Waves, bumps, and patterns in neural field theories - CiteSeerX

Waves, bumps, and patterns in neural field theories S Coombes∗ April 13, 2005

Abstract

on the proportion of neurons becoming activated per

unit time in a given volume of model brain tissue conNeural field models of firing rate activity have had sisting of randomly connected neurons, Beurle was a major impact in helping to develop an under- able to analyse the triggering and propagation of large standing of the dynamics seen in brain slice prepara- scale brain activity. However, this work only dealt tions. These models typically take the form of integro- with networks of excitatory neurons with no refracdifferential equations. Their non-local nature has led tory or recovery variable. It was Wilson and Cowan in to the development of a set of analytical and numeri- the 1970’s [4, 5] who extended Beurles work to include cal tools for the study of waves, bumps and patterns, both inhibitory and excitatory neurons as well as rebased around natural extensions of those used for lo- fractoriness. For a fascinating historical perspective cal differential equation models. In this paper we on this work we refer the reader to the recent article by present a review of such techniques and show how re- Cowan [6]. Further work, particularly on pattern forcent advances have opened the way for future studies mation, in continuum models of neural activity was of neural fields in both one and two dimensions that pursued by Amari [7, 8] under natural assumptions can incorporate realistic forms of axo-dendritic inter- on the connectivity and firing rate function. Amari actions and the slow intrinsic currents that underlie considered local excitation and distal inhibition which bursting behaviour in single neurons. is an effective model for a mixed population of interacting inhibitory and excitatory neurons with typical

1

cortical connections (commonly referred to as Mexi-

Introduction

can hat connectivity). Since these seminal contributions to dynamic neural field theory similar models

The multi-scale properties of spatio-temporal neural

have been used to investigate EEG rhythms [9], visual

activity leads naturally to some interesting mathemat-

hallucinations [10, 11], mechanisms for short term

ical challenges, in terms of both modelling strategies

memory [12, 13] and motion perception [14]. The sorts

and subsequent analysis. Since the number of neu-

of dynamic behaviour that are typically observed in

rons and synapses in even a small piece of cortex is

neural field models includes, spatially and temporally

immense a popular modelling approach has been to

periodic patterns (beyond a Turing instability) [10, 15],

take a continuum limit and study neural networks in

localised regions of activity (bumps and multi-bumps)

which space is continuous and macroscopic state vari-

[12, 16] and travelling waves (fronts, pulses, target

ables are mean firing rates. Perhaps the first attempt

waves and spirals) [17, 18, 19].

at developing a continuum approximation of neural activity can be attributed to Beurle [1] in the 1950’s

mentally using multi-electrode recordings and imag-

and later by Griffith [2, 3] in the 1960’s. By focusing ∗ Department

of

Nottingham,

of

Mathematical

Nottingham,

NG7

Sciences, 2RD,

ing methods. In particular it is possible to electri-

University UK.

In the latter case

corresponding phenomena may be observed experi-

cally stimulate slices of pharmacologically treated tis-

email:

sue taken from the cortex [20, 21, 22], hippocampus

[email protected]

1

[23] and thalamus [24]. In brain slices these waves ters. Moreover, localised bump solutions are simican take the form of spindle waves seen at the on- larly handled by recognising them as standing pulse set of sleep [25], the propagation of synchronous dis- waves. The extension of the standard model to incorcharge during an epileptic seizure [26] and waves of porate space-dependent delays, arising from axonal excitation associated with sensory processing [27]. In- and dendritic communication delays is the subject of terestingly, spatially localised bumps of activity have section 7. As well as describing the conditions under been linked to working memory (the temporary stor- which these models may be reduced to a PDE descripage of information within the brain) in prefrontal cor- tion, we review the effect of such delays on the onset tex [28, 29], representations in the head-direction sys- of a dynamic Turing instability. In section 8 we retem [30], and feature selectivity in the visual cortex, turn to the starting point for the derivation of a firing where bump formation is related to the tuning of a rate model and show how it is also possible to carry particular neuron’s response [31].

forward slow ionic currents into a firing rate descrip-

In this paper we present a review of neural field the- tion of neural tissue. In illustration we construct a firories of Wilson-Cowan and Amari type and describe ing rate model that incorporates a slow T-type calcium the mathematical techniques that have been used in current, IT , known to be important in the bursting retheir analysis to date. For the purposes of exposition sponse of thalamo-cortical relay cells. Moreover, for a we shall stick to single population models, though purely inhibitory network and a Heaviside firing rate all of what we say can be easily taken over to the function we show how to construct so-called lurchcase of two or more populations, such as discussed ing pulses, often seen in simulations of more detailed in [32]. In section 2 we introduce the standard integro- biophysical networks expressing IT . Some non-trivial differential equation (IDE) for a scalar neural field and consequences of working with neural fields in two didiscuss the conditions under which this reduces to a mensions are discussed in section 9. In particular we local partial differential equation (PDE) model. More- show that the stable so-called dimple bump that can over, we show how the IDE model may be written as be found in a one dimensional model does not have a purely integral equation. The integral framework is a stable two dimensional analogue. Finally in section convenient for certain types of analysis, such as cal- 10 we discuss some of the open challenges relating to culating the onset of a Turing instability. We briefly il- the further development and analysis of neural field lustrate this in section 3. Next in section 4 we move on theories. to a study of travelling wave solutions and show how techniques from the reaction-diffusion literature may

2

be used to provide estimates of wave speed. Interestingly for the choice of a Heaviside firing rate function

Mathematical framework

In many continuum models for the propagation of

wave speeds can be calculated exactly. We describe

electrical activity in neural tissue it is assumed that the

this procedure for travelling front solutions in section

synaptic input current is a function of the pre-synaptic

5. Moreover, borrowing from ideas first developed

firing rate function [5]. These infinite dimensional dy-

in the PDE community we also show how to anal-

namical systems are typically variations on the form

yse wave stability using an Evans function approach.

[32]

In section 6 we consider a slightly more general set

1 ∂ u(x, t) = −u + α ∂t

of neural field equations that incorporate modulatory terms. These models support travelling pulses, as

Z ∞ −∞

dyw(y) f ◦ u(x − y, t).

(1)

well as fronts. Generalising the techniques used for Here, u(x, t) is interpreted as a neural field representthe study of fronts we show how to determine pulse ing the local activity of a population of neurons at speed and stability as a function of system parame- position x ∈ R. The second term on the right represents the synaptic input, with f ◦ u interpreted as the 2

R

Re

firing rate function. The strength of connections be-

ikx

w(x)dx, of w(x) has a simple rational polynomial

tween neurons separated by a distance y is denoted structure we may exploit the convolution property of w(y) = w(| y|) (assuming a spatially homogeneous and

(3) to write the equation for ψ (x, t) as a PDE. To il-

isotropic system), and is often referred to as the synap- lustrate this consider the choice w(x) = e−|x| /2, with FT[w](k) = (1 + k2 )−1 . In this case taking the Fourier

tic footprint.

