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Chaos, Special Focus Issue on Nonlinear Dynamics in Cognitive and Neural Systems.

Controlling the onset of traveling pulses in excitable media

by nonlocal spatial coupling and time-delayed feedback

Felix M. Schneider, Eckehard Scholl, and Markus A. DahlemInstitut fur Theoretische Physik, Technische Universitat Berlin, Hardenbergstraße 36, D-10623 Berlin, Germany

The onset of pulse propagation is studied in a reaction-diffusion (RD) model with control byaugmented transmission capability that is provided either along nonlocal spatial coupling or by time-delayed feedback. We show that traveling pulses occur primarily as solutions to the RD equationswhile augmented transmission changes excitability. For certain ranges of the parameter settings,defined as weak susceptibility and moderate control, respectively, the hybrid model can be mapped tothe original RD model. This results in an effective change of RD parameters controlled by augmentedtransmission. Outside moderate control parameter settings new patterns are obtained, for examplestep-wise propagation due to delay-induced oscillations. Augmented transmission constitutes asignaling system complementary to the classical RD mechanism of pattern formation. Our hybridmodel combines the two major signalling systems in the brain, namely volume transmission andsynaptic transmission. Our results provide insights into the spread and control of pathologicalpulses in the brain.

Traveling pulses are of fundamental importance

in neuroscience. They propagate not only infor-

mation along nerve fibers, but are also related to

pathological phenomena. Examples are cell depo-

larizations that lead to a temporary complete loss

of normal cell functions in migraine and stroke.

This state spreads in cortical tissue via chemi-

cal signals that diffuse through the extracellular

space. We study these spreading depolarization

pulses in a standard reaction-diffusion model and

suggest to augment the transmission capabilities

such that they reflect the cortical structural and

functional connectivity. With this new modal-

ity, we investigate control of emerging spread of

pathological states in the brain.

1. INTRODUCTION

Within the last years control of complex dynamics hasevolved as one of the central issues in applied nonlin-ear science [1]. Major progress has been made in neu-roscience, among other areas, by extending methods ofchaos control, in particular time-delayed feedback [2], tospatio-temporal patterns [3, 4, 5], and by developing ap-plications in the field of biomedical engineering [6, 7].In this study, control is introduced to suppress spatio-temporal pattern formation. Our emphasis is on under-standing the recruitment of cortical tissue into dysfunc-tional states by traveling pulses of pathological activity,in particular, on internal cortical circuits that provideaugmented transmission capabilities and that can pre-vent such events. Our long-term aim is to design strate-gies that either support the internal cortical control ormimic its behavior by external control loops and trans-late these methods into applications.

There is growing experimental evidence that particu-lar spatio-temporal pulse patterns in the human cortex,

called cortical spreading depression, cause transient neu-rological symptoms during migraine [8, 9]. Similar pulsesoccur after stroke, called periinfarct depolarization, andcontribute to the loss of potentially salvageable tissue,i. e., tissue at risk of infarction [10]. These dysfunctionalstates of the cortex are also referred to as spreading depo-

larizations (SD) to point out the nearly complete depo-larizations of cortical cells and its spread as the commonaspect in these patterns.

SD is usually called a cortical wave, not pulse, whichmight cause some confusion. In many cases these twoterms can be used interchangeably. We adhere to a pre-cise mathematical terminology, referring to a travelingpulse as a localized event with a spatial profile havinga single front and back while a wave usually refers to aperiodic spatial profile. SD is a pulse in this terminology,the strict use of which is necessary because the spatio-temporal pattern formation of SD occurs in cortical tissuebeing only weakly susceptible to SD [11]. The definitionof weak susceptibility (Sec. 2) depends on a bifurcationfor which one must strictly distinguish between solitaryand periodic wave forms.

The pulse of SD extends in the cortex over several cen-timeters with a remarkably slow speed of several mil-limeter per minute. Accordingly, the mathematical de-scription of SD considers large-scale neuronal activity inpopulations of neurons rather than ion channels in themembrane of nerve fibers, and the spatial coupling isprovided by volume transmission, which is essentially adiffusion process. The first mathematical model of SDwas proposed by Hodgkin and Grafstein [12] based onbistable K+ dynamics and K+ diffusion. This model,however, describes only the front dynamics of SD. Wesuggest (Sec. 3) two extensions to this model to studythe onset of SD. The first is a necessary extension to ob-tain onset behavior in the Hodgkin-Grafstein model. Weinclude in a generic form a recovery process for the pulsetrailing edge. In a second step, we extend this model fur-

2

ther by nonlocal and time-delayed signal transmission tostudy the effect of internal control provided as a feedbackto the traveling pulse. Results and conclusions are givenin Sec. 4 and Sec. 5, respectively.

2. CLASSIFICATIONS OF EXCITABILITY IN

LOCAL ELEMENTS AND EXTENDED MEDIA

Excitability, as a property of a single element, is basedon threshold behavior and therefore requires a nonlin-ear process with a stable fixed point. If the system issufficiently perturbed from this fixed point, it returns af-ter a large excursion in phase space, emitting a spike[13]. If excitable elements are locally interconnected, anew behavior can emerge, namely the capacity to prop-agate a sustained pulse through this spatially extendedsystem. This emergent property defines a medium as be-ing excitable, also termed an active medium. Excitableelements and excitable media differ in their response tosuperthreshold stimulation. The response of an excitableelement to a superthreshold stimulation will eventuallyend in the stable fixed point value of the steady stateof this element. In contrast, the response of an ex-citable medium, which is initially in the homogeneoussteady state, to a superthreshold stimulation results inapproaching a new attractor, the pulse solution. Thisdifferent behavior of local elements and spatial media in-dicates that there are also different ways to classify ex-citability in elements and media.In Sec. 2.1 definitions are provided for excitability of

local elements with a focus on neural systems. Some ofour results concerning control of spatial-temporal pat-terns can be explained by considering the effect of con-trol on the local dynamics (Sec. 4.2.1). Furthermore, wewill consider local dynamics with the aim to extend theHodgkin-Grafstein model (Sec. 3). However, our main fo-cus is on spatial excitability (Sec. 2.2) and its control bynonlocal and time-delayed feedback described in Sec. 4.1and Sec. 4.2, respectively. For a thorough treatment oflocal dynamics, in particular in neural systems, see for ex-ample Ref. [14] or for more complex discharge patterns,like bursting, Ref. [15].

2.1. Local excitability

A generic mechanism of local excitability requires acertain configuration of trajectories in the phase spaceof a single excitable element (inset of Fig. 1). This con-figuration usually results from the parameter vicinity ofan oscillatory regime whose large amplitude limit cycleis suddenly destructed [15].The parameter space is schematically depicted in

Fig. 1, where the white area corresponds to the oscilla-tory regime, and the colored areas mark various regimesof excitability and non-excitability. If parameters set-tings are in the excitable regime (to the right of the thick

blue dashed line), a rest state (fixed point) is the onlyattractor. There exists also a threshold close to this reststate, e.g. a ”sharp” separatrix or trajectory in phasespace (thin dashed line, in the inset). Trajectories start-ing on the near side of the threshold (green dotted) ap-proach the rest state directly (subthreshold response),while those starting on the far side (red solid) perform alarge excursion in phase space, guided by the ghost of adestructed large amplitude limit cycle, before returningto the rest state (superthreshold response). Note that werefer to excitable elements and not to neurons becausealso a population of neurons described in a firing ratemodel can behave like an excitable element.

