{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:51:13Z","timestamp":1760237473598,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,5,11]],"date-time":"2020-05-11T00:00:00Z","timestamp":1589155200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["IIS-1845836"],"award-info":[{"award-number":["IIS-1845836"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["EB-026946"],"award-info":[{"award-number":["EB-026946"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Brain dynamics can exhibit narrow-band nonlinear oscillations and multistability. For a subset of disorders of consciousness and motor control, we hypothesized that some symptoms originate from the inability to spontaneously transition from one attractor to another. Using external perturbations, such as electrical pulses delivered by deep brain stimulation devices, it may be possible to induce such transition out of the pathological attractors. However, the induction of transition may be non-trivial, rendering the current open-loop stimulation strategies insufficient. In order to develop next-generation neural stimulators that can intelligently learn to induce attractor transitions, we require a platform to test the efficacy of such systems. To this end, we designed an analog circuit as a model for the multistable brain dynamics. The circuit spontaneously oscillates stably on two periods as an instantiation of a 3-dimensional continuous-time gated recurrent neural network. To discourage simple perturbation strategies, such as constant or random stimulation patterns from easily inducing transition between the stable limit cycles, we designed a state-dependent nonlinear circuit interface for external perturbation. We demonstrate the existence of nontrivial solutions to the transition problem in our circuit implementation.<\/jats:p>","DOI":"10.3390\/e22050537","type":"journal-article","created":{"date-parts":[[2020,5,11]],"date-time":"2020-05-11T12:26:30Z","timestamp":1589199990000},"page":"537","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Birhythmic Analog Circuit Maze: A Nonlinear Neurostimulation Testbed"],"prefix":"10.3390","volume":"22","author":[{"given":"Ian D.","family":"Jordan","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794, USA"},{"name":"Institute for Advanced Computing Science, Stony Brook University, Stony Brook, NY 11794, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4255-7750","authenticated-orcid":false,"given":"Il Memming","family":"Park","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794, USA"},{"name":"Institute for Advanced Computing Science, Stony Brook University, Stony Brook, NY 11794, USA"},{"name":"Department of Neurobiology and Behavior, Stony Brook University, Stony Brook, NY 11794, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.physrep.2014.02.007","article-title":"Control of multistability","volume":"540","author":"Pisarchik","year":"2014","journal-title":"Phys. Rep."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"8378","DOI":"10.3182\/20110828-6-IT-1002.00749","article-title":"Mechanisms for coexistence of two limit cycles in a biochemical model1","volume":"44","author":"Chaves","year":"2011","journal-title":"IFAC Proc. Vol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"6917","DOI":"10.1073\/pnas.79.22.6917","article-title":"Birhythmicity, chaos, and other patterns of temporal self-organization in a multiply regulated biochemical system","volume":"79","author":"Decroly","year":"1982","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"6171","DOI":"10.1063\/1.441007","article-title":"Chaos in the Belousov\u2013Zhabotinskii reaction","volume":"74","author":"Hudson","year":"1981","journal-title":"J. Chem. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"056206","DOI":"10.1103\/PhysRevE.83.056206","article-title":"Extreme multistability in a chemical model system","volume":"83","author":"Ngonghala","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"042226","DOI":"10.1103\/PhysRevE.94.042226","article-title":"Control of birhythmicity through conjugate self-feedback: Theory and experiment","volume":"94","author":"Biswas","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"244101","DOI":"10.1103\/PhysRevLett.91.244101","article-title":"Experimental observation of multifrequency patterns in arrays of coupled nonlinear oscillators","volume":"91","author":"In","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"164501","DOI":"10.1103\/PhysRevLett.93.164501","article-title":"Multistability and Memory Effect in a Highly Turbulent Flow: Experimental Evidence for a Global Bifurcation","volume":"93","author":"Ravelet","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"6188","DOI":"10.1103\/PhysRevE.60.6188","article-title":"Multistability and symmetry breaking in the two-dimensional flow around a square cylinder","volume":"60","author":"Shiau","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"031916","DOI":"10.1103\/PhysRevE.75.031916","article-title":"Inherent multistability in arrays of autoinducer coupled genetic oscillators","volume":"75","author":"Koseska","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"043118","DOI":"10.1063\/1.4898795","article-title":"Networks of coupled circuits: From a versatile toggle switch to collective coherent behavior","volume":"24","year":"2014","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/S0022-460X(03)00740-5","article-title":"Interaction of Hopf and period doubling bifurcations of a vibro-impact system","volume":"275","author":"Ding","year":"2004","journal-title":"J. Sound Vib."