Paul C. Bressloff and Stefanos E. Folias- Front Bifurcations in an Excitatory Neural Network

8/3/2019 Paul C. Bressloff and Stefanos E. Folias- Front Bifurcations in an Excitatory Neural Network

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FRONT BIFURCATIONS IN AN EXCITATORY NEURAL NETWORK

PAUL C. BRESSLOFF AND STEFANOS E. FOLIAS

SIAM J. A PPL. M ATH . c 2004 Society for Industrial and Applied MathematicsVol. 65, No. 1, pp. 131151

Abstract. We show how a one-dimensional excitatory neural network can exhibit a symmetry

breaking front bifurcation analogous to that found in reaction diffusion systems. This occurs ina homogeneous network when a stationary front undergoes a pitchfork bifurcation leading to bidi-rectional wave propagation. We analyze the dynamics in a neighborhood of the front bifurcationusing perturbation methods, and we establish that a weak input inhomogeneity can induce a Hopf instability of the stationary front, leading to the formation of an oscillatory front or breather. Wethen carry out a stability analysis of stationary fronts in an exactly solvable model and use this toderive conditions for oscillatory fronts beyond the weak input regime. In particular, we show howwave propagation failure occurs in the presence of a large stationary input due to the pinning of astationary front; a subsequent reduction in the strength of the input then generates a breather viaa Hopf instability of the front. Finally, we derive conditions for the locking of a traveling front to amoving input, and we show how locking depends on both the amplitude and velocity of the input.

Key words. traveling waves, neural networks, cortical models, front bifurcations, inhomoge-neous media

AMS subject classication. 92C20

DOI. 10.1137/S0036139903434481

1. Introduction. Nonlinear integro-differential equations of the form

su (x, t )

t= u(x, t ) +

w(x x )f (u(x , t ))dx v (x, t ) + I (x),

1

v(x, t )t

= v(x, t ) + u(x, t )(1.1)have arisen as continuum models of one-dimensional cortical tissue [1, 12], in whichu(x, t ) is a neural eld that represents the local activity of a population of excitatoryneurons at position x R , I (x) is an external input current, s is a synaptic timeconstant (assuming rst-order synapses), f (u) denotes the output ring rate function,and w(x x ) is the strength of connections from neurons at x to neurons at x. Thedistribution w(x) is taken to be a positive, even function of x. The neural eldv(x, t ) represents some form of negative feedback mechanism such as spike frequencyadaptation or synaptic depression, with , determining the relative strength and rateof feedback. If additional nonlocal terms in v are introduced, then v represents insteadthe activity of a population of inhibitory neurons [17, 1]. The nonlinear function f isusually taken to be a smooth sigmoid function

f (u) =1

1 + e (u )(1.2)

with gain and threshold . The units of time are xed by setting s = 1; a typicalvalue of s is 10 msec. It can be shown [12] that there is a direct link between the above

model and experimental studies of wave propagation in cortical slices where synap-tic inhibition is pharmacologically blocked [4, 7, 18]. Since there is strong vertical

Received by the editors September 8, 2003; accepted for publication (in revised form) May 12,2004; published electronically September 24, 2004. This research was supported by NSF grant DMS-0209824.

http://www.siam.org/journals/siap/65-1/43448.html Department of Mathematics, University of Utah, 155 South 1400 East 233 JWB, Salt Lake City,

UT 84112 ([email protected], [email protected]).

131


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132 PAUL C. BRESSLOFF AND STEFANOS E. FOLIAS

coupling between cortical layers, it is possible to treat a thin cortical slice as aneffective one-dimensional medium. Analysis of the model provides valuable informa-tion regarding how the speed of a traveling wave, which is relatively straightforwardto measure experimentally, depends on various features of the underlying cortical

circuitry.A number of previous studies have considered the existence and stability of trav-eling wave solutions of (1.1) in the case of a uniform input I , which is equivalent toa shift in the threshold . In particular, it has been shown that in the absence of any feedback ( = 0), the resulting scalar network can support the propagation of traveling fronts [5, 10], whereas traveling pulses tend to occur when there is signicantnegative feedback [17, 1, 12]. In this paper, we show that such feedback can also havea nontrivial effect on the propagation of traveling fronts. This is due to the occurrenceof a symmetry breaking front bifurcation analogous to that found in reaction diffusionsystems [14, 8, 16, 9, 2, 15, 13, 11]. We begin by deriving conditions for the existence of traveling wavefronts in the case of a homogeneous network (section 2). We then carryout a perturbation expansion in powers of the wavespeed c to show that a stationaryfront can undergo a supercritical pitchfork bifurcation at a critical rate of negativefeedback, leading to bidirectional front propagation (section 3). As in the case of reaction diffusion systems, the front bifurcation acts as an organizing center for a va-riety of nontrivial dynamics including the formation of oscillatory fronts or breathers.We show how the latter can occur through a Hopf bifurcation from a stationary frontin the presence of a weak stationary input inhomogeneity (section 4). Finally, weanalyze the existence and stability of stationary fronts in an exactly solvable model,which is obtained by taking the high gain limit of the sigmoid function f such that f (u) = H (u ), where H is the Heaviside function (section 5). As brieyreported elsewhere [3], the exactly solvable model allows us to study oscillatory frontsbeyond the weak input regime. Rather than perturbing about the homogeneous case,we now consider a large input amplitude for which wave propagation failure occursdue to the pinning of a stationary front. A subsequent reduction in the amplitudeof the input then induces a Hopf instability, leading to the formation of a breather.We conclude our analysis of the exactly solvable model by deriving conditions for thelocking of a traveling front to a moving input, and we show how locking depends onboth the amplitude and speed of the input.

