Highly synchronized noise-driven oscillatory
behavior of a FitzHugh–Nagumo ring
with phase-repulsive coupling
Gonzalo Izús∗,†, Roberto Deza∗ and Alejandro Sánchez∗,†
∗Departamento de Física, Facultad de Ciencias Exactas y Naturales,
Universidad Nacional de Mar del Plata,
Deán Funes 3350, 7600 Mar del Plata, Argentina.
†Member, CONICET
Abstract. We investigate a ring of N FitzHugh–Nagumo elements coupled in phase-repulsive
fashion and submitted to a (subthreshold) common oscillatory signal and independent Gaussian
white noises. This system can be regarded as a reduced version of the one studied in [Phys. Rev.
E 64, 041912 (2001)], although externally forced and submitted to noise. The noise-sustained
synchronization of the system with the external signal is characterized.
Keywords: synchronization, signal transduction, chemical waves, neuroscience
PACS: 05.45.Xt, 87.16.Xa, 87.18.Pj, 87.19.La
INTRODUCTION
In Ref. [1]—through comparison with the synchronization patterns arising in two-
dimensional arrays of FitzHugh–Nagumo (FHN) elements with phase-repulsive linear
nearest-neighbor coupling—the authors were able to conclude that intracellular cal-
cium oscillations in cultures of human epileptic astrocytes do interact, since the phases
of nearby oscillating astrocytes maintain a nontrivial relationship. It is a fortunate fact
that the (space-independent) FHN model is one of the very few multicomponent systems
for which a nonequilibrium potential (NEP) has been found [2, 3], since NEPs allow in
general for a deep insight on the dynamical mechanisms leading to pattern formation and
other phenomena where fluctuations play a constructive role [4]. The (albeit minimal)
extension of the result in Refs. [2, 3] towards extended systems carried out in this work
is however enough to shed light on the dynamical cause of the conclusion in Ref. [1]: a
dynamical symmetry breakdown takes place because the phase-repulsive coupling min-
imizes the corresponding NEP. When the system is externally forced with a frequency
less than the typical inverse deterministic time the cycle duplicates, breaking down into
an “excited” phase and an “inhibited” one. These phases force neighbor elements to al-
ternate with the one in between, thus creating a nontrivial phase relationship between
nearby oscillating elements.
The system we consider is sketched in Fig. 1: a ring of N = 256 identical FHN el-
ements with phase-repulsive nearest-neighbor coupling and submitted to a (subthresh-
old) common oscillatory signal and independent Gaussian white noises ξui(t), ξvi(t)
with 〈ξm(t)ξn(t ′)〉 = 2ηδmnδ (t − t ′), m,n = 1, . . . ,2N. The set of equations governing
ξ iI    (t)+ (t)
FIGURE 1. Sketch of the system and of its response Ac(t).
its dynamics is
u̇i = ac ui (1−u2i )− vi +Sg(t)−D(ui+1 +ui−1)+ r1 ξui(t)+ r2 ξvi(t) (1)
v̇i = ε (β ui− vi +C)+ r3 ξui(t)+ r4 ξvi(t), i = 1, . . . ,N, uN+1 = u1.
where ε = 0.01 is the ratio between the relaxation rates of ui and vi, β = 0.01, ac = 0.06
and C = 0.02 is a suitable constant to set the rest point in Fig. 2a. D = 0.01 is the
phase-repulsive coupling constant, and the ri (which determine the transport matrix) are
r1 = 0.998×102, r2 = 0.499×101, r3 = 0.998, r4 = 0.499×10−1. Moreover, taking the
Milshtein integration step as dt = 5×10−3, we estimate the typical inverse deterministic
time as 0.838× 10−3 and so we take the excitation frequency Ω0 as a fraction of that
value (typically 0.1–0.4). Given that, Sg(t) = 0.0275sinΩ0t.
