Okay, so we come to the third talk of today and the last talk, which so in the previous

talk we saw that we had networks of neurons and in the second talk we had the main feed

and then in this talk we will focus on the microscopic scale so we will just look at

one neuron that is excited by some bounded noise. So experimentally it has been seen in the lab that

some neurons that have the same physiological features which are better by some sort of

I mean some synaptic input that are either the same or maybe have some similar distributions

but the neurons are observed to encode completely different activities and this question has been

a bit interesting in the experimental community. So in this talk we are going to look at some

ways for which neurons can encode information using noise. There have been some mechanisms

that have been developed so there's what they call stochastic resonance, covariance resonance

and so on but in this talk we will focus just on two and the goal of this talk is first of all to

wonder so we'll focus on two of them so focus on certain stochastic resonance and inverse

stochastic resonance which are two phenomena that encode kind of opposite information so

I will explain what they are. So the goal of this talk is first of all to independently establish

the mathematical conditions that are necessary for neurons to encode information to serve

in this stochastic resonance which are the assets of s and inverse stochastic resonance

and then with that understanding we will try to see how the neuron could switch from encoding one

type of information that is due to self-induced stochastic resonance and the other which is due

to inverse stochastic resonance in the same noise limits or like in having the same distributions.

Okay so here's the output of the outline of my talk so I will present the Shizuoka-Nagumo

model that was used in the first talk so it comes again here the stochastic version of it

and then we look at serving the stochastic resonance what it is what are what are the

conditions necessary and then inverse stochastic resonance we have some of it and then we look at

some of the problems and future research. Okay so here we have a stochastic Shizuoka-Nagumo model

which is a slow fast dynamical system so you can see here in this case here it is we have

what an epsilon parameter here which is very important parameter it is what's called a singular

parameter it's small so you can write it just gonna go more in this way where here you have

some sort of notificative noise and motion which is written here on the slow timescale tau which

you can use the definition of epsilon to scale even to the fast timescale t so these two dynamical

equations are topological equivalents so trajectories of the first one you can always find

homomorphism that translate the trajectory of the first first dynamical system to the other so you

could choose to study the first one or the second one depending on what you want but

we will see later on that the singular parameter plays a very important role in studying this type

of behavior so for the Shizuoka-Nagumo we have equation one here the electrophores are given by

this one which is a bit different from what we saw in the first talk in the first talk we realized

that the second in the second vector field was was just a v plus k but here we enrich the dynamics

by adding more terms of course which preserve the spiking activity of the Shizuoka-Nagumo and you

you see why this is very very important and then v here is the membrane potential that is the that

is the variable that in course information that we use to store current variable I will not get

into the biology of those but I was just want to show that w is a slow variable because epsilon

here is a small parameter so the velocity vector field of g here is actually slow dynamics and then

of course we have two other important parameters a here and then c here is the quarter mission one

for bifurcation parameter and then we have our multiplicative noise and our additive noise that

can be either this term in the slow time scale or just sigma in the past time scale as you might

know in stochastic processes the banyan motion rescale that's quite out of time but in this talk

of course for simplicity we will consider just additive noise and then the o vector b is just

a stochastic integral with respect to our banyan motion here defined on certain probability space

and for simplicity again we are going to use bounded noise yeah yeah so I mean the banyan

motion is bounded noise so that's what we're going to focus on this on this one current work

that we're piercing on the archive in which we use unbounded noise which is quite interesting but