The major advantage of the exactly solvable model is that it allows us to explicitlydetermine the existence and stability of stationary and traveling fronts in the presenceof external inputs, without any restrictions on the size of the input. However, it hasthe disadvantage of restricting the nonlinear function f to be a step function. This isless realistic than the smooth nonlinearity (1.2), which matches quite well the inputoutput characteristics of populations of neurons. The lack of smoothness also makes itdifficult to carry out a nonlinear analysis in order to determine whether or not the Hopf instability is supercritical, for example. As we show in this paper, such an analysiscan be carried out for smooth f provided that the input amplitude is sufficiently

weak. The fact that the nonlocal integro-differential equation (1.1) exhibits behaviorsimilar to a reactiondiffusion system might not be surprising, particularly given thatfor the exponential weight distribution w(x) = e | x | , equation (1.1) can be reducedto a PDE of the reactiondiffusion type. It is important to emphasize, however,that our results hold for a more general class of weight distribution w(x) for whicha corresponding (nite-order) PDE cannot be constructed. Hence, the analysis is anontrivial extension of known results for reactiondiffusion equations.


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FRONT BIFURCATIONS IN AN EXCITATORY NEURAL NETWORK 133

2. Traveling fronts in a homogeneous network. In this section we inves-tigate the existence of traveling front solutions of (1.1) for homogeneous inputs bycombining results on scalar networks [5] with an extension of the analysis of frontbifurcations in nonscalar reactiondiffusion equations [8, 2].

2.1. The scalar case. The existence of traveling front solutions in scalar, ho-mogeneous networks was previously analyzed by Ermentrout and McLeod [5]. Theiranalysis can be applied to a scalar version of (1.1) obtained by taking sothat v = u and setting I (x) = h with h a constant input. This leads to the scalarintegro-differential equation

u (x, t )t

= (1 + )u(x, t ) +

w(x x )f (u(x , t ))dx h.(2.1)

Without loss of generality we choose h such that = 0 in the sigmoid function (1.2).The weight distribution w is assumed to be a positive, even, continuously differentiablefunction of x with unit normalization

w(y)dy = 1. Suppose that the function

F h, (u) = f (u) (1 + )u h(2.2)has precisely three zeros at u = U (h, ), U 0(h, ) with U < U 0 < U + and F h, (U ) 0.

To leading order in , u is independent of so that we can explicitly solve for v

according tov(x, t ) = v0(x)e t + u(x, t )(1 e t ).(2.6)

Thus after an initial transient of duration tO( 1), the eld v adiabatically followsthe eld u, with the latter evolving according to the scalar equation (2.1). It followsthat in the large regime there exists a unique traveling wave solution of the fullsystem with ( u(x, t ), v(x, t )) = ( U (x ct), V (x ct)) such that ( U, V ) (U , U ) as and c = c(h, ), U = U (h, ). The front is stable in the large regimeprovided that the solution of the corresponding scalar equation is stable, which isfound to be the case numerically. If h, = 0, then the front is stationary and persistsfor all but may become unstable as is reduced.

2.3. The regime 0 < 1. In the small regime, additional front solutions

can be constructed that connect the two xed points ( u, v ) = ( U (h, ), U (h, )).This follows from the observation that the neural eld v remains approximately con-stant on the length scale over which u varies, that is, within the transition layer of the front. Suppose that the system is prepared in the down state ( U , U ) and isperturbed on its left-hand side to induce a transition to the upper state ( U + , U + ). Inthis case v U within the transition layer, and this generates a front propagatingto the right whose speed is approximately given by (2.3) with h h + U , thatis, c = c(h + U , 0). If, on the other hand, the system is prepared in the up state(U + , U + ) and is perturbed on its right-hand side to induce a transition to the downstate ( U , U ), then a left-propagating front is generated with c = c(h + U + , 0).Note from (2.4) that

h + U ,0 > h, + U +

U (u U )du, h + U + ,0 < h, +

U +

U (u U + )du

(2.7)

so that h + U ,0 > h, > h + U + ,0 . Hence, the existence of fronts propagating inopposite directions clearly holds when h, are chosen such that h, = 0.

3. Front bifurcation. The above analysis suggests that if h, = 0, then atsome critical rate of feedback = c , a pair of counterpropagating fronts bifurcate


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from a stationary front. Moreover, all the front solutions have the same asymptoticbehavior ( U (), V ()) (U , U ) as . Following along lines analogous toHagberg and Meron [8], we carry out a perturbation expansion in powers of the speedc about this critical point, and we show that the stationary solution undergoes a

pitchfork bifurcation.First, set I (x) = h and ( u(x, t ), v(x, t )) = ( U (x ct), V (x ct)) in (1.1) toobtain the pair of equations

cU = U + wf (U ) V,cV = [V + U ],(3.1)

where U = dU/d and denotes the convolution operator,

wU =

w( )U ( )d .(3.2)

Suppose that and h are xed such that h, = 0, and denote the corresponding

stationary solution by ( U, V ). Expand the elds U, V as power series in c:U () = U () + cU 1() + c2U 2() + ,V () = V () + cV 1() + c2V 2() + .(3.3)

Note that the higher order terms U n (), V n (), n 1, should all decay to zero as , since the stationary solution already has the correct asymptotic behavior.Also expand according to = c + c1 + c22 + .(3.4)

