week 4 - the kuramoto oscillator model Flashcards
(18 cards)
What are uses of the connectome
Providing structural information that can be implemented as part of large scale computational models
Provide an important tool for mechanistic modelling and interpretation of human functional brain data
What creates oscillation in the brain? Give examples
Synchronized neuron firing from a population of neural firing
This give rise to oscillations in a wide range of frequencies. These different frequencies are associated with different cognitive functions
E.g alpha waves, gamma waves
Oscillations allow for temporal and spatial coordination of different brain processes
What is the communication through coherence hypothesis?
Synchronisation is key for communication between brain regions or populations of neurons
Coherehence between the osscilation in the sending group and the recieving group is also essential for communication between brain regions
When the oscillations are synchronised between brain regions, there is a window of opportunity for communication. This is so the message can reach the destination when it’s in its most excitable state.
When the information reaches the neuron in peak excitability, the information transmits
what is synchronisation in coupled systems?
Synchronisation in a coupled system is when individual units adjust their dynamics to behave in a coordinated or temporally aligned way due to their interactions.
What is the general idea behind dynamic whole brain modelling
structural connectome + model = simulated activity
This can then be paired with empirical recordings of activity to compare themodel with empirical data
This comparison can be used by any analytical method for time series data, e.g static or dynamic connectivity or graph theory
Describe the kuramotor oscillator model equation
(Print equation). Check equation with online
model terms:
phase of node(i) (fraction of the cycle at any point in time)
rate of change (how much does the phase change in a small increment of time
is equal to:
Frequency (how many cycles does the oscillatory node make for a time point ). E.g 1 Hz is one phase cycle per second
+
k = coupling parameter - relative importance of the frequency vs. the interactions. If K is zero, theres no interaction between oscillators and the phase change simply depends on natural frequency. If K is very high, it dwarfs the intrinsic frequency, and the activity of the oscillators will purely depend on the interaction between other oscillator.
(optional * Cij)
*
interactions = the influence of the oscillators on each other (interactions between oscilattors depend on the Sin of the phase difference between them)
+
n external noise
D = relative delays between oscillators
Tau = average length/velocity - scales all delays
Additional note: The entire equation is time dependent, meaning that the phases themself are a function of time
what happens if K is high
K determines relative importance of the interactions between oscillators vs. the intrinsic frequency.
If K is high, only the interactions influences the activity of the oscillators, this likely means that the nodes synchronise very quickly
explain the interactions explained by Sin 0
The interaction term is the sin of the phase difference between the two oscillators.
So if the two oscillators are nearly at the same phase, Sin 0 is 0, meaning the two oscillators are not influencing each other much at all. If they are further away, the Sin of the difference between the two phases is bigger, and so they have a bigger influence on each other, pulling each other closer together
pi radians = 180 degrees
What happens when you add Cij and how does it relate to connectome
A parameter is added that is multiplied by the interaction term, that varies the relative coupling of oscillators, or the relative influence of oscillators on each other
So when Cij is 1, the influence of all oscillators on each other is identical
Allows you to bring in extra oscillators, that influence each other in different levels
You an represent this coupling as a matrix. You can even try and edit Cij so the matrix of Cij is equal to a structural connectome
what is the difference between k and c?
C = how strongly is a pair of oscillators coupled
K = essentially scales all of this matrix up or down
so K applies to all coupling, but C refers to the relative coupling between oscillators
K allows you to tune the model to a point that it might behave in an interesting way
How to take into account the time taken for information to travel between regions
Add in a delay matrix into the interaction term
This delay matrix is typically fibre lengths
So interaction is the Sin of the Phase difference between oscillators, dependent on time minus the delay between oscillators
What is Wi
The intrinsic frequnecy of oscilaltions
You can make this biologically plausible, e.g 60 Hz for Gamma
Kuramoto model assumptions
- Coupling is week
- Oscillators are nearly identical
- interactions depend sinusoidally on the relationsip between oscillators
What is the kuramoto order parameter (Rt)
The length of the average vector between oscillators at any one time point.
It is a way to quantify coherence/Synchrony
The maximum value is 1. If the two oscillators are fully synchronised then the average vector is 1 and thats the max order parameter number.
The min is 0, if the two oscillators are maximally unsynchronised
The smaller the order parameter, the less synchronised
So when the system is highly synchronised, you have a high value
What is syncrony?
The average order parameter over time
What is metastability?
The standard deviation over time of the order parameter
This is one signature of metastability, but actually there are lots of signatures of metastability
How did varying k change the properties in cabral et al’s study
Cabral et al used the kuramato model with delays to simulate the fMRI time series. They added modelled fMRI time series processing steps.
RESULTS
At low coupling K, the system showed low synchrony and low metastability
At high coupling K, the system showed high coupling and low metastabiltiy
At mid level couplign K, the synchronisation was variable and the metastability was high
At the point that the model best matched empirical data, subsets of nodes were synchronised. So there was synchoronisation within large network modules, but not between them. This corresponds to intrinsic connectivity networks (resting state networks, e.g the DMN and DAN)
