000130063 001__ 130063 000130063 005__ 20240122171020.0 000130063 0247_ $$2doi$$a10.1016/j.cnsns.2019.105100 000130063 0248_ $$2sideral$$a115709 000130063 037__ $$aART-2020-115709 000130063 041__ $$aeng 000130063 100__ $$0(orcid)0000-0002-8089-343X$$aBarrio, R.$$uUniversidad de Zaragoza 000130063 245__ $$aSpike-adding structure in fold/hom bursters 000130063 260__ $$c2020 000130063 5060_ $$aAccess copy available to the general public$$fUnrestricted 000130063 5203_ $$aSquare-wave or fold/hom bursting is typical of many excitable dynamical systems, such as pancreatic or other endocrine cells. Besides, it is also found in a great variety of fast-slow systems coming from other neural models, chemical reactions, laser dynamics, and so on. We focus on the spike-adding process and its connection with the homoclinic structure of the system. The creation of new fast spikes on a bursting neuron is an important phenomenon as it increases the duty cycle of the neuron. Here we mainly work with the Hindmarsh-Rose neuron model, a prototype of fold/hom bursting, but also with the pancreatic ß-cell model, where, as already known from the literature, homoclinic bifurcations play an important role in bursting dynamics. Based on several numerical simulations, we present a theoretical scheme that provides a complete scenario of bifurcations involved in the spike-adding process and their connection with the homoclinic bifurcations on the parametric space. The global scheme explains the different phenomena of the spike-adding processes presented in literature (continuous and chaotic processes after Terman analysis) and moreover, it also indicates where each kind of spike-adding process occurs. Different elements are involved in the theoretical scheme, such as homoclinic isolas, canard orbits, inclination and orbit flip codimension-two bifurcation points and several pencils of period doubling and fold bifurcations, all of them illustrated with different numerical techniques. Some of these bifurcations needed in the process may be not visible on some numerical simulations because the organizing points are in different parametric planes due to the high dimension of the whole parameter space, but their effects are present. Therefore, we introduce a mechanism of the spike-adding process in fold/hom bursters in the whole space of parameters, even if apparently no role is played by the “far-away” homoclinic bifurcations. This fact is illustrated showing how the theoretical scheme provides a theoretical explanation to the different interspike-interval bifurcation diagrams (IBD) that have appeared in the literature for different models. 000130063 536__ $$9info:eu-repo/grantAgreement/ES/DGA-FEDER/E24-17R$$9info:eu-repo/grantAgreement/ES/DGA/LMP124-18$$9info:eu-repo/grantAgreement/ES/MICINN/PGC2018-096026-B-I00$$9info:eu-repo/grantAgreement/ES/MICINN/MTM2014-56953-P$$9info:eu-repo/grantAgreement/ES/MICINN/MTM2015-64095-P$$9info:eu-repo/grantAgreement/ES/MICINN/MTM2017-87697-P$$9info:eu-repo/grantAgreement/ES/UZ/UZCUD2019-CIE-04 000130063 540__ $$9info:eu-repo/semantics/openAccess$$aby-nc-nd$$uhttp://creativecommons.org/licenses/by-nc-nd/3.0/es/ 000130063 590__ $$a4.26$$b2020 000130063 591__ $$aMATHEMATICS, INTERDISCIPLINARY APPLICATIONS$$b11 / 108 = 0.102$$c2020$$dQ1$$eT1 000130063 591__ $$aMATHEMATICS, APPLIED$$b5 / 265 = 0.019$$c2020$$dQ1$$eT1 000130063 591__ $$aPHYSICS, MATHEMATICAL$$b3 / 55 = 0.055$$c2020$$dQ1$$eT1 000130063 591__ $$aPHYSICS, FLUIDS & PLASMAS$$b2 / 34 = 0.059$$c2020$$dQ1$$eT1 000130063 591__ $$aMECHANICS$$b23 / 135 = 0.17$$c2020$$dQ1$$eT1 000130063 592__ $$a1.159$$b2020 000130063 593__ $$aApplied Mathematics$$c2020$$dQ1 000130063 593__ $$aNumerical Analysis$$c2020$$dQ1 000130063 593__ $$aModeling and Simulation$$c2020$$dQ1 000130063 655_4 $$ainfo:eu-repo/semantics/article$$vinfo:eu-repo/semantics/acceptedVersion 000130063 700__ $$aIbáñez, S. 000130063 700__ $$aPérez, L. 000130063 700__ $$0(orcid)0000-0002-5701-1670$$aSerrano, S.$$uUniversidad de Zaragoza 000130063 7102_ $$12005$$2595$$aUniversidad de Zaragoza$$bDpto. Matemática Aplicada$$cÁrea Matemática Aplicada 000130063 773__ $$g83 (2020), 105100 [15 pp]$$pCommun. nonlinear sci. numer. simul.$$tCommunications in Nonlinear Science and Numerical Simulation$$x1007-5704 000130063 8564_ $$s18629351$$uhttps://zaguan.unizar.es/record/130063/files/texto_completo.pdf$$yPostprint 000130063 8564_ $$s1945614$$uhttps://zaguan.unizar.es/record/130063/files/texto_completo.jpg?subformat=icon$$xicon$$yPostprint 000130063 909CO $$ooai:zaguan.unizar.es:130063$$particulos$$pdriver 000130063 951__ $$a2024-01-22-15:24:26 000130063 980__ $$aARTICLE