Entropy field decomposition for analysis in a dynamic system
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:
receiving in a computer processor signal data associated with the dynamic system, wherein the signal data comprises temporal variations in a plurality of voxels; and
causing the computer processor to perform the steps of:
generating a coupling matrix by ranking optimal paths between voxels according to path entropy;
determining entropy spectrum pathway (ESP) eigenvalues and eigenvectors for the coupling matrix;
constructing an information Hamiltonian using the ESP eigenvalues and eigenvectors, wherein the information Hamiltonian includes a diagonal matrix of ESP eigenvalues and interaction terms constructed with the ESP eigenvalues and eigenvectors;
estimating mode amplitudes corresponding to spatially and temporally interacting modes of the information Hamiltonian; and
generating an output comprising a display of the mode amplitudes, wherein the mode amplitudes correspond to space and time localization patterns indicative of connectivity within the dynamic system.
2. The method of claim 1 , wherein generating the coupling matrix comprises determining interactions between locations i and j within a voxel according the relationship Q ij =e −γ ij , where Q is the coupling matrix and γ ij are Lagrange multipliers defining the interactions corresponding to local potentials based on a function of space-time locations in the voxel.
3. The method of claim 2 , wherein an eigenvector φ associated with an eigenvalue Δ k for the coupling matrix generates a transition probability from location j to location i of a k th path according to a relationship
p
ijk
=
Q
ji
λ
k
ϕ
i
(
k
)
ϕ
j
(
k
)
.
4. 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.
5. The method of claim 4 , wherein the medical instrumentation uses an anatomical imaging method comprising magnetic resonance imaging (MRI) or computed tomography (CT).
6. The method of claim 4 , wherein the living tissue is a brain, and wherein the anatomical imaging method comprises functional imaging of the brain.
7. The method of claim 6 , wherein the space and time localization patterns correspond to functional tractography and functional eigentracts.
8. The method of claim 1 , wherein the dynamic system is a network.
9. 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
†
Λ
a
k
+
∑
n
=
1
∞
1
n
!
∑
k
1
K
…
∑
k
n
K
Λ
~
k
1
…
k
n
(
n
)
a
k
1
…
a
k
n
,
where matrix Λ is the diagonal matrix Diag {λ 1 , . . . , Δ K } composed of eigenvalues of a noise corrected coupling matrix, a 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.
10. The method of claim 9 , wherein the mode amplitudes are determined according to the relationship
Λ
a
k
=
(
j
k
-
∑
n
=
1
∞
1
n
!
∑
k
1
K
…
∑
k
n
K
Λ
~
kk
1
…
k
n
(
n
+
1
)
a
k
1
…
a
k
n
)
.
11. A system for determining connectivity within a dynamic system represented by complex spatio-temporal data, the system comprising:
one or more computer processors configured to receive signal data associated with the dynamic system, the signal data comprising temporal variations in a plurality of voxels, and execute operations on the signal data comprising:
generating a coupling matrix by ranking optimal paths between voxels according to path entropy;
determining entropy spectrum pathway (ESP) eigenvalues and eigenvectors for the coupling matrix;
constructing an information Hamiltonian using the ESP eigenvalues and eigenvectors, wherein the information Hamiltonian includes a diagonal matrix of ESP eigenvalues and interaction terms constructed with the ESP eigenvalues and eigenvectors;
estimating mode amplitudes corresponding to spatially and temporally interacting modes of the information Hamiltonian; and
generating an output comprising a display of the mode amplitudes, wherein the mode amplitudes correspond to space and time localization patterns indicative of connectivity within the dynamic system.
12. The system of claim 11 , wherein generating the coupling matrix comprises determining interactions between locations i and j within a voxel according the relationship Q ij =e −γ ij , where Q is the coupling matrix and γ ij are Lagrange multipliers defining the interactions corresponding to local potentials based on a function of space-time locations in the voxel.
13. The system of claim 12 , wherein an eigenvector φ associated with an eigenvalue Δ k for the coupling matrix generates a transition probability from location j to location i of a k th path according to a relationship
p
ijk
=
Q
ji
λ
k
ϕ
i
(
k
)
ϕ
j
(
k
)
.
14. The system of claim 11 , wherein the dynamic system is living tissue and the signal is generated by medical instrumentation configured for detecting biological activity.
15. The system of claim 14 , wherein the medical instrumentation uses an anatomical imaging method comprising magnetic resonance imaging (MRI) or computed tomography (CT).
16. The system of claim 14 , wherein the living tissue is a brain, and wherein the anatomical imaging method comprises functional imaging of the brain.
17. The system of claim 16 , wherein the space and time localization patterns correspond to functional tractography and functional eigentracts.
18. The system of claim 11 , wherein the dynamic system is a network.
19. The system of claim 11 , wherein the information Hamiltonian is of the form
H
(
d
,
a
k
)
=
-
j
k
†
a
k
+
1
2
a
k
†
Λ
a
k
+
∑
n
=
1
∞
1
n
!
∑
k
1
K
…
∑
k
n
K
Λ
~
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, a 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.
20. The system of claim 19 , wherein the mode amplitudes are determined according to the relationship
Λ
a
k
=
(
j
k
-
∑
n
=
1
∞
1
n
!
∑
k
1
K
…
∑
k
n
K
Λ
~
kk
1
…
k
n
(
n
+
1
)
a
k
1
…
a
k
n
)
.