THE NONEQUILIBRIUM POTENTIAL
Excitable dynamics can be conceptually decomposed into two phases, a fluctuation-
dominated one and a deterministic one. It would be highly desirable to find a Lyapunov
function, since it greatly simplifies the dynamical analysis. However, the existence of
non-variational (or conserving) components in the phase-space flow is a hint that the
integrability conditions fail for the purely deterministic system. This apparently insur-
mountable drawback was partially solved two decades ago by Graham and collaborators
(see references in [2]) who defined the NEP for Langevin-type dynamics as the zero-
noise limit of the logarithm of the stationary probability density function (pdf). The
extra freedom in the choice of the transport matrix can render in some cases the prob-
lem integrable. That is precisely the case for the space-independent FitzHugh–Nagumo
model in its bistable and excitable regimes [2, 3]. This approach can be generalized to
extended systems and the NEP associated to Eq.(1) (in the adiabatic limit, i.e. for slow
FIGURE 2. a) Phase-space excursions in excitable regime (the nullclines are indicated in dashed line);
b) Time evolution of u (dashed line), v (dotted line), and the NEP (full line) during a phase-space
excursion. The scales of v and the NEP were adjusted for better comparison.
FIGURE 3. a) NEP in excitable regime; b) Stationary pdf in excitable regime.
signal) is [5]
Φ(t) = Φ{t,ui(t),vi(t)}=
(v2i −2β uivi−2Cvi)+
(β u2i +2Cui)
u2i −
u4i +Sg(t)ui
uiui+1
, (2)
FIGURE 4. a) NEP in bistable regime; b) Stationary pdf in bistable regime.
FIGURE 5. Time evolution of u for two neighbor neurons.
which must obey the integrability condition βλ1 +λ2/ε = 2λ [2].
Figure 2b depicts (in full line) the time evolution of the NEP during the phase-space
excursion starting at the upper initial condition in Fig. 2a, together with that of u (dashed
line) and v (dotted line). We remark that in Figs. 2a and 2b there is no noise and Φ(t) is
the Lyapunov functional of the deterministic dynamics. Figures 3a and 3b (respectively
4a and 4b) are 3D and contour plots of the NEP and the corresponding stationary pdf for
the excitable (respectively bistable) regime.
RESULTS FOR THE COUPLED SYSTEM
Synchronization between the coupled system and the external signal is observed above
some noise-intensity threshold. Figure 5 is a plot of the time evolution of ui (full line),
together with that of ui+1 (dashed line) for a given neuron i, showing their phase relation
to the signal (dotted line)1 According to Fig. 2, we may call “active” those cells i for
which ui(t) exceeds some threshold value uth. Because of the coupling, as one neuron
becomes active, it inhibits the activation of its nearest neighbors. The perfect alternance
seen in the figure may fail because of the noise, a necessary ingredient for the activation.
A detail of the alternance can be seen in Fig. 6 for an N′= 21 subset of the ring. Figure
6a shows a situation (snapshot) of poor synchronization, in which only two neurons are
active; Fig. 6b exhibits a case of a “kink” in the synchronized configuration, induced by
the fact that noises are local. Note that the kinks break locally the observed coherence,
and the complete history of the time evolution can be followed as a record of activity
(see Fig. 7).
1 The signal has been augmented in about two orders of magnitude and shifted to aid the sight.
FIGURE 6. Two snapshots of {ui} showing different degrees of synchrony.
FIGURE 7. Synchronization of the ring. White corresponds to activation and black to inhibition.
Horizontal dimension corresponds to time and vertical to space.
A measure of “activity” for the whole ring is
Ac(t) =
θ [ui(t)−uth]. (3)
In perfect synchrony, Ac = 0.5. Note that since the signal is subthreshold for the coupled
system, Ac = 0 below threshold. Figure 8a depicts the activity as a function of time
for a fixed noise intensity, showing again its phase relationship with the signal (dashed
line). In Fig. 8b we show the NEP for the whole ring as a function of time, together
with the (scaled) signal for reference. We remark that the observed dynamical symmetry
breakdown decreases the Lyapunov function of the whole ring with respect to that of the
homogeneous state, providing the route to stable synchronization.
A global estimator of synchronization can be defined as
∫ t f
0 Ac(t)dt
0.5N t f
. (4)
Figure 9a is a plot of Ga as a function of the noise intensity. The existence of a
threshold value of noise intensity and of a saturation effect can be clearly seen. The
noise intensities are low enough not to degrade the excitable dynamics.
