# American Institute of Mathematical Sciences

August  2017, 14(4): 821-842. doi: 10.3934/mbe.2017045

## A chaotic bursting-spiking transition in a pancreatic beta-cells system: observation of an interior glucose-induced crisis

 1 Instituto Superior de Engenharia de Lisboa - ISEL, Department of Mathematics, Rua Conselheiro Emídio Navarro 1, 1949-014 Lisboa, Portugal 2 Center for Mathematical Analysis, Geometry and Dynamical Systems, Mathematics Department, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

* Corresponding author

Received  September 2016 Revised  December 09, 2016 Published  February 2017

Fund Project: This work was partially funded by FCT/Portugal through UID/MAT/04459/2013

Nonlinear systems are commonly able to display abrupt qualitative changes (or transitions) in the dynamics. A particular type of these transitions occurs when the size of a chaotic attractor suddenly changes. In this article, we present such a transition through the observation of a chaotic interior crisis in the Deng bursting-spiking model for the glucose-induced electrical activity of pancreatic $β$-cells. To this chaos-chaos transition corresponds precisely the change between the bursting and spiking dynamics, which are central and key dynamical regimes that the Deng model is able to perform. We provide a description of the crisis mechanism at the bursting-spiking transition point in terms of time series variations and based on certain amplitudes of invariant intervals associated with return maps. Using symbolic dynamics, we are able to accurately compute the points of a curve representing the transition between the bursting and spiking regimes in a biophysical meaningfully parameter space. The analysis of the chaotic interior crisis is complemented by means of topological invariants with the computation of the topological entropy and the maximum Lyapunov exponent. Considering very recent developments in the literature, we construct analytical solutions triggering the bursting-spiking transition in the Deng model. This study provides an illustration of how an integrated approach, involving numerical evidences and theoretical reasoning within the theory of dynamical systems, can directly enhance our understanding of biophysically motivated models.

Citation: Jorge Duarte, Cristina Januário, Nuno Martins. A chaotic bursting-spiking transition in a pancreatic beta-cells system: observation of an interior glucose-induced crisis. Mathematical Biosciences & Engineering, 2017, 14 (4) : 821-842. doi: 10.3934/mbe.2017045
##### References:
 [1] J. Aguirre, E. Mosekilde and M. A. F. Sanjuán, Analysis of the noise-induced bursting-spiking transition in a pancreatic $β$-cell model ,Pysical Review E, 69 (2004), 041910, 16pp. doi: 10.1103/PhysRevE.69.041910. Google Scholar [2] I. Atwater, C. M. Dawson, A. Scott, G. Eddlestone and E. Rojas, The Nature of the Oscillatory Behaviour in Electrical Activity from Pancreatic Beta-cell ,Georg Thieme, New York, 1980.Google Scholar [3] C. A. S. Batista, A. M. Batista, J. A. C. de Pontes, R. L. Viana and S. R. Lopes, Chaotic phase synchronization in scale-free networks of bursting neurons,Phys. Rev. E, 76 (2007), 016218, 10pp. doi: 10.1103/PhysRevE.76.016218. Google Scholar [4] C. A. S. Batista, E. L. Lameu, A. M. Batista, S. R. Lopes, T. Pereira, G. Zamora-López, J. Kurths and R. L. Viana. Phase synchronization of bursting neurons in clustered small-world networks ,Phys. Rev. E ,86 (2012), 016211. doi: 10.1103/PhysRevE.86.016211. Google Scholar [5] R. Bertram and A. Sherman, Dynamical complexity and temporal plasticity in pancreatic beta cells, J. Biosci., 25 (2000), 197-209. Google