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Yu S, Tuo H, Yao B, Zhang H, Liu F. Expression pattern of Arc in the hippocampus of a rat model of epilepsy and depression comorbidity. Brain Res Bull 2025; 223:111267. [PMID: 40023450 DOI: 10.1016/j.brainresbull.2025.111267] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/06/2024] [Revised: 02/19/2025] [Accepted: 02/21/2025] [Indexed: 03/04/2025]
Abstract
BACKGROUND Two key factors associated with the comorbidity of epilepsy and depression (EAD), activity regulated cytoskeletal protein (Arc) and homer protein homolog 1 (Homer1), were previously identified by our group through bioinformatics methods (Yu et al., 2022). The expression of Arc and Homer1 were verified through animal experiments. METHODS Six-week-old male specific pathogen-free grade Sprague Dawley rats (weighing 200 ± 20 g) received intraperitoneal injection of lithium chloride (LiCl)-pilocarpine for status epilepticus (SE) induction. SE was terminated after 30 min by intraperitoneal injection of diazepam, and spontaneous SE in rats was monitored by video for 2 weeks. The control group (Con group) was injected with an equal dose of sterile normal saline. Subsequently, EAD rats (EAD group) were selected from rat models of LiCl-pilocarpine-induced chronic epilepsy according to the immobility time of the forced swimming test on day 14 after LiCl-pilocarpine induced epilepsy. The remaining rats were included in the epilepsy group (EP group). Depression-like behaviors were evaluated using sucrose preference, open-field, and forced swimming tests. Body weight, sucrose preference percentage, the total distance of the open-field test, the average speed, the number of upright times, and the immobility time of the forced swimming test were assessed 14 and 28 days after LiCl-pilocarpine induced epilepsy. Rats in the EAD and EP groups were monitored by video for 2 weeks, and the frequency, grade, and duration of chronic spontaneous epileptic seizures were recorded. Epileptic seizures were compared between the EAD and EP groups. The expression of activity-regulated cytoskeletal protein (Arc) and Homer protein homolog 1 (Homer1) in the hippocampus of each group was detected by real-time quantitative PCR and western blot analysis. The fluorescence intensity of Arc in the hippocampus of each group was detected by immunofluorescence (IF) assay. RESULTS Compared with the Con and EP groups, rats in the EAD group exhibited a decreased body weight on day 28, a significant decrease in sucrose preference percentage on days 14 and 28, significantly extended immobility time, and significantly reduced total travel, average speed, the number of upright times. No significant differences in the number, grade, and duration of seizures were observed between the EAD and EP groups. Meanwhile, the expression level of Arc in the hippocampus was significantly decreased in the EAD group compared with the Con and EP groups; however, the expression level of Homer1 showed no significant change. IF results showed that Arc was mainly expressed in the cytoplasm, and the fluorescence intensity of Arc in hippocampal CA1, DG, and CA3 was lower in the EAD group than in the Con and EP groups. CONCLUSIONS The expression of Arc in the hippocampal tissue of EAD rats is significantly decreased, suggesting that Arc is associated with EAD.
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Affiliation(s)
- Shiqian Yu
- Department of Pediatrics, Renmin Hospital of Wuhan University, Wuhan, China
| | - Hu Tuo
- Department of Pediatrics, Renmin Hospital of Wuhan University, Wuhan, China.
