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GHOSTS OF BUMP ATTRACTORS IN STOCHASTIC NEURAL

FIELDS: BOTTLENECKS AND EXTINCTION

ZACHARY P. KILPATRICK∗

Abstract. We study the eﬀects of additive noise on stationary bump solutions to spatially

extended neural ﬁelds near a saddle-node bifurcation. The integral terms of these evolution equations

have a weighted kernel describing synaptic interactions between neurons at diﬀerent locations of the

network. Excited regions of the neural ﬁeld correspond to parts of the domain whose fraction of

active neurons exceeds a sharp threshold of a ﬁring rate nonlinearity. As the threshold is increased,

these a stable and unstable branch of bump solutions annihilate in a saddle node bifurcation. Near

this criticality, we derive a quadratic amplitude equation that describes the slow evolution of the even

mode as it depends on the distance from the bifurcation. Beyond the bifurcation, bumps eventually

become extinct, and the time it takes for this to occur increases for systems nearer the bifurcation.

When noise is incorporated, a stochastic amplitude equation for the even mode can be derived, which

can be analyzed to reveal bump extinction time both below and above the saddle-node.

Key words. Stochastic partial diﬀerential equations, Langevin equation, Perturbation theory,

Amplitude qquations, Saddle-node bifurcation

1. Introduction. Continuum neural ﬁelds are a well-accepted model of spa-

tiotemporal neuronal activity evolving within in vitro and in vivo brain tissue [6,10].

Wilson and Cowan initially introduced these nonlocal integrodiﬀerential equations to

model activity of neuronal populations in terms of mean ﬁring rates [41]. While they

discount the intricate dynamics of neuronal spiking, these models are capable of qual-

itatively capturing a wide range of phenomena such as propagating activity waves

observed in disinhibited slice preparations [23,33–35]. Neural ﬁeld models exhibit a

wide variety of spatiotemporal dynamics including traveling waves, Turing patterns,

stationary pulses, breathers, and spiral waves [14,16]. A distinct advantage of uti-

lizing these continuum equations to model large-scale neural activity is that many

analytical methods for studying their behavior can be adapted from nonlinear partial

diﬀerential equations (PDEs) [6]. Recently, several authors have explored the impact

of stochasticity on spatiotemporal patterns in neural ﬁelds [8,25,27] by employing

techniques originally used to study stochastic front propagation in reaction-diﬀusion

systems [36]. Typically, the approach is to perturb about a linearly stable solution

of the deterministic system, under the assumption of weak noise. However, some re-

cent eﬀorts have been aimed at understanding the impact of noise on patterns near

bifurcations [25,28].

In this work, we are particularly interested in how noise interacts with stationary

pulse (bump) solutions near a saddle-node bifurcation at which a branch of stable

bumps and a branch of unstable bumps annihilate [1]. Bumps are commonly utilized

as a model of persistent and tuned neural activity underlying spatial working memory

[18,42]. This activity tends to last for a few seconds, after which it is extinguished,

to allow for subsequent memories to be formed [20]. One proposed mechanism for

terminating persistent activity is a strong and brief global inhibitory signal, which

would drive the system from the stable bump state to a stable uniform quiescent

state [9]. In terms of neural ﬁeld and spiking models, this can be thought of as

momentarily raising the ﬁring threshold of the system, essentially driving it beyond

the saddle-node bifurcation from which the stable bump emerges.

∗Department of Mathematics, University of Houston, Houston, Texas 77204, USA

(zpkilpat@math.uh.edu). This author is supported by an NSF grant (DMS-1311755).

1

arXiv:1505.06257v1 [nlin.PS] 23 May 2015

We focus on a scalar neural ﬁeld model that supports stationary bump solutions

for appropriate choices of parameters and constituent functions [1,10]:

∂u(x, t)

∂t =−u(x, t) + ZΩ

w(x−y)f(u(y, t))dy(1.1)

where u(x, t) is the total synaptic input arriving to location xand time t, and w(x−y)

describes the strength (amplitude) and polarity (sign) of synaptic connections from

neurons at location yto neurons at location x. We assume w(x) is an even-symmetric

function w(x) = w(−x) with a bounded integral RΩw(x)dxover the spatial domain

x∈Ω=(−x∞, x∞). The nonlinearity f(u) is a ﬁring rate function, which we take to

be the sigmoid [41]

f(u) = 1

1+e−η(u−κ),(1.2)

and we also ﬁnd it useful to take the high gain limit η→ ∞, in which case:

f(u) = H(u−θ) = 1 : u > θ,

0 : u < θ, (1.3)

allowing for analytical tractability in several of our calculations.

Amari was the ﬁrst to analyze Eq. (1.1) in detail, showing that when f(u) is de-

ﬁned to be a Heaviside function (1.3), the network supports stable stationary bump

solutions when the weight function w(x) is a lateral inhibitory (Mexican hat) distri-

bution satisfying: (i) w(x)>0 for x∈[0, x0) with w(x0) = 0; (ii) w(x)<0 for

x∈(x0, x∞); (iii) w(x) is decreasing on [0, x0]; and (iv) w(x) has a unique minimum

on [0, x∞) at x=x1with x1> x0and w(x) strictly increasing on (x1, x∞) [1]. Based

on restrictions (i)-(iv), Amari made use of the integral of the weight function

W(x)≡Zx

0

w(y)dy(1.4)

to prove some of the main results of his seminal work. For instance, it is clear that

W(0) = 0 and W(x) = −W(−x) based on the above assumptions. Moreover, there

will be a single maximum of the function W(x) on the interval (0, x∞) given at

x=x0, i.e. Wmax = maxxW(x) = W(x0), due to conditions (i) and (ii), and

w(x0) = 0. When θ < W (x0) there are two bump solutions: one stable and one

unstable (up to translation symmetry), and when θ > W (x0) there are no bump

solutions to (1.1). When θ=θc≡W(x0), there is a single marginally stable bump

solution. It is at this point that the two branches (stable and unstable) of bump

solutions meet and annihilate in a saddle-node bifurcation (Fig. 2.1). Dynamics of

(1.1) for values of θbeyond this saddle-node bifurcation evolve to quasi-stationary

solutions resembling the ghost of the bump at θc, lasting for a period of time inversely

related to p|θ−θc|[39]. A principled exploration of these dynamics (section 2) is

one of the primary goals of this paper.

As mentioned, the neural ﬁeld equation (1.1) in the absence of noise has been

analyzed extensively [1,10,14]. We expand upon these previous studies by also ex-

ploring the impact of noise on stationary bump solutions to (1.1) near a saddle-node

bifurcations (section 3). Additive noise is incorporated, so that the evolution of the

neural ﬁeld is now described by the spatially extended Langevin equation [4,6,25,30]:

du(x, t) = −u(x, t) + ZΩ

w(x−y)f(u(y, t))dydt+dW(x, t),(1.5)

2

|x−y|

01234

w(|x−y|)

-0.4

-0.2

0

0.2

0.4

0.6

w(2ac) = 0

A

x

01234

W(x)

0

0.1

0.2

0.3

0.4

0.5

0.6

B

θc: saddle-node bifurcation

θc<θ: no equilibria

2acθc>θ: two equilibria

2au2as

Fig. 2.1.Saddle-node bifurcation of bumps in (1.1) with a Heaviside ﬁring rate function (1.3).

