This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright
Author's personal copy
Physica D 238 (2009) 1101–1112
Contents lists available at ScienceDirect
Physica D journal homepage: www.elsevier.com/locate/physd
Wave fronts in inhomogeneous neural field models H. Schmidt a,∗ , A. Hutt b , L. Schimansky-Geier a a
Institut für Physik, Humboldt Universität zu Berlin, Newtonstr. 15, 12489 Berlin, Germany
b
LORIA, 615 Rue du Jardin Botanique, 54506 Vandoeuvre-lés-Nancy Cedex, France
article
abstract
info
Article history: Received 28 February 2008 Received in revised form 16 December 2008 Accepted 26 February 2009 Available online 26 March 2009 Communicated by A. Doelman
In this work we investigate the influence of inhomogeneities on wave front propagation in an excitatory neural field model describing synaptic activity in the absence of delays. This allows the derivation of the spatial (and hence temporal) behaviour of the front velocity under the assumption that the front is approximatively homogeneous in a very small time window. With this assumption we can also derive the spatiotemporal behaviour of the membrane potential analytically in the vicinity of the wave front. In addition to this we investigate stationary solutions such as standing wave fronts and localised activity (socalled bumps) to determine the existence condition of travelling and standing fronts. Numerical results are included to point out the accordance of theory and simulation. © 2009 Elsevier B.V. All rights reserved.
Keywords: Travelling wave fronts Stationary solutions Inhomogeneous neural fields
1. Introduction The study of complex systems has attracted much attention in recent decades. Prominent examples for such systems are fluids [1,2], heterogeneous solids [3], biological [4], and neural systems [5,6]. As regards the latter, system interactions traverse temporal and spatial scales of several orders. For instance, single neuron spikes evolve over some milliseconds while cognitive information is processed by populations of thousands of neurons on a time scale of 50 ms and longer. These multi-scale interactions in space and time represent a great challenge for theoretical models. To study information processing in the brain requires the choice of a high level description that takes into account important temporal and spatial features from a lower description level, i.e. from a more microscopic level. Such a high level description may involve neural populations, which have been shown to reflect functional entities, e.g. in working memory [7], in vision [8–10,51,52] and in motor coordination [11,12]. The present work investigates activity propagation in spatially extended, one-dimensional neural populations, socalled neural fields, focusing on inhomogeneities associated with spatially periodic modulation. Many neural field models considered to date are homogeneous and isotropic [13–18]. This means that the synaptic connectivity or the neuron density is invariant under the transformation x → x + a of the space coordinate x. These models are in good accordance with slice preparations taken from certain cortical areas [19–21], i.e. they reproduce experimental results such as the propagation velocity of externally induced waves. However, neural structures have been found experimentally which exhibit strong inhomogeneities. We mention the mammalian visual cortex showing ocular dominance and orientation selectivity [22,9,10,23], the macaque monkey visual cortex [24] and the monkey prefrontal cortex [25]. Inhomogeneities have been investigated theoretically in recent studies [26–30]. Recent work by Bressloff [26] has shown that spatially periodic inhomogeneities affect the average propagation velocity of wave fronts up to wave propagation failure for large enough inhomogeneity parameters. The same effect is also known for reaction-diffusion equations [31–33]. The general expression for the evolution of the synaptic activity in an inhomogeneous neural field reads
τ
∂ V (x, t ) = −V (x, t ) + ∂t
Z
∞
dyKˆ (x, y)S [V (x − y, t )], −∞
where τ is the synaptic time constant. In accordance with [26] we choose the special case Kˆ (x, y) = K (y)A(x − y):
∂ τ V (x, t ) = −V (x, t ) + ∂t ∗
Z
∞
dyK (y)A(x − y)S [V (x − y, t )]. −∞
Corresponding author. Tel.: +49 30 29047487. E-mail address:
[email protected] (H. Schmidt).
0167-2789/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2009.02.017
(1)
Author's personal copy
1102
H. Schmidt et al. / Physica D 238 (2009) 1101–1112
Here A(x − y) may be interpreted as the neuron density or synapse density in the neuronal population. The present work assumes excitatory homogeneous synaptic interactions for the kernel K (x) = (1/2σ ) exp(−|x|/σ ). In the case of spatially periodic modulation then we take A(x) to be a periodic function, while A(x) = const reflects spatially homogeneous media. In addition it is important to note that Eq. (1) neglects constant feedback delays and space-dependent delays, i.e. the system’s transmission speed is infinite. Further S [V (x, t )] represents the mean firing rate function of neurons depending on V . The present work chooses S [V ] to the Heaviside step function, i.e. S [V ] = Θ [V − θ], where Θ is the Heaviside step function and θ is a constant threshold. The right-hand side of Eq. (1) gives the synaptic input depending on x and t. A closer look at the model equation (1) reveals a scale-invariance in time and a reduction of scales in space from two to one. Mathematically new scales t /τ → τ and x/σ → x yield
ˆ (x, t ) = LV
∞
Z
dy
e−|y|
−∞
2
A(x − y)S [V (x − y, t )],
