Bifurcations and canards in the FitzHugh-Nagumo system: a tutorial in fast-slow dynamics

Alexandre A. P. Rodrigues; Bruno F. F. Gon\c{c}alves; Isabel S. Labouriau

REVIEW 3 major objections 4 minor 33 references

Bifurcations and canards in the FitzHugh-Nagumo system: a tutorial in fast-slow dynamics

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper aims to prove that the fast-slow FitzHugh-Nagumo system combines relaxation oscillations, Hopf-induced periodic solutions, homoclinic orbits, and canards, with explicit formulas for the oscillation period and the canard…

desk verdict Useful tutorial with a real sign error in the canard-location formula and an unsupported claim about rigorous homoclinic verification. read the letter →

arxiv 2411.11209 v3 pith:SEDL5U73 submitted 2024-11-18 math.DS

Bruno F. F. Gonçalves , Isabel S. Labouriau , Alexandre A. P. Rodrigues This is my paper · ORCID

classification math.DS MSC 34E1534C2637G1537C2992C20

keywords FitzHugh-Nagumosystemfast-slowsystemscanardscanardexplosionrelaxationoscillationsHopfbifurcationhomoclinicsingularperturbationtheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This tutorial-style paper seeks to prove, rather than merely observe, that the two-variable FitzHugh-Nagumo fast-slow system (derived from the classic four-variable nerve-impulse model) exhibits three intertwined dynamical mechanisms: relaxation oscillations, periodic orbits born at Hopf bifurcations, and canard orbits that connect them. The authors work out the singular limit, apply the standard persistence theory for slow manifolds, and then check bifurcation conditions by hand. They also provide quantitative formulas, including a closed-form estimate for the relaxation-oscillation period and a canard parameter value in terms of the small time-scale ratio. If the proofs hold, the paper converts a qualitative picture of excitable dynamics into concrete parameter predictions for a classical mathematical-neuroscience model.

What carries the argument

The central objects are the cubic critical manifold $C_0 = \{(x,y): y = 4x - x^3\}$, the explicit slow flow in the fast variable $x$ given by $\dot x = (b x^3 + (1-4b)x + c)/(4-3x^2)$, and the canard normal form (27) around a regular singular fold. A regular singular fold is a fold point of the critical manifold that is simultaneously an equilibrium of the slow flow, with both the $x$-dependence and the parameter-dependence of the slow equation nonzero there. The normal form expresses the system near such a point as $x' = -y l_1 + x^2 l_2 + \varepsilon l_3$ and $y' = \varepsilon(\pm x l_4 - \lambda l_5 + y l_6)$, and its coefficients $l_1,\dots,l_6$ determine the constants $A$ and $B$ that give the Hopf value $\lambda_H = -B\varepsilon$ and the canard value $\lambda_c = -(B+A)\varepsilon$. The paper's computations fix these coefficients in both parameter slices, obtaining $A=-3/8$ in Example 6.1 and $A<0$ in Example 6.2, which is what fixes the direction of the Hopf bifurcation and the location of the canard explosion.

What would settle it

For $\varepsilon=0.5$, the paper's own numerical figure places rapid canard-length growth near $c \approx 1.150077$, whereas the formula $\bar c_c = -3\varepsilon/8$ with $\bar c = 2/\sqrt{3} - c$ puts the canard at $c \approx 1.342$; continuing the periodic orbits of system (23) numerically and locating the actual headless-canard value of $c$ would settle which prediction is correct.

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Extended reading notes

Core claim

The central claim is that the FitzHugh-Nagumo fast-slow system (5), with fast equation $x' = -y + 4x - x^3$ and slow equation $y' = \varepsilon(x - by - c)$, is a fully worked case study of geometric singular perturbation theory. In the singular limit $\varepsilon = 0$, the paper proves the existence of a relaxation cycle whose period is estimated by $T_{\gamma_0} \approx 2\int_{4/\sqrt{3}}^{2/\sqrt{3}} \frac{4-3x^2}{bx^3+(1-4b)x+c}\,dx$. For $0<\varepsilon\ll1$, it argues that the regular system retains this cycle with $O(\varepsilon)$ period error. In the parameter slice $b=0$, it identifies a subcritical Hopf bifurcation at $c_H = 2/\sqrt{3}$ and, via the canard normal form, derives the canard location $\bar c_c = -3\varepsilon/8$ in the translated coordinate $\bar c = c_H - c$. In the slice $c=0$, it finds a pitchfork at $b=1/4$, a Hopf point at $b_H^\varepsilon = (-4+\sqrt{16+3\varepsilon})/\varepsilon$, and a homoclinic bifurcation that destroys the unstable periodic orbit; numerically, for $\varepsilon=0.5$, this occurs at $b \approx 0.36932$.

