0:59

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.

2:42

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?

4:34

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.

6:25

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.

8:05

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.

11:12

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.

18:37

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.

20:23

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.

22:23

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.

25:42

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.

28:50

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.