S Coombes et al- Modeling electrocortical activity through improved local approximations of integral neural field equations

8/3/2019 S Coombes et al- Modeling electrocortical activity through improved local approximations of integral neural field eq

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Modeling electrocortical activity through improved local approximations of integral

neural field equations

S Coombes1 and N A Venkov1 and L Shiau2 and I Bojak3 and D T J Liley4 and C R Laing51School of Mathematical Sciences, University of Nottingham, NG7 2RD, UK.

2Department of Mathematics, University of Houston, Houston, TX 77058, USA.3Department of Cognitive Neuroscience, Radboud University Nijmegen Medical Centre,

Postbus 9101, 6500 HB Nijmegen, The Netherlands.4

Brain Sciences Institute, Swinburne University of Technology, PO Box 218, Victoria 3122, Australia.5Institute of Information and Mathematical Sciences, Massey University,

Private Bag 102-904, North Shore Mail Centre, Auckland, New Zealand.(Dated: October 3, 2007)

Neural field models of firing rate activity typically take the form of integral equations with space-dependent axonal delays. Under natural assumptions on the synaptic connectivity we show howone can derive an equivalent partial differential equation (PDE) model that properly treats theaxonal delay terms of the integral formulation. Our analysis avoids the so-called long-wavelengthapproximation that has previously been used to formulate PDE models for neural activity in twospatial dimensions. Direct numerical simulations of this PDE model show instabilities of the ho-mogeneous steady state that are in full agreement with a Turing instability analysis of the originalintegral model. We discuss the benefits of such a local model and its usefulness in modeling electro-cortical activity. In particular we are able to treat patchy connections, whereby a homogeneousand isotropic system is modulated in a spatially periodic fashion. In this case the emergence ofa lattice-directed traveling wave predicted by a linear instability analysis is confirmed by thenumerical simulation of an appropriate set of coupled PDEs.

PACS numbers: 87.10.+e

I. INTRODUCTION

In many regions of mammalian neocortex, synapticconnectivity patterns follow a laminar arrangement, withstrong vertical coupling between layers. Consequentlycortical activity is considered as occurring on a two di-

mensional plane, with the coupling between layers en-suring near instantaneous vertical propagation. Hence,it is highly desirable to obtain solutions to fully planarneural field models. The most popular Wilson-Cowan[1] or Amari [2] style neural field models are typicallywritten in the language of integro-differential or purelyintegral equations (see [35] for recent reviews). In re-cent years there has been a growing interest in neuralfield models where the communication between differentparts of the domain is delayed due to the finite conduc-tion speed of action potentials [610]. The advent of thiswork can be traced back to that of Nunez [11]. How-ever, the set of mathematical techniques for the analysis

of non-local models with space-dependent delays is notyet as thoroughly developed as it is for local partial dif-ferential equation (PDE) models. As discussed in [12]a local PDE model would offer a number of advantagesover its non-local counterpart, allowing the use of i) pow-erful techniques from nonlinear PDE theory, ii) standardnumerical techniques for the solution of PDEs, and iii)a more numerically straightforward analysis of the ef-fects of spatial inhomogeneities. To date progress in thisarea has been made by Jirsa and Haken [13] for neuralfield models in one spatial dimension with axonal delays,and by Laing and Troy [14] in two spatial dimensions

for models lacking axonal delays. In both cases integraltransform techniques are exploited and a PDE descrip-tion is obtained only when the integral models under con-sideration are defined by spatio-temporal kernels whoseFourier transform has a rational polynomial structure.It is the goal of this paper to address the physiologicallyimportant case of a model in two spatial dimensions with

axonal delays and to obtain an equivalent PDE model.Previous work on this problem by Liley et al. [12] hasshown that for synaptic connectivity functions that falloff exponentially with distance, there is an equivalent lo-cal model consisting of an infinite set of PDEs involvingfractional derivative terms. Although not particularlyuseful in its own right this system can be approximatedby a single hyperbolic PDE. This PDE has been shownto provide a so-called long-wavelength approximation tothe underlying integral model, and equations of this typehave been used in several EEG modeling studies (see forexample [1517]).