Load-bearing premise

The canard-location formula rests on the assumption that a single translation of coordinates and parameter puts the system into the exact normal form whose canard expansion is known, with no additional rescaling; if that identification is off by a scaling or a sign, the formula $\bar c_c = -3\varepsilon/8$ does not apply to the original parameter $c$.

Editorial extensions

If this is right

For $0<\varepsilon\ll1$, the regular relaxation cycle persists with period $T_{\gamma_0} + O(\varepsilon)$, so formula (14) gives a quantitative prediction for the spike interval in the nerve-impulse model.
In the slice $b=0$, the subcritical Hopf bifurcation sits exactly at $c_H = 2/\sqrt{3}$, and stable small-amplitude cycles exist for $c$ just below this value, with the canard value indicating where the cycle expands into a large relaxation loop.
In the slice $c=0$, the pitchfork at $b=1/4$, the supercritical Hopf at $b_H^\varepsilon$, and the homoclinic bifurcation at $b \approx 0.36932$ for $\varepsilon=0.5$ organize the birth, growth, and destruction of the unstable periodic orbit.
The normal-form coefficient $A=-3/8$ in Example 6.1 fixes the bifurcation as supercritical in the translated parameter and subcritical in the original parameter $c$, confirming the shape of the numerically observed branch of periodic solutions.
Since all conditions of the persistence theorem and the canard normal form are checked explicitly, the paper positions the FitzHugh-Nagumo system as a template in which every step from singular limit to canard explosion is made explicit.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit that its quoted $\bar c_c = -3\varepsilon/8$ conflicts in sign with the normal-form formula $\lambda_c = -(B+A)\varepsilon$ when $B=0$ and $A=-3/8$; resolving this sign or rescaling issue would change the quantitative canard prediction.
The same normal-form computation could be applied to the two coupled FitzHugh-Nagumo systems mentioned in Section 7, potentially predicting which coupling strengths produce mixed-mode oscillations without full numerical simulation.
The period integral (14) can be evaluated in closed form for other $(b,c)$ parameter pairs, turning the singular limit into testable frequency-amplitude relations for excitable cells.
A natural continuation is to track the homoclinic value $b_{\rm Hom}^\varepsilon$ as $\varepsilon \to 0$ and check whether it approaches $3/8$ with a scaling consistent with canard explosion theory.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a genuinely useful tutorial on GSPT for the FitzHugh-Nagumo system, but the headline quantitative canard claim, the formula in Example 6.1, has the wrong sign, and the paper overstates what it proves about the homoclinic orbit.

What is actually new: not the phenomena—relaxation oscillations and canards in FH-N are well established—but the explicit period integral, the first-order slow manifold expansion, the normal-form computation for the singular Hopf/canard point, and a clean bifurcation analysis of both parameter cases. These are worked out carefully and would save a student a lot of time. The singular analysis in Section 3 and the Hopf/pitchfork calculations in Section 5 checked out.

The soft spot is Example 6.1. The stress-test is right: with l5 = -1 and λ = ¯c, Theorem 4 gives λ_c = -(B + A)ε = +3/8ε, not -3/8ε. The paper's own numerics (c ≈ 1.150 and 1.154, both below cH) support the corrected sign—canards at positive ¯c. So the qualitative picture is fine, but the only quantitative canard-location prediction in the paper is printed with the wrong sign and needs correction. Note also the O(ε^{3/2}) term is large at the simulated ε, so the numerical evidence doesn't independently confirm the coefficient, but the sign definitely agrees with the corrected formula.

Second, the introduction says all conditions for a homoclinic orbit are 'rigorously checked', but the homoclinic bifurcation in Section 5 is only observed numerically. The local Hopf and pitchfork results are proved, and the canard conditions are checked via Theorem 4, but the homoclinic claim is not a proof. That should be reworded.