There are several natural choices for the firing rate transform of (3) yields function, the simplest being a Heaviside step function.

FT[ψ ](k, t) =

In this case a neuron fires maximally (at a rate set by

1 FT[ f ◦ u](k, t). 1 + k2

(5)

its absolute refractory period) or not at all, depending Cross multiplying by 1 + k2 and inverting (rememberon whether or not synaptic activity is above or below ing that FT[ψ ](k) = ikψ ), gives x

some threshold. In a statistical mechanics approach (1 − ∂xx )ψ (x, t) = f ◦ u(x, t).

to formulating mean-field neural equations this all or nothing response is replaced by a smooth sigmoidal

(6)

Hence, the evolution of u is described by the pair of

form [4, 33]. For an arbitrary firing rate response the

coupled partial differential equations, (2) and (6). By

model (1) is naturally written in the form

exploiting the local PDE structure that can be obtained

(2) with such special choices it is possible to use many of the standard tools from dynamical systems analysis where ψ (x, t) is given by the second term on the right to study solutions of inherently non-local neural field hand side of (1) and Q = (1 + α−1 ∂t ). The linear difmodels. For example, in a co-moving frame travelferential operator Q is used to model the dynamics asling wave solutions are given by a system of ordinary sociated with first order synaptic processing and can differential equations (ODEs), with travelling fronts easily be generalised to represent higher order synapand pulses viewed as global (heteroclinic and homotic processing [34]. It is convenient to write ψ (x, t) in clinic) connections. Standard shooting and numerical the form continuation techniques (both numerical and analytiψ (x, t) = (w ⊗ f )(x, t), (3) cal) may then be brought to bear in their construction where ⊗ represents a spatial convolution: [18, 34]. Qu(x, t) = ψ (x, t),

(w ⊗ f )(x, t) =

Z ∞ −∞

w(y) f (x − y, t)dy.

Another common choice of w(x) in the study of

(4)

neural field models is that of a Mexican hat function,

Numerical simulations of (2) using the integral equa- such as w(x) = (1 − |x|)e−|x| /4 (perhaps more properly tion (3) with sigmoidal f , show that such systems sup- called a wizard hat function, because of its cusp at the port unattenuated travelling waves as a result of lo- origin [35]). In this case FT[w](k) = k2 /(1 + k2 )2 , and a calised input [5]. We note that the time independent similar argument to that above gives solutions of (2) are given by u(x) = ψ (x). For this class

(1 − ∂xx )2 ψ (x, t) = −[ f ◦ u(x, t)]xx .

of solutions Amari [8] was able to find localised stable

(7)

pulses that are bistable with a homogeneous steady Time-independent solutions of (7) are solutions of the state, assuming a Heaviside firing rate function and fourth order ODE: (1 − dxx )2 u(x, t) = −[ f ◦ u(x)]xx (usMexican hat connectivity. Subsequently Kishimoto ing Qu(x) = ψ (x)). Interestingly, numerical solution and Amari showed that such solutions also exist for of such systems typically yield single and multi-bump a smooth sigmoidal function (at least in the high gain structures (regions of localised activity) [12, 13]. Morelimit) [16]. over, the governing equations are now known to posFor certain (special) choices of w(x) it is possi- sess a Hamiltonian structure [12]. The extensive use ble to re-cast the equation for ψ (x, t) in a more lo-

of local PDE methods (particularly those for fourth

cal form [17]. If the Fourier transform, FT[w](k) = order reversible systems) for studying such localised 3

structures can be found in the work of Laing and Troy fined by [36] and Krisner [37]. A detailed numerical analysis

u = κ f (u),

of localised time-independent solutions to equation

where κ =

(7) with a sigmoidal form of firing rate function can

R

R w(y)dy.

(11)

Note that for positive weight

kernels it is quite common to normalise them such

be found in [34]. Here, it is shown that this particu-

R

= 1. We linearise about the steady state by letting u(x, t) → u + u(x, t) so that f (u) → f (u) + tions, and that such localised multi-bumps are lost (in f 0 (u)u to obtain favour of global patterns) when a stable N-bump and an unstable (N + 2)-bump coalesce. u = γη ∗ w ⊗ u, γ = f 0 (u). (12) that

lar fourth order system admits multiple bump solu-

Apart from waves and bumps, neural field models

R w(y)dy

This has solutions of the form eλt eikx , so that λ = λ(k)

are also known to support the formation of globally

is given implicitly by the solution to

periodic patterns [10]. Such patterns can emerge beyond a so-called Turing bifurcation point. To develop

γ LT[η ](λ)FT[w](k) − 1 = 0,

(13)

this, and other techniques, in full generality, it is convenient to use the language of Green’s functions and where LT[η ](λ) is the Laplace transform of η (t): write Qη (t) = δ (t),

LT[η ](λ) =

(8)

Z ∞

η (s)e−λs ds.

(14)

0

where η (t) is the Green’s function of the linear dif-

The uniform steady state is linearly stable if Re λ(k)
γc , λ(kc ) > 0 and this pattern grows with time. In fact there will typ-

t

η (s) f (x, t − s)ds.

0

(10) ically exist a range of values of k ∈ (k1 , k2 ) for which

terns. We do this using a Turing instability analysis.

λ(k) > 0, signalling a set of growing patterns. As the patterns grow, the linear approximation breaks down and nonlinear terms dominate behaviour. (iv) The saturating property of f (u) tends to create patterns with √ finite amplitude, that scale as k − kc close to bifurcation and have wavelength 2π/kc . (v) If kc = 0 then we would have a bulk instability resulting in the formation of another homogeneous state. Note that if Im λ(kc ) 6= 0, then the homogeneous solution would be time-periodic.

One solution of the neural field equation (1) is the spa-

Since the Fourier transform of Mexican hat type

tially uniform resting state u(x, t) = u for all x, t, de-

functions, which represent short-range excitation and

The distributed delay kernel η (t) can be chosen so as best to describe the response of a given synapse.

