Nonlinear Dynamics (2006) 44: 231–242

DOI: 10.1007/s11071-006-1978-5 c© Springer 2006

Types of Bifurcations of FitzHugh–Nagumo Maps

JORGE DUARTE1,∗, LUIS SILVA2, and J. SOUSA RAMOS3

1Departamento de Engenharia Quımica, Seccao de Matematica, Instituto Superior de Engenharia de Lisboa, Rua ConselheiroEmıdio Navarro 1, 1949-014 Lisboa, Portugal; 2Departamento de Matematica, Universidade de Evora, Rua Romao Ramalho,59, 7000-671 Evora, Portugal; 3Departamento de Matematica, Instituto Superior Tecnico, Av. Rovisco Pais, 1, 1049-001Lisboa, Portugal; ∗Author for correspondence (e-mail:jduarte@deq.isel.ipl.pt)

(Received: 15 September 2004; accepted: 20 March 2005)

Abstract. The FitzHugh–Nagumo-like systems are of fundamental importance to the understanding of the qualitative nature of

nerve impulse propagation. Our work provides a numerical investigation of bifurcations associated with a family of piecewise

differentiable canonical maps for a planar FitzHugh–Nagumo system. We describe the bifurcation structure of the maps with the

variation of the parameters.

Key words: excitable systems, FitzHugh-Nagumo maps, bifurcations, route to chaos

1. Introduction

The behavior of neural cells is often explored by examining the response of their cell membrane potential

to the injection of an external current.

The FitzHugh–Nagumo model governs the initiation of the nerve impulse propagation and can be

derived from a simplified model of the cell membrane. Here the cell (or membrane patch) consists of

three components: A capacitor representing the membrane capacitance, a current–voltage device for the

fast current, and a resistor, inductor and battery in series for the recovery current. In the 1960s, Nagumo,

a Japanese electrical engineer, built this circuit using a tunnel diode [1], thereby attaching his name to

the system.

In the present paper, we consider a periodically forced FitzHugh–Nagumo system depending on three

real parameters A, θ , and T defining a piecewise constant forcing.

The presence of the three parameters leads to a large variety of dynamics, each of them responsible

for a specific physiological function. For physiologists it is highly desirable to have a global view

of all possible qualitatively distinct responses of the FitzHugh–Nagumo model for several values of

the parameters. This reduces to the knowledge of the bifurcation diagram. For further information

concerning the physiological significance of bifurcations that occur in the FitzHugh–Nagumo model,

the reader is referred to the book [2]. Bifurcations play an important role in the study of the qualitative

behavior of biological systems. For instance, the idea that human disease may be associated with

bifurcations in the dynamics of living organisms was explicitly made by Mackey and Glass [3]. They

introduced the notion of dynamical disease to denote abnormal dynamics in physiological systems

associated with changes in its parameters.

Our goal in this paper is to describe the bifurcations performed by a special family of piecewise

differentiable maps associated with a singular system. We illustrate some bifurcations with phase space

plots and regimes of temporal behavior of the fast dynamical variable.

232 J. Duarte et al.

2. Description of the Model

The model is a planar piecewise linear system for which the equations are given by⎧⎪⎪⎨⎪⎪⎩ε

dv

dt= f (v) − w + ψ(t) ≡ F(v, w, t)

dw

dt= v − δw ≡ G(v, w)

(1)

Here f : R −→ R is the continuous, piecewise linear function given by

f (v) =

⎧⎪⎨⎪⎩−α1v + γ1 for v < vl

α2v − γ2 for vl ≤ v < vr

−α3v + γ3 for v ≥ vr

where αi and γi , i = 1, 2, 3, are fixed positive constants and 0 < vl < vr. Without loss of generality

we assume that γ1 = 0. The forcing function ψ : R −→ R is a piecewise constant periodic function of

period T , defined by

ψ(t) ={

0 for mT ≤ t < (m + θ )T

A for (m + θ )T ≤ t < (m + 1)T

m = 0, ±1, ±2, . . .

where A > 0 and θ ∈ [0, 1]. In the context of biophysical models of excitable cells, this forcing

corresponds to the injection of an external depolarizing current during a fraction (1−θ )T of each cycle.