The way in which, by changing a bifurcation parame-ter, the dynamics of the local element changes from ex-citable to oscillatory behavior, that is, the way the largeamplitude limit cycle is built up, is used to classify thetype of excitability in a single excitable element. This can

LCH

SNIPER

SN

oscillatory

wea

kly

susc

eptib

le

Ghost

FP

S

excitable

∂M∂C ∂S∂P∂R

FIG. 1: Parameter space of excitable systems illustratinga universal scheme for classifications of both local and spa-tial excitability. Bifurcation lines for excitable elements of anactive media are marked by thick blue (dashed) lines. Tran-sitions from the excitable (magenta, red, orange) to the oscil-latory regime (white) occur usually either through a saddle-node infinite period bifurcation (SNIPER) causing excitabil-ity of type I, or via a Hopf bifurcation (H) causing excitabilityof type II. Between the SNIPER and a further saddle-node(SN) bifurcation line (magenta) three fixed points (stablenode, unstable focus, and saddle) exist in the local excitableelements. The thick white solid lines ∂C, ∂M , and ∂R markbifurcations of active media where the spatio-temporal pat-tern in 2D changes (complex, meandering, and rigid rotatingspiral patterns, to the left of ∂C, ∂M and ∂R, respectively).At ∂P the medium becomes non-excitable. Sustained pulsesdo not exist in the yellow regime but there is a transientpropagating pulse. To the right of ∂S (green) the transientactivation radius becomes zero. The insets (thin lines) indi-cate the corresponding configuration of trajectories in phasespace, with limit cycle (LC), sharp separatrix (S), and fixedpoint (FP).

3

retracting collapsing

particle−like

parameter space

borderline case

not spreadingspiral−shaped

weak excitabilityweak susceptibility

control∂P∂R ∂S

FIG. 2: (Color online) Scheme of the classification of ex-citability according to spatio-temporal patterns (spiral-wave,particle-like, retracting, collapsing, no spread) in 2D: wavefront at two instances in time (t1: black, t2: gray) and tra-jectory of open wave ends (dashed). This scheme illustratesthe effect of control on the propagation boundary (∂P ) by ad-ditional nonlocal and time-delayed transmission capabilities.∂P is controlled by shifting the bifurcation in parameter space(horizontal (multi-colored) line). This affects the regime ofweak susceptibility centered around ∂P and bounded by therotor boundary ∂R and the spreading boundary ∂S.

happen in different ways, two of which are usually distin-guished [14]. Type I excitability obtains its characteristicfeatures from a saddle-node infinite period (SNIPER) bi-furcation: the frequency of the emerging periodic orbittends to zero, while the amplitude starts with a finitevalue. Type II excitability is caused by a Hopf bifur-cation, which is characterized by the features that theperiodic orbit emerges with zero amplitude and nonzerofrequency. In practice, the main characteristic for type IIexcitability is, however, the onset frequency, because forthis type a canard explosion renders the zero amplitudepractically invisible.

2.2. Onset of pulse propagation in spatial

excitability

The important criterion for spatial excitability is theexistence of a traveling pulse solution. In contrast to anexcitable local element and its corresponding phase spaceconfiguration (inset in Fig. 1), excitability in active me-dia is based on bistability. A superthreshold stimulationtakes the system from the homogeneous steady state intothe basin of attraction of the pulse solution. The classi-fication of spatial excitability in active media is based onthis configuration in phase space. For example, the pri-mary rough classification into excitable and non-excitablemedia is based on the existence and non-existence, re-spectively, of the pulse as the most basic spatio-temporalpattern that sustainedly propagates in space. This prop-agation boundary is called ∂P [16]. The focus in thisstudy is on the onset of excitability in active media at∂P .The propagation boundary ∂P is determined in the pa-

rameter space of a co-moving frame. In this frame, ∂Pis caused by a saddle-node bifurcation above which trav-eling pulse solutions exist. As active media are usuallybased on a reaction-diffusion mechanism, this bifurcationis computed in a parabolic partial differential equation(PDE). The details of this computation can be found intextbooks [17] and have been further related to the phe-nomenon of SD in the cortex in Ref. [3]. Therefore, wejust sketch the basic idea. The propagation boundary ina parabolic PDE is obtained by searching pulse profilesas stationary solutions in a co-moving frame. The profileequation in the co-moving frame must tend to the fixedpoint value of the original system for ξ → ±∞, with ξbeing the spatial coordinate in the co-moving frame. Atraveling pulse is thus equivalent to the existence of ahomoclinic orbit satisfying a system of ordinary differen-tial equations, namely the profile equation system, withappropriate boundary conditions.In Sec. 4 we describe how the locus of the propaga-

tion boundary ∂P in the parameter space is controlledby nonlocal spatial coupling and time-delayed feedbackby a shift to new parameter values without adding a dis-tinctly new character to the spatio-temporal patterns, asschematically illustrated in Fig. 2. Since the propagationboundary is essentially a feature of an active mediumwith one spatial dimension, we can limit our main in-vestigation to one-dimensional systems. Yet, to get apicture of the patterns that arise in the neighborhoodof ∂P , we will end Sec. 2 by a brief review of patternsin higher spatial dimensions, and provide definitions ofweak excitability and weak susceptibility.

2.3. Weak excitability and weak susceptibility

The variety of qualitatively different spatio-temporalpatterns in higher excitable regimes, i. e., beyond ∂P to-wards the oscillatory regime (to the left of ∂P in Fig. 1),provides the foundation for a classification of spatial ex-citability in active media. This is in so far analogousto the classification into type I and II of excitable localelements (Sec. 2.1) as both classifications are built onthe patterns that emerge. However, the classification oflocal excitability defines different types of mechanisms,whereas the classification of spatial excitability charac-terizes rather the degree of excitability.Excitability is described by ordinary differential equa-

tions in local elements and, by partial differential equa-tions in spatial media. Roughly speaking, the additionalindependent spatial variable allows for more complex pat-terns, due to the fact that the spatial dimension rendersthe phase space infinite-dimensional. In fact, the pat-terns can become even more complex if spatial dimen-sions are increased from one to two (for example spiralwaves in retinal spreading depression [18]) or three di-mensions (Winfree turbulence of scroll waves in cardiacfibrillation [19]).Close to the propagation boundary, the complexity of

4

emerging patterns is largely independent of the numberof spatial dimensions. This fact is also paraphrased asweak excitability [20]. This term is used for active mediato indicate that either no reentrant patterns occur (de-scribed by the rotor boundary ∂R [16], see Figs. 1 and2) or that the rotation period is large enough, so thatthe front of the pulse does not interact with its refrac-tory back. The onset of interactions between front andrefractory back in a reentrant wave pattern is describedby the meandering boundary ∂M , see Fig. 1. To theleft of this boundary, the core of a freely rotating spiralwave performs a meandering pattern [21, 22], whereas tothe right of ∂M the spiral core follows a rigidly circularrotation [23](Fig. 2). Changing excitability parametersfurther, the spiral core can start to perform more com-plex manoeuvres to avoid the refractory zone (complexboundary ∂C), see Fig. 1.Patterns of spreading depression in chicken retina are

observed in vitro in the complex regime to the left of ∂C[18], but in human brain tissue they occur close to ∂R.It was predicted that the window of cortical excitabilitylies between ∂R and ∂P [24] and this seems to be con-firmed by a functional magnetic resonance imaging studyin migraine, mapping spatio-temporal patterns of symp-tom reports onto the folded cortical surface [25]. Thesepatterns are similar to particle-like or retracting wavesegments that occur between ∂R and ∂P [26, 27, 28].Transient patterns can also be observed at even lowerexcitability, limited by the spreading boundary ∂S [29](Fig. 2). Since the regime of retracting wave segmentsis not identical with weak excitability (absence of front-back interactions) it was called weak susceptibility basedon a susceptibility scale that can also be operationallydefined in experimental systems [25]. The control of ex-citability in media being weakly susceptible to patternformation can be investigated in a model with one spa-tial dimension, because ∂P is essentially a property of a1D parabolic PDE.

3. REACTION-DIFFUSION WITH

AUGMENTED TRANSMISSION CAPABILITY

The generic framework to model spatial excitability isa reaction-diffusion system of activator-inhibitor type. Infact, already a single species model has the capacity topropagate fronts [30]. Such a model was originally sug-gested for SD by Hodgkin and Grafstein [12] (Sec. 3.1).We extend this generic model by choosing appropriate in-hibitor dynamics (Sec. 3.2) to obtain pulses with the on-set saddle-node bifurcation ∂P and introduce augmentedtransmission capabilities (Sec. 3.3).

