So before I go into the development of ideas based on mathematics of dendrites,

I want to repeat something that I already mentioned in the first lesson.

I want to repeat the importance of mathematics in understanding complicated

systems. So again, I want to discuss why model

mathematically, why use mathematics as a tool to model complicated systems like

the brain. So this is again, Lord Kelvin, and Lord

Kelvin said something again, very important.

He said, I'm never content until I have constructed a mathematical model of what

I'm studying. If I succeed in making one, I understand,

otherwise I do not. So the basic claim here is really, that

if you want to describe mathematics eh, eh, physical system like the brain or any

other complicated physical, physical system, words and graphs and data

collection is not enough. Basically, you have to approach it with a

very rigorous, very systematic mathematical approach in order to

compactly describe the system. I already showed you the Hodgkin-Huxley

model. I hope you are convinced that this

Hodgkin-Huxley model really made a jump, conceptual jump in understanding the

spike. The spike was always there, people

recorded the spike, including Hodgkin's and Huxley.

But eventually, only after they wrote the mathematics of the spikes, we can very,

very clearly say that we understand the spike.

So, let me summarize three highlights or three basic aspects of why to model

mathematically, because there are levels of why using mathematics.

Okay. So let me say a few words about why

modeling in general, mathematically, and in particular, why, why taking into

account details. We'll discuss later the details.

But I want to discuss the general notion of modeling or theory and also discuss

the issue of details. So, so, what, what are the, in general,

three aspects of reasons for doing mathematical modeling of complex systems?

The first thing, and the very clear thing is that you want somehow to interpret

your, your details, your experimental findings.

So you find something experimental and you want to have some interpretation of

this experiment. And not only interpretation, but you also

want to have some predictions. So you take all that you have, like in

the case of Hodgkin-Huxley for the squid axon, you take a lot of experimental

results. And you want to give this experiment some

meaning, some interpretations in order to cross from the details that you measured

from the microscopic details of, in case of Hodgkin and Huxley, the ions, the

membranes, conductances, and so on, to the macroscopic phenomena of the spine.

So you want to have this interpretation of how the details, how the experiments

explained the phenomena, and not only that.

What kind of predictions can you do using the model that you build?

So the purpose of a good model is not only to replicate in a compact way the

experiments, but also to provide some predictions.

And indeed, Hodgkin-Huxley did predict the refractory period.

They did predict, and we did not discuss it, the spike velocity within the axon.

They predict aspects that were not directly pointing to the model.

So interpretation and predictions are a very important part of a good model.

The other part of the good model is the issue of finding key biophysical

parameters. So, the emphasis is on the word, key.