(A) Diﬀerence of Gaussians weight function w(x)=e−x2

−Ae−x2/σ2has a Mexican hat proﬁle

since A= 0.4<1and σ= 2 >1. The critical bump half-width acat the saddle-node satisﬁes

the relation w(2ac)=0. (B) The weight function integral (1.4) determines the bump half-widths

a. When θis below the critical threshold θcat the saddle-node, there are two stationary bump

solutions to (1.1): one stable asand one unstable au. When θ > θc, there are zero equilibria, but

the dynamics of (1.1) are slow in the bottleneck near Uc(x).

where the term dW(x, t) is the increment of a spatially varying Wiener process with

spatial correlations deﬁned by hdW(x, t)i= 0 and dW(x, t)dW(y, s)i=C(x−y)δ(t−

s)dtdsand describes the amplitude of the noise, assumed to be weak (1). The

function C(x−y) describes the spatial correlation in each noise increment between

two points x, y ∈Ω.

2. Slow bump extinction in the deterministic system. We begin by ex-

amining the dynamics of stationary bump solutions near a saddle-node bifurcation,

where a stable and unstable branch of solutions annihilate. Our initial analysis fo-

cuses on the noise-free case W(x, t)≡0, allowing us to derive an amplitude equation

that approximates the evolution of the bump height. Linearization of bumps in (1.1)

typically reveals that they are marginally stable to translating perturbations, so the

overall stability is typically characterized by the stability to even perturbations that

expand/contract the bump [14]. Our analysis will emphasize the region of parameter

space near where bumps are marginally stable to even perturbations.

2.1. Existence and stability of bumps. We now brieﬂy review existence and

stability results for stationary bump solutions to the neural ﬁeld equation (1.1). These

results are analogous to those presented in [1,27,40]. For transparency, we focus on

the case of a Heaviside ﬁring rate function (1.3). This allows us to cast bump stability

in terms of a ﬁnite dimensional set of equations, focusing on the evolution of the two

edge interfaces of the bump [1,13]. Assuming a stationary solution u(x, t) = U(x),

we ﬁnd (1.1) requires

U(x) = ZΩ

w(x−y)H(U(y)−θ)dy. (2.1)

Given a unimodal bump solution U(x), without loss of generality, we can ﬁx the center

and peak of the bump to be at the origin x= 0. In the case of even-symmetric bumps

U(x) = U(−x) [1], we will have the conditions for the bump half-width a:U(x)> θ

for x∈(−a, a), U(x)< θ for x∈Ω\[−a, a], and U(±a) = θ. In this case, (2.1)

3

becomes

U(x) = Za

−a

w(x−y)dy=Zx+a

x−a

w(y)dy=Zx+a

0

w(y)dy−Zx−a

0

w(y)dy.

By utilizing the integral function (1.4), we can write the even-symmetric solution

U(x) = W(x+a)−W(x−a).(2.2)

To determine the half-width a, we require the threshold conditions U(±a) = θof the

solution (2.2) to yield

U(a) = W(2a) = Z2a

0

w(y)dy=θ.

Note that when θ < Wmax = maxxW(x), there will be a stable and unstable bump

solution to (1.1). When θ=θc≡Wmax , there is a single marginally stable bump

solution Uc(x) to (1.1), as illustrated in Fig. 2.1B. Diﬀerentiating W(2a) by its ar-

gument yields W0(2ac) = w(2ac)≡0 as an implicit equation for the half-width acat

this criticality. Utilizing the notation of Amari condition (i), we have that ac=x0/2.

Note, the relation w(2ac) = 0 is explicitly solvable for acfor several typical lateral in-

hibitory type weight functions. For instance, in the case of the diﬀerence of Gaussians

w(x)=e−x2−Ae−x2/σ2on x∈(−∞,∞) [1], we have ac=σpln(1/A)/2√σ2−1

and θc=√π

2[erf(2ac)−Aσerf(2ac/σ)]. For the “wizard hat” w(x) = (1 −|x|)e−|x|on

x∈(−∞,∞) [11], we have ac= 1/2 and θc= e−1. For a cosine weight w(x) = cos(x)

on the periodic domain x∈[−π , π] [27], we have ac=π/4 and θc= 1.

To characterize the stability of bump solutions to (1.1), we will study the evolution

of small smooth perturbations ε¯

ψ(x, t) (ε1) to stationary bumps U(x) by utilizing

the Taylor expansion u(x, t) = U(x) + ε¯

ψ(x, t) + O(ε2). By plugging this expansion

into (1.1) and truncating to O(ε), we can derive an equation whose solutions constitute

the family of eigenfunctions associated with the linearization of (1.1) about the bump

solution U(x). We begin by truncating (1.1) to O(ε) assuming uis given by the above

expansion and that the nonlinearity f(u) is given by the Heaviside function (1.3), so

∂¯

ψ(x, t)

∂t =−¯

ψ(x, t) + ZΩ

w(x−y)H0(U(y)−θ)¯

ψ(y, t)dy, (2.3)

and we can diﬀerentiate the Heaviside function, in the sense of distributions, by noting

H(U(x)−θ) = H(x+a)−H(x−a), so

δ(x+a)−δ(x−a) = dH(U(x)−θ)

dx=H0(U(x)−θ)U0(x),

which we can rearrange to ﬁnd

H0(U(x)−θ) = δ(x+a)−δ(x−a)

U0(x)=1

|U0(a)|(δ(x+a) + δ(x−a)) .(2.4)

Upon applying the identity (2.4) to (2.3), we have

∂¯

ψ(x, t)

∂t =−¯

ψ(x, t) + γw(x+a)¯

ψ(−a, t) + w(x−a)¯

ψ(a, t),(2.5)

4

where γ−1=|U0(a)|=w(0) −w(2a). One class of solutions, such that ¯

ψ(±a, t) =

¯

ψ(±a, 0) = 0, lies in the essential spectrum of the linear operator that deﬁnes (2.5).