In section II we introduce the class of neural field popu-lation models that we study in this paper. Next in sectionIII we derive the equivalent PDE model, and compareit to the model obtained using the long-wavelength ap-proximation. In section IV we present a Turing instabil-ity analysis of the original integral model. Importantlywe show that numerical simulations of the PDE modelare in precise agreement with the behavior predicted atthe onset of a Turing instability. The case of spatiallymodulated synaptic connectivity is treated in section V.Finally, in section VI we discuss natural extensions of thework in this paper.


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II. INTEGRAL NEURAL FIELD MODEL

We consider planar neural field models that incorpo-rate delayed synaptic interactions between distinct neu-ronal populations where the activity of synapses in pop-ulation a induced by activity in population b can be writ-ten

uab = ab ab. (1)

Here uab = uab(r, t), r = (r, ) (r R+, [0, 2), t R+), and a and b label functionally homogeneous neu-ronal populations. The activity variable uab(r, t) can beinterpreted as a spatially averaged synaptic activity cen-tered about r. The symbol represents a temporal con-volution in the sense that

( )(r, t) =

t0

ds(s)(r, t s). (2)

The variable ab(r, t) describes the presynaptic input topopulation a arriving from population b, which we write

as

ab(r, t) =

R2

drwab(r, r)fbhb(r

, t|rr|/vab). (3)

The function ab(t) (with ab(t) = 0 f o r t < 0) rep-resents a normalised synaptic filter, whilst wab(r, r

)is a synaptic footprint describing the anatomy of net-work connections. One common choice for the synap-tic filter is the so-called delayed difference of exponen-tials: ab(t) = (t ab; ab, ab), where (t; , ) =(1/ 1/)1[et et ](t) and ab is a mean synap-tic processing delay between populations a and b. Here,(t) is the Heaviside step function. In the absence ofdetailed anatomical data it is common practice to con-sider cortico-cortical connectivity functions to be homo-geneous and isotropic so that wab(r, r

) = wab(|r r|).

The function fa represents the firing rate of population a,and vab is the mean synaptic axonal velocity along a fibreconnecting population b to population a. For conductionvelocities in the range 1.57 m/s (typical of white matteraxons) axonal delays are significant over scales rangingfrom a single cortical area (of spatial scale 10mm) up tothe scale of inter-hemispherical collosal connections.

In a Wilson-Cowan or Amari style neural field modelthe variables ha are taken to be of the form ha =

b uab+

h0a

, with h0a

a constant drive term. In more sophisticatedmodels of EEG activity, such as in the work of Liley etal. [12], ha is interpreted as the average soma membranepotential of a population and chosen to obey a nonlinearequation of the form

(1 + at)ha =b

ab(ha)uab + h0a. (4)

Here the activity dependent functions ab weight the con-tributions from the various contributing neuronal pop-ulations, and take into account the shunting nature ofsynaptic interactions (see [12] for details).

III. EQUIVALENT PDE MODEL

The numerical solution of the neural field model de-fined in section II is challenging for two reasons in partic-ular. The first being that the non-local presynaptic inputterm (3) is defined by an integral over a two-dimensionalspatial domain, and the second that it involves an ar-gument that is delayed in time. In fact since this delayterm is space-dependent it requires keeping a memory ofall previous synaptic activity. One of the key motiva-tions of our work is to circumvent the huge numericaloverheads in simulating such a delayed non-local system.

