Noise-induced toroidal excitability in neuron model

Noise-induced toroidal excitability in neuron model

Journal Pre-proof

Noise-induced toroidal excitability in neuron model L. Ryashko, E. Slepukhina PII: DOI: Reference:

S1007-5704(19)30390-9 https://doi.org/10.1016/j.cnsns.2019.105071 CNSNS 105071

To appear in:

Communications in Nonlinear Science and Numerical Simulation

Please cite this article as: L. Ryashko, E. Slepukhina, Noise-induced toroidal excitability in neuron model, Communications in Nonlinear Science and Numerical Simulation (2019), doi: https://doi.org/10.1016/j.cnsns.2019.105071

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.

Highlights • Influence of random noise on the 3D Hindmarsh-Rose neuron model is studied. • Oscillations in the parameter region of torus canards are analyzed. • Noise-induced transition from amplitude-modulated spiking to bursting is observed. • The stochastic system exhibits anti-coherence and coherence resonances. • Explanation for the mechanism of the observed stochastic phenomena is suggested.

1

Noise-induced toroidal excitability in neuron model L. Ryashkoa , E. Slepukhinaa,b,∗ a Department

of Theoretical and Mathematical Physics, Ural Federal University, Lenin ave., 51, Ekaterinburg, 620000, Russian Federation b Institute of Applied Mathematics and Statistics, University of Hohenheim, Schloss Hohenheim 1, Stuttgart, 70599, Germany

Abstract We study the stochastic Hindmarsh–Rose neuron model in the torus canards region of the parameter space. We show that noise can transform the torus canard into the large amplitude torus. This corresponds to the noise-induced transition from amplitude-modulated spiking regime to the bursting one. This phenomenon is confirmed by qualitative changes of both amplitude and frequency characteristics. We investigate this phenomenon with the help of interspike intervals statistics. The onset of noise-induced spiking–bursting transition is accompanied by the increase of the mean values of interspike intervals. Moreover, the anti-coherence resonance reflecting the growth of variability of interspike intervals is observed. On the contrary, the distribution of burst duration values for noise-generated bursting oscillations shows the coherence resonance. We suggest an explanation for the observed stochastic phenomena by the specificity of the deterministic portrait and the high excitability of the system in the zone of torus canards. Keywords: Hindmarsh–Rose neuron model, Random disturbances, Torus canards, Amplitude-modulated spiking, Bursting, Noise-induced excitability, Anti-coherence resonance, Coherence resonance

∗ Tel.:

+49 711 45922149 Email address: [email protected] (E. Slepukhina)

Preprint submitted to Communications in Nonlinear Science and Numerical SimulationOctober 23, 2019

1. Introduction Excitability, the ability of a neuron to change electrical potential on a its membrane abruptly (i.e. generate a spike), is known to be one of the essential properties of a nerve cell. Intrinsic mechanisms of excitability and its types were studied by means of mathematical models, nonlinear dynamics and theory of bifurcations [1, 2, 3]. Traditionally a number of mathematical models looked into the dynamics of excitability in terms of transitions between the equilibrium and the limit cycle (e.g. models with Hopf, SNIC bifurcations etc.). Transitions between different types of oscillations is another important topic in analysis of neuronal behavior. One of scenarios of transformation from small amplitude oscillations to large amplitude relaxation oscillations is related to existence in a system of special limit cycles called canards [4, 5]. This fast transition (canard explosion) occurs within a very narrow range of a control parameter. In neuronal dynamics, this phenomenon is also explained by excitable behavior of a cell [6]. Recently, a new type of excitable regime associated with so called torus canards was discovered in neuron models [7, 8]. Torus canards are a threedimensional generalized case of the two-dimensional (limit cycle) canards. In neuronal models, torus canard explosion leads from a tonic spiking regime (modeled by a limit cycle) to a special type of bursting (modeled by an invariant torus) through amplitude-modulated spiking oscillations described by torus canards solutions [9, 10]. The mechanism of appearance of torus canards in terms of slow and fast manifolds was explained in [8, 10]. Due to many factors neuronal activity has a stochastic character. Therefore, an interaction of neuronal excitability and stochastisity is an important subject for research. It is widely known that random disturbances in nonlinear systems can cause qualitatively new dynamic regimes, which are not observed in absence of noise. In particular, stochastic neuron models can exhibit such phenomena as coherence resonance [11, 12, 13], stochastic resonance [14, 15], noiseinduced bursting [16, 17, 18], stochastic generation of mixed-mode oscillations

