{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T11:02:10Z","timestamp":1780570930400,"version":"3.54.1"},"reference-count":59,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,19]],"date-time":"2025-02-19T00:00:00Z","timestamp":1739923200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Sciences and Engineering Council of Canada (NSERC) Discovery","award":["210977"],"award-info":[{"award-number":["210977"]}]},{"name":"Natural Sciences and Engineering Council of Canada (NSERC) Discovery","award":["259113"],"award-info":[{"award-number":["259113"]}]},{"name":"NSERC Horizons","award":["210977"],"award-info":[{"award-number":["210977"]}]},{"name":"NSERC Horizons","award":["259113"],"award-info":[{"award-number":["259113"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Interactions between excitatory and inhibitory neurons in the cerebral cortex give rise to different regimes of activity and modulate brain oscillations. A prominent regime in the cortex is the inhibition-stabilized network (ISN), defined by strong recurrent excitation balanced by inhibition. While theoretical models have captured the response of brain circuits in the ISN state, their connectivity is typically hard-wired, leaving unanswered how a network may self-organize to an ISN state and dynamically switch between ISN and non-ISN states to modulate oscillations. Here, we introduce a mean-rate model of coupled Wilson-Cowan equations, link ISN and non-ISN states to Kolmogorov-Sinai entropy, and demonstrate how homeostatic plasticity (HP) allows the network to express both states depending on its level of tonic activity. This mechanism enables the model to capture a broad range of experimental effects, including (i) a paradoxical decrease in inhibitory activity, (ii) a phase offset between excitation and inhibition, and (iii) damped gamma oscillations. Further, the model accounts for experimental work on asynchronous quenching, where an external input suppresses intrinsic oscillations. Together, findings show that oscillatory activity is modulated by the dynamical regime of the network under the control of HP, thus advancing a framework that bridges neural dynamics, entropy, oscillations, and synaptic plasticity.<\/jats:p>","DOI":"10.3390\/e27020215","type":"journal-article","created":{"date-parts":[[2025,2,19]],"date-time":"2025-02-19T05:34:26Z","timestamp":1739943266000},"page":"215","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Control of Inhibition-Stabilized Oscillations in Wilson-Cowan Networks with Homeostatic Plasticity"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0520-5975","authenticated-orcid":false,"given":"Camille","family":"Godin","sequence":"first","affiliation":[{"name":"School of Psychology, University of Ottawa, 156 Jean-Jacques Lussier, Ottawa, ON K1N 6N5, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1837-687X","authenticated-orcid":false,"given":"Matthew R.","family":"Krause","sequence":"additional","affiliation":[{"name":"Department of Neurology and Neurosurgery, Montreal Neurological Institute, McGill University, Montreal, QC H3A 2B4, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9433-8675","authenticated-orcid":false,"given":"Pedro G.","family":"Vieira","sequence":"additional","affiliation":[{"name":"Department of Neurology and Neurosurgery, Montreal Neurological Institute, McGill University, Montreal, QC H3A 2B4, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christopher C.","family":"Pack","sequence":"additional","affiliation":[{"name":"Department of Neurology and Neurosurgery, Montreal Neurological Institute, McGill University, Montreal, QC H3A 2B4, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2457-7173","authenticated-orcid":false,"given":"Jean-Philippe","family":"Thivierge","sequence":"additional","affiliation":[{"name":"School of Psychology, University of Ottawa, 156 Jean-Jacques Lussier, Ottawa, ON K1N 6N5, Canada"},{"name":"Brain and Mind Research Institute, University of Ottawa, 451 Smyth Rd., Ottawa, ON K1H 8M5, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1146\/annurev-neuro-071714-034002","article-title":"Plasticity of Cortical Excitatory-Inhibitory Balance","volume":"38","author":"Froemke","year":"2015","journal-title":"Annu. Rev. Neurosci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1016\/j.neuron.2009.03.021","article-title":"Beyond Poisson: Increased spike-time regularity across primate parietal cortex","volume":"62","author":"Maimon","year":"2009","journal-title":"Neuron"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1926","DOI":"10.1126\/science.1099745","article-title":"Neuronal Oscillations in Cortical Networks","volume":"304","author":"Draguhn","year":"2004","journal-title":"Science"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3870","DOI":"10.1523\/JNEUROSCI.18-10-03870.1998","article-title":"The variable discharge of cortical neurons: Implications for connectivity, computation, and information