Third, minor: the tutorial occasionally falls into 'we leave the proof to references' for Fenichel and Hopf, which is fine, but it should not be described as fully self-contained. The self-citation [13] in the discussion is benign.

Bottom line: the paper is a solid pedagogical contribution, not a research advance. With the sign fix and a more accurate claim about the homoclinic orbit, it would be a good fit for a journal that publishes expository/tutorial work. I would send it to a referee and expect a revise-and-resubmit.

Referee Report

3 major / 4 minor

Summary. The paper is a tutorial-style study of the two-dimensional FitzHugh-Nagumo fast-slow system (5). It develops the critical-manifold geometry and slow flow, proves the existence of a relaxation-type periodic orbit in the singular limit, estimates its period in Eq. (14), applies Fenichel theory to obtain a first-order slow-manifold expansion, computes Hopf and pitchfork bifurcations in the two parameter cases b=0 and c=0, and uses the Krupa-Szmolyan normal form to identify canard and singular Hopf bifurcations. All theoretical statements are accompanied by numerical simulations. The paper advertises an analytic proof of relaxation oscillations, Hopf-induced periodic solutions, and a rigorous check of the conditions for homoclinic orbits and canards in the FH-N system.

Significance. The paper is a useful pedagogical resource: the critical-manifold analysis, the period integral (14), and the linearized Hopf/pitchfork computations are explicit, parameter-free, and independently checkable. The numerical illustrations are extensive and well matched to the qualitative geometric picture. However, the quantitative canard-location formula in Example 6.1 is affected by a sign error in the normal-form identification, and the claimed equivalence in Example 6.2 is not exact; both issues concern the advertised rigorous check of canard conditions. The underlying geometric singular-perturbation narrative appears sound, and the errors are local and fixable.

major comments (3)

[Section 6, Example 6.1, Eq. (28) and Theorem 4] The text sets l1=1, l2=-2*sqrt(3)-x, l3=0, l4=1, l5=-1, l6=0 and lambda=cbar. In the normal form (27), y'=epsilon(+/-x l4 - lambda l5 + y l6); substituting l5=-1 gives y'=epsilon(x+lambda), so lambda must equal cbar. Then A=(-0+3(-1)-0+0)/8=-3/8 and B=0, and Theorem 4 gives lambda_c=-(B+A)epsilon=+3/8 epsilon+O(epsilon^{3/2}). The printed value -3/8 epsilon has the wrong sign. It cannot be repaired by setting lambda=-cbar without changing l5 to +1. Moreover, Figure 10 reports canard explosion near c=1.150077 for epsilon=0.5 and c=1.153794 for epsilon=0.1, both below c_H=2/sqrt(3), so cbar>0 there; this is consistent with +3/8 epsilon, not -3/8 epsilon. Since this is the paper's headline quantitative canard prediction, the sign error must be corrected.
[Section 6, Example 6.2, Eq. (29)] The stated change of variables (xbar,ybar,lambda)=(x-x+, y-8x+/3, b-3/8) does not transform system (24) into the printed system (29). Direct substitution gives xbar'=-ybar-xbar^3-3x+ xbar^2 and ybar'=epsilon[xbar-lambda(ybar+8x+/3)-3/8 ybar]; the term -epsilon lambda(8x+/3) is omitted in (29). The system identified in the normal form is therefore not equivalent to the original system. The omission is consequential: with the printed l6=-3/8, Theorem 4 gives lambda_H=-B epsilon=+3/16 epsilon, whereas the exact Hopf value in Section 5, b_H=(-4+sqrt(16+3epsilon))/epsilon, expands as b_H=3/8-9epsilon/512+O(epsilon^2), so lambda_H=-9epsilon/512+O(epsilon^2). The two asymptotics have opposite signs. The application of Theorem 4 in Example 6.2 needs to be redone with the correct transformed system and an appropriately rescaled parameter.
[Section 5(ii) and Example 6.2] The statement that a supercritical Hopf bifurcation at E+ 'generates an unstable periodic solution' is internally contradictory under the standard terminology used in the same section for the c-Hopf case. A supercritical Hopf bifurcation produces a stable periodic orbit, while an unstable periodic orbit corresponds to a subcritical bifurcation. The text and Figure 8 locate the unstable periodic solutions for b>b_H, where the equilibrium is stable; this description should be reconciled with the normal-form application in Example 6.2.