The dynamical variables of the FitzHugh–Nagumo model are v and w. The variable v represents the

voltage (the difference of the membrane potential from its resting state) and is called the excitationvariable. The variable w represents the effects of changes in ion-channel conductances that tend to

return the membrane potential to its resting level. This variable is called the recovery variable. We

emphasize that the system is governed by two distinct dynamics: A fast one (associated to v) and a

slow one (associated to w). We assume that ε is very small. The assumption ε � 1 often appears in the

literature in the form ε → 0. Since the small parameter multiplies the derivative dv/dt , the system is

called singularly perturbed [4].

The choice of a piecewise linear system is preferable for two reasons:

(a) piecewise linear models possess explicit solutions; and

(b) the essential feature of the cubic nonlinearity in the FitzHugh–Nagumo system is its “N” shape and

this is reproduced by the piecewise linear form.

Figure 1 shows the nullclines of the system (1), Table 2 gives the analytic definitions of the lines, and

Table 1 gives the definitions of the special points shown in that figure.

The system (1) is an autonomous system in each of the time intervals [mT, (m + θ )T ) and [(m +θ )T, (m + 1)T ). The v-nullcline is given by w = f (v) when t ∈ [mT, (m + θ )T ) and by w =f (v) + A when t ∈ [(m + θ )T, (m + 1)T ). The continuity of f implies that γ2 = (α1 + α2)vl and

γ3 = (α2 +α3)vr − (α1 +α2)vl. Like in [5], we assume that α2vr − γ2 = α2vr − (α1 +α2)vl > 0, which

implies that f (r ) > 0. We also assume that 1δ

> α2. Then G = 0 intersects F = 0 at exactly one point

when the forcing is off and exactly one point when it is on.

Types of Bifurcations of FitzHugh–Nagumo Maps 233

Table 1. The coordinates of labeled points in Figure 1, for A = 0.6.

Formula Values W

Q1 vl(1, −α1) (0.2, −0.4) −0.4

Q2 (0, 0)T (0, 0) 0

Q3 − (α2vr−γ2+A)α1

(1, −α1) −(0.6 + A)(0.5, −1.0) 1.2

Q4 (vl, −α1vl + A) (0.2, −0.4 + A) 0.2

Q5 A/(1 + δα1)(δ, 1) A(0.25, 0.5) 0.3

Q6 ( −(α2vr−γ2)α1

, α2vr − γ2 + A) (−0.3, 0.6 + A) 1.2

Q7 ( (γ3+α1vl)α3

, −α1vl) (2.1, −0.4) −0.4

Q8 (vr, α2vr − γ2) (1.2, 0.6) 0.6

Q9 ( (γ3+α1vl+A)α3

, −α1vl) (4.2 + A, −0.4) −0.4

Q10 (vr, α2vr − γ2 + A) (1.2, 0.6 + A) 1.2

Q11(A+γ3)(1+δα3) (δ, 1) ( (A+1.8)

3.0 ,(A+1.8)

1.5 ) 1.6

Q12γ3

(1+δα3) (δ, 1) (0.6, 1.2) 1.2

Q13 ( (γ3+α1vl−A)α3

, −α1vl + A) (2.2 − A, −0.4 + A) 0.2

Q14 ( (γ2+γ3−α2vr)α3

, α2vr − γ2) (1.2, 0.6) 0.6

Table 2. The analytic definitions of the lines shown in Figure 1.