Introducing Gab(r, t) = Gab(r, t) (where r = |r|) with

Gab(r, t) = wab(r)(t r/vab), (5)

allows us to re-write (3) as

ab(r, t) =

ds

R2

drGab(|rr|, t s)b(r

, s), (6)

where a = faha. Importantly the right hand side of (6)has a convolution structure. Introducing the 3D Fouriertransform according to

(r, t) =1

(2)3

R3

dkd(k, )ei(kr+t), (7)

then we find that Gab(k, ) = Gab(k, ), (k = |k|) where

Gab(k, ) = 2

0

wab(r)J0(kr)reir/vabdr. (8)

Here J0(z) = (2)120

deiz cos is the Bessel functionof the first kind of order zero. We recognize (8) as theHankel transform of wab(r)e

ir/vab .If Gab(k, ) can be represented in the form

Rab(k2, i)/Pab(k2, i) then we have thatPab(k2, i)ab(k, ) = Rab(k2, i)b(k, ). By identi-fying k2 2 and i t, then a formal inverseFourier transform will yield a local model in terms ofthe operators 2 and t. However, unless the functionsPab and Rab are polynomial in their arguments then theinterpretation of functions of these operators is unclear.To illustrate this we revisit the common choice:

wab(r) = w0abe

r/ab/(2), (9)

for which

Gab(k, ) = w0ab

Aab()(A2ab() + k

2)3/2, (10)

where Aab() = ab1+i/vab. Introducing the operator

Aab:

Aab =

1

ab+

1

vabt

2, (11)

then the problem arises as how to interpret [Aab2]3/2.In the long-wavelength approximation one merely ex-pands Gab(k, ) around k = 0 for small k, yielding a


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nice rational polynomial structure which is then ma-nipulated as described above to give the PDE:

Aab 3

22

ab = w0abb. (12)

We refer to (12) as the long-wavelength model. Higherorder approximations can be obtained by expanding to

higher powers in k [18], although all resulting higher or-der PDE models will still be long-wavelength approxima-tions.

To obtain a PDE model that side-steps the need tomake the long-wavelength approximation we use the ob-servation that er can be fitted with a two-parameterfunction of the form [19]

E(r) = (21 22)1 [K0(r/1) K0(r/2)] , (13)

where K0 is the modified Bessel function of the secondkind of order zero. The prefactor (21

22)1 ensures a

common normalisation for er and E(r).

The form of equation (13) motivates the approxi-mation er/eir/v (21 22)1[K0(Aab;1()r)

K0(Aab;2()r)], where Aab;() is obtained under thereplacement of by in Aab(). This form is particu-larly useful since K0(ar) has a simple Hankel transformgiven by 1/(a2 + k2). For this choice Gab(k, ) takes theform

w0ab21

22

1

(A2ab;1() + k2)

1

(A2ab;2() + k2)

. (14)

In this case we obtain the PDE model:

(Aab;1 2)(Aab;2

2)ab = w0

ab

Babb, (15)

where

Bab =1

2122

2ab

1 +

212abvab(1 + 2)

t

. (16)

For want of a better name we shall refer to (15) asthe rational model. Note that the long-wavelengthmodel can also be obtained using the approximationer L(r) = 2K0(

2/3r)/3. However, this approxi-

mation is poor for both small and large values of r sincelimr0, L(r)/e

r = . Note that limr0 E(r) = (21 22)

1 ln(1/2), and so is well behaved at the origin. A

plot of E(r) (rational model) and L(r) (long-wavelengthmodel) is shown in Fig. 1. The long-wavelength modelis recovered from the rational model with the choice(1, 2) = (

3/2, 0).

We note that the formulation of the rational model(and indeed the long-wavelength model) provides onlyan approximation to the original relationship Gab(r, t) =wab(r)(t r/vab). As a result synaptic activity doesnot just arrive at t = r/vab. This is best seen byderiving Gab(r, t) for the choice (14). Writing the in-verse transform of 1/(A2ab() + k

2) as Hab(r, t) (calcu-lated in Appendix A) we have that Gab(r, t) = (21

0 1 2 3 4 50

0.5

1

r

E(r)

L(r) long-wavelength model

rational model

FIG. 1: (Color online). A plot ofE(r) (rational model) with

(1, 2) = (p

3/2, ), and L(r) (long-wavelength model) isshown together with a plot of er. In this illustrative plotwe have chosen as the root of 21

2 ln(1/), so as to fixE(0) = 1.