In this case, ¯

ψ(x, t) = ¯

ψ(x, 0)e−t, so perturbations of this type to not contribute

to any instabilities of the stationary bump U(x) [21]. Assuming separable solutions

¯

ψ(x, t) = b(t)ψ(x), we can characterize the remaining solutions to (2.5). In this case,

b0(t) = λb(t), so b(t)=eλt and

(λ+ 1)ψ(x) = γ[w(x+a)ψ(−a) + w(x−a)ψ(a)] .(2.6)

Solutions to (2.6) that do not satisfy the condition ψ(±a)≡0 can be separated into

two classes: (i) odd ψ(a) = −ψ(−a) and (ii) even ψ(a) = ψ(−a). This is due to the

fact that the equation (2.6) implies the function ψ(x) is fully speciﬁed by its values

at x=±a. Thus, we need only concern ourselves with these two points, yielding the

two-dimensional linear system

(λ+ 1)ψ(−a) = γ[w(0)ψ(−a) + w(2a)ψ(a)] (2.7a)

(λ+ 1)ψ(a) = γ[w(2a)ψ(−a) + w(0)ψ(a)] .(2.7b)

For odd solutions ψ(a) = −ψ(−a), the eigenvalue

λo=−1 + γ[w(0) −w(2a)] = −1 + w(0) −w(2a)

w(0) −w(2a)= 0,

reﬂecting the fact that (1.1) is translationally symmetric, so bumps are marginally

stable to perturbations that translate their position. Even solutions ψ(a) = ψ(−a)

have associated eigenvalue

λe=−1 + γ[w(0) + w(2a)] = −1 + w(0) + w(2a)

w(0) −w(2a)=2w(2a)

w(0) −w(2a).

Thus, when θ < θc, the wide bump as> acwill be linearly stable to expand-

ing/contracting perturbations since w(2as)<0 due to Amari’s condition (ii) [1]. The

narrow bump au< acis linearly unstable to such perturbations since w(2au)>0

due to condition (i). When θ=θc, we have w(2ac) = 0 so that λe= 0 and

|U0(±ac)|=w(0).

In anticipation of our derivations of amplitude equations, we deﬁne the eigenfunc-

tions at the criticality θ=θc. Utilizing the fact that |U0(±ac)|=w(0) and the linear

system (2.7a), we have that the odd eigenfunction at the bifurcation is

ψo(x) = 1

w(0) [w(x−ac)−w(x+ac)] ,(2.8)

and the even eigenfunction is

ψe(x) = 1

w(0) [w(x−ac) + w(x+ac)] .(2.9)

Note, this speciﬁes that ψe(±a) = ψo(a) = −ψ0(−a) = 1. Furthermore, we will ﬁnd

it useful to compute the derivatives

ψ0

o(x) = 1

w(0) [w0(x−a)−w0(x+a)] ,

5

which is even (ψ0

o(−ac) = ψ0

o(ac)), and

ψ0

e(x) = 1

w(0) [w0(x−a) + w0(x+a)] ,

which is odd (ψ0

e(−ac) = −ψ0

e(ac)). Lastly, we note that we will utilize the fact

that, for even symmetric functions, w0(0) = 0, so ψ0

o(±ac) = ψ0

e(ac) = −ψ0

e(−ac) =

w0(2ac)/w(0).

2.2. Saddle-node bifurcation of bumps. Motivated by the above linear sta-

bility analysis, we now carry out a nonlinear analysis in the vicinity of the saddle-node

bifurcation from which the stable and unstable branches of stationary bumps emanate.

Speciﬁcally, we will perform a perturbation expansion about the bump solution Uc(x)

at the critical threshold value θc. We therefore deﬁne θ=θc+µε2,ε1, so that µis

a bifurcation parameter determining the distance of θfrom the saddle-node bifurca-

tion point. As demonstrated above, the linear stability problem for Uc(x) reveals two

zero eigenvalues λo=λe= 0 associated with the odd ψoand even ψeeigenfunctions

(2.8) and (2.9), respectively. Our analysis employs the ansatz:

u(x, t) = Uc(x) + εAe(τ)ψe(x) + ε2Ao(t)ψo(x) + ε2u2(x, τ ) + O(ε3),(2.10)

where τ=εt is a temporal rescaling that reﬂects the vicinity of the system to a

saddle-node bifurcation associated with the expanding/contracting eigenmode ψe[39].

Similar expansions have been utilized in the analysis of front bifurcations in reaction-

diﬀusion systems [3,37] and neural ﬁeld models [5]. Upon plugging (2.10) into (1.1)

and expanding in orders of ε, we ﬁnd that at O(1), we simply have the stationary

bump equation (2.1) at θ=θc. Proceeding to O(ε), we ﬁnd

0 = Ae(τ)ZΩ

w(x−y)H0(Uc(y)−θc)ψe(y)dy−ψe(x),

so we can use (2.4) to write

0 = Ae(τ)1

w(0)(w(x+a)ψe(−a) + w(x−a)ψe(a)) −ψe(x).(2.11)

The right hand side of (2.11) vanishes due to the formula for the even (2.9) eigen-

function associated with the stability of the bump Uc(x). At O(ε2), we obtain an

equation for higher order term u2:

L[Aoψo+u2] =A0

eψe+A0

oψo+µZΩ

w(x−y)H0(Uc(y)−θc)dy(2.12)

−A2

e

2ZΩ

w(x−y)H00(Uc(y)−θc)ψe(y)2dy,

where Lis the non-self-adjoint linear operator

Lu(x) = −u(x) + ZΩ

w(x−y)H0(Uc(y)−θc)u(y)dy. (2.13)

Both ψo(x) and ψe(x) lie in the nullspace N(L), as demonstrated in the previous

section by identifying solutions to (2.3). Thus, the ψoterms on the left hand side of

(2.12) vanish. We can ensure a bounded solution to (2.12) exists by requiring that the

6

right hand side be orthogonal to all elements of the nullspace of the adjoint operator

L∗. The adjoint is deﬁned with respect to the L2inner product

hLu, vi=ZΩ

[Lu(x)] v(x)dx=ZΩ

u(x) [L∗v(x)] dx=hu, L∗vi.(2.14)

Thus, we ﬁnd

L∗v(x) = −v(x) + H0(Uc(x)−θc)ZΩ

w(x−y)v(y)dy, (2.15)

deﬁned in the sense of distributions under the L2inner product given in (2.14). It is

straightforward to show that ϕo:= H0(Uc−θc)ψoand ϕe:= H0(Uc−θc)ψelie in the

nullspace of L∗. Components of N(L∗) are deﬁned by the equation

v(x) = H0(Uc(x)−θc)ZΩ

w(x−y)v(y)dy. (2.16)