3

[19], noise-induced suppression of oscillations [20], noise-induced chaotic behavior [18], noise-induced transitions between tonic spiking and bursting regimes [21], stochastic toroidal bursting [22, 23]. In this paper, we study noise-induced changes in the dynamics of the threedimensional neuron model of the Hindmarsh–Rose type [24, 25, 9] in the torus canards region of the parameter space. In the following, we show that in this zone, random disturbances can transform the torus canard into the large amplitude torus. This corresponds to the noise-induced transition from amplitudemodulated spiking regime to the bursting one. We investigate this phenomenon with the help of interspike intervals statistics. The intrinsic mechanism of noiseinduced bursting in the torus canard region is explained by peculiarities of the deterministic phase portrait of the system.

2. Deterministic dynamics Consider the following deterministic neuron model of the Hindmarsh–Rose (HR) type [24, 25]: x˙

= sax3 − sx2 − y − bz

y˙

= ϕ(x2 − y)

z˙

= r(sαx + β − kz),

(1)

where the variable x is the transmembrane potential, the variables y and z describe the gating dynamics of the potassium and the calcium ion channels respectively. Here, we examine the dynamics of the system (1) varying the parameter β which determines the equilibrium potential for the calcium. Other parameter values are fixed as in [9]: a = 0.5, b = 10, k = 0.2, s = −1.95, α = −0.1, ϕ = 1, r = 10−5 .

Fig. 1 the bifurcation diagram for the system (1) is presented. The system has one equilibrium point, which is stable for β < β1 ≈ −0.1927 and unstable for β > β1 . At the point β1 , the supercritical Andronov–Hopf bifurcation occurs,

4

x

x

1

1

0.5

0.5

β1

0

β2

−0.2

−0.18

−0.16

β2

β

−0.1605

−0.1604

−0.1603

−0.1604

−0.1603

β

(a) −3

−3

x 10

x 10

z

z

−1

−1

−2

−2

β1 −0.2

β2 −0.18

−0.16

β

β

2

−0.1605

β

(b) Figure 1: Bifurcation diagram: stable (black dash-dotted) and unstable (black dotted) equilibria, maximal and minimal values of x-coordinates (a) and z-coordinates (b) along stable (blue solid) and unstable (blue dashed) limit cycles, maximal and minimal values of x-coordinates (a) and z-coordinates (b) along invariant tori (red solid). The right column is a blow-up of the critical region where the torus canard explosion happens.

and for a very narrow parameter range a stable limit cycle is an attractor of the system. Close to the point β1 , the system undergoes the torus bifurcation, and in the region −0.1927 . β < β2 ≈ −0.16026 the system possesses a stable invariant torus. As the parameter passes the point β = β2 , the backward torus bifurcation occurs, and for β > β2 a stable limit cycle becomes an attractor. The emergence of the invariant torus in this model occurs with the torus canard explosion [9]. This phenomenon is characterized by an abrupt change of the size and form of the torus (see the enlarged fragment of the bifurcation diagram in Fig. 1 and Fig. 2). The limit cycles in the zone β > β2 describe uni-

5

y

10-3

z

1

x 1

-1

0.5

0.5 1

-2 0

0.5

1

x

0.5

0

0.5

1

x

0

0

5

t

0 10000

(a) y

10-3

z

1

x 1

-1

0.5

0.5 1

-2 0

0.5

1

x

0.5

0

0.5

1

x

0

0

0 10000

5

t

(b) Figure 2: Deterministic invariant tori for: (a) β = −0.16045 (torus canard), (b) β = −0.162 (large amplitude torus).

form amplitude spiking oscillations. The torus bifurcation at the point β = β2 leads to the small amplitude modulated spiking regime, represented by the torus canards (see Fig. 2a). Near the point β ≈ −0.16047 the amplitude modulation increases significantly. The further decrease of β results to the transition from the amplitude-modulated spiking to the bursting regime, modeled by the large amplitude tori (see Fig. 2b). A sharp increase in amplitudes of tori is observed for both x- and z-coordinates (see Fig. 1a and b). 3. Stochastic deformation of invariant tori Consider the stochastically forced model: x˙

