Effects of asymmetry in an output function on the pinning of rotating waves in a ring neural...

Effects of asymmetry in an output function on the pinning of rotatingwaves in a ring neural oscillator with asymmetric bidirectionalcoupling and self-coupling

Yo Horikawa n

Faculty of Engineering, Kagawa University, Takamatsu 761-0396, Japan

a r t i c l e i n f o

Article history:Received 19 September 2013Received in revised form17 February 2014Accepted 5 March 2014Communicated by J. Torres

Keywords:Ring neural networkRing oscillatorTraveling wavePinningSelf-coupling

a b s t r a c t

Effects of asymmetry in a sigmoidal output function of neurons on rotating waves in a ring of neuronswith asymmetric bidirectional coupling and self-coupling were studied. Propagation of wave frontsin rotating waves failed, i.e., the pinning of rotating waves occurred not only when self-coupling wasexcitatory but also when self-coupling was inhibitory, in which there were unsaturated neurons at wavefronts. Conditions for the pinning of wave fronts were derived by using a piecewise linear outputfunction. The pinning conditions depended on whether bidirectional coupling was excitatory orinhibitory in the presence of asymmetry in an output function.

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1. Introduction

A ring of coupled neurons with sigmoidal input–output rela-tions, which is referred to as a ring neural network, has attractedmuch attention. A sigmoidal neuron is a firing rate model of aneuron or neural assembly [1–3] and it is widely used in artificialneural networks. A ring of unidirectionally coupled sigmoidalneurons can show stable oscillations when the number of inhibi-tory connections is odd [4]. The oscillations of the states ofneurons are due to a rotating wave propagating in a ring. Such aring is qualitatively the same as a ring oscillator, which is a closedloop of inverters and buffers, and this type of a network is widelyused as a variable-frequency oscillator [5]. Although the struc-ture of a ring neural network is simple, its properties have beenstudied as a basic model of a recurrent neural network [6,7] andmathematically as a cyclic feedback system [8]. The formation ofspatiotemporal patterns in one-dimensional and two-dimensionalarrays of piecewise linear neurons with local coupling and theirapplication to signal processing have been studied as a cellularneural network [9–14]. Further, a lot of work has been carried outon effects of delays on spatiotemporal patterns in a ring ofsigmoidal neurons, e.g., see references in [15]. Although most of

their studies are restricted to rings of small numbers (from two tosix), studies on rings of general (not specific) numbers of neuronshave been carried out [16–18]. It has been shown that delays causelong-lasting rotating waves and oscillations in a ring of unidir-ectionally coupled neurons [19–21]. It has also been shown thattransient rotating waves in a ring with unidirectional coupling aredynamically metastable even in the absence of delays, i.e. theirduration increases exponentially with the number of neurons [22].Effects of inertial terms [23], spatiotemporal noise and asymmetryin a sigmoidal output function of neurons [24] on metastabledynamical transient rotating waves in a ring of unidirectionallycoupled neurons have then been studied.

Such propagating waves in rings of coupled systems havedrawn a lot of interest in various fields and have been widelystudied. The existence and stability of traveling waves in latticedynamical systems and coupled map lattices have been studiedextensively and we just mention some review papers [25–27].Concerning neural networks, rings of synaptically coupled spikingneurons have been employed as models of central pattern gen-erators in the central nervous system, e.g., for early work [28,29].Then, a lot of work has been carried out on rings of various kindsof neuron models, e.g., see references in [15,30]. Discrete-timedynamics of ring neural networks has also been studied [31,32].Further, wave propagation in a large population of neurons inthe brain, e.g., the cortex, hippocampus and thalamus, havebeen examined by neural field models, which are described by

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n Corresponding author. Tel.: þ81 87 864 2211; fax: þ81 87 864 2262.E-mail address: [email protected]

Please cite this article as: Y. Horikawa, Effects of asymmetry in an output function on the pinning of rotating waves in a ring neuraloscillator with asymmetric bidirectional coupling and..., Neurocomputing (2014), http://dx.doi.org/10.1016/j.neucom.2014.03.036i

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integro-differential equations with sigmoidal functions of neuralactivity; for review [33,34]. Apart from the nervous system andneural networks, studies on traveling waves as splay-phase syn-chronized oscillations in rings of coupled nonlinear oscillators (vander Pol oscillators) date back to [35–37]. Then, a lot of work onrings of coupled various kinds of oscillators has been carried out inelectronics, physics and biology; for recent work, e.g., overdampedDuffing oscillators [38,39], the Stuart–Landau oscillators [40–42]and delayed-feedback optoelectronic oscillators [43]. For example,rings of chaotic oscillators have been examined in relation to chaossynchronization. Then, not only periodic rotating waves but alsoquasiperiodic, chaotic and hyperchaotic rotating waves have beenobserved in rings of unidirectionally coupled Chua's circuits [44–48],Lorenz systems [44,46,47,49–52] and Duffing oscillators [53].Transient chaotic rotating waves have also observed in rings ofunidirectionally coupled Bonhoeffer–van der Pol oscillators [54]and Lorenz systems [55]. Long-lasting metastable dynamicaltransient rotating waves, which emerge in a ring of unidirection-ally coupled sigmoidal neurons as mentioned above, have beenfound in rings of unidirectionally coupled various systems: over-damped Duffing oscillators [56], cubic maps [57], parametricoscillators [58], Lorenz systems [55] and Bonhoeffer–van der Polmodels [30]. For practical applications, these metastable dynami-cal rotating waves have been reported in a ring of ferromagneticcores [59] and a generalized repressilator model, which is one ofgenetic regulatory networks [60,61], as well as traffic jams in a car-following model for a traffic flow problem [62].

In this paper, we consider a ring of sigmoidal neurons with asym-metric bidirectional coupling and self-coupling as well as asymmetryin an output function. We then study conditions for the pinning(propagation failure) of wave fronts, which seem not to have beenexamined. A ring with unidirectional coupling is special structureand bidirectional coupling and self-coupling are generally included inneural networks. When bidirectional coupling or self-coupling exists,the pinning of rotating waves can occur, i.e. the propagation of theirwave fronts fails and rotating waves change into steady states. Suchpinning of wave propagation is commonly seen in spatially discretecoupled systems [63–68].

When a sigmoidal output function is symmetric and piecewiselinear, conditions for the pinning of wave fronts in the presence ofexcitatory (positive) self-coupling have been derived [12,13]. Wavefronts are pinned when the strength of excitatory self-coupling islarger than asymmetry in the strength of bidirectional coupling.Excitatory self-coupling tends to keep the states of neuronsunchanged. Then the stability of steady states increases and wavefronts are hard to propagate. It has also been shown that thepinning of rotating waves occurs even when self-coupling isinhibitory (negative) [15]. Wave fronts in rotating waves and theirpinned states have neurons the sigmoidal output of which is notsaturated, which correspond to steady states obtained for sym-metric bidirectional coupling [10,12]. Inhibitory self-couplingtends to change the signs of the states of neurons and thus makeswave fronts propagate. However, the states of neurons at wavefronts become unsaturated so that pinning can occur.

Rotating waves, which are caused by qualitatively the samemechanism as those in a ring of coupled sigmoidal neurons andshow metastable transient dynamics, have also been shown toemerge in a ring of synaptically coupled Bonhoeffer–van der Polmodels [30], i.e., spiking neuron models. Adjacent neurons arecoupled unidirectionally with slow inhibitory synapses. In itssteady state, neurons in a firing state and a resting state arealternately located. The rotating waves take the form of propagat-ing oscillations, in which successive two neurons are in the samestates (resting–resting or firing–firing) and the location of theinconsistency propagates in the direction of coupling. The firingand resting states of a spiking neuron correspond to a positive and

negative steady states of a sigmoidal neuron, respectively. The twostates of a spiking neuron differ qualitatively and the differencecan be modeled by asymmetry in the output of a sigmoidalneuron. Thus, effects of asymmetry in a sigmoidal neuron are ofimportance in order to study the pinning of rotating waves in ringsof such spiking neurons with asymmetric bidirectional couplingand self-coupling.

We derived conditions for the pinning of wave fronts in thepresence of asymmetry in a sigmoidal output function by usinga piecewise linear output function. There were multiple separatedpinned regions in a plane of the strengths of asymmetric bidirec-tional coupling and self-coupling, in which the numbers of unsatu-rated neurons at wave fronts were different with each other.Effects of asymmetry in an output function depended on whetherbidirectional coupling was excitatory or inhibitory. When bidirec-tional coupling was excitatory, pinned regions shifted monotoni-cally along the axis of the strength of asymmetric bidirectionalcoupling as asymmetry in an output function increased. Wavefronts were pinned when asymmetry in an output function wascompensated with asymmetric bidirectional coupling. When bidir-ectional coupling was inhibitory, changes in pinned regions weremore complicated and depended on the parity of the number ofunsaturated neurons at wave fronts. When the number of unsa-turated neurons at a wave front was odd, pinned regions shiftedalong the axis of the strength of asymmetric bidirectional couplinglike the shifts for excitatory coupling, but in a different manner.When the number of unsaturated neurons was even, the size ofpinned regions changed along the axis of the strength of self-coupling. Then rotating waves were still always pinned whenbidirectional coupling was symmetric. Conditions for the pinningalso changed when one of positive and negative steady states wasunsaturated, in which only wave fronts with even numbers ofunsaturated neurons existed.

In the rest of the paper, a model equation of a ring of sigmoidalneurons with asymmetric bidirectional coupling, self-couplingand an asymmetric sigmoidal output function is introduced inSection 2. The bifurcations of its steady states and limit cyclesare explained and the patterns of rotating waves are shown.In Sections 3 and 4, rings with excitatory and inhibitory bidirec-tional coupling are dealt with, respectively. It is shown that thereare wave fronts having neurons in unsaturated states when self-coupling is inhibitory. Then, conditions for the pinning of wavefronts are derived by using a piecewise linear output function.Finally, conclusion and future work are given in Section 5.In Appendix A, the existence and stability of steady solutions toassociated linear differential equations for unsaturated neuronsare shown.

2. A ring of sigmoidal neurons and rotating waves

We consider the following ring of sigmoidal neurons withasymmetric bidirectional coupling and self-coupling.

dxn=dt ¼ �xnþcð1þdÞ

2f ðgxn�1Þþ

cð1�dÞ2

f ðgxnþ1Þþsf ðgxnÞf ðxÞ ¼ f1�exp½�2x=ð1�e2Þg=f1=ð1þeÞþexp½�2x=ð1�e2Þ�=ð1�eÞgð1rnrN; xn7N ¼ xn; g40; da0; �1oeo1Þ ð1Þwhere xn is the state of the nth neuron, f is an output function of aneuron with an asymmetric shift e, g is an output gain, c is thestrength of bidirectional coupling between adjacent neurons, d isasymmetry in bidirectional coupling, and s is the strength of self-coupling. A periodic boundary condition is imposed so that a totalof N neurons make a closed loop. The asymptotic values of f atinfinity of x are f(x)-1þe (40) (x-1) and f(x)-�1þe (o0)(x-�1), while f(x)¼tanh(x) when e¼0. The origin (xn¼0

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(1rnrN)) is always a steady state of Eq. (1) since f(0)¼0. Theeigenvalues λk of the Jacobian matrix evaluated at the origin aregiven by

λk ¼ �1þg½sþc cos ð2kπ=NÞþ icd sin ð2kπ=NÞ� ð0rkoNÞ ð2Þ

since f'(0)¼1. The eigenvalues are located on an ellipse withcenter at (�1þgs, 0), horizontal axis of 2g|c| and vertical axis of2g|c|d in the complex plane. When bidirectional coupling isasymmetric (da0), a limit cycle is generated from the origin asg increases in the following manner.