Substitute these expansions into (3.1) and Taylor expand the nonlinear function f (U )about U :

f (U ) = f (U ) +n 1

f n (U U )n , f n = 1n! dn f

dU n U = U .(3.5)

Collecting all terms at successive orders of c then generates a hierarchy of equations forthe perturbative corrections U n , V n . The lowest order equation recovers the conditionsfor a stationary solution:

(1 + )U + h = wf (U ),V = U.(3.6)

At order c we have

U =

U 1 + w

[f 1U 1]

V 1 ,

V = c[V 1 + U 1] + c[V + U ].(3.7)The term V 1 in the rst line can be eliminated using the second. Since V = U , wethus nd that

MU 1 = c 1 U , V 1 = U 1 +

U c

,(3.8)


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where Mis the linear operator

MU = (1 + )U + w[f 1U ].(3.9)

Since the functions U n (), V n () decay to zero as , we will assume that Macts on the space L2(R ) and introduce the generalized inner product

U |V =

f (U ())U ()V ()d(3.10)

for all U, V L2(R ). With respect to this space, Mis self-adjoint and has the null

vector U 1 :

MU = MU = 0 .(3.11)Applying the Fredholm alternative to (3.8) then gives the solvability condition

U |U c 1 = 0 .(3.12)

Since f (U ()) > 0 for all , it follows that U |U > 0 and thus c = . This inturn means that MU 1 = 0 and hence U 1 = AU for some constant A. Since U is thegenerator of uniform translations, we are free to choose the origin such that A = 0.Under this choice,

U 1 = 0 , V 1 =U c

.(3.13)

At order c2 we obtain

U

1=

MU

2+ [

V

2+ U

2] + w

[f 2U 2

1],

V 1 = c[V 2 + U 2] + 1[V 1 + U 1] + 2[V + U ].(3.14)Substituting for V 2 + U 2 in the rst line, taking V = U , = c , and using equation(3.13) then gives

MU 2 =1c

U 1U , V 2 = U 2 +12c

(U 1U ).(3.15)

Applying the Fredholm alternative to (3.15) yields the solvability condition

U |U = 1 U |U .(3.16)In order to evaluate the inner product U

|U , we use the result

(1 + )d2U d2

=

w( )

d2f (U ( ))d 2

d ,(3.17)

1 We could equally well proceed by taking the standard inner product U |V = U ()V ()d .The adjoint of M is then given by M U = (1 + )U + f 1 w U , which has the null vector f 1 U where f 1 = f (U ).


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which follows from differentiating (3.6) with respect to and using the asymptoticproperties of w. Then

U

|U =

f (U ())U ()U ()d

=

df (U ())d

U ()d

=1

1 +

df (U ())d

w( )d2f (U ( ))

d 2d d

=1

1 +

df (U ())d

w ( )df (U ( ))

dd d

= 0 ,(3.18)

since w () is an odd function of . Hence, 1 = 0 and

MU 2 =U c , V

2 = U 2 +U 2c .(3.19)

At order c3 we obtain

U 2 = MU 3 + [V 3 + U 3] + 2w[f 2U 1U 2] + w[f 3U 31 ],V 2 = c[V 3 + U 3] + 1[V 2 + U 2] + 2[V 1 + U 1] + 3[V + U ].(3.20)

Substituting for V 2 + U 2 in the rst line, taking V = U , = c , 1 = 0, and using(3.13) and (3.19) then gives

MU 3 =12c

U 2cU , V 3 = U 3 +13c

(U + 2c U 2 2cU ).(3.21)

Applying the Fredholm alternative to (3.21) yields the solvability condition

2 =U |U

c U |U < 0.(3.22)

The sign of 2 can be determined using (3.17),

U |U =

f (U ())U ()U ()d

=

df (U ())d

U ()d

=

d

2

f (U ())d2 U ()d

= 1

1 +

d2f (U ())d2

w( )d2f (U ( ))

d 2d d

< 0,(3.23)

since w() is an even, monotonically decreasing function of ||. Hence 2 < 0.


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Combining these various results, we nd that

U () = U () + O(c2),V () = U () +

c

cU () +

O(c2),(3.24)

and

= c + c22 + O(c3).(3.25)Equation (3.25) implies that the stationary front undergoes a pitchfork bifurcation,which is supercritical since 2 < 0. (This assumes of course that the stationary frontis stable for > c . This can be conrmed numerically, and also proven analyticallyin the high gain limit; see section 5.) Close to the bifurcation point the shape of thepropagating fronts is approximately the same as the stationary front, except that therecovery variable V is shifted relative to U by an amount proportional to the speedc, that is,

U () U (), V () U ( + c/ c).(3.26)An analogous result was previously obtained for reactiondiffusion equations [8]. Itis important to emphasize that the occurrence of a pitchfork bifurcation from a sta-tionary front does not require any underlying inection symmetries of the nonlinearfunction f (see also [2]). We only require that the scalar equation (2.1) supports astationary front for appropriate choices of h, . The fact that the weight distribu-tion w(x) is even means that there must be a pitchfork bifurcation from a stationarysolution rather than a transcritical bifurcation as in the case of a nonsymmetric w.