Labels Equations

L−1 w = −α1v, v ≤ vl

L+1 w = −α1v + A, v ≤ vl

L−2 w = α2v − γ2, vl < v ≤ vr

L+2 w = α2v − γ2 + A, vl < v ≤ vr

L−3 w = −α3v + γ3, v > vr

L+3 w = −α3v + γ3 + A, v > vr

-1 0 1 2 3 4

0

1

2

3

4

Q2

L1+

L1−

L2+

L2−

L3+L3

−Q1

Q2

Q3

Q4

Q5

Q6

Q7

Q8

Q9

Q10

Q11

Q12

Q13Q14

W

V

G=0

F=0

Figure 1. A schematic of the nullclines of (1). The analytic definition of the lines is given in Table 2 and the coordinates of the

labeled points are given in Table 1. The second column of Table 1 gives the coordinates for the standard values of the parameters,

which are α1 = 2, α2 = α3 = 1, γ2 = 0.6, and γ3 = 1.8. For these parameters A1 = 0.4, A2 = 0.8, A3 = 1.8, vl = 0.2, and

vr = 1.2.

234 J. Duarte et al.

As Othmer and Watanabe have shown in [5], in the singular limit ε = 0, the system (1) reduces to

dw

dt=

{−λi (w − w∗

i ), (v, w) ∈ L−i

−λi (w − (w∗i + A∗

i )), (v, w) ∈ L+i

(2)

where w∗i ≡ γi

1+δαiand A∗

i ≡ A1+δαi

, i = 1, 3.

The w∗i (resp., w∗

i + A∗i ) are the w-coordinates of the rest points (resp., virtual rest points, i.e.,

the intersection of the extension of L±i with G = 0). From Figure 1 and Table 1, we obtain: w∗

1 =0, w∗

1 + A∗1 = W5, w

∗3 = W12, w

∗3 + A∗

3 = W11.

Let T ±i (w0, w) denote the time to reach w from w0 while remaining on L±

i . Then for i = 1, 3

T −i (w0, w) = 1

λiln

[w0 − w∗

i

w − w∗i

](3)

and

T +i (w0, w) = 1

λiln

[w0 − (w∗

i + A∗)

w − (w∗i + A∗)

]. (4)

We direct our attention to the study of the local dynamics of the singular system (2) (that is, when ε = 0),

associated to the piecewise linear model (1) under step function. Note that solutions are constrained to

the lines L±i when ε = 0.

A solution of the singular system (2) is presented in Figure 2, and a typical regime of temporal

behavior of the v-component of the solution of (2) is plotted in Figure 3.

3. Discrete Dynamics: Bifurcations and Routes to Chaos

Families of iterated maps on the interval provide a simple and helpful context, which allow us to study

interesting properties of several dynamical systems. As in [5] and [6], we are going to use a return

-0.5 0 0.5 1 1.5 2 2.5 3

-0.5

0

0.5

1

1.5

2

Figure 2. Solution of (2) in the phase plane for (A, θ, T ) = (0.6, 0.4, 4.0).

Types of Bifurcations of FitzHugh–Nagumo Maps 235

2 4 6 8 10

-0.5

0

0.5

1

1.5

2

2.5

3

Figure 3. v-Component of the solution as a function of time for (A, θ, T ) = (0.6, 0.4, 4.0).

map derived from the singular system (2). In order to facilitate comparison of the maps for different

combinations of the parameters A, θ , and T , we map the w-coordinate into the unit interval via the

transformation

x =

⎧⎪⎨⎪⎩w

W8 + W10

if (v, w) ∈ L−1 , w ≥ 0

1 − w

W8 + W10

if (v, w) ∈ L−3 , w ≥ 0.

This mapping takes the segment of L−3 lying between Q8 and the v-axis and appends it to L−

1 (truncated

above w10) and rescales the result. Thus, the segment of L−1 lying between the origin and the intersection

of the horizontal line with w-coordinate W4 is mapped into the interval [0, x4], and so on.