(2)
which is independent of the synaptic time scale τ and the spatial range of synaptic interactions σ . In Eq. (2), Lˆ is the differential operator Lˆ = 1 + ∂∂t and K (y) = represents the synaptic connectivity, and we have shifted A(x) → A(x/σ ). It is well known that for bistable media wave fronts can exist that interconnect two stable states of the system. In the case of neural media these states are the active, i.e. V 6= 0, and the quiescent state V = 0. We define the wave front position by V (ζ (t ), t ) = θ and choose the boundary conditions V (x < ζ (t ), t ) > θ and V (x > ζ (t ), t ) < θ . In more physical terms the cortical tissue is active for V (x, t ) < ζ (t ) and quiescent for V (x, t ) > ζ (t ). First we consider wave fronts in homogeneous media for a better understanding of wave propagation and its principles (Section 2). The subsequent section discusses wave fronts in inhomogeneous media and shows a novel analytical approach to obtain the front speed and the spatiotemporal evolution of the wave front for general functions A(x). This approach is based on the assumption that the characteristics of this wave front are approximately the same as those of a homogeneous wave front when considering a very small time window. In Section 4 explicit results are shown for periodic inhomogeneities recovering results in [26]. Further, under our assumptions we derive the time and space-dependence of the speed and the front position extending previous studies. The next section investigates various types of stationary solution, such as global solutions, standing wave fronts, and especially bumps (Section 5). Conditions for wave propagation failure are obtained. Moreover, a linear stability analysis is performed for standing wave fronts and standing bumps. The final section gives conclusions and closes the work. 2. Wave fronts in homogeneous neural fields Wave fronts in homogeneous media are well studied [34–40]. Due to the homogeneity we may expect a constant propagation velocity c and corresponding evolution equation reads
ˆ (x, t ) = A LV
∞
Z
dyK (y)S [V (x − y, t )],
(3)
−∞
where A is the global neuron density. To obtain solutions for wave fronts we replace the differential operator Lˆ in Eq. (3) by its Greens function η(s) = e−s and apply the co-moving frame transformation ξ = x − ct to gain V (ξ , t ) = A
∞
Z
dsη(s)
∞
Z
dyK (y)S (V (ξ − y + cs, t − s) − θ ).
(4)
−∞
0
Due to the homogeneous character of Eq. (4) we may assume Vt = 0 and consider henceforth stationary solutions q(ξ ) of Eq. (4): q(ξ ) = A
∞
Z
dsη(s)
Z
∞
dyK (y)S (q(ξ − y + cs) − θ ).
(5)
−∞
0
For simplicity we consider the case c > 0. Using the boundary condition q(0) = θ and considering the front shape introduced in Section 1, Eq. (5) re-casts to q(ξ ) = A
∞
Z
dsη(s) 0
Z
∞ ξ +cs
dyK (y).
(6)
Since K (y) ≥ 0 ∀ y ∈ R it is obvious that q(χ) > q(ξ ) for all χ < ξ and hence the threshold θ is crossed just once. In other words the front conditions as presented in Section 1 are fulfilled. Moreover solving Eq. (6) leads to [35] q(ξ ≥ 0) =
1
1
21+c
Ie−ξ ,
q(ξ < 0) = (1 − eξ /c )I +
(7) 1 2
1 1−c
(eξ /c − eξ ) +
1 1+c
eξ /c
I,
where I = A/2 is the synaptic input which reaches neurons at ξ = 0 because A/2 = A the well-known result 1 A −θ , c= θ 2
(8)
R∞ 0
dyK (y). Then the condition q(0) = θ yields
see [40] for details. Since A/2 may be interpreted as the synaptic input at the position ξ = 0 and c is linear in A we may say that the synaptic input is driving the wave front forward. For the inhomogeneous case the synaptic input depends on x, hence the propagation velocity is no longer constant. We engage this problem in the next section.
Author's personal copy
H. Schmidt et al. / Physica D 238 (2009) 1101–1112
1103
3. Wave fronts in inhomogeneous media In contrast to the previous section, now the synaptic input is not constant but space-dependent. To gain the solution V (ξ , t ) or even V (x, t ) similar to Eq. (6), first we seek the solution for the spatial dependence of the propagation velocity c (x). To this end we define the position of the front ζ (t ) by V (ζ (t ), t ) = θ and without loss of generality the initial condition is chosen to ζ (0) = 0, i.e. V (0, 0) = θ . R wave t Then ζ (t ) = 0 c˜ (t 0 ) dt 0 , where c˜ (t ) is the time dependent expression of the velocity. c (x) and c˜ (t ) are linked via the initial condition, i.e. c (ζ (t )) = c˜ (t ). To cope with the problem of the non-constant front speed, previous studies applied perturbation theory [26,27,30]. In the present work, we choose a different approach. Here the key idea is the calculation of the front speed in a small neighbourhood of a specific spatial location x = x0 = ζ (t0 ), i.e. in the small time interval {t = t0 , t = t0 + ∆t } and a small spatial patch where V (x0 , t0 ) = θ . We suppose that the introduced conditions for a wave front hold. For small enough ∆t it is reasonable to assume c (x) = c (x0 ) =const, i.e. we assume homogeneity in this time interval, and hence apply the homogeneous co-moving frame transformation ξ = x − x0 − c (x0 )tˆ given that tˆ = t − t0 . Furthermore, the assumption of homogeneity allows us to set Vt = 0, since the shape of the wave front does not change significantly within the considered time interval. The limit ∆t → 0, x → x0 or equivalently ξ + c (x0 )tˆ → 0 will reproduce the solution of c (x0 ). Inserting the co-moving frame transformation into Eq. (2) and taking account of Vt = 0 leads to the stationary equation
1 − c (x0 )
∂ ∂ξ
q(ξ + x0 + c (x0 )tˆ) =
Z
∞
d yK (y)A(ξ + x0 + c (x0 )tˆ − y)S [q(ξ + x0 + c (x0 )tˆ − y)].