FIGURE 8. a) Ac vs t for high synchronization; b) Time evolution of the NEP
FIGURE 9. a) Ga vs η ; b) C vs η .
Numerical simulations indicate that the coherence of firing decreases with the noise
intensity although the global activity (representative of global estimators) keeps the
order of magnitude. To quantify this phenomena we have calculated the normalized self-
correlation C = 〈uiui+2〉 as a function of the noise intensity η . As we show in Fig. 9b
the system shows a kind of “stochastic resonance in coherence” that cannot be inferred
from measures of global activity.
CONCLUSIONS
We have investigated the noise-induced synchronization with an external signal of a ring
of phase-repulsively coupled FHN elements. We have derived the exact NEP of the ex-
tended system and the observed symmetry breakdown was related with the Lyapunov-
functional properties of the NEP. We remark that the same conclusion holds qualita-
tively for the work in Ref. [1]. Although the observed phenomenon is noise-sustained
and global activity increases with noise intensity, a degradation of coherence can be
appreciated.
ACKNOWLEDGMENTS
Financial support from CONICET, ANPCyT and the National University of Mar del
Plata is acknowledged.
REFERENCES
1. G. Balázsi, A. Cornell-Bell, A. B. Neiman, and F. Moss, Phys. Rev. E 64, 041912 (2001).
2. G. Izús, R. Deza, and H. S. Wio. Phys. Rev. E 58, 93–98 (1998).
3. G. Izús, R. Deza, and H. S. Wio. Comp. Phys. Comm. 121–122, 406–407 (1999).
4. H. S. Wio, in Fourth Granada lectures in computational physics, P. L. Garrido and J. Marro, Eds.;
LNP 493 (Springer-Verlag, Berlin, 1997), p. 135.
5. A. Sánchez, G. Izús, and R. Deza, in preparation.

  We investigate a ring of $N$ FitzHugh--Nagumo elements coupled in
\emph{phase-repulsive} fashion and submitted to a (subthreshold) common
oscillatory signal and independent Gaussian white noises. This system can be
regarded as a reduced version of the one studied in [Phys. Rev. E \textbf{64},
041912 (2001)], although externally forced and submitted to noise. The
noise-sustained synchronization of the system with the external signal is
characterized.

Highly synchronized noise-driven oscillatory
behavior of a FitzHugh–Nagumo ring
with phase-repulsive coupling
Gonzalo Izús∗,†, Roberto Deza∗ and Alejandro Sánchez∗,†
∗Departamento de Física, Facultad de Ciencias Exactas y Naturales,
Universidad Nacional de Mar del Plata,
Deán Funes 3350, 7600 Mar del Plata, Argentina.
†Member, CONICET
Abstract. We investigate a ring of N FitzHugh–Nagumo elements coupled in phase-repulsive
fashion and submitted to a (subthreshold) common oscillatory signal and independent Gaussian
white noises. This system can be regarded as a reduced version of the one studied in [Phys. Rev.
E 64, 041912 (2001)], although externally forced and submitted to noise. The noise-sustained
synchronization of the system with the external signal is characterized.
Keywords: synchronization, signal transduction, chemical waves, neuroscience
PACS: 05.45.Xt, 87.16.Xa, 87.18.Pj, 87.19.La
INTRODUCTION
In Ref. [1]—through comparison with the synchronization patterns arising in two-
dimensional arrays of FitzHugh–Nagumo (FHN) elements with phase-repulsive linear
nearest-neighbor coupling—the authors were able to conclude that intracellular cal-
cium oscillations in cultures of human epileptic astrocytes do interact, since the phases
of nearby oscillating astrocytes maintain a nontrivial relationship. It is a fortunate fact
that the (space-independent) FHN model is one of the very few multicomponent systems
for which a nonequilibrium potential (NEP) has been found [2, 3], since NEPs allow in
general for a deep insight on the dynamical mechanisms leading to pattern formation and
other phenomena where fluctuations play a constructive role [4]. The (albeit minimal)
extension of the result in Refs. [2, 3] towards extended systems carried out in this work
is however enough to shed light on the dynamical cause of the conclusion in Ref. [1]: a
dynamical symmetry breakdown takes place because the phase-repulsive coupling min-
imizes the corresponding NEP. When the system is externally forced with a frequency
less than the typical inverse deterministic time the cycle duplicates, breaking down into
an “excited” phase and an “inhibited” one. These phases force neighbor elements to al-
ternate with the one in between, thus creating a nontrivial phase relationship between
nearby oscillating elements.