When c40, the eigenvalue with the largest real part isλ0¼�1þg(sþc). The origin is stable when go1/(sþc) and it isdestabilized at g¼1/(sþc) when sþc40 through a supercriticalpitchfork bifurcation for e¼0. A pair of stable spatially uniformsteady states (xn40 and xno0 (1rnrN)) is then generated.When ea0, a pitchfork bifurcation breaks up into a transcriticalbifurcation at the origin and a saddle-node bifurcation of agenerated unstable steady state. There is a pair of stable spatiallyuniform steady states for g41/(sþc). A further increase ing causes a supercritical Hopf bifurcation of the origin at g¼1/[sþc cos (2π/N)] when sþc cos (2π/N)40 and NZ3. The gener-ated unstable limit cycle is a traveling wave rotating in a ring withthe wave number k¼1.

When co0 and the number of neurons is even (N¼2M), theeigenvalue with the largest real part is also real: λN/2¼�1þg(sþ |c|).The origin is destabilized at g¼1/(sþ |c|) when sþ |c|40 through asupercritical pitchfork bifurcation irrespective of e. A pair of gener-ated stable steady states has the form of xn40 for odd (even) n andxno0 for even (odd) n. The origin causes a supercritical Hopfbifurcation at g¼1/[sþ |c| cos (2π/N)] when sþ |c| cos (2π/N)40 andNZ4 so that an unstable limit cycle (a traveling wave with the wavenumber k¼N/2–1) is generated. In both cases, Eq. (1) becomes anunstable ring oscillator, in which the states of neurons oscillateaccording to the passage of the wave fronts. It has been shown thatthe traveling wave can be stabilized for |d|4 |c| when e¼0 [12,14],and for s4 |c| when e¼0 and gb1 [15].

When co0 and the number of neurons is odd (N¼2Mþ1), theeigenvalue with the largest real part in Eq. (2) is a pair of complexconjugate: λ(N71)/2¼–1þg{sþc cos [(N71)π/N]þ icd sin [(N71)π/N]}. The origin is stable when go1/{sþc cos [(N71)π/N]} (¼1/[sþ |c| cos (π/N)]). A stable limit cycle is first generated from theorigin through a supercritical Hopf bifurcation at g¼1/[sþ |c| cos(π/N)] when sþ |c| cos (π/N)40, which is a traveling wave rotatingin a ring with the wave number k¼(N�1)/2. Eq. (1) then becomesa stable ring oscillator.

Fig. 1 shows examples of spatiotemporal patterns of rotatingwaves in Eq. (1) with c¼1.0 and N¼20 (a transient rotating wave) (a),c¼�1.0 and N¼20 (an unstable traveling wave) (b), and c¼�1.0and N¼11 (a stable traveling wave) (c). The values of otherparameters are g¼10.0, d¼0.5, s¼0.2 and e¼0.1. A top panelshows a time course of the state x1(t) of the first neuron.A middle panel shows a spatiotemporal patterns of the states xn(t)(1rnrN), in which black and white regions correspond to positiveand negative signs, respectively. A bottom panel showsa snapshot of xn(t) (1rnrN) at t¼20.0. They were obtained withcomputer simulation of Eq. (1) using the Runge–Kutta method with atime step 0.001. The initial conditions were xn¼cþs (1rnr3),xn¼�(cþs) (4rnrN) (a); xn¼(�1)n(|c|þs) (1rnrN/2),xn¼(�1)nþ1(|c|þs) (N/2þ1rnrN) (b); and xn�N(0, 0.12)(1rnrN) (Gaussian random) (c).

When c40 (a), the states of neurons are divided into twopositive and negative bumps. We refer to the boundary betweenbumps with xno0 and xnþ140 as a kink and the boundary withxn40 and xnþ1o0 as an antikink. Both propagate rightward, butthe kink propagates slower than the antikink so that the states of

all neurons become positive eventually (one of stable spatiallyuniform states). When co0 and N is even (b), the states ofneurons have positive and negative signs alternately except for apair of successive two states of the same sign. We refer toinconsistency with xno0 and xnþ1o0 as a kink and inconsistencywith xn40 and xnþ140 as an antikink. They propagate rightwardin the same speed by varying between a kink and an antikinkwhen they are equally separated, i.e., distance between themis N/2. However, this traveling wave is unstable and a kink andan antikink collapse eventually when their separation is unequal.The states of neurons then become positive and negative alter-nately without inconsistencies, which is a stable steady state.When co0 and N is odd (c), the states of neurons are positiveand negative alternately except for one inconsistency (successivetwo states of the same sign), which propagates rightward.This inconsistency never disappears and a ring shows a stabletraveling wave.

Fig. 1. Spatiotemporal patterns of the rotating waves in Eq. (1) with c¼1.0 andN¼20 (a transient rotating wave) (a), c¼�1.0 and N¼20 (an unstable travelingwave) (b), and c¼�1.0 and N¼11 (a stable traveling wave) (c), where g¼10.0,d¼0.5, s¼0.2 and e¼0.1. Time course of the state x1(t) of the first neuron (toppanel); spatiotemporal patterns of the states xn(t) (1rnrN), black: positive,white: negative; snapshot of xn(t) (1rnrN) at t¼20.0 (bottom panel).

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When asymmetry in bidirectional coupling is small (d-0),these kinks or antikinks cannot propagate, i.e., they are pinned.In the following, we derive conditions for the pinning of kinks andantikinks by using the following piecewise linear function fL(x) forf(x) in Eq. (1).

f LðxÞ ¼ �1þe ðxo�1þeÞx ð�1þeoxo1þeÞ1þe ðx41þeÞ ð3Þ

The steady states and the traveling waves exist in the sameparameter range as noted above. When a sigmoidal outputfunction is symmetric (e¼0), conditions for the pinning of kinksand antikinks in the s–d plane have been derived [15]. Whena sigmoidal output function is asymmetric (ea0), conditions fortheir pinning depend on whether bidirectional coupling is excita-tory (c40) or inhibitory (co0). We deal with rings with excita-tory and inhibitory bidirectional coupling in Sections 3 and 4,respectively.

3. Excitatory bidirectional coupling

In this section, we deal with a ring of neurons with excitatorybidirectional coupling (c40) in Eq. (1). We let s �1/g4–c cos(2π/N) (Re[λ1]¼–1þg[sþc cos (2π/N)]40) so that a pair of stablespatially uniform steady states and an unstable traveling waveexists. The values of spatially uniform steady states of Eq. (1) withf(x)¼ fL(x) are given by xeþ¼(cþs)(1þe) (4(1þe)/g) andxe�¼(cþs)(�1þe) (o(�1þe)/g). A rotating wave or its pinnedsteady state consists of a kink and antikink connecting positiveand negative bumps (Fig. 1(a)). Shapes of a kink and antikinkdepend on the values of coupling strength c, d and s as well as anasymmetric shift e in f(x). Fig. 2 shows spatial patterns of kinks andantikinks in Eq. (1) with g¼100.0, c¼1.0 and e¼0.1. In (a) (s¼0.5,d¼0.1 (kink), �0.1 (antikink)), the output of neurons xn and xnþ1

composing a kink (or an antikink) is saturated, i.e., f(gxn)E–1þe

(xno(–1þe)/g) and f(gxnþ1)E1þe (xnþ14(1þe)/g). Note thatthe location of xn (the leftmost neuron in a kink or an antikink) isdenoted by n. In (b) (s¼�0.3, d¼0.1 (kink), �0.1 (antikink)), theoutput of one neuron xnþ1 in a kink (or an antikink) is unsatu-rated, i.e., f(gxnþ1)Egxnþ1 ((�1þe)/goxnþ1o(1þe)/g), which isalmost zero since 1/g¼0.01. There are two and three unsaturatedneurons in kinks (or antikinks) in (c) (s¼�0.6, d¼0.05 (kink),–0.05 (antikink)) and (d) (s¼�0.75, d¼0.05 (kink), �0.05 (anti-kink)), respectively. It will be shown that such kinks and antikinksincluding unsaturated neurons exist when s �1/go0. We referto a kink and an antikink with m unsaturated neurons (xnþ j

(1r jrm)) as a type-m kink and a type-m antikink, respectively.In the following, conditions for the pinning of kinks are derived byusing a piecewise linear function fL(x) in Eq. (3) for f(x). Pinningconditions for antikinks are obtained by changing the sign ofasymmetry in bidirectional coupling (d-�d) or by changing thesign of asymmetry in an output function (e-�e).

3.1. Type-0 kink

We consider a condition for the pinning of a type-0 kink. Letxn�1o(�1þe)/g, xno(�1þe)/g, xnþ14(1þe)/g and xnþ24(1þe)/g with fL(gxn�1)¼ fL(gxn)¼�1þe and fL(gxnþ1)¼fL(gxnþ2)¼1þe so that xn and xnþ1 compose a type-0 kink(Fig. 2(aL)). The states change as

dxn=dt ¼ �xnþce�cdþsð�1þeÞ ¼ �xnþx0ndxnþ1=dt ¼ �xnþ1þce�cdþsð1þeÞ ¼ �xnþ1þx0nþ1 ð4Þ

where x0n and x0nþ1 are stable steady states. The kink cannotpropagate when xn and xnþ1 remain negatively and positivelysaturated, respectively. A condition for the pinning of a type-0 kinkis thus given by

x0noð�1þeÞ=g 2 cðe�dÞþs0ð�1þeÞo0

x0nþ14 ð1þeÞ=g 2 cðe�dÞþs0ð1þeÞ40 ðs0 ¼ s�1=gÞ ð5Þ

In the following, we let s'¼s�1/g. A condition for the pinningof a type-0 antikink is derived in the same manner, and it is givenby letting d-�d or e-�e in Eqs. (4) and (5) as noted above.

Fig. 3 shows the pinned regions of kinks and antikinks in thes'–d plane (s'¼s�1/g) for c¼1.0 and e¼0.1. The boundaries(x0n ¼ ð�1þeÞ=g, x0nþ1 ¼ ð1þeÞ=g) between propagating andpinned regions of a type-0 kink and antikink are plotted with

Fig. 2. Spatial patterns of type-m kinks and antikinks in a ring of excitatorybidirectional coupling in Eq. (1) with g¼100.0, c¼1.0 and e¼0.1; m¼0, s¼0.5,d¼70.1 (a), m¼1, s¼�0.3, d¼70.1 (b), m¼2, s¼�0.6, d¼70.05 (c), m¼3,s¼�0.75, d¼70.05 (d). Left: d40 for kinks; right: do0 for antikinks.