| | - Baozhen Yao
- Department of Pediatrics, Renmin Hospital of Wuhan University, Wuhan, China
| | - Haiju Zhang
- Department of Pediatrics, Renmin Hospital of Wuhan University, Wuhan, China
| | - Fang Liu
- Department of Pediatrics, Renmin Hospital of Wuhan University, Wuhan, China
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Dai J, Lin Q, Ye L, Chen X, Li Z, Lu C, Chen M, Ba H, Sun J, Cai J. Temporal Trends in Serum Homer1 Levels and Their Prognostic Implications in Aneurysmal Subarachnoid Hemorrhage: A Prospective Cohort Study. Int J Gen Med 2025; 18:567-584. [PMID: 39911298 PMCID: PMC11796440 DOI: 10.2147/ijgm.s508325] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/24/2024] [Accepted: 01/23/2025] [Indexed: 02/07/2025] Open
Abstract
Background Homer scaffold protein 1 (homer1) may harbor neuroprotective effects against acute brain injury. This study aimed to investigate the prognostic role of serum homer1 in human aneurysmal subarachnoid hemorrhage (aSAH). Methods A total of 209 patients with aSAH and 100 controls were encompassed in this prospective cohort study. Serum homer1 levels were quantified at admission in all patients, on post-aSAH days 1, 3, 5, 7, 10, and 14 in 83 patients and at recruitments in controls. The modified Fisher scale (mFisher) and World Federation of Neurological Surgeons Scale (WFNS) were used for severity assessment. Glasgow Outcome Scale (GOS) scores of 1-3 at post-aSAH 90 days indicated poor prognosis. Results Serum homer1 levels of patients were abruptly elevated at admission, peaked at day 3, and afterwards decreased from day 5 until day 14 after aSAH, and were markedly higher during 14 days than those of controls. Serum homer1 levels were linearly and independently correlated with WFNS scores, mFisher scores, continuous GOS scores, ordinal GOS scores, poor prognosis risk and delayed cerebral ischemia (DCI) likelihood. DCI partially mediated association of serum homer1 levels with poor prognosis. The prognosis model was composed of the four independent predictors, that is serum homer1 levels, DCI, WFNS scores and mFisher scores. As demonstrated by a series of statistical methods, the model had a good performance. Conclusion Serum homer1 levels are significantly elevated in acute phase after aSAH, and are strongly related to heightened bleeding intensity, poor 90-day prognosis and DCI. Nevertheless, associational mechanism of serum homer1 and poor prognosis mediated by DCI needs to be further deciphered.
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Affiliation(s)
- Junxia Dai
- Department of Neurosurgery, Laboratory of Pan-Vascular Disease Management Center, The Dingli Clinical College of Wenzhou Medical University, The Wenzhou Central Hospital, Wenzhou, 325000, People’s Republic of China
| | - Qun Lin
- Department of Neurosurgery, Laboratory of Pan-Vascular Disease Management Center, The Dingli Clinical College of Wenzhou Medical University, The Wenzhou Central Hospital, Wenzhou, 325000, People’s Republic of China
| | - Liangzhi Ye
- Department of Neurosurgery, Laboratory of Pan-Vascular Disease Management Center, The Dingli Clinical College of Wenzhou Medical University, The Wenzhou Central Hospital, Wenzhou, 325000, People’s Republic of China
| | - Xiaoxiang Chen
- Department of Neurosurgery, Laboratory of Pan-Vascular Disease Management Center, The Dingli Clinical College of Wenzhou Medical University, The Wenzhou Central Hospital, Wenzhou, 325000, People’s Republic of China
| | - Zhiwei Li
- Department of Neurosurgery, Laboratory of Pan-Vascular Disease Management Center, The Dingli Clinical College of Wenzhou Medical University, The Wenzhou Central Hospital, Wenzhou, 325000, People’s Republic of China
| | - Chuan Lu
- Department of Neurosurgery, Laboratory of Pan-Vascular Disease Management Center, The Dingli Clinical College of Wenzhou Medical University, The Wenzhou Central Hospital, Wenzhou, 325000, People’s Republic of China
| | - Maohua Chen
- Department of Neurosurgery, Laboratory of Pan-Vascular Disease Management Center, The Dingli Clinical College of Wenzhou Medical University, The Wenzhou Central Hospital, Wenzhou, 325000, People’s Republic of China
| | - Huajun Ba
- Department of Neurosurgery, Laboratory of Pan-Vascular Disease Management Center, The Dingli Clinical College of Wenzhou Medical University, The Wenzhou Central Hospital, Wenzhou, 325000, People’s Republic of China
| | - Jun Sun
- Department of Neurosurgery, Laboratory of Pan-Vascular Disease Management Center, The Dingli Clinical College of Wenzhou Medical University, The Wenzhou Central Hospital, Wenzhou, 325000, People’s Republic of China
| | - Jianyong Cai
- Department of Neurosurgery, Laboratory of Pan-Vascular Disease Management Center, The Dingli Clinical College of Wenzhou Medical University, The Wenzhou Central Hospital, Wenzhou, 325000, People’s Republic of China