(9)
−∞
Although we have no explicit dependence on time as long as we restrict ourselves on the interval t ∈ {t0 , t0 + ∆t } it is important to note that this time interval can be chosen at each time point t0 and each spatial location x0 and thus our analytical treatment is valid for all time and space. Since we aim to investigate the front at x = x0 , t = t0 and approximate the fronts evolution in the small time interval t − t0 = ∆t, it is valid to assume ξ + c ∆t ≈ 0. Then the Taylor-expansion of q with respect to ξ up to the linear term yields q(ξ + x0 ) = q(x0 ) + ξ
! ∂ q(ξ + x0 ) . ∂ξ ξ =0
Here we have absorbed c ∆t into ξ . Hence, the left-hand side of Eq. (9) gives
∂ 1−c ∂ξ
q(ξ + x0 ) =
∂ 1−c ∂ξ
q(ξ + x0 )
∂ + ξ q(ξ + x0 ) ∂ξ ξ =0
ξ =0
! ∂ . q(ξ + x0 ) ∂ξ ξ =0
∂ − ξc ∂ξ
Since we are free to choose ξ the terms independent of ξ and linear in ξ are separable yielding two separate equations. We show below that one equation is conditional on the other. Similarly the Taylor expansion of the righthand side of Eq. (9) yields ∞
Z
d yK (y)A(ξ + x0 − y)S [q(ξ + x0 − y)] = −∞
∞
Z ξ
+ξ
d yK (y)A(ξ + x0 − y)
Z
+ξ ξ =0
Z
∞
∞
d yK (y)A(x0 − y)S [q(x0 − y)] 0
−∞
d yK (y)A0 (x0 − y)S [q(x0 − y)]
−∞
! ∂ + O(ξ 2 ). q(ξ + x0 − y) ∂ξ ξ =0
Again we encounter terms up to first order in ξ . The lower limit of y is due to the cut-off by the Heaviside function. Separating the orders in ξ , the zero-order terms give
Z ∞ ∂ q(ξ + x0 )|ξ =0 − c q(ξ + x0 ) = d yK (y)A(ξ + x0 − y) ∂ξ ξ ξ =0 ξ =0
(10)
and the first-order terms read
∂ ∂ q(ξ + x0 ) −c ∂ξ ∂ξ ξ =0
! Z ∞ ∂ ∂ q(ξ + x0 ) = d yK (y)A(ξ + x0 − y)S [q(ξ + x0 − y)] . ∂ξ ∂ξ −∞ ξ =0 ξ =0
(11)
We observe that Eq. (11) may be obtained by differentiating Eq. (10) and hence the first order terms in Eq. (11) vanish if Eq. (10) holds. Consequently it is sufficient that Eq. (10) holds which yields the exact result of the wavespeed. Hence, we have to solve
∂ q(ξ + x0 ) − c q(ξ + x0 ) = ∂ξ
∞
Z
d yK (y)A(ξ + x0 − y),
ξ
(12)
and set ξ = 0 to obtain c (x0 ) at spatial location x0 . This is achieved by considering ξ > 0 and re-writing the right-hand side of Eq. (12) as ∞
Z ξ
d yK (y)A(ξ + x0 − y) =
∞
Z
d yK (y + ξ )A(x0 − y) 0
= e−ξ
∞
Z
d yK (y)A(x0 − y) = e−ξ I (x0 ). 0
Author's personal copy
1104
H. Schmidt et al. / Physica D 238 (2009) 1101–1112
The last step is valid due to K (y) =
1 −|y| e . 2
The homogeneous solution of Eq. (12) gives
qh (ξ + x0 ) = C¯ eξ /c and method of variation of the constant yields the solution of Eq. (12) q(ξ + x0 ) = e−ξ
1 1+c
I (x0 ).
(13)
Finally ξ = 0 yields 1
q(x0 ) = θ =
I (x0 )
1+c
and 1
c (x) =
θ
(I (x) − θ ) ,
(14)
R∞
for general spatial locations x. Here I (x) = 0 d yK (y)A(x − y) is the synaptic input at the position of the wave front when it reaches x. The specific choice A(x − y) = A reproduces the solution for wave fronts in homogeneous media (cf. Section 2). Moreover we observe that c (x) may vanish for locations x = x˜ . Then there exist stationary solutions, i.e. standing wave fronts, with the boundary condition q(˜x) = θ . This case is studied in Section 5. Further, it is not obvious that the conditions for a wave front as presented in Section 1 hold in inhomogeneous media. Therefore, it is necessary to investigate whether the implied shape conditions for the travelling front V (xhζ (t ))iθ and V (x > ζ (t )) < θ are consistent with the results obtained. A detailed mathematical analysis of this problem exceeds the object of this work, hence we leave it to future work and give a less strict explanation for a special case. Considering K (x) > 0, A(x) > 0, and c (x) > 0 for all x ∈ R, the time derivative of the right-hand side of Eq. (2) is strictly positive. Since the temporal operator Lˆ is of first order it does not allow oscillatory behaviour in time then, thus Vt (x0 , t ) > 0 for any x0 ∈ R. With c (x) > 0 for all x ∈ R we have that the wave front passes by a certain location x0 at time t0 , thus V (x0 , t0 ) = θ . With Vt (x0 , t ) > 0 for any t ∈ R we find that the condition for a wave front holds. Having the initial condition V (0, 0) = θ the time that the wave front needs to reach the position ζ (t ) is given by ζ
Z t = 0
dx0 c (x0 )
.