Fig. 3. Pinned regions of kinks (solid lines) and antikinks (dashed lines) in a ring ofexcitatory bidirectional coupling in Eq. (1) with fL(x), c¼1.0 and e¼0.1 in the s'–dplane (s'¼s�1/g).






solid and dashed lines, respectively. Eq. (5) corresponds to theright-hand side region of the two lines crossing the d-axis at (s',d)¼(0, e) for a kink or (s', d)¼(0, �e) for an antikink, which isdenoted by PK0 for a kink or PA0 for an antikink. The pinned regionsof a kink and antikink are reflection across the s'-axis (d¼0) witheach other. Two bumps (a pair of a kink and an antikink) remainonly in an overlapping region of PK0 and PA0 (|d|o(1�e)s'/c�e).A kink propagates in the direction of the ascending order of n (1-N)(rightward in Fig. 2) in the region over two lines (x0noð�1þeÞ=g,x0nþ1o ð1þeÞ=g). That is, xnþ1 becomes unsaturated and thenbecomes negatively saturated so that a kink continues to propa-gate. Strictly, we have to consider a change in an unsaturated xnþ1,but xnþ1 does not remain unsaturated under x0no ð�1þeÞ=gbecause of the monotonicity of f(x). In the same manner, a kinkpropagates in the inverse direction (N-1) (leftward in Fig. 2) inthe region below two lines (x0n4 ð�1þeÞ=g, x0nþ14ð1þeÞ=g).Thus, a kink propagates when one of inequalities in Eq. (5) isunsatisfied and the other is satisfied. Then, in the left-hand side oftwo lines (x0n4ð�1þeÞ=g, x0nþ1o ð1þeÞ=g; (1þe)s'/cþeodo(–1þe)s'/cþe) in s'o0, i.e., when both inequalities in Eq. (5) areunsatisfied, both states (xn, xnþ1) move to be unsaturated anda type-1 kink can be generated.

3.2. Type-1 kink

We then consider conditions for the existence and pinning of atype-1 kink, which has one unsaturated neuron. As noted above, anecessary condition for the existence of a type-1 kink is given byreversing both inequalities in a condition for the pinning of a type-0 kink Eq. (5). Thus the existence condition is x0n4 ð�1þeÞ=g andx0nþ1o ð1þeÞ=g. Then, let xn�1 and xn be negatively saturated, letxnþ2 and xnþ3 be positively saturated, and let xnþ1 be unsaturated(fL(gxnþ1)¼gxnþ1) so that xn, xnþ1 and xnþ2 compose a type-1 kink(Fig. 2(bL)). The unsaturated state xnþ1 changes as

dxnþ1=dt ¼ �xnþ1þce�cdþs gxnþ1 ð6Þ

The above existence condition for a type-1 kink agrees with thecondition that Eq. (6) has a stable unsaturated steady solution x1nþ1satisfying

x1nþ1 ¼ cðd�eÞ=ðs0gÞ; �1þeogx1nþ1o1þe ðs0 ¼ s�1=go0Þð7Þ

Note that the solution x1nþ1 is stable since s'o0.When xnþ1 is unsaturated (xnþ1¼x1nþ1), the states xn and xnþ2

change as

dxn=dt ¼ �xnþcð1þdÞð�1þeÞ=2þc2ð1�dÞðd�eÞ=ð2s0Þþsð�1þeÞ ¼ �xnþx1n

dxnþ2=dt ¼ �xnþ2þc2ð1þdÞðd�eÞ=ð2s0Þþcð1�dÞð1þeÞ=2þsð1þeÞ ¼ �xnþ2þx1nþ2 ð8Þ

A condition for the pinning of a type-1 kink is thus given by

x1no ð�1þeÞ=g 2 cð1þdÞð�1þeÞ=2þc2ð1�dÞðd�eÞ=ð2s0Þþs'ð�1þeÞo0

x1nþ24 ð1þeÞ=g 2 c2ð1þdÞðd�eÞ=ð2s'Þþcð1�dÞð1þeÞ=2þs0ð1þeÞ40 ð9Þ

The boundaries (x1n ¼ ð�1þeÞ=g, x1nþ2 ¼ ð1þeÞ=g) of pinnedregions of a type-1 kink and antikink are plotted in Fig. 3 withsolid and dashed curves crossing the d-axis at (s', d)¼(0, e) and(0, �e), respectively. Each pinned region of a type-1 kink and antikinkis bounded by two curves, which is denoted by PK1 and PA1,respectively. A kink propagates in the direction of the ascendingorder of n (1-N) in the region over two curves (x1no ð�1þeÞ=g,x1nþ2o ð1þeÞ=g), while it propagates in the inverse direction

(N-1) in the region below two curves (x1n4ð�1þeÞ=g, x1nþ24ð1þeÞ=g). In the left-hand side of two curves (x1n4 ð�1þeÞ=g,x1nþ2o ð1þeÞ=g), both states (xn, xnþ2) move to be unsaturated,and type-m kinks (mZ2) can be generated.

3.3. Type-m kink

Next, we consider conditions for the existence and pinningof a type-m kink (mZ2). Assume that a type-(m�1) kink xnþ j

(0r jrm) exists, where xnþ j (1r jrm�1) are unsaturated.A necessary condition for the existence of a type-m kink is thatboth xn and xnþm move to be unsaturated. An expression for thisexistence condition is described later.

Then, let xn�1 and xn negatively saturated, let xnþmþ1 andxnþmþ2 positively saturated, and let xnþ j (1r jrm) be unsatu-rated (Fig. 2(cL, dL) for m¼2, 3) so that xnþ j (0r jrmþ1)compose a type-m kink. Changes in the states of m unsaturatedneurons are described as

dxnþ1=dt ¼ �xnþ1þcð1þdÞ

2ð�1þeÞþcð1�dÞ

2gxnþ2þsgxnþ1

dxnþ j=dt ¼ �xnþ jþcð1þdÞ

2gxnþ j�1þ

cð1�dÞ2

gxnþ jþ1þsgxnþ j

ð2r jrm�1Þ

dxnþm=dt ¼ �xnþmþcð1þdÞ2

gxnþm�1þcð1�dÞ

2ð1þeÞþsgxnþm ð10Þ

A necessary condition for the existence of a type-m kink isequivalent to the condition that Eq. (10) has a stable unsaturatedsteady solution xmnþ j (1r jrm) satisfying

cð1þdÞ2

ð�1þeÞþcð1�dÞ2

gxmnþ2þs0gxmnþ1 ¼ 0

cð1þdÞ2

gxmnþ j�1þcð1�dÞ

2gxmnþ jþ1þs0gxmnþ j ¼ 0 ð2r jrm�1Þ

cð1þdÞ2

gxmnþm�1þcð1�dÞ

2ð1þeÞþs0gxmnþm ¼ 0

�1þeogxmnþ jo1þe ð1r jrmÞ ð11Þ

Since Eq. (11) is satisfied when xnþ1 or xnþm becomes unsatu-rated under the condition that the other m�1 neurons areunsaturated (a type-(m�1) kink exists), the existence conditionis intrinsically given by gxmnþ14�1þe and gxmnþmo1þe. Notethat the stability of a solution to Eq. (10) depends on the largesteigenvalue of the coefficient matrix in Eq. (10), and the stabilitycondition is shown in Appendix A.

Under Eq. (11) (–1þeogxnþ j¼gxmnþ jo1þe (1r jrm)), thestates xn and xnþmþ1 change as



2gxmnþ1þsð�1þeÞ ¼ �xnþxmn

dxnþmþ1=dt ¼ �xnþmþ1þcð1þdÞ

2gxmnþm

þcð1�dÞ2

ð1þeÞþsð1þeÞ ¼ �xnþmþ1þxmnþmþ1

ð12ÞA condition for the pinning of a type-m kink is thus given by

xmn o ð�1þeÞ=g 2cð1þdÞ


2gxmnþ1þs0ð�1þeÞo0

xmnþmþ14 ð1þeÞ=g 2cð1þdÞ

2gxmnþmþcð1�dÞ

2ð1þeÞþs0ð1þeÞ40

ð13ÞIn Fig. 3, the boundaries (xmn ¼(�1þe)/g, xmnþmþ1¼(1þe)/g) of

pinned regions of type-m kinks and antikinks for 2rmr4 areplotted with solid and dashed curves, respectively. Each pinnedregion of a type-m kink (PKm) and antikink (PAm) is boundedby two curves. Two boundary curves extend to (s', d)¼(�c, c)






or (�c, �c), and the left-hand side region of the two curves satisfya necessary condition for the existence of a type-(mþ1) kink orantikink. The leftmost point of a pinned region of a type-m kink(antikink) is connected to the rightmost point of a pinned region ofa type-(mþ1) kink (antikink). The pinned regions for kinks andantikinks shift upward (d40) and downward (do0), respectively.There are no overlap between the pinned regions of a type-m kinkand antikink for mZ2 so that a pair of pinned ones cannot existwhen e¼0.1.

A necessary condition for the existence of a type-(mþ1) kink isgiven by reversing inequalities in Eq. (13): xmn 4 ð�1þeÞ=g,xmnþmþ1o ð1þeÞ=g so that both states move to be unsaturated.A necessary condition for the existence of a type-m kink is thengiven by reversing inequalities in Eq. (13) with m-m�1: xm�1

n o(–1þe)/g and xm�1

nþm4(1þe)/g, where xm�1n and xm�1

nþm are stableunsaturated solutions to Eq. (10) with m-m�1.

Up to m¼N/2–2 pinned regions of type-m kinks and antikinksexist in a ring of N neurons (N¼2MZ4) since a stable pinned statehas at least two positively saturated and two negatively saturatedneurons when s'oc [10,12]. When an output function of a neuronis symmetric (e¼0), the pinned regions of kinks and antikinks arethe same and are symmetric with respect to the s'-axis (d¼0) [15],which are shown in Fig. A1 in the Appendix A. The pinningconditions are also the same for excitatory (c40) and inhibitory(co0) bidirectional coupling. It has been shown that a pinnedtype-m kink and antikink exist in –|c| cos [π/(mþ2)]os'o–|c| cos[π/(mþ1)] (mZ1) on the s'-axis (d¼0) when e¼0 [10,12]. Then,the pinned region of a type-(m�1) kink connects to that of a type-mkink at (s', d)¼(–|c| cos [π/(mþ1)], 0) (mZ1).

3.4. Equivalent pinning condition

A condition for the pinning of a type-(m�1) kink is equivalentto the condition that Eq. (10) for m unsaturated neurons in a type-m kink has an unstable unsaturated steady solution xmnþ j(�1þeogxmnþ jo1þe) (1r jrm). This pinning condition is given by Eq.(11) (intrinsically gxmnþ14�1þe and gxmnþmo1þe) for mZ2,which is apparently the same as the existence condition for atype-m kink, but the unsaturated solution must be unstable. Whenm¼1, the condition is given by Eq. (7) with s'40. There is anunstable unsaturated steady solution x1nþ1 to Eq. (6) in the pinnedregion of a type-0 kink bounded by two lines ((�1þe)s'/cþeodo(1þe)s'/cþe (s'40)) in Fig. 3. Steady solutions toEq. (10) and their stability are shown in Appendix A.