3.1. Hodgkin-Grafstein model

A single species model of bistability has the capacityto propagate one state into the other if these bistable

dynamics of the species, called u, are coupled by diffusion

∂u

∂t= f(u) − v +

∂2u

∂x2, (1)

f(u) = u−u3

3. (2)

Here v is a bifurcation parameter. In these non-dimensional equations the diffusion coefficient, whichmerely scales space, is set to unity.Eqs. (1)-(2) were proposed by Hodgkin as a model for

spreading depression based on the bistable K+ dynam-ics suggested by Grafstein [12]. The front profile of thespecies u and its speed can be calculated analytically forthe cubic polynomial form of f(u), i.,e., the generic formof bistable dynamics (see for example the textbook ref-erences [31, 32]). In the context of control one shouldthink of the two stable states as one being a physiologi-cal (healthy) state and the other one being a pathological(depolarized) state, and the latter invading the former,which shall be prevented by control.As is shown in Sec. 4, control of the onset of propaga-

tion depends essentially on the interaction of the healthystate with the pulse front via the augmented transmissioncapability. However, in the system defined by Eqs. (1)-(2) there does not exist a boundary like ∂P (Sec. 2.2).In other words, there is no abrupt onset of the capacityto propagate fronts with finite speed. Instead, the het-eroclinic orbit, which corresponds to the front profile ina co-moving frame, exists for the whole bistable regime,in particular also for arbitrarily slow velocities of the co-moving frame including zero (standing front) [31]. Toobtain propagation onset behavior, we need to add in-hibitor dynamics, i. e., a second recovery species for theHodgkin-Grafstein model.

3.2. Extension by inhibitor dynamics

With a proper choice of a second species whose dy-namics is coupled to u, the local system can change frombistable to excitable dynamics, which is needed to ob-serve and compute ∂P . The natural choice is to use thebifurcation parameter v of the Hodgkin-Grafstein schemeas the additional species, namely as an inhibitor :

∂v

∂t= εg(u, v). (3)

The rate equation g(u, v) of the inhibitor determines thedynamics in the refractory phase of the pulse, where itrecovers the initial healthy state that was recruited intothe pathological state of species u. In this two-speciesmodel, u is called the activator. Inhibitor dynamics usu-ally changes on a slower time scale ε ≪ 1.There are several models in the neuroscience litera-

ture obeying this structure in Eqs. (1)-(3), both modelswith local excitability of type I [31, 33] and of type II[34, 35, 36]. These models are, however, models of action

5

potentials describing the propagation of normal electro-physiological activity along single nerve fibers. Thereforethese models do not relate directly to SD, because SD iscaused by large-scale neuronal activity [37, 38]. There-fore the analogy to models as in Refs. [31, 33, 34, 35, 36]is mainly formal in the mathematical structure but notin its biological interpretation. For example, in theHodgkin-Grafstein Eqs. (1)-(2) the activator is K+ whilein action potential models it is the membrane potential.Models exhibiting type I excitability are also capable

to show type II behavior for a certain parameter regimebut the reverse is not necessarily true. This suggeststhat type II behavior is more generic, because it can beobserved in both types of models and only requires arate function g(u, v) that is linear in both arguments.We introduce inhibitor dynamics as

g(u, v) = u+ β − γv. (4)

With this choice, the local excitable dynamics is of typeII. The parameter β and γ determine the precise con-figuration of trajectories in phase space that lead to ex-citability and are thus, in addition to ε, the only twoparameters that control excitability of type II in a nor-mal form model.Eqs. (1)-(4) correspond to the well studied FitzHugh-

Nagumo (FHN) systems, which is, in fact, used as aparadigm for excitable systems [13]. Pattern formation ina net of FitzHugh-Nagumo elements under the influenceof time-delayed feedback was investigated with the aimto increase the coherence of noise-induced wave patterns[39, 40].

3.3. Augmented transmission capability

We extend the FHN model by introducing augmentedtransmission capability into Eqs. (1)-(4) as a feedbackloop [41]

(

∂tuε−1∂tv

)

=

(

f(u)− vg(u, v)

)

+D

(

uv

)

+ H

(

uv

)

, (5)

where D is the local diffusion operator, and H representsthe augmented transmission capability. In Eqs. (1)-(4)we have considered diffusion in the activator species uonly:

D =

(

∇2 00 0

)

. (6)

The augmented transmission capability

H = K F (7)

is described by the control strength K and the controlmatrix

F =

(

Fuu Fuv

Fvu Fvv

)

, (8)

+time delay

space shift

t − τx ± δ

K s

excitable RD system

F

(

u

v

)

D

(

u

v

)

=

(

f(u) − v

εg(u, v)

)

+

FIG. 3: Diagrammatic view of control by augmented trans-mission (CAT).

whose elements Fij are operators which represent threeindividual steps of the control by augmented transmis-sion (CAT), namely (i) selecting a species j whose trans-mission capability is augmented, (ii) creating the controlforce from this species, and (iii) feeding this control forceback into the dynamical variable i, see Fig. 3.Formally, this can be represented by splitting

Fij = AijF into the components of a coupling matrix A,which represents the coupling scheme, e.g.,

Auu =

(

1 00 0

)

, Auv =

(

0 10 0

)

,

Avu =

(

0 01 0

)

, Avv =

(

0 00 1

)

, (9)

and an operator F that creates the type of control signal.The control signal F is generated from the input vari-

able s = u or s = v by time-delayed or nonlocal feedback.For example, in time-delayed feedback as introduced byPyragas [2], the operator F creates the difference betweenthe time-delayed signal s(t− τ) and its current counter-part s(t). We study this type and two nonlocal types ofcoupling. The control force F of these different types, asa function of the signal s, is described in detail in Sec. 4.1and 4.2. It should be noted that in Eq. (5), H representscortical circuits and neurovascular coupling [3], that is,internal control loops.The four coupling matrices in Eq. (9) lead to four

transmission pathways, also termed coupling schemes:two self-couplings uu and vv, and two cross-couplingsvu and uv, corresponding to the indices ij in Eq. (8).Note that other coupling schemes, for example diagonalcoupling or coupling with a rotation matrix [5, 42, 43],can also be investigated within this framework.

4. SUPPRESSION OF PULSE PROPAGATION

We perform simulations of the RD system inEqs. (1)-(4) in one spatial dimension with RD parametersettings (ε=0.1, β=0.85, γ=0.5) for the weakly excitableregime close to the propagation boundary ∂P (Sec. 2.2).

6

The RD system is extended by three types of couplingin the framework of Eq. (5), i.e., two types of nonlocalspatial coupling (isotropic and anisotropic) and one typeof time-delayed feedback. In the context of spreadingdepolarizations, nonlocal spatial and local time-delayedcouplings represent neural structural and functional con-nectivity and neurovascular feedback in the cortex. Thisis discussed in more detail in Ref. [3]. For each typeof coupling there are four principal coupling schemes,two self-coupling schemes (uu, vv) and two cross-couplingschemes (uv, vu), defined by the coupling matrix A inEq. (9).We initialize each simulation with a stable pulse profile

solution of the RD system in Eqs. (1)-(4) to investigatethe effect of control on RD excitability. This is accom-plished by testing whether this specific initial conditionof the free system (RD only) is in the basin of attrac-tion of the homogeneous steady state of Eq. (5), i. e., thecontrolled system (RD-CAT). There are two CAT param-eters: The control gain K in Eq. (7), and a spatial (δ)or temporal (τ) control scale, repectively, which will beintroduced in the following sections. For a large range ofthese two CAT parameters, we determine whether pulsepropagation is suppressed or not. The propagation issuppressed if the excitation dies out, so that the systemapproaches the homogeneous steady state. In this case,CAT is considered successful. In the reverse case, anysustained spatio-temporal pattern that evolves from theinitial conditions (free pulse solution), after the CAT is”switched on”, is considered an unsuccessful control sincethe activity is not completely suppressed and the homo-geneous steady state is not reached.For the following simulations we adopt an active

medium with a spatial extension of L = 160. As spa-tial increment in the discretized Laplacian D we takeδx = 0.2. All simulations are run for 2000 time unitsand use an Euler forward algorithm with discretizationδt = 0.00125. The spatial and temporal width of the free-running activator pulse, ∆x and ∆t, respectively, serveas reference space and time scales [3].

