We present a novel functional holography (FH) analysis devised to study the dynamics of task-performing dynamical networks. The latter term refers to networks composed of dynamical systems or elements, like gene networks or neural networks. The new approach is based on the realization that task-performing networks follow some underlying principles that are reflected in their activity. Therefore, the analysis is designed to decipher the existence of simple causal motives that are expected to be embedded in the observed complex activity of the networks under study. First we evaluate the matrix of similarities (correlations) between the activities of the network’s components. We then perform collective normalization of the similarities (or affinity transformation) to construct a matrix of functional correlations. Using dimension reduction algorithms on the affinity matrix, the matrix is projected onto a principal three-dimensional space of the leading eigenvectors computed by the algorithm. To retrieve back information that is lost in the dimension reduction, we connect the nodes by colored lines that represent the level of the similarities to construct a holographic network in the principal space. Next we calculate the activity propagation in the network (temporal ordering) using different methods like temporal center of mass and cross correlations. The causal information is superimposed on the holographic network by coloring the nodes locations according to the temporal ordering of their activities. First, we illustrate the analysis for simple, artificially constructed examples. Then we demonstrate that by applying the FH analysis to modeled and real neural networks as well as recorded brain activity, hidden causal manifolds with simple yet characteristic geometrical and topological features are deciphered in the complex activity. The term “functional holography” is used to indicate that the goal of the analysis is to extract the maximum amount of functional information about the dynamical network as a whole unit.