To show ϕo, ϕe∈ N(L∗), we simply plug these formulas into (2.16) to ﬁnd

H0(Uc(x)−θc)ψj(x) = H0(Uc(x)−θc)ZΩ

w(x−y)H0(Uc(y)−θc)ψj(y)dy,

for j=o, e, which is true due to the fact that ψoand ψelie in N(L). Thus, we will

impose solvability of (2.12) by taking the inner product of both sides of the equation

with respect to ϕo:= H0(Uc−θc)ψoand ϕe:= H0(Uc−θc)ψeyielding

0 = ϕj, A0

eψe+A0

oψo+µw ∗H0(Uc−θc)−A2

e

2w∗H00(Uc−θc)ψ2

e,(2.17)

for j=o, e, where we have deﬁned the convolution w∗F=RΩw(x−y)F(y)dy. Due to

odd-symmetry, terms of the form hH0(Uc−θc)ψj, ψki,j6=k, vanish. In a similar way,

the term hH0(Uc−θc)ψo, w ∗H0(Uc−θc)ivanishes due to odd-symmetry. Isolating the

temporal derivatives A0

jin (2.17), we ﬁnd that the amplitudes Aj(j=o, e) satisfy

the following fast-slow system of nonlinear diﬀerential equations

dAo

dt=ϕo, w ∗H00(Uc−θc)ψ2

e

2hϕo, ψoiAe(τ)2,(2.18a)

dAe

dτ=−µhϕe, w ∗H0(Uc−θc)i

hϕe, ψei+ϕe, w ∗H00(Uc−θc)ψ2

e

2hϕe, ψeiAe(τ)2.(2.18b)

With the system (2.18) in hand, we can determine the long term dynamics of

the amplitudes as the bifurcation parameter µis varied. We begin by computing the

constituent components of the right hand sides, using properties of the eigenfunctions

ψoand ψe. To start, we will compute the second derivative H00(Uc−θc), which appears

in the coeﬃcient of the quadratic term A2

e. Diﬀerentiating the function H(Uc(x)−θc)

twice with respect to x, using the chain and product rule, we ﬁnd the following formula

d2H(Uc(x)−θc)

dx2= (U0

c(x))2H00(Uc(x)−θc) + U00

c(x)H0(Uc(x)−θc)

= (U0

c(x))2H00(Uc(x)−θc) + U00

c(x)

U0

c(x)

dH(Uc(x)−θc)

dx,

7

where we have applied the identity (2.4) for the ﬁrst derivative H0(U−θ). Rearranging

terms, we ﬁnd that

H00(Uc−θc) = 1

U0

c(x)2

d2H(Uc(x)−θc)

dx2−U00

c(x)

|U0

c(a)|3[δ(x+ac) + δ(x−ac)] .(2.19)

We can further specify the formula (2.19) by diﬀerentiating dH(Uc−θc)

dx=δ(x+ac)−

δ(x−ac) with respect to xto yield

d2H(Uc−θc)

dx2=δ0(x+ac)−δ0(x−ac),

where δ0(x−x0) is deﬁned, in the sense of distributions, for any smooth function F(x)

by using integration-by-parts [26]:

ZΩ

δ0(x−x0)F(x)dx=−ZΩ

δ(x−x0)F0(x)dx=−F0(x0).

Furthermore, we note that the spatial derivative |U0

c(±ac)|=w(0) and U00

c(x) =

w0(x+ac)−w0(x−ac). Even symmetry of w(x) mandates that w0(x) = −w0(−x)

and w0(0) = 0, so U00

c(±ac) = w0(2ac). Thus, we can at last write

H00(Uc−θc) = δ0(x+ac)−δ0(x−ac)

w(0)2−w0(2ac) [δ(x+ac) + δ(x−ac)]

w(0)3.(2.20)

Computing the inner products in (2.18) then simply amounts to evaluating the inte-

grals in the sense of distributions. First, we use (2.4) to note

hϕj, ψji=ZΩ

ψj(x)2H0(Uc(x)−θc)dx=γψj(ac)2+ψj(−ac)2=2

w(0),

for j=o, e, since ψe(±ac) = ψo(ac) = −ψo(−ac) = 1. Furthermore,

hϕe, w ∗H0(Uc−θc)i=ZΩZΩ

w(x−y)ψe(x)H0(Uc(x)−θc)H0(Uc(y)−θc)dydx

=γ2ZΩ"X

a=±ac

w(x+a)#ψe(x)"X

a=±ac

δ(x+a)#dx

=γ2[ψe(ac) + ψe(−ac)] ·[w(0) + w(2ac)] = 2

w(0),(2.21)

where we have utilized ψe(±ac) = 1 and w(2ac)≡0. Finally, we compute the

quadratic terms using the identity (2.20), starting with

hϕo, w ∗H00(Uc−θc)ψ2

ei=ZΩZΩ

w(x−y)ϕo(x)H00(Uc(y)−θc)ψe(y)2dydx

=γX

a=±ac

ψo(a)ZΩ

w(a−y)H00(Uc(y)−θc)ψe(y)2dy,

(2.22)

8

and we note that individual terms under the integral from the sum deﬁning (2.20) are

ZΩ

w(−ac−y)δ0(y+ac)ψe(y)2dy=w0(0)ψe(−ac)2−2w(0)ψ0

e(−ac)ψe(−ac)

= 2w0(2ac),

ZΩ

w(ac−y)δ0(y+ac)ψe(y)2dy=w0(2ac)ψe(−ac)2−2w(2ac)ψ0

e(−ac)ψe(−ac)

=w0(2ac),

ZΩ

w(−ac−y)δ0(y−ac)ψe(y)2dy=w0(−2ac)ψe(ac)2−2w(2ac)ψ0

e(ac)ψe(ac)

=−w0(2ac),

ZΩ

w(ac−y)δ0(y−ac)ψe(y)2dy=w0(0)ψe(ac)2−2w(0)ψ0

e(ac)ψe(ac)

=−2w0(2ac),

for the terms involving the distributional derivative δ0(x−x0), whereas the terms

involving δ(x−x0) are

ZΩ

w(−ac−y)δ(y+ac)ψe(y)2dy=w(0)ψe(−ac)2=w(0),

ZΩ

w(ac−y)δ(y+ac)ψe(y)2dy=w(2ac)ψe(−ac)2= 0,

ZΩ

w(−ac−y)δ(y−ac)ψe(y)2dy=w(2ac)ψe(ac)2= 0,

ZΩ

w(ac−y)δ(y−ac)ψe(y)2dy=w(0)ψe(ac)2=w(0).

Thus, each integral term

ZΩ

w(−ac−y)H00(Uc(y)−θc)ψe(y)2dy=2w0(2ac)

w(0)2(2.23)

ZΩ

w(ac−y)H00(Uc(y)−θc)ψe(y)2dy=2w0(2ac)

w(0)2.(2.24)

Finally, using the fact that ψo(a) = −ψo(−a) = 1, we ﬁnd that the two terms in the

sum of (2.22) cancel and the integral vanishes. Thus, hϕo, w ∗H00(Uc−θc)ψ2

ei= 0,

so Ao(t)≡¯

Aois constant. On the other hand, computing the quadratic coeﬃcient in

the equation for Ae, we have

hϕe, w ∗H00(Uc−θc)ψ2

ei=ZΩZΩ

w(x−y)ϕe(x)H00(Uc(y)−θc)ψe(y)2dydx

=γX

a=±ac

ψe(a)ZΩ

w(a−y)H00(Uc(y)−θc)ψe(y)2dy.