4.1. Pulse suppression by nonlocal spatial coupling

In this section we present results on two types of spatialcoupling.

4.1.1. Isotropic coupling

Nonlocal isotropic spatial coupling is defined as

F (s) = s(x+ δ, t) + s(x− δ, t)− 2s(x, t) (10)

where s = u or s = v.Regimes in which reaction-diffusion pulses are sup-

pressed by additional nonlocal connections are calculatedfor the parameters δ andK and represented by gray areas

in Fig. 5. For the two self-coupling schemes uu and vv,shown in Fig. 5 (a) and (d), respectively, the sign of thegain parameter K determines the effect of the nonlocalconnection. Pulse propagation can only be suppressed forK > 0. For the two cross-coupling schemes uv and vu,shown in Fig. 5 (b) and (c), respectively, the sign of thegain parameter K changes at δ ≈ ∆x for pulse suppres-sion, and these two cross-coupling schemes show similarcontrol domains with respect to reflection K → −K.

The sign of K in the control domains of the self-coupling schemes (Fig. 5 (a) and (d)) can be qualitativelyunderstood by considering the effect of nonlocal connec-tions upon the homogeneous steady state in the limitδ → 0. For K > 0, this limit corresponds to diffusivelycoupled elements. In general, the homogeneous steadystate is stabilized by diffusion against small inhomoge-neous perturbations. In the same way, a local pertur-bation is leveled by a nonlocal connection in the form ofEq. (10) for self-coupling. In the diffusion limit, however,K would increase the diffusion coefficient, which causesthe pulses to become broader. Yet, a qualitative changein the dynamics cannot occur by changing the diffusioncoefficient. Therefore the effect of suppressing pulses de-pends on the nonlocal character of the connection, whichmust extend at least over a distance of about 20% of thepulse width ∆x.

The control domains of the cross-coupling schemes(Fig. 5 (b)-(c)) are qualitatively different from self-coupling. K changes its sign at δ ≈ ∆x within onescheme and accordingly the situation of pulse suppres-sion is more complex. There is a long-range regime(δ > ∆x) and a short-range regime (δ < ∆x). The signof K for pulse suppression in the long-range regime de-pends mainly on the backward connection (solid arrowin Fig. 4(a)), as will be shown in the next section. Inthe short-range regime, the effect of the forward connec-tion (dotted arrow in Fig. 4(a)) seems to dominate andreverse the effect observed in the long-range regime.

u / v

u*

v*

δ δ

space x

u*

v*

space x τ

t- τ t

u*

v*

space x τ

t- τ t

u*

v*

space x τ

t- τ ta b

FIG. 4: (Color online) Schematic diagram illustrating nonlo-cal spatial coupling and time-delayed feedback. (a) Nonlocalspatial coupling: isotropic coupling (dashed and solid arrow),and anisotropic backward coupling (solid arrow). (b) Time-delayed feedback: backward coupling of traveling pulses.Pulse profiles are shown in red (solid) and green (dashed)for the activator and inhibitor, respectively. Dotted: time-delayed profiles.

7

0 0.5

1 1.5

2 2.5

3δ/

∆x

(a) uu

0 0.5

1 1.5

2 2.5

3δ/

∆x

(a) uu

0 0.5

1 1.5

2 2.5

3δ/

∆x

(a) uu

0

5

10

15

20

25

δ

(b) uv

0

5

10

15

20

25

δ

(b) uv

0 0.5

1 1.5

2 2.5

3

-1 -0.5 0 0.5 1

δ/∆

x

K

(c) vu

0 0.5

1 1.5

2 2.5

3

-1 -0.5 0 0.5 1

δ/∆

x

K

(c) vu

-1 -0.5 0 0.5 1 0

5

10

15

20

25

δ

K

(d) vv

-1 -0.5 0 0.5 1 0

5

10

15

20

25

δ

K

(d) vv

-1 -0.5 0 0.5 1 0

5

10

15

20

25

δ

K

(d) vv

FIG. 5: Control planes of isotropic spatial coupling spannedby the two CAT parameters: control gain factor K and non-local space scale δ normalized to pulse width ∆x (left scale)and in spatial units (right scale). (a) Activator self-couplingscheme uu, (b) cross-couplings uv (inhibitor signal fed back toactivator rate equation) and (c) vu (reverse), and (d) inhibitorself-coupling vv. Suppression of pulse propagation is markedby gray control domains. Parameters: ε = 0.1, β = 0.85,γ = 0.5. ∆x = 8.65.

4.1.2. Anisotropic backward coupling

Nonlocal connections can also be introduced in onlyone direction

F (s) = s(x± δ, t)− s(x, t). (11)

This directed connectivity would correspond toanisotropic nonlocal coupling in a two dimensionalexcitable medium. One reason to investigate this typeof connectivity is to obtain a better understanding ofthe results in the previous section by separating effectsof forward and backward connections (Fig. 4 (a)).Moreover, the functional and structural connectivity ofthe cortex is realistically modeled as an anisotropic (andalso inhomogeneous) medium due to the patchy natureof nonlocal horizontal cortical connections. Whileanisotropies in the cortical connections are usuallyconsidered to merely cause variations in wave speed indifferent directions, inhomogeneities are known to causewave propagation failure [44]. In contrast, our focusis on the change in excitability by anisotropic nonlocalcoupling that leads to suppression of wave propagation.We limit your investigation to the backward connec-

tion. This corresponds to the plus (minus) sign inEq. (11) for pulses propagating in the positive (nega-tive) x direction (solid arrow in Fig. 4(a)). We choosethe backward connection because this type of couplingshares many properties with local time-delayed feedbackcoupling, as we will show by comparing the results fromthis section with those of the next section (Sec. 4.2).

The control domain of the uu self-coupling scheme(Fig. 6 (a)) is very similar to the isotropic one shownin Fig. 5 (a). This indicates that in the uu schemeof isotropic coupling the backward connection accountsfor the main contribution to suppression of wave prop-agation. However, we want to note that both controltypes (isotropic and anisotropic) with the same couplingscheme uu can differ in their efficiency within some partsof the gray control domain (not shown). The efficiencyrefers to the length a pulse travels after the connectivityis ”switched on”. This transient effect was investigatedin detail in [3, 29], whereas in this study we concentrateon an understanding of the size and shape of control do-mains for the different schemes and types of coupling(Figs. 5-7).Pulse propagation is not suppressed by the vv self-

coupling scheme for anisotropic backward coupling forany control parameter within the investigated range(Fig. 6(d)). This, however, does not mean that the pulsepropagates essentially unchanged. Any sustained spatio-temporal pattern other than the homogeneous steadystate that evolves from the free initial pulse solution—after the connectivity is ”switched on”—is considered asunsuccessful pulse suppression. In fact, for the vv schemeof anisotropic coupling in the expected control domain ofcorresponding Fig. 5 (d), the homogeneous steady statebecomes unstable and stationary spatial patterns emergeafter the initial pulse is suppressed.The control domains of the cross-coupling schemes

(Fig. 6 (b)-(c)) are similar to Fig. 5 (b)-(c) in the long-range regime, except for a failure of suppression in theisotropic uv scheme for K > 0.2. The long-range regimefor anisotropic backward cross-coupling extends over thenearly complete spatial scale of δ (except for very smallδ). Thus, K does not change its sign. Also, there is aperfect reflection symmetric pattern in the control planeswith respect to the axis K = 0 for the two cross-coupling

0 0.5

1 1.5

2 2.5

3

δ/∆

x

(a) uu

0 0.5

1 1.5

2 2.5

3

δ/∆

x

(a) uu

0

5

10

15

20

25

δ

(b) uv

0

5

10

15

20

25

δ

(b) uv

0 0.5

1 1.5

2 2.5

3

-1 -0.5 0 0.5 1

δ/∆

x

K

(c) vu

-1 -0.5 0 0.5 1 0

5

10

15

20

25

δ

K

(d) vv

-1 -0.5 0 0.5 1 0

5

10

15

20

25

δ

K

(d) vv

FIG. 6: Control planes of anisotropic backward spatial cou-pling. Parameters and notation as in Fig. 5.