(15)
This equation defines the time-space relation t = t (ζ ) and the inverse function ζ (t ) gives the position of the wave front depending on time. In turn the derivative of ζ (t ) with respect to t yields the time dependent velocity c˜ (t ). In other words Eq. (15) defines the position of the front with respect to time. Now let us go one step further and attempt to describe the spatio-temporal evolution of the wave front. To this end we attach the inhomogeneous system at a certain location denoted by x or ξ and argue that for the specific space-dependent front speed c (x) in the small patch about x the system may be viewed as homogeneous with the synaptic input (current) which refers to x or ξ . This assumption is justified by the fact that the synaptic response is not affected by the inhomogeneity. Consequently, wave fronts obey the solutions ξ (7), (8) well-known for homogeneous neural fields in this small patch. In the homogeneous case we can identify t 0 = c , where t 0 is the time needed by the wave front to reach the considered position x = ξ . Hence, considering Eqs. (7) and (8) we obtain V˜ (t 0 ≥ 0) =
1
1
0
21+c
Ie−ct ,
0
V˜ (t 0 < 0) = (1 − et )I +
1
(16)
2
1 1−c
0
0
(et − ect ) +
1 1+c
et
0
I,
where V˜ denotes the temporal characteristics of the membrane potential at a certain position. The important step towards spatio-temporal solutions for the inhomogeneous case is the use of the relation t = t (ζ ) according to Eq. (15). For reasons of clarity we rewrite t in this relation by tζ , i.e. tζ = t (ζ ). We may apply this relation to t 0 = t 0 (ξ ) by taking
account of the condition t 0 (ξ = 0) = 0, for q(ξ = 0) = θ and consequently V˜ (t 0 = 0) = θ . Hence we obtain the relation t 0 (ξ ) = tζ (ξ +ζ (t ))− tζ (ζ (t )). This relation assigns t 0 to any spatial location and any point in time. It is nonlinear due to the inhomogeneity. Since c and I depend on x as well we write c = c (ξ + ζ (t )) and I = I (ξ + ζ (t )) to obtain a consistent solution and gain subsequently V = V (ξ , ζ (t )) = V (ξ , t ). In the next section we give an example for periodic inhomogeneities and show that the analytical results obtained agree well with numerical calculations. 4. Wave fronts for periodic inhomogeneities To illustrate the previous analytical results presented in the last section, let us focus to the periodic inhomogeneity A(x) = 1 + a cos
x
+ φ0
a, > 0,
(17)
that has been studied previously in [26]. With the space-dependent speed defined in Eq. (14) and φ = x/ + φ0 we obtain
c (φ) =
1 + a√
2 +1
2θ
cos φ
− 2θ ,
(18)
Author's personal copy
H. Schmidt et al. / Physica D 238 (2009) 1101–1112
1105
Fig. 1. The velocity of the wave front depending on φ (left) and on t (right), for = a = 1 and θ =
1 . 12
Fig. 2. The dependence of c¯ on for four different values of a (from top to bottom: a = 0.5, a = 0.833, a = 1, a = 1.5), and for θ = 1/12. The critical value for wave propagation failure (for → ∞) is a∞ = 0.833 (second curve from top).
Fig. 3. The critical amplitude a for wave propagation failure, depending on for five different values of θ (from top to bottom: θ = 0, θ = 1/12, θ = 1/6, θ = 1/3, θ = 11/24).
cf. Fig. 1, left panel. At first, this expression allows us to derive the average velocity c¯ = 2π /T , where T =
c¯ =
1 2θ
s (1 − 2θ )2 − a2
2 +1
R 2π 0
d φ 0 c −1 (φ 0 ). We obtain
(19)
2
which confirms the result obtained previously by a perturbative ansatz including the higher order corrections [26]. The author used as the perturbation parameter and showed that the higher order corrections affect the average velocity. The result including only first order terms significantly deviates from Eq. (19). Further c¯ = 0 in Eq. (19) yields the critical parameters for wave propagation failure. Fig. 2 shows the average speed c¯ with respect to p for different values of a. We observe the critical values ac = (1 − 2θ ) 1 + 1/ 2 below which wave propagation failure may occur, see Fig. 3). Moreover we can derive the time dependence of the position of the wave front. By virtue of the systems periodicity, it is appropriate to replace the spatial coordinate by the phase φ leading to t (φ) or φ(t ). The time t (φ) is found to t (φ) =
4θ D
" arctan
B + (1 − 2θ ) tan[ D
φ+arctan[] 2
]
#
" − arctan
[] ] B + tan[ arctan 2
D
#! ,
(20)
Author's personal copy
1106
H. Schmidt et al. / Physica D 238 (2009) 1101–1112
Fig. 4. The left plot shows the result of the simulation for = a = 1 and θ = (where V (ζ (t )) = θ ). The space and time steps have been set to ∆x = ∆t = right plot. We see that the calculation is in accordance with the simulation.
1 . Only values V 12 1 in the left plot. 20
≥ θ have been plotted. The right plot shows the theoretical result of ζ (t ) Thus, a distance of 100 in the left plot corresponds to distance of 5 in the
with the abbreviations B = a√
2 + 1
,
D=
p
(1 − 2θ )2 − B2 .
The inverse function of Eq. (20) then yields φ(t )
φ(t ) = 2 arctan
D tan
Dt 4θ
0 + arctan BD − B − arctan , 1 − 2θ
(21)
where B0 = B + tan 21 arctan[] . In addition multiplying both sides of (21) by we obtain the position of the wave front
ζ (t ) = 2 arctan
D tan
Dt 4θ
0 + arctan BD − B − arctan 1 − 2θ
+ 2π × integerpart[t /T − t˜ ] − 2π × Θ [−t /T + t˜ ].