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Le Belle JE, Condro M, Cepeda C, Oikonomou KD, Tessema K, Dudley L, Schoenfield J, Kawaguchi R, Geschwind D, Silva AJ, Zhang Z, Shokat K, Harris NG, Kornblum HI. Acute rapamycin treatment reveals novel mechanisms of behavioral, physiological, and functional dysfunction in a maternal inflammation mouse model of autism and sensory over-responsivity. BIORXIV : THE PREPRINT SERVER FOR BIOLOGY 2024:2024.07.08.602602. [PMID: 39026891 PMCID: PMC11257517 DOI: 10.1101/2024.07.08.602602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 07/20/2024]
Abstract
Maternal inflammatory response (MIR) during early gestation in mice induces a cascade of physiological and behavioral changes that have been associated with autism spectrum disorder (ASD). In a prior study and the current one, we find that mild MIR results in chronic systemic and neuro-inflammation, mTOR pathway activation, mild brain overgrowth followed by regionally specific volumetric changes, sensory processing dysregulation, and social and repetitive behavior abnormalities. Prior studies of rapamycin treatment in autism models have focused on chronic treatments that might be expected to alter or prevent physical brain changes. Here, we have focused on the acute effects of rapamycin to uncover novel mechanisms of dysfunction and related to mTOR pathway signaling. We find that within 2 hours, rapamycin treatment could rapidly rescue neuronal hyper-excitability, seizure susceptibility, functional network connectivity and brain community structure, and repetitive behaviors and sensory over-responsivity in adult offspring with persistent brain overgrowth. These CNS-mediated effects are also associated with alteration of the expression of several ASD-,ion channel-, and epilepsy-associated genes, in the same time frame. Our findings suggest that mTOR dysregulation in MIR offspring is a key contributor to various levels of brain dysfunction, including neuronal excitability, altered gene expression in multiple cell types, sensory functional network connectivity, and modulation of information flow. However, we demonstrate that the adult MIR brain is also amenable to rapid normalization of these functional changes which results in the rescue of both core and comorbid ASD behaviors in adult animals without requiring long-term physical alterations to the brain. Thus, restoring excitatory/inhibitory imbalance and sensory functional network modularity may be important targets for therapeutically addressing both primary sensory and social behavior phenotypes, and compensatory repetitive behavior phenotypes.
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Bhandare A, Haley M, Torrico Anderson V, Domingos LB, Lopes M, Corrêa SAL, Wall MJ. ArcKR expression modifies synaptic plasticity following epileptic activity: Differential effects with in vitro and in vivo seizure-induction protocols. Epilepsia 2024; 65:2152-2164. [PMID: 38804501 DOI: 10.1111/epi.17981] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/16/2024] [Revised: 03/25/2024] [Accepted: 03/28/2024] [Indexed: 05/29/2024]
Abstract
OBJECTIVES Pathological forms of neural activity, such as epileptic seizures, modify the expression pattern of multiple proteins, leading to persistent changes in brain function. One such protein is activity-regulated cytoskeleton-associated protein (Arc), which is critically involved in protein-synthesis-dependent synaptic plasticity underlying learning and memory. In the present study, we have investigated how the expression of ArcKR, a form of Arc in which the ubiquitination sites have been mutated, resulting in slowed Arc degradation, modifies group I metabotropic glutamate receptor-mediated long-term depression (G1-mGluR-LTD) following seizures. METHODS We used a knock-in mice line that express ArcKR and two hyperexcitation models: an in vitro model, where hippocampal slices were exposed to zero Mg2+, 6 mM K+; and an in vivo model, where kainic acid was injected unilaterally into the hippocampus. In both models, field excitatory postsynaptic potentials (fEPSPs) were recorded from the CA1 region of hippocampal slices in response to Schaffer collateral stimulation and G1-mGluR-LTD was induced chemically with the group 1 mGluR agonist DHPG. RESULTS In the in vitro model, ArcKR expression enhanced the effects of seizure activity and increased the magnitude of G1-mGluR LTD, an effect that could be blocked with the mGluR5 antagonist MTEP. In the in vivo model, fEPSPs were significantly smaller in slices from ArcKR mice and were less contaminated by population spikes. In this model, the amount of G1-mGluR-LTD was significantly less in epileptic slices from ArcKR mice as compared to wildtype (WT) mice. SIGNIFICANCE We have shown that expression of ArcKR, a form of Arc in which degradation is reduced, significantly modulates the magnitude of G1-mGluR-LTD following epileptic seizures. However, the effect of ArcKR on LTD depends on the epileptic model used, with enhancement of LTD in an in vitro model and a reduction in the kainate mouse model.
