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Souza DLM, Gabrick EC, Protachevicz PR, Borges FS, Trobia J, Iarosz KC, Batista AM, Caldas IL, Lenzi EK. Adaptive exponential integrate-and-fire model with fractal extension. Chaos 2024; 34:023107. [PMID: 38341761 DOI: 10.1063/5.0176455] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/13/2023] [Accepted: 01/08/2024] [Indexed: 02/13/2024]
Abstract
The description of neuronal activity has been of great importance in neuroscience. In this field, mathematical models are useful to describe the electrophysical behavior of neurons. One successful model used for this purpose is the Adaptive Exponential Integrate-and-Fire (Adex), which is composed of two ordinary differential equations. Usually, this model is considered in the standard formulation, i.e., with integer order derivatives. In this work, we propose and study the fractal extension of Adex model, which in simple terms corresponds to replacing the integer derivative by non-integer. As non-integer operators, we choose the fractal derivatives. We explore the effects of equal and different orders of fractal derivatives in the firing patterns and mean frequency of the neuron described by the Adex model. Previous results suggest that fractal derivatives can provide a more realistic representation due to the fact that the standard operators are generalized. Our findings show that the fractal order influences the inter-spike intervals and changes the mean firing frequency. In addition, the firing patterns depend not only on the neuronal parameters but also on the order of respective fractal operators. As our main conclusion, the fractal order below the unit value increases the influence of the adaptation mechanism in the spike firing patterns.
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Affiliation(s)
- Diogo L M Souza
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Enrique C Gabrick
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
- Department of Physics, Humboldt University Berlin, Newtonstraße 15, 12489 Berlin, Germany
- Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany
| | | | - Fernando S Borges
- Department of Physiology and Pharmacology, State University of New York Downstate Health Sciences University, Brooklyn, New York 11203, USA
- Center for Mathematics, Computation, and Cognition, Federal University of ABC, 09606-045 São Bernardo do Campo, SP, Brazil
| | - José Trobia
- Department of Mathematics and Statistics, State University of Ponta Grossa, 84030-900 Ponta Grossa, Brazil
| | - Kelly C Iarosz
- University Center UNIFATEB, 84266-010 Telêmaco Borba, PR, Brazil
| | - Antonio M Batista
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
- Institute of Physics, University of São Paulo, 05508-090 São Paulo, SP, Brazil
- Department of Mathematics and Statistics, State University of Ponta Grossa, 84030-900 Ponta Grossa, Brazil
| | - Iberê L Caldas
- Institute of Physics, University of São Paulo, 05508-090 São Paulo, SP, Brazil
| | - Ervin K Lenzi
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
- Departament of Physics, State University of Ponta Grossa, Av. Gen. Carlos Cavalcanti 4748, Ponta Grossa 84030-900, PR, Brazil
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Gabrick EC, Brugnago EL, de Souza SLT, Iarosz KC, Szezech JD, Viana RL, Caldas IL, Batista AM, Kurths J. Impact of periodic vaccination in SEIRS seasonal model. Chaos 2024; 34:013137. [PMID: 38271628 DOI: 10.1063/5.0169834] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/28/2023] [Accepted: 12/26/2023] [Indexed: 01/27/2024]
Abstract
We study three different strategies of vaccination in an SEIRS (Susceptible-Exposed-Infected-Recovered-Susceptible) seasonal forced model, which are (i) continuous vaccination; (ii) periodic short-time localized vaccination, and (iii) periodic pulsed width campaign. Considering the first strategy, we obtain an expression for the basic reproduction number and infer a minimum vaccination rate necessary to ensure the stability of the disease-free equilibrium (DFE) solution. In the second strategy, short duration pulses are added to a constant baseline vaccination rate. The pulse is applied according to the seasonal forcing phases. The best outcome is obtained by locating intensive immunization at inflection of the transmissivity curve. Therefore, a vaccination rate of 44.4% of susceptible individuals is enough to ensure DFE. For the third vaccination proposal, additionally to the amplitude, the pulses have a prolonged time width. We obtain a non-linear relationship between vaccination rates and the duration of the campaign. Our simulations show that the baseline rates, as well as the pulse duration, can substantially improve the vaccination campaign effectiveness. These findings are in agreement with our analytical expression. We show a relationship between the vaccination parameters and the accumulated number of infected individuals, over the years, and show the relevance of the immunization campaign annual reaching for controlling the infection spreading. Regarding the dynamical behavior of the model, our simulations show that chaotic and periodic solutions as well as bi-stable regions depend on the vaccination parameters range.