(22)
The last two summands have been added to guarantee a continuous and monotonically increasing function for ζ (t ) and t˜ obeys the relation Dt˜
+ arctan
B0
π
=
.
4θ D 2 The variable ζ (t ) given by Eq. (22) represents the location of the fronts threshold crossing. Fig. 4 compares this analytical crossing point with the numerical results gained from simulations of Eq. (2) and we observe good accordance. The time derivative of Eq. (22) yields the time-dependent front speed c˜ (t ) =
D2 (1 − 2θ )
1 2θ (1 − 2θ)
2
cos2
E (t ) + (D sin E (t ) − B cos E (t ))2
,
with the abbreviation Dt B0 E (t ) = + arctan . 4θ D (Fig. 1) shows c˜ (t ) and we observe that it has not the simple sinusoidal shape as c (φ), since the wave front obviously passes areas of high RT velocity faster than areas of low velocity. In addition, we mention that the mean front speed can be found by c¯ = 0 dt c˜ (t ). Taking into account the previous results, it is possible to compute the spatio-temporal dynamics of the membrane potential. At first we have to modify Eq. (20) to obtain t 0 (ξ ) with t 0 (0) = 0. To this end φ is replaced by ξ / and the phase ζ (t )/ is added in Eq. (20) such that this boundary condition holds. More illustratively, we are moving along the curve t (φ) given by Eq. (20) which is possible due to the introduction of the phase shift ζ (t )/ . Then the distance ξ gives the time that the wave front needs to pass this distance. We gain t 0 (ξ ) =
4θ D
−
" arctan
4θ D
B + (1 − 2θ ) tan[
2
]
#
D
" arctan
ξ /+ζ /+arctan[]
B + (1 − 2θ ) tan[ D
ζ /+arctan[] 2
]
# + T × integerpart[ξ /2π ] − T × Θ [−ξ ].
(23)
Author's personal copy
H. Schmidt et al. / Physica D 238 (2009) 1101–1112
1107
Fig. 5. The theoretical result for the evolution of the wave front for = a = 1 and θ =
1 . 12
The left plot shows V (ξ , φ), and the right plot shows V (ξ , t ).
Fig. 6. The simulation of V (x, t ) (left) and the analytical result for V (ξ , t ) for = a = 1 and θ =
1 . 12
Note that in the right panel the wave front is pinned at ξ = 0 since we
have the boundary condition V (ξ = 0) = θ . The size of the space and time steps of the simulation are ∆x = ∆t =
Fig. 7. The front V (ξ , t ) for a = 1, θ =
1 , 12
1 . 20
and = 0.3, = 1.3.
The last two summands guarantee that t 0 is monotonically increasing. We may now insert Eq. (23) into Eq. (16). To obtain a consistent solution, we replace φ by ξ / + ζ / in c (φ) and I (φ) yielding V (ξ , ζ ) and V (ξ , φ). Further inserting Eq. (22) into V (ξ , ζ ), essentially one obtains V (ξ , t ). Figs. 5–7 illustrate the fronts subject to various parameters. 5. Stationary solutions for periodic inhomogeneities In the previous section, we detected wave propagation failure yielding stationary solutions. The present section investigates such solutions in some detail. The stationary general solution of Eq. (2) obeys V (x) =
Z
∞
dyK (y)A(x − y)S [V (x − y)], −∞
while specific solutions results from the appropriate choice of boundary conditions.
Author's personal copy
1108
H. Schmidt et al. / Physica D 238 (2009) 1101–1112
Fig. 8. Possible parameters θ , (left, with a = 1) and θ , a (right, with = 1) for standing wave fronts (grey area). The thick line denotes the upper boundary for bistability. Below the grey area wave fronts exist with c > 0, above the grey area and below the thick line (for small , a) wave fronts exist with c < 0. Parameters between the thin lines support bump-like solutions, yet above the thick line are only single-peak solutions possible.
5.1. Global stationary solutions There are two globally stationary solutions which obey S [V (x)] = 0 ∀x ∈ R or S [V (x)] = 1 ∀x ∈ R. The first one has the trivial solution V (x) = 0, and the solution for the latter reads V (x) =
∞
Z
dyK (y)A(x − y). −∞
For periodic inhomogeneity (17) we find that V (x) = 1 + a
x 2 cos + φ0 . +1 2
In addition, the condition V (x) > θ determines the bistability condition and thus the existence for the wave front. Since the lower boundary of the cosine is −1, the condition for the upper boundary of the firing threshold θ reads
θ =1−a
2 . +1 2
The lower boundary is obviously given by θ = 0. Hence the bistability condition for inhomogeneous media reads 0 θ and V (x > 0) < θ and hence S [V (x < 0)] = 1, S [V (x > 0)] = 0. This will lead to standing wave fronts given by V (x) =
∞
Z
dyK (y)A(x − y).
(25)
x
Considering again the periodic inhomogeneity (17) and the boundary condition V (0) = θ we find
θ=
1
1+a
2
2 2 +1
cos φ0 +
1
sin φ0
.
