Dynamical differential covariance (DDC)

Reference: https://www.pnas.org/doi/abs/10.1073/pnas.2117234119
All scripts were carefully annotated. Below is a quick summary.

Connectivity estimation:

• estimators.m: Common estimators including covariance <x,x>, Precision matrix, nonlinear averaging matrix <R(x),x>, dCov computed by symmetric difference quotient
• dCov_numerical.m: dCov calculated by different numerical estimations of the time derivative
• derivative_123.m: Supporting file for dCov_numerical.m
• DDC calculation: matrix product of dCov and <x,x>^{-1} or <R(x),x>^{-1}
• L2 regularized DDC calculation: dCov_linear_Reg.m

PSD Bootstrapping (evaluate significance of connections)

• PSDBootstrap_example.m: generate null timeseries under the null hypothesis that each time series is independent; calculate the null FC matrices
• PSDBootstrap_Getmdl.m: fit the best AR model to the order of q for each time series

Network simulations:

• Linear_simulation.m: simulation of linear stochastic systems
• Nonlinear_simulation.m: simulation of nonlinear stochastic systems
• LIF_network_YC.m: Leaky-integrate-and-Fire network simulation
• simulate_reduced_wong_wang.ipynb: Reduced Wong-Wang simulation

Performance evaluation:

• FC_GT_ROC.m: classification sensitivity for LIF network recovery
• c_sensitivity_YC.m: c-sensitivity calculation

HCP dataset

• rs-fMRI time series data available through https://www.humanconnectome.org
• dMRI dataset available through https://www.eneuro.org/content/8/1/ENEURO.0416-20.2020/tab-article-info
• ATLAS_HCP_dMRIparcel_to_IC100node_connectivity.m: convert parcel-level dMRI strength to IC-level dMRI strength

Demo (MATLAB syntax):

For HCP ICA preprocessed time series:

 TR = 0.72; 
 [T, N] = size(V); % number of timepoints x number of nodes
 V_obs = zscore(V)
 [dCov1, dCov2,~,~] = dCov_numerical(V_obs,TR);
 [Cov,Precision,B,~] = estimators(V_obs,prctile(V_obs(:),50),TR);
 Delta_L = dCov2*Precision
 Delta_ReLU = dCov2*B