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"999","DOI":"10.1063\/1.1586551","article-title":"Chaotic cluster itinerancy and hierarchical cluster trees in electrochemical experiments","volume":"13","author":"Kiss","year":"2003","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/0025-5564(74)90031-5","article-title":"The existence of persistent states in the brain","volume":"19","author":"Little","year":"1974","journal-title":"Math. Biosci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1038\/nn.4497","article-title":"Dynamic models of large-scale brain activity","volume":"20","author":"Breakspear","year":"2017","journal-title":"Nat. Neurosci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"A1","DOI":"10.1016\/j.conb.2017.09.009","article-title":"Editorial overview: Computational neuroscience","volume":"46","author":"Fairhall","year":"2017","journal-title":"Curr. Opin. Neurobiol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1016\/j.conb.2014.01.008","article-title":"Neural circuits as computational dynamical systems","volume":"25","author":"Sussillo","year":"2014","journal-title":"Curr. Opin. Neurobiol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.conb.2019.06.007","article-title":"Cortical computations via metastable activity","volume":"58","author":"Fontanini","year":"2019","journal-title":"Curr. Opin. Neurobiol."},{"key":"ref_19","unstructured":"Zhao, Y., and Park, I.M. (2016). Interpretable Nonlinear Dynamic Modeling of Neural Trajectories. arXiv."},{"key":"ref_20","first-page":"E6192","article-title":"Attention model of binocular rivalry","volume":"114","author":"Li","year":"2017","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1038\/nature11129","article-title":"Neural population dynamics during reaching","volume":"487","author":"Churchland","year":"2012","journal-title":"Nature"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1038\/nature12742","article-title":"Context-dependent computation by recurrent dynamics in prefrontal cortex","volume":"503","author":"Mante","year":"2013","journal-title":"Nature"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.1016\/j.neuron.2017.02.019","article-title":"Navigating the Neural Space in Search of the Neural Code","volume":"93","author":"Jazayeri","year":"2017","journal-title":"Neuron"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/0167-9457(96)00003-6","article-title":"Dynamics of movement disorders","volume":"15","author":"Wagenaar","year":"1996","journal-title":"Hum. Mov. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1109\/TBME.2003.810705","article-title":"Epileptic seizure prediction and control","volume":"50","author":"Iasemidis","year":"2003","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"e33250","DOI":"10.7554\/eLife.33250","article-title":"A transient cortical state with sleep-like sensory responses precedes emergence from general anesthesia in humans","volume":"7","author":"Lewis","year":"2018","journal-title":"Elife"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/j.conb.2013.10.007","article-title":"Large-scale brain dynamics in disorders of consciousness","volume":"25","author":"Schiff","year":"2014","journal-title":"Curr. Opin. Neurobiol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"R898","DOI":"10.1016\/j.cub.2014.08.002","article-title":"Focusing Brain Therapeutic Interventions in Space and Time for Parkinson\u2019s Disease","volume":"24","author":"Little","year":"2014","journal-title":"Curr. Biol."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Hocker, D., and Park, I.M. (2019). Myopic control of neural dynamics. PLoS Comput. Biol., 15.","DOI":"10.1371\/journal.pcbi.1006854"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2359","DOI":"10.1007\/s11071-017-3808-3","article-title":"Numerical analyses and experimental validations of coexisting multiple attractors in Hopfield neural network","volume":"90","author":"Bao","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_31","first-page":"20170111","article-title":"Qualitative models and experimental investigation of chaotic NOR gates and set\/reset flip-flops","volume":"474","author":"Rahman","year":"2018","journal-title":"Proc. R. Soc. Math. Phys. Eng. Sci."},{"key":"ref_32","unstructured":"Zhao, Y., Nassar, J., Jordan, I., Bugallo, M., and Park, I.M. (2019). Streaming Variational Monte Carlo. arXiv."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Cho, K., van Merrienboer, B., Gulcehre, C., Bahdanau, D., Bougares, F., Schwenk, H., and Bengio, Y. (2014). Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation. arXiv.","DOI":"10.3115\/v1\/D14-1179"},{"key":"ref_34","unstructured":"Jordan, I.D., Sokol, P.A., and Park, I.M. (2019). Gated recurrent units viewed through the lens of continuous time dynamical systems. arXiv."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.physleta.2007.04.108","article-title":"An electronic implementation for Liao\u2019s chaotic delayed neuron model with non-monotonous activation function","volume":"369","author":"Duan","year":"2007","journal-title":"Phys. Lett. A"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Wilmshurst, T.H. (2001). Analog Circuit Techniques: With Digital Interfacing, Elsevier.","DOI":"10.1016\/B978-075065094-6\/50001-4"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/5\/537\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:27:48Z","timestamp":1760174868000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/5\/537"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,11]]},"references-count":36,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2020,5]]}},"alternative-id":["e22050537"],"URL":"https:\/\/doi.org\/10.3390\/e22050537","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2020,5,11]]}}}