An intuitive explanation for this pinning condition is as follows.When m¼1, if the unstable unsaturated solution x1nþ1 to Eq. (6)exists, xnþ1 can be either positively or negatively saturated.A pinned type-0 kink is generated accordingly. In other words,neither positively nor negatively saturated xnþ1 can be unsatu-rated since there is an unstable unsaturated solution. When mZ2,if an unstable unsaturated steady solution to Eq. (10) exists(�1þeogxmnþ jo1þe, 1r jrm), either xnþ1 becomes negativelysaturated or xnþm becomes positively saturated. A pinned type-(m�1)kink (xnþ j (2rjrm) or (1rjrm�1)) is generated accordingly.

4. Inhibitory bidirectional coupling

In this section, we deal with a ring of neurons with inhibitorybidirectional coupling (co0) in Eq. (1) with f(x)¼ fL(x). When thenumber of neurons is even (N¼2M), a stable steady state consist-ing of alternate positive and negative states of neurons, e.g., xn40for odd n and xno0 for even n, exists when s' (¼s�1/g)4–|c|(λN/2¼–1þg(s�c)40). When the output of all neurons is saturated,the values of positive and negative states in the steady stateare given by xiþ¼c(�1þe)þs(1þe)¼ |c|þsþe(s� |c|) and xi�¼c

(1þe)þs(�1þe)¼–(|c|þs)þe(s� |c|), respectively. A conditionthat the steady states remain saturated is given by xiþ4(1þe)/gand xi–o(�1þe)/g, which leads to

s04cð1�jejÞ=ð1þjejÞ ¼ s0uo0 ð14ÞWhen s'os'u (o0), one of the positive and negative steady

states is unsaturated as

xiuþ ¼ cð1�eÞ=ðs0gÞ ð0oxiuþ o ð1þeÞ=gÞxis� ¼ cgxiuþ þsð�1þeÞ ¼ ð�1þeÞðc2=s0 �sÞo ð�1þeÞ=gðs0 ¼ s�1=gÞ for e40 ð15Þ

xiu� ¼ �cð1þeÞ=ðs0gÞ ðð�1þeÞ=goxiu� o0Þxisþ ¼ cgxiu� þsð1þeÞ ¼ ð1þeÞð�c2=s'þsÞ4 ð1þeÞ=gðs0 ¼ s�1=gÞ for eo0 ð16Þ

We let s'4c cos (2π/N) (Re[λN/271]¼–1þg[s�c cos (2π/N)]40)for even N and let s'4c cos (π/N) (Re[λ(N71)/2]¼–1þg[s�c cos(π/N)]40) for odd N so that a traveling wave exist. A propagatingwave or its pinned steady state consists of two inconsistencies(a kink and an antikink) when the number N of neurons is even(Fig. 1(b)) or one inconsistency (a kink or an antikink) when N isodd (Fig. 1(c)) Since a kink and antikink interchange by propagat-ing over one neuron, conditions for their pinning can be dealt withirrespective of the parity of N. As a ring with excitatory bidirec-tional coupling in Section 3, kinks and antikinks with unsatu-rated neurons exist when s'o0. We refer to inconsistency (xnþ j

(0r jrmþ1)) with m unsaturated neurons (xnþ j (1r jrm)) as atype-m kink when xno(–1þe)/g (o0) and a type-m antikinkwhen xn4(1þe)/g (40). Further, pinning conditions for kinks andantikinks also depend on whether both positive and negativesteady states are saturated (s'4s'u) or one of them is unsaturated(s'os'u). We then refer to a kink and antikink with m unsaturatedneurons as type-um when s'os'u.

Fig. 4 shows spatial patterns of kinks and antikinks forg¼100.0, c¼�1.0. The location of xn (the leftmost neuron in akink or an antikink) is denoted by n. When both steady states aresaturated (e¼0.1, s'uE�0.818), both type-m kinks and antikinksexist as m¼0 (a), 1 (b), 2 (c) and 3 (d). In contrast with a ring withexcitatory bidirectional coupling (c40), the signs of the statesof the leftmost (xn) and rightmost (xnþmþ1) neurons in type-minconsistency are different when m is odd. When e40 and thepositive steady state is unsaturated (0ogxiuþo1þe), only type-um kinks with even m exist as m¼0 (e), 2 (f) and 4 (g) (e¼0.5,s'u¼�1/3), and antikinks cannot exist. When eo0 and thenegative steady state is unsaturated (�1þeogxiu–o0), onlytype-um antikinks with even m exist and kinks cannot exist inthe same manner. It is because the states of neurons cannot bepositively saturated when e40 and cannot be negatively satu-rated when eo0.

In the following, conditions for the pinning of the propagationof type-m inconsistencies (mZ0) are derived by using fL(x) inEq. (3) for f(x). Pinning conditions for type-m antikinks areobtained from pinning conditions for type-m kinks by changingthe sign of asymmetry in an output function (e-�e). Pinningconditions for type-um antikinks are the same as those for type-um kinks. These pinning conditions depend on the parity of thenumber m of unsaturated neurons because of the difference in theshape of inconsistency.

4.1. Type-0 inconsistency

First, we consider type-0 inconsistency when the steadystates are saturated (s' (¼s�1/g)4s'u). Let xn�14(1þe)/g, xno(�1þe)/g, xnþ1o(�1þe)/g and xnþ24(1þe)/g with fL(gxn)¼fL(gxnþ1)¼�1þe and fL(gxn�1)¼ fL(gxnþ2)¼1þe so that xn and






xnþ1 compose a type-0 kink (Fig. 4(aL)). The states change as

dxn=dt ¼ �xnþceþcdþsð�1þeÞ ¼ �xnþxk0n

dxnþ1=dt ¼ �xnþ1þce�cdþsð�1þeÞ ¼ �xnþ1þxk0nþ1 ð17Þ

A kink cannot propagate when both xn and xnþ1 remainnegatively saturated. A condition for the pinning of a type-0 kinkis thus given by

xk0n o ð�1þeÞ=g 2 cðeþdÞþs0ð�1þeÞo0

xk0nþ1o ð�1þeÞ=g 2 cðe�dÞþs0ð�1þeÞo0 ð18ÞA condition for the pinning of a type-0 antikink is obtained in

the same manner as

xa0n 4 ð1þeÞ=g 2 cðe�dÞþs0ð1þeÞ40

xa0nþ14 ð1þeÞ=g 2 cðeþdÞþs0ð1þeÞ40 ð19ÞInequalities in the right-hand side of arrows in Eq. (19) are also

obtained by letting e-–e in those in Eq. (18). Fig. 5 shows pinnedregions of kinks (a) and antikinks (b) in the s'–d plane (s'¼s�1/g)for c¼�1.0 and e¼0.1 (s'uE�0.818). The two boundary lines

(xk0n ¼ ð�1þeÞ=g and xk0nþ1 ¼ ð�1þeÞ=g) for a type-0 kink cross thes'-axis at (s', d)¼(ce/(1�e), 0), which are plotted with solid lines inFig. 5(a), and the crossing point moves left as e-1 and right ase-�1. The pinned region (PK0) of a type-0 kink expands (reduces)as e increases (decreases). On the other hand, the boundaries(xa0n ¼ ð1þeÞ=g and xa0nþ1 ¼ ð1þeÞ=g) for a type-0 antikink cross thes'-axis at (s', d)¼(�ce/(1þe), 0), which are plotted with solid linesin Fig. 5(b), and its pinned region (PA0) reduces (expands) ase increases (decreases). In contrast with a ring of excitatorybidirectional coupling (Fig. 3), the pinned region PA0 is includedin PK0 so that both a kink and an antikink are pinned in PA0. In PK0and outside PA0, an antikink propagates by one neuron, changesinto a kink and is pinned. Each pinned region of type-0 incon-sistency is symmetric with respect to the s'-axis (d2�d). It isbecause changing the sign of d is equivalent to changing xn and xnþ1

(type-0 inconsistency is symmetric with respect to n when d¼0),i.e., a shift in xn due to d is equivalent to a shift in xnþ1 due to �d.

Next, we consider type-u0 inconsistency when one of positive andnegative steady states is unsaturated (s'os'u (o0)). We let e40so that the positive steady state is unsaturated (0oxiuþo(1þe)/g,xis–o(�1þe)/g). Let xn�1¼xnþ2¼xiuþ , xn�2¼xnþ3¼xis� , xno(�1þe)/g and xnþ1o(�1þe)/g (Fig. 4(eL)). The states (xn andxnþ1) in a type-u0 kink change as

dxn=dt ¼ �xnþcð1þdÞgxiuþ =2þcð1�dÞð�1þeÞ=2þsð�1þeÞ¼ �xnþð�1þeÞ½�c2ð1þdÞ=ð2s'Þþcð1�dÞ=2þs� ¼ �xnþxuk0n

Fig. 4. Spatial patterns of type-m kinks and antikinks in a ring of inhibitorybidirectional coupling in Eq. (1) with g¼100.0 and c¼–1.0. Steady states aresaturated for e¼0.1; m¼0, s¼0.3, d¼70.1 (a), m¼1, s¼–0.1, d¼70.1 (b), m¼2,s¼–0.55, d¼70.01 (c), m¼3, s¼�0.73, d¼70.04 (d). Left: d40 for kinks; right:do0 for antikinks. Positive steady state is unsaturated for e¼0.5; m¼0, s¼�0.4,d¼0.05 (e), m¼2, s¼�0.6, d¼0.05 (f), m¼4, s¼�0.85, d¼0.01 (g).

Fig. 5. Pinned regions of kinks (a) and antikinks (b) in a ring of inhibitorybidirectional coupling in Eq. (1) with fL(x), c¼�1.0 and e¼0.1 (s'uE�0.818)in the s'–d plane (s'¼s�1/g).






dxnþ1=dt ¼ �xnþ1þcð1þdÞð�1þeÞ=2þcð1�dÞgxiuþ =2þsð�1þeÞ

¼ �xnþ1þð�1þeÞ½cð1þdÞ=2�c2ð1�dÞ=ð2s'Þþs�¼ �xnþ1þxuk0nþ1 ð20Þ

A pinning condition for a type-u0 kink is given by

xuk0n o ð�1þeÞ=g 2 ð�1þeÞðcþs0Þ½2s0 �cð1þdÞ�=ð2s0Þo02 2s04cð1þdÞ

xuk0nþ1o ð�1þeÞ=g 2 ð�1þeÞðcþs0Þ½2s0 �cð1�dÞ�=ð2s0Þo0

2 2s04cð1�dÞ ð21Þwhere we use �1þeo0, cþs'o0 and s'o0. Eq. (21) is indepen-dent of e and thus a pinning condition for a type-u0 antikink foreo0 is the same as Eq. (21). Fig. 6 shows pinned regions of kinksand antikinks in the s'–d plane for c¼�1.0 and e¼0.5 (s'u¼�1/3).A pinned type-u0 kink or antikink exists in the triangular regionbounded by the line s'¼s'u and two lines (2s'¼c(1þd), 2s'¼c(1�d)) crossing the s'-axis at (s', d)¼(c cos (π/3), 0) (¼(�0.5, 0)),which is denoted by PU0. In the right-hand side of the line (s'4s'u),a type-0 kink (PK0) with saturated neurons exists between two linescrossing the d-axis at (s', d)¼(0, 7e) (¼(0, 70.5)) (Eq. (18)). When|e|¼1/3 and s'u¼c/2, these four lines of the boundaries of PU0 andPK0 in Eqs. (18) and (21), respectively, cross the s'-axisat (s', d)¼(s'u, 0) (¼(c/2, 0)). Thus a pinned region of type-0inconsistency does not enter the region of s'oc/2.