= sax3 − sx2 − y − bz + εw, ˙

y˙

= ϕ(x2 − y)

z˙

= r(sαx + β − kz), 6

(2)

y

y 1

1

0.5

0.5

0

0.5

1

x

0

−3

z

z

−1

−1

−1.5

−1.5

−2

−2

0

0.5

1

x

0.5

1

x

−3

x 10

0.5

1

x

x 10

0

(a)

(b)

Figure 3: Stochastic deformation of invariant tori for β = −0.16045: (a) ε = 10−6 , (b) ε = 10−5 .

where w is a standard Wiener process with E(w(t) − w(s)) = 0, E(w(t) − w(s))2 = |t − s|, and the value ε is a noise intensity. The solution of the

stochastic differential equations is considered in Ito sense. In this paper, we focus on the region of the parameter space −0.16047 < β < 0.16026, where the original deterministic system (1) exhibits torus canards. First, we fix the value β = −0.16045. Here, the canard type invariant torus is the attractor of the deterministic system. It reproduces the amplitudemodulated spiking regime. Fig. 3 shows the stochastic trajectories starting from this deterministic torus. For sufficiently small noise levels, the random trajectories localize near the deterministic torus, and the dynamic regime remains spiking (see Fig. 3a for ε = 10−6 ). When the noise intensity is enhanced, stochastic trajectories deviate far from the deterministic torus, and the formation of the large amplitude torus is observed (see Fig. 3b for ε = 10−5 ). This corresponds to the noise-induced transition from amplitude-modulated spiking regime to the bursting one. With a further increase of the noise intensity, one can observe

7

x

x

x

x

1

1

1

1

0.5

0.5

0.5

0.5

0

t

0 10000

0

(a)

0

t

0 10000

(b)

0

t

0 10000

t

0 10000

(c)

(d)

Figure 4: Time series x(t) for β = −0.16045: (a) ε = 10−6 , (b) ε = 10−5 , (c) ε = 10−4 , (d) ε = 10−3 .

that the number of bursts per time interval increases, and the duration of active phase in bursting regime decreases (see Fig. 4). Let us study these noise-induced changes in dynamics of the system (2) with the help of interspike intervals statistics. Fig. 5 shows the mean value m = hτ i, √ h(τ −m)2 i and the coefficient of variation (CV), CV = , of interspike intervals τ m for different parameter β values from the torus canards zone under increasing noise intensity. One can observe that the mean ISI almost does not change for small levels of noise and corresponds to the period of spiking oscillations. However, the further increase of the noise intensity results in the sharp rise of the mean ISI due to the appearance of long intervals of quiescence in the noise-generated bursting regime. The CV shows a sharp increase of its values

14

= -0.16045 = -0.1604 = -0.1603

8 6

12

4

10

= -0.16045 = -0.1604 = -0.1603

2 8 10-6

10-5

10-4

0 10-6

10-3

(a)

10-5

10-4

(b)

Figure 5: ISI statistics: a) mean value, b) coefficient of variation.

8

10-3

N

l

β = −0.16045 β = −0.1604 β = −0.1603

60 50

4

40

10

30 20

β = −0.16045 β = −0.1604 β = −0.1603

3

10

10

0 −6 10

−5

−4

10

10

−3

10

ε

−6

10

−5

−4

10

(a)

10

−3

10

ε

(b)

Figure 6: ISI statistics: a) mean number of bursts per time interval t = 100000, b) mean duration of active phase (solid) and quiescence interval (dashed).

when the noise intensity becomes greater than some threshold level (e.g. the CV increases from 0 to 9 for β = −0.1603 (green curve) while ε increases from 10−5 to 10−4 ). These changes of the CV indicate the noise-induced increase

of variability of ISIs (anti-coherence resonance [26]): along with short intervals in spiking phase, long intervals of quiescence appear. This effect also indicates the transition from spiking to bursting. With the further enhancement of the noise intensity, the CV decreases, which corresponds to the stabilization of the system coherence. The plots of m(ε) and CV (ε) allow us to estimate threshold values of noise intensity related to the onset of the noise-induced spiking–bursting transition. Note that these critical values decrease with an approach of the parameter β to the boundary of torus canards zone (β ≈ −0.16047). Let us study temporal characteristics of noise-induced bursting oscillations in more detail. Fig. 6 shows that the average number of bursts per a fixed time interval t = 105 increases (see Fig. 6a) while the average duration of active phase shortens (see Fig. 6b), as was observed on time series (see Fig. 4). Note that the average duration of quiescent phase changes insignificantly with a rise of noise level (see Fig. 6b). One can see that when noise reaches some level, these temporal character-

