Data assimilation for heterogeneous networks: the consensus set

Abstract

Data assimilation in dynamical networks is intrinsically challenging. A method is introduced for the tracking of heterogeneous networks of oscillators or excitable cells in a nonstationary environment, using a homogeneous model network to expedite the accurate reconstruction of parameters and unobserved variables. An implementation using ensemble Kalman filtering to track the states of the heterogeneous network is demonstrated on simulated data and applied to a mammalian brain network experiment. The approach has broad applicability for the prediction and control of biological and physical networks.

FIG. 1

The six parameters estimated by the EnKF converge over time to consensus values. The network's mean parameter values are denoted by closed circles. Here both the voltage and recovery variables u and a are observed.

FIG. 2

(Color online) The noisy u_ij input data and the output a_ij, shown as the black dashed line, from the consensus method at a typical grid point. The black solid line is the actual value of a_ij, for comparison. The red (light gray) dashed line is the filtered u_ij returned by EnKF.

FIG. 3

(Color online) (a) The noisy u_ij data and the output a_ij from the consensus method (black dashed line) at one of the grid points. The black solid line is the actual value of a_ij. The reconstructed a_ij is magnified by 1 / γ=2. (b) Consensus parameters from the EnKF. Compared with Fig. 1, c₂ has been replaced by γc₂=−0.32 (lowest trace).

FIG. 4

(Color online) (a) Snapshot of the transmembrane voltage optical intensities u_ij from the neural tissue. (b) The experimental u_ij plotted as blue (dark gray) points, the filtered u_ij plotted as red (light gray) points, and the output a_ij from the consensus method (black dashed line) at one of the grid points. The reconstructed a_ij has no physical significance; it is created by the EnKF to reconstruct a compatible dynamical state in the homogeneous network model. (c) Decrease in the EnKF innovations show stability of the method's convergence.