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Affiliation(s)
- Enrique C Gabrick
- Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany
- Department of Physics, Humboldt University Berlin, Newtonstraße 15, 12489 Berlin, Germany
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Eduardo L Brugnago
- Institute of Physics, University of São Paulo, 05508-090 São Paulo, SP, Brazil
| | - Silvio L T de Souza
- Federal University of São João del-Rei, Campus Centro-Oeste, 35501-296 Divinópolis, MG, Brazil
| | - Kelly C Iarosz
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
- University Center UNIFATEB, 84266-010 Telêmaco Borba, PR, Brazil
| | - José D Szezech
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
- Department of Mathematics and Statistics, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Ricardo L Viana
- Institute of Physics, University of São Paulo, 05508-090 São Paulo, SP, Brazil
- Department of Physics, Federal University of Paraná, 81531-980 Curitiba, PR, Brazil
| | - Iberê L Caldas
- Institute of Physics, University of São Paulo, 05508-090 São Paulo, SP, Brazil
| | - Antonio M Batista
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
- Institute of Physics, University of São Paulo, 05508-090 São Paulo, SP, Brazil
- Department of Mathematics and Statistics, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany
- Department of Physics, Humboldt University Berlin, Newtonstraße 15, 12489 Berlin, Germany
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Borges FS, Protachevicz PR, Souza DLM, Bittencourt CF, Gabrick EC, Bentivoglio LE, Szezech JD, Batista AM, Caldas IL, Dura-Bernal S, Pena RFO. The Roles of Potassium and Calcium Currents in the Bistable Firing Transition. Brain Sci 2023; 13:1347. [PMID: 37759949 PMCID: PMC10527161 DOI: 10.3390/brainsci13091347] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/14/2023] [Revised: 09/13/2023] [Accepted: 09/15/2023] [Indexed: 09/29/2023] Open
Abstract
Healthy brains display a wide range of firing patterns, from synchronized oscillations during slow-wave sleep to desynchronized firing during movement. These physiological activities coexist with periods of pathological hyperactivity in the epileptic brain, where neurons can fire in synchronized bursts. Most cortical neurons are pyramidal regular spiking (RS) cells with frequency adaptation and do not exhibit bursts in current-clamp experiments (in vitro). In this work, we investigate the transition mechanism of spike-to-burst patterns due to slow potassium and calcium currents, considering a conductance-based model of a cortical RS cell. The joint influence of potassium and calcium ion channels on high synchronous patterns is investigated for different synaptic couplings (gsyn) and external current inputs (I). Our results suggest that slow potassium currents play an important role in the emergence of high-synchronous activities, as well as in the spike-to-burst firing pattern transitions. This transition is related to the bistable dynamics of the neuronal network, where physiological asynchronous states coexist with pathological burst synchronization. The hysteresis curve of the coefficient of variation of the inter-spike interval demonstrates that a burst can be initiated by firing states with neuronal synchronization. Furthermore, we notice that high-threshold (IL) and low-threshold (IT) ion channels play a role in increasing and decreasing the parameter conditions (gsyn and I) in which bistable dynamics occur, respectively. For high values of IL conductance, a synchronous burst appears when neurons are weakly coupled and receive more external input. On the other hand, when the conductance IT increases, higher coupling and lower I are necessary to produce burst synchronization. In light of our results, we suggest that channel subtype-specific pharmacological interactions can be useful to induce transitions from pathological high bursting states to healthy states.