Scholar [6] T. R. Chay and J. Keizer, Minimal model for membrane oscillations in the pancreatic beta-cel, Biophys. J., 42 (1983), 181-190. Google Scholar [7] T. R. Chay, Chaos in a three-variable model of an excitable cell, Physica D, 16 (1984), 233-242. Google Scholar [8] L. O. Chua, M. Komuro and T. Matsumoto, The double scroll family, IEEE Trans. Circuits Syst., 32 (1985), 797-818. doi: 10.1109/TCS.1985.1085791. Google Scholar [9] B. Deng, A mathematical model that mimics the bursting oscillations in pancreatic $β$-cells, Math. Biosciences, 119 (1994), 241-250. doi: 10.1016/0025-5564(94)90078-7. Google Scholar [10] B. Deng, Glucose-induced period-doubling cascade in the electrical activity of pancreatic $β$-cells, J. Math. Biol., 38 (1999), 21-78. doi: 10.1007/s002850050141. Google Scholar [11] J. Duarte, C. Januário and N. Martins, Topological entropy and the controlled effect of glucose in the electrical activity of pancreatic beta-cells, Physica D, 238 (2009), 2129-2137. doi: 10.1016/j.physd.2009.08.010. Google Scholar [12] J. Duarte, C. Januário and N. Martins, Explicit series solution for a glucose-induced electrical activity model of pancreatic cells, Chaos, Solitons & Fractals, 76 (2015), 1-9. doi: 10.1016/j.chaos.2015.02.029. Google Scholar [13] J. Duarte, C. Januário, C. Rodrigues and J. Sardany és, Topological complexity and predictability in the dynamics of a tumor growth model with Shilnikov's chaos ,Int. J Bifurcation Chaos, 23 (2013), 1350124, 12pp. doi: 10.1142/S0218127413501241. Google Scholar [14] H. Fallah, Symmetric fold / super-Hopf bursting, chaos and mixed-mode oscillations in Pernarowski model Int.,J Bifurcation Chaos, 26 (2016), 1630022, 14pp. doi: 10.1142/S0218127416300226. Google Scholar [15] Y.-S. Fan and T. R. Chay, Crisis Transitions in excitable cell models, Chaos, Solitons & Fractals, 3 (1993), 603-615. Google Scholar [16] Y.-S. Fan and T. R. Chay, Crisis and topological entropy, Physical Review E, 51 (1995), 1012-1019. doi: 10.1103/PhysRevE.51.1012. Google Scholar [17] L. E. Fridlyand, N. Tamarina and L. H. Philipson, Bursting and calcium oscillations in pancreatic beta cells: Specific pacemakers, Am J Physiol Endocrinol Metab., 299 (2010), E517-E532. Google Scholar [18] J. M. González-Miranda, Observation of a continuous interior crisis in the Hindmarsh-Rose neuron model, Chaos, 13 (2003), 845.Google Scholar [19] J. M. González-Miranda, Complex bifurcations structures in the Hindmarsh-Rose neuron model, Int. J Bifurcation Chaos, 17 (2007), 3071-3083. doi: 10.1142/S0218127407018877. Google Scholar [20] J. M. González-Miranda, Nonlinear dynamics of the membrane potential of a bursting pacemaker cell, Chaos, 22 (2012), 013123.Google Scholar [21] J. M. González-Miranda, Pacemaker dynamics in the full Morris-Lecar model, Commun. Nonlinear Sci Numer Simulat, 19 (2014), 3229-3241. doi: 10.1016/j.cnsns.2014.02.020. Google Scholar [22] H.-G. Gu, B. Jia and G.-R. Chen, Experimental evidence of a chaotic region in a neural pacemaker, Physics Letters A, 377 (2013), 718-720. Google Scholar [23] H. Gu, B. Pan and G. Chen, Biological experimental demonstration of bifurcations from bursting to spiking predicted by theoretical models, Nonlinear Dyn., 78 (2014), 391-407. doi: 10.1007/s11071-014-1447-5. Google Scholar [24] S. Jalil, I. Belykh and A. Shilnikov. Spikes matter for phase-locked bursting in inhibitory neurons ,Phys. Rev. E,85 (2012), 036214. doi: 10.1103/PhysRevE.85.036214. Google Scholar [25] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems ,Cambridge University Press, 1995. doi: 10.1017/CBO9780511809187. Google Scholar [26] J. P. Lampreia and J. S. Ramos, Symbolic