4. The effect of a weak input inhomogeneity. Now suppose that both and h are allowed to vary. We then expect a codimension 2 cusp bifurcation in whichthe pitchfork bifurcation unfolds into a saddle-node bifurcation, with the stationaryfront replaced by a traveling front in the large regime. More interestingly, as inthe case of reactiondiffusion systems [16, 9, 2], the pitchfork bifurcation acts as anorganizing center for a variety of dynamical phenomena, including the formation of breathers due to the presence of a weak input inhomogeneity or due to curvature (inthe case of two spatial dimensions). These breathers consist of periodic reversals inpropagation that can be understood in terms of a dynamic transition between the pairof counterpropagating fronts that is induced by the weak intrinsic perturbation. Sucha transition involves an interaction between a translational degree of freedom and anorder parameter that determines the direction of propagation. In order to unravelthis interaction, it is necessary to extend the perturbation analysis of section 3 alonglines analogous to previous treatments of reactiondiffusion systems [16, 9, 2].

Suppose that the system (1.1) undergoes a pitchfork bifurcation from a stationarystate when = c = and I (x) = h. Introduce the small parameter according toc =

2

and introduce a weak input inhomogeneity by taking I (x) = h + 3

(x).Since any fronts are slowly propagating, we rescale time according to = t so that(1.1) becomes

u (x, )

= u(x, t ) +

w(x x )f (u(x , ))dx v(x, ) h + 3(x),

v(x, )

= ( c + 2 ) [v(x, ) + u(x, )] .(4.1)


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Motivated by (3.24), we introduce the ansatz that, sufficiently close to the pitchforkbifurcation, the solutions of (4.1) can be expanded in the form

u(x, ) = U (x p( )) + 2u2(x, ) + 3u3(x, ) + ,v(x, ) = U (x p( )) + a( )c U (x p( )) +

2v2(x, ) + 3v3(x, ) + .(4.2)Here p is identied with the translational degree of freedom, whereas a representsthe order parameter associated with changes in propagation direction. Note that a isassumed to evolve on a slower time scale than p. We now substitute the ansatz (4.2)into (4.1) and expand in powers of along lines similar to the perturbation calculationof section 3.

At order we nd that

p = a,(4.3)

where p = dp/d . At order 2 we obtain the pair of equations

Mu2 = a2U c

, v2 = u2 + a2U 2c

(4.4)

after setting p = a. The solvability condition for (4.4) is automatically satised. Atorder 3 we have

u 2

= Mu3 + [v3 + u3] + ,v2

+U a

c= c[v3 + u3]a

U c

(4.5)

with = . Using (4.4), the following equation for u3 is obtained:

Mu3 =12c

a3U a cU a cU .(4.6)

Applying the Fredholm alternative to (4.6) yields an amplitude equation for a:

a = a + a3U |U

c U |U c

U |U |U

.(4.7)

Finally, rescaling p, a , and , we obtain the pair of equations

pt = a,

a t = ( c )a +U

|U

c U |U a3

cU

|

U |U .(4.8)Note that U = U (x p), so that the nal coefficient on the right-hand side of (4.8)will be p-dependent in the case of an inhomogeneous input = (x).

Cusp bifurcation for homogeneous inputs. It is clear from (4.8) that when = 0we recover the pitchfork bifurcation of a stationary front as found in section 3. Inparticular, for < c there are three constant speed solutions of (4.8) such that


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a t = 0 , P t = a = c, corresponding to an unstable stationary front and a pair of stablecounterpropagating fronts with speeds

c = (c )c

U

|U

| U |U |.(4.9)If is nonzero but constant, on the other hand, the nal term on the right-handside of (4.8) reduces to the constant coefficient c(f (U + ) f (U )) / U |U , and thepitchfork bifurcation unfolds to a saddle-node bifurcation. There are two saddle-node lines in the ( , )-plane corresponding to the condition dG(a)/da = 0, wherea t = G(a):

sn = 2

33(c )3/ 2

1/ 2c

U |U 3/ 2(f (U + ) f (U )) | U |U |1/ 2

,(4.10)

and the corresponding speed along these lines is

csn = (c )c U |U 3| U |U |.(4.11)Hopf bifurcation for a weak inhomogeneity. The introduction of a weak input

inhomogeneity can lead to a Hopf instability of the stationary front. We shall illustratethis by considering the particular example of the step inhomogeneity

(x) = s/ 2 if x 0,s/ 2 if x > 0

(4.12)

with s > 0. For such an input we nd that

U | =s2[2f (U ( p)) f (U + ) f (U )].(4.13)

Recall from section 2 that when h = 0 .5 the homogeneous network with f given by(1.2) supports a stationary front solution for which U = 0.5/ (1 + ), and U (0) = 0such that f (U + ) + f (U ) = 2 f (0). Hence, (4.8) has a xed point at p = 0 , a = 0.Linearization about this xed point shows that there is a Hopf bifurcation of thestationary front at = c with Hopf frequency

H = s cf (0) |U (0) |U |U

.(4.14)

The supercritical or subcritical nature of the Hopf bifurcation can then be determinedby evaluating higher order terms in a, p . However, this is complicated by the factthat we do not have an analytical expression for the stationary front solution U , incontrast to the case of a reactiondiffusion equation with a cubic nonlinearity [2].(Note that, as in the case of reactiondiffusion equations [2], one can develop a moreintricate perturbation analysis that takes into account O(2) inhomogeneities andcorresponding shifts in the Hopf bifurcation point. Here we have followed a simplerapproach in order to illustrate the basic ideas underlying the perturbative treatmentof the integro-differential equation (1.1).)


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5. Exactly solvable model. We now consider the high gain limit , forwhich (1.2) reduces to f (u) = H (u ), where H is the Heaviside function H (u) = 1if u > 0 and H (u) = 0 if u 0. The advantage of using a threshold nonlinearity isthat explicit analytical expressions for front solutions can be obtained, which allowsus to derive conditions for the Hopf instability of a stationary front without anyrestrictions on the size of the input inhomogeneity. Numerical simulations of thefull system establish that the bifurcation is supercritical and that it generates anoscillatory modulation of the stationary front in the form of a breather [3]. (For acorresponding analysis of reactiondiffusion equations, see Prat and Li [13].)