9

1.2 1 0.8 0.6 0.4

= -0.16045 = -0.1604 = -0.1603

10 -6

10 -5

10 -4

10 -3

Figure 7: Coefficient of variation for burst duration in dependence on noise intensity.

istics of generated bursts stabilize and change slightly as the noise intensity increases further. Fig. 4 also shows that for small noise intensity bursts are generated irregularly with different duration. On the contrary, for greater noise intensity (ε = 10−3 ), the stochastic bursting oscillations seem almost uniform periodic with constant burst duration. Let us consider the coefficient of variation CV (ε) for burst duration in dependence on noise intensity. These statistics are plotted in Fig. 7 for different values of parameter β. These plots show that the function CV (ε) for burst duration has a minimum value which means that the system exhibits coherence resonance. Due to the coherence resonance, the generated bursts are almost regular. Note that the coherence resonance critical values of noise intensity are almost the same for different values of parameter β and equal ε∗ ≈ 0.0007. The noise-induced deformation of tori is characterized not only by changes in frequency characteristics, but also by changes of the amplitude of oscillations. In Fig. 8 z-coordinates of random trajectories started from a torus canard for two values of parameter β = −0.16045 and β = −0.1603 in dependence on noise intensity ε are plotted. It shows that for a small noise, trajectories concentrate near the small amplitude deterministic torus. When a noise intensity reaches some critical level, an abrupt amplification of amplitudes of oscillations is observed. This sharp increase of amplitudes of oscillations for tori attractors is one of 10

10 -3

10 -3

-1

-1

-1.5

-1.5

-2

-2

10 -6

10 -5

10 -4

10 -6

10 -5

(a)

10 -4

(b)

Figure 8: z-coordinates of stochastic trajectories in dependence on noise intensity ε for a) β = −0.16045, b) β = −0.1603.

attributes of a torus canard explosion leading to a transition from amplitudemodulated spiking regime to the bursting one in deterministic systems. One can see that noise can induce this transition for values of parameter for which only amplitude-modulated spiking mode is observed in the deterministic case. Thus, noise can move a bifurcation value related to this transition. This is confirmed on stochastic bifurcation diagrams plotted in Fig. 9. It shows the plots of the minimum values of x-coordinate for the stochastic trajectories in dependence on the parameter β for several fixed values of the noise intensity. These plots can be considered as stochastic bifurcation diagrams (see Fig. 1a for deterministic

x 0.4 0.3 0.2

ε=0

0.1

ε = 10−5

ε = 10−6 ε = 5×10−5 −0.1604

−0.1603

β

Figure 9: Stochastic bifurcation diagrams: average minimum values of x-coordinates depending on the parameter β for different values of ε.

11

(a)

(b)

(c)

Figure 10: Points in the Poincare section (in the projection on xOz plane) a) for the deterministic system with β = −0.16045 (red) and β = −0.162 (black); for the stochastic system

with β = −0.16045 and b) ε = 10−6 , c) ε = 10−5 .

bifurcation diagram). For the small noise intensity (ε = 10−6 ) the curve of the minimum values of x-coordinate slightly deviates from the one calculated for the deterministic system. One can observe that as the noise intensity increases, this deviation becomes more significant, so that large amplitude oscillations are observed for greater values of β. Thus, the value of the parameter β corresponding to the transition from torus canards to large amplitude tori shifts to the right, and accordingly, the zone of large amplitude tori expands. An appearance of an invariant torus can be demonstrated with the help of Poincare sections. Let us use a Poincare section constructed as follows. Consider three points of an large amplitude torus: two of them belong to the “inside” part of the torus, and the third one lies on the “outside” surface of the torus. The chosen section is the plane passing through these points. Fig. 10 shows points of intersection of phase trajectories with the chosen Poincare section for two