(2.25)

The integrals in (2.25) are identical to those in (2.22), so it is straightforward to

compute, using (2.23) and (2.24) that

hϕe, w ∗H00(Uc−θc)ψ2

ei=γ2w0(2ac)

w(0)2+2w0(2ac)

w(0)2=4w0(2ac)

w(0)3.

9

Thus, we can at last compute all the terms in (2.18), specifying that

dAo

dt= 0,(2.26a)

dAe

dτ=−µ−|w0(2ac)|

w(0)2Ae(τ)2,(2.26b)

where we have noted the fact that w0(2ac)<0 due to Amari’s conditions (iii) and

(iv) on the weight function w(x) [1].

Equation (2.26a) reﬂects the translational symmetry of the original neural ﬁeld

equation (1.1), so bumps are neutrally stable to translating perturbations ψoregard-

less of the bifurcation parameter µ. On the other hand, as the bifurcation parameter

µis changed, the dynamics of the even eigenmode ψereﬂect the relative distance to

the saddle-node bifurcation at which point bumps are marginally stable to expand-

ing/contracting perturbations. When µ < 0, there are two ﬁxed points of equation

(2.26b) at Ae=±w(0)p|µ/w0(2ac)|, corresponding to the pair of emerging station-

ary bump solutions which are wider (+) and narrower (−) than the critical bump Uc.

As expected, the wide bump is linearly stable since a linearization of (2.26b) yields

λ+=−p|µ·w0(2ac)|/w(0) <0, and the narrow bump is linearly unstable since

λ−= +p|µ·w0(2ac)|/w(0) >0 [1,14]. Crossing through the subcritical saddle-node

bifurcation, we ﬁnd that for µ≡0, there is a single ﬁxed point Ae≡0, which is

marginally stable, since λ0= 0.

Lastly, note when µ > 0, there are no ﬁxed points of the diﬀerential equa-

tion (2.26b). However, starting at the initial condition Ae(0) = 0 (correspondingly

u(x, 0) = Uc(x)), we ﬁnd that the dynamics of the amplitude Ae(τ) are strongly de-

termined by the ghost of the ﬁxed point at Ae= 0 [39]. Note in Fig. 2.2Athat the

transient bump retains a shape much like that of the critical bump for an appreciable

period of time before extinguishing. Trajectories of the full system (1.1) evolve more

slowly when the distance to the bifurcation |θ−θc|=|µ|ε2is smaller. Solving for

Ae(τ) in this speciﬁc case and reverting the the original time coordinate t=τ /ε, we

ﬁnd

Ae(t) = −w(0)√µ

p|w0(2ac)|tan εpµ· |w0(2ac)|t/w(0).(2.27)

Thus, the residence time tbin the bottleneck, or neighborhood of the ghost of the ﬁxed

point Ae= 0, is given by the amount of time it takes for Ae(t) to traverse to some set

value. Of course, this is dependent on the bifurcation parameter µ. For illustration,

we examine how long it takes until Ae(tb) = −1. Using the formula (2.27), it is

straightforward to ﬁnd that

tb=w(0)

εpµ· |w0(2ac)|tan−1 p|w0(2ac)|

w(0)√µ!.(2.28)

We compare this formula to the results of numerical simulations in Fig. 2.2B, utilizing

the diﬀerence of Gaussians weight function w(x) = e−x2−Ae−x2/σ2on x∈(−∞,∞).

Comparisons are made by noting that when Ae(tb) = −1, then u(x, t)≈Uc(x)−

εψe(x), so that the peak of the activity proﬁle will be

u(0, tb)≈Uc(0) −εψe(0) = W(ac)−W(−ac)−2w(ac)ε

w(0) .

10

t

0 2 4 6

u(0, t)

0

0.1

0.2

0.3

0.4

0.5

0.6 Uc(0)

Uc(0) −εψe(0)

tb

B

bottleneck

t

0 2 4 6 8 10 12

u(0, t)

0

0.1

0.2

0.3

0.4

0.5

0.6

ε= 0.2

ε= 0.1

ε= 0.07

C

ε

0 0.05 0.1 0.15 0.2

tb

0

5

10

15

20

D

Fig. 2.2.Slow passage of bumps on x∈(−∞,∞)when w(x)=e−x2

−Ae−x2/σ2. (A) Slow

passage of a transient bump by the ghost of the critical solution Uc(x)when θ=θc+ε2for ε= 0.1

(µ= 1), A= 0.4, and σ= 2. (B) The peak of the bump u(0, t)slowly decreases in amplitude until

breaking down quickly in the vicinity of Ae(t) = −1. Note the theoretical formula for the amplitude

(solid line) given by (2.27) matches the numerical simulation (dashed line) in the slow passage

region. (C) Amplitude of the even mode Ae(t)slowly decreases with time. The duration of the

bottleneck increases as the distance to the bifurcation is decreased by reducing ε. (D) Comparison

of the theory (solid) given by (2.28) to the numerically computed (dots) duration in the bottleneck

(the crossing Ae(tb) = −1).

t

0 10 20 30

u(0, t)

0

0.5

1

1.5

ε= 0.2

ε= 0.1

ε= 0.05

ε

0 0.05 0.1 0.15 0.2

tb

0

5

10

15

20

25

30

Fig. 2.3.Slow passage of a bump on x∈[−π, π]for a cosine weight function w(x) = cos(x).

(A) Amplitude of the even mode Ae(t)slowly decreases with time. Numerical simulations (dashed

lines) of (1.1) are compared to the trajectory √2(1 −εtan(εt)). (B) The duration of bottleneck

increases as the distance to the bifurcation is decreased. Simulations (dots) are well ﬁt by the theory

tb=π/[4ε].

11

Notice in Fig. 2.2C,Dthat, as predicted, the time spent in the bottleneck increases

as the amplitude of the small parameter εis decreased. The attracting impact of

the ghost is stronger when the parameters of the system lie closer to the bottleneck.

For further comparison, we consider the case w(x) = cos(x) in Fig. 2.3. In this case

the constituent functions ac=π/4, w(0) = 1, and w(2ac) = −1. Furthermore, by

setting µ= 1 the formulas for the amplitude (2.27) and residence time (2.28) simplify

considerably to Ae(t) = −tan(εt) and tb=π/[4ε].