5.1. Traveling fronts (homogeneous case). We begin by deriving exact trav-eling front solutions of (1.1) for f (u) = H (u ) and a homogeneous input I (x) = 0.That is, we seek a solution of the form u(x, t ) = U (), = x ct, c > 0, such thatU (0) = , U () < for > 0 and U () > for < 0. Setting v(x, t ) = V (), we thenhave

cU () + U () =

0

w(

)d

V (),(5.1)

c

V () = V () + U ().(5.2)Differentiating the rst equation and substituting into the second, we obtain a second-order ODE with boundary conditions at = 0 and :

c2U () + c[1 + ]U () [1 + ]U () = cw() W (),U (0) = ,

U () = U ,(5.3)where

W () =

w(y)dy.(5.4)

Here U are the homogeneous xed point solutions

U + =1

1 + , U = 0 .(5.5)

We have used the fact that w has unit normalization, W ()

w(y)dy = 1. Itfollows that a necessary condition for the existence of a front solution is < U + .

In order to establish the existence of a traveling front, we solve the boundaryvalue problem in the domains

0 and

0 and match the solutions at =

0. For further mathematical convenience, we take the weight distribution to be anexponential function

w(x) =1

2de| x | /d ,(5.6)

where d determines the range of the synaptic interactions. We x the spatial scale bysetting d = 1; a typical value of d is 1 mm. We rst consider the case of right-moving


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waves (c > 0). On the domain 0, the particular solution is U > () = e , with related to the speed c according to the self-consistency condition =

c +

2(c2

+ c[1 + ] + [1 + ])

, c

0.(5.7)

In the domain 0 the solution consists of complementary and particular parts:U < () = A+ e + + A e + Ae + U + ,(5.8)

where

=12c

1 + (1 + )2 4(1 + ) .(5.9)The coefficient Ais obtained by direct substitution into the differential equation forU , whereas the coefficients A are determined by matching solutions at the boundary = 0, that is, U < (0) = and U < (0) = . Thus we nd

A= c 2(c2 c[1 + ] + [1 + ]),(5.10)

A+ = U + + ( 1)A (1 + )

+ ,(5.11)

A = + U + + (1 + )A+ (1 + + )

+ .(5.12)

In the limit 0 we recover the standard result for an excitatory network withoutfeedback [5]:

U () =

12(c + 1)

e for > 0,

1 + ( 1)e/c +1

2(c 1)e e/c for < 0

(5.13)

with

=1

2(c + 1), c 0.(5.14)

A similar analysis can be carried out for left-moving waves. Now the speed c isdetermined by the particular solution in the domain 0, which takes the formU < () =

e + U + with = (1 + ) 1

. This leads to the self-consistency

condition

= c

2(c2 c[1 + ] + [1 + ]), c 0.(5.15)

The existence of traveling front solutions can now be established by nding posi-tive real solutions of (5.7) and negative real solutions of (5.15). For concreteness, wewill assume that the threshold is xed and determine the solution branches as a


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0 0.5 1 1.5-1

-0.5

0

0.5

1

0 0.5 1 1.5

-2

-1

0

1

0 0.5 1 1.5

-0.5

0

0.5

1

c

= 1.0

= 1.5 = 0.75

Fig. 5.1 . Plot of wavefront speed c as a function of for various values of and a xed threshold = 0 .25: (i) 2 (1 + ) = 1 , (ii) 2 (1 + ) > 1, (iii) 2 (1 + ) < 1. Stable (unstable)branches are shown as solid (dashed) curves.

function of the feedback parameters , with 1/ 1 > > 0. The roots of (5.7)and (5.15) can be written explicitly as

c =1

2

1 +

1

2 1 +

1

2

2

4 1 +

1

2

(5.16)

and

c =12

1 + 1

2 1 + 12 2 4 1 + 12

.(5.17)

Using the fact that sign 1 + 12 = sign 1 + 12 , we nd that there arethree bifurcation scenarios, as shown in Figure 5.1:(i) If 2(1 + ) = 1, then there exists a stationary front for all . At a critical

value of the stationary front undergoes a pitchfork bifurcation, leading tothe formation of a left- and a right-moving wave. This is the high gain limit

of the front bifurcation analyzed in section 3 for smooth f .(ii) If 2(1 + ) > 1, then there is a single left-moving wave for all . There alsoexists a pair of right-moving waves that annihilate in a saddle-node bifurcationat a critical value of that approaches zero as 0.(iii) If 2(1+ ) < 1, then there is a single right-moving wave for all . There alsoexists a pair of left-moving waves that annihilate in a saddle-node bifurcationat a critical value of that approaches zero as 0.

5.2. Stability analysis of stationary fronts (inhomogeneous case). Sta-tionary front solutions of (1.1) with f (u) = H (u ) in the case of an inhomogeneousinput I (x) satisfy the equation

(1 + )U (x) =

x 0

w(x x )dx + I (x).(5.18)

Suppose that I (x) is a monotonically decreasing function of x. Since the system is nolonger translation invariant, the position of the front is pinned to a particular locationx0 , where U (x0) = . Monotonicity of I (x) ensures that U (x) > for x < x 0 andU (x) < for x > x 0 . The center x0 satises

(1 + ) =12

+ I (x0)(5.19)


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under the normalization

0 w(y)dy = 1 / 2. Equation (5.19) implies that in contrastto the homogeneous case, there exists a stationary front over a range of thresholdvalues (for xed ); changing the threshold simply shifts the position of the centerx0 . In the particular case of the exponential weight distribution (5.6), we have

(1 + )U (x) =

ex 0 x

2+ I (x), x > x 0 ,

1 ex x 0

2+ I (x), x < x 0 .

