In this study, the Hénon map was analyzed using quantifiers from information theory in order to compare its dynamics to experimental data from brain regions known to exhibit chaotic behavior. The goal was to investigate the potential of the Hénon map as a model for replicating chaotic brain dynamics in the treatment of Parkinson’s and epilepsy patients. The dynamic properties of the Hénon map were compared with data from the subthalamic nucleus, the medial frontal cortex, and a q-DG model of neuronal input–output with easy numerical implementation to simulate the local behavior of a population. Using information theory tools, Shannon entropy, statistical complexity, and Fisher’s information were analyzed, taking into account the causality of the time series. For this purpose, different windows over the time series were considered. The findings revealed that neither the Hénon map nor the q-DG model could perfectly replicate the dynamics of the brain regions studied. However, with careful consideration of the parameters, scales, and sampling used, they were able to model some characteristics of neural activity. According to these results, normal neural dynamics in the subthalamic nucleus region may present a more complex spectrum within the complexity–entropy causality plane that cannot be represented by chaotic models alone. The dynamic behavior observed in these systems using these tools is highly dependent on the studied temporal scale. As the size of the sample studied increases, the dynamics of the Hénon map become increasingly different from those of biological and artificial neural systems.

1.
L.
Pezard
and
J.-L.
Nandrino
, “
Dynamical paradigm in psychopathology: ‘Chaos theory,’ from physics to psychiatry
,”
L’Encéphale
27
,
260
268
(
2001
), available at http://hdl.handle.net/20.500.12210/39357.
2.
L.
Zhang
, “Design and implementation of neural network based chaotic system model for the dynamical control of brain stimulation” (2017).
3.
A. L.
Goldberger
, “
Fractal variability versus pathologic periodicity: Complexity loss and stereotypy in disease
,”
Perspect. Biol. Med.
40
,
543
561
(
1997
).
4.
M. C.
Mackey
and
L.
Glass
, “
Oscillation and chaos in physiological control systems
,”
Science
197
,
287
289
(
1977
).
5.
M. C.
Mackey
and
J. G.
Milton
, “
Dynamical diseases
,”
Ann. N.Y. Acad. Sci.
504
,
16
32
(
1987
).
6.
E.
Pereda
,
R. Q.
Quiroga
, and
J.
Bhattacharya
, “
Nonlinear multivariate analysis of neurophysiological signals
,”
Prog. Neurobiol.
77
,
1
37
(
2005
).
7.
R.
Falahian
,
M. M.
Dastjerdi
,
M.
Molaie
,
S.
Jafari
, and
S.
Gharibzadeh
, “
Artificial neural network-based modeling of brain response to flicker light
,”
Nonlinear Dyn.
81
,
1951
1967
(
2015
).
8.
H.
Korn
and
P.
Faure
, “
Is there chaos in the brain? II. Experimental evidence and related models
,”
C. R. Biol.
326
,
787
840
(
2003
).
9.
L.
Zhang
, “Hénon map chaotic system critical points analysis and classification for the dynamic control of brain stimulation,” in 2017 IEEE XXIV International Conference on Electronics, Electrical Engineering and Computing (INTERCON) (IEEE, 2017), pp. 1–4.
10.
L.
Zhang
, “Hénon map chaotic system analysis and vhdl-based fixed-point fpga implementation for brain stimulation,” in 2017 IEEE 30th Canadian Conference on Electrical and Computer Engineering CCECE (IEEE, 2017).
11.
Z. E.
Patai
,
T.
Foltynie
,
P.
Limousin
,
H.
Akram
,
L.
Zrinzo
,
R.
Bogacz
, and
V.
Litvak
, “
Conflict detection in a sequential decision task is associated with increased cortico-subthalamic coherence and prolonged subthalamic oscillatory response in the beta band
,”
bioRxiv
(
2022
), see https://www.biorxiv.org/content/early/2022/02/10/2020.06.09.141713.full.pdf.
