seen in the cell body at the output site. Very important. Now, I want to go into using these intuitions using these mathematics and try to understand now or look at the neuron, not as a biophysical element, as I just did, but as a computational element, as I just said, that we want to jump from the biophysics into the computation at the single cell level.