3

Turing instability analysis

We now describe how a spatially homogeneous state can become unstable to spatially heterogeneous perturbations, resulting in the formation of periodic pat-

4

long range inhibition, are peaked away from the ori- Note that standing waves (with c = 0), are defined gin they are capable of supporting a Turing instabil- by Qu(x, t) = q(x) so that q(x) = ψ (x). It is conveity. An example of such a function is w(x) = e−|x| − nient to regard the bumps of spatially localised timee−|x|/2 /2 (a wizard hat). Another classic example is a independent solutions that we have mentioned earlier as standing waves with speed c = 0.

difference of Gaussians. Generalising this approach to two dimensions is

For sigmoidal firing rate functions it is generally

straight forward. Near a bifurcation point we would possible to arrange for the system to have three hoexpect spatially heterogeneous solutions of the form mogeneous steady states, u1 < u2 < u3 . In this case it eλt eikc ·r for some kc = (k1 , k2 ), and r ∈ R2 . For a given is natural to look for travelling front solutions with kc = |kc | there are an infinite number of choices for

q(−∞) = u3 and q(∞) = u1 , which connect u1 and

k1 and k2 . It is therefore common to restrict attention

u3 (which are stable to homogeneous perturbations).

to doubly periodic solutions that tessellate the plane. Arguing in analogy to techniques used for estimatThese can be expressed in terms of the basic symmetry ing front speeds for reaction-diffusion equations we groups of hexagon, square and rhombus. Solutions consider the linearised equations of motion around can then be constructed from combinations of the ba- the fixed points. In this case we are led to consider sic functions eikc R·r , for appropriate choices of the basis systems with linear firing rate functions of the form vectors R. Details of this programme, and the non-

f (u) = γ u, which give rise to exponential solutions

linear analysis necessary in order to correctly select

u(ξ ) ∼ eλξ . It is easily established that λ is the solu-

which of the modes will stably appear are discussed tion to L(c, λ) = 0 where in [10, 15, 32, 38, 39]. For a recent discussion of how L(c, λ) = γ LT[η ](−cλ)FT[w](−iλ) − 1.

to treat spatio-temporal pattern formation in systems with heterogeneous connection topologies (more real-

(19)

If the temporal and spatial kernels, η (t) and w(x),

istic of real cortical structures) we refer the reader to

are both normalised to unity, then we see that that

[40, 41].

L(c, 0) = γ − 1. For λ < 0, L(c, λ) is a monotonically decreasing function of c with limc→∞ L(c, λ) = −1. More-

over, we have that Z ∞ ∂ L(c, λ) = γ c sη (s)ds > 0, (20) Waves in the form of travelling fronts and pulses have ∂λ λ=0 0 now been observed in a variety of slice preparations 2 2 [20, 21, 22, 42, 43]. To establish properties of waves for c > 0, and that ∂ L/∂ λ(c, λ) > 0 for all λ and c > 0

4

Travelling waves

in a model neural system it is convenient to introduce (i.e. L(c, λ) is a convex function of λ). Following Diekthe coordinate ξ = x − ct and seek functions U(ξ, t) = mann [44, 45], we introduce a minimum propagation speed, c∗ , as

u(x − ct, t) that satisfy (9). In the (ξ, t) coordinates, the integral equation (9) reads U(ξ, t) =

Z ∞ −∞

dyw(y)

Z ∞

c∗ = inf{c | L(c, λ) = 0, for some λ < 0}.

(21)

dsη (s)

0

× f ◦ U(ξ − y + cs, t − s).

Now, choosing γ > 1, a minimum with c > 0 can only (16) occur for λ < 0 (using the convexity of L(c, λ) and

A travelling wave, with speed c, is a stationary solu- the fact that ∂ L/∂λ(c, 0) > 0). Consider, for example the case of an exponential synaptic footprint w(x) = tion U(ξ, t) = q(ξ ) (independent of t), that satisfies q(ξ ) =

Z ∞

η (z)ψ (ξ + cz)dz,

e−|x| /2 and an exponential synapse η (t) = αe−αt . In

(17)

this case we have from (19) that c = c(λ), where

0

with

ψ (ξ ) =

Z ∞ −∞

w(y) f (q(ξ − y))dy.

  γ α 1− , c(λ) = λ 1 − λ2

(18) 5

(22)

for |λ| < 1. Since the value of λ for which c0 (λ) = 0 the threshold is crossed and not directly on the shape is independent of α we immediately see that c∗ is lin- of u. Apart from allowing an explicit construction of ear in α. For a general nonlinear firing rate c∗ is still travelling waves this choice also allows for a direct expected to be a good predictor of wave speed if the calculation of wave stability via the construction of an linearisation at u = u satisfies f (u) < f 0 (u)u [44, 45].

Evans function [51]. Although often chosen for math-

For sigmoidal firing rate functions it has been ematical reasons the Heaviside function may be reshown by Ermentrout and McLeod that there exists garded as a natural restriction of sigmoidal functions a unique monotone travelling wave front for posi- to the regime of high gain. Importantly, numerical tive spatially decaying synaptic connections [17]. In- simulations show that many of the qualitative propdeed, there are now a number of results about exis- erties of solutions in the high gain limit are retained tence, uniqueness and asymptotic stability of waves with decreasing gain [17, 18, 34]. for IDEs, such as can be found in [8, 46, 47, 48]. Other work on the properties of travelling fronts, and in par-

5

ticular speed as a function of system parameters can be found in [18, 34, 49]. Note also that a formal link

Fronts in a scalar integral model

In this section we introduce the techniques for con-

between travelling front solutions in neural field the-

structing the Evans function with the example of trav-

ories and travelling spikes in integrate-and-fire net-

elling front solutions to (9). A more detailed discus-

works can be found in [50].

sion of the construction of the Evans function for neuThe linear stability of waves is obtained by writing ral field theories can be found in [51]. We look for travU(ξ, t) = q(ξ ) + u(ξ )eλt , and Taylor expanding (16), to elling front solutions such that q(ξ ) > h for ξ < 0 and obtain the eigenvalue equation u = L u: u(ξ ) =

Z ∞ −∞

dyw(y)

Z ∞

× u(ξ − y + cs).

q(ξ ) < h for ξ > 0. It is then a simple matter to show that

dsη (s)e−λs f 0 (q(ξ − y + cs))

ψ (ξ ) =

0

(23)

Z ∞

w(y)dy.