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Affiliation(s)
- Amol Bhandare
- School of Life Sciences, University of Warwick, Coventry, UK
| | - Maisy Haley
- School of Life Sciences, University of Warwick, Coventry, UK
| | | | - Luana B Domingos
- Bradford School of Pharmacy and Medical Sciences, University of Bradford, Bradford, UK
| | - Marcia Lopes
- Bradford School of Pharmacy and Medical Sciences, University of Bradford, Bradford, UK
| | - Sonia A L Corrêa
- Bradford School of Pharmacy and Medical Sciences, University of Bradford, Bradford, UK
- Department of Life Sciences, Faculty of Science and Engineering, Manchester Metropolitan University, Manchester, UK
| | - Mark J Wall
- School of Life Sciences, University of Warwick, Coventry, UK
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Deng L, Jiang H, Lin J, Xu D, Qi A, Guo Q, Li PP, Wang X, Liu JS, Fu X, Li P. Clock knockout in inhibitory neurons reduces predisposition to epilepsy and influences anxiety-like behaviors in mice. Neurobiol Dis 2024; 193:106457. [PMID: 38423191 DOI: 10.1016/j.nbd.2024.106457] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/23/2023] [Revised: 02/21/2024] [Accepted: 02/25/2024] [Indexed: 03/02/2024] Open
Abstract
Epilepsy is a brain disorder affecting up to 1 in 26 individuals. Despite its clinical importance, the molecular mechanisms of epileptogenesis are still far from clarified. Our previous study showed that disruption of Clock in excitatory neurons alters cortical circuits and leads to generation of focal epilepsy. In this study, a GAD-Cre;Clockflox/flox mouse line with conditional Clock gene knockout in inhibitory neurons was established. We observed that seizure latency was prolonged, the severity and mortality of pilocarpine-induced seizure were significantly reduced, and memory was improved in GAD-Cre;Clockflox/flox mice. We hypothesize that mice with CLOCK knockout in inhibitory neurons have increased threshold for seizure, opposite from mice with CLOCK knockout in excitatory neurons. Further investigation showed Clock knockout in inhibitory neurons upregulated the basal protein level of ARC, a synaptic plasticity-associated immediate-early gene product, likely through the BDNF-ERK pathway. Altered basal levels of ARC may play an important role in epileptogenesis after Clock deletion in inhibitory neurons. Although sEPSCs and intrinsic properties of layer 5 pyramidal neurons in the somatosensory cortex exhibit no changes, the spine density increased in apical dendrite of pyramidal neurons in CLOCK knockout group. Our results suggest an underlying mechanism by which the circadian protein CLOCK in inhibitory neurons participates in neuronal activity and regulates the predisposition to epilepsy.
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Affiliation(s)
- Lu Deng
- Department of Geriatrics, The Second Affiliated Hospital and Yuying Children's Hospital of Wenzhou Medical University, Wenzhou 325000, Zhejiang Province, China
| | - Hong Jiang
- Department of Neurology, The Second Affiliated Hospital and Yuying Children's Hospital of Wenzhou Medical University, Wenzhou 325000, Zhejiang, China
| | - Jingjing Lin
- Department of Neurology, The Second Affiliated Hospital and Yuying Children's Hospital of Wenzhou Medical University, Wenzhou 325000, Zhejiang, China
| | - Di Xu
- Department of Geriatrics, The Second Affiliated Hospital and Yuying Children's Hospital of Wenzhou Medical University, Wenzhou 325000, Zhejiang Province, China
| | - Ailin Qi
- Department of Neurology, The Second Affiliated Hospital and Yuying Children's Hospital of Wenzhou Medical University, Wenzhou 325000, Zhejiang, China
| | - Qing Guo
- Department of Neurology, The Second Affiliated Hospital and Yuying Children's Hospital of Wenzhou Medical University, Wenzhou 325000, Zhejiang, China
| | - Ping-Ping Li
- Department of Neonatology, The Second Affiliated Hospital and Yuying Children's Hospital of Wenzhou Medical University, Wenzhou 325000, Zhejiang Province, China
| | - Xinshi Wang
- Department of Neurology, The First Affiliated Hospital of Wenzhou Medical University, Shangcai Village, Ouhai District, Wenzhou, Zhejiang Province, China
| | - Judy S Liu
- Department of Neurology, Department of Molecular Biology, Cell Biology, and Biochemistry, Brown University, Providence, RI 02903, USA.
