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Okay. Good morning everybody and thanks, Marius for inviting me to this nice mini workshop.

I will talk about population dynamics. So finite size neural populations, and with a focus on on short term plasticity.

Okay, so, and fundamental question is, how does, how does our brain work. So that's here you see a cartoon of how information may flow through brain but this is of course just doesn't tell us much.

So, we want to understand better how the brain works we need to look more closely let's say and some little piece of cortex.

And there we have a huge network of recurrently connected neurons, neurons are densely packed, we have several 10,000 of neurons in one cubic millimeter and a lot of of cables so so the brain tissue is a highly complex system.

But that's not, not all also the single cells, the neurons are very complex machineries that that receive inputs from 10,000 of other neurons and make a nonlinear processing step and and generate the most important signal in the brain namely the action potentials

or to cause spikes.

So then fundamental question is how can we understand the collective neural activity so that emerges from thousands of of interacting neurons.

So one way that researchers have used to understand this network dynamics, the brain is to build a microscopic model, and to measure physiological parameters.

So let's say the parameters of single neurons, the connectivity between different neuron types, but also as we saw in the previous talk maybe more large scale connectivity.

Then also measure the properties of the synapses or the synaptic dynamics, and also estimate the number of neurons in a given location and of a given cell type.

And then using such parameters one could come up with cortical micro circuit model that, yeah, that needs to be simulated on a computer of course because this contains a lot of neurons so here is one classical example, the model of

Portiaz and Diesman which contains 80,000 leaky integrative fire neurons organized in in four cortical layers, each layer containing excitatory and inhibitory cells.

And this can be simulated on a supercomputer and one obtains really the spikes of each cell.

And then we may see some pattern emerging in this huge network. But in order to understand how new activity emerges on this population level.

It is actually quite complicated, because this model is so high dimensional that it's difficult to understand why a certain synchronization event emerges. So it's difficult to understand how the neurons work together.

Okay, so we may want to simplify the model and one approach would be to coarse grain the dynamics and have a more low dimensional description.

How can, how could we do this? So one approach would be that we look at these spike trains, suppose we have these eight populations that we saw in these different layers.

And then for each neuron we have the spike, the spikes, and this can be mathematically described by a spike train, by Dirac Delta spike train, or as a derivative of the count process of neuron i in some given population alpha.

And then to coarse grain, we may take a time slice and count the total number of spikes in this time slice.

Yeah, so, so let this, this total number be z alpha. And then we could define the population activity as the sum over all, all the spike trains in a given population alpha.

And divide by the number of neurons in that population and alpha. And that would be, if you have a small but finite time step delta t, you could think of the total number of spikes in this time bin here, divided by number of neurons and divided by this time step.

So this is an estimation of the empirical firing rate.

Okay, so we can measure or, or extract this population activity from such a microscopic simulation where we simulate each neuron individually.

So, that could be done for, for example, this, this model.

Or, what is the goal of this talk is to find a model, a so called mesoscopic model on the population level, where we only simulate population activities directly, and this directly gives us these population activities without the need to simulate each neuron individually.

Of course, you can imagine that such a microscopic, sorry, sorry, mesoscopic model is much more efficient.

You would only need to simulate eight populations instead of 80,000 neurons.

So, one classical way to describe this populations would be via so called firing rate models.

And one would just posit that activity of given population alpha is maybe defined as follows. So this is one instance of a firing rate model.

And says, you take some nonlinear function of some variable H, which is, can be regarded as an input potential or membrane potential that is given by this first order differential equation that basically integrates up the synaptic input.

So these are the, the sum over all population activities, weighted by some synaptic efficacy J alpha beta.

And, yeah, that is a very popular model to mean field model to describe populations of neurons. And it is nice because it's, it's low dimensional it's in fact one population is just one dimensional described by one dimensional dynamics that's mathematically tractable.

And it is used a lot for describing neural computations on the population level.

However, this firing rate model has also a lot of challenges or first of all it's a, it's a holistic model, it does not really connect to the underlying microscopic model.

And for this reason, because on a microscopic level the neurons work with spikes.

This model cannot, for example, deal with spike synchronization.

So here would be an example that shows synchronization of spikes in response to an external stimulus so we have to step current here and then neurons are forced basically to spike and if one looks at the population activity over many neurons.

Never would see that there is a high firing rate just after the spike onset, but then.

So neurons are get all synchronized by this by the step. And then they all go together into some refractory period.

So, until they recover again there will be a second peak and then we get some, some oscillation. So this oscillation can be easily obtained by a network of integrate and fire spiking your own models, but not with a with this simple first order firing rate model.

So the first order firing rate model would just predict exponential relaxation towards the steady state.

Another problem of this firing rate model is that it is unclear how to include fluctuations due to the finite number of neurons. So first of all, in, in the cortex, if uncounts, the number of neurons of a given cell type in a given cortical layer