First, we consider a necessary condition for the existence oftype-1 inconsistency for s'4s'u (Fig. 4(b)). Let xno(�1þe)/g andxnþ1o(�1þe)/g, which compose a type-0 kink as shown in Fig. 4(aL). A type-1 kink is generated if xnþ1 becomes unsaturated,while a type-1 antikink is generated if xn becomes unsaturated.Then, let xn4(1þe)/g and xnþ14(1þe)/g, which compose a type-0antikink as shown in Fig. 4(aR). A type-1 kink is generated if xnbecomes unsaturated, while a type-1 antikink is generated if xnþ1

becomes unsaturated. That is, a kink and antikink interchangeswhen xn becomes unsaturated. A type-1 kink is thus generatedwhen xk0nþ14 ð�1þeÞ=g and xa0n o ð1þeÞ=g, while a type-1 anti-kink is generated when xk0n 4 ð�1þeÞ=g and xa0nþ1oð1þeÞ=g.These inequalities give necessary conditions for the existence oftype-1 inconsistency. Their boundary lines cross the s-axis at(s', d)¼(0, e) for a type-1 kink and at (s', d)¼(0, �e) for a type-1antikink. The lines xk0nþ1 ¼ ð�1þeÞ=g and xa0n ¼ ð1þeÞ=g (s'o0) are

plotted in Fig. 5(a) with dashed lines, and the lines xk0n ¼ ð�1þeÞ=gand xa0nþ1 ¼ ð1þeÞ=g (s'o0) are plotted in Fig. 5(b) with dashedlines. Type-1 inconsistency can exist in the left-hand side of theselines. Note that these lines are extensions to the boundary lines forthe pinned regions of a type-0 kink and antikink in s'o0. Then,the lower boundary of the existence condition for a type-1 kinkand the upper boundary of the pinned region of type-0 kink arethe same in ce/(1�e)os'o0.

Then, let xn�14(1þe)/g, xno(�1þe)/g, (�1þe)/goxnþ1o(1þe)/g (unsaturated, fL(gxnþ1)¼gxnþ1), xnþ24(1þe)/g andxnþ3o(�1þe)/g so that xn, xnþ1 and xnþ2 compose a type-1 kink(Fig. 4(bL)). When s’ (¼s�1/g)o0, the state xnþ1 changes as

dxnþ1=dt ¼ �xnþ1þce�cdþsgxnþ1 ð22ÞThe above existence condition for a type-1 kink (xk0nþ14

ð�1þeÞ=g, xa0n oð1þeÞ=g) is equivalent to the condition thatEq. (22) has a stable unsaturated steady solution xk1nþ1 satisfying

xk1nþ1 ¼ cðd�eÞ=ðs0gÞ; �1þeogxk1nþ1o1þe ðs0 ¼ s�1=go0Þð23Þ

Note that Eqs. (22) and (23) are the same as Eqs. (6) and (7) forc40, but the existence condition is different since the sign of c isnegative. When xnþ1 is unsaturated (xnþ1¼xk1nþ1), the states xn andxnþ2 change as

dxn=dt ¼ �xnþcð1þdÞð1þeÞ=2þc2ð1�dÞðd�eÞ=ð2s0Þþsð�1þeÞ ¼ �xnþxk1n

dxnþ2=dt ¼ �xnþ2þc2ð1þdÞðd�eÞ=ð2s0Þþcð1�dÞð�1þeÞ=2þsð1þeÞ ¼ �xnþ2þxk1nþ2 ð24Þ


xk1n o ð�1þeÞ=g 2 cð1þdÞð1þeÞ=2þc2ð1�dÞðd�eÞ=ð2s0Þþs0ð�1þeÞo0

xk1nþ24 ð1þeÞ=g 2 c2ð1þdÞðd�eÞ=ð2s0Þþcð1�dÞð�1þeÞ=2þs0ð1þeÞ40 ð25Þ

under the existence condition, xk0nþ14(�1þe)/g and xa0n o(1þe)/g),equivalently Eq. (23). A condition for the pinning of a type-1antikink is derived in the same manner: xa1n 4(1þe)/g and xa1nþ2o(�1þe)/g with �1þeogxa1nþ1¼�c(dþe)/s'o1þe, where xa1n ,xa1nþ1 and xa1nþ2 are steady states for an antikink. This condition isgiven not only by letting e-�e but also by letting d-�d ininequalities in the right-hand side of arrows in Eq. (25).

The boundaries (xk1n ¼(�1þe)/g, xk1nþ2¼(1þe)/g) and(xa1n ¼(1þe)/g, xa1nþ2¼(�1þe)/g) of pinned regions for a kink andantikink are plotted in Fig. 5 with solid curves, which cross thed-axis at (s', d)¼(0, e) and (0, –e), respectively. Both boundarycurves cross the s'-axis at (s', d)¼(c cos (π/3), 0) (¼(–0.5, 0)) and(ce/(1�e), 0) (E(�0.11, 0)). Pinned regions for a type-1 kink(PK1 in (a)) and antikink (PA1 in (b)) are bounded by two curves,which are denoted by PK1 and PA1, respectively. These pinnedregions are reflection across the s-axis with each other. Theregions for a type-1 kink and antikink shift upward (d40) anddownward (do0), respectively, as e increases in a similar mannerto those for excitatory bidirectional coupling (c40). It is becausespatial patterns of a type-1 kink and antikink are reverse withrespect to n, i.e., the values of xnþ j (0r jr2) in a kink is equal tothe value of xnþ2� j (0r jr2) in an antikink when d¼0 as thosefor c40. The difference is that the boundaries are always fixed at(s', d)¼(c/2, 0) so that both a type-1 kink and antikink are pinnedwhen d¼0 in c/2os'oce/(1�e).

It should be noted that not all the region of the pinningcondition for a type-1 kink (Eq. (25)) is included in the region forthe existence of type-1 kink (Eq. (23)), i.e., a part of the region ofEq. (25) lies outside the region of Eq. (23) (below the line xk1nþ1 ¼ð�1þeÞ=g or xk0nþ1 ¼ ð�1þeÞ=g), i.e., c(e�d)þs'(�1þe)40).

Fig. 6. Pinned regions of kinks and antikinks in a ring of inhibitory bidirectionalcoupling in Eq. (1) with fL(x), c¼�1.0 and e¼0.5 (s'uE�1/3) in the s'–d plane(s'¼s�1/g).






In Fig. 6, two boundary curves for the pinned region of a type-1kink (PK1) in Eq. (25) are also plotted with solid lines in the right-handside of s'¼s'u (¼�1/3). Most of the pinned region of a type-1 kink isbelow the boundary line for the existence condition for a type-1 kinkin Eq. (23) xk1nþ1 ¼ ð�1þeÞ=g (c(e�d)þs'(�1þe)40). This boundaryline is also the boundary for the pinned region of a type-0 kink (PK0),and there is a type-0 pinned kink below the line. This overlap of thepinned regions of a type-0 kink and a type-1 kink means that a type-1kink is pinned (xk1n o(�1þe)/g) even if f(xk1nþ1)¼gxk1nþ1o�1þe.

When s'u¼c/2 (e¼1/3), a curve for the lower boundary of apinned type-1 kink (xk1n ¼ ð�1þeÞ=g) for Eq. (25) is tangent to thes'-axis at (s', d)¼(c/2, 0) (¼(�0.5, 0)). Then a type-1 kink cannotpinned when dr0 so that the pinned regions of a type-1 kinkand antikink do not overlap. When s'u4c/2 (s'¼�1/34�0.5¼c/2in Fig. 6), the boundary line (�1þe¼gxk1nþ1) for Eq. (23), theboundary curve (xk1nþ2 ¼ ð1þeÞ=g) for Eq. (25) for a type-1 kink andthe boundary line (2s04cð1�dÞ) for Eq. (21) for a type-u0 kinkcross at (s', d)¼(s'u, 1�2s'u/c) (¼(�1/3, 1/3) in Fig. 6) on the lines'¼s'u (a dashed line). Thus a type-1 kink does not exist for s'os'usince there is a type-u0 kink.


First, we consider type-2 inconsistency when the steady statesare saturated (s'4s'u, Fig. 4(c)). As type-1 inconsistency, each type-2inconsistency is generated from both a type-1 kink and antikink.A type-1 kink changes into a type-2 kink if xnþ2 becomes unsatu-rated and changes into a type-2 antikink if xn becomes unsatu-rated. A type-1 antikink changes into a type-2 antikink if xnþ2

becomes unsaturated and changes into a type-2 kink if xn becomesunsaturated. Thus, a type-2 kink is generated when xk1nþ2oð1þeÞ=g and xa1n o ð1þeÞ=g, while a type-2 antikink is generatedwhen xk1n 4 ð�1þeÞ=g and xa1nþ2o ð�1þeÞ=g. These inequalitiesgive necessary conditions for the existence of type-2 inconsis-tency. Their boundaries cross the s'-axis at (s', d)¼(c cos (π/3), 0)(¼(c/2, 0)¼(�0.5, 0)), which are plotted with dashed curves fors'o�0.5 in Fig. 5. Type-2 inconsistency can exist in the left-handside regions of these curves. Note that a curve of the upper (lower)boundary of the existence condition for a type-2 kink in s'o�0.5is an extension to a curve of the lower (upper) boundary for thepinned region of a type-1 antikink (kink) in s'4�0.5.

