0:02

In this final part of the lecture, we're going to continue to go beyond

Hodgkin-Huxley, but this time, in this direction.

We're going to be going back to biophysical reality and addressing the

issue of geometry. How do the complexities of neuronal shape

and structure affect our computation? In the first lecture we invested heavily

in understanding the spike generation process in a patch of membre here at the

easy end. But its a little embarrassing to zoom out

and look at real neurons which have a truly extraordinary range of beauty and

complexity in the geometry of the dendritic abras.

So for moving toward building biophysically realistic models of neuro

processing. It would be good to know how these

structures can contribute to the processing of information.

0:46

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.

6:28

Now let's put in a brief pause and see how that behaves at T equals 0 we put in

a spike of input at X equals 0. Here's what happens to it, the input gets

broader, spreads out spatially and also decays in time.

This is a lot like diffusion. If you spray a pulse of perfume,

somewhere in a room, and were able to watch what happened to it, it would do

something similar. It would spread out with the same spatial

profile. But for the perfume, there's always the

same amount of perfume. If you were to add up all the molecules

of perfume in the smeared out blob, it would be the same as you started with.

For the voltage, that's not the case. The total voltage signal is decaying away

in time, because of that first order part in the differential equation.

That is, the total voltage is decaying exponentially, just as it did in the RC

circuit. So what are we going to see if we sit

some distance away, and observe the change in voltage?

That's what's shown in this figure. So these are different curves that plot

the voltage that's observed at different locations.

X equals 0, x equals .5, x equals 1. These are all in units of the space

constant of the membrane. At different times, and so sitting at x

equal 0, we see an exponential decay. As we sit a little further away we're

going to see that. That, that pulse of voltage change first

rise, then seal the decay away again. So you can see how rapidly the signal

decays as a function of space. So at about, at one space constant you

still see a reasonably large Deflection in voltage that's caused by that pulse.

But at two space constance away, the signal, the size of the, of the pulse

that we see there is down to five percent of the original.

8:55

So for some of you who like to see general solutions here's how the voltage

responds to a pulse of input at time T equals zero and position X equals zero

looks at position X and time T. We can see that this solution is made up

of two parts. So here's the diffusive spread for this

Gaussian profile. That's very similar to the way things

spread diffusively. And mutliplying that, there's this

pre-factor that has an exponential decay. So, knowing the solution, one could take

some arbitrary pattern of inputs, decompose it into pulses, as we, as we

put in here, at different central locations.

Say, T prime and X prime, and add together a weighted sum of this solution

form. We've centered up those differnt

locations and times. So now we know how to find solutions for

a very long pass of cable with a fixed radius.

In fact doesn't get us very far in dealing with the real neuron, because of

two things, the inter current branching of dendrites and the fact that many

dendrites are not passive but are active. That is, they have ion channels in them.

So it quickly becomes very difficult to solve anything analytically.

The path forward is by dividing the dendritic arbor into what's called

compartments. So here's an example, one can approximate

the dendritic arbor as a coupled set of compartments.

So these, we're going to break this dendrite into little sub regions In which

the radius and the ion channel density is taken to be constant.

Each compartment will then have an equation that only depends on the time

derivative of the voltage, and not on x. The spatial dependence is incorporated by

coupling each compartment together. So, so [INAUDIBLE] Rall device is a

helpful way to approximate complex dendritic trees of.

Let's consider a branch that divides into 2 daughters.

It turns out that if the diameters of these two branches have this particular

relationship to the diameter of their mother so if the if the diameter of 1 of

them raises to 3 halves plus the diameter of the other also raised to the 3 halves

is equal to the diameter of the mother raised to the 3 halves.

What that means is that these two branches are impedance matched to this

branch, and one can simply accumulate them all together into one long branch of

the same, of the same diameter as the mother.

So one needs to thus compute the effective electrotonic length of these

two additional branch elements. And extend the original cable by that

much. So you can see that one can continue to

do this iteratively. If the same property holds here, lennox

can extend that branch out to an effective branch of this, of D2 diameter.

And then one can agormate those two together.

Until one eventually has a single, a single cable coming out of the soma.

So it turns out that this condition on the diameters it approximately satisfied

by real dendrites and even when it's not exact the resulting approximation is

often quiet accurate. So the role models are very useful for

passive membranes But it doesn't address the issue of ion channels which make the

problem nonlinear. Furthermore ion channels densities often

very along dendrites which can lead to a lot of interesting effects that one might

like to explore. so here's the full approach, given the

geometry and the ion channel density of the dendritic tree One can divide it

where the properties are approximately constant.

12:36

One can then write down equations for the membrane potential in each compartment

individually. So let's say compartment 1 we represent

here in terms of a, a similar second model that we saw for the passive

membrane we're going to give that the voltage V1 and now right down an equation

for V1. We can do the same for compartment two

and compartment three, here. These equations will be similar to the

passive membrane equations we looked at for Hodgkin-Huxley, but with the

individual ion conductances, membrane resistance and capacitance set

appropriately for each piece of the cable.