(26)
This equation determines the parameters for the stationary solutions. Eq. (26) coincides with our previous result (18) for c (φ) = 0. Moreover 1 − a√
2 + 1
≤ 2θ ≤ 1 + a √ , 2 + 1
gives stationary solutions and the bistability condition (24) essentially leads to the condition for standing wave fronts
2 ≤ 2θ ≤ Min 1 + a √ , 2 − 2a 2 . +1 2 + 1 2 + 1
1 − a√
The parameter regime of this condition is shown in Fig. 8. Moreover these stationary solutions occur via a saddle-node bifurcation if c (φ) < 0. One may choose a, , or θ as a bifurcation parameter. Fig. 9 shows such bifurcations for the bifurcation parameter a, and the phase dependent velocity which determines the stability of the stationary states.
Author's personal copy
H. Schmidt et al. / Physica D 238 (2009) 1101–1112
1109
Fig. 9. Left: The saddle-node bifurcation of the stationary states for several firing thresholds (from top to bottom: θ = 0.15, θ = 0.25, θ = 0.35, and θ = 0.45) and =1. The bifurcation parameter is a. Steady states on the left (right) of the vertical line are stable (unstable). Right: c (φ) for = 1, θ = 0.15, and several amplitudes (from top to bottom at φ = 4: a = 0.5, a = 1, and a = 1.5). Intersections with the c = 0-line mark steady states.
To gain the stability conditions of the front, we perform a linear stability analysis by perturbing the stationary solution such that V˜ (x, t ) = V (x) + u(x, t ), u(x, t ) = u(x)eλt . After inserting this ansatz into Eq. (2) we obtain
(1 + λ)u(x) =
Z
∞
dyK (x − y)A(y)S 0 [V (y) − θ ]u(y), −∞
where S 0 [V (y) − θ ] = sgn(V 0 (y))δ(V (y) − θ ). For a profound introduction into stability analysis of neural fields with boundary conditions, such as travelling waves, see e.g. [41]. Further we use the transformation z = V (y) to gain
(1 + λ)u(x) =
Z
∞
dzK (x − V −1 (z ))A(V −1 (z )) −∞
δ(z − θ) u(V −1 (z )). |V 0 (V −1 (z ))|
After the evaluation of this integral the boundary condition V −1 (θ ) = 0 and the choice x = 0 leads to the self-consistent equation in u and the characteristic equation K (0)A(0)
(1 + λ) =
|V 0 (0)|
that is called Evans function [41–43] or stability index function [44]. |V 0 (0)| can be easily derived from Eq. (25). After some transformations we finally obtain
λ ∝ cos φ0 − sin φ0 , i.e. the stationary solutions are stable for arctan < φ0 < π + arctan , and unstable otherwise. 5.3. Local activity The finding that inhomogeneities cause wave propagation failure (i.e. stationary solutions) for firing thresholds which do not produce standing wave fronts in homogeneous media, see Eq. (26), suggests that it might be possible to produce stable local activity (so-called bumps) without inhibition. This is indeed the fact, as we will see in the following. The homogeneous model presented in Section 2 produces standing wave fronts just for θ = A/2. There have been several ways to produce bumps: one may use oscillatory synaptic connectivities [45,46], dynamic firing thresholds [47], inhomogeneous inputs within a local recovery model [48], or the two population model [49,50]. Our aim is to show that bumps can also occur due to an inhomogeneous distribution of the neuron density (or synaptic connectivity, respectively). To obtain symmetric solutions we choose V (−∆) = V (∆) = θ with V (−∆ < x < ∆) > θ which implies φ0 = 0 or φ0 = π , respectively, and gain the stationary solution V (x) =
∆
Z
−∆
dyK (y)A(x − y).
(27)
The width of the stationary solutions are found again by solving V (∆) = θ , see Fig. 10), left panel illustrating the relation of the threshold θ and the bump width ∆. The linear stability analysis is performed in the same way as in the previous subsection, except for the fact that we have to take into account the two boundary conditions V −1 (θ ) = −∆ and V −1 (θ ) = ∆. Then self-consistent solutions stipulate
u(−∆) u(∆)
=
A(−∆, λ) A(∆, λ)
B(−∆, λ) B(∆, λ)
u(−∆) u(∆)
with the coefficients 1 A(x) = K (x + ∆)A(∆)|V 0 (∆)|−1 , 1+λ B(x) = A(x − 2∆).
= Aˆ (λ)
u(−∆) u(∆)
Author's personal copy
1110
H. Schmidt et al. / Physica D 238 (2009) 1101–1112
Fig. 10. The left plot shows the necessary firing threshold to obtain stationary solutions of the width 2∆. The right plot shows the eigenvalues λ+ (upper curve) and λ− (lower curve) depending on ∆. Parameters: a = 1, = 1, φ0 = 0.
Fig. 11. The four most narrow stationary solutions are shown for θ = 0.3, a = 1, = 1, and φ0 = 0. Dotted lines denote stable solutions, the dots are the results of simulations.
Fig. 12. Firing threshold (left) and eigenvalues (right) depending on ∆ for φ0 = π (a = 1, = 1).
The necessary condition for nontrivial solutions in (28) is E (λ) = |Aˆ (λ) − I| = 0 leading to
λ+/− = (K (0) ± K (2∆))A(∆) − |V 0 (∆)|.