12.
M.
Frank
, “
Hold your horses: A dynamic computational role for the subthalamic nucleus in decision making
,”
Neural Netw.
19
,
1120
36
(
2006
).
13.
Y. M.
Vachez
and
M. C.
Creed
, “
Deep brain stimulation of the subthalamic nucleus modulates reward-related behavior: A systematic review
,”
Front. Hum. Neurosci.
14
,
578564
(
2020
).
14.
B.
Seymour
,
M.
Barbe
,
P.
Dayan
,
T.
Shiner
,
R.
Dolan
, and
G.
Fink
, “
Deep brain stimulation of the subthalamic nucleus modulates sensitivity to decision outcome value in Parkinson’s disease
,”
Sci. Rep.
6
,
32509
(
2016
).
15.
M. C.
Biagioni
,
K.
Sharma
,
H. A.
Migdadi
, and
A.
Cucca
, “Non-invasive neuromodulation therapies for Parkinson’s disease,” in Parkinson’s Disease—Understanding Pathophysiology and Developing Therapeutic Strategies (InTech, 2018).
16.
K.-H. S.
Chen
and
R.
Chen
, “
Invasive and noninvasive brain stimulation in Parkinson’s disease: Clinical effects and future perspectives
,”
Clin. Pharmacol. Ther.
106
,
763
775
(
2019
).
17.
Y.
Scharf
, “
A chaotic outlook on biological systems
,”
Chaos, Solitons Fractals
95
,
42
47
(
2017
).
18.
W.
Xingyuan
and
L.
Chao
, “
Researches on chaos phenomenon of EEG dynamics model
,”
Appl. Math. Comput.
183
,
30
41
(
2006
).
19.
J.
Humplik
and
G.
Tkačik
, “
Probabilistic models for neural populations that naturally capture global coupling and criticality
,”
PLoS Comput. Biol.
13
,
e1005763
(
2017
).
20.
M.
Hénon
, “
A two-dimensional mapping with a strange attractor
,”
Commun. Math. Phys.
50
,
69
77
(
1976
).
21.
C.
Bandt
and
B.
Pompe
, “
Permutation entropy: A natural complexity measure for time series
,”
Phys. Rev. Lett.
88
,
174102
(
2002
).
22.
S.-I.
Amari
,
H.
Nakahara
,
S.
Wu
, and
Y.
Sakai
, “
Synchronous firing and higher-order interactions in neuron pool
,”
Neural Comput.
15
,
127
142
(
2003
).
23.
J. H.
Macke
,
M.
Opper
, and
M.
Bethge
, “The effect of pairwise neural correlations on global population statistics,” Max-Planck Institute for Biological Cybernetics, Technical Report No. 183 (2009).
24.
J. H.
Macke
,
M.
Opper
, and
M.
Bethge
, “An analytically tractable model of neural population activity in the presence of common input explains higher-order correlations and entropy,” http://arxiv.org/abs/arXiv:1009.2855v2 (g-bio.nc) (2010).
25.
J. H.
Macke
,
P.
Berens
,
A. S.
Ecker
,
A. S.
Tolias
, and
M.
Bethge
, “
Generating spike trains with specified correlation coefficients
,”
Neural Comput.
21
,
397
423
(
2009
).
26.
L.
Montangie
and
F.
Montani
, “
Higher-order correlations in common input shapes the output spiking activity of a neural population
,”
Physica A
471
,
845
861
(
2017
).
27.
L.
Montangie
, “Modelos minimales y teoría de la información de poblaciones neuronales,” Ph.D. thesis (Universidad Nacional de La Plata, 2017).
28.
O.
Rosso
and
C.
Masoller
, “
Detecting and quantifying temporal correlations in stochastic resonance via information theory measures
,”
Eur. Phys. J. B
69
,
37
43
(
2009
).
29.
F.