8

0

0.5

1

1.5

2τ/

∆t

(a) uu

0

0.5

1

1.5

2τ/

∆t

(a) uu

0

0.5

1

1.5

2τ/

∆t

(a) uu

0

0.5

1

1.5

2τ/

∆t

(a) uu

0

5

10

15

20

τ

(b) uv

0

5

10

15

20

τ

(b) uv

0

5

10

15

20

τ

(b) uv

0

5

10

15

20

τ

(b) uv

0

0.5

1

1.5

2

-1 -0.5 0 0.5 1

τ/∆

t

K

(c) vu

0

0.5

1

1.5

2

-1 -0.5 0 0.5 1

τ/∆

t

K

(c) vu

0

0.5

1

1.5

2

-1 -0.5 0 0.5 1

τ/∆

t

K

(c) vu

0

0.5

1

1.5

2

-1 -0.5 0 0.5 1

τ/∆

t

K

(c) vu

-1 -0.5 0 0.5 1 0

5

10

15

20

τ

K

(d) vv

-1 -0.5 0 0.5 1 0

5

10

15

20

τ

K

(d) vv

-1 -0.5 0 0.5 1 0

5

10

15

20

τ

K

(d) vv

0

0.2

0.4

5 10t

I(t)

FIG. 7: Control planes of time-delayed feedback couplingspanned by the two CAT parameters: control gain factor K

and time delay τ normalized to pulse width ∆t (left scale)and in temporal units (right scale). (a) Activator self-couplingscheme uu, (b) cross-couplings uv (inhibitor signal fed back toactivator rate equation) and (c) vu (reverse), and (d) inhibitorself-coupling vv. Suppression of pulse propagation is markedby gray control domains. The black lines denote saddle-nodebifurcations. The dashed lines mark the values of controlparameters for which the u-amplitude of a single local FHNsystem, stimulated in u by I(t) (shown as inset in (d)), isdecreased by 10% compared to the u-amplitude of the uncon-trolled system. Parameters as in Fig. 5, ∆t = 10.73.

schemes.The reason for the reflection symmetry in the pattern

of control domains for cross-coupling schemes and, fur-thermore, the explanation for the observed signs of thegain parameter K for the control domains of all couplingschemes for anisotropic backward coupling are given inSec. 4.3 together with the explanation of some of the re-sults we have obtained for local time-delayed feedbackcontrol (Pyragas feedback), which will be described inthe next section.

4.2. Pulse suppression by local time-delayed

feedback

In case of local time-delayed feedback the control forceF is given by

F (s) = s(x, t− τ) − s(x, t), (12)

where τ is the delay time. Note the formal similarity toEq. (11). This control method was first introduced byPyragas [2] for chaos control. In Fig. 4 (b), this controlmethod is illustrated for the uu coupling scheme: at eachspatial location x the activator u at time t receives thesignal from the same location but at the past time t− τ .That means particularly for the dynamics of the front

that the deviation from the homogeneous fixed point isfed back.The domains of successful control, i.e., pulse suppres-

sion, in the (K, τ) plane are shown as gray areas in Fig. 7.For successful control, the pulse dies out and the systemreturns to the homogeneous steady state, as shown ex-emplarily in the space-time plot of Fig. 8 (b).For time-delayed feedback a domain of successful con-

trol exists for each coupling scheme. As for the spatialcoupling schemes in Fig. 5 and 6, the two cross-couplingschemes uv and vu show a reflection symmetry with re-spect to the axis K = 0. This symmetry is explainedby the effective parameters introduced in Sec. 4.3. Cou-pling schemes that feed back the signal to the activatorcan suppress pulses for positive K. For coupling schemesthat feed back the signal to the inhibitor successful con-trol is possible for negative values of K. This is simi-lar to the results of anisotropic backward coupling (cf.Fig. 6). This similarity is due to the similarity of thesignals the pulse receives through the feedback. In bothcases the wave front gets the feedback from the homo-geneous steady state which is ahead of the wave, in atemporal or spatial sense, respectively.The main differences between the control domains of

time-delayed feedback and spatial anisotropic couplingare that in case of time-delayed feedback the control do-mains are limited towards large values of τ (upper bor-ders of gray domains in Fig. 7 (a-d)), and, moreover, thatthere is a control domain also for vv, the inhibitor self-coupling scheme. The limitation of the control domainstowards large values of τ is caused by the formation oftracking patterns [45] (Fig. 8), which is further explainedin Sec. 4.2.2. These patterns emerge by delay-induced os-cillations. The local dynamics becomes bistable due to asaddle-node bifurcation (black solid lines, Fig. 7). Thisis demonstrated in the next section.

4.2.1. Effect of time-delayed feedback in a local excitablesystem

In this section we investigate the effect of time-delayedfeedback on a single local excitable element. There aretwo main effects. First, depending on the sign of the con-trol parameter K, the activator amplitude is reduced,which selects the location of the control domain in thecontrol plane with respect to the axis K = 0 (Fig. 7).Furthermore, we show that a minimum amplitude reduc-tion of about 10% is needed to suppress pulse propaga-tion when these elements form an active medium (lowerbounds of the control domains). Second, for too large avalue of τ , the local dynamics becomes bistable (upperbounds of the control domains).To obtain qualitative insight into how time-delayed

feedback operates, we perform numerical simulations inorder to compare trajectories starting from the same ini-tial conditions with and without time-delayed feedback.Figure 9 shows exemplary superthreshold phase space ex-

9

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

u

0

20

40

60

80

100

40 60 80 100 120 140

t

x

-2

0

2

0

20

40

60

80

40 60 80 100

t

x

TAR

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

u

0

20

40

60

80

100

120

0 20 40 60 80 100

t

x

a

b

c

FIG. 8: Pattern formation for ε = 0.1, β = 0.85 and γ = 0.5:(a): pulse propagation without control; (b): suppressed pulsevia nonlocal coupling (K = 0.2, δ = 0.58∆x). The horizontalline at t = 25 marks the onset of control. The distance thatthe transient excitation spreads after control is activated be-fore the medium relaxes to the homogeneous fixed defines thetransient activation radius (TAR) or tissue at risk. (c): delay-induced oscillatory tracking pattern (K = 0.5, τ = 2.8∆t,onset of control at t = 21)

cursions with and without (K = 0) time-delayed feed-back (solid and dashed trajectories, respectively). Thesystem is initialized in the interval [t0 − τ ; t0[ (historyfunction) with the fixed point value (u = u∗ and v = v∗)and for t0 with a superthreshold value (u = u∗ +0.5 andv = v∗).For coupling schemes that feed back into the activator

rate equation, i. e., uu and uv, the control force acts par-allel to the u axis. In Fig. 9 (a) the trajectories with andwithout control for the uu coupling scheme are plotted.The direction of the control force is denoted by the hori-

zontal arrow. In order to reduce the amplitude in u, thecontrol force has to be directed towards the fixed point,i.e. it has to be negative for u > u∗ and v > v∗. This isthe case for t < τ if K > 0, because the history functionis initialized as (u∗, v∗).

For the vu and vv coupling scheme the control forceacts parallel to the v-axis. In Fig. 9 (b) the trajectorieswith and without control are shown for vv coupling. Thedirection of the control force is denoted by the verticalarrow. To get lower amplitudes of u the control force hasto be directed towards the opposite direction of the fixedpoint. This is the case for t < τ if K < 0.