minor comments (4)

[Example 6.2] The value x+=4/3 is incorrect: for b=3/8, x+=sqrt(4-1/b)=sqrt(4-8/3)=2/sqrt(3). The correct value also gives l2=-2sqrt(3)-xbar rather than l2=-4-xbar.
[Abstract and Introduction] There is a recurring typo 'Hodgin-Huxley' where 'Hodgkin-Huxley' is intended; this occurs in the abstract and in Section 1.
[Figure 10 caption] The caption contains a typo: 'with §epsilon = 0.1' should read 'with epsilon = 0.1'.
[Theorem 4 and Example 6.1] The sign convention for lambda and l5 in the normal form (27) deserves an explicit remark, since the confusion between lambda=cbar and lambda=-cbar is at the root of the sign error in Example 6.1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claims are derived from explicit calculations and external standard theorems, not from fitted inputs or self-citations.

full rationale

The derivation chain is self-contained in the relevant sense. The relaxation-oscillation period estimate (14) is obtained by direct integration of the singular slow flow (11), which is itself derived from the defining equations f(x,y,0)=0 and ẏ=x−by−c; no fitted parameter is used and the integral is not assumed from the conclusion. The Hopf-bifurcation statements in Section 5 follow from explicit Jacobian eigenvalue computations and the standard Hopf theorem. The canard-location claims in Examples 6.1 and 6.2 are applications of the external Krupa–Szmolyan normal-form theorem (Theorem 4), with the coefficients l1,...,l6 read off from the translated systems (28) and (29); the numerical simulations in Figure 10 are illustrative and are not inputs to the analytic identification. The only self-citation, ref. [13], appears in the future-work discussion of coupled FH-N systems and is not load-bearing for any theorem or prediction in this paper. The sign inconsistency in Example 6.1—where the stated values λ=c̄ and l5=-1, together with Theorem 4, point to c̄_c=+3/8ε rather than the printed −3/8ε—is a correctness or consistency issue, not a circularity: the canard value is not obtained by fitting the answer or by assuming the conclusion. Similarly, the homoclinic orbit is supported numerically rather than analytically, which concerns rigor or completeness, not circularity. No step reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard GSPT theorems (Fenichel, Hopf, Krupa-Szmolyan) and on the specific identification of the normal form in Example 6.1. No free parameters are fitted to data and no new entities are introduced. The numerical simulations are illustrations, not inputs to the derivation.

assumptions (5)

standard math Fenichel's theorem: for a compact normally hyperbolic critical manifold, there is an O(ε)-close slow manifold Sε with the same stability.
Invoked in Section 4 as Theorem 1 to connect singular and regular dynamics; the paper does not prove it.
standard math Hopf bifurcation theorem (Theorem 3): eigenvalues crossing the imaginary axis with nonzero speed yield a limit cycle.
Used in Section 5 for cases b=0 and c=0; the paper cites [19,27] and applies the condition dα/dc ≠ 0.
standard math Krupa-Szmolyan theorem (Theorem 4) for singular folds and canard explosion, including the formula λc = -(B+A)ε.
The canard analysis in Section 6 rests entirely on this theorem and its normal form (27).
domain assumption The vector fields f and g are C^2 polynomials, and the critical manifold is a cubic; this is a domain assumption from the FH-N model.
Used throughout to justify Hartman-Grobman and Fenichel applicability; satisfied by (5).
ad hoc to paper The change of coordinates in Example 6.1 puts (23) exactly into the normal form (27) with λ = c̄ and l5 = -1, without additional parameter rescaling.
This identification is load-bearing for the canard location formula; the paper asserts it in Example 6.1 but does not show the transformation details.