dynamics of bimodal maps, Portugal. Math., 54 (1997), 1-18. Google Scholar [27] S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method ,CRC Press, Chapman and Hall, Boca Raton, FL, 2004. Google Scholar [28] S. J. Liao and Y. Tan, A general approach to obtain series solutions of nonlinear differential equations, Stud. Appl. Math., 119 (2007), 297-354. doi: 10.1111/j.1467-9590.2007.00387.x. Google Scholar [29] S. Liao, Advances in the Homotopy Analysis Method, World Scientific Publishing Co, 2014. doi: 10.1142/8939. Google Scholar [30] A. J. Tan and M. A. Lieberman, Regular and Chaotic Dynamics, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4757-2184-3. Google Scholar [31] A. Markovic, T. L. O. Stozer, M. Gosak, J. Dolensek and M. Marhl, Progressive glucose stimulation of islet beta cells reveals a transition from segregated to integrated modular functional connectivity patterns, Scientific Reports, 5 (2015), 7845. doi: 10.1038/srep07845. Google Scholar [32] T. Matsumoto, T. L. O. Chua and M. Komuro., The double scroll family, IEEE Trans. Circuits Syst., 32 (1985), 797-818. doi: 10.1109/TCS.1985.1085791. Google Scholar [33] J. Milnor and W. Thurston, On iterated maps of the interval, Lect. Notes in Math., 1342 (1988), 465-563. doi: 10.1007/BFb0082847. Google Scholar [34] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63. Google Scholar [35] S. E. Newhouse, D. Ruelle and F. Takens, Occurrence of strange Axiom A attractors near quasiperiodic flows on T$^{m}$, $m≥3$, Commun. Math. Phys., 64 (1978), 35-40. doi: 10.1007/BF01940759. Google Scholar [36] E. Ott, Chaos in Dynamical Systems ,Cambridge University Press, Cmabridge, UK, 2002. doi: 10.1017/CBO9780511803260. Google Scholar [37] T. Parker and L. O. Chua, Practical Numerical Algorithms for Chaotic Systems ,Springer-Verlag, 1989. doi: 10.1007/978-1-4612-3486-9. Google Scholar [38] M. G. Pederson, E. Mosekilde, K. S. Polonsky and D. S. Luciani, Complex Patterns of Metabolic and Ca$^{2+}$ entrainment in pancreatic islets by oscillatory glucose, Biophysical Journal, 105 (2013), 29-39. Google Scholar [39] K. Ramasubramanian and M. S. Sriram, A Comparative study of computation of Lyapunov spectra with different algorithms, Physica D, 139 (2000), 72-86. doi: 10.1016/S0167-2789(99)00234-1. Google Scholar [40] J. Rinzel, Ordinary and Partial Differential Equations, Springer, New York, 1985. Google Scholar [41] G. A. Rutter and D. J. Hodson, Minireview: Intraislet regulation of insulin secretion in humans, Molecular Endocrinology, 27 (2013), 1984-1995. doi: 10.1210/me.2013-1278. Google Scholar [42] A. Sherman, P. Carroll, R. M. Santos and I. Atwater, Glucose Dose Response of Pancreatic $β$-cells: Experimental and Theoretical Results, Transitions in Biological Systems, Eds Pienum, New York, 1990.Google Scholar [43] A. Stozer, M. Gosak, J. Dolensek, M. Perc, M. Marhl and M. S. Rupnik, Functional Connectivity in Islets of Langerhans from Mouse Pancreas Tissue Slices, PLoS Comput. Biol. , 9 (2013), e1002923.Google Scholar [44] A. Stozer, M. Gosak, J. Dolensek, M. Perc, M. Marhl and M. S. Rupnik, Functional connectinity in islets of Langerhans from mouse pancreas tissue slices, PLOS Comp. Biol. ,9 (2013), e1002923.Google Scholar [45] J. Wang, S. Liu and X. Liu, Quantification of synchronization phenomena in two reciprocally gap-junction coupled bursting pancreatic $β$-cells, Chaos, Solitons & Fractals, 68 (2014), 65-71. Google Scholar [46] J. Wang, S. Liu and X. Liu, Bifurcation and firing patterns of the pancreatic $β$-cell ,Int. J Bifurcation Chaos, 9 (2015), 1530024, 11pp. doi: 10.1142/S0218127415300244. Google Scholar