coding","volume":"18","author":"Shadlen","year":"1998","journal-title":"J. Neurosci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1002\/(SICI)1098-1063(1996)6:2<149::AID-HIPO6>3.0.CO;2-K","article-title":"Theta phase precession in hippocampal neuronal populations and the compression of temporal sequences","volume":"6","author":"Skaggs","year":"1996","journal-title":"Hippocampus"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1077","DOI":"10.1016\/j.neuron.2013.08.019","article-title":"Attentional modulation of cell-class-specific gamma-band synchronization in awake monkey area v4","volume":"80","author":"Vinck","year":"2013","journal-title":"Neuron"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"12337","DOI":"10.1523\/JNEUROSCI.3680-12.2013","article-title":"Feedforward inhibition underlies the propagation of cholinergically induced gamma oscillations from hippocampal CA3 to CA1","volume":"33","author":"Zemankovics","year":"2013","journal-title":"J. Neurosci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1038\/nrn2578","article-title":"Extracting information from neuronal populations: Information theory and decoding approaches","volume":"10","author":"Quiroga","year":"2009","journal-title":"Nat. Rev. Neurosci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1038\/nrn2315","article-title":"Regulation of spike timing in visual cortical circuits","volume":"9","author":"Tiesinga","year":"2008","journal-title":"Nat. Rev. Neurosci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1250","DOI":"10.1523\/JNEUROSCI.1623-09.2010","article-title":"Gamma-phase shifting in awake monkey visual cortex","volume":"30","author":"Vinck","year":"2010","journal-title":"J. Neurosci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"e54875","DOI":"10.7554\/eLife.54875","article-title":"Inhibition stabilization is a widespread property of cortical networks","volume":"9","author":"Sanzeni","year":"2020","journal-title":"eLife"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1585","DOI":"10.1137\/S003613990240814X","article-title":"A Periodically Forced Wilson\u2014Cowan System","volume":"63","author":"Pollina","year":"2003","journal-title":"SIAM J. Appl. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4382","DOI":"10.1523\/JNEUROSCI.17-11-04382.1997","article-title":"Paradoxical effects of external modulation of inhibitory interneurons","volume":"17","author":"Tsodyks","year":"1997","journal-title":"J. Neurosci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1152\/jn.1968.31.3.337","article-title":"Relations between unit activity and evoked potentials in prepyriform cortex of cats","volume":"31","author":"Freeman","year":"1968","journal-title":"J. Neurophysiol."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Ledoux, E., and Brunel, N. (2011). Dynamics of networks of excitatory and inhibitory neurons in response to time-dependent inputs. Front. Comput. Neurosci., 5.","DOI":"10.3389\/fncom.2011.00025"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0006-3495(72)86068-5","article-title":"Excitatory and Inhibitory Interactions in Localized Populations of Model Neurons","volume":"12","author":"Wilson","year":"1972","journal-title":"Biophys. J."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Zou, X., and Wang, D.-H. (2016). On the Phase Relationship between Excitatory and Inhibitory Neurons in Oscillation. Front. Comput. Neurosci., 10.","DOI":"10.3389\/fncom.2016.00138"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1016\/j.neuron.2009.02.005","article-title":"Balanced amplification: A new mechanism of selective amplification of neural activity patterns","volume":"61","author":"Murphy","year":"2009","journal-title":"Neuron"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"578","DOI":"10.1016\/j.neuron.2009.03.028","article-title":"Inhibitory stabilization of the cortical network underlies visual surround suppression","volume":"62","author":"Ozeki","year":"2009","journal-title":"Neuron"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"830","DOI":"10.1109\/JPROC.2014.2313113","article-title":"Regulating Cortical Oscillations in an Inhibition-Stabilized Network","volume":"102","author":"Jadi","year":"2014","journal-title":"Proc. IEEE Inst. Electr. Electron. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Krishnakumaran, R., Raees, M., and Ray, S. (2022). Shape analysis of gamma rhythm supports a superlinear inhibitory regime in an inhibition-stabilized network. PLoS Comput. Biol., 18.","DOI":"10.1371\/journal.pcbi.1009886"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2477","DOI":"10.1162\/NECO_a_00786","article-title":"Periodic Forcing of Inhibition-Stabilized Networks: Nonlinear Resonances and Phase-Amplitude Coupling","volume":"27","author":"Veltz","year":"2015","journal-title":"Neural Comput."