22)1w0ab[Hab;1(r, t) Hab;2(r, t)], which gives

Gab(r, t) = w0ab

vab(vabt r)

2

v2abt2 r2

(21 22)1

evabt/(1ab) evabt/(2ab)

.

(17)

Although the shape ofG(r, t) is consistent with our phys-ical expectations, namely a positive decaying pulse be-

yond t = r/vab, we have as yet not fixed the parameters(1, 2) of the rational model. To do this we could try andapproximate the complex function (10) of the full modelusing (14). However, a simpler approach is to considerfitting a projection of (10) that describes the linear stabil-ity of the homogeneous steady state. We do this in thenext section. Here we show that the dynamic patternforming instability borders for the rational model are incloser quantitative agreement to those of the full modelthan the long-wavelength model. In this sense we arguethat the rational model is an improvement over the long-wavelength model (which it recovers as a special case).

We can also interpret the above in terms of a dis-

tance dependent distribution of velocities qab(v, r) for thespreading of synaptic activity by writing

Gab(r, t) = wab(r)

0

dv qab(v, r)(t r/v)

= wab(r)v2

rqab(v, r)

v=r/t

, (18)

with the normalization0

dv qab(v, r) = 1. The originaldefinition (8) is recovered for a single conduction velocityqab(v, r) = (v vab). Rearranging (18) gives the velocity


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distribution as

qab(v, r) =r

v2Gab(r,r/v)

wab(r). (19)

Using (17) we find that, for both the rational and long-wavelength model, the distribution qab(v, r) has a peakat v = vab as expected, as well as one for small r and

v. This second peak however is strongly localized andhence merely introduces an insignificant overall delay toa traveling pulse.

IV. TURING INSTABILITY ANALYSIS

Here we explore the stability of the homogeneoussteady state for the choice ha =

b uab + h

0a. The

extension of this analysis to include the Liley modelgiven by (4) is straightforward. Let ha(r, t) = h

ssa de-

note the homogeneous steady state, defined by hssa =

b Wabfb(hssb ) + h0a, where Wab = R2 drwab(r). Lin-earizing around this solution and considering perturba-tions of the form ha(r, t) = hae

teikr, gives the systemof equations

ha =b

ab()Gab(k, i)bhb. (20)Here ab() = 0 dsesab(s) is the Laplace transformof ab(t) and a = f

a(h

ssa ). Demanding non-trivial so-

lutions yields an equation for the continuous spectrum = (k) in the form E(k, ) = 0, where E(k, ) =det(D(k, ) I), and

[D(k, )]ab = ab()Gab(k, i)b. (21)Note that for a delayed difference of exponentials functionwe have simply that

ab() = eab(1 + /ab)(1 + /ab)

. (22)

An instability occurs when for the first time there arevalues of k at which the real part of is nonnegative. ATuring bifurcation point is defined as the smallest valueof some order parameter for which there exists some non-zero kc satisfying Re ((kc)) = 0. It is said to be static

if Im ((kc)) = 0 and dynamic if Im ((kc)) c = 0.The dynamic instability is often referred to as a Turing-Hopf bifurcation and generates a global pattern withwavenumber kc, which moves coherently with a speedc = c/kc, i.e. as a periodic traveling wave train. Gener-ically one expects to see the emergence of doubly peri-odic solutions that tessellate the plane, namely travelingwaves with hexagonal, square or rhombic structure. Ifthe maximum of the dispersion curve is at k = 0 thenthe mode that is first excited is another spatially uni-form state. If c = 0, we expect the emergence of ahomogeneous limit cycle with temporal frequency c.