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##### References:
 [1] J. Aguirre, E. Mosekilde and M. A. F. Sanjuán, Analysis of the noise-induced bursting-spiking transition in a pancreatic $β$-cell model ,Pysical Review E, 69 (2004), 041910, 16pp. doi: 10.1103/PhysRevE.69.041910. Google Scholar [2] I. Atwater, C. M. Dawson, A. Scott, G. Eddlestone and E. Rojas, The Nature of the Oscillatory Behaviour in Electrical Activity from Pancreatic Beta-cell ,Georg Thieme, New York, 1980.Google Scholar [3] C. A. S. Batista, A. M. Batista, J. A. C. de Pontes, R. L. Viana and S. R. Lopes, Chaotic phase synchronization in scale-free networks of bursting neurons,Phys. Rev. E, 76 (2007), 016218, 10pp. doi: 10.1103/PhysRevE.76.016218. Google Scholar [4] C. A. S. Batista, E. L. Lameu, A. M. Batista, S. R. Lopes, T. Pereira, G. Zamora-López, J. Kurths and R. L. Viana. Phase synchronization of bursting neurons in clustered small-world networks ,Phys. Rev. E ,86 (2012), 016211. doi: 10.1103/PhysRevE.86.016211. Google Scholar [5] R. Bertram and A. Sherman, Dynamical complexity and temporal plasticity in pancreatic beta cells, J. Biosci., 25 (2000), 197-209. Google Scholar [6] T. R. Chay and J. Keizer, Minimal model for membrane oscillations in the pancreatic beta-cel, Biophys. J., 42 (1983), 181-190. Google Scholar [7] T. R. Chay, Chaos in a three-variable model of an excitable cell, Physica D, 16 (1984), 233-242. Google Scholar [8] L. O. Chua, M. Komuro and T. Matsumoto, The double scroll family, IEEE Trans. Circuits Syst., 32 (1985), 797-818. doi: 10.1109/TCS.1985.1085791. Google Scholar [9] B. Deng, A mathematical model that mimics the bursting oscillations in pancreatic $β$-cells, Math. Biosciences, 119 (1994), 241-250. doi: 10.1016/0025-5564(94)90078-7. Google Scholar [10] B. Deng, Glucose-induced period-doubling cascade in the electrical activity of pancreatic $β$-cells, J. Math. Biol., 38 (1999), 21-78. doi: 10.1007/s002850050141. Google Scholar [11] J. Duarte, C. Januário and N. Martins, Topological entropy and the controlled effect of glucose in the electrical activity of pancreatic beta-cells, Physica D, 238 (2009), 2129-2137. doi: 10.1016/j.physd.2009.08.010. Google Scholar [12] J. Duarte, C. Januário and N. Martins, Explicit series solution for a glucose-induced electrical activity model of pancreatic cells, Chaos, Solitons & Fractals, 76 (2015), 1-9. doi: 10.1016/j.chaos.2015.02.029. Google Scholar [13] J. Duarte, C. Januário, C. Rodrigues and J. Sardany és, Topological complexity and predictability in the dynamics of a tumor growth model with Shilnikov's chaos ,Int. J Bifurcation Chaos, 23 (2013), 1350124, 12pp. doi: 10.1142/S0218127413501241. Google Scholar [14] H. Fallah, Symmetric fold / super-Hopf bursting, chaos and mixed-mode oscillations in Pernarowski model Int.,J Bifurcation Chaos, 26 (2016), 1630022, 14pp. doi: 10.1142/S0218127416300226. Google Scholar [15] Y.-S. Fan and T. R. Chay, Crisis Transitions in excitable cell models, Chaos, Solitons & Fractals, 3 (1993), 603-615. Google Scholar [16] Y.-S. Fan and T. R. Chay, Crisis and topological entropy, Physical Review E, 51 (1995), 1012-1019. doi: 10.1103/PhysRevE.51.1012. Google Scholar [17] L. E. Fridlyand, N. Tamarina and L. H. Philipson, Bursting and calcium oscillations in pancreatic beta cells: Specific pacemakers, Am J Physiol Endocrinol Metab., 299 (2010), E517-E532. Google Scholar [18] J. M. González-Miranda, Observation of a continuous interior crisis in the Hindmarsh-Rose neuron model, Chaos, 13 (2003), 845.Google Scholar [19] J. M. González-Miranda, Complex bifurcations structures in the Hindmarsh-Rose neuron model, Int. J Bifurcation Chaos, 17 (2007), 3071-3083. doi: 10.1142/S0218127407018877. Google Scholar [20] J. M. González-Miranda, Nonlinear dynamics of the membrane potential of a bursting pacemaker cell, Chaos, 22 (2012), 013123.Google Scholar [21] J. M. González-Miranda, Pacemaker dynamics in the full Morris-Lecar model, Commun. Nonlinear Sci Numer Simulat, 19 (2014), 3229-3241. doi: 10.1016/j.cnsns.2014.02.020. Google Scholar [22] H.