(28)
We see that λ+ > λ− for all ∆ > 0. Hence the important condition for stability is λ+ < 0. For large enough ∆ Eq. (28) recovers the eigenvalue for the standing wave front under the transformation x + ∆ → x. It can be shown that λ+ = dθ / d∆ = dV (∆)/ d∆. This is expected since the flank of a bump can be approximately considered as a standing wave front, and θ is the necessary threshold to keep c = 0. Since θ is chosen to be constant, the increase (decrease) V (∆) about ¯ implies c > 0 (c < 0) for ∆ > ∆ ¯ and c < 0 (c > 0) for ∆ < ∆ ¯ and hence ∆ ¯ is unstable (stable). the steady state ∆ Fig. 11 shows an odd number of peaks of the steady state and a global maximum at x = 0 for φ0 = 0. In contrast φ0 = π yields stationary solutions with an even number of peaks and a local minimum at x = 0 (cf. Figs. 12 and 13), simply by adding π to the arguments of the sines and cosines in Eqs. (27) and (28). We have to distinguish single-peak solutions and multi-peak solutions regarding their existence. In the case of multi-peak solutions the bistability condition has to hold due to the local minima occurring between the peaks. These minima do not arise for single-peak solutions, hence bistability is not crucial for their existence (cf. Figs. 8 and 11). 6. Conclusions In this work we make use of the model proposed in [26] that incorporates periodic inhomogeneities described by a function A(x/). This function may be interpreted as a modulation of the density of neurons or the synaptic connectivity. This work derives analytically the spatio-temporal dynamics of a travelling wave front. In contrast to previous studies, we have not applied a perturbation theory but gained the solutions by a novel analytical treatment. This novel approach considers homogeneity in a small neighbourhood of a spatial location. Although the investigation of the accuracy of this approach is left to future work, the analytical results are in good accordance with numerical findings.
Author's personal copy
H. Schmidt et al. / Physica D 238 (2009) 1101–1112
1111
Fig. 13. The most narrow stable solution for θ = 0.3, a = 1, = 1, and φ0 = π .
Moreover, we point out that the model under study neglects constant and space-dependent delays. Hence the work leaves open the problem of wave propagation within an inhomogeneous and delayed neural field. The reason for the expected problems is the necessary prior knowledge of the time dependent velocity c˜ (t ). In addition to the propagation of wave fronts we have investigated the existence of stable local activity within this inhomogeneous field model without inhibition. The analytical treatment comprises the stability analysis of the stationary solutions and hence gives stability conditions for the model parameters. As already mentioned, bumps are linked to working memory. In this context working memory modules play a central role. Such modules are groups of neurons that code for one specific feature. Indeed, there is physiological evidence that working memory modules in the frontal lobe are organised periodically in separated clusters similar to orientation columns in the visual cortex [7]. Though the assumption of sinusoidal inhomogeneities might not quite meet physiological circumstances, their mathematical analysis proves that bumps can exist in purely excitatory, inhomogeneous models. An advanced model could include inhomogeneities described by step functions to meet the cluster-like character of the working memory modules. It could also include additional inhibition, since modules coding for oppositional features interact in an inhibitory manner [7], and may consider time-dependent external inputs. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]
G. Nicolis, I. Prigogine, Self-Organization in Non-Equilibrium Systems: From Dissipative Structures to Order Through Fluctuations, J. Wiley and Sons, New York, 1977. M.C. Cross, P.C. Hohenberg, Pattern formation outside of equilibrium, Rev. Modern Phys. 65 (3) (1993) 851–1114. S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Springer, Berlin, 2002. J.D. Murray, Mathematical Biology, Springer, Berlin, 1989. P.L. Nunez, Electric Fields of the Brain: The Neurophysics of EEG, Oxford University Press, Oxford, 2006. H. Haken, Principles of Brain Functioning, Springer, Berlin, 1996. P.S. Goldman-Rakic, Cellular basis of working memory, Neuron 14 (1995) 477–485. B. Ermentrout, J.D. Cowan, A mathematical theory of visual hallucination patterns, Biol. Cybern. 34 (1979) 137–150. D.H. Hubel, T.N. Wiesel, Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex, J. Physiol. (Lond.) 160 (1962) 106–154. D.H. Hubel, T.N. Wiesel, Receptive fields and functional architecture of monkey striate cortex, J. Physiol. (Lond.) 195 (1968) 215–243. V.K. Jirsa, P. Fink, P. Foo, J.A.S. Kelso, Parametric stabilization of biological coordination: A theoretical model, J. Biol. Phys. 26 (2000) 85–112. T.D. Frank, A. Daffertshofer, C.E. Peper, P.J. Beek, H. Haken, Towards a comprehensive theory in brain activity: Coupled oscillator systems under external forces, Physica D 144 (2000) 62–86. H.R. Wilson, J.D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons, Biophys. J. 12 (1972) 1–24. H.R. Wilson, J.D. Cowan, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetik 13 (1973) 55–80. S. Amari, Homogeneous nets of neuron-like elements, Biol. Cybern. 