ξ

(24)

The choice of origin, q(0) = h, gives an implicit equa-

A travelling wave is said to be linearly stable if tion for the speed of the wave as a function of system Re (λ) < 0 for λ 6= 0. Since we are concerned with sys- parameters. tems where the real part of the continuous spectrum

The construction of the Evans function begins with

has a uniformly negative upper bound, it is enough an evaluation of (23). Using the identity to determine the location of the normal spectrum for δ (ξ ) d Θ(q(ξ ) − h) = 0 , wave stability [51]. In general the normal spectrum of dq |q (0)|

(25)

the operator obtained by linearising a system about we arrive at the expression Z ∞ its travelling wave solution may be associated with u(0) u(ξ ) = dyw(y)η (−ξ/c + y/c)e−λ(y−ξ )/c . c|q0 (0)| −∞ the zeros of a complex analytic function, the so-called (26) Evans function. This was originally formulated by From this equation we may generate a self-consistent Evans [52] in the context of a stability theorem about equation for the value of the perturbation at ξ = 0, excitable nerve axon equations of Hodgkin-Huxley simply by setting ξ = 0 on the left hand side of (26). type. The extension to integral models is far more reThis self-consistent condition reads cent [51, 53, 54, 55, 56, 57]. Z ∞ u(0) dyw(y)η (y/c)e−λ y/c , (27) u(0) = Throughout the rest of this paper we shall focus on c|q0 (0)| 0 the particular choice of a Heaviside firing rate func- remembering that η (t) = 0 for t ≤ 0. Importantly this

tion, f (u) = Θ(u − h) for some threshold h. The Heav- has a non-trivial solution if E (λ) = 0, where Z ∞ iside function is defined by Θ(x) = 1 for x ≥ 0 and is 1 E ( λ ) = 1 − dyw(y)η (y/c)e−λ y/c . zero otherwise. In this case ψ depends only on where c|q0 (0)| 0 6

(28)

We identify (28) with the Evans function for the trav- connectivities. For example with the choice w(x) = elling front solution of (9). It can be shown that i) the (1 − a|x|)e−|x| , we have that  Evans function is only real-valued if the eigenvalue e−ξ (1 − a − aξ ) parameter λ is real, ii) the complex number λ is an ψ (ξ ) = 2(1 − a) − eξ (1 − a + aξ ) eigenvalue of the operator L if and only if E (λ) = 0,

ξ≥0

.

(34)

ξ h, requires the choice exactly equal to the order of the zero of the Evans R R w(y)dy = 2(1 − a) > h. Plots of (33) and (34) are function [51]. Also, from translation invariance, λ = 0 shown in Fig. 1. is an eigenvalue (with eigenfunction q0 (ξ )), so that

E (0) = 0.

ψ(ξ) 1

A common choice for the synaptic response function is η (t) = αe−αt . In this case the condition q(0) = h gives an implicit expression for the speed of the wave

0.5

in the form [58] h=

κ − LT[w](α/c). 2

(29)

0

Moreover, the Evans function takes the explicit form

E (λ) = 1 −

LT[w]((α + λ)/c)) , LT[w](α/c)

-10

(30)

where we have made use of the fact that E (0) = 0.

-5

0

5

ξ

10

Figure 1: A plot of the travelling wave solution ψ (ξ )

As an example it is illustrative to consider w(x) = for an exponential synaptic footprint w(x) = e−|x| /2 e−|x| /2, with Laplace transform LT[w](α) = (1 + (solid curve) and a wizard hat w(x) = (1 − |x|/2)e−|x|

α)−1 /2. The speed of the front is determined from (29) (dashed curve). For both examples κ = R w(y)dy = 1/2. Note that the wizard hat footprint leads to a nonas 1 − 2h c=α , (31) monotone shape for the travelling front. 2h which we observe is linear in α (as in the earlier example for a linear firing rate function). Using (30) the Evans function is easily calculated as R

E (λ) =

6

λ . c+α+λ

In real cortical tissues there are an abundance of

The equation E (λ) = 0 only has the solution λ = 0.

metabolic processes whose combined effect is to mod-

We also have that E 0 (0) > 0, showing that λ = 0 is a

ulate neuronal response. It is convenient to think of

simple eigenvalue. Hence, the travelling wave front

these processes in terms of feedback mechanisms that

for this example is linearly stable. Assuming c > 0

modulate synaptic currents. Such feedback may act

the travelling front (17) is given in terms of (24) which takes the explicit form   1 e−ξ ψ (ξ ) = 2 1 − 1 eξ 2

Recovery and lateral inhibition

(32)

to decrease activity in the wake of a travelling front so as to generate travelling pulses (rather than fronts).

ξ≥0 ξ 0 on the lower solution branch of Fig.

4.

Hence, det J < 0, and from (60), we see that this solu-

q(x) is the stationary bump solution.

tion is a saddle. On the upper branch det J > 0 and a Hopf bifurcation occurs when TrJ = 0, which is ex-

solution of the form u(xi (t), t) = h for i = 1, 2, such that pected to occur with increasing α. A plot of the fixed x1 < x2 and u(x, t) > h for x ∈ (x1 (t), x2 (t)) at time t and point of the system of equations (56) as well as the u(x, t) < h otherwise. Differentiation of this defining maximum amplitude of oscillation for periodic orbits arising at a Hopf bifurcation are plotted in Fig. 8. expression for a time-dependent bump gives ux (xi , t)

This numerical solution of (56) shows that the Hopf

dxi + ut (xi , t) = 0. dt

(55) 8

This equation can be used to obtain the evolution of the bump-width ∆(t) = x2 (t) − x1 (t). The expressions

∆ 6

for ut and ux are naturally obtained by differentiating (36) (remembering that f is a Heaviside). For example, if we choose Q = 1 + α−1 ∂t , Qa = 1 + ∂t , then the

4

bump evolves according to y d∆ 2 dt dz dt dy dt dz1 dt

2

= α(−h + W(∆) − z), = −z + gWa (∆),

0 0

0.04

0.08

h

0.12

= α(− y + w(0) − w(∆) − gz1 ), = −z1 + wa (0) − wa (∆),

(56)

where we identify y = ux (x1 , t) = −ux (x2 , t). Here x

1 w(y)dy = [1 − e−x ], 2 0 Z x 1 Wa (x) = wa (y)dy = [1 − e−x/σa ]. 2 0 W(x) =

Z

Denoting the fixed point by an expression for

∆∗

model with lateral inhibition and nonlinear recovery. Here σa = 2, g = 1 and α = 4. Solid (dashed) lines are

stable (unstable). Open circles denote the maximum (57) amplitude of unstable periodic orbits emerging from (58) (59)

(∆∗ , z∗ , y∗ , z∗1 ),

Figure 8: Pulse width as a function of threshold h in a

a sub-critical Hopf bifurcation.

bifurcation is sub-critical, with no emerging stable or-

we recover bits. In fact the emergent unstable periodic orbit is de-

identical to earlier, i.e. W(∆∗ ) − stroyed in a collision with the unstable lower branch

gWa (∆∗ ) = h, which is equivalent to (54). Hence, the

of fixed points. Pinto and Ermentrout have suggested 12

that this is the reason why direct numerical simula- tion, provided that the two dimensional FT of K(x, t) tions (just beyond the point of instability) do not show has a rational structure. For an axonal delay the choice w(x) = e−|x| /2 gives rise to a type of damped inhomo-

stable breathing bumps [60]. Although this approach

is useful in predicting qualitative behaviour of the full geneous wave equation: equations of motion, it is not particularly useful in

[(v + ∂t )2 − v2 ∂xx ]ψ (x, t) = [v2 + v∂t ] f ◦ u(x, t). (63)

providing accurate estimates of the critical values of α and h necessary to see a dynamical instability. Unfortunately, there is not an accurate agreement with the

This equation was first derived by Jirsa and Haken [67, 70] and has been studied intensively in respect to

point of instability calculated using the exact Evans

the brain-behaviour experiments of Kelso et al. [71].

function approach and that of the Hopf bifurcation in

Similar equations have been presented in [72, 73, 74,

the kinematic theory of bump dynamics.