For computational purposes it is convenient to splitthe dispersion relation into real and imaginary parts andwrite = + i to obtain

ER(, ) = 0, EI(, ) = 0, (23)

where ER(, ) = Re E(k, + i) and EI(, ) =Im E(k, + i). Solving the system of equations (23)

gives us a curve in the plane (, ) parameterized by k.A static bifurcation may thus be identified with the tan-gential intersection of = () along the line = 0 atthe point = 0. Similarly a dynamic bifurcation is iden-tified with a tangential intersection at = 0. This isequivalent to tracking points where () = 0, given bythe equation kEREI kEIER = 0.

For example, consider two populations, one excitatoryand one inhibitory, and use the labels a {E, I}, withw0EE,IE > 0 and w

0II,EI < 0. In this case

E(k, ) = [1

IIGIII][1

EEGEEE ]

EIIEGEIGIEIE , (24)where Gab = Gab(k, i) and ab = ab(). For sim-plicity we shall set aI = aI = 1 and aE = aE = and ignore synaptic delays by taking ab = 0. In neocor-tex the extent of excitatory connections WaE is broaderthan that of inhibitory connections WaI, and so we takeaE > aI. Again for simplicity we set aI = I = 1 andaE = E . For a common choice of firing rate functionfa = f we may also set a = . Finally we focus ononly a single axonal conduction velocity and set vab = v.In Fig. 2 we show a plot of the critical curves in the(v, ) plane above which the homogeneous steady stateis unstable to dynamic instabilities with kc = 0 (bulk os-

cillations) and kc = 0 (traveling waves). The upper panelin Fig. 2 shows results for the full model (defined by (9)),the middle panel that for the long-wavelength model, andthe lower is that for the rational model. We note thatthere are no qualitative differences between the modelsin the sense that, at the linear level, all models supportHopf and Turing-Hopf instabilities, with a switch fromone to the other with increasing v. One obvious differ-ence is that with an appropriate choice of (1, 2) the ra-tional model is in far better quantitative agreement withthe full model. To test the predictions of our linear sta-bility analysis and to compare the nonlinear behaviourof the two models we resort to direct numerical simula-

tions. Necessarily this requires the choice of a firing ratefunction.

In all direct numerical simulations of the PDE modelswe take the sigmoidal form

f(h) =1

1 + eh. (25)

Here is a gain parameter. Without loss of generalitywe set the steady state value of hE,I to be zero, giving = f(0) = /4. The predictions of the linear stabilityanalysis are found to be in excellent agreement with the


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0

5

10

15

20

0 2 4 6 8 10 1 2v

Hopf

Turing-Hopf

0

5

10

15

20

0 2 4 6 8 10 1 2

Turing-Hopf

Hopf

v

0

5

10

15

20

0 2 4 6 8 10 1 2

Turing-Hopf

Hopf

v

FIG. 2: Critical curves showing the instability thresholds fordynamic instabilities in the (v, ) plane. Top: Full model.Middle: Long-wavelength model. Bottom: Rational model

with (1, 2) = (0.6, 0.6). Parameters = 1, w0

EE = w0

IE = 1,w0II = w

0

EI = 4, h0

E = h0

I = 0 and E = 2.

behavior of the PDE models. Figure 3 shows a patternseen in the long-wavelength model, beyond the Turing-Hopf bifurcation, while Figure 4 shows a pattern seen inthe rational model, also beyond the Turing-Hopf bifur-cation. For both models parallel moving stripes are verycommonly seen beyond the Turing-Hopf bifurcation, par-ticularly for small domains, but a variety of other pat-

1.98

1.99

2.00

2.01

2.02

FIG. 3: (Color online). Left to right, top to bottom: snap-shots of a periodic pattern for the long-wavelength model,each 1/4 of a period later than the previous one. uEE isshown. Domain is 30 30. v = 12, = 20. Other parameters

are as in Fig. 2.

