So, here's what we learnt to model, a compact cell, and here's what real

neurons really look like. Here's even quite a simple version.

So, as with the complexities of ion channel dynamics, what is the appropriate

level of description of a single neuron that's necessary to understand brain

operation? Because we don't yet know the answer to

this, and there probably is not one answer to this, it's important to pursue

models with many different approaches. So, here I'll be introducing you to the

techniques that one can use to handle dendrites, and some ideas about what they

may contribute to computation. So let's start by looking to what extent

dendrites feel what's going on in the soma.

So here, this is an impulse point that's being put in at the soma.

Let's see what that input looks like when it reaches the dendrite.

You can see that it's both delayed, it's reduced in amplitude, and it's broader

Similarly if we put an input here, add in the dendrite, and we look at what happens

at the soma in response to that input, you also see that it's much reduced in

size and it's broadened out. Furthermore, how thin the dendrite is

affects how big a voltage change you could make with a given amount of current

input. The thinner the dendrite the larger the

voltage change but generally the further away the the more that input gets

filtered and attenuated. This tells us the inputs that come along

different parts of the dendrite can have very different effects and very different

influence on firing at the soma. As you can image this can have a

tremendous impact on the information that is integrated and representated by the

receiving neurons The theoretical basis for understanding voltage propogation in

dendrites and axons is cable theory, which was developed by Kelvin in quite a

different context. The voltage, V, is now a function of both

space and time, which means that we're now dealing with partial, rather than

ordinary differential equations. So here's the setup.

We now think about a tube of membrane with sides have the same properties as

our membrane patch. They have both capacitance and

resistance. So we see little elements that look a lot

like our, like our previous patch model, but now they distributed down a cable.

There's additionally the resistance of the cable interior.

Current can flow along the cable as well as through it.

So generally we're not going to worry about the external medium here.

We'll just take it to be infinitely conducting with a resistance of zero.

So the cable equation for a passive membrane, we're not going to deal with

ion channels for now, is derived by considering the changes in current as a

function of space. The current down the cable will be driven

by steps in voltage as a function of x. So, if we have a voltage difference

between two points in the membrane, that's going to drive a current down the

membrane. Of course, current is also passing out of

the membrane. That's the im that we modeled previously.

Now when one puts those 2 things togteher dealing with the way current flows out of

the membrane and the way that it flows down the membrane 1 obtains an equation

that is actually 2nd order in space so it has a 2nd derivative with respect to

space. So this half of the equation you 'll

recognize that we've seen before of the passive membrane now we have an

additional term that, that includes a spacial derivative.

So, some of you will recognize that this equation is not unlike the equation that

describes diffusion or heat propagation, but it has this additional term, so this

part looks like diffusiion, has this additional term that's linear in v.

You might remember that when we rewrote the RC second equation for the passive

membrane to find the time constant of the membrane that gave us sort of the

fundamental time scale of its dynamics. So there's something very similar in this

case too we can rewrite the equation in this form where we bring together all the

dimensional quantities. This will ask us to read off the natural

time scale so going with this time derivative there's a a time.

Constant which is our tow M and now when we look at the, the spacial derivative

this has units of 1 over space squared and there's a space constant out the

front lambda that carries the typical spacial scale from the coefficient of the

space derivative. That's given by the square root of the

ratio of the membrane resistance divided by the internal resistance.