The system we consider is sketched in Fig. 1: a ring of N = 256 identical FHN el-
ements with phase-repulsive nearest-neighbor coupling and submitted to a (subthresh-
old) common oscillatory signal and independent Gaussian white noises ξui(t), ξvi(t)
with 〈ξm(t)ξn(t ′)〉 = 2ηδmnδ (t − t ′), m,n = 1, . . . ,2N. The set of equations governing
ξ iI    (t)+ (t)
FIGURE 1. Sketch of the system and of its response Ac(t).
its dynamics is
u̇i = ac ui (1−u2i )− vi +Sg(t)−D(ui+1 +ui−1)+ r1 ξui(t)+ r2 ξvi(t) (1)
v̇i = ε (β ui− vi +C)+ r3 ξui(t)+ r4 ξvi(t), i = 1, . . . ,N, uN+1 = u1.
where ε = 0.01 is the ratio between the relaxation rates of ui and vi, β = 0.01, ac = 0.06
and C = 0.02 is a suitable constant to set the rest point in Fig. 2a. D = 0.01 is the
phase-repulsive coupling constant, and the ri (which determine the transport matrix) are
r1 = 0.998×102, r2 = 0.499×101, r3 = 0.998, r4 = 0.499×10−1. Moreover, taking the
Milshtein integration step as dt = 5×10−3, we estimate the typical inverse deterministic
time as 0.838× 10−3 and so we take the excitation frequency Ω0 as a fraction of that
value (typically 0.1–0.4). Given that, Sg(t) = 0.0275sinΩ0t.
THE NONEQUILIBRIUM POTENTIAL
Excitable dynamics can be conceptually decomposed into two phases, a fluctuation-
dominated one and a deterministic one. It would be highly desirable to find a Lyapunov
function, since it greatly simplifies the dynamical analysis. However, the existence of
non-variational (or conserving) components in the phase-space flow is a hint that the
integrability conditions fail for the purely deterministic system. This apparently insur-
mountable drawback was partially solved two decades ago by Graham and collaborators
(see references in [2]) who defined the NEP for Langevin-type dynamics as the zero-
noise limit of the logarithm of the stationary probability density function (pdf). The
extra freedom in the choice of the transport matrix can render in some cases the prob-
lem integrable. That is precisely the case for the space-independent FitzHugh–Nagumo
model in its bistable and excitable regimes [2, 3]. This approach can be generalized to
extended systems and the NEP associated to Eq.(1) (in the adiabatic limit, i.e. for slow
FIGURE 2. a) Phase-space excursions in excitable regime (the nullclines are indicated in dashed line);
b) Time evolution of u (dashed line), v (dotted line), and the NEP (full line) during a phase-space
excursion. The scales of v and the NEP were adjusted for better comparison.
FIGURE 3. a) NEP in excitable regime; b) Stationary pdf in excitable regime.
signal) is [5]
Φ(t) = Φ{t,ui(t),vi(t)}=
(v2i −2β uivi−2Cvi)+
(β u2i +2Cui)
u2i −
u4i +Sg(t)ui
uiui+1
, (2)
FIGURE 4. a) NEP in bistable regime; b) Stationary pdf in bistable regime.
FIGURE 5. Time evolution of u for two neighbor neurons.
which must obey the integrability condition βλ1 +λ2/ε = 2λ [2].
Figure 2b depicts (in full line) the time evolution of the NEP during the phase-space
excursion starting at the upper initial condition in Fig. 2a, together with that of u (dashed
line) and v (dotted line). We remark that in Figs. 2a and 2b there is no noise and Φ(t) is
the Lyapunov functional of the deterministic dynamics. Figures 3a and 3b (respectively
4a and 4b) are 3D and contour plots of the NEP and the corresponding stationary pdf for
the excitable (respectively bistable) regime.