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1016\/j.conb.2009.07.005","article-title":"Spontaneous and driven cortical activity: Implications for computation","volume":"19","author":"Ringach","year":"2009","journal-title":"Curr. Opin. Neurobiol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"e2200621119","DOI":"10.1073\/pnas.2200621119","article-title":"Paradoxical self-sustained dynamics emerge from orchestrated excitatory and inhibitory homeostatic plasticity rules","volume":"119","author":"Seay","year":"2022","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"892","DOI":"10.1038\/36103","article-title":"Activity-dependent scaling of quantal amplitude in neocortical neurons","volume":"391","author":"Turrigiano","year":"1998","journal-title":"Nature"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Srinivasan, R., Thorpe, S., and Nunez, P.L. (2013). Top-Down Influences on Local Networks: Basic Theory with Experimental Implications. Front. Comput. Neurosci., 7.","DOI":"10.3389\/fncom.2013.00029"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Krause, M.R., Vieira, P.G., Thivierge, J.-P., and Pack, C.C. (2022). Brain stimulation competes with ongoing oscillations for control of spike timing in the primate brain. PLoS Biol., 20.","DOI":"10.1371\/journal.pbio.3001650"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00422-009-0328-3","article-title":"The Wilson\u2013Cowan model, 36 years later","volume":"101","author":"Destexhe","year":"2009","journal-title":"Biol. Cybern."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Li, X., Li, Z., Yang, W., Wu, Z., and Wang, J. (2022). Bidirectionally regulating gamma oscillations in Wilson-Cowan model by self-feedback loops: A computational study. Front. Syst. Neurosci., 16.","DOI":"10.3389\/fnsys.2022.723237"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Ponce-Alvarez, A., and Deco, G. (2024). The Hopf whole-brain model and its linear approximation. Sci. Rep., 14.","DOI":"10.1038\/s41598-024-53105-0"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"103105","DOI":"10.1063\/1.5111803","article-title":"Linear response in neuronal networks: From neurons dynamics to collective response","volume":"29","author":"Cessac","year":"2019","journal-title":"Chaos"},{"key":"ref_32","first-page":"15629","article-title":"Universality and individuality in neural dynamics across large populations of recurrent networks","volume":"2019","author":"Maheswaranathan","year":"2019","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"626","DOI":"10.1162\/NECO_a_00409","article-title":"Opening the black box: Low-dimensional dynamics in high-dimensional recurrent neural networks","volume":"25","author":"Sussillo","year":"2013","journal-title":"Neural Comput."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"402","DOI":"10.1016\/j.neuron.2014.12.026","article-title":"The stabilized supralinear network: A unifying circuit motif underlying multi-input integration in sensory cortex","volume":"85","author":"Rubin","year":"2015","journal-title":"Neuron"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"482","DOI":"10.1016\/j.neuron.2008.02.005","article-title":"Inhibition, spike threshold, and stimulus selectivity in primary visual cortex","volume":"57","author":"Priebe","year":"2008","journal-title":"Neuron"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"113130","DOI":"10.1063\/5.0111131","article-title":"Charbonneau Key role of neuronal diversity in structured reservoir computing","volume":"32","author":"Thivierge","year":"2022","journal-title":"Chaos"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Berberich, S., Pohle, J., Pollard, M., Barroso-Flores, J., and K\u00f6hr, G. (2017). Interplay between global and pathway-specific synaptic plasticity in CA1 pyramidal cells. Sci. Rep., 7.","DOI":"10.1038\/s41598-017-17161-z"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1523\/JNEUROSCI.17-02-00597.1997","article-title":"Pathway-Specific Synaptic Plasticity: Activity-Dependent Enhancement and Suppression of Long-Term Heterosynaptic Facilitation at Converging Inputs on a Single Target","volume":"17","author":"Schacher","year":"1997","journal-title":"J. Neurosci."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1016\/j.neuron.2019.08.031","article-title":"Cortical circuit dynamics are homeostatically tuned to criticality in vivo","volume":"104","author":"Ma","year":"2019","journal-title":"Neuron"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1394","DOI":"10.1016\/j.neuron.2014.04.045","article-title":"Optimal control of transient dynamics in balanced networks supports generation of complex movements","volume":"82","author":"Hennequin","year":"2014","journal-title":"Neuron"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1399","DOI":"10.1152\/jn.00732.2015","article-title":"Inhibitory stabilization and visual coding in cortical circuits with multiple interneuron subtypes","volume":"115","author":"Rosenbaum","year":"2016","journal-title":"J. Neurophysiol."