We denominate FitzHugh–Nagumo maps the three-parameters return maps that are obtained by

computing the trajectories of points on the interval [0, 1], for one period or cycle of the forcing. By a

return map we mean a map from the unit interval to itself that maps a point into its image after time T(one cycle of the forcing). With this procedure the interval [0, 1] is decomposed into a disjoint union

of characteristic intervals (see [5] and [6] for more details). The study of representative return maps

that arise for several values of the parameters provides us a deeper comprehension of the qualitative

behavior. If we consider the invariant regions of the return maps for special values of the parameters when

0.8 < A ≤ 1.8, we obtain the following three families of canonical maps, which are characterized by

(a) three branches and two turning points (one relative maximum and one relative minimum);

(b) four branches and three turning points (one relative maximum and two relative minima);

(c) two branches and one turning point (one relative maximum).

In this paper, we restricted our attention to the first family of maps (quoted as f ), which has three

branches and two turning points: c1 the relative maximum and c2 the relative minimum (Figure 4).

In the theory of the asymptotic behavior of maps of the interval into itself under iteration, one of the

questions of most interest is the determination of the limit sets of points for a map f : I −→ I , where

I = [0, 1]. The limit set of x ∈ I is the set of limit points of the sequence { f n(x)}n∈N (the superscript

denotes repeated composition). If we are interested in the change of limit sets of the map f : I −→ I

236 J. Duarte et al.

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

xn+1

xn

Figure 4. The computed return map for (A, θ, T ) = (1.1, 0.1, 1).

as one parameter μ is varied, we write fμ(x) or f (μ, x). Of particular interest are the following orbit

types:

(a) Period-1 orbit or 1-cycle, that is, a fixed point x∗ of f (μ, x), which is a solution of the equation

x = f (μ, x). The fixed point is stable when | ∂ f (μ,x)∂x |x∗ < 1.

(b) Period-n orbit or n-cycle. Each point xi of the n-cycle is a fixed point of f n(μ, x), i.e., xi =f n(μ, xi ), for i = 0, 1, . . . , n − 1. We have the stability condition | ∂ f n (μ,xi )

∂x | = ∏n−1i=0 | f ′(μ, xi )| <

1. This condition remains the same for all points xi in the n-cycle, i.e., they all acquire or lose

stability at the same time.

The maps can display various qualitative types of asymptotic behavior for different values of the

parameters A, θ , and T : Steady states, periodic cycles of different lengths, and chaos. The qualitative

changes of behavior as a parameter μ is varied are called bifurcations, and the parameter values at

which they occur are called bifurcation points. An important goal in studying the return maps is to

understand the bifurcations that can appear for several values of the parameters. Instead of formulating

general statements at this point, we are going to discuss typical cases that occur in the model. To see

the long-term behavior for different values of T at once, we plot the orbit diagram, which shows only

the attractive orbits. Figure 5 plots the system’s attractive orbits as a function of T (cycle of the forcing)

for A = 1.1 and θ = 0.1.

The last numerical results raise many important questions. We attempt to identify types of bifurcations

that can occur with the variation of the parameters. To start with, we consider the variation of T , namely

when T decreases from 1.25 to 0.2. Since the family of return maps is obtained by numerically computing

the trajectories of a certain number of points on the interval [0, 1], we are going to rely on graphical

and numerical arguments to describe the qualitative behavior.

The family of maps is continuous and piecewise differentiable (the derivative does not exist at the

turning points). At these points the left derivative, ddx f−, and the right derivative, d

dx f+, of f do not

have the same value.

We shall be concerned with those bifurcation values that involve periodic orbits. Assume f n(μ, x∗) =x∗ and that n is the smallest integer for which this is true. We say then that x∗ is periodic with prime

Types of Bifurcations of FitzHugh–Nagumo Maps 237

Figure 5. Images under the return map of the initial point x0 = 0.5 vs. the parameter T for θ = 0.1 and A = 1.1. The blue and

green lines represent the turning points c1 and c2 as functions of the parameter.

Figure 6. The second-iterate map for (A, θ, T ) = (1.1, 0.1, 1.094 . . . ).

period n. The local bifurcation theory allows us to assign a stability label (s for stable, u for unstable)

to points according to whether the absolute value of ddx f n(μ, x∗) = λ(x∗) is smaller or larger than 1.