Then, we consider a type-2 kink, i.e., let xnþ1 and xnþ2 beunsaturated, xno(�1þe)/g and xnþ3o(�1þe)/g as shown inFig. 4(cL). The unsaturated states change as

dxnþ1=dt ¼ �xnþ1þcð1þdÞð�1þeÞ=2þcð1�dÞgxnþ2=2þsgxnþ1

dxnþ2=dt ¼ �xnþ2þcð1þdÞgxnþ1=2þcð1�dÞð�1þeÞ=2þsgxnþ2

ð26ÞThe above existence condition for a type-2 kink (xk1nþ2o

ð1þeÞ=g, xa1n o ð1þeÞ=g) agrees with the condition that Eq. (26)has a stable unsaturated steady solution xk2nþ1 and xk2nþ2 satisfying

cð1þdÞð�1þeÞ=2þcð1�dÞgxk2nþ2=2þs0gxk2nþ1 ¼ 0

cð1þdÞgxnþ1=2þcð1�dÞð�1þeÞ=2þs0gxk2nþ2 ¼ 0 ðs0 ¼ s�1=go0Þ

�1þeogxk2nþ1o1þe; �1þeogxk2nþ2o1þe ð27Þ

Since a type-2 kink is generated from a type-1 kink or antikink,the existence condition in Eq. (27) is intrinsically given by gxk2nþ1o1þe and gxk2nþ2o1þe, which correspond to xa1n o ð1þeÞ=g andxk1nþ2o ð1þeÞ=g, respectively, in the above existence conditionfor a type-2 kink. Under Eq. (27) (�1þeogxnþ j¼gxk2nþ jo1þe(j¼1, 2)), the states xn and xnþ3 change as

dxn=dt ¼ �xnþcð1þdÞð1þeÞ=2þcð1�dÞgxk2nþ1=2

þsð�1þeÞ ¼ �xnþxk2n

dxnþ3=dt ¼ �xnþ3þcð1þdÞgxk2nþ2=2þcð1�dÞð1þeÞ=2þsð�1þeÞ ¼ �xnþ3þxk2nþ3 ð28Þ


xk2n o ð�1þeÞ=g 2 cð1þdÞð1þeÞ=2þcð1�dÞgxk2nþ1=2þs0ð�1þeÞo0

xk2nþ3o ð�1þeÞ=g 2 cð1þdÞgxk2nþ2=2þcð1�dÞð1þeÞ=2þs0ð�1þeÞo0

ð29ÞConditions for the existence and pinning of a type-2 antikink

are derived in the same manner. They are obtained by lettinge-�e in the conditions for a type-2 kink.

In Fig. 5, each pinned region of a type-2 kink (PK2 in (a)) andantikink (PA2 in (b)) is bounded by two solid curves, which cross s'-axis at (s', d)¼(c/2, 0) (¼(�0.5, 0)) commonly and at (s', d)E(�0.80, 0) (a) and (s', d)E(�0.63, 0) (b), respectively. When e¼0,two curves cross s'-axis at (s', d)¼(c cos (π/3), 0) (¼(�0.5, 0)) and(c cos (π/4), 0) (E(�0.71, 0)). The regions are symmetric withrespect to the s'-axis (d¼0) since type-2 inconsistency is sym-metric with respect to n when d¼0 as type-0 inconsistency. Thepinned region of a type-2 kink expands (a) while the pinnedregion of a type-2 antikink reduces (b) as e-1, which is a similarchange to the pinned region of type-0 inconsistency. The pinnedregion of a type-2 antikink disappears at e¼1/3 (s'u¼c/2) whenthe boundary curves are tangent to the s'-axis at (s', d)¼(c/2, 0),at which an overlap of PK1 and PA1 on the s'-axis disappears. This isdue to the emergence of a type-u2 kink, which is shown below.

Next, we consider type-u2 inconsistency when one of positiveand negative steady states is unsaturated (s'os'u). Since type-u1inconsistency does not exist, type-u2 inconsistency is generatedfrom type-u0 inconsistency. We let e40 so that the positive steadystate is unsaturated (�1þeogxiuþo1þe, xis–o(�1þe)/g).We then consider a type-u0 kink (xno(�1þe)/g, xnþ1o(�1þe)/g). A necessary condition for the existence of a type-u2kink is that both xn and xnþ1 increase to be unsaturated, i.e.,their steady states in Eq. (21) satisfy xuk0n 4 ð�1þeÞ=g andxuk0nþ14 ð�1þeÞ=g. Thus, the existence condition is given by rever-sing both inequalities in Eq. (21), i.e., 2s'oc(1þd) and 2s'oc(1�d), which corresponds to the right-hand side region of twolines crossing the s'-axis at (s', d)¼(c/2, 0) (¼(�0.5, 0)) in Fig. 6.

Then we let xnþ1 become unsaturated and let xn�1¼xiuþ ,xno(�1þe)/g, (�1þe)/goxnþ1o(1þe)/g), (�1þe)/goxnþ2o(1þe)/g, xnþ3o(–1þe)/g and xnþ4¼xiuþ so that xnþ j (0r jr3)make a type-u2 kink (Fig. 4(f)). The unsaturated states xnþ1

and xnþ2 in the type-u2 kink change according to Eq. (26), andtheir stable steady states are given by xk2nþ1 and xk2nþ2 in Eq. (27),respectively. The above existence condition for a type-u2 kink(2s'oc(1þd) and 2s'oc(1�d), in s'oc/2) is also equivalent to thecondition that Eq. (26) has this stable unsaturated steady solution.However, it is intrinsically given by gxk2nþ14�1þe and gxk2nþ24�1þe since these states cannot become positively saturated incontrast with a type-2 kink. The saturated states xn and xnþ3 thenchange as

dxn=dt ¼ �xnþcð1þdÞgxiuþ =2þcð1�dÞgxk2nþ1=2

þsð�1þeÞ ¼ �xnþxuk2n

dxnþ3=dt ¼ �xnþ3þcð1þdÞgxk2nþ2=2þcð1�dÞgxiuþ =2þsð�1þeÞ ¼ �xnþ3þxuk2nþ3 ð30Þ

The type-u2 kink is pinned when their steady states (xuk2n , xuk2nþ3)satisfy

xuk2n o ð�1þeÞ=g 2 cð1þdÞgxiuþ =2þcð1�dÞgxk2nþ1=2þs0ð�1þeÞo0

xuk2nþ3o ð�1þeÞ=g 2 cð1þdÞgxk2nþ2=2þcð1�dÞgxiuþ =2þs0ð�1þeÞo0

ð31ÞIt can be shown that inequalities in Eq. (31) have common

factors �1þe and cþs' as in Eq. (21) and thus Eq. (31) is






independent of e. They are also reflections across the s'-axis (d¼0)with each other since a type-u2 kink and antikink are symmetricwith respect to n when d¼0.

In Fig. 6, the pinned region of type-u2 inconsistency (PU2) isbounded by two solid curves, which cross s'-axis at (s', d)¼(c cos(π/3), 0) (¼(�0.5, 0)) and at (s', d)¼(c cos (π/5), 0) (E(�0.809, 0)).A pinned type-u2 inconsistency appears at (s', d)¼(c cos (π/5), 0)rightwards when s'u¼c cos (π/5) (e¼(1�cos (π/5))/(1þcos (π/5))(E0.106)). When c cos (π/5)os'uoc cos (π/3), i.e., (1�cos (π/5))/(1þcos (π/5)) (E0.106)oeo(1�cos (π/3))/(1þcos (π/3)) (¼1/3),the boundaries of the pinned regions for a type-2 kink (Eq. (29))and a type-u2 kink (Eq. (31)) cross with each other at s'¼s'u.A type-2 and type-u2 kinks exist for s'uos'oc cos (π/3) and c cos(π/5)os'os'u, respectively, so that a type-2 kink cannot existfor s'oc cos (π/5). Also, the pinned region of a type-2 antikink(Eq. (29) with e-�e) disappears when s'u4c cos (π/3) (e41/3)and then only a type-u2 kink exists.

4.4. Type-m inconsistency

We then consider type-m inconsistency (mZ3) when thesteady states are saturated (s'4s'u (Fig. 4(d) for m¼3). In a type-m kink (xnþ j (0r jrmþ1)) with unsaturated xnþ j (1r jrm),the state xnþmþ1 is positively saturated when m is odd, while it isnegatively saturated when m is odd. Then, conditions for theexistence and pinning of type-m inconsistency depend on theparity of the number m of unsaturated neurons. Further, a type-mkink is generated from a type-(m�1) kink (xnþ j (0r jrm); xn andxnþm: saturated) when xnþm becomes unsaturated and froma type-(m�1) antikink when xn becomes unsaturated. In the samemanner, a type-m antikink is generated from a type-(m�1) kinkwhen xn becomes unsaturated and from a type-(m�1) antikinkwhen xnþm becomes unsaturated. Thus, a kink and antikinkinterchange with each other as xn becomes unsaturated. Thisinterchange also makes conditions for the existence and pinningof inconsistency rather complicated than those for excitatorybidirectional coupling (c40).

Let xn�14(1þe)/g, xno(�1þe)/g, and let xnþ j (1r jrm) beunsaturated. Then, let xnþmþ14(1þe)/g, xnþmþ2o(�1þe)/g forodd m while let xnþmþ1o(�1þe)/g, xnþmþ24(1þe)/g for evenm so that xnþ j (0r jrmþ1) compose a type-m kink. Theunsaturated states in a type-m kink change as

dxnþ1=dt ¼ �xnþ1þ cð1þdÞ2 ð�1þeÞþ cð1�dÞ

2 gxnþ2þsgxnþ1


2gxnþ j�1þ

cð1�dÞ2


ð2r jrm�1Þ


gxnþm�1

þcð1�dÞ2

ðð�1Þm�1þ eÞþsgxnþm ð32Þ

A necessary condition for the existence of a type-m kink isequivalent to the condition that Eq. (32) has a stable unsaturatedsteady solution xkmnþ j (1r jrm) satisfying

cð1þdÞ2

ð�1þeÞþcð1�dÞ2

gxkmnþ2þs0gxkmnþ1 ¼ 0

cð1þdÞ2

gxkmnþ j�1þcð1�dÞ

2gxkmnþ jþ1þs0gxkmnþ j ¼ 0 ð2r jrm�1Þ

cð1þdÞ2

gxkmnþm�1þcð1�dÞ

2ðð�1Þm�1þeÞþs0gxkmnþm ¼ 0

�1þeogxmnþ jo1þe ð1r jrmÞ ð33Þ

Since Eq. (33) is satisfied when xnþ1 or xnþm becomes unsaturatedunder the condition that the other m�1 neurons are unsaturated

(a type-(m�1) kink or antikink exists), the existence condition isintrinsically given by gxkmnþ1o1þe and gxkmnþm4 �1þe for odd m,while gxkmnþ1o1þe and gxkmnþmo1þe for even m. In the samemanner, a necessary condition for the existence of a type-mantikink is equivalent to the condition that the following Eq. (34)has a stable unsaturated steady solution xamnþ j (1r jrm).

dxnþ1=dt ¼ �xnþ1þcð1þdÞ

2ð1þeÞþcð1�dÞ

2gxnþ2þsgxnþ1


2gxnþ j�1þ

cð1�dÞ2


ð2r jrm�1Þ


gxnþm�1þcð1�dÞ

2ðð�1Þmþ eÞþsgxnþm

ð34ÞUnder Eq. (33) (�1þeogxnþ j¼gxkmnþ jo1þe (1r jrm)), the

states xn and xnþmþ1 change as



2gxkmnþ1þsð�1þeÞ ¼ �xnþxkmn

dxnþmþ1=dt ¼ �xnþmþ1þcð1þdÞ

2gxkmnþm

þcð1�dÞ2

ðð�1ÞmþeÞþsðð�1Þm�1þeÞ

¼ �xnþmþ1þxkmnþmþ1 ð35ÞA condition for the pinning of a type-m kink is thus given by

xkmn o ð�1þeÞ=g 2cð1þdÞ


2gxkmnþ1þs0ð�1þeÞo0

xkmnþmþ14ð1þeÞ=g 2cð1þdÞ

2gxkmnþmþcð1�dÞ

2ð1þeÞ

þs0ð1þeÞ40 ðm : oddÞ

xkmnþmþ1oð�1þeÞ=g 2cð1þdÞ

2gxkmnþmþcð1�dÞ

2ð1þeÞ

þs0ð�1þeÞo0 ðm : evenÞ ð36ÞA condition for the pinning of a type-m antikink is derived in

the same manner, which is given by letting e-�e in inequalitiesin the right-hand side of arrows in Eq. (36).