Furthermore, there'll be also two tons that couple the compartment with it's

neighbors. The current input from the neighboring

compartments. Which depends on the voltage difference

between the two compartments and a fixed coupling conductance.

So here for example, let me write down an equation for for v two.

We're going to need to include a current that's coming from, from compartment one

that's going to go like g one, two multiply it by V1 minus v2.

And similarly, there'll be a current that comes into compartment one from

compartment two that's going to have a different coupling conductance and the

opposite voltage difference. These fixed turns, these coupling

conductances, depend on the area of the connection, and whether or not the

compartments straddle a branching point. So if you notice, that in general, these

coupling conductances are not necessarily symmetric, which is why there are 2

values at each of these connections. So there are many models like these that

have been built from microscopic reconstructions of single neurons.

And a great many have been made publicly available on the ModelDB site maintained

at Yale. So if you're in the mood to go explore a

dendritic forest, there's plenty out there, so don't forget your adventure

hat. So what do den-, so what do dendrites add

to neuronal computation? There are many proposals for ways in

which the filtering and active properties of dendrites can work, to shift the way

in which incoming information is Clearly, where an input arrives on the tree can

influence the effective strength of the input because of the passer properties.

Interestingly, it's been found that in the hippocampus, neuronal dendrites have

solved this problem. So that when inputs arrive at the suma/g,

they have a very similar shape no matter where they come in.

This amazing property is known as synaptic scaling.

14:58

Filtering through the dendritic tree on the way to the sonar also determines

whether a sequence of successive inputs is integrated to build up to potentially

drive the spike or not. Where two different inputs enter the

dendritic tree can also make a huge difference in how they interact with each

other. For instance, if two inputs come in on

separate branches. They contribute independently.

While if they are on the same branch, they can sum either sublinearly or super

linearly which leads to amplification. Another very important property is that

thanks to their ion channels, dendrites can generate spikes, generally calcium

spikes. [INAUDIBLE] This leads to the possibility

that one could use coincidence of inputs, driving spikes in the dendrites.

Along with back propagating spikes from the soma back to the dendrites to drive

synaptic plasticity. This is a topic you'll be hearing much

more about in the next lectures. So, let's close out by looking at two

ideas for how dendrites might perform a computational role.

The experimental evidence supporting these mechanisms is somewhat mixed, but

the fundamental ideas stand as interesting conceptual models.

16:05

First, here's a wonderful example where the propagation of signals through cables

is thought to help out in carrying out a computation.

Nuclei in the auditory brain stem are responsible for sound localization, the

ability that we all have to locate where a sound is coming from.

The que that's thought to be used is the timing difference in the arrival of a

sound at our two years. The sound arise at the two ears at

slightly different times, and the signal those travels through the left and right

auditory pathways at slightly different times.

Imagine that these two signals are piped into the population of neurons The

thresholds are set such that they can only fire, when coincidence signals from

two different inputs arrive at the same time.

Each neuron receives the two inputs with a delay caused by traveling different

distances along the dendrites. The neuron that fires the most is the one

for which the relative timing delays, due to the timing difference between the two

ears Is compensated for by the dendritic delay.

This mechanism turns a tiny difference into a place code where by the label of

the neuron that fires, indicates the timing delay, which can then be

translated into the spatial location of the sound.

17:19

Here's a final example Neurons in the retina show direction cell activity.

That is, they respond to individual stimulus moves in one direction, and is

suppressed when it moves in the other. So how might such direction cell activity

begin constructed at the single neuron level?

Imagine that inputs from different spatial locations are coming in to a

dendrite at locations along the dendrite, arranged as in space.

As the dendrite receives a sequence of activations, you can see that if they

receive that input first, at the far end of the dendrite, and it begins to travel

toward the soma, then another input comes in, closer to the soma.

Then the influence of these two inputs can sum and build up as more and more

inputs arrive. So that the net input crosses some

threshold. On the other hand if the nearby location

is stimulated first, then the next one and then the next one and then the first

inputs will die away by the time the later ones arrived.

And you can see how, how that, how that voltage signal at the soma might behave.

So this idea was first proposed by Rall. While it may not fully explain direction

selectivity and retinal ganglion cells, the general idea does predict that the

firing probability of the neuron should be sensitive to the sequence of inputs

along the dendrite. Let's say these inputs occur in some

sequence. One could then scramble the order, and

see whether the firing of the neuron can distinguish those inputs.

Michael Hausser's lab carried out this experiment by simulating sequences of

synaptic inputs into single neurons and found that indeed different input

sequences were discriminable. Okay so here's where we wrap up my

section of the course. I hope you've enjoyed this brief

introduction into the electrical basis of neural signaling in the brain.

Roshesh is going to take it from here, to guide you through the ways in which these

basic cellular components get wired up together, to produce the amazing variety

amount of behaviors that I know the systems are capable of, through

experience and through learning. Have fun.