Analytical estimation of the network statistics of the Hopf whole-brain model

1704 

Abstract Submission 

Adrián Ponce-Alvarez¹, Gustavo Deco²

¹Polytechnic University of Catalonia, Barcelona, Spain, ²Pompeu Fabra University, Barcelona, Spain

Adrián Ponce-Alvarez  
Polytechnic University of Catalonia
Barcelona, Spain

Gustavo Deco  
Pompeu Fabra University
Barcelona, Spain

Whole-brain models are useful to understand the emergence of collective activity among brain regions. These models combine connectivity matrices (connectomes) with local node dynamics, noise, and transmission delays. Diverse choices for the local dynamics have been proposed. Among them, nonlinear oscillators corresponding to a Hopf bifurcation have been used to study collective phase and amplitude dynamics in different brain states (e.g. Deco et al., 2017; López-González et al., 2021). However, estimating the network statistics of such system requires long simulations, impeding the exploration of different model parameters. Here, we studied the linear fluctuations of this model to analytically estimate its stationary statistics, i.e. the covariances in the temporal and frequency domains.

Hopf network:
The whole-brain dynamics are obtained by coupling the local dynamics of N nodes interconnected through a coupling matrix C representing anatomical connections. Here, we used a publicly available human diffusion MRI connectome from the Human Connectome Project (HCP) (Smith et al. 2013). The state variables of the network, z_j (for j=1,...,N), are given by the complex stochastic differential equations:

dz_j/dt = F_j(z₁,z₂,...,z_N) = (a_j + iω_j)z_i - |z_j|² + gΣ_kC_jk(z_k-z_j) + η_j,

where i is the complex unit, g is a global scaling of connectivity C, ω_j is the intrinsic frequency of node j, η_j represents noise, and a_j is the bifurcation parameter of node j. In isolation, a node undergoes noise-driven oscillations if a_j < 0 or self-sustain oscillations if a_j > 0. Node parameters can be homogenous (same for all nodes) (Deco et al. 2017) or heterogeneous (different for different nodes, i.e., hierarchy) (López-González et al. 2021).

In the case of delayed interactions, the coupling term becomes: gΣ_kC_jk[z_k(t-d_kj)-z_j(t)], where d_kj is the time-delay of the interaction between nodes j and k.

Linearization:
In the case of weak noise and small non-linearities, one can estimate the network statistics using a linear approximation, without the need of long, computationally-costly stochastic simulations. For this, we studied the linear fluctuations δz around the fixed point (z₁,...,z_N) = (0,...,0), which dynamics are governed by the Jacobian matrix J, given by the partial derivatives of the system at the fixed point: J_jk = ∂F_j/∂z_k. We showed that the Jacobian matrix determines the statistics of the system and that delays can be treated in the Fourier space.

We showed that the stationary instantaneous and lagged covariance matrices, the cross-spectrum, and the power spectral densities (PSDs) of the model can be obtained through algebraic operations including the Jacobian matrix. This can be done both in the homogeneous and the heterogeneous cases, and also in the presence of time delays. We illustrated the method by fitting human rs-fMRI signals from the HCP in the model's parameter space.

Using a linear approximation, we derived the network statistics of the Hopf whole-brain model. This can be done in the most general form of the model, namely in the delay-coupled heterogeneous case (allowing to study temporal and spatial hierarchies). The estimated statistics can be used to track changes in brain state, e.g.,low-level states of consciousness, anesthesia, sleep, etc., or to evaluate the effect of lesions in the connectome. Finally, the linear approximation of delay-coupled model derived here can represent a valuable tool to study the PSDs and cross-spectrum of MEG, which are well-stablished methods for FC analysis in the frequency domain.

Connectivity (eg. functional, effective, structural) ²

fMRI Connectivity and Network Modeling ¹

Other Methods

Computational Neuroscience

FUNCTIONAL MRI

Other - Whole-brain model