2.3. Amplitude equations for smooth nonlinearities. Our nonlinear anal-

ysis in the case of Heaviside nonlinearities f(u)≡H(u−θ) made extensive use of the

speciﬁc form of the distributional derivatives. Inner products with these functions

lead to dynamical equations focused on a ﬁnite number of discrete points in space,

rather than over the spatial continuum x∈Ω. Here, we show it is straightforward

to extend this analysis to the case of arbitrary smooth nonlinearities f(u). There

are several detailed analyses of stationary bumps in neural ﬁeld with smooth ﬁring

rate, showing a similar bifurcation structure to that presented in Fig. 2.1: a stable

and an unstable branch of bump solutions annihilate in a saddle-node bifurcation as

the threshold of the ﬁring rate function is increased. We refrain from such a detailed

analysis here and refer the reader to these works [12,15,27,29,31,40]. Again, deﬁning

θ=θc+µε2,ε1, so µdetermines the distance of θfrom the bifurcation and on

which side of θcit lies. Following our previous analysis, we utilize the ansatz (2.10)

and rescale time τ=ετ. In this case, ψo(x) and ψe(x) will still be odd and even

eigenmodes associated with the linear stability of stationary bump solutions to (1.1).

At the criticality θ≡θc, their associated eigenvalues will be λo=λe≡0, as in the

case of Heaviside ﬁring rates [40]. Expanding (1.1) in orders of εusing the ansatz

(2.10) yields at O(ε):

0 = Ae(τ)ZΩ

w(x−y)f0(Uc(y))ψe(y)dy−ψe(x).(2.29)

The right hand side of (2.29) will vanish as long as ψe(x) lies in the null space of the

non-self-adjoint linear operator

Lu(x) = −u(x) + ZΩ

w(x−y)f0(Uc(y))u(y)dy, (2.30)

deﬁning the linear stability of the critical bump solution to the stationary equation

Uc(x) = RΩw(x−y)f(Uc(y))dy. Of course, this condition must be satisﬁed for the

system (1.1) to lie at a saddle-node bifurcation at θ≡θc. At O(ε2), the equation for

the higher order term u2is

L[Aoψo+u2] =A0

eψe+A0

oψo+µZΩ

w(x−y)f0(Uc(y))dy(2.31)

−A2

e

2ZΩ

w(x−y)f00(Uc(y))ψe(y)2dy,

where Lis the linear operator (2.30). The ψoterms on the left of (2.31) vanish since

ψo∈ N(L). We can show that ϕo:= f0(Uc)ψoand ϕe:= f0(Uc)ψelie in the nullspace

of the adjoint operator N(L∗) where

L∗v(x) = −v(x) + f0(Uc(x)) ZΩ

w(x−y)v(y)dy

12

under the L2inner product (2.14). Elements of N(L∗) satisfy the equation

v(x) = f0(Uc(x)) ZΩ

w(x−y)v(y)dy. (2.32)

Plugging the formulas for ϕoand ϕeinto (2.32), we ﬁnd

f0(Uc(x))ψj(x) = f0(Uc(x)) ZΩ

w(x−y)f0(Uc(y))ψj(y)dy,

for j=o, e, which must be satisﬁed since ψo, ψe∈ N(L). Imposing solvability of

(2.31), we ﬁnd that

0 = f0(Uc)ψj, A0

e(τ)ψe+A0

o(t)ψo+µw ∗f0(Uc)−A2

e

2w∗f00(Uc)ψ2

e,

for j=o, e. After canceling odd terms and isolating the derivatives A0

j, we ﬁnd the

amplitudes Ajsatisfy the system:

dAo

dt=hϕo, w ∗f00(Uc)ψ2

ei

2hϕo, ψoiAe(τ)2,(2.33a)

dAe

dτ=−µhϕe, w ∗f0(Uc)i

hϕe, ψei+hϕe, w ∗f00(Uc)ψ2

ei

2hϕe, ψeiAe(τ)2.(2.33b)

We can derive the coeﬃcients in the system (2.33) by computing the inner prod-

ucts therein. To do so, we must choose a speciﬁc nonlinearity, such as the sigmoid

(1.2), and a weight kernel. For illustration, we consider the cosine kernel w(x) on

the ring x∈Ω = [−π, π] with periodic boundaries. As shown in previous stud-

ies, the bump solution Uc(x) = Accos xwhile the eigenmodes ψo(x) = sin(x) and

ψe(x) = cos(x) [22,27,40]. Since Lψj≡0 for j=o, e, this means

sin(x) = Zπ

−π

cos(x−y)f0(Accos(y)) sin(y)dy= sin xZπ

−π

sin2(y)f0(Accos y)dy,

where we have used cos(x−y) = cos xcos y+ sin xsin y, and

cos(x) = Zπ

−π

cos(x−y)f0(Accos(y)) cos(y)dy= cos xZπ

−π

cos2(y)f0(Accos y)dy,

so that we can write

Zπ

−π

sin2(y)f0(Accos y)dy≡1,Zπ

−π

cos2(y)f0(Accos y)dy≡1.(2.34)

The identities (2.34) allow us to compute

hϕo, ψoi=Zπ

−π

f0(Accos(y)) sin(y)2dy= 1,

and

hϕe, ψei=Zπ

−π

f0(Acsin(y)) cos(y)2dy= 1.

13

Furthermore,

hϕo, w ∗f00(Uc)ψ2

ei=Zπ

−π

f00(Uc(y))ψe(y)2Zπ

−π

cos(x−y)f0(Accos(y)) sin(y)dxdy

=Zπ

−π

f00(Uc(y)) cos(y)2sin(y)dy= 0,(2.35)

where the last equality holds due to the integrand being odd. Thus, the equation

(2.33a) reduces to A0

o(t) = 0, so Ao(t)≡¯

Ao. Now, we can calculate the coeﬃcients

of the Aeamplitude equation. First by utilizing the fact that Rπ

−πw(x−y)ϕe(y)dy=

ψe(x), we can compute

hϕe, w ∗f0(Uc)i=Zπ

−π

f0(Accos(x)) cos(x)dx=hϕe,1i.(2.36)

Lastly, we can simplify the integrals in the quadratic term by again making use of the

identity Rπ

−πw(x−y)ϕe(y)dy=ψe(x), so

hϕe, w ∗f00(Uc)ψ2

ei=Zπ

−π

f00(Accos(x)) cos3(x)dx=hf00(Uc), ψ3

ei.(2.37)

so we can simplify (2.33b) to

dAe

dτ=−µhϕe,1i+1

2hf00(Uc), ψ3

eiAe(τ)2.(2.38)

3. Stochastic neural ﬁelds near the saddle-node. We now study the impact

of stochastic forcing near the saddle-node bifurcation of bumps. Our analysis utilizes

the spatially extended Langevin equation with additive noise (1.5). Guided by our

analysis of the deterministic system (1.1), we will utilize an expansion in the small

parameter ε, which determines the distance of the system from the saddle-node. To

formally derive stochastic amplitude equations, we must specify the scaling of the

noise amplitude as it relates to the small parameter ε, as this will determine the

level of the perturbation hierarchy wherein the noise term dWwill appear. We opt for

the scaling =ε5/2, as this introduces a nontrivial interaction between the nonlinear

amplitude equation for Aeand the noise, but random perturbations do not shift the

location of the bifurcation as in [2,25].