17 (1975) 211–220. S. Amari, Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. Cybern. 27 (1977) 77–87. J.L. van Hemmen, Continuum limit of discrete neuronal structures: Is cortical tissue an ‘excitable’ medium?, Biol. Cybern. 91 (6) (2004) 347–358. A. Hutt, F.M. Atay, Analysis of nonlocal neural fields for both general and gamma-distributed connectivities, Physica D 203 (2005) 30–54. R.D. Chervin, P.A. Pierce, B.W. Connors, Periodicity and directionality in the propagation of epileptiform discharges across cortex, J. Neurophys. 60 (1988) 1695–1713. D. Golomb, Y. Amitai, Propagating neuronal discharges in neocortical slices: Computational and experimental study, J. Neurophysiol. 78 (1997) 1199–1211. J.Y. Wu, L. Guan, Y. Tsau, Propagating activation during oscillations and evoked responses in neocortical slices, J. Neurosci. 19 (1999) 5005–5015. D.H. Hubel, T.N. Wiesel, Functional architecture of macaque monkey visual cortex, Proc. Roy. Soc. B 198 (1977) 1–59. D.H. Hubel, T.N. Wiesel, Sequence regularity and geometry of orientation columns in the monkey striate cortex, J. Comp. Neurol. 158 (1974) 295–306. J.S. Lund, A. Angelucci, P.C. Bressloff, Anatomical substrates for functional columns in macaque monkey primary visual cortex, Cerebral Cortex 12 (2003) 15–24. J.B. Levitt, D.A. Lewis, T. Yishioka, J.S. Lund, Topography of pyramidal neuron intrinsic connections in macaque monkey prefrontal cortex (Areas 9 and 46), J. Comp. Neurol. 338 (1993) 360–376. P.C. Bressloff, Traveling fronts and wave propagation failure in an inhomogeneous neural network, Physica D 155 (2001) 83–100. P.C. Bressloff, Bloch waves, periodic feature maps and cortical pattern formation, Phys. Rev. Lett. 89 (2002) 088101. M.R. Qubbaj, V.K. Jirsa, Neural field dynamics with heterogeneous connection topology, Phys. Rev. Lett. 98 (2007) 238102. V.K. Jirsa, J.A.S. Kelso, Spatiotemporal pattern formation in neural systems with heterogeneous connection topologies, Phys. Rev. E 62 (6) (2000) 8462–8465. Z.P. Kilpatrick, S.E. Folias, P.C. Bressloff, Traveling pulses and wave propagation failure in inhomogeneous neural media, SIAM J. Appl. Dyn. Sys. 7 (2008) 161–185. W. Ebeling, A. Engel, L. Schimansky-Geier, Ch. Zülicke, Dynamics of fronts, nuclei and patterns in 2D random media, Physica D 49 (1991) 170–176. J.P. Keener, Homogenization and propagation in the bistable equation, Physica D 136 (2000) 1–17. G.S. Medvedev, T.J. Kaper, N. Kopell, A reaction-diffusion system with periodic front dynamics, SIAM J. Appl. Math. 60 (2000) 1601–1638. G.B. Ermentrout, J.B. McLeod, Existence and uniqueness of travelling waves for a neural network, Proc. Roy. Soc. Edinburgh 123A (1993) 461–478. P.J. Pinto, G.B. Ermentrout, Spatially structured activity in synaptically coupled neuronal networks: I. Travelling fronts and pulses, SIAM J. Appl. Math. 62 (2001) 206–225. A. Hutt, Effects of nonlocal feedback on traveling fronts in neural fields subject to transmission delay, Phys. Rev. E 70 (2004) 052902. F.M. Atay, A. Hutt, Neural fields with distributed transmission speeds and long-range feedback delays, SIAM J. Appl. Dyn. Syst. 5 (4) (2006) 670–698. A. Hutt, F.M. Atay, Effects of distributed transmission speeds on propagating activity in neural populations, Phys. Rev. E 73 (2006) 021906. M. Enculescu, A note on traveling fronts and pulses in a firing rate model of a neuronal network, Physica D 196 (2004) 362–386. S. Coombes, Waves, bumps, and patterns in neural field theories, Biol. Cybern. 93 (2005) 91–108. S. Coombes, M.R. Owen, Evans functions for integral neural field equations with Heaviside firing rate function, SIAM J. Appl. Dyn. Sys. 34 (2004) 574–600. L. Zhang, On stability of traveling wave solutions in synaptically coupled neuronal networks, Differential Integral Equations 16 (5) (2003) 513–536. B. Sandstede, Evans functions and nonlinear stability of travelling waves in neuronal network models, Internet J. Bifur. Chaos 17 (2007) 2693–2704. L. Zhang, How do synaptic coupling and spatial temporal delay influence traveling waves in nonlinear nonlocal neuronal networks?, SIAM J. Appl. Dyn. Syst. 6 (2007) 597–644. C.R. Laing, W.C. Troy, B. Gutkin, G.B. Ermentrout, Multiple bumps in a neuronal model of working memory, SIAM J. Appl. Math. 63 (2002) 62–97.
Author's personal copy
1112 [46] [47] [48] [49] [50] [51] [52]
H. Schmidt et al. / Physica D 238 (2009) 1101–1112 C.R. Laing, W.C. Troy, PDE methods for nonlocal models, SIAM J. Appl. Dyn. Syst. 2 (2003) 487–516. S. Coombes, M.R. Owen, Bumps, breathers, and waves in a neural network with spike frequency adaptation, Phys. Rev. Lett. 94 (2005) 148102. S.E. Folias, P.C. Bressloff, Breathing pulses in an excitatory neural network, SIAM J. Appl. Dyn. Syst. 3 (2004) 378–407. S. Coombes, G.J. Lord, M.R. Owen, Waves and bumps in neuronal networks with axo-dendritic synaptic interactions, Physica D 178 (2003) 219–241. C.R. Laing, S. Coombes, The importance of different timings of excitatory and inhibitory pathways in neural fields models, Network 17 (2005) 151–172. C.L. Colby, J.R. Duhamel, M.E. Goldberg, Oculocentric spatial representation in parietal cortex, Cerebral Cortex 5 (1995) 470–481. R. Ben-Yishai, L. Bar-Or, H. Sompolinsky, Theory of orientation tuning in visual cortex, Proc. Natl. Acad. Sci. USA 92 (1995) 3844–3848.