RESULTS FOR THE COUPLED SYSTEM
Synchronization between the coupled system and the external signal is observed above
some noise-intensity threshold. Figure 5 is a plot of the time evolution of ui (full line),
together with that of ui+1 (dashed line) for a given neuron i, showing their phase relation
to the signal (dotted line)1 According to Fig. 2, we may call “active” those cells i for
which ui(t) exceeds some threshold value uth. Because of the coupling, as one neuron
becomes active, it inhibits the activation of its nearest neighbors. The perfect alternance
seen in the figure may fail because of the noise, a necessary ingredient for the activation.
A detail of the alternance can be seen in Fig. 6 for an N′= 21 subset of the ring. Figure
6a shows a situation (snapshot) of poor synchronization, in which only two neurons are
active; Fig. 6b exhibits a case of a “kink” in the synchronized configuration, induced by
the fact that noises are local. Note that the kinks break locally the observed coherence,
and the complete history of the time evolution can be followed as a record of activity
(see Fig. 7).
1 The signal has been augmented in about two orders of magnitude and shifted to aid the sight.
FIGURE 6. Two snapshots of {ui} showing different degrees of synchrony.
FIGURE 7. Synchronization of the ring. White corresponds to activation and black to inhibition.
Horizontal dimension corresponds to time and vertical to space.
A measure of “activity” for the whole ring is
Ac(t) =
θ [ui(t)−uth]. (3)
In perfect synchrony, Ac = 0.5. Note that since the signal is subthreshold for the coupled
system, Ac = 0 below threshold. Figure 8a depicts the activity as a function of time
for a fixed noise intensity, showing again its phase relationship with the signal (dashed
line). In Fig. 8b we show the NEP for the whole ring as a function of time, together
with the (scaled) signal for reference. We remark that the observed dynamical symmetry
breakdown decreases the Lyapunov function of the whole ring with respect to that of the
homogeneous state, providing the route to stable synchronization.
A global estimator of synchronization can be defined as
∫ t f
0 Ac(t)dt
0.5N t f
. (4)
Figure 9a is a plot of Ga as a function of the noise intensity. The existence of a
threshold value of noise intensity and of a saturation effect can be clearly seen. The
noise intensities are low enough not to degrade the excitable dynamics.
FIGURE 8. a) Ac vs t for high synchronization; b) Time evolution of the NEP
FIGURE 9. a) Ga vs η ; b) C vs η .
Numerical simulations indicate that the coherence of firing decreases with the noise
intensity although the global activity (representative of global estimators) keeps the
order of magnitude. To quantify this phenomena we have calculated the normalized self-
correlation C = 〈uiui+2〉 as a function of the noise intensity η . As we show in Fig. 9b
the system shows a kind of “stochastic resonance in coherence” that cannot be inferred
from measures of global activity.
CONCLUSIONS
We have investigated the noise-induced synchronization with an external signal of a ring
of phase-repulsively coupled FHN elements. We have derived the exact NEP of the ex-
tended system and the observed symmetry breakdown was related with the Lyapunov-
functional properties of the NEP. We remark that the same conclusion holds qualita-
tively for the work in Ref. [1]. Although the observed phenomenon is noise-sustained
and global activity increases with noise intensity, a degradation of coherence can be
appreciated.
ACKNOWLEDGMENTS
Financial support from CONICET, ANPCyT and the National University of Mar del
Plata is acknowledged.
REFERENCES
1. G. Balázsi, A. Cornell-Bell, A. B. Neiman, and F. Moss, Phys. Rev. E 64, 041912 (2001).
2. G. Izús, R. Deza, and H. S. Wio. Phys. Rev. E 58, 93–98 (1998).
3. G. Izús, R. Deza, and H. S. Wio. Comp. Phys. Comm. 121–122, 406–407 (1999).
4. H. S. Wio, in Fourth Granada lectures in computational physics, P. L. Garrido and J. Marro, Eds.;
LNP 493 (Springer-Verlag, Berlin, 1997), p. 135.
5. A. Sánchez, G. Izús, and R. Deza, in preparation.