1.98

1.985

1.99

1.995

2

2.005

2.01

2.015

2.02

FIG. 4: (Color online). Left to right, top to bottom: snap-shots of a periodic pattern for the rational model, each 1/4 ofa period later than the previous one. uEE is shown. Domainis 30 30. v = 12, = 15. Other parameters are as in Fig. 2.

terns such as those shown here are also possible, i.e. bothsystems have multiple attractors. Therefore, based uponnumerics alone it is not possible to make the statementthat there is a qualitative difference between the typesof possible patterns in the two models. A short discus-sion of the numerical techniques used in this paper canbe found in Appendix B.


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V. SPATIAL MODULATION

It is now known that the neocortex has a crystallinemicro-structure at the millimeter length scale, so thatthe assumption of isotropic connectivity has to be re-vised (for a recent discussion see [20]). For example, invisual cortex it has been shown that long range horizontalconnections (extending several millimeters) tend to linkneurons having common functional properties (as definedby their feature maps). Since the feature maps (for ori-entation preference, spatial frequency preference and oc-ular dominance) are approximately periodic this leads topatchy connections that break continuous rotation sym-metry (but not necessarily continuous translation sym-metry). With this in mind we introduce a periodicallymodulated spatial kernel of the form

wPab(r, r) = wab(|r r

|)Jab(r r), (26)

where Jab(r) varies periodically with respect to a regu-lar planar lattice L. Note that the patchy kernel wPab is

homogeneous, but not isotropic. Following recent workof Robinson [21] on patchy propagators we show how toobtain an equivalent PDE model for an integral neuralfield equation with a spatial kernel given by (26).

First we exploit the periodicity of Jab(r) and representit with a Fourier series:

Jab(r) =q

Jqabeiqr. (27)

The vectors q are the reciprocal lattice vectors of theunderlying lattice L, and Jqab are Fourier coefficients

given by (2)2 R2 dreiqrJab(r), with J

qab = (J

q

ab)

(where denotes complex-conjugation). In this caseab(k, ) = GPab(k, )b(k, ), where

GPab(k, ) =q

JqabGab(|k q|, ), (28)

and Gab(k, ) is given by (8). We may then writeab(r, t) =

q J

q

abq

ab(r, t), where q

ab(k, ) = Gab(|k q|, )b(k, ). Choosing Gab(k, ) according to (14) wesee that qab(r, t) satisfies

(Aab;1 2q)(Aab;2

2q)

q

ab = w0abBabb, (29)

where q = ( iq). Hence, we have an infinite setof PDEs for the complex amplitudes qab indexed by the

reciprocal lattice vectors q. Since qab = (q

ab) then

ab(r, t) =

q Jq

abq

ab(r, t) R as required. Assumingthat there is a natural cut-off in q, then we need onlyevolve a finite subset of these PDEs to see the effectsof patchy connections on solution behavior. Note alsothat the Turing instability analysis for the patchy modelis identical to that of the isotropic model under the re-placement of Gab by (28) in (21), so that now dependson the direction as well as the magnitude of k. For themode selected by the Turing mechanism all other modes

0

2

4

6

0

2

4

6

5

0

d.

0

2

4

6

0

2

4

6

8

6

4

2

0

0

2

4

6

0

2

4

6

8

6

4

2

0

0.2

b.

0

2

4

6

0

2

4

6

8

6

4

2

0

0.2

a.

c.

0.2

-1

Re(kx,k

y)

FIG. 5: (Color online). The dispersion surfaces Re (k) forfixed parameters: a. d = (unmodulated model), v = 4, =5; b. d = 4, v = 4, = 15; c. v = 4, d = 2, = 20; d.v = 10, d = 4, = 50. The values of are chosen so that thehomogeneous steady state is just unstable. The peaks in thesurfaces are pinned by the lattice wavevectors k1,2, |k1,2| =2/d.


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generated by discrete rotations of the reciprocal latticewill also be selected. Thus periodic patchy connectionsfavour the generation of periodic patterns.