A necessary condition for the existence of a type-(mþ1) kink isgiven by reversing inequalities in Eq. (36) so that both xn andxnþmþ1 in a type-m kink move to be unsaturated. Necessaryconditions for the existence of a type-m kink and antikink areobtained in the same manner with m-m�1. The existencecondition for a type-m kink is that both xn in a type-(m�1)antikink and xnþm in a type-(m�1) kink move to be unsaturated,

i.e., gxaðm�1Þn o1þe and gxkðm�1Þ

nþm 4�1þe for odd m; while

gxaðm�1Þn o1þe and gxkðm�1Þ

nþm o1þe for even m. The existencecondition for a type-m antikink is that both xn in a type-(m�1)kink and xnþm in a type-(m�1) antikink move to be unsaturated,

i.e., gxkðm�1Þn 4�1þe and gxaðm�1Þ

nþm o1þe for odd m; while

gxkðm�1Þn 4�1þe and gxaðm�1Þ

nþm 4�1þe for even m.In Fig. 5(a), a pinned type-3 kink (Eq. (36) withm¼3) exists in a

small region (PK3) bounded by the upper-left part of the pinnedregion of a type-2 kink, which locates just left-hand side ofs'¼c cos (π/5) (E�0.809). Its boundaries cross the s'-axis at(s', d)¼(c cos (π/5), 0) (E(�0.809, 0)) and the upper boundarycurve of a type-2 kink at (s', d)E(�0.707, 0.050), which areplotted with solid curves. In the same manner as a type-1 kink,a lower part of the pinned region of a type-3 kink in Eq. (36) lieswithin the pinned region of a type-2 kink. The upper boundary ofthe pinned region of a type-2 kink gives a lower boundary of theexistence condition for a type-3 kink in Eq. (33) (gxkmnþmo1þe).Au upper boundary of the existence condition for a type-3 kink(gxkmnþ14�1þe) is plotted with a dashed line, which connects (s',






d)E(�0.818, 0.228) and (�0.707, 0.050). In Fig. 5(b), boundariesfor the existence and pinning conditions for a type-3 antikink(Eqs. (33) and (36) with e-�e) are plotted with solid and dashedcurves, respectively, which are reflection across the s'-axis withthose of a type-3 kink in Fig. 5(a). An upper region in the pinnedregion of a type-3 antikink also lies outside of the existence regionof a type-3 antikink. A pinned type-4 kink exists in a narrow rangein s'u (E�0.818)os'oc cos (π/5) (E�0.809), which is scarcelyvisible. The pinned region connects to a pinned region of type-u4kinks at s'¼s'u (E�0.818) so that a type-4 kink continuouslychanges into a type-u4 kink in s'os'u.

4.5. Type-um inconsistency

Finally, we consider type-um inconsistency (mZ4) when oneof positive and negative steady states is unsaturated (s'os'u)(Fig. 4(g) for m¼4). We again let e40 and consider a type-um

kink. Let a type-u(m�2) kink exist with unsaturated xnþ j¼xkðm�2Þnþ j

(1r jrm�2) and negatively saturated xn and xnþm�1, where

xkðm�2Þnþ j is a stable unsaturated solution to Eq. (32) with m-m�2

given by Eq. (33) with m-m�2. A type-um kink is generatedwhen both saturated xn and xnþm�1 increase to be unsaturated sothat one of them becomes unsaturated. The states xn and xnþm�1

change as

dxn=dt ¼ �xnþcð1þdÞgxiuþ =2þcð1�dÞgxkðm�2Þnþ1 =2

þsð�1þeÞ ¼ �xnþxukðm�2Þn

dxnþm�1=dt ¼ �xnþm�1þcð1þdÞgxkðm�2Þnþm�2=2

þcð1�dÞgxiuþ =2þsð�1þeÞ¼ �xnþm�1þxukðm�2Þ

nþm�1 ð37Þ

Thus a necessary condition for the existence of a type-um kinkis given by

xukðm�2Þn oð�1þeÞ=g 2 cð1þdÞgxiuþ =2

þcð1�dÞgxkðm�2Þnþ1 =2þs0ð�1þeÞo0

xukðm�2Þnþm�1 o ð�1þeÞ=g 2 cð1þdÞgxkðm�2Þ

nþm�2=2

þcð1�dÞgxiuþ =2þs0ð�1þeÞo0 ð38ÞThis existence condition is also equivalent to the condition that

Eq. (32) for m unsaturated states has a stable unsaturated steadysolution given by Eq. (33). However, it is intrinsically given bygxkmnþ24�1þe and gxkmnþm�14�1þe (mZ4) for s'os'u sincexkmnþ1 and xkmnþm cannot be positively saturated in contrast witha type-m kink.

Then, let xnþ j (0r jrmþ1) compose a type-um kink with munsaturated neurons xnþ j¼xkmnþ j (1r jrm ) given by Eq. (33). Theunsaturated states xn and xnþmþ1 change in the same manner asEq. (37), and a condition for the pinning of a type-um kink is thattheir steady states (xukmn , xukmnþmþ1) remains unsaturated.

xukmn oð�1þeÞ=g 2 cð1þdÞgxiuþ =2þcð1�dÞgxkmnþ1=2þs0ð�1þeÞo0

xukmnþmþ1o ð�1þeÞ=g 2 cð1þdÞgxkmnþm=2

þcð1�dÞgxiuþ =2þs0ð�1þeÞo0 ð39ÞIn Figs. 5 and 6, a pinned region of a type-u4 kink (PU4) is

bounded by two solid curves, which cross the s'-axis at (s', d)¼(c cos (π/5), 0) (E(�0.809, 0)) and (c cos (π/7), 0) (E(�0.901, 0)).Two dashed curves connecting (s', d)¼(c cos (π/5), 0) and (c, 7c)are the boundaries of the existence condition for a type-u4 kink(Eq. (38) with m¼4). They are extensions to the boundary curvesof a pinned type-u2 kink for Eq. (31) in s'oc cos (π/5). Also two

dashed curves connecting (s', d)¼(c cos (π/7), 0) and (c, 7c) givethe boundaries of the existence condition for a type-u6 kink(Eq. (38) with m¼6), which are extensions to the boundary curvesof a pinned type-u4 kink for Eq. (39) in s'oc cos (π/7). It can beshown that a pinned region of a type-um kink exists aroundthe s'-axis between c cos (π/(mþ3))os'oc cos (π/(mþ1)), whichis symmetric with respect to the s'-axis. The pinned region of atype-m kink (m: even) expands leftward along the s'-axis froms'¼c cos (π/(mþ2)) up to s'¼c cos (π/(mþ3)) as e increases from0 to (1�cos (π/(mþ3))/(1þcos (π/(mþ3)) (s'u increases from c toccos(π/(mþ3)). When s'u4c cos (π/(mþ3), a type-m kink changesinto a type-um kink at s'¼s'u. Then, only type-um kinks existin s'os'u.

4.6. Equivalent pinning condition

In the same manner as explained in Section 3.4, a condition forthe pinning of type-(m�1) inconsistency (mZ1) is given byconditions for the existence of unstable unsaturated steady solu-tions xkmnþ j (1r jrm) to Eq. (32) for a type-m kink and xamnþ j(1r jrm) to Eq. (34) for a type-m antikink. These conditionsare intrinsically given by gxkmnþ1o1þe and (�1)m�1gxkmnþm4(�1)m�1[(�1)mþe] for a kink, while gxamnþ14�1þe and(�1)mgxamnþm4(�1)m[(�1)m�1þe] for an antikink. Althoughthese conditions are apparently the same as the existence condi-tions for stable unsaturated solutions to Eqs. (32) and (34), i.e., theexistence conditions for a type-m kink and antikink, but unstableunsaturated solutions exist in different regions in the s'–d plane.Intuitively, if these unstable unsaturated solutions to Eqs. (32) and (34)exist, both xnþ1 and xnþm can become saturated. Accordingly, oneof them becomes saturated and pinned type-m inconsistency isgenerated. However, a kink and antikink interchange with eachother when xnþ1 becomes saturated. Then, the boundariesgxkmnþ1¼1þe and gxkmnþm¼(�1)mþe for an unstable unsaturatedsolution to Eq. (32) for a type-m kink give the boundaries for thepinned regions of a type-(m�1) antikink and kink, respectively.In other words, the equivalent pinning condition for a type-(m�1)kink is composed of one inequality in the existence condition foran unstable unsaturated solution to Eq. (32) for a type-m kink((�1)m�1gxkmnþm4(�1)m�1[(�1)mþe]) and one inequality inthe existence condition for an unstable unsaturated solution toEq. (34) for a type-m antikink (gxamnþ14�1þe). The existenceconditions for unstable unsaturated solutions to Eqs. (32) and (34)are shown in Appendix A.

A similar equivalent pinning condition exists for a type-umkink for s'os'u. The pinning condition Eq. (21) for a type-u0 kink isequivalent to the condition that Eq. (26) for unsaturated neuronsin a type-u2 kink has an unstable unsaturated steady solution fors'os'u. However, this equivalent condition is intrinsically given bygxk2nþ14�1þe and gxk2nþ24�1þe since these states cannot bepositively saturated in contrast with a type-2 kink. When bothconditions hold, both xk2nþ1 and xk2nþ2 can become negativelysaturated, and then one of them becomes saturated so thata pinned type-u0 kink is generated. In the same manner, the pinningcondition Eq. (39) with m-m�2 for a type-u(m�2) kink (mZ4)is also equivalent to the condition that Eq. (32) for m unsaturatedneurons in a type-um kink has an unstable unsaturated steadysolution satisfying Eq. (33). Also, this equivalent condition isintrinsically given by gxkmnþ24�1þe and gxkmnþm�14�1þe fors'os'u since the states cannot be positively saturated in contrastwith a type-m kink. Note that two inequalities, gxkmnþ1o1þe andgxkmnþmo1þe, in the existence condition for the unstable unsatu-rated solution to Eq. (32) for even m still give a pinning conditionfor type-(m�1) inconsistency for s'4s'u.






5. Conclusion and future work

The pinning (propagation failure) of rotating waves in a ring ofcoupled neurons with asymmetric bidirectional coupling (c(1þd)/2,c(1�d)/2) and self-coupling (s) as well as an asymmetric sigmoidaloutput function f(x) was studied. Conditions for pinning of wavefronts (a kink and antikink; inconsistency for co0) were obtainedby using a piecewise linear output function fL(x) in Eq. (3). Thepinning of wave fronts with no unsaturated neurons (type-0)occurred in the presence of excitatory self-coupling. This pinningcondition was given by s'¼s�1/g4 |d|(40) when f(x) was sym-metric [12,13], where g is an output gain. In addition, wave frontswith m unsaturated neurons (type-m) existed and their pinningoccurred when self-coupling was inhibitory (s'o0). These pinnedstates of type-m wave fronts corresponded to steady states ina ring of neurons with symmetric bidirectional coupling (d¼0)[10,12].