75], where the linearisation of such equations (about Returning our attention to the results described a homogeneous steady state) has been used in the above for dynamic instabilities of localised bumps, it interpretation of EEG spectra. As regards the set would appear that the question of how to generate of full nonlinear integral equations one obvious constable breathing solutions in a neural field model is sequence of introducing an axonal delay is that the an interesting one. One way to generate such solu- speed of a travelling wave must be slower than that of tions has been found that relies upon the inclusion the action potential, i.e. c < v. The calculation of wave of localised inputs [65], breaking the homogeneous speed and stability for a Heaviside firing rate function structure of the network. The use of unimodal inputs is easily generalised to the case of finite v and is demeans that this mechanism does not require a Mexi- scribed in [18, 34, 51]. For the case of the exponential can hat connectivity to either generate bumps or sta- synaptic footprint chosen above and an exponential ble breathing bumps (merely just a positive footprint synaptic response, η (t) = αe−αt , it is possible to obtain such as a spatially decaying exponential). However, a closed form expression for the speed, c, of a front in it is also possible to find stable breathing solutions in terms of the speed of an action potential, v, as a homogeneous model with Mexican hat connectivity that incorporates a dynamic firing threshold [66].

7

c=

v(2h − 1) . 2h − 1 − 2hv/α

(64)

Note that we recover equation (31) in the limit v → ∞

Space-dependent delays

as expected. The techniques used in section 6 may also be adapted to construct travelling pulse solutions,

In the presence of space-dependent delays, it is natu-

and indeed this has recently been done by Enculescu

ral for ψ (x, t) to take the slightly more general form

ψ (x, t) =

Z ∞Z ∞ −∞ −∞

[76]. This calculation is easily reproduced (although

K(x − y, t − s) f ◦ u(y, s)dyds. (62)

we do not do so here) and we have used this to make a plot of front and pulse speed as a function of v in

A model with space-dependent axonal delays may Fig. 9. We note that the pulse (if it exists) always travbe obtained by choosing K(x, t) = w(x)δ (t − |x|/v) els slower than the front. Interestingly an examination [34, 67], where v is the finite speed of action potential of the Evans function for each solution (adapting the propagation. Alternatively a model of dendritic de- calculations in sections 5 and 6) shows that the front lays studied intensively by Bressloff [68, 69] is recov- is always stable whilst the pulse is always unstable. ered with the choice K(x, t) = w(x)g(t). Here g(t) is the

Hence, it is not possible to change the stability of a

Green’s function of the cable equation with a synapse front or pulse by varying v. However, the affect of at a fixed (dendritic) distance from the cell body. As varying v can have a far more profound effect on the in section 2 the (double) convolution structure of this stability of a homogeneous steady state, as we now equation may be exploited to obtain a PDE formula- discuss. 13

0.4

w(x) = (|x| − 1)e−|x| and alpha function synaptic re-

c

sponse η (t) = α2 te−αt . Proceeding as in section 3 we

front

0.3

linearise around a homogeneous solution and consider perturbations of the form eλt eikx . In this case the dispersion relation for λ = λ(k) takes the modified

0.2

form b γ LT[η ](λ)FT[w](k) − 1 = 0,

pulse

0.1

(65)

b = w(x)e−λ|x|/v . Compared to (13) (obtained where w(x) in the absence of space-dependent delays), equation

0 0

0.2

0.4

v

b (65) is not separable in the sense that FT[w](k) is not

0.6

just a function of k, but is also a function of λ. It

Figure 9: The speed of a travelling front (solid line)

is natural to decompose λ in the form λ = ν + iω and equate real and imaginary parts of (65) to ob-

and a travelling pulse (dashed line) as a function of

tain two equations for ν and ω . If we write these

action potential velocity in a model with space dependent axonal delays. Here the synaptic footprint is exponential, w(x) = e−|x| /2, and the synaptic response function is also exponential, η (t) = αe−αt . The firing rate function is a Heaviside, f (u) = Θ(u − h). Note that the pulse travels slower than the front. Moreover, an

in the form G(ν, ω ) = 0 and H(ν, ω ) = 0, then the simultaneous solution of these two equations gives the pair (ν (k), ω (k)), so that we may parametrically express ω = ω (ν ). An example of such a plot is shown in Fig. 10. Here, it can be seen that for a fixed value

examination of the Evans function for both solution

2.5

types shows that the front is always stable, and the

ω

pulse is always unstable. Parameters are α = 1 and

2

h = 0.25.

γ < γc(v)

γ > γc(v)

γ = γc(v)

1.5 In section 3, we showed that static Turing instabilities can arise for Mexican hat connectivities in the absence of space-dependent delays. However, when working with (62) it is possible for dynamic Turing

1 -0.04

0

ν

0.04

instabilities to occur. These were first found in neural field models by Bressloff [68] for dendritic de- Figure 10: Continuous spectrum for a scalar neural lays and more recently by Hutt et al. [77] for ax- field model with an inverted wizard hat synaptic footonal delays. Both these studies suggest that a com- print, axonal delays (of speed v = 1) and an alpha bination of inverted Mexican hat connectivity with a function synapse (with α = 1). On the left γ = 7 < space-dependent delay may lead to a dynamic instability. Indeed the choice of inverted Mexican hat is natural when considering cortical tissue and remembering that principal pyramidal cells i) are often enveloped by a cloud of inhibitory interneurons, and ii) that long range cortical connections are typically

γc (v) and on the right γ = 9 > γc (v). For γ < γc (v) the continuous spectrum lies in the left hand complex plane and the homogeneous solution is stable. For γ > γc (v) part of the continuous spectrum lies in the right hand complex plane and the homogeneous solution is unstable.