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Affiliation(s)
- Fernando S. Borges
- Department of Physiology and Pharmacology, State University of New York Downstate Health Sciences University, Brooklyn, NY 11203, USA
- Center for Mathematics, Computation and Cognition, Federal University of ABC, São Bernardo do Campo 09606-045, Brazil
| | | | - Diogo L. M. Souza
- Graduate Program in Science, State University of Ponta Grossa, Ponta Grossa 84010-330, Brazil
| | - Conrado F. Bittencourt
- Graduate Program in Science, State University of Ponta Grossa, Ponta Grossa 84010-330, Brazil
| | - Enrique C. Gabrick
- Graduate Program in Science, State University of Ponta Grossa, Ponta Grossa 84010-330, Brazil
| | - Lucas E. Bentivoglio
- Graduate Program in Science, State University of Ponta Grossa, Ponta Grossa 84010-330, Brazil
| | - José D. Szezech
- Graduate Program in Science, State University of Ponta Grossa, Ponta Grossa 84010-330, Brazil
- Department of Mathematics and Statistics, State University of Ponta Grossa, Ponta Grossa 84030-900, Brazil
| | - Antonio M. Batista
- Graduate Program in Science, State University of Ponta Grossa, Ponta Grossa 84010-330, Brazil
- Department of Mathematics and Statistics, State University of Ponta Grossa, Ponta Grossa 84030-900, Brazil
| | - Iberê L. Caldas
- Institute of Physics, University of São Paulo, São Paulo 05508-090, Brazil
| | - Salvador Dura-Bernal
- Department of Physiology and Pharmacology, State University of New York Downstate Health Sciences University, Brooklyn, NY 11203, USA
- Center for Biomedical Imaging and Neuromodulation, The Nathan S. Kline Institute for Psychiatric Research, Orangeburg, NY 10962, USA
| | - Rodrigo F. O. Pena
- Department of Biological Sciences, Florida Atlantic University, Jupiter, FL 33458, USA
- Stiles-Nicholson Brain Institute, Florida Atlantic University, Jupiter, FL 33458, USA
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Gabrick EC, Sayari E, Souza DLM, Borges FS, Trobia J, Lenzi EK, Batista AM. Fractal and fractional SIS model for syphilis data. Chaos 2023; 33:093124. [PMID: 37712917 DOI: 10.1063/5.0153122] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/04/2023] [Accepted: 08/25/2023] [Indexed: 09/16/2023]
Abstract
This work studies the SIS model extended by fractional and fractal derivatives. We obtain explicit solutions for the standard and fractal formulations; for the fractional case, we study numerical solutions. As a real data example, we consider the Brazilian syphilis data from 2011 to 2021. We fit the data by considering the three variations of the model. Our fit suggests a recovery period of 11.6 days and a reproduction ratio (R0) equal to 6.5. By calculating the correlation coefficient (r) between the real data and the theoretical points, our results suggest that the fractal model presents a higher r compared to the standard or fractional case. The fractal formulation is improved when two different fractal orders with distinguishing weights are considered. This modification in the model provides a better description of the data and improves the correlation coefficient.