Effects of asymmetry in a sigmoidal output function f(x) on thepinning of wave fronts depended on whether bidirectional cou-pling was excitatory or inhibitory. When bidirectional couplingwas excitatory (c40), a pinning condition for a type-m wave front(a kink or antikink) was that saturated leftmost (xn) and rightmost(xnþmþ1) neurons in a type-m wave front remain saturated. Incontrast, the condition that both saturated neurons in a type-mwave front move to be unsaturated gave an existence condition fora type-(mþ1) wave front. Thus, the existence condition for a type-(mþ1) wave front was given by reversing both inequalities in thepinning condition for a type-m wave front. This existence condi-tion agreed with a condition for the existence of mþ1 stableunsaturated neurons in a type-(mþ1) wave front. The pinningcondition for a type-m wave front was also equivalent to acondition for the existence of mþ1 unstable unsaturated neuronsin a type-(mþ1) wave front. Then, the pinned regions in the s'–dplane were simply shifted along the d-axis as an asymmetricshift e in f(x) increased (Fig. 3). These shifts were due to competi-tion between asymmetry d in coupling and asymmetry e in f(x).

When bidirectional coupling was inhibitory (co0), however,effects of asymmetry in f(x) depended on the parity of the numberm of unsaturated neurons in inconsistency. This dependencecame from the difference in the spatial form of inconsistency,which was symmetric (xn¼xnþmþ1) for even m or antisymmetric(xn¼�xnþmþ1) for odd m with respect to n when d¼0.In addition, a kink and antikink interchanged as a saturatedleftmost neuron (xn) became unsaturated, i.e., a type-m kink(antikink) changed into a type-(mþ1) antikink (kink). Then, theabove existence and pinning conditions for a kink and antikinkalso interchanged. The pinning condition derived with saturatedneurons and the existence condition derived with unsaturatedneurons held for each of a kink and antikink. However, the reverseof one inequality in the existence condition derived with saturatedneurons (both move to be unsaturated) for a type-m kink (anti-kink) gave one inequality in the existence condition for a type-(mþ1) antikink (kink). Also, the reverse of one inequality in thepinning condition derived with unsaturated neurons (an unstablesolution exists) for a type-(mþ1) kink (antikink) gave one inequal-ity in the pinning condition for a type-m antikink (kink). As aresult, the pinned regions of type-m wave fronts were shiftedalong the d-axis due to e when m was odd, while the pinnedregions expanded or reduced along the s'-axis when m was even.One of a kink and antikink was always pinned when d¼0 eventhough asymmetry in f(x) was large (|e|-1). Further, one ofpositive and negative steady states was unsaturated whens'os'u¼c(1� |e|)/(1þ |e|) (o0), equivalently when |e|4(c�s')/(cþs'). Then only kinks or antikinks with even numbers ofunsaturated neurons existed, and their pinned regions weresymmetric with respect to the s'-axis.

When an output function is a smooth sigmoidal function, it canbe shown that type-m kinks and antikinks for large m are lostwhen |c|g is not large (E1). Only type-0 and typ-1 kinks andantikinks exist when c¼1.0 and g¼2.0, for instance. However,type-m kinks and antikinks for large m are generated when g islarge, e.g., g¼100, and pinning due to inhibitory self-couplingremains.

As noted in Section 1, it has been shown that metastable dynamicaltransient rotating waves emerge in a ring of synaptically coupledBonhoeffer–van der Pol models [30], i.e., spiking neuron models. Sincetwo states (resting and firing) of a spiking neuron are differentqualitatively, a spiking neuron can be modeled by an asymmetricsigmoidal neuron. Then, the results obtained in this paper will beapplicable to such rings consisting of biologically more plausiblespiking neurons. The pinning of rotating waves in rings of spikingneurons due to asymmetric bidirectional coupling and self-coupling isone of future areas of interest.

Acknowledgment

The author would like to acknowledge valuable discussionswith Dr. H. Kitajima.

Appendix A. Solutions to associated linear differentialequations

As explained in Section 3.4, a condition for the pinning ofa type-(m�1) kink for excitatory bidirectional coupling (c40)is given by a condition for the existence of an unstable unsaturatedsteady solution xmnþ j (1r jrm) to Eq. (10) given by Eq. (11) for thestates of m unsaturated neurons. Also, a condition for the pinningof type-(m�1) inconsistency for inhibitory bidirectional coupling(co0) is given by conditions for the existence of an unstableunsaturated steady solution xkmnþ j (1r jrm) to Eq. (32) for a kinkand Eq. (34) for an antikink as explained in Section 4.6. Theexistence and stability of unsaturated solutions to the lineardifferential equations for unsaturated neurons are shown.

First, we consider an unstable unsaturated solution to Eq. (10)for c40. A coefficient matrix in Eq. (10) is tridiagonal, and itseigenvalues μj(m) (1r jrm) are given by [69]

μjðmÞ ¼ g s0 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2ð1�d2Þ

qcos ½jπ=ðmþ1Þ�

� �ð1r jrmÞ ðA1Þ

which are real when |d|o1. A steady solution to Eq. (10) isunstable when the largest eigenvalue is positive (μ140). Form¼1 (Eq. (6)), μ1(m)¼s' and the steady solution is unstable whens'40. For mZ2, the unstable region of the steady solution in thes'–d plane is the inside of the ellipse: s'2/{c2 cos2 [π/(mþ1)]}þd2¼1. In Fig. A1(a), the boundaries μ1(m)¼0 for 1rmr5 in thes'–d plane are plotted with thick solid lines, which connect to thes'-axis at (s', d)¼(�c cos [π/(mþ1)], 0), where c¼1.0. Numericalcalculation can show that a stable and unstable unsaturated steadysolutions to Eq. (10) for mZ2 exist in an unbounded region in theleft-hand side of the ellipse (μ1(m)¼0) and a bounded regioninside of the ellipse, respectively. Their existence regions connectwith each other at a point on the ellipse, at which the solution ismarginally stable. The region of the unstable unsaturated solutionis located inside of the ellipses μ1(m)¼0 and outside of μ1(m�1)¼0 (μ1(1)¼0 is s'¼0). Thus, the leftmost (rightmost) point of theregion of the unstable unsaturated solution to Eq. (10), whichcorresponds to the pinned region of a type-(m�1) kink orantikink, is located on the ellipse μ1(m)¼0 (μ1(m�1)¼0). InFig. A1(a), boundaries for the regions of unstable unsaturatedsolutions to Eq. (10) (1rmr5) for e¼0.0, 0.1, 0.5 and 0.8 are plotted






with thin solid, dashed, dotted and dash-dotted lines, respectively.They correspond to the pinned regions of type-m kinks for0rmr4. For each value of e, there are connected five regionsand the regions of type-0 and type-1 kinks connects to the d-axisat (s', d)¼(0, e).

Next, we consider unstable unsaturated solutions to Eqs. (32)and (34) for co0. Coefficient matrices in Eqs. (32) and (34) are thesame as that to Eq. (10) and their eigenvalues are given by Eq. (A1).Then, the stability conditions for solutions to Eqs. (32) and (34) arethe same as that for c40. However, the signs of saturated neurons(xn and xnþmþ1) on both sides of inconsistency are different (thesame) for odd (even) m, and then changes in the regions ofunstable unsaturated solutions to Eq. (32) for a kink and (34) foran antikink depend on the parity of the number m of unsaturatedneurons. Fig. A1(b) shows boundaries for the regions of unstableunsaturated solutions to Eq. (32) (solid lines) and Eq. (34) (dottedlines) (1rmr3) for c¼�1.0 and e¼0.1. (Note that one ofboundaries in each region is extended in order to show a boundaryfor a corresponding pinned region of inconsistency as explainedbelow.) When m is odd, the leftmost point of each unstableunsaturated solution is also shifted on the ellipse μ1(m)¼0, whileits rightmost point is fixed at (s', d)¼(� |c| cos (π/m), 0) for mZ3.Then, the regions for Eqs. (32) and (34) are reflection across the s'-axis with each other. When m is even, the leftmost point of theregion of each unstable unsaturated solution is fixed at (s', d)¼(� |c| cos [π/(mþ1)], 0), while its rightmost point is shifted along thes'-axis from s'¼� |c| cos (π/m) rightward (leftward) for e40 (eo0)for Eq. (32), and vise versa for Eq. (34). Then, the regions forEq. (32) with e and (34) with –e are the same and they are symmetricwith respect to the s'-axis. Further, one boundary for the region ofthe unstable unsaturated solution corresponds to one boundaryfor the pinned region of a type-(m�1) kink, while the otherboundary corresponds to that for a type-(m�1) antikink, sincea kink and antikink interchange with m when xn becomes unsatu-

rated. Consequently, a pinned region of type-(m�1) inconsistencyfor evenm can expand outside of the ellipse μ1(m)¼0 although theregion of the unstable solution remains its inside. In Fig. A1(b), theupper boundaries for the pinned regions of type-m kinks(0rmr2) correspond to the upper boundaries for the regionsof unstable unsaturated solutions to Eq. (32) (solid lines) whiletheir lower boundaries correspond to the lower boundaries for theregions of unstable unsaturated solutions to Eq. (34) (dotted lines).For the pinned regions of type-m antikinks (0rmr2), their upperboundaries correspond to those to Eq. (34) (dashed lines) and theirlower boundaries correspond to those to Eq. (32) (solid lines).When s'os'u, the existence condition for an unstable unsaturatedsolution to Eq. (32) for even m changes from gxkmnþ1o1þe andgxkmnþmo1þe to gxkmnþ24�1þe and gxkmnþm�14�1þe. This exis-tence condition is independent of e and its region corresponds tothe pinned region of a type-u(m�2) kink shown as PU(m�2) inFigs. 5 and 6.

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Fig. A1. Boundaries μ1(m)¼0 (1rmr5) of stable/unstable solutions to Eq. (10)with c¼1.0 (thick solid lines) in the s'–d plane (the same for Eqs. (32) and (34) withc¼�1.0). Regions of unstable unsaturated solutions to Eq. (10) (1rmr5) forc¼1.0; e¼0.0 (a thin solid line), 0.1 (a dashed line), 0.5 (a dotted line) and 0.8(a dot-dashed line) (a). Regions of unstable unsaturated solutions to Eq. (32) (solidlines) and Eq. (34) (dashed lines) (1rmr3) for c¼�1.0 and e¼0.1 (b).



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Yo Horikawa is a professor in the Faculty of Engineer-ing at Kagawa University, Japan. He has receivedB. Eng., M. Eng. and Ph.D. (Eng.) degrees in mathematicalengineering and information physics from the Univer-sity of Tokyo in 1983, 1985 and 1994, respectively. Hisresearch interests include nonlinear dynamical systemsand statistical pattern recognition.















































































































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