excitatory. We now illustrate the possibility of a dynamic Turing instability for a model with axonal de- of v there is a critical value of γ = γc (v) such that for lays by considering an inverted wizard hat function 14

γ < γc (v), the continuous spectrum lies in the left hand

complex plane, whilst for γ > γc (v) part of the spectrum lies in the right hand complex plane. The Tur-

γ

ing bifurcation point is defined by the smallest non-

12

zero wave number kc that satisfies Re (λ(kc )) = 0. It

Dynamic Turing Patterns

is said to be static if Im (λ(kc )) = 0 and dynamic if Im (λ(kc )) ≡ ωc 6= 0. A static bifurcation may then be identified with the tangential intersection of ω = ω (ν )

8

and ν = 0 at ω = 0. Similarly a dynamic bifurcation is identified with a tangential intersection with ω 6= 0. The integral transforms in (65) are easily calculated as LT[η ](λ) =

(1 + λ/α)−2

0

1

3

v

5

and Figure 11: The critical curve for a dynamic Turing instability in a neural field with axonal delays and an inverted wizard-hat connectivity, with an alpha func-

−2λ/v[(1 + λ/v)2 − k2 ] − 4k2 (1 + λ/v) , 2  (1 + λ/v)2 + k2 (66) so that we may rewrite (65) as a sixth order polynomial in λ; ∑6n=0 an λn = 0 where the coefficients an = an (k, v, α, γ ) are given in the appendix. Hence, the functions G(ν, ω ) and H(ν, ω ) may also be written as polynomials in (ν, ω ). For the calculation of a dynamic Turing instability we are required to track points in parameter space for which ν 0 (ω ) = 0. By differentiating G(ν, ω ) = 0 = H(ν, ω ) with respect to ω we see that this is equivalent to tracking points where Gk Hω − Hk Gω = 0 (itself another polynomial equation). Beyond a dynamic Turing instability we expect the growth of travelling patterns of the form ei(ωc t+kc x) . A plot of the critical curve γ = γc (v) for a dynamic Turing instability (with kc 6= 0) is shown in Fig. 11. Here, it can be seen that with increasing γ (the gradient of the firing rate at the homogeneous steady state) a dynamic instability is first met for v ∼ 1. Direct numerical simulations (not shown) of the full model show excellent agreement with the predictions of the dynamic Turing instability analysis. To determine the conditions under which one might see a standing wave (arising from the interaction of a left and right travelling wave), it is necessary to go beyond linear analysis and determine the evolution of mode amplitudes. The techniques to do this in (one dimensional) neural field theories are nicely described by Curtu and Ermentrout [78].

tion synaptic response (and α = 1). Above the curve

b , λ) = w(k

γ = γc (v), the homogeneous steady state is unstable, leading to the growth of travelling patterns of the form ei(ωc t+kc x) .

8

Neural field equations with slow ionic currents

In the type of continuum models we have considered so far it has been assumed that the synaptic input current is a function of the pre-synaptic firing rate function. To see how this might arise consider a one dimensional continuum of spiking single neurons with synaptic input at position x given by u(x, t) = η ∗ w ⊗

∑ δ (t − Tm (x)).

(67)

m∈Z

This models the effect of an idealised action potential (delta-Dirac function) arriving at a synapse and initiating a postsynaptic current at time T m . If the synaptic response is on a slower time scale than that of the mean interspike-interval (T m − T m−1 ) and fluctuations around the mean are small, then it is natural to replace the spike train in (67) with a (smooth) firing rate function (see for example [79, 80]). To illustrate how one might go about deriving this firing rate function we consider an integrate-and-fire process for the evolution of a cell membrane voltage given by

15

C

N ∂v = −g L (v − v L ) + ∑ Ik + u. ∂t k=1

(68)

Ca2+ entering the neuron via T-type Ca2+ channels

Here, C is a membrane capacitance, g L a constant

leakage conductance, and v L a constant leakage re- causes a large voltage depolarisation known as the versal potential. The Ik represent a set of slow in- low-threshold Ca2+ spike (LTS). Conventional action trinsic ionic currents, which typically have the form potentials mediated by fast Na+ and K+ currents often p

q

Ik = gk mk k hkk (vk − v) where pk , qk ∈ Z, mk and hk are ride on the crest of an LTS resulting in a burst response gating variables that satisfy differential equations, vk

(i.e., a tight cluster of spikes). If the neuron is hyper-

is the reversal potential of the kth channel and the gk

polarised below vr , the low-threshold current deinac-

are a set of constant conductances. From now on we tivates (with a time constant of τ − ). In this situation shall refer all voltages to v L . An action potential is said release from inhibition results in a post inhibitory reto occur whenever v reaches some threshold h. The set bound response consisting of an LTS and a cluster of 2of action potential firing times are defined by T m (x) = inf{t | v(x, t) ≥ h ; t ≥ T m−1 (x) + τ R }.

10 spikes. This type of dynamical behaviour is known (69)

to play an important role within the context of thalamocortical oscillations [25]. When neurons can fire

Here τ R represents an absolute refractory period. Im- via post inhibitory rebound it is also well known that mediately after a firing event the system undergoes this can lead to lurching waves of activity propagating Assuming

through an inhibitory network [82]. A lurching wave

that the dynamics for v is much faster than that

does not travel with a constant profile, (i.e., there is no

of u and any intrinsic currents (equivalent to tak-

travelling wave frame) although it is possible to iden-

ing τ ≡ C/ g L → 0 in (68)), then v equilibrates to

tify a lurching speed. Rather, the propagating wave

a discontinuous reset such that v → 0.

it’s steady-state value, which we denote by vss = recruits groups of cells in discrete steps. The leadvss (u, m1 , . . . , m N , h1 , . . . , h N ). Moreover, we may com- ing edge of active cells inhibits some cluster of cells pute the firing rate of the IF process as f = f (vss ), ahead of it (depending on the size of the synaptic footprint). Inhibited cells (ahead of the wave) must wait where until they are released from inhibition before they can, C . (70) in turn, fire. The first mathematical analysis of this gL phenomenon can be attributed to Terman et al. [83]. So a neural field model that respects the presence of These authors work with a slightly more complicated intrinsic ionic currents should be written as version of the slow IT current than considered here f (v) =

1
Θ(v − h), τ R + τ ln v/(v − h)

τ=

(71) and treat conductance based models (rather than firing rate). Using techniques from geometric singular Note that in the absence of any slow intrinsic currents perturbation theory they derive explicit formulas for we obtain the standard model u = η ∗ w ⊗ f ◦ u, since when smooth and lurching waves exist and also detervss = u (after choosing units such that g L = 1). mine the effect of network parameters on wave speed. To demonstrate the enormous impact the inclusion However, this work relies partly on numerically deteru = η ∗ w ⊗ f ◦ vss .