Firstly, amongst the bifurcations of the family of maps, that occur when T decreases from 1.25 to 0.2,

we quote the well-known period-doubling bifurcation (PDB) at T = 1.094 . . . (see Figure 6). These

maps do not undergo a succession of period-doubling bifurcations that lead to chaos. We do not have a

period-doubling cascade in periodic windows like in the family of quadratic maps xn+1 = axn(1 − xn).

For details about the period-doubling bifurcation, see ref. [7].

At T = 1.0, the attractor is a period-4 orbit, as indicated by the four branches (Figure 5). We analyze

the stability of a cycle reducing the problem to a question about the stability of a fixed point. The graph

of a fourth-iterate map (Figure 7) provides insight about the stability of the 4-cycle.

The four fixed points at which the slope of the map has absolute value less than 1 corresponds to a

stable period-4 orbit. In contrast, the slope exceeds 1 in the remaining fixed points (one period-1 orbit

and one period-2 orbit).

238 J. Duarte et al.

Figure 7. The fourth-iterate map for (A, θ, T ) = (1.1, 0.1, 1).

Figure 8. Sketch (not to scale) of the bifurcation diagram of the Type A maps for A = 1.1, θ = 0.1, and T ∈ (0.94 . . . , 1.25 . . . ).

It is assigned a stability label: s for stable, u for unstable. There are two special values of T , which are t1 = 0.942318 . . . and

t2 = 0.94883 . . . .

Figure 8 shows a partial bifurcation diagram of the return maps. As T decreases, a pair of repulsive

orbits is created at T = t2. The graph of the fourth-iterate map is helpful to understand the birth of these

orbits. We denominate this type of bifurcation unstable-unstable saddle-node bifurcation (SNBuu). In

the literature, a saddle-node bifurcation is also called tangent bifurcation.

In order to facilitate the analysis about the stability of the fixed points, we present Figures 9 and

10, which represent the sketch of the fourth-iterate map at these points when T = t2 and T = t1,

respectively.

We emphasize the creation of pairs of repulsive orbits at T = t2 and the occurrence of a stable–unstableinverted saddle-node bifurcation (ISNBsu) at T = t1, that is, the destruction of pairs stable–unstable

orbits. The marginal case of stability, when fixed points are points of nondifferentiability, requires

Types of Bifurcations of FitzHugh–Nagumo Maps 239

1s2u3u

4u

5s 6u7u

8u

9u

10u 11s

12u

13s 14u

15u

Figure 9. Sketch of the fourth-iterate map at the fixed points when T = t2.

1s

4u

2u3u

5s 6u7u

8u

9u

10u

11s

12u

13s14u

15u

Figure 10. Sketch of the fourth-iterate map at the fixed points when T = t1.

special investigation. This happens at a fixed parameter μ∗. The change of the number and stability of

solutions takes place at μ∗.

We are going to describe the above two types of tangent bifurcations: The unstable–unstable saddle-node bifurcation and the stable–unstable inverted saddle-node bifurcation.

We concentrate on the maps f (μ, x), with μ = T . However, the following description carries over

to the maps f (A, x) and f (θ, x). As illustrated in Figure 9 (see in parallel Figure 8), at the fixed points

defined by the pairs of segments (2u, 3u), (6u, 7u), (9u, 10u) and (14u, 15u), we have for μ∗ = t2:

1. f n(μ∗, x∗) = x∗;

2. | ∂∂x f n

−(μ∗, x∗)| > 1 and∣∣ ∂∂x f n

+(μ∗, x∗)∣∣ > 1.

When μ = T ∈ (t1, t2), the diagonal line intersects all the segments of the graph. More precisely,

there is a small region in μx-plane centered at (μ∗, x∗), {(μ, x) : μ∗ − η < μ < μ∗ + η and x∗ − ε <

240 J. Duarte et al.

-0.5 0 0.5 1 1.5 20.8

1

1.2

1.4

1.6

w

v

Figure 11. Phase space plot of the dynamical variables v and w for (A, θ, T ) = (1.1, 0.1, 1.0).

