Table 1.
Fixed parameters for large-scale circuit model.
Fig 1.
Schematic diagram of reconstruction.
(A) Temporal Reconstruction. Neural activity S(t) is transformed into a Jacobian matrix via Dynamical Differential Covariance (DDC) which provides an unbiased estimation of network coupling Jacobian Jij. The Jacobian matrix (EC) here can be further divided into effective heterogeneity hi and asymmetric SC Cij. (B) Symmetric SC , is considered as structural constraints for spatial reconstruction. (C) Spatial reconstruction further separates effective heterogeneity hi (top) and asymmetric SC Cij (bottom) following the temporal reconstruction and structural constraints. (D) The effective heterogeneity hi represents an example calculated using Eq. 6. (Bottom) The systematic difference between ground truth asymmetric SC and symmetric SC approximation, with asymmetry level
, representing that this empirical SC is symmetry-dominated but regulated by asymmetric connections.
is calculated as the element-wise correlation between the upper and lower triangular matrices.
Fig 2.
Robustness of reconstruction in effective heterogeneity and asymmetry.
(A) The relative errors between ground truth and estimation of EC (Jacobian in Eq. 9) as functions of the global coupling strength G. (B and C) The relative errors of effective heterogeneity hi (B) and asymmetric SC Cij (C) revealed by the reconstruction method as functions of G. Red line represents reconstruction considering both heterogeneity and asymmetry. Blue line is baseline comparison ignoring asymmetry (B). Yellow line is baseline comparison ignoring heterogeneity (C). (D) The relative errors of Jacobian as function of noise strength σ. (E) The relative errors of asymmetric SC Cij (blue) and regional heterogeneity hi (red) as functions of σ. The colored lines show the mean relative errors across 10 simulations, with shaded areas indicating one standard deviation from the mean. Simulation length 50,000s.
Table 2.
Expressions of effective heterogeneity hi across different models.
Fig 3.
Performance in reconstruction of detailed parameters.
(A) Ground truth features of Model A. The regional recurrent strength wi is rescaled from empirical anatomical heterogeneity [43], while the external input Ii is set to decrease with hierarchy to be consistent with previous work [36]. (B) The relative errors in Model A between ground truth and estimation of wi (blue) and Ii (red) as functions of G. Vertical dash lines represent different G values: G = 0.1 (red square), G = 0.8 (blue round) and G = 1.3 (yellow triangle). (C and D) Derivative of firing rate at stable states S* for each ROI under three different global coupling strengths corresponding to B. Firing rate change of each ROIs under high global coupling are located at nonlinear region (D, yellow triangle). The colored lines show the mean value across 10 simulations, with shaded areas indicating one standard deviation from the mean. Simulation length 50,000s.
Fig 4.
Reconstruction performance of Model B and Model C.
(A) The relative errors of reconstruction of the asymmetric SC (blue) and regional heterogeneity hi (red) of Model B as functions of G. (B) The relative errors between ground truth and estimation of τi (blue) and firing threshold bi (red) of Model B as functions of G. (C) Illustration of reconstruction procedure with localized excitation-inhibition interactions (Model C). The full model consists of an excitatory and an inhibitory population
for each region (
), and only excitatory populations have inter-region connections:
represents the directed connection from region j to region i, with Cij denotes the SC and
denotes the effective heterogeneity of excitatory population.
represents the local recurrent strength of the excitatory population of region i. Dash square represents that only the activity of excitatory populations is observed for reconstruction.
represents the effective local recurrent strength reconstructed from excitatory activity and hi is the effective heterogeneity absorbing the influence from inhibitory population. (D) Reconstruction example of Model C while only excitatory activity is observed. Ground truth Jeff and hi are derived in Eqs 11 and 13. Dash line represents y = x.
Fig 5.
Reconstruction performance with respect to the sampling interval of observations.
(A and B) Reconstruction and prediction of exponential scaling strongly match across sampling steps. (A) Element-wise comparison between estimated Exponential Jacobian from sampled neural activity and analytical Jacobian JT across different sampling resolution from 0.01s (unsampled, red) to 1s (100 steps, blue). Dash line represents y = x. (B) Correlation of
and JT (blue),
and
(red) across different sampling step n. (C) The relative errors between ground truth and estimation of asymmetric SC (blue) and effective heterogeneity hi (red) estimated from
as functions of sampling steps. The colored lines show the mean relative errors across 10 simulations, with shaded areas indicating one standard deviation from the mean. Simulation length 50,000s.
Table 3.
Fixed parameters for large-scale excitatory-inhibitory circuit model (Model C).