In the following the reduction of amplitude u of a sin-gle local FHN system is investigated in order to obtainthe lower boundaries (dashed) of the control domains inFig. 7. To excite the system a stimulation current I(t)(inset in Fig. 7(d)) is added to the activator u. Choos-ing for each coupling scheme the proper sign of controlgain K to reduce the activator amplitude, time-delayedfeedback is activated and the maximum amplitude umax

is observed. By performing a bisection method for eachvalue ofK, the proper τ is detected that reduces umax by10% from the original amplitude after stimulation with-out feedback (K = 0). The dashed lines in Fig. 7 markthe positions where the u-amplitude is reduced by 10%for a fixed stimulation. These lines form the lower bound-aries of the control domains.

Also the upper boundaries of the control domains canbe understood by investigating the single system withfeedback. For largeK and τ the system becomes bistablein the sense that in addition to the stable fixed point astable limit cycle occurs. This limit cycle appears in asaddle-node bifurcation of limit cycles, as was also foundin a system of two delay-coupled FHN-systems [46, 47].The limit cycle emerges if (i) τ is sufficient large to allowfor recovery to the fixed point, and (ii) the feedback andhence the gain parameter K is strong enough to push thesystem beyond the threshold.

Again with the help of a bisection method the saddle-node bifurcation lines for the single system are deter-mined in the K-τ plane. In Fig. 7 they are plotted asblack solid lines. These lines mark the boundaries wherethe local dynamics becomes bistable and control fails.Only in case of uu coupling the boundary of the con-trol domain deviates appreciably from this bifurcationline. This is due to the stabilizing effect of diffusion thatdamps out local oscillations. Since in the interspace thesingle system is bistable and thus the medium is ableto perform homogeneous oscillations, the difference be-tween the upper boundary of the control domain and thebifurcation line depends on the initial conditions that arechosen to locally stimulate the medium.

Investigating a single FHN-element with time-delayedfeedback provides a qualitative understanding of the dy-namics of the controlled spatial system: The sign of Kand the form of the control domains can be understood.Thus the control of local excitability provides control ofthe global spatial excitability. This will be quantified

10

in Sec. 4.3. In the next section, we present the spatio-temporal patterns that emerge above the upper bound-ary of the control domain, i.e., the bistable domain of thesingle local element.

4.2.2. Delay-induced oscillatory pattern formation

As we have seen in the previous section, the local FHNdynamics with time-delayed feedback becomes bistablefor too large values of K and τ . Due to this local bista-bility, in the spatially extended system delay induced os-cillatory pattern formation occurs (Fig. (8 c)): After alocal stimulation the medium performs local oscillationsthat spread slowly: Within each local oscillation the exci-tation spreads a short way, and thus gradually new partsin the neighborhood become excited. The area of oscil-latory excitation grows slowly all over the medium. Wehave observed these patterns for each coupling schemebeyond the upper boundary of the control domain.This effect will again play an important role in sec-

tion 4.3.2, that focuses on the propagation boundaries in(ε, β) - parameter space.

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1 0 1 2

v

u

with controlwithout control

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1 0 1 2

v

u

with controlwithout control

(a)

(b)

FIG. 9: Phase portraits of a local FHN system with (solid tra-jectory) and without (dashed trajectory) time-delayed feed-back control. (a) uu coupling: K=0.03 (b) vv coupling: K=-0.7. Parameters: ε = 0.1, β = 0.85, γ = 0.5, τ = 5, nodiffusion. The solid arrow denotes the action of the controlforce.

4.3. Description through effective parameters

In this section we further invastigate the two controlschemes of spatial anisotropic backward coupling and lo-cal time-delayed feedback to achieve an analytic approx-imation. In Sec. 4.3.1 we use this to describe the changeof excitability and in Sec. 4.3.2 we investigate how thisis reflected in the parameter space.For our approximation we assume that the front dy-

namics governs the location of the propagation boundary∂P . Therefore we focus on the part of control that actson the front dynamics. For the coupling types of time-delayed feedback and nonlocal anisotropic coupling thefront of the pulse receives the feedback of the homoge-neous steady state. The main idea is to replace the time-delayed (in Eq. (12)) or the space shifted (in Eq. (11))quantities by their fixed point values (u(x, t − τ) =u(x+ δ, t) = u∗ and v(x, t− τ) = v(x+ δ, t) = v∗). Thistransforms the controlled system, after some simple al-gebraic manipulations, into the form of the uncontrolledsystem (K=0), but with new effective values of the pa-

rameters ε(K), β(K) and γ(K).For each coupling scheme the obtained effective param-

eters and the corresponding transformations are shown inthe Table I. Since the delayed or shifted quantities arereplaced by the fixed point values, the obtained effective

parameters ε, β and γ only depend on K and not on τor δ.Although the transformations look rather complex

there is a common feature for vv and the two cross-coupling schemes vu and uv: The change of the parame-ters is such that the fixed point value of the system doesnot change, i.e., for these three coupling types the fixedpoint after the transformation to the system with effec-tive parameters has the same homogeneous fixed pointas the system without control. Since for the uu couplingscheme the transformation is more complex, the fixedpoint in the transformed equations differs from that inthe original equations, i.e., the intersection of the null-clines changes its relative position with respect to the cu-bic nullcline. Note that this does not mean that the fixedpoint of the original system changes by adding control.Due to the non-invasivity of all investigated control typesthe fixed point does not change. However, the transfor-mation that converts the original system with control intoa new system without control but effective parameters,is in case of uu coupling not invariant for the fixed point(u∗, v∗).The effects of parameter changes in the FHN system

are well known [3, 29]. As rule of thumb one can keep inmind that the larger ε,β or γ, the lower the excitabilityof the system.In the following the K dependence is discussed:

In case of vv coupling ε does not change. For K < 0 onlyβ becomes larger while γ decreases. However, the changeof β dominates, i.e., the excitability decreases for K < 0.For vu coupling ε and β increase for K < 0, while

γ decreases. In that case the influence of changing ε

11

TABLE I: Effective parameters and transformations for cou-pling schemes uu, uv, vu,vv.

uu uv vu vv

t t1−K t t t

x x√1−K

x x x

u u√1−K

u u u

v(v−Ku∗)√1−K

3 v −K(v∗ − v) v v

ε ε(1−K)2 (1 +K)ε (1−K)ε ε

β(β−γKu∗)

√1−K

β−Ku∗

1+Kβ+Ku∗

1−K β +Kv∗

γ γ(1−K) γ1+K

γ1−K γ +K

dominates, i.e., excitability decreases for K < 0.For uu and uv coupling ε and β increase for K > 0,

while γ decreases. The influence of ε is dominating, andtherefore the excitability decreases for positive values ofK.For the vv and vu schemes the inhibitor receives a feed-

back. In these cases by simply rearranging the inhibitorequation the original form without control, but with thenew effective parameters, can be obtained, and thereforeno transformation in time and space or of u and v vari-able has to be performed. For the two coupling schemes,for which the activator receives a feedback (uu and uv)the equations need to be transformed in order to retainthe original form without additional control force. Incase of uv only the v variable is transformed, whereasin case of uu coupling both dynamic variables u and vand also time and space are transformed. However, thisdoes not change the qualitative dynamics of the system,since transformations in u and v variable and in time andspace correspond to rescaling only.For the two cross-coupling schemes the effective pa-

rameters are symmetric with respect to K → −K. Thissymmetry of the cross-coupling schemes was observed inthe control domains of all control types (Figs. 5, 6 and7).