| | - Xiaoqin Fu
- Department of Neonatology, The Second Affiliated Hospital and Yuying Children's Hospital of Wenzhou Medical University, Wenzhou 325000, Zhejiang Province, China.
| | - Peijun Li
- Department of Geriatrics, The Second Affiliated Hospital and Yuying Children's Hospital of Wenzhou Medical University, Wenzhou 325000, Zhejiang Province, China; Department of Neurology, The Second Affiliated Hospital and Yuying Children's Hospital of Wenzhou Medical University, Wenzhou 325000, Zhejiang, China; Institute of Brain Science and Brain-inspired Research, Shandong First Medical University & Shandong Academy of Medical Sciences, 250117, Jinan, Shandong, China.
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Chen Y, Wang X, Xiao B, Luo Z, Long H. Mechanisms and Functions of Activity-Regulated Cytoskeleton-Associated Protein in Synaptic Plasticity. Mol Neurobiol 2023; 60:5738-5754. [PMID: 37338805 DOI: 10.1007/s12035-023-03442-4] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/18/2023] [Accepted: 06/10/2023] [Indexed: 06/21/2023]
Abstract
Activity-regulated cytoskeleton-associated protein (Arc) is one of the most important regulators of cognitive functions in the brain regions. As a hub protein, Arc plays different roles in modulating synaptic plasticity. Arc supports the maintenance of long-term potentiation (LTP) by regulating actin cytoskeletal dynamics, while it guides the endocytosis of AMPAR in long-term depression (LTD). Moreover, Arc can self-assemble into capsids, leading to a new way of communicating among neurons. The transcription and translation of the immediate early gene Arc are rigorous procedures guided by numerous factors, and RNA polymerase II (Pol II) is considered to regulate the precise timing dynamics of gene expression. Since astrocytes can secrete brain-derived neurotrophic factor (BDNF) and L-lactate, their unique roles in Arc expression are emphasized. Here, we review the entire process of Arc expression and summarize the factors that can affect Arc expression and function, including noncoding RNAs, transcription factors, and posttranscriptional regulations. We also attempt to review the functional states and mechanisms of Arc in modulating synaptic plasticity. Furthermore, we discuss the recent progress in understanding the roles of Arc in the occurrence of major neurological disorders and provide new thoughts for future research on Arc.
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Affiliation(s)
- Yifan Chen
- Department of Neurology, Xiangya Hospital, Central South University, Changsha, 410008, Hunan, China
- Xiangya School of Stomatology, Central South University, Changsha, 410008, Hunan, China
| | - Xiaohu Wang
- State Key Laboratory of Cardiovascular Disease, Fuwai Hospital, National Center for Cardiovascular Diseases, Chinese Academy of Medical Sciences and Peking Union Medical College, Beijing, China
| | - Bo Xiao
- Department of Neurology, Xiangya Hospital, Central South University, Changsha, 410008, Hunan, China
- National Clinical Research Center for Geriatric Disorders, Xiangya Hospital, Central South University, Changsha, 410008, Hunan, China
- Clinical Research Center for Epileptic Disease of Hunan Province, Central South University, Changsha, Hunan, People's Republic of China, 410008
| | - Zhaohui Luo
- Department of Neurology, Xiangya Hospital, Central South University, Changsha, 410008, Hunan, China.
- National Clinical Research Center for Geriatric Disorders, Xiangya Hospital, Central South University, Changsha, 410008, Hunan, China.
- Clinical Research Center for Epileptic Disease of Hunan Province, Central South University, Changsha, Hunan, People's Republic of China, 410008.
| | - Hongyu Long
- Department of Neurology, Xiangya Hospital, Central South University, Changsha, 410008, Hunan, China.
- National Clinical Research Center for Geriatric Disorders, Xiangya Hospital, Central South University, Changsha, 410008, Hunan, China.
- Clinical Research Center for Epileptic Disease of Hunan Province, Central South University, Changsha, Hunan, People's Republic of China, 410008.
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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: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [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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