(5.20)

If the stationary front is stable, then it will prevent wave propagation. Stabilityis determined by writing u(x, t ) = U (x) + p(x, t ) and v(x, t ) = V (x) + q(x, t ) withV (x) = U (x) and expanding (1.1) to rst-order in ( p, q):

p(x, t )t

= p(x, t ) +

w(x x )H (U (x )) p(x , t )dx q(x, t ),

1

q(x, t )t

= q(x, t ) + p(x, t ).(5.21)

We assume that p, qL2(R ). The spectrum of the associated linear operator is found

by taking p(x, t ) = e t p(x) and q(x, t ) = e t q(x). Using the identity

dH (U (x))dU

=(x x0)|U (x0)|

(5.22)

we obtain the equation

( + 1) p(x) =w(x

x0)

|U (x0)| p(x0) p (x) + .(5.23)

Equation (5.23) has two classes of solution. The rst consists of any function p(x)such that p(x0) = 0, for which =

(0) , where

(0) = (1 + ) (1 + )2 4(1 + )2 .(5.24)Note that (0) belong to the essential spectrum since they have innite multiplicity.The second class of solution is of the form p(x) = Aw (x x0), A = 0, for which isgiven by the roots of the equation

+ 1 + +

= 12|U (x0)|

.(5.25)

Since

U (x0) =1

1 + I (x0)

12

,(5.26)


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it follows that = , where

= 2 4(1 )(1 + )2(5.27)with

= 1 + (1 + )(5.28)and

=1

1 + 2 D, D = |I (x0)|.(5.29)

We have used the fact that I (x0) 0. The eigenvalues determine the discretespectrum.5.3. Hopf bifurcation to a breathing front. Equation (5.27) implies that the

stationary front is locally stable, provided that > 0 or, equivalently, the gradient of the inhomogeneous input at x0 satises

D > D c 12

1 +

.(5.30)

Since D 0, it follows that the front is stable when < , that is, when the feedback issufficiently weak or fast. On the other hand, if > , then there is a Hopf bifurcationat the critical gradient D = Dc . The corresponding critical Hopf frequency is

H = 2D c(1 + )2Dc + 1 = ( ).(5.31)Note that the frequency depends only on the size and rate of the negative feedbackbut is independent of the details of the synaptic weight distribution and the size of theinput. This should be contrasted with the corresponding Hopf frequency in the caseof a smooth nonlinearity f and a weak step-inhomogeneity; see (4.14). The latterdepends on the input amplitude and the form of the stationary solution U , whichitself depends on the weight distribution w.

In order to investigate the nature of solutions around the Hopf bifurcation point,we consider the particular example of a smooth ramp inhomogeneity

I (x) = s2

tanh( x ),(5.32)

where s is the size of the step and determines its steepness. A stationary front willexist provided that

s > s |1 2(1 + )|.(5.33)The gradient D = s sech2(x 0)/ 2 depends on x0 , which is itself dependent on and through (5.19). Using the identity sech 2x = 1 tanh 2 x, it follows that

D = 2s

s2 s2 .(5.34)


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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

s

= 0.5 = 1.0 = 1.5

Fig. 5.2 . Stability phase diagram for a stationary front in the case of a step input I (x) =

s tanh( x )/ 2, where is the steepness of the step and s its height. Hopf bifurcation lines (solid curves) in s parameter space are shown for various values of . In each case the stationary front is stable above the line and unstable below it. The shaded area denotes the region of parameter spacewhere a stationary front solution does not exist. The threshold = 0 .25 and = 0 .5.

Combining (5.30) and (5.34), we obtain an expression for the critical value of s thatdetermines the Hopf bifurcation points:

sc =1

2

1 +

+ 1 + 2

+ 4 s2 2

.(5.35)

The critical height sc is plotted as a function of for various values of and xed, in Figure 5.2. Note that in the homogeneous case ( s = 0) a stationary solutionexists only at the particular value of given by = 1 / (2)1. This solution is stablefor > and unstable for < , which is consistent with the pitchfork bifurcationshown in Figure 5.1. Close to the front bifurcation = , the Hopf bifurcationoccurs in the presence of a weak input inhomogeneity, which is the case consideredin section 2. Now, however, it is possible to determine the bifurcation curve withoutany restrictions on the size of the input.

Numerically solving the full system of equations (1.1) for a step input I (x), ex-ponential weights w(x), and threshold nonlinearity f (u) = H (u ) shows that theHopf bifurcation is supercritical, in which there is a transition to a small amplitudebreather whose frequency of oscillation is approximately equal to the Hopf frequencyH . As the input amplitude s is reduced beyond the Hopf bifurcation point, the am-plitude of the oscillation increases until the breather itself becomes unstable and thereis a secondary bifurcation to a traveling front. This is illustrated in Figure 5.3, whichshows a space-time plot of the developing breather as the input amplitude is slowlyreduced. Note that analogous results have been obtained for pulses in the presenceof stationary Gaussian inputs, where a reduction in the input amplitude induces aHopf bifurcation to a pulse-like breather [3, 6]. Interestingly, the localized breathercan itself undergo a secondary instability leading to the periodic emission of travelingwaves. In one dimension such waves consist of pairs of counterpropagating pulses,whereas in two dimensions the waves are circular target patterns [6].