3.1. Stochastic amplitude equation for bumps. Motivated by our quanti-

tative analysis in the noise-free case, we rescale time in the stochastic term of (1.5)

using τ=εt, so

du(x, t) = −u(x, t) + ZΩ

w(x−y)f(u(y, t))dydt+ε2dˆ

W(x, τ ),(3.1)

where d ˆ

W(x, τ ) := √εdW(x, ε−1τ) is a rescaled version of the Wiener process dW

that is independent of ε[19]. We then apply the ansatz (2.10) once again and take

Heaviside ﬁring rate functions (1.3), thus ﬁnding (2.11) at O(ε). The O(ε) equation

is satisﬁed due to the fact that ψe∈ N(L), where Lis the linear operator given by

(2.30). Finally, proceeding to O(ε2), we ﬁnd

L[Aoψo+u2] dt=dAeψe+ dAoψo+µZΩ

w(x−y)H0(Uc(y)−θc)dydt(3.2)

−A2

e

2ZΩ

w(x−y)H00(Uc(y)−θc)ψe(y)2dydt+ d ˆ

W .

14

As before, the ψoterms on the left vanish since Lψo≡0, and we ensure a bounded

solution to (3.2) exists by requiring the inhomogeneous part is orthogonal to ϕo, ϕe∈

N(L∗), where L∗is the adjoint linear operator given by (2.15). Taking inner products

yields

0 = hϕj,dAe(τ)ψe(x)+dAo(t)ψo(x) + µw ∗H0(Uc−θc)dt(3.3)

−Ae(τ)2

2w∗H00(Uc−θc)ψ2

edt+ d ˆ

W,

for j=o, e. Isolating temporal derivatives, we ﬁnd the amplitudes Ao(t) and Ae(τ)

obey the following pair of nonlinear stochastic diﬀerential equations

dAo(t) =hϕo, w ∗H00(Uc−θc)ψ2

ei

2hϕo, ψoiAe(τ)2dt−hϕo,dˆ

Wi

hϕo, ψoi(3.4a)

dAe(τ) = −µhϕe, w ∗[H0(Uc−θc)]i

hϕe, ψei+hϕe, w ∗H00(Uc−θc)ψ2

ei

2hϕe, ψeiAe(τ)2(3.4b)

−hϕe,dˆ

Wi

hϕe, ψei.

Utilizing the formulas for H0(Uc−θc) (2.4) and H00(Uc−θc) (2.19) we derived in the

previous section, we can simplify the expressions in (3.4). Additionally, we make use

of the fact that

dˆ

Wo(τ) := −hϕo,dˆ

Wi

hϕo, ψoi=−1

2hψo(−ac)d ˆ

W(−ac, τ ) + ψo(ac)d ˆ

W(ac, τ )i

=dˆ

W(−ac, τ )−dˆ

W(ac, τ )

2,

dˆ

We(τ) := −hϕe,dˆ

Wi

hϕe, ψei=−1

2hψe(−ac)d ˆ

W(−ac, τ ) + ψe(ac)d ˆ

W(ac, τ )i

=−dˆ

W(ac, τ )+dˆ

W(−ac, τ )

2.

Utilizing the fact that hdˆ

W(x, τ )d ˆ

W(y, τ 0)i=C(x−y)δ(τ−τ0)dτdτ0, it is straightfor-

ward to compute the variances hˆ

Wo(τ)2i=Doτ= (C(0)−C(2ac))t/2 and hˆ

We(τ)2i=

Deτ= (C(0) + C(2ac))t/2. Clearly, for spatially ﬂat correlation functions C(x)≡¯

C,

noise will have no impact on the odd amplitude Ao(t) since Do≡0. Thus, (3.4)

becomes

dAo(t) = √εdWo(t),(3.5a)

dAe(τ) = −µdτ−|w0(2ac)|

w(0)2Ae(τ)2dτ+ d ˆ

We(τ),(3.5b)

where we have converted the noise term in (3.5a) back to the original time coordinate:

dWo(t) = d ˆ

Wo(εt)/√ε[19]. Note that in equation (3.5a), we essentially recover the

diﬀusion approximation of the translating mode of the bump hAo(t)2i=εDot, which is

analyzed in [27]. Equation (3.5b) is a stochastic amplitude equation, so that the noise

term dWeis projected onto the direction of the neutrally stable even perturbation ψe.

15

t

0 20 40 60 80 100

maxxu(x, t)

0

0.5

1

1.5

2

Uc(x)−εψe(0)

B

Fig. 3.1.Noise-induced extinction of bumps in the stochastic neural ﬁeld (1.5) on x∈[−π, π]

for a cosine weight w(x) = cos(x). (A) A single realization of the equation (1.5) with the initial

condition u(x, 0) = Uc(x) = √2 cos(x)leads to a stochastically wandering bump that eventual ly

crosses a separatrix at t≈70, leading to extinction. The noise-free system possesses a stable

bump solution since µ=−0.2<0;ε= 0.4. (B) The large deviation can easily be detected by

tracking maxxu(x, t), which departs the bottleneck of the noise-free system, whose lower bound lies

at maxx[Uc(x)−εψe(0)] = √2(1 −ε).

3.2. Metastability and bump extinction. To analyze the one-dimensional

nonlinear SDE (3.5b), we further rescale the equation by setting A:= |w0(2ac)|

w(0)2Ae:

dA(t) = −m+A(t)2dt+ d ˆ

W(t),(3.6)

where m:= |w0(2ac)|

w(0)2µ. Thus, the eﬀective diﬀusion coeﬃcient of the rescaled noise

term is hˆ

W(τ)2i=Dτ =w0(2ac)2(C(0) + C(2ac))t/ 2w(0)4. Note the rescaled

equation (3.6) has an eﬀective potential [32,39]:

V(A) = A3

3+mA, (3.7)

the derivative V0(A) of which yields the deterministic part of the right hand side. As

the bifurcation parameter mis varied, the potential exhibits a minimum (at A=√m)

and a maximum (at A=−√m) when m < 0, a saddle point (at A= 0) when m= 0,

and no extrema for m > 0 (Fig. 3.2A). For all parameter values m, the state of the

stochastic system (3.6) will eventually escape to the limit A→ −∞ as τ→ ∞. Such

trajectories were observed in the noise-free system in the case m > 0, as demonstrated

in Fig. 2.2 of the previous section. However, we show here that noise qualitatively

alters the dynamics of the system, so its state will not remain in the vicinity of the

stable attractor (at A=√m) when m < 0.

As before, we study the problem of bump extinction using the stochastic am-

plitude equation (3.6) in the case m > 0. We show that the noise decreases the

average amount of time until an extinction event will occur. For clarity, we as-

sume the initial condition A(0) = 0 (correspondingly u(x, 0) = Uc(x)). We take

the bottleneck to be the region Ae∈[−1,1], which in the rescaled variable is A∈

[−|w0(2ac)|/w(0)2,|w0(2ac)|/w(0)2]. The residence time τbin the bottleneck is given

by the amount of time it takes for Ato escape this region. We can determine the

statistics of τbby considering it as a ﬁrst passage time problem.