Montani
and
O. A.
Rosso
, “
Entropy-complexity characterization of brain development in chickens
,”
Entropy
16
,
4677
4692
(
2014
).
30.
O.
Rosso
and
C.
Masoller
, “
Detecting and quantifying stochastic and coherence resonances via information-theory complexity measurements
,”
Phys. Rev. E
79
,
040106(R)
(
2009
).
31.
K.
Keller
and
H.
Lauffer
, “
Symbolic analysis of high-dimensional time series
,”
Int. J. Bifurcat. Chaos
13
,
2657
2668
(
2003
).
32.
A. A. B.
Pessa
and
H. V.
Ribeiro
, “
Ordpy: A Python package for data analysis with permutation entropy and ordinal network methods
,”
Chaos
31
,
063110
(
2021
).
33.
D.
Feldman
and
J.
Crutchfield
, “
Measures of statistical complexity: Why?
,”
Phys. Lett. A
238
,
244
252
(
1998
).
34.
C.
Shannon
and
W.
Weaver
,
The Mathematical Theory of Communication
(
University of Illinois Press
,
Champaign, IL
,
1949
), ISBN-10: 0252725484.
35.
M.
Martín
,
A.
Plastino
, and
O.
Rosso
, “
Generalized statistical complexity measures: Geometrical and analytical properties
,”
Physica A
369
,
439
462
(
2006
).
36.
R.
López-Ruiz
,
H.
Mancini
, and
X.
Calbet
, “
A statistical measure of complexity
,”
Phys. Lett. A
209
,
321
326
(
1995
).
37.
P.
Lamberti
,
M.
Martín
,
A.
Plastino
, and
O.
Rosso
, “
Intensive entropic non-triviality measure
,”
Physica A
334
,
119
131
(
2008
).
38.
I.
Grosse
,
P.
Bernaola-Galván
,
P.
Carpena
,
R.
Román-Roldán
,
J.
Oliver
, and
H.
Stanley
, “
Analysis of symbolic sequences using the Jensen-Shannon divergence
,”
Phys. Rev. E
65
,
041905
(
2002
).
39.
D.
Feldman
,
C.
McTague
, and
J.
Crutchfield
, “
The organization of intrinsic computation: Complexity-entropy diagrams and the diversity of natural information processing
,”
Chaos
18
,
043106
(
2008
).
40.
R.
Fisher
, “
On the mathematical foundations of theoretical statistics
,”
Philos. Trans. R. Soc. London, Ser. A
222
,
309
368
(
1922
).
41.
B.
Frieden
,
Science from Fisher Information: A Unification
(
Cambridge University Press
,
Cambridge
,
2004
).
42.
A.
Mayer
,
C.
Pawlowski
, and
H.
Cabezas
, “
Fisher information and dynamic regime changes in ecological systems
,”
Ecol. Modell.
195
,
72
82
(
2006
).
43.
O.
Rosso
,
L.
De Micco
,
A.
Plastino
, and
H.
Larrondo
, “
Info-quantifiers’ map-characterization revisited
,”
Physica A
389
,
249
262
(
2010
).
44.
F.
Olivares
,
A.
Plastino
, and
O.
Rosso
, “
Ambiguities in the Bandt-Pompe’s methodology for local entropic quantifiers
,”
Physica A
391
,
2518
2526
(
2012
).
45.
F.
Olivares
,
A.
Plastino
, and
O.
Rosso
, “
Contrasting chaos with noise via local versus global information quantifiers
,”
Phys. Lett. A
376
,
1577
1583
(
2012
).
46.
K.
Zografos
,
K.
Ferentinos
, and
T.
Papaioannou
, “
Discrete approximations to the Csiszár, Renyi, and Fisher measures of information
,”
Can. J. Stat.
14
,
355
366
(
1986
).
47.
L.
Pardo
,
D.
Morales
,
K.
Ferentinos
, and
K.