5 10 15 20 25 30t

-0.5

0

0.5

1

1.5

2

2.5V

Figure 12. Time sequence of membrane potentials for (A, θ, T ) = (1.1, 0.1, 1.0).

-0.5 0 0.5 1 1.5 2

0.8

1

1.2

1.4

1.6

w

v

Figure 13. Phase space plot of the dynamical variables v and w for (A, θ, T ) = (1.1, 0.1, 1.942318 . . . ).

x < x∗ + ε, η, ε > 0}, with the following properties concerning μ∗ = t2 and the fixed points quoted:

(i) if μ ∈ (t2, t2 + η), then f n(μ, x) has no fixed points in (x∗ − ε, x∗ + ε);

(ii) if μ ∈ (t2 − η, t2), then f n(μ, x) has two fixed points in (x∗ − ε, x∗ + ε), both repelling.

This bifurcation is called unstable–unstable saddle-node bifurcation. This type of bifurcation cannot

occur for the differentiable maps, since the derivative at the tangent point would be equal to one.

Types of Bifurcations of FitzHugh–Nagumo Maps 241

5 10 15 20 25 30t

-0.5

0

0.5

1

1.5

2

2.5V

Figure 14. Time sequence of membrane potentials for (A, θ, T ) = (1.1, 0.1, 1.942318 . . . ).

Figure 15. Images under the return map of the initial point x0 = 0.5 vs. the parameter A for θ = 0.1 and T = 0.8. The blue and

green lines represent the turning points c1 and c2 as functions of the parameter.

Figure 16. Images under the return map of the initial point x0 = 0.5 vs. the parameter θ for A = 1.1 and T = 0.8. The blue and

green lines represent the turning points c1 and c2 as functions of the parameter.

In Figure 10, at the fixed points defined by the pairs of segments (1s, 2u), (5s, 6u), (10u, 11s), and

(13s, 14u), we have for μ∗ = t1:

1. f n(μ∗, x∗) = x∗;

2. | ∂∂x f n

−(μ∗, x∗)| < 1 and | ∂∂x f n

+(μ∗, x∗)| > 1 or | ∂∂x f n

−(μ∗, x∗)| > 1 and | ∂∂x f n

+(μ∗, x∗)| < 1.

When μ = T < t1, the diagonal line does not intersect those pairs of segments. There is a small region

in μx-plane centered at (μ∗, x∗), {(μ, x) : μ∗ − η < μ < μ∗ + η and x∗ − ε < x < x∗ + ε, η, ε > 0},with the following properties concerning μ∗ = t1 and the fixed points quoted above:

242 J. Duarte et al.

(i) if μ ∈ (t1 − η, t1), then f n(μ, x) has no fixed points in (x∗ − ε, x∗ + ε),

(ii) if μ ∈ (t1, t1 + η), then f n(μ, x) has two fixed points in (x∗ − ε, x∗ + ε), one attracting and one

repelling.

This bifurcation is called stable–unstable inverted saddle-node bifurcation and it is responsible for

the interruption of the period-doubling cascade in periodic windows.

In order to illustrate the bifurcations, we exhibit examples of phase plane representations and regimes

of temporal behavior of the fast dynamical variable (Figures 11–14).

It is important to notice that this pattern of bifurcations are common to the three bifurcation diagrams.

Indeed, it occurs when we plot the bifurcation diagram as a function of A and as a function of θ (Figures 15

and 16).

In this note, we have provided a contribution for the detailed description of qualitative changes of

behavior associated with a family of piecewise differentiable singular maps for a FitzHugh–Nagumo

system.

Acknowledgements

This work was partially supported by PRODEP, Medida 5 - Accao 5.3 and by FCT/POCTI/FEDER.

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