Let p(A, t) be the probability density for the stochastic process A(t) given the

16

A

-2 -1 0 1 2

V(A)

-2

-1

0

1

2

node

saddle

m=−1

m= 0

m= 1

A

m

-0.4 -0.2 0 0.2 0.4

¯

tb

0

5

10

15

20

25

30

35

B

Fig. 3.2.(A) Potential function (3.7) associated with the stochastic amplitude equation (3.6)

has zero (m > 0); one (m≡0); or two (m < 0) extrema - associated with equilibria of ˙

A=−m−A2.

When m < 0, crossing the saddle point requires stochastic forcing. (B) Mean time ¯

tbuntil bump

extinction is approximated by a mean ﬁrst passage time problem of the stochastic amplitude equation

(3.6). Numerical simulations (circles) of the full system (1.5) are well approximated by this theory

(line) given by (3.12) for ε= 0.6.

initial condition A(0) = A0. Then the corresponding Fokker-Planck equation is given

∂p

∂t =∂(m+A2)p(A, t)

∂A +D

2

∂2p(A, t)

∂A2≡ −∂J (A, t)

∂A ,(3.8)

where

J(A, t) = −D

2

∂p(A, t)

∂A −(m+A2)p(A, t),(3.9)

and p(A, 0) = δ(A−A0). We focus on the three diﬀerent scenarios discussed above.

First, if m < 0, there there is a single stable ﬁxed point of the deterministic equation

˙

A=−m−A2at A=√mand a single unstable ﬁxed point at A=−√m. The basin of

attraction of A=√mis given by the interval (−√m, ∞). When D > 0, ﬂuctuations

can induce rare transitions on exponentially long timescales whereby A(t) crosses the

point A=−√m, leaving the basin of attraction. For the non-generic case m= 0, the

timescale of departure scales algebraically [38]. When m > 0, noise simply modulates

the ﬂows of the deterministic equation ˙

A=−m−A2, leading to an average speed-up

in the departure from the bottleneck. In general, we consider solving the ﬁrst passage

time problem as an escape from the domain (−α, ∞) where α:= |w0(2ac)|

w(0)2(equivalently

where Ae=−1) [19]. To do so, we impose an absorbing boundary condition at −α:

p(−α, t) = 0. Now let T(A) denote the stochastic ﬁrst passage time for which (3.6)

ﬁrst reaches the point −α, given it started at A∈(−α, ∞). The ﬁrst passage time

distribution is related to the survival probability that the system has not yet reached

−α:

S(t)≡Z∞

−α

p(A, t)dA,

which is S(t) := Pr(t>T(A)), so the ﬁrst passage time density is [19]

F(t) = −dS

dt=−Z∞

−α

∂p

∂t (A, t)dA.

17

Substituting for the expression for ∂p/∂t using the Fokker-Planck equation (3.8) and

the formula for the ﬂux (3.9) shows

F(t) = Z∞

−α

∂J (A, t)

∂A dA=−J(−α, t),

where we have utilized the fact that limA→∞ J(A, t) = 0. Thus, the ﬁrst passage time

density F(t) can be interpreted as the total probability ﬂux through the absorbing

boundary at A=−α. To calculate the mean ﬁrst passage time T(A) := hT(A)i, we

use standard analysis to associate T(A) with the solution of the backward equation

[19]:

−(m+A2)dT

dA+D

2

d2T

dA2=−1,(3.10)

with the boundary conditions T(−α) = 0 and T0(∞) = 0. Solving (3.10) yields the

closed form solution

T(A) = 2

DZA

−αZ∞

y

φ(z)

φ(y)dzdy, (3.11)

where

φ(A) = exp 2 [V(−α)−V(A)]

D,

and V(x) is the potential function (3.7). Explicit expressions for the integral (3.11)

can be found in some special cases [32,38]. For our purposes, we simply integrate

(3.11) numerically to generate theoretical relationships between the mean ﬁrst passage

time and model parameters. For comparison, we focus on the case the weight function

w(x) = cos(x) and the correlations C(x) = cos(x), so that Uc(x) = √2 cos(x), ac=π

4,

w(0) = 1, w0(2ac) = −1, C(0) = 1, and C(2ac) = 0. Therefore, α= 1, m=µ,

D= 1/2 This allows us to write the formula (3.11) at A= 0 as

T(0) = 4 Z0

−1Z∞

y

exp 4z3−y3

3+µ(z−y)dzdy. (3.12)

Lastly, note that by rescaling time t=ετ, we have that the mean ﬁrst passage time

in units of twill be ¯

tb=T(0)/ε. We compare our theory (3.12) with the results of

numerical simulations of the full stochastic neural ﬁeld (1.5) in Fig. 3.2B.

4. Discussion. We have developed a weakly nonlinear analysis for saddle-node

bifurcations of bumps in deterministic and stochastic neural ﬁeld equations. While

most of our analysis has focused upon Heaviside ﬁring rate functions, we have also

demonstrated the techniques can easily be extended to arbitrary smooth nonlinear-

ities. Our main ﬁnding is that even symmetric eigenmodes, associated with linear

stability of bumps, can be described by quadratic amplitude equations in the vicinity

of the saddle-node. For deterministic neural ﬁelds, this low dimensional approxima-

tion can be used to approximate the trajectory and lifetime of bumps as they slowly

extinguish. To do so, we focused on the initial time epoch in the bottleneck surround-

ing the ghost of the critical bump Uc(x). In stochastic neural ﬁelds with appropriate

noise scaling, a stochastic amplitude equation for the even mode of the bump can be

18

derived. We then cast the lifetime of the bump in terms of a mean ﬁrst passage time

problem of the reduced system.

Our work extends a variety of recent studies that have derived low-dimensional

nonlinear approximations of neural ﬁeld pattern dynamics in the vicinity of bifur-

cations [5,7,17,25,27,28]. As in our work, most of these previous studies derived

approximations where the location of the bifurcation was unaﬀected by noise terms.

On the other hand, Hutt et al. showed that noise can in fact shift the position of

Turing bifurcations in neural ﬁelds, and the amplitude of the bifurcation threshold

shift was proportional to the noise variance [25]. Note, it was necessary in our work

to apply a speciﬁc noise scaling (ε5/2), as compared to the distance from criticality

(ε2), in order for the noise to simply appear as a modiﬁcation of the even mode am-

plitude equation. Were we to have selected noise of larger amplitude, this could have

induced bifurcation shifts analogous to that found in [25]. Another potential future

direction would be to consider the impact of axonal propagation delays [24] on the

dynamics close to the saddle-node. As demonstrated in this work, the neural ﬁeld

(1.1) is quite sensitive to small perturbations near criticality, so delays may serve to

alter the duration of the bottleneck or even shift the saddle-node bifurcation point.

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