of extra slow currents can have, consider a single in- mined properties of the single cell model. trinsic ionic current (N = 1 in (68)) with p1 = q1 = 1, Here we show how an exact analysis of lurching m1 = Θ(vss − vr ) and h1 = r given by waves can be performed when the IT current dedr scribed above is incorporated into a firing rate model. τ (vss ) = −r + r∞ (vss ), (72) dt Taking v1  v and introducing gr = g1 v1 , then vss = with r∞ (v) = Θ(vr − v) and τ (v) = τ − Θ(v − vr ) +

τ + Θ(v

vss (u, r) is given by the solution to

− v). This is a minimal model for the slow vss = u + gr rΘ(vss − vr ), (73) T-type calcium current IT [81]. The slow variable r represents the deinactivation of the low-threshold choosing units such that g L = 1. For mathematical Ca2+ conductance. When this conductance is evoked, convenience we work with the Heaviside firing rate r

16

function f (v) = Θ(v − h) (obtained from (70) in the

0.2

limit τ → 0 with units such that τ R =1) and consider

vss

a purely inhibitory network with

1

2

3 h

0.1

1 Θ(σ − |x|). (74) 2σ We denote the size of a cluster involved in a lurch by L. w(x) = −

∆h 0

For simplicity we shall only consider lurching pulses

vr -0.1

t

∆ TL

∆

t

4 TL

Figure 13: A plot of the analytical solution for a lurch-

h

h

ing pulse, with an α function synaptic response, η (t) =

x

α2 te−αt . The lines labelled 1, 2, and 3 represent trajectories from neurons in adjacent clusters, with cluster 1 firing first. It is assumed that clusters can only fire once through rebound. In this example α = σ = gr = 1, τ + = 2, τ − = 10, h = .1, and vr = −0.05. The numerical solution of the system of defining equations gives (L, TL , ∆, ∆h ) = (0.5, 3.21, 2.39, 1.07). Note that for clarity only partial trajectories are plotted.

L

TL

3 TL

2 TL

∆ vr

Figure 12: A diagram of an idealised solitary lurching

pulse showing the four unknowns that parameterise for x ∈ (0, L) and t > 0, where the solution. Here L represents the size of a cluster, TL the period of the lurch, ∆h the time spent firing and

Q(t, a) =

η (t − s)ds,

0

∆ the duration of inhibition where the rebound vari-

W(x) =

Z x+ L x

able r is increasing. Grey regions indicate where the

w(y)dy.

(76)

The full solution is defined by periodic extension such

system is firing.

that u(x + L, t + TL ) = u(x, t). Hence, using (73), we have a closed form expression for vss in terms of the

where consecutive active clusters are adjacent to each other. We suppose that to a first approximation neurons for x ∈ (0, L) are simultaneously released from inhibition and start firing at time t = TL . The next group with x ∈ (L, 2L) fires when t = 2TL . We define the firing duration of a cluster as ∆h (i.e., the time spent above h) and the duration of inhibition (time spent below vr before release) as ∆. An illustration of this type of lurching pulse is shown in Figs. 12 and 13. An analysis of this type of solution has been given in [84]. Here it was shown that a lurching wave takes the simple (separable) form u(x, t) = W(x)Q(t, min(t, ∆h )),

a

Z

(75) 17

four unknowns L, TL , ∆ and ∆h . Note that if 2L
h for r ≤ a and is zero otherwise.

it is hard to find closed form expressions for λm it is a simple matter to obtain them numerically.

Hence, from (78) q(r) =

Z 2π Z 0

a

w(|r − r0 |)r0 dr0 dθ.

10 a 8

(80)

0

This is readily evaluated using a 2D Fourier transform (equivalent to a Hankel transform) of w(r), which we

6

write in the form Z ∞

(81)

4

Here Jν (x) is the Bessel function of the first kind, of

2

w(r) =

0

e w(k)J 0 (rk)kdk .

order ν and

a+

a-

0 e = w(k)

Z R2

eik·r w(r)dr.

0

(82)

0.05

0.1

0.15

h

0.2

Following [65] it may then be shown that substitution Figure 15: Bump radius a as a function of threshold h. of (81) into (80) gives Note that for h < h the bump solution on the branch D

q(r) = 2π a

Z ∞ 0

Using the fact that

f 0 (u)

e w(k)J 0 (rk)J1 (ak)dk .

= δ (r −

a)/|q0 (a)|

(83)

a+ has a dimple. The point where h = h D on the upper branch is indicated by the filled black circle.

means that

(79) reduces to

An evaluation of the bump solution (83) in closed

aLT[η ](λ) u(r, θ ) = |q0 (a)|

Z 2π

(84) form is typically only possible for special choices of e w(r). In fact it is easier to choose forms of w(k) (the where a0 = (a, θ 0 ). Following [36] and [65] we look for 2D Fourier transform of w(r)) that allow the use of w(|r − a0 |)u(a, θ 0 )dθ 0 ,

0

solutions of the form u(r, θ ) = um (r)eimθ , where m ∈ Z. known integral formulas involving products of Bessel In this case the radial component of the eigenfunction functions. From the analysis of one dimensional stasatisfies

tionary solutions we would expect to obtain bump um (r) = um (a)

Z aLT[η ](λ) 2π

|q0 (a)|

dθ cos(mθ )

solutions for a radially symmetric kernel of the form w(r) = e−r − e−r/2 /2. Since this two dimensional Mex-

0

√ × w( r2 + a2 − 2ra cos θ ),

(85) ican hat function does not have a simple Hankel transform we make use of the approximation R where we have exploited the fact that 02π w(|r − 1 −r 2 a0 |) sin(mθ )dθ = 0. Hence, radial perturbations away e ≈ (87) (K0 (r) − K0 (2r)) ≡ E(r), 2π 3π from the border of the bump are completely determined by the perturbation at the bump edge (as in where Kν (x) is the modified Bessel function of the secthe one dimensional case). Setting r = a in (85) gen- ond kind. For computational simplicity we now work erates an implicit expression for the discrete spec- with the explicit choice w(r) = E(r) − E(r/2)/4. The trum λ = λm , where λm is the solution to Em (λ) ≡ factor of 4 enforces the balance condition 19

R

R2

w(|r|)dr =

The bump radius is determined by the condition

2

q(a) = h. In Fig.