Zografos
, “
Discretization problems on generalized entropies and R-divergences
,”
Kybernetika
30
,
445
460
(
1994
).
48.
M.
Madiman
,
O.
Johnson
, and
I.
Kontoyiannis
, “Fisher information, compound Poisson approximation, and the Poisson channel,” in IEEE International Symposium on Information Theory, ISIT (IEEE, Nice, 2007), pp. 976–980.
49.
P.
Sanchez-Moreno
,
J.
Dehesa
, and
R.
Yanez
, “Discrete densities and Fisher information,” in Proceedings of the 14th International Conference on Difference Equations and Applications (Uğur-Bahçeşehir University Publishing Company, Istanbul, 2009), pp. 291–298.
50.
F.
Pennini
and
A.
Plastino
, “
Reciprocity relations between ordinary temperature and the Frieden-Soffer Fisher temperature
,”
Phys. Rev. E
71
,
047102
(
2005
).
51.
M.
Zanin
and
F.
Olivares
, “
Ordinal patterns-based methodologies for distinguishing chaos from noise in discrete time series
,”
Commun. Phys.
4
,
190
(
2021
).
52.
O.
Rosso
,
H.
Larrondo
,
M.
Martin
,
A.
Plastino
, and
M.
Fuentes
, “
Distinguishing noise from chaos
,”
Phys. Rev. Lett.
99
,
154102
(
2007
).
53.
R.
Bogacz
,
V.
Litvak
,
A.
Oswal
,
S.
Little
,
D.
Pedrosa
,
D.
Herz
,
V.
Dayal
, and
E.
Zita Patai
, “Human LFP recordings from STN during sequential conflict task,” University of Oxford, https://data.mrc.ox.ac.uk/data-set/human-lfp-recordings-stn-during-sequential-conflict-task (2020).
54.
B.
Frauscher
,
N.
von Ellenrieder
,
R.
Zelmann
,
I.
Doležalová
,
L.
Minotti
,
A.
Olivier
,
J.
Hall
,
D.
Hoffmann
,
D. K.
Nguyen
,
P.
Kahane
,
F.
Dubeau
, and
J.
Gotman
, “
Atlas of the normal intracranial electroencephalogram: Neurophysiological awake activity in different cortical areas
,”
Brain
141
,
1130
1144
(
2018
).
55.
B.
Frauscher
,
N.
von Ellenrieder
,
R.
Zelmann
,
C.
Rogers
,
D. K.
Nguyen
,
P.
Kahane
,
F.
Dubeau
, and
J.
Gotman
, “
High-frequency oscillations in the normal human brain
,”
Ann. Neurol.
84
,
374
385
(
2018
).
56.
N.
von Ellenrieder
,
J.
Gotman
,
R.
Zelmann
,
C.
Rogers
,
D. K.
Nguyen
,
P.
Kahane
,
F.
Dubeau
, and
B.
Frauscher
, “
How the human brain sleeps: Direct cortical recordings of normal brain activity
,”
Ann. Neurol.
87
,
289
301
(
2020
).
57.
S.
Panzeri
,
R. S.
Petersen
,
S. R.
Schultz
,
M.
Lebedev
, and
M. E.
Diamond
, “
The role of spike timing in the coding of stimulus location in rat somatosensory cortex
,”
Neuron
29
,
769
777
(
2001
).
58.
F.
Montani
,
A.
Kohn
,
M. A.
Smith
, and
S. R.
Schultz
, “
The role of correlations in direction and contrast coding in the primary visual cortex
,”
J. Neurosci.
27
,
2338
2348
(
2007
).
59.
M. A.
Montemurro
,
S.
Panzeri
,
M.
Maravall
,
A.
Alenda
,
M. R.
Bale
,
M.
Brambilla
, and
R. S.
Petersen
, “
Role of precise spike timing in coding of dynamic vibrissa stimuli in somatosensory thalamus
,”
J. Neurophysiol.
98
,
1871
1882
(
2007
).
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