4.3.1. Change of excitability

In this section we clarify the influence of the controlparameterK on the excitability of the system, within theapproximation of effective parameters. In particular theinfluence of β and γ is investigated, since they changefor all control schemes in opposite ways. For vv couplingthe influence of β dominates and for the other couplingschemes the influence of ε is decisive.For the three coupling schemes vv, vu, and uv the

transformations from the original system to that with ef-fective parameters leave the fixed point invariant, i.e. thefixed point values with respect to the transformed vari-ables u and v are the same as with respect to the original

one (u and v). Since the fixed point depends only onβ and γ, the condition that the fixed point remains thesame forces the effective parameters β and γ to change si-multaneously. Thus they are not independent from eachother. The parameters −β and 1

γ define the intersection

with the u axis and the slope of the v-nullcline of theFHN system, respectively. Thus, for a smaller slope ofthe v-nullcline and unchanged fixed point, γ has to beincreased and β decreased and vice versa. For the threecases of uv, vu and vv the dependence between β and γyields the condition

β(γ) = β + (γ − γ)v∗. (13)

It is not intuitively clear how excitability changes. In-creasing γ while respecting the invariant fixed point con-dition given by Eq.(13), decreases β, which yields twoopposing effects on excitability. In the following it isshown that under the condition in Eq. (13) the influence

of β dominates.To investigate the influence of simultaneously changing

β and γ with unchanged fixed point, numerical simula-tions of a single FHN systems with effective parametersare performed under the condition in Eq. (13). To excitethe system the stimulation current I(t) (Inset in Fig.10)is added to the activator u. The response of the ac-tivator u is determined in dependence of γ(K). Notethat these simulations are done without feedback. Inorder to deduce the influence of β and γ upon excitabil-ity, ε = ε = 0.1 is not changed during the simulations.Thus the chosen parameters are equivalent to the effec-tive parameter set of vv coupling. For the other systemparameters β = 0.85 and γ = 0.5 are chosen.In Fig. 10 the maximum amplitude umax of the acti-

vator is plotted versus γ(K). For γ(K) < −0.05 the re-sponse of the system to the stimulation I(t) has a smallamplitude. For γ(K) > −0.05 the amplitude of the sys-tem blows up in a very tiny parameter range. After thisblow-up the amplitude remains for all γ(K) > −0.05 atabout the same level.This blow-up resembles the canard explosion of the

FHN system near the Hopf bifurcation. The differenceis that a canard transition usually characterizes the fastblow-up of limit cycles, whereas in case of an excitablesystem a limit cycle does not exist, but only its ghost isvisible from the excited trajectories. However, the resultis that increasing γ(K) with simultaneously adjusting βaccording to Eq. (13) increases the response to a stimu-lus. We conclude that excitability increases. Thereforeit follows that the influence of decreasing β dominates.Considering the ranges of γ(K) that result for the dif-

ferent coupling schemes by varying K in the interval[−1, 1] the impact of each effective parameter on the cou-pling schemes can be determined.For the vv coupling scheme the γ(K) is in the interval

[−0.5, 1.5] for K ∈ [−1, 1]. The excitation of the loopleads to the canard like blow-up close to γ(K) = −0.05,which is equivalent to Kvv = −0.55. Hence we estimatefor Kvv = −0.55 a change of excitability. This is in

12

-0.5

0

0.5

1

1.5

2

-0.5 0 0.5 1 1.5 2 2.5 3

um

ax

γ~

vvuv vu /

0

0.2

0.4

5 10t

I(t)

FIG. 10: Maximum amplitude umax of activator u afterstimulation with I(t)(inset: solid line). System parameters:

ε=0.1, β=β(K), γ=γ(K). The bars below the x-axis markthe ranges for the effective parameter γ(K) for vv couplingand the two cross-coupling schemes uv and vu. K is variedin the range [−1, 1].

good agreement with the onset of successful control fortime-delayed feedback in the spatially extended system(cf. Fig. 7 d)).For the two cross-coupling schemes uv and vu the

change of ε dominates. γ(K) lies in the interval [0.25,∞[.The values of γ for these two schemes are beyond thecanard-like transition, in the region of large amplitudes.For the investigated values of K the system does notundergo the canard-like transition from large to smallamplitudes and hence the influence of γ and β on theexcitability is small. Therefore the influence of ε is de-cisive. To verify this assumption we estimate the valueof Kuv/vu which is necessary to reach the propagationboundary ∂P by only considering ε. The propagationboundary ∂P for β = 0.85 and γ = 0.5 is reached forε = 0.1123. For larger ε a stable pulse does not ex-ists. Since the simulations were performed for ε = 0.1,one can compute the values of Kuv/vu needed to movefrom ε = 0.1 to ε = 0.1123, neglecting the influence ofβ and γ. In the to cross-coupling schemes this yieldsKuv/vu = ±0.123, which is in very good agreement withthe results from the simulations, where successful sup-pression is found for

∣

∣Kuv/vu

∣

∣ > 0.12.Performing the same calculation for the uu coupling

assuming that also in that case the influence of ε dom-inates, ε = 0.1123 is reached for K = 0.056, whereas inthe simulations already for K > 0.03 successful controlwas observed, which is still of the correct order.

4.3.2. Shift of propagation boundary

Above we have introduced effective parameters by re-placing the time-delayed or space-shifted quantities inEq. (12) and Eq. (11) respectively, and have investigatedthe influence on excitability for one set of system param-eters (ε = 0.1, β = 0.85, γ = 0.5) and variable controlparameter (K, δ and τ). In this section the influenceof the effective parameters on the propagation boundary

∂P in (ε, β)-parameter space (for γ = 0.5) is compared tothe one obtained through simulations with time-delayedfeedback. Therefore |K| = 0.2 and τ = 0.5∆t is chosen,where ∆t is the temporal pulse width of the activator u.The sign of the control parameter K is chosen such thatthe excitability is decreased, namely negative for vv andvu and positive for uu and uv.

The propagation boundaries are obtained in two differ-ent ways. Those with time-delayed feedback are obtainedby simulating the full equations and performing a bisec-tion method. Those propagation boundaries with effec-tive parameters are obtained by continuation of homo-clinic orbits (stationary pulse profiles) in the co-movingframe [3, 48].

In Fig. 11 the propagation boundaries ∂P are shown.Those obtained with effective parameters are the fulllines separating different colors. The white dashed curvesdisplay the propagation boundaries for the full systemwith time-delayed feedback. The boundaries are in goodagreement for vv coupling and the cross-coupling schemesuv and vu. For uu coupling the two propagation bound-ary differs from each other. The reason is that the effec-tive parameters are only exact for large τ .

The inset shows propagation boundaries of the uu cou-pling. The full line represents that obtained by effectiveparameters. The different thin lines represent the prop-agation boundaries obtained by simulating the full sys-tem with different time delays. In the range of smallε for increasing τ the displayed propagation boundariesof uu coupling converge towards the propagation bound-ary of the effective parameters. The propagation bound-ary with τ = 30∆t perfectly fits the one obtained witheffective parameters for ε < 0.04. However, for largerε, this boundary (dotted) diverges sharply from the one

0.6

0.8

1

1.2

1.4

1.6

0.04 0.08 0.12 0.16

β

ε

τ=0.5∆t

0.6

0.7

0.8

0.9

1

1.1

0.02 0.04 0.06 0.08 0.1

β

ε

uu vvvu

uv/

uu

10

30

3015

5

FIG. 11: Shift of propagation boundary ∂P : full lines: ef-fective parameters, dashed lines: with time-delayed feedbackτ = 0.5∆t. Inset: propagation boundary for uu coupling, fullline: with effective parameters, dashed lines: with differentdelay times τ = 5∆t, τ=10∆t, τ=15∆t, τ=30∆t (τ is givenin units of the pulse width ∆t). Parameters: γ = 0.5, K = 0.2for uu and uv, K = −0.2 for vu and vv.

13

with effective parameters. This is caused by the delay-induced bifurcation that occurs for large K and τ , seeSection 4.2.1. Fig. 8 (c) shows the originating patternsthat are already described in Section 4.2.2. After a lo-cal stimulation the time-delayed feedback suppresses theemerging pulse, as predicted by the effective parameters.But another effect occurs: the feedback is strong enoughto create a delay-induced oscillation. This is due to thebistability of the local system, where in addition to thestable fixed point a stable and an unstable limit cycleare born in a saddle-node bifurcation. Each of the exci-tations can spread for a small distance until it is againsuppressed by the feedback. Step by step this patternpropagates and grows.

