Entropy field decomposition for analysis in a dynamic system
A method for analysis of complex spatio-temporal data within a dynamic system that includes spatial positions and fields, at least a portion of which are interacting, includes determining values of mean field at every spatial position, determining spatio-temporal eigenmodes in spatial-frequency space assuming interacting fields, and determining spatial and temporal interactions between the eigenmodes. The resulting display indicates space/time localization patterns that are indicative of connectivity within the dynamic system.
1. A method for determining connectivity within a dynamic system represented by complex spatio-temporal data, the method comprising, in a computer processor:
inputting signal data corresponding to the dynamic system into the computer processor, the signal data comprising points within a lattice;
ranking a plurality of optimal paths within the lattice according to path entropy, wherein the rankings are arranged in a coupling matrix;
using eigenvalues and eigenvectors from the coupling matrix to construct an information Hamiltonian;
estimating mode amplitudes corresponding to spatially and temporally interacting modes of the information Hamiltonian, wherein the mode amplitudes correspond to space and time localization patterns; and
generating an output comprising a display of space and time localization patterns that are indicative of connectivity within the dynamic system.
2. The method of claim 1 , wherein the dynamic system is living tissue and the signal is generated by medical instrumentation configured for detecting biological activity in the living tissue.
3. The method of claim 2 , wherein the medical instrumentation uses an anatomical imaging method comprising magnetic resonance imaging (MRI) or computed tomography (CT).
4. The method of claim 3 , wherein the living tissue is a brain, and wherein the anatomical imaging method comprises functional imaging of the brain.
5. The method of claim 4 , wherein the space and time localization patterns correspond to functional tractography and functional eigentracts.
6. The method of claim 1 , wherein the dynamic system is a network.
7. The method of claim 1 , wherein the information Hamiltonian is of the form
H
(
d
,
a
k
)
=
-
j
k
†
a
k
+
1
2
a
k
†
Aa
k
+
∑
n
=
1
∞
1
n
!
∑
k
1
K
…
∑
k
n
K
A
¨
k
1
-
k
n
(
n
)
a
k
1
…
a
k
n
,
where matrix Λ is the diagonal matrix Diag{λ 1 , . . . , λ K } composed of eigenvalues of a noise corrected coupling matrix, α k is an amplitude of the kth mode, and j k is the amplitude of the kth mode in the expansion of the source j.
8. The method of claim 7 , wherein the mode amplitudes are determined according to the relationship
Aa
k
=
(
j
k
-
∑
n
=
1
∞
1
n
!
∑
k
1
K
…
∑
k
n
K
A
⋒
kk
1
-
k
n
(
n
+
1
)
a
k
1
…
a
k
n
)
.
9. The method of claim 6 , wherein the network is a root system architecture of a plant and the connectivity comprises water transport properties and global structure of the root system.
10. The method of claim 6 , wherein the network is an internet or communications system.
11. The method of claim 1 , wherein the signal data corresponding to points within a lattice are derived from historical weather records.