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Affiliation(s)
- Enrique C Gabrick
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Elaheh Sayari
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Diogo L M Souza
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Fernando S Borges
- Department of Physiology and Pharmacology, State University of New York Downstate Health Sciences University, Brooklyn, New York 11203, USA
| | - José Trobia
- Center for Mathematics, Computation, and Cognition, Federal University of ABC, 09606-045 São Bernardo do Campo, SP, Brazil
| | - Ervin K Lenzi
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
- Department of Physics, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Antonio M Batista
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
- Center for Mathematics, Computation, and Cognition, Federal University of ABC, 09606-045 São Bernardo do Campo, SP, Brazil
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5
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Borges FS, Protachevicz PR, Souza DLM, Bittencourt CF, Gabrick EC, Bentivoglio LE, Szezech JD, Batista AM, Caldas IL, Dura-Bernal S, Pena RFO. The Role of Potassium and Calcium Currents in the Bistable Firing Transition. bioRxiv 2023:2023.08.16.553625. [PMID: 37645875 PMCID: PMC10462112 DOI: 10.1101/2023.08.16.553625] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 08/31/2023]
Abstract
Healthy brains display a wide range of firing patterns, from synchronized oscillations during slowwave sleep to desynchronized firing during movement. These physiological activities coexist with periods of pathological hyperactivity in the epileptic brain, where neurons can fire in synchronized bursts. Most cortical neurons are pyramidal regular spiking cells (RS) with frequency adaptation and do not exhibit bursts in current-clamp experiments ( in vitro ). In this work, we investigate the transition mechanism of spike-to-burst patterns due to slow potassium and calcium currents, considering a conductance-based model of a cortical RS cell. The joint influence of potassium and calcium ion channels on high synchronous patterns is investigated for different synaptic couplings ( g syn ) and external current inputs ( I ). Our results suggest that slow potassium currents play an important role in the emergence of high-synchronous activities, as well as in the spike-to-burst firing pattern transitions. This transition is related to bistable dynamics of the neuronal network, where physiological asynchronous states coexist with pathological burst synchronization. The hysteresis curve of the coefficient of variation of the inter-spike interval demonstrates that a burst can be initiated by firing states with neuronal synchronization. Furthermore, we notice that high-threshold ( I L ) and low-threshold ( I T ) ion channels play a role in increasing and decreasing the parameter conditions ( g syn and I ) in which bistable dynamics occur, respectively. For high values of I L conductance, a synchronous burst appears when neurons are weakly coupled and receive more external input. On the other hand, when the conductance I T increases, higher coupling and lower I are necessary to produce burst synchronization. In light of our results, we suggest that channel subtype-specific pharmacological interactions can be useful to induce transitions from pathological high bursting states to healthy states.
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Affiliation(s)
- Fernando S Borges
- Department of Physiology and Pharmacology, State University of New York Downstate Health Sciences University, Brooklyn, New York, USA
- Center for Mathematics, Computation, and Cognition, Federal University of ABC, 09606-045 São Bernardo do Campo, SP, Brazil
| | | | - Diogo L M Souza
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Conrado F Bittencourt
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Enrique C Gabrick
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Lucas E Bentivoglio
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - José D Szezech
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
- Department of Mathematics and Statistics, State University of Ponta Grossa, Ponta Grossa, Brazil
| | - Antonio M Batista
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
- Department of Mathematics and Statistics, State University of Ponta Grossa, Ponta Grossa, Brazil
| | - Iberê L Caldas
- Institute of Physics, University of São Paulo, 05508-090 São Paulo, SP, Brazil
| | - Salvador Dura-Bernal
- Department of Physiology and Pharmacology, State University of New York Downstate Health Sciences University, Brooklyn, New York, USA
- Center for Biomedical Imaging and Neuromodulation, The Nathan S. Kline Institute for Psychiatric Research, New York, USA
| | - Rodrigo F O Pena
- Department of Biological Sciences, Florida Atlantic University, Jupiter, Florida, USA
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Sayari E, Gabrick EC, Borges FS, Cruziniani FE, Protachevicz PR, Iarosz KC, Szezech JD, Batista AM. Analyzing bursting synchronization in structural connectivity matrix of a human brain under external pulsed currents. Chaos 2023; 33:033131. [PMID: 37003788 DOI: 10.1063/5.0135399] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/18/2022] [Accepted: 02/27/2023] [Indexed: 06/19/2023]
Abstract
Cognitive tasks in the human brain are performed by various cortical areas located in the cerebral cortex. The cerebral cortex is separated into different areas in the right and left hemispheres. We consider one human cerebral cortex according to a network composed of coupled subnetworks with small-world properties. We study the burst synchronization and desynchronization in a human neuronal network under external periodic and random pulsed currents. With and without external perturbations, the emergence of bursting synchronization is observed. Synchronization can contribute to the processing of information, however, there are evidences that it can be related to some neurological disorders. Our results show that synchronous behavior can be suppressed by means of external pulsed currents.