15 we plot the bump radius as a

function of firing threshold. This clearly has the same a-

1 λ0

trend as seen in the one dimensional case (cf Fig. 4). As in the one dimensional case we find two types of solution; one with q00 (0) < 0 for h > h D and the other with q00 (0) > 0 for h < h D . Examples are shown in the

0

insets of Fig.

a+ 0

0.05

0.1

0.15

h

a+ λ2 a-

-1 0.05

0.1

0.15 h

0.2

Figure 16: A plot of λ0 and λ2 along the solution curve of Fig.

15. We note that λ1 = 0 for all points on

the curve a = a(h). Hence, although solutions on a+ are stable to radial perturbations, for h < h D dimpled solutions are unstable to perturbations of the form u2 (r) cos(2θ ).

0, although this is not strictly necessary for the generation of bump solutions. Using the fact that the Hankel transform of K0 (pr) is H p (k) = (k2 + p2 )−1 we may write e = w(k)

2 3π

  5 1 − H2 (k) + H1 (k) − H1/2 (k) . 4 4

(88)

10

Substitution into (83) leads to integrals of the form Z ∞ J0 (rk)J1 (ak) 0

k 2 + p2

dk ≡ L p (a, r).

p 0

Discussion

(89) Although it is clear that there are an increasing number of powerful mathematical techniques to choose

Integrals of this type are given by [65, 88]   1 I1 (pa)K0 (pr) r≥a L p (a, r) = p ,  1 − 1 I (pr)K (pa) r < a ap2

15

q00 (0)

= 0, defining the transition from dimpled to non-dimpled solutions at h = h D . In the one dimensional case no instabilities were found on the upper branch where a = a+ . However, we shall now show that in two dimensions there is the possibility of an instability on the upper branch precisely at the point where h = h D . Consider, for example, an exponential synaptic time course η (t) = e−t . In this case LT[η ](λ)−1 = 1 + λ and the condition for stability is simply that λm < 0 for all m, where λm = −1 + µm . In Fig. 16 we plot λ0 and λ2 along the solution branch of Fig. 15 (λ1 is identically zero by rotation invariance). Hence, although the bump on the branch with a = a+ is stable to radial perturbations (since λ0 < 0 on a+ ), it is not stable to perturbations with m = 2. Indeed λ2 crosses through zero precisely at the point h = h D on a+ , signalling the fact that dimple solutions are unstable. From the shape of the eigenfunction u2 (r) cos 2θ , plotted in Fig. 17, we would expect the bump to split in two as h is decreased through h D . This result is confirmed in [89], where a further discussion of both bump and ring instabilities can be found. we have plotted the point at which

0.2

0

0

15. On the upper branch of Fig.

from when studying neural field equations it is still true that such studies would benefit from contribu(90) tions of a more fundamental nature. Much of the dis-

1

cussion in this paper has revolved around either mak-

which allows us to compute (83) as ing links between integral models and PDEs, or work  ing with the Heaviside firing rate function. In the for4a 5 1 q(r) = − L2 (a, r) + L1 (a, r) − L1/2 (a, r) . (91) mer case this merely side-steps the need to develop 3 4 4 20

more, one must remember that the mean firing rate assumption neglects the precise details of spiking activity and as such does not take into account the effects of temporal correlations between firing events. Indeed, direct numerical simulations of spiking neural field models have uncovered a number of interesting bifurcations and dynamical phenomena, that would be ruled out in a corresponding firing rate model, e.g. [93, 61, 94, 80]. A program of work that addresses the above issues is currently underway, and will be reported on else-

Figure 17: A plot showing the shape of the function u2 (r) cos(2θ ) on the branch a+ when h = h D . The peaks

where.

of this function occur at r = a. Beyond the instability point a one bump solution splits into two pieces.

Acknowledgements I would like to thank Markus Owen for many inter-

general techniques for the study of nonlinear integral equations. In contrast the latter case does allow for analysis in an integral framework, but at the expense of being able to choose an arbitrary (and perhaps more

esting discussions held during the completion of this work. I would also like to acknowledge ongoing support from the EPSRC through the award of an Advanced Research Fellowship, Grant No. GR/R76219.

realistic) firing rate function. However, it is pleasing to note that some exact results for the existence and stability of bumps have recently been obtained for non-Heaviside firing rate functions with a piecewiselinear nature [35, 90]. For smooth firing rate functions techniques from singular perturbation theory, such as reviewed in [32], have also been useful for moving away from the Heaviside limit. Ideally however, one would like to call upon a set of new techniques that would allow the numerical continuation of solutions to nonlinear integral equations, as is commonly done for solutions to nonlinear ODEs using packages like AUTO [91]. Besides the obvious mathematical challenges of dealing with dynamic neural fields, particularly in two spatial dimensions, there are also issues to do with incorporating more biologically realistic features. We have already hinted at how to incorporate the effects of passive dendritic structures and slow ionic currents in sections 7 and 8. However, it is also important to remember that real neural tissue is anisotropic and inhomogeneous, and that the neural field equations presented here must be modified to reflect this, as in the work of Bressloff [92]. Further21

Appendix

[10] G B Ermentrout and J D Cowan. A mathematical theory of visual hallucination patterns. Biological Cybernetics, 34:137–150, 1979.

The coefficients a0 , . . . , a6 used in section 7 are given explicitly by

[11] P C Bressloff, J D Cowan, M Golubitsky, P J Thomas, and M Wiener. Geometric visual hallucinations, Eu-

a0 = (1 + k2 )2 + 4γ k2 ,

(92)

clidean symmetry and the functional architecture of

[α(2 + γ ) + (1 + k2 )v] striate cortex. Philosophical Transactions of the Royal Soa1 = 2(1 + k2 ) , (93) αv ciety London B, 40:299–330, 2001. 2α2 (k2 + 2γ + 3) + 8α(1 + k2 )v + (1 + k2 )2 v2 a2 = , [12] C R Laing, W C Troy, B Gutkin, and G B Ermentrout. α2 v2 Multiple bumps in a neuronal model of working mem(94)

α2 (2 + γ ) + 2α(3 + k2 )v + 2(1 + k2 )v2 , a3 = 2 α2 v3 α2 + 8αv + 2(3 + k2 )v2 a4 = , α2 v4 α + 2v a5 = 2 2 4 , α v 1 a6 = 2 4 . α v

ory. SIAM Journal on Applied Mathematics, 63:62–97,

(95)

2002. [13] C R Laing and W C Troy.

(96)

Two bump solutions of

Amari-type models of working memory. Physica D, 178:190–218, 2003.

(97)

[14] M A Geise. Neural Field Theory for Motion Perception.

(98)

Kluwer Academic Publishers, 1999. [15] P Tass. Cortical pattern formation during visual hallucinations. Journal of Biological Physics, 21:177–210, 1995.

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AUTO97

continuation and bifurcation software for ordinary differential equations.

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