5. DISCUSSION AND CONCLUSION

The increasing interest in both computational investi-gations of brain functioning [49] and control of complexdynamics [1] has led us to combine methodologies andconcepts from both fields to investigate the pathogenesisand potential treatment of brain disorders [50]. This is-sue sets the context for our studies. However, we expectthat our results on controlling traveling pulses can findalso applications in other fields of biomedical engineering,since traveling pulses occur in many biological systems[51, 52]. Furthermore, in this study we consider modelsand methods of generic type, i.e., on the one side, westudy pattern formation in reaction-diffusion systems ofactivator-inhibitor type, namely the FitzHugh-Nagumosystem. On the other side stands a universal methodof chaos control, that is, time-delayed feedback (Pyragascontrol [2]), which is used to control the reaction-diffusionpatterns. In addition, we consider nonlocal spatial cou-pling as a control method. Nonlocal spatial coupling andtime-delayed feedback have, as we have shown in Sec. 4,a common mechanism that underlies the control of trav-eling pulses.The discussion will focus on two issues. Firstly, the

comparison of time-delayed feedback and nonlocal spatialcoupling, in particular the predictive power of the effec-tive parameters that can be introduced in the same man-ner in both cases (Sec. 5.1). Secondly, the time-delayedfeedback and nonlocal spatial coupling will be consid-ered from a broader perspective as augmented transmis-sion capabilities in reaction-diffusion systems with par-ticular respect to the corresponding cortical structures(Sec. 5.2).

5.1. Time-delayed feedback and nonlocal spatial

coupling

The control force F (s) in Eq. (11), i. e., the force in theanisotropic nonlocal type of coupling with backward con-nections, can be directly compared to the control forceF (s) in Eq. (12), which is time-delayed feedback (Pyra-

gas control) applied to each element in the active medialocally. By going from Eq. (12) to Eq. (11) nonlocal con-nections are introduced simply by changing the positionof the shift operator from the first to the second argu-ment of signal s(x, t). In other words, Pyragas control istranslated from the temporal to the spatial domain, asillustrated in Fig. 4. This is compatible because the pulseis stationary in the co-moving frame and thus the speedof the pulse relates space to time scales. If δ and τ arenormalized to the pulse width in the spatial and tempo-ral domain, ∆x and ∆t, respectively, the common effectof nonlocal coupling and time-delayed feedback on trav-eling pulses is reflected in similar locations of the controldomains in the control planes in Fig. 6 and 7.

The analogy between nonlocal coupling and time-delayed feedback, of course, oversimplifies the situation.However, the use of both types of coupling has been pro-posed for control of spatio-temporal chaos in spatiallyextended systems based on the idea of stabilization ofunstable periodic patterns embedded in spatio-temporalchaos [53]. Using both types of coupling simultaneously,it has been demonstrated through numerical analysisthat unstable roll patterns in a transversely extendedthree-level laser model can be stabilized. The motivationfor this combined approach to control unstable periodicpatterns lies in the noninvasive character of both non-local coupling and time-delayed feedback if the unstableperiodic patterns is approached.

Our motivation to study nonlocal coupling and time-delayed feedback is somewhat different from that of chaoscontrol, because we do not want to stabilize unstableperiodic patterns. We investigate the control of trav-eling pulses by comparing nonlocal coupling with time-delayed feedback using both types separately. We sug-gest a method to predict the effect of the gain factorK on excitability by identifying the controlled systemwith the free system with effective parameters, based onthe idea that the effect of control makes its main con-tribution upon the front dynamics. Under this assump-tion, we suggest to replace in both Eq. (11) and Eq. (12)the shifted quantities by their fixed point values, i. e.,s(x, t− τ) = s(x+ δ, t) = s∗, where the signal s was cho-sen to be either u or v and s∗ is the corresponding fixedpoint value.

Despite its simplicity, this method yields accurate pre-dictions for traveling pulses invading the homogeneoussteady state. The resulting equations of effective param-eters, given in Tab. I, describe by simple algebraic rela-tions how control changes excitability (Fig. 2). However,in which direction excitability changes with changing Kin the four coupling schemes has to be investigated byevaluating the combined effect on all three effective pa-rameters ε, β and γ (Fig. 10).

The noteworthy feature of our method is that the ef-fective parameters describe a shift in the excitability ofthe local elements. By investigating time-delayed feed-back in a single local element it is shown that the onset ofpulse suppression as well as the boundedness of the con-

14

trol domains for large K and τ can be explained by thelocal dynamics. Furthermore, transferring the results oftime-delayed feedback to the coupling type of anisotropicnonlocal backward coupling one finds the same depen-dences.

5.2. Reaction-diffusion with augmented

transmission as a hybrid model for the cortex

Control introduces augmented transmission capabili-ties in the reaction-diffusion model (Fig. 3). The result-ing hybrid model in Eq. (5) combines the two major sig-nalling systems in the brain, namely local coupling bydiffusion, termed volume transmission, and nonlocal cou-

pling in the spatial domain described by Eqs. (10) and(11) or in the temporal domain by Eq. (12). The aug-mented transmission capabilities are typical for synaptictransmission and neurovascular coupling. For example,the change of sign in the gain parameter K for pulsesuppression (Fig. 5 b,c) is reminiscent of the Mexican-hat type functional and structural connectivity patternin the cortex. Also time-delays of the order of sec-onds, that is, the order of the width of the pulse profile∆t in spreading depolarizations, can occur in synaptictransmission if metabotropic ion channels are involved,like metabotropic glutamate receptors, which have in-creased open probabilities in the range of seconds aftertheir activation. Moreover, typical latencies of this or-der result from the neurovascular coupling. Therefore,the augmented transmission capabilities represent inter-nal neural circuity that is complementary to the vol-ume transmission introduced in the original Hodgkin-Grafstein Eqs. (1)-(2) as a model for spreading depolar-izations.The first hybrid model for spreading depolarizations

that also combines the two major signalling systems inthe brain has been studied by Reggia and Montgomery[54]. Potassium dynamics was modeled by a quadraticrate function (cf. Eq. (2)) and coupled to a neural net-

work that mimics cortical dynamics and sensory maporganization. The key result of this model is that atthe leading edge of the potassium pulse, the elsewherelargely uniform neural activity was replaced by a pat-tern of small, irregular patches and lines of highly activeelements, which could explain neurological symptoms inmigraine. However, there is no feedback of the networkactivity to the potassium reaction-diffusion pulse. There-fore this hybrid models can not address the questions ofthe controversially discussed Hodgkin-Grafstein mecha-nism [55, 56] because the augmented transmission ca-pabilities work only one way. In a more general con-text, such hybrid models are similar to creation of spatio-temporal networks in addressable excitable media, whichare studied in the chemical Belousov-Zhabotinskii reac-tion [57].

Volume transmission described by the Hodgkin-Grafstein mechanism was long thought to be the mainfactor that causes the propagation in spreading depo-larizations [55] although synaptic transmission and gapjunction coupling were suggested to provide an alterna-tive mechanism [56, 58]. Hybrid models can address thesecontroversial issue and may help to provide insight intothe spread and control of pathological pulses in the brain.Our emphasis is on understanding the role of internal cor-tical circuits that provide augmented transmission capa-bilities and can prevent such events. However, a long-term biomedical engineering therapeutic aim is also todesign strategies that either support the internal corticalcontrol or mimic its behavior by external control loopsand translate these methods into applications.

Acknowledgements

This work was supported by DFG in the framework ofSfb 555. The authors would like to thank Martin Gassel,Erik Glatt, Yuliya Dahlem, Steven Schiff, Ken Showalterand Hugh Wilson for fruitful discussions.

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