12

−3

z

−3

x 10

z

x 10

−1 −2 −1 0

5000

t

0

5000

t

x 1 0.5

−2 0

0.5

1

0

x

(a)

(b)

Figure 11: Deterministic phase portrait for β = −0.16045 a) phase trajectories starting from different initial points in the projection on xOz plane; b) corresponding time series x(t) and z(t). The red trajectory starts in the subthreshold zone and approaches the deterministic torus monotonously, while the blue trajectory which starts in the suprathreshold zone goes far from the attractor before approaching it.

solutions of the deterministic system (the torus canard for β = −0.16045 and the large amplitude torus for β = −0.162, see Fig. 10a) and for the stochastic trajectories for β = −0.16045 with two values of noise intensity (Fig. 10b,c). The points in the Poincare section of the deterministic torus canard for β = −0.16045 form two closed invariant curves. With an decrease of the parameter β these two curves join and form a larger closed invariant curve of a more complicated shape (see Fig. 10a (black) for β = −0.162). For the stochastic solutions for β = −0.16045, when the noise intensity is sufficiently small, the points in Poincare section are located close to ones in the deterministic case (see Fig. 10b). With an increase of the noise, the points in the Poincare section localize both close and far from the deterministic solution forming a structure similar to the one formed by points in the Poincare section for the large amplitude deterministic torus for β = −0.162 (see Fig. 10c). The stochastic deformation of tori can be related to a specificity of deterministic dynamics in the zone of canard-type tori. In this zone the system is excitable, this means that its behavior significantly changes with slight changes

13

of initial conditions. Consider different initial points near the deterministic torus canard. One can observe that a character of a transition process depends on a deviation from the deterministic attractor (see Fig. 11). If an initial point is close enough to the torus, then a trajectory monotonously approaches it. If the initial deviation is greater than a certain threshold, the trajectory first goes far from the torus to the zone of large values of z-coordinates, passing the unstable equilibrium, and then slowly approaches the torus, making turns of large amplitudes around some surface. Thus, in the phase space, one can distinguish two sets of initial points corresponding to two different transient modes (these points form subthreshold and suprathreshold zones), and a boundary separating them (pseudoseparatrix). Due to noise a trajectory can ”jump” across the pseudoseparatrix into the suprathreshold zone, where large amplitude tori are formed.

Conclusion We studied the influence of noise on the Hindmarsh–Rose neuron model in the parameter zone of torus canards. We showed that under random disturbances, the torus canard transforms into the large amplitude torus. This corresponds to the noise-induced transition from amplitude-modulated spiking oscillations to the bursting ones. These qualitative changes in dynamics are confirmed by both amplitude and frequency characteristics. Plots of amplitude of stochastic oscillations showed the abrupt amplification when the noise intensity reaches some critical value. This threshold noise level decreases as the parameter approaches the bifurcation value corresponding to a transition to bursting in the deterministic system. The phenomenon of noise-induced spiking–bursting transition is also accompanied with the increase of the mean values of interspike intervals. Moreover, the anti-coherence resonance reflecting the increase of variability of interspike intervals is observed. On the contrary, the distribution of burst duration values for noise-generated bursting oscillations showed the coherence resonance. The stochastic bifurcation diagrams exhibited the expansion

14

of the parameter region corresponding to bursting oscillations with an increase of a noise level. We showed that the noise-induced bursting can be related to specificity of the deterministic portrait in the torus canards zone, namely, the existence of sub- and suprathreshold transient regimes in the vicinity of the deterministic torus due to excitability of the system.

Acknowledgments The work was supported by Russian Science Foundation (N 16-11-10098).

References [1] A. L. Hodgkin, The local electric changes associated with repetitive action in a non-medullated axon, J Physiol 107 (2) (1948) 165–181. [2] J. Rinzel, G. B. Ermentrout, Analysis of neural excitability and oscillations, Methods in Neuronal Modeling / ed. by C. Koch, I. Segev (Cambridge : MIT Press, 1989) 135–169. [3] E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, MIT Press, Cambridge, 2007. [4] E. Benoit, J.-L. Callot, F. Diener, M. Diener, Chasse au canard, Collectanea Mathematica 31–32 (1–3) (1981) 37–119. [5] J. Guckenheimer, Homoclinic orbits of the Fitzhugh-Nagumo equation: Bifurcations in the full system, SIAM J. Appl. Dyn. Syst. 9 (2010) 138. [6] E. K. Ers¨ oz, M. Desroches, C. Mirasso, S. Rodrigues, Anticipation via canards in excitable systems, Chaos: An Interdisciplinary Journal of Nonlinear Science 29 (1) (2019) 013111. [7] M. A. Kramer, R. D. Traub, N. J. Kopell, New dynamics in cerebellar Purkinje cells: torus canards, Phys. Rev. Lett. 101 (2008) 068103.