-G. Gu, B. Jia and G.-R. Chen, Experimental evidence of a chaotic region in a neural pacemaker, Physics Letters A, 377 (2013), 718-720. Google Scholar [23] H. Gu, B. Pan and G. Chen, Biological experimental demonstration of bifurcations from bursting to spiking predicted by theoretical models, Nonlinear Dyn., 78 (2014), 391-407. doi: 10.1007/s11071-014-1447-5. Google Scholar [24] S. Jalil, I. Belykh and A. Shilnikov. Spikes matter for phase-locked bursting in inhibitory neurons ,Phys. Rev. E,85 (2012), 036214. doi: 10.1103/PhysRevE.85.036214. Google Scholar [25] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems ,Cambridge University Press, 1995. doi: 10.1017/CBO9780511809187. Google Scholar [26] J. P. Lampreia and J. S. Ramos, Symbolic dynamics of bimodal maps, Portugal. Math., 54 (1997), 1-18. Google Scholar [27] S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method ,CRC Press, Chapman and Hall, Boca Raton, FL, 2004. Google Scholar [28] S. J. Liao and Y. Tan, A general approach to obtain series solutions of nonlinear differential equations, Stud. Appl. Math., 119 (2007), 297-354. doi: 10.1111/j.1467-9590.2007.00387.x. Google Scholar [29] S. Liao, Advances in the Homotopy Analysis Method, World Scientific Publishing Co, 2014. doi: 10.1142/8939. Google Scholar [30] A. J. Tan and M. A. Lieberman, Regular and Chaotic Dynamics, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4757-2184-3. Google Scholar [31] A. Markovic, T. L. O. Stozer, M. Gosak, J. Dolensek and M. Marhl, Progressive glucose stimulation of islet beta cells reveals a transition from segregated to integrated modular functional connectivity patterns, Scientific Reports, 5 (2015), 7845. doi: 10.1038/srep07845. Google Scholar [32] T. Matsumoto, T. L. O. Chua and M. Komuro., The double scroll family, IEEE Trans. Circuits Syst., 32 (1985), 797-818. doi: 10.1109/TCS.1985.1085791. Google Scholar [33] J. Milnor and W. Thurston, On iterated maps of the interval, Lect. Notes in Math., 1342 (1988), 465-563. doi: 10.1007/BFb0082847. Google Scholar [34] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63. Google Scholar [35] S. E. Newhouse, D. Ruelle and F. Takens, Occurrence of strange Axiom A attractors near quasiperiodic flows on T$^{m}$, $m≥3$, Commun. Math. Phys., 64 (1978), 35-40. doi: 10.1007/BF01940759. Google Scholar [36] E. Ott, Chaos in Dynamical Systems ,Cambridge University Press, Cmabridge, UK, 2002. doi: 10.1017/CBO9780511803260. Google Scholar [37] T. Parker and L. O. Chua, Practical Numerical Algorithms for Chaotic Systems ,Springer-Verlag, 1989. doi: 10.1007/978-1-4612-3486-9. Google Scholar [38] M. G. Pederson, E. Mosekilde, K. S. Polonsky and D. S. Luciani, Complex Patterns of Metabolic and Ca$^{2+}$ entrainment in pancreatic islets by oscillatory glucose, Biophysical Journal, 105 (2013), 29-39. Google Scholar [39] K. Ramasubramanian and M. S. Sriram, A Comparative study of computation of Lyapunov spectra with different algorithms, Physica D, 139 (2000), 72-86. doi: 10.1016/S0167-2789(99)00234-1. Google Scholar [40] J. Rinzel, Ordinary and Partial Differential Equations, Springer, New York, 1985. Google Scholar [41] G. A. Rutter and D. J. Hodson, Minireview: Intraislet regulation of insulin secretion in humans, Molecular