17/21


-20 200space x (in units of d)

0

100

180

t i m e

t ( i n u n

i t s o

f

)

0

0.8

0.6

0.4

0.2

activity u

Fig. 5.3 . Breather-like solution arising from a Hopf instability of a stationary front due to a slow reduction in the size s of the step input inhomogeneity (5.32) . Here = 0 .5, = 0 .5, = 1 , = 0 .25.The input amplitude s = 2 at t = 0 and s = 0 at t = 180 . The amplitude of the oscillation steadily grows until it destabilizes at s 0.05, leading to the generation of a traveling front.

5.4. Locking to a moving input. We conclude our analysis of the exactlysolvable model by considering the effects of a moving input stimulus. This is inter-esting from a number of viewpoints. First, introducing a persistent stationary inputinto an in vitro cortical slice can damage the tissue, whereas a moving input (at leastif it is localized) will not. Second, in vivo inputs into the intact cortex are typicallynonstationary, as exemplied by inputs to the visual cortex induced by moving visualstimuli. We consider the particular problem of whether or not a traveling front canlock to a step-like input I (x) = I 0 (x vt) traveling with constant speed v, where

(x) =

1, x > 0,0, x = 0 ,

+1 , x < 0.

Such a front moves at the same speed as the input but may be shifted in space relativeto the input.

We proceed by introducing the traveling wave coordinate = x vt and derivingexistence conditions for a front solution U () satisfying U () 0 as , U () (1 + ) 1 as , and U (0) = . Substituting into (1.1) gives

vU () = U () + 0

w( )d V () + I 0 (),(5.36)

vV () = (V () + U ()) .(5.37)Setting W () =

w()d, we can rewrite this pair of equations in the matrix form

LS vU U V vV + U V

= N E0 ,(5.38)

where

S = ( U, V )T , N E () = W ( 0) + I 0 ().(5.39)


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We use variation of parameters to solve this linear equation. The homogeneous prob-lem LS = 0 has the two linearly independent solutions,

S+ () =

m+ 1exp(+ ),(5.40)

S () =

m 1exp( ),(5.41)

where

=m

v, m =

12

1 + (1 )2 4 .By variation of parameters we dene

S() = [S+ ()|S ()]a()b() ,

where [A |B ] denotes the matrix whose rst column is dened by the vector A andwhose second column is dened by the vector B . ThenLS = v

[S+ ()|S ()]a()b()

1

[S+ ()|S ()]

a()b()

= v[S+ ()|S ()]

a()b() ,(5.42)

since LS = 0. Thus (5.38) reduces to

[S+ ()|S ()]

a()b() =

1v

N E0 .(5.43)

The matrix [ S+ ()

|S ()] is invertible. Introducing the vector-valued functions

Z+ () =1 m exp(+ ),(5.44)

Z () = 1 m+ exp( ),(5.45)

we have

[S+ |S ][Z+ |Z ]T = [Z+ |Z ]T [S+ |S ] = (m+ m )I,where I denotes the identity matrix. Multiplying (5.43) by [ Z+ |Z ]T nally yields therst-order equation

a()b() = 1v (m+ m )[Z+ ()|Z ()]T N E ()0 .(5.46)

In order to solve (5.46) we need to specify the sign of v. First, suppose that v > 0,which corresponds to a right-moving front. Integrating over the interval [ , ) gives

a()b() =

ab

+1

v (m+ m )

[Z+ ()|Z ()]T

N E ()0 d,


19/21


where a , b denote the values of a, b at . Since we seek a bounded solution S(),we must require that a = b = 0. Hence the solution isa()

b()=

1

v (m+ m )

[Z+ ()

|Z ()]T

N E ()

0d,

so that

S() =1

v (m+ m )[S+ ()|S ()]

[Z+ ()|Z ()]T

N E ()0 d.(5.47)

Further simplication occurs by introducing the functions

M () =1v

1m+ m

e ( ) N E ()d.

We can then express the solution for ( U (), V ()) as follows:

U () = (1 m )M + () (1 m+ )M (),(5.48)V () = 1(m+ 1)(1 m ) [M + () M ()] .(5.49)

To ensure that such a front exists we require that U (0) = , i.e.,

= (1 m )M + (0) (1 m+ )M (0).(5.50)Taking w(x) = e | x | / 2 so that

W () =

1 12

e , < 0,

12

e , 0,we can calculate M (0) explicitly as

M (0) =1

(m + m )1

2(v + m ) 1

m F (0) ,

where

F (0) =

I 0(2e 0 1), 0 < 0,I 0 , 0 0.

The case of a left-moving front for which v < 0 follows along similar lines byintegrating (5.46) over ( , 0]:

U () = ( m 1)M + () (m+ 1)M (),(5.51)

V () = 1(m+ 1)(1 m ) M + () M () ,(5.52)


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0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

i n p u

t a m p

l i t u

d e

I 0

-2 -1input velocity v

1 20

Fig. 5.4 . Locking of a traveling front to a moving step input with velocity v and amplitude I 0 .Other parameter values are = 1 , = 0 .1, = 0 .25. Unshaded regions show where locking can occur in the (v, I 0 )-plane. When I 0 = 0 there are three front solutions corresponding to a stationary front ( v = 0 ) and two counterpropagating fronts, which is consistent with the front bifurcation shown in Figure 5.1. Each of these solutions forms the vertex of a distinct locking region whose width increases monotonically with I 0 so that ultimately the locking regions merge.

where

M (0) =1

(m + m )12

m 2vm (v m )

1m

G(0)

and

G(0) =

I 0 , 0 < 0,I 0(1 2e 0 ), 0 0.