For example, consider a square lattice with length-scale d. The generators of the reciprocal lattice arek1 = 2/d(1, 0) and k2 = 2/d(0, 1). Now chooseJab(r) = [cos(k1 r) + cos(k2 r)]/2. In this case

Jqab = [(q k1) + (q + k1) + (q k2) + (q + k2)]/4,and we need only consider two coupled complex PDEs(indexed by k1,2).

In Fig. 5 we plot the dispersion surfaces Re (k),k = (kx, ky), for parameters selected just beyond theinstability of the homogeneous steady state. In the limitd we recover the unmodulated model. For finite dwe find that each lattice wavevector k1,2 introduces ashifted copy of the peak of the dispersion surface fromthe unmodulated case (Fig. 5a). When these peaks arewidely separated (for lattice spacing d 3) the interac-tion between them is weak and the bifurcation param-

eter portrait is expected to be analogous to that of theunmodulated model (Fig. 2) (at least up to a factor of4 coming from the particular choice of Jqab above). InFig. 6 we plot the bifurcations for the modulated model.Compared to the unmodulated case the Hopf bifurcationis transformed to a Turing-Hopf bifurcation with criticalwavevectors coinciding with those of the lattice and inde-pendent of the axonal velocity v. This is associated withthe central peaks at k1,2 in Fig. 5c crossing through zerofrom below. With increasing v the dominant bifurcationis also of Turing-Hopf type. However, in this case it is aring of wavevectors surrounding k1,2 that go unstablefirst, as in Fig. 5d. In both cases this suggests the emer-

gence of traveling waves aligned to the lattice size anddirection, which are indeed observed in direct numericalsimulations. We shall refer to these as lattice-directedtraveling waves. In the regime 3 d 6 four wavevec-tors become unstable with |kx| = |ky|, as in Fig. 5b , andfor d 6 the system is effectively that of the unmodu-lated case described by Fig. 5a.

In Fig. 7 we plot the speed of a traveling wave at theTuring-Hopf bifurcation at v = 1. The speed of the waveis seen to increase almost linearly with the spacing of thesquare lattice, d. This reflects the fact that for smalld, the emergent frequency c is independent of d and

kc coincides with |k1,2|. The linear analysis predictingemergent wave speed is found to be in excellent agree-ment with direct numerical simulations. Figure 8 showsa lattice-directed traveling wave created in the Turing-Hopf bifurcation shown in Fig. 6, for v = 1.

In Fig. 9 we show a pattern generated in the Turing-Hopf bifurcation for the system with a periodically-modulated kernel for v = 10. It emerged from thespatially uniform state at the value of predicted fromFig. 6. The pattern is compatible with the wavenumberk for which the spatially uniform state is unstable.

0

20

40

60

80

0 2 4 6 8 10 1 2

Turing-Hopf

(central peak)

Turing-Hopf

(surrounding ring)

v

FIG. 6: Critical curves showing the Turing-Hopf instabilitiesin the (v, ) plane for the rational model (15), with a peri-odically modulated kernel. In this example the underlyinglattice is square, with spacing d. Parameters are as in Fig. 2with d = 1.

0 0.5 1 1.5 2.0 2.5

c

d

0.5

0.3

0.4

0.2

0.1

0

FIG. 7: Speed (c = c/kc) of a lattice-directed traveling waveat the Turing-Hopf bifurcation shown in Fig. 6 for v = 1 as afunction of square lattice spacing d. The speed of the wave isseen to increase almost linearly with d. Here v = 1 and otherparameters are as in Fig. 2. The circles denote the resultsfrom direct numerical simulations.