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Affiliation(s)
- Elaheh Sayari
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Enrique C Gabrick
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Fernando S Borges
- Department of Physiology and Pharmacology, State University of New York Downstate Health Sciences University, Brooklyn, New York 11203, USA
| | - Fátima E Cruziniani
- Department of Physics, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | | | - Kelly C Iarosz
- University Center UNIFATEB, 84266-010 Telêmaco Borba, PR, Brazil
| | - José D Szezech
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Antonio M Batista
- Graduate Program in Science, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
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Borges FS, Gabrick EC, Protachevicz PR, Higa GSV, Lameu EL, Rodriguez PXR, Ferraz MSA, Szezech JD, Batista AM, Kihara AH. Intermittency properties in a temporal lobe epilepsy model. Epilepsy Behav 2023; 139:109072. [PMID: 36652897 DOI: 10.1016/j.yebeh.2022.109072] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 10/05/2022] [Revised: 12/22/2022] [Accepted: 12/26/2022] [Indexed: 01/18/2023]
Abstract
Neuronal synchronization is important for communication between brain regions and plays a key role in learning. However, changes in connectivity can lead to hyper-synchronized states related to epileptic seizures that occur intermittently with asynchronous states. The activity-regulated cytoskeleton-associated protein (ARC) is related to synaptic alterations which can lead to epilepsy. Induction of status epilepticus in rodent models causes the appearance of intense ARC immunoreactive neurons (IAINs), which present a higher number of connections and conductance intensity than non-IAINs. This alteration might contribute to abnormal epileptic seizure activity. In this work, we investigated how IAINs connectivity influences the firing pattern and synchronization in neural networks. Firstly, we showed the appearance of synchronized burst patterns due to the emergence of IAINs. Second, we described how the increase of IAINs connectivity favors the appearance of intermittent up and down activities associated with synchronous bursts and asynchronous spikes, respectively. Once the intermittent activity was properly characterized, we applied the optogenetics control of the high synchronous activities in the intermittent regime. To do this, we considered that 1% of neurons were transfected and became photosensitive. We observed that optogenetics methods to control synchronized burst patterns are effective when IAINs are chosen as photosensitive, but not effective in non-IAINs. Therefore, our analyses suggest that IAINs play a pivotal role in both the generation and suppression of highly synchronized activities.
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Affiliation(s)
- F S Borges
- Department of Physiology and Pharmacology, State University of New York Downstate Health Sciences University, Brooklyn, NY, USA; Center for Mathematics, Computation, and Cognition, Federal University of ABC, São Bernardo do Campo, SP, Brazil.
| | - E C Gabrick
- Graduate in Science Program - Physics, State University of Ponta Grossa, Ponta Grossa, PR, Brazil
| | - P R Protachevicz
- Institute of Physics, University of São Paulo, São Paulo, SP, Brazil
| | - G S V Higa
- Center for Mathematics, Computation, and Cognition, Federal University of ABC, São Bernardo do Campo, SP, Brazil; Institute of Chemistry, University of São Paulo, São Paulo, SP, Brazil
| | - E L Lameu
- Snyder Institute for Chronic Diseases, University of Calgary, Calgary, AB, Canada
| | - P X R Rodriguez
- Center for Mathematics, Computation, and Cognition, Federal University of ABC, São Bernardo do Campo, SP, Brazil; Faculty of Medicine, University of Bonn, Bonn, Germany
| | - M S A Ferraz
- Center for Mathematics, Computation, and Cognition, Federal University of ABC, São Bernardo do Campo, SP, Brazil
| | - J D Szezech
- Graduate in Science Program - Physics, State University of Ponta Grossa, Ponta Grossa, PR, Brazil; Department of Mathematics and Statistics, State University of Ponta Grossa, Ponta Grossa, PR, Brazil
| | - A M Batista
- Graduate in Science Program - Physics, State University of Ponta Grossa, Ponta Grossa, PR, Brazil; Institute of Physics, University of São Paulo, São Paulo, SP, Brazil; Department of Mathematics and Statistics, State University of Ponta Grossa, Ponta Grossa, PR, Brazil
| | - A H Kihara
- Center for Mathematics, Computation, and Cognition, Federal University of ABC, São Bernardo do Campo, SP, Brazil.
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