This leads to the following threshold condition for v < 0:

= ( m 1)M + (0) (m+ 1)M (0).(5.53)We can now numerically solve (5.50) and (5.53) to determine the range of input

velocities v and input amplitudes I 0 for which locking occurs. For the sake of illustra-tion, we assume the threshold condition 2 (1 + ) = 1 and take < . This ensuresthat, in the absence of any input, there exists an unstable stationary front and apair of stable counterpropagating waves (see Figure 5.1). The continuation of thesestationary and traveling fronts as I 0 increases from zero is shown in Figure 5.4. Since2(1+ ) = 1, equations (5.50) and (5.53) are equivalent under the interchange v vand 0 0 . This implies that the locking regions are symmetric with respect tov. For nonzero v the traveling front is shifted relative to the input such that 0 < 0when v > 0 and 0 > 0 when v < 0. In other words, the wave is dragged by the input.

Figure 5.4 determines where locking can occur but not whether the resultingtraveling wave is stable or unstable. Indeed, the stability analysis of traveling frontsis considerably more involved than that of stationary fronts. Nevertheless, we expectthat for sufficiently small I 0 the locking regions around the counterpropagating frontsare stable, whereas the central region containing the stationary front is unstable. Onthe other hand, since > , we know that the stationary front is stable for largeinputs I 0 and undergoes a Hopf bifurcation as I 0 is reduced. This suggests that theHopf bifurcation point at v = 0 lies on a Hopf curve within the locking region so that


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a traveling front locked to a moving input can also be destabilized as the strength of the input is reduced (or as the input velocity changes relative to the intrinsic velocityof waves in the homogeneous network). Recently, Zhang [19] analyzed the asymptoticstability of traveling wave solutions of (1.1) in the case of homogeneous inputs by

deriving the associated Evans function and evaluating it in the singular limit 1.In future work we will extend this analysis to the case of inhomogeneous inputs andnite , thus determining the stability of the locking regions shown in Figure 5.4. Wewill also construct corresponding locking regions for traveling pulses in the presenceof moving Gaussian inputs, and numerically explore the types of oscillatory solutionsbifurcating from these waves.

Acknowledgment. We would like to thank Yue-Xian Li (University of BritishColumbia) for many helpful discussions regarding his work on wavefront instabilitiesin reactiondiffusion equations.

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[3] P. C. Bressloff, S. E. Folias, A. Prat, and Y.-X. Li , Oscillatory waves in inhomogeneousneural media , Phys. Rev. Lett., 91 (2003), article 178101.

[4] R. D. Chervin, P. A. Pierce, and B. W. Connors , Periodicity and directionality in the prop-agation of epileptiform discharges across neocortex , J. Neurophysiol., 60 (1988), pp. 16951713.

[5] G. B. Ermentrout and J. B. McLeod , Existence and uniqueness of travelling waves for a neural network , Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), pp. 461478.

[6] S. E. Folias and P. C. Bressloff , Breathing pulses in an excitatory neural network , SIAMJ. Appl. Dynam. Systems, to appear.

[7] D. Golomb and Y. Amitai , Propagating neuronal discharges in neocortical slices: Computa-tional and experimental study , J. Neurophysiol., 78 (1997), pp. 11991211.

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[9] A. Hagberg, E. Meron, I. Rubinstein, and B. Zaltzman , Controlling domain patterns far from equilibrium , Phys. Rev. Lett., 76 (1996), pp. 427430.

[10] M. A. P. Idiart and L. F. Abbott , Propagation of excitation in neural network models ,Network, 4 (1993), pp. 285294.

[11] Y.-X. Li , Tango waves in a bidomain model of fertilization calcium waves , Phys. D, 186 (2003),pp. 2749.

[12] D. J. Pinto and G. B. Ermentrout , Spatially structured activity in synaptically coupled neu-ronal networks : I. Traveling fronts and pulses , SIAM J. Appl. Math., 62 (2001), pp. 206225.

[13] A. Prat and Y.-X. Li , Stability of front solutions in inhomogeneous media , Phys. D, 186(2003), pp. 5068.

[14] J. Rinzel and D. Terman , Propagation phenomena in a bistable reaction-diffusion system ,SIAM J. Appl. Math., 42 (1982), pp. 11111137.

[15] J. E. Rubin , Stability, bifurcations and edge oscillations in standing pulse solutions to an inhomogeneous reaction-diffusion system , Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999),pp. 10331079.

[16] P. Schutz, M. Bode, and H.-G. Purwins , Bifurcations of front dynamics in a reaction-diffusion system with spatial inhomogeneities , Phys. D, 82 (1995), pp. 382397.

[17] H. R. Wilson and J. D. Cowan , A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue , Kybernetik, 13 (1973), pp. 5580.

[18] J.-Y. Wu, L. Guan, and Y. Tsau , Propagating activation during oscillations and evoked responses in neocortical slices , J. Neurosci., 19 (1999), pp. 50055015.

[19] L. Zhang , On stability of traveling wave solutions in synaptically coupled neuronal networks ,Differential Integral Equations, 16 (2003), pp. 513536.

Documents

Paul C. Bressloff and Stefanos E. Folias- Front Bifurcations in an Excitatory Neural Network