VI. CONCLUSIONS

Neural field models of firing rate activity have hada major impact in helping to develop an understand-ing of the dynamics of EEG [12, 22, 23]. In this pa-per we have shown how to write down an equivalentPDE model for a neural field model in two spatial di-mensions with a particular form of decaying connectivityand a space-dependent axonal delay. Importantly thishas avoided the so-called long-wavelength approximationthat has been used in many EEG models to date. Di-rect numerical simulations of the equivalent PDE model


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0

0.5

1

1.5

2

0

0.5

1

1.5

2

0.05

0

0.05

xy

uEE

FIG. 8: Turing-Hopf pattern for patchy propagation. uEE isshown at one instant in time. v = 1, = 10, d = 1. Thespeed is 0.182 in the x direction. Other parameters are asin Fig. 2.

6

4

2

0

2

4

6

x 103

FIG. 9: (Color online). Left to right, top to bottom: snap-shots of a periodic pattern for the rational model with patchyconnections, each 1/4 of a period later than the previous one.uEE is shown. The domain is 7 7, v = 10, = 50, d = 1and other parameters as in Fig. 2.

have been shown to be consistent with a linear instabilityanalysis of the original integral neural field model. More-over, we have extended our approach to allow for patchyconnections and used simulations of an appropriate set ofcoupled PDEs to confirm the existence of lattice-directedtraveling waves.

A number of natural extensions of the work presentedhere are possible. The first concerns pattern selection;linear stability analysis alone cannot distinguish whichof the doubly periodic solutions (hexagon, square, rhom-bus) will be excited first. To do this requires a further

weakly nonlinear analysis. Techniques to do this for inte-gral models in one spatial dimension with axonal delayshave recently been developed in [24], and are naturallygeneralized to two spatial dimensions. To further copewith patchy connections one may well be able to bor-row from techniques developed for the study of ampli-tude equations in anisotropic PDE models [25]. Anotherextension would be to use the ideas presented here to

discover if axonal delays have any significant effect onthe existence and stability of other patterns seen in two-dimensional neural fields such as spiral waves [26] andspatially-localised bumps [14, 27]. The treatment of dis-tributed transmission speeds [2830] is another impor-tant area, as is the extension to heterogeneous connec-tion topologies (more realistic of real cortical structures)[20, 31] and addressing parameter heterogeneity (describ-ing more realistic physiological scenarios) [17]. All ofthese are topics of ongoing activity and will be reportedupon elsewhere.

Acknowledgments

SC would like to acknowledge support from the EPSRCthrough the award of an Advanced Research Fellowship,Grant No. GR/R76219. The work of LS was supportedin part by NSF Grant DMS-0244529. IB is supported bygrant DP0209218 from the Australian Research Council.

Appendix A

We calculate Hab(r, t) as an inverse Fourier transform

[32], namely

Hab(r, t) =1

(2)3

R3

dkdei(kr+t)

(1/ab + i/vab)2 + k2

=vabe

vabt/ab

2

0

dk sin(kvabt)J0(kr)

=vabe

vabt/ab

2

1v2abt

2 r2(vabt r). (30)

Appendix B

Here we provide some details on the simulation strat-egy for equations of the form (15). By defining someauxilliary variables and applying the chain rule this canbe written as four first-order (in time) PDEs. For ourchoice of , equation (1) can be written

1

abab

2

t2+

ab + ab

abab

t+ 1

uab = ab(t ab).

(31)Note that for simplicity we set ab = 0 in our simulations.

To solve an equation like (29) we let qab = eiqr

qab,


9/9

9

which converts (29) to

(Aab;1 2)(Aab;2

2)qab = eiqrw0abBabb, (32)and then proceed as above. The square domains werediscretised with a regular grid and the spatial deriva-

tives were approximated using finite differences. Peri-odic boundary conditions were used. The resulting ODEswere integrated using ode45 in Matlab with default tol-erances. Figure 3 had a discretisation of 60 60, Fig. 4was 51 51, Fig. 8 was 20 20 and Fig. 9 used 50 50.

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S Coombes et al- Modeling electrocortical activity through improved local approximations of integral neural field equations