15

[8] G. N. Benes, A. M. Barry, T. J. Kaper, M. A. Kramer, J. Burke, An elementary model of torus canards, Chaos 21 (2011) 023131. [9] J. Burke, M. Desroches, A. M. Barry, T. J. Kaper, M. A. Kramer, A showcase of torus canards in neuronal bursters, Journal of Mathematical Neuroscience 2 (3) (2012) 1–30. [10] H. Ju, A. B. Neiman, A. L. Shilnikov, Bottom-up approach to torus bifurcation in neuron models, Chaos: An Interdisciplinary Journal of Nonlinear Science 28 (10) (2018) 106317. [11] A. S. Pikovsky, J. Kurths, Coherence resonance in a noise-driven excitable system, Phys. Rev. Lett. 78 (5) (1997) 775–778. [12] B. Lindner, J. Garcia-Ojalvo, A. Neiman, L. Schimansky-Geier, Effects of noise in excitable systems, Physics Reports 392 (2004) 321–424. [13] B. Lindner, L. Schimansky-Geier, Analytical approach to the stochastic Fitzhugh–Nagumo system and coherence resonance, Phys. Rev. E 60 (6) (1999) 7270–7276. [14] A. Longtin, Autonomous stochastic resonance in bursting neurons, Phys. Rev. E 55 (1) (1997) 868–876. [15] J. Baltanas, J. Casado, Noise-induced resonances in the Hindmarsh–Rose neuronal model, Phys. Rev. E 65 (2002) 041915. [16] P. Hitczenko, G. S. Medvedev, Bursting oscillations induced by small noise, SIAM J. Appl. Math. 69 (2009) 1359. [17] P. Channell, I. Fuwape, A. B. Neiman, A. L. Shilnikov, Variability of bursting patterns in a neuron model in the presence of noise, J. Comput. Neurosci. 27 (2009) 527. [18] I. Bashkirtseva, S. Fedotov, L. Ryashko, E. Slepukhina, Stochastic bifurcations and noise-induced chaos in 3D neuron model, International Journal of Bifurcation and Chaos 26 (12) (2016) 1630032. 16

[19] E. Slepukhina, Stochastic sensitivity analysis of noise-induced mixed-mode oscillations in Morris-Lecar neuron model, Mathematical Modeling of Natural Phenomena 12 (4) (2017) 74–90. [20] I. Bashkirtseva, A. B. Neiman, L. Ryashko, Stochastic sensitivity analysis of noise-induced suppression of firing and giant variability of spiking in a Hodgkin–Huxley neuron model, Phys. Rev. E 91 (2015) 052920. [21] I. Bashkirtseva, L. Ryashko, E. Slepukhina, Methods of stochastic analysis of complex regimes in the 3D Hindmarsh–Rose neuron model, Fluctuation and Noise Letters 17 (1) (2018) 1850008. [22] L. B. Ryashko, E. S. Slepukhina, Noise-induced torus bursting in the stochastic Hindmarsh-Rose neuron model, Physical Review E 96 (2017) 032212. [23] I. Bashkirtseva, L. Ryashko, E. Slepukhina, Stochastic generation and deformation of toroidal oscillations in neuron model, International Journal of Bifurcation and Chaos 28 (6) (2018) 1850070. [24] J. L. Hindmarsh, R. M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc R Soc Lond B Biol Sci 221 (1222) (1984) 87–102. [25] K. Tsaneva-Atanasova, H. M. Osinga, T. Riess, A. Sherman, Full system bifurcation analysis of endocrine bursting models, J. Theor. Biol. 264 (4) (2010) 1133–1146. [26] A. M. Lacasta, F. Sagues, J. M. Sancho, Coherence and anticoherence resonance tuned by noise, Phys. Rev. E 66 (2002) 045105.

17