Endocrinology, 27 (2013), 1984-1995. doi: 10.1210/me.2013-1278. Google Scholar [42] A. Sherman, P. Carroll, R. M. Santos and I. Atwater, Glucose Dose Response of Pancreatic $β$-cells: Experimental and Theoretical Results, Transitions in Biological Systems, Eds Pienum, New York, 1990.Google Scholar [43] A. Stozer, M. Gosak, J. Dolensek, M. Perc, M. Marhl and M. S. Rupnik, Functional Connectivity in Islets of Langerhans from Mouse Pancreas Tissue Slices, PLoS Comput. Biol. , 9 (2013), e1002923.Google Scholar [44] A. Stozer, M. Gosak, J. Dolensek, M. Perc, M. Marhl and M. S. Rupnik, Functional connectinity in islets of Langerhans from mouse pancreas tissue slices, PLOS Comp. Biol. ,9 (2013), e1002923.Google Scholar [45] J. Wang, S. Liu and X. Liu, Quantification of synchronization phenomena in two reciprocally gap-junction coupled bursting pancreatic $β$-cells, Chaos, Solitons & Fractals, 68 (2014), 65-71. Google Scholar [46] J. Wang, S. Liu and X. Liu, Bifurcation and firing patterns of the pancreatic $β$-cell ,Int. J Bifurcation Chaos, 9 (2015), 1530024, 11pp. doi: 10.1142/S0218127415300244. Google Scholar
(a) Attractor of the system (1) for $\rho =$0.59625. (b) Bursting orbit of $V(t)$ with $\rho =$0.4. (c) Spiking orbit of $V(t)$ with $\rho =$0.8. In all situations $\epsilon =$0.08
Samples of time series of the three model variables before the interior glucose-induced crisis, with $\rho =$0.58, and after the interior glucose-induced crisis, with $\rho =$0.64. In all situations $\epsilon =$0.08
Characterization of the Deng bursting system given by Eqs. 1. (a) The maximum Lyapunov exponent. (b) Bifurcation diagram computed using the successive maxima of $C(t)$, considering $\rho \in$[0.42, 0.642] and $\epsilon =$0.08. (c) Estimate of the size of the attractor given by the range of variation of $\Delta _{C}( \rho )$. (d) The function $\Delta _{C}( \rho )$ and (e) its derivative $\frac{d\Delta _{C}}{d \rho }$, both displayed in a narrow interval arround the transition point, $\rho ^{\ast }$
Representation of functions $\Lambda _{C}( \rho )$ for different values of $\epsilon$: $\epsilon$=0.05, $\epsilon$=0.06, $\epsilon$=0.07, $\epsilon$ =0.08, $\epsilon$=0.09 and $\epsilon$=1.0, considering $\rho \in$[0.42, 0.642]
Representation of function $S( \rho )$ for $\rho \in$[0.42, 0.642] and $\epsilon$=0.08
(a) Time series of the variable $V$. (b) Time series of the variable $C$ (each local maximum of $C(t)$ corresponds to a peak in the time series of the membrane voltage $V(t)$). (c) Iterated map constructed from the successive local maxima of variable $C$. In all situations $\rho =$0.5927 and $\epsilon =$0.08
(a) The extrema of invariant intervals I=$\left[ f^{2}(c),f(c) \right]$, for $\epsilon$=0.08 and $\rho \in$[0.42, 0.642]. (b) Amplitudes of the invariant intervals as a function of $\rho$, $\Lambda _{I}\left( \rho \right)$, for $\epsilon$=0.08 and $\rho \in$[0.42, 0.642]
Curve of critical points in the $( \rho , \epsilon )$ -plane separating bursting from spiking dynamics, for $\rho \in$ [0.580, 0.6004] and $\epsilon \in$[0.05, 0.1]
Dynamics along the crisis transition curve. (a) Values of the largest Lyapunov exponent and values of the Topological entropy. (b) Typical bifurcation diagram with $\rho$ as control parameter
(a) Variation of the triggering interval $J_{ \rho }$ with the control parameter $\rho$. (b) Bursting orbit of the derived analytical solution (with $h$= -0.660746), for $\rho =$0.59439 and $\delta =$0.02815. (c) Continuous spiking orbit of the derived analytical solution (with $h$= -0.73926), for $\rho =$0.59439 and $\delta$= -0.02184
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