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1147","DOI":"10.1016\/j.neuron.2017.08.014","article-title":"Synaptic Mechanisms of Feature Coding in the Visual Cortex of Awake Mice","volume":"95","author":"Adesnik","year":"2017","journal-title":"Neuron"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1038\/nature11526","article-title":"A neural circuit for spatial summation in visual cortex","volume":"490","author":"Adesnik","year":"2012","journal-title":"Nature"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1016\/j.neuron.2017.06.019","article-title":"Network-Level Control of Frequency Tuning in Auditory Cortex","volume":"95","author":"Kato","year":"2017","journal-title":"Neuron"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"e48622","DOI":"10.7554\/eLife.48622","article-title":"Spatiotemporal constraints on optogenetic inactivation in cortical circuits","volume":"8","author":"Li","year":"2019","journal-title":"Elife"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1038\/s41583-020-00390-z","article-title":"Inhibitory stabilization and cortical computation","volume":"22","author":"Sadeh","year":"2021","journal-title":"Nat. Rev. Neurosci."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1109\/TAC.1972.1100089","article-title":"Harmonic entrainment of van der Pol oscillations: Phaselocking and asynchronous quenching","volume":"17","author":"Dewan","year":"1972","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"93","DOI":"10.31887\/DCNS.2014.16.1\/ffroehlich","article-title":"Endogenous and exogenous electric fields as modifiers of brain activity: Rational design of noninvasive brain stimulation with transcranial alternating current stimulation","volume":"16","year":"2014","journal-title":"Dialogues Clin. Neurosci."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"2514183X18793515","DOI":"10.1177\/2514183X18793515","article-title":"Rational design of transcranial alternating current stimulation: Identification, engagement, and validation of network oscillations as treatment targets","volume":"2","author":"Kurmann","year":"2018","journal-title":"Clin. Transl. Neurosci."},{"key":"ref_50","first-page":"827353","article-title":"Entrainment and spike-timing dependent plasticity\u2014A review of proposed mechanisms of transcranial alternating current stimulation Frontiers in Systems","volume":"16","author":"Vogeti","year":"2022","journal-title":"Neuroscience"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1146\/annurev-neuro-090919-022842","article-title":"Synaptic Plasticity Forms and Functions","volume":"43","author":"Magee","year":"2020","journal-title":"Annu. Rev. Neurosci."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"eaaz2747","DOI":"10.1126\/sciadv.aaz2747","article-title":"Dose-dependent effects of transcranial alternating current stimulation on spike timing in awake nonhuman primates","volume":"6","author":"Johnson","year":"2020","journal-title":"Sci. Adv."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"5747","DOI":"10.1073\/pnas.1815958116","article-title":"Transcranial alternating current stimulation entrains single-neuron activity in the primate brain","volume":"116","author":"Krause","year":"2019","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Hashimoto, Y., and Yotsumoto, Y. (2018). The amount of time dilation for visual flickers corresponds to the amount of neural entrainments measured by EEG. Front. Comput. Neurosci., 12.","DOI":"10.3389\/fncom.2018.00030"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"2785","DOI":"10.1093\/cercor\/bhab381","article-title":"Suppression of low-frequency gamma oscillations by activation of 40-Hz oscillation","volume":"32","author":"Sugiyama","year":"2022","journal-title":"Cereb. Cortex"},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"W\u00e4lti, M.J., B\u00e4chinger, M., Ruddy, K.L., and Wenderoth, N. (2019). Steady state responses in the somatosensory system interact with endogenous beta activity. bioRxiv.","DOI":"10.1101\/690495"},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Krause, M.R., Vieira, P.G., and Pack, C.C. (2023). Transcranial electrical stimulation: How can a simple conductor orchestrate complex brain activity?. PLoS Biol., 21.","DOI":"10.1371\/journal.pbio.3001973"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"9891","DOI":"10.1523\/JNEUROSCI.1200-21.2021","article-title":"Developmental Regulation of Homeostatic Plasticity in Mouse Primary Visual Cortex","volume":"41","author":"Wen","year":"2021","journal-title":"J Neurosci."},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Galanis, C., and Vlachos, A. (2020). Hebbian and Homeostatic Synaptic Plasticity-Do Alterations of One Reflect Enhancement of the Other?. Front. Cell Neurosci., 14.","DOI":"10.3389\/fncel.2020.00050"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/2\/215\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:37:49Z","timestamp":1760027869000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/2\/215"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,19]]},"references-count":59,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["e27020215"],"URL":"https:\/\/doi.org\/10.3390\/e27020215","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,19]]}}}