Textbook Group meeting for Parr, Pezzulo, Friston 2022 .

all right welcome back everyone we're in the first discussion for chapter8 so does anyone want to begin with like just any thought or question on eight any quote or or part and we'll just see where it goes H yeah I was I'm was curious about the role of attractors um in the yeah how how how the the role of attractors in the um generative let me let me try to find the exact quote section yeah just basically the the Ro of fixed point attractors in in the generative process anyone with a thought on that all right there's a lot to say about a tractor behavior in dynamical systems overall and that's kind of where we're at one way to think about it is if there wasn't some kind of attractor point or set or limit cycle or something like that that variable would like never converge to aible stationerity so if something's oscillatory it could be thought of as like converging to a cyclic attractor if something is kind of like dampening like a spring that's like a fix Point attractor plus dampening or it could be like just oscillating around a fix point of tractor so it's like dynamical systems are of certain types analytically and they can be converging and dampening converging oscillating limit Cycles all these are like different outcomes or possibilities for dynamical systems but to kind of pull back um a step so chapter 8 follows on chapter 7 which shows the discrete time formalism so in chapter 7 discrete time steps in the simulation are explicitly modeled like we have time 67 time 68 Etc chapter eight is going to show a totally different complimentary treatment of time for generative models which is the continuous time formalism in this chapter is about like unpacking a lot of the differences from the discret time and one of the key differences is like a different take on the relationship between observations and hidden States so in the um discreet time setting and figure 4.3 is like the Rosetta Stone with the top discrete time model and the bottom continuous time so in the partially observable marov process setting the a matrix is kind of going both ways between the hidden State and the observation here kind of in line with the continuous time formulation being like uh closer to some Physics based uh formula there's there's two core uh equations there's observations y same as observations here and then x dot so the um derivative on hidden States so derivative on hidden States here and basically each of these are mapped out so there's like a kind of regularity of observations or a function of observations and a noise function and then there's also an underlying generator and this is kind of coming from fris and at all's earlier work with SPM where why are like all of the sensor data for the neuroimaging like fmri or EEG and then x dot is the rates of change of the underlying neural activity that's related to the observations that are coming in so that's kind of the that could describe any G and any F so then they by 8.3 get to a kind of simple generative uh model where there's a fix Point attractor so here's the what it looks like to have a fixed Point attractor so this is basically when X is less than V the set point the rate of change is positive so it goes up and then when X is higher than the set point the rate of change is negative so it goes down so it's kind of like a first order thermostat with a set point encoded in the underlying Dynamics like this any thoughts on that or does anyone like want to add more pieces to this about attractors or anything yeah I think um the the attractors become like significantly more interesting whenever we move into the the next uh sections with like the the luck voltera Dynamics and the lawen system just like trying to map them to like other phenomena like I really like the um the bird song example I guess this is around 100 page 163 um where we're introduced to the concept of generalized synchrony and so the birds are basically like their internal states are starting to synchronize as…