1.
J.
Ihmels
,
R.
Levy
, and
N.
Barkai
, “
Principles of transcriptional control in the metabolic network of Saccharomyces cerevisiae
,”
Nat. Biotechnol.
22
(1),
86
92
(
2003
).
2.
L. T.
Vernon
,
I.
Syed
,
C.
Berger
,
R.
Grzesczcuk
,
J.
Milton
,
R. K.
Erickson
,
P.
Cogen
,
E.
Berkson
, and
J. P.
Spire
, “
Identification of the sensory/motor area and pathologic regions using ECoG coherence
,”
Electroencephalogr. Clin. Neurophysiol.
106
,
30
39
(
1998
).
3.
M. A. B.
Brazier
,
Cluster Analysis
(
Edward Arnold
, London,
1993
), p.
256
.
4.
I.
Baruchi
and
E.
Ben-Jacob
, “
Functional holography of recorded neuronal networks activity
,”
Neuroinformatics
2
,
333
352
(
2004
).
5.
P. E.
Hoffman
and
G. G.
Grinstein
,
A Survey of Visualizations for High-Dimensional Data Mining
(
Morgan Kaufmann Publishers Inc.
,
2002
), pp.
47
82
.
6.
I.
Baruchi
,
V. L.
Towle
, and
E.
Ben-Jacob
, “
Functional holography of complex networks activity from cultures to the human brain
,”
Complexity
10
,
38
51
(
2005
).
7.
E. W.
Weinstein
, “
Singular value decomposition
,” from MATHWORLD
8.
R.
Segev
,
M.
Benveniste
,
E.
Hulata
,
N.
Cohen
,
A.
Palevski
,
E.
Kapon
,
Y.
Shapira
, and
E.
Ben-Jacob
, “
Long term behavior of lithographically prepared in vitro neuronal networks
,”
Phys. Rev. Lett.
88
,
118102
(
2002
).
9.
R.
Segev
,
I.
Baruchi
,
E.
Hulata
, and
E.
Ben-Jacob
, “
Hidden neuronal correlations in cultured networks
,”
Phys. Rev. Lett.
92
,
118102
(
2004
).
10.
H.
Kamioka
,
E.
Maeda
,
Y.
Jimbo
,
H. P. C.
Robinson
, and
A.
Kawana
, “
Spontaneous periodic synchronized bursting during formation of mature patterns of connections in cortical cultures
,”
Neurosci. Lett.
206
,
109
112
(
1996
).
11.
N.
Raichman
,
V.
Volman
, and
E.
Ben-Jacob
, “Collective plasticity and individual stability in cultured neuronal networks,”
Neurocomputing
(to be published).
12.
R. K.
Otnes
and
L.
Enochson
,
Applied Time Series Analysis
(
Wiley - Interscience
, New York,
1978
).
13.
V.
Volman
,
I.
Baruchi
,
E.
Persi
, and
E.
Ben-Jacob
, “
Generative modeling of regulated activity in cultured neuronal networks
,”
Physica A
335
,
249
278
(
2004
);
E.
Persi
,
D.
Horn
,
V.
Volman
,
R.
Segev
, and
E.
Ben-Jacob
, “
Modeling of synchronized bursting events—The importance of inhomogeneity
,”
Neural Comput.
16
,
2577
2595
(
2004
).
[PubMed]
14.
V.
Volman
,
I.
Baruchi
, and
E.
Ben-Jacob
, “
Manifestation of function-follow-form in cultured neuronal networks
,”
Phys. Biol.
2
,
98
110
(
2005
).
15.
T. I.
Netoff
and
S. J.
Schiff
, “
Decreased neuronal synchronization during experimental seizures
,”
J. Neurosci.
22
,
7297
7307
(
2002
).
16.
V. L.
Towle
,
F.
Ahmad
,
M.
Kohrman
,
K.
Hecox
, and
S.
Chkhenkeli
,
Electrocorticographic Coherence Patterns of Epileptic Seizures in Epilepsy as a Dynamic Disease
, edited by
P.
Jung
and
J.
Milton
(
Springer
, Berlin,
2002
).
17.
W. K.
Doyle
and
D. D.
Spencer
, “
Anterior temporal resections
,” in
Epilepsy: A Comprehensive Textbook
, edited by
J.
Engel
, Jr.
and
T. A.
Pedley
(
Lippincott-Raven
,
1998
), Vol.
2
, pp.
1807
1817
.
18.
S. A.
Chkhenkeli
,
V. L.
Towle
,
J. G.
Milton
, and
J.-P.
Spire
, “
Multitarget stereotactic surgery of intractable epilepsy
,”
Abstracts of the XIII Congress of ESSFN
(
Freiburg
, Germany,
1998
), Vol.
4
, p.
21
.
19.
P. A.
Bandettini
,
C. T. W.
Moonen
, and
G. K.
Aguirre
,
Functional MRI
, 1st ed. (
Springer-Verlag
, Berlin,
1999
).
20.
V.
Volman
,
I.
Baruchi
, and
E.
Ben-Jacob
,
Self-regulated homoclinic chaos in neural networks activity
.
8th Experimental Chaos Conference
, edited by
S.
Boccaletti
 et al (
American Institute of Physics
, Melville, NY,
2004
), pp.
197
209
.
21.
E.
Hulata
,
I.
Baruchi
,
R.
Segev
,
Y.
Shapira
, and
E.
Ben-Jacob
, “
Self-regulated complexity in cultured neuronal networks
,”
Phys. Rev. Lett.
92
,
198105
1
198105
4
(
2004
).
22.
E.
Ben-Jacob
, “
Bacterial self-organization: Co-enhancement of complexification and adaptability in a dynamic environment
,”
Philos. Trans. R. Soc. London, Ser. A
361
,
1283
1312
(
2003
).
23.
A.
Ayali
,
E.
Fuchs
,
Y.
Zilberstein
,
O.
Shefi
,
E.
Hulata
,
I.
Baruchi
, and
E.
Ben Jacob
, “
Contextual regularity and complexity of neuronal activity: from stand-alone cultures to task-performing animals
,”
Complexity
9
,
25
32
(
2004
).
24.
B.
Stevens
,
S.
Porta
,
L. L.
Haak
,
V.
Gallo
, and
R. D.
Fields
, “
Adenosine: A neuron-glial transmitter promoting myelination in the CNS in response to action potentials
,”
Neuron
36
,
855
868
(
2002
).
25.
P. R.
Laming
,
H.
Kimelberg
,
S.
Robinson
,
A.
Salm
,
N.
Hawrylak
,
C.
Müller
,
B.
Roots
, and
K.
Ng
, “
Neuronal-glial interactions and behavior
,”
Neurosci. Biobehav Rev.
24
,
295
340
(
2000
).
26.
C. E.
Stout
,
J. L.
Constantin
,
C. G.
Naus
, and
A. C.
Charles
, “
Intercellular calcium signaling in astrocytes via ATP release through connexin hemichannels
,”
J. Biol. Chem.
277
,
10482
10488
(
2002
).
27.
E. M.
Izhikevich
, “
Which model to use for cortical spiking neurons?
,”
IEEE Trans. Neural Netw.
15
,
1063
1070
(
2004
).
28.
E.
Ben-Jacob
and
H.
Levine
, “
Physical schemata underlying biological pattern formation - examples, issues and strategies
,”
Phys. Biol.
1
,
14
22
(
2004
).
29.
C.
Morris
and
H.
Lecar
, “
Voltage oscillations in the barnacle giant muscle fiber
,”
Biophys. J.
35
,
193
213
(
1981
).
30.
L. F.
Abbott
and
T.
Kepler
, “
Model neurons: from Hodgkin-Huxley to Hopfield
,”
Statistical Mechanics of Neural Networks
(
Springer-Verlag
, Berlin,
1990
).
31.
M.
Abeles
,
Corticonics
(
Cambridge University Press
, Cambridge,
1991
).
32.
M.
Tsodyks
,
A.
Uziel
, and
H.
Markram
, “
Synchrony generation in recurrent networks with frequency-dependent synapses
,”
J. Neurosci.
20
,
RC50
(
2000
).
34.
W. J.
Krzanowski
,
Principles of Multivariate Analysis–A user’s Perspective
(
Oxford University Press
, Oxford,
1990
), pp.
89
102
.
35.
See EPAPS Document No. E-CHAOEH-16-045601 for Appendices A–D. This document can be reached via a direct link in the online article's HTML reference section or via the EPAPS homepage (http://www.aip.org/pubservs/epaps.html).

Supplementary Material

You do not currently have access to this content.