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Pith. "Pith review of Bifurcations and canards in the FitzHugh-Nagumo system: a tutorial in fast-slow dynamics." pith.science (2026). https://pith.science/paper/SEDL5U73

@misc{pith2026241111209,
  author       = {Pith},
  title        = {Pith review of: Bifurcations and canards in the FitzHugh-Nagumo system: a tutorial in fast-slow dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SEDL5U73}},
  note         = {Machine review of arXiv:2411.11209}
}

abstract

In this article, we study the FitzHugh-Nagumo $(1,1)$--fast-slow system where the vector fields associated to the slow/fast equations come from the reduction of the Hodgin-Huxley model for the nerve impulse. After deriving dynamical properties of the singular and regular cases, we perform a bifurcation analysis and we investigate how the parameters (of the affine slow equation) impact the dynamics of the system. The study of codimension one bifurcations and the numerical locus of canards concludes this case-study. All theoretical results are numerically illustrated.

Figures

Figures reproduced from arXiv: 2411.11209 by the authors.

**Figure 1.** (a) - Critical manifold of the system (5): Attracting (solid red) and repelling (dashed red) regions; Fold points (black dots). (b) - Geometrical argument for the existence of at least one and at most three equilibrium points in the system (5). When considering a (1,1)–fast-slow system, since there is only one fast variable, we have Dxf = ∂f ∂x. For convenience, we use the notation φσ when referring to the partial d… view at source ↗

**Figure 2.** Examples of Cases (i)–(iv). Critical manifold in red, attracting regions in solid line and repelling region in dashed line. Line y˙ = 0 in blue. Fast trajectories in yellow and slow trajectories in green. Points of interest/equilibrium/fold in black. Initial point, A, outlined in black. (ii) The point B is not an equilibrium, but there exists a stable equilibrium C = (xc, yc) ∈ C0L \ B; (iii) The point B is not an … view at source ↗

**Figure 3.** Example of case (v) with parameters b = 0.2 and c = 0. Periodic solution γ0 composed of two slow trajectories (green) and two fast trajectories (yellow) with a period Tγ0 ≈ 3.61 time units (τ ). The transition from the slow flow to the fast one occurs at the fold points P± = ± 2/ √ 3, 16/3 √ 3 . Similarly, the transition from the fast flow to the slow one happens at the points F = −4/ √ 3, 16/3 √ 3 and D = … view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: (a) - Trajectory of the system (5) with parameters b = 0 = c, started at A ≈ (−2.8, 1.64) and with ε ∈ {0, 0.1, 0.5, 1} (violet, blue, green, yellow). Critical manifold (red) and unstable equilibrium point (blue). (b) - Dynamics of the trajectories of (a) in the coordi…

**Figure 5.** Figure 5: Evolution of the real and imaginary parts of the eigenvalues of the Jacobian matrix of system (23), evaluated at the equilibrium point E, as a function of the parameter c (λ + ε with solid line and λ − ε with dashed line) for different values of ε, ε = 1 (green), ε = 0…

**Figure 6.** Figure 6: (a), (b) and (c) Bifurcation diagrams: length A of the periodic solutions of system (23) with ε = 0.5 as a function of the parameter c with different levels of magnification. Equilibria correspond to A = 0. The graphs in (b) and (c) are zooms of the red and magenta re…

**Figure 7.** Figure 7: Evolution of the real and imaginary parts of the eigenvalues of the Jacobian matrix of the system (24), evaluated at the equilibrium point, as function of the parameter b (λ + ε with solid line, λ − ε with dashed line, and Hopf bifurcation point) for different values o…

**Figure 8.** Figure 8: (a) - Bifurcation of the equilibria of the system (24) as a function of the parameter b, with ε = 0.5. Equilibrium point E0 is always unstable (dashed line). Equilibria E− and E+ are unstable for 1/4 < b < bH 0.5 ≈ 0.3666 and stable b > bH 0.5 (solid line). Bifurcation…

**Figure 9.** Figure 9: (a) - (x, y) plane of the system (24) for ε = 0.5 and b = b Hom 0.5 ≈ 0.36932. C0 and equilibrium points in red. Homoclinic orbit (black), trajectories whose initial condition is within the region bounded by the homoclinic curve (yellow) and trajectories whose initial …

**Figure 10.** Figure 10: (a) - Periodic solutions of the system (23) for several values of c, with ε = 0.5. Critical manifold in red, attracting regions (solid line) and repelling region (dashed line), fold points (black), canards without head (yellow), canards with head (green), and limit cy…

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