Hopf model

202303181050
Status:
Tags: whole-brain model Neuroimaging Hopf oscillator

A Hopf whole-brain model captures the dynamics emerging from mutual interactions between connected brain regions using the anatomical structural connectivity & weighted by the GEC (generative effective connectivity).

There are many variations of the model, but the standard used by Kringelbach appears to be a 62 coupled dynamical regions from the Mindboggle-modified Desikan-Killany parcellation with 62 cortical regions per hemisphere.

The local dynamics of each region are described by the normal form of a supercritical Hopf bifurcation → Landau-Stuart oscillator (canonical model for studying transition from noisy to oscillatory dynamics).

Implementation

Dynamics of an uncoupled node $n$ are given by a set of coupled dynamical equations which describe the normal form of a supercritical Hopf bifurcation in Cartesian coordinates:

$\frac{d{x}_n}{dt}=[a_n-x_n^2-y_n^2]x_n-{\omega }_n{y}_n+\beta {\eta }_n(t)$

$\frac{d{y}_n}{dt}=[a_n-x_n^2-y_n^2]y_n+{\omega }_n{x}_n+\beta {\eta }_n(t)$

with additive Gaussian noise ${\eta }_n(t)$ and SD $\beta$. This normal form has a supercritical bifurcation $a_n=0$ so if $a_n>0$ then the system engages in a stable limit cycle with frequency $f_n={\omega }_n/2\pi$. For $a_n<0$, local dynamics are in a stable fied point representing low activity noisy state.

The intrinsic frequency ${\omega }_n$ of each node is in the 0.008-0.08Hz band.

Intrinsic frequencies are estimated from the data as given by the averaged peak frequency of the narrowband BOLD signals of each brain region. The best fit was obtained with $a_n=-0.02$ in the paper referenced.

Whole-brain dynamics

TO model whole-brain dynamics, we require modeling the coupling which is included by adding diffusive coupling term which represents the input received in region $n$ from every other region $p$, and is weighted by the corresponding GEC $G_{np}$. This is modeled using common difference coupling, which approximates the simplest (linear) part of a general coupling function:

$\frac{d{x}_n}{dt}=[a_n-x_n^2-y_n^2]x_n-{\omega }_n{y}_n+{\sum }_{p=1}^N{G}_{np}(x_p-x_n)$

$\frac{d{y}_n}{dt}=[a_n-x_n^2-y_n^2]x_n-{\omega }_n{x}_n+{\sum }_{p=1}^N{G}_{np}(y_p-y_n)$

Updating GEC

optimizing GEC between brain regions was performed by comparing output of model with empirical measures of forward & reversed time-shifted correlations & empirical FC.

this is done with a heuristic gradient algorithm, where to ensure only positive values all values are transformed into a mutual information measure which assumes Gaussianity:

$G_{ij}=G_{ij}+ϵ(F{S}_{ij}^{empirical}-F{S}_{ij}^{model})-{ϵ}^{‘}([F{S}_{forward,ij}^{empirical}(T)-F{S}_{reversal,ij}^{empirical}(T)]-[F{S}_{forward,ij}^{model}(T)-F{S}_{reversal,ij}^{model}(T)])$
where $F{S}_{ij}$ measure is based on nonshifted $F{C}_{ij}$ as mutual information measure obtained by:

$F{S}_{ij}=-\frac{1}{2}log[1-(F{C}_{ij}{)}^2]$

The model is run repeatedly with the updated GEC until fit converges toward a stable value. The model should be initialized using the anatomical connectivity obtained with probabilistic tractography from dMRI and only update known existing connections from this matrix in either hemisphere.

• Exception: algorithm also updates homolog connections b/w the same regions in either hemisphere, since tractography is less accurate when accounting for this connectivity.
• $ϵ=0.0005$ and $ϵ=0.0001$ until algorithm converges.
• For each iteration, model results = average over as many simulations as there are participants

Functional hierarchical organization

$G_{in}(i)={\sum }_j{G}_{ij}$
$G_{out}(i)={\sum }_i{G}_{ji}$
$G_{total}(i)=G_{in}(i)+G_{out}(i)$

Asymmetry of diff. optimized GEC matrices

${\sum }_j|G_{ij}-Gji|$ → sum of differences between the matrix and its transposed.

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References