Propagating interfaces in a two-layer bistable neural ... - Le2i - CNRS

Propagating interfaces in a two-layer bistable neural network V.B. Kazantsev, V.I. Nekorkin Institute of Applied Physics of Russian Academy of Science 46 Uljianov Str., 603950 Nizhny Novgorod, Russia

S. Morfu, J.M. Bilbault and P. Marqui´ e Laboratoire LE2I UMR Cnrs 5158, Aile des Sciences de l’ing´enieur, Universit´e de Bourgogne, BP 47870, 21078 Dijon Cedex, France

Running title: Propagating interfaces Abstract

The dynamics of propagating interfaces in a bistable neural network is investigated. We consider the network composed of two coupled 1D lattices and assume that they interact in a local spatial point (pin contact). The network unit is modeled by the FitzHugh-Nagumo-like system in a bistable oscillator mode. The interfaces describe the transition of the network units from the rest (unexcited) state to the excited state where each unit exhibits periodic sequences of excitation pulses or action potentials. We show how the localized inter-layer interaction provides an “excitatory” or “inhibitory” action to the oscillatory activity. In particular, we describe the interface propagation failure

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and the initiation of spreading activity due to the pin contact. We provide analytical results, computer simulations and physical experiments with two-layer electronic arrays of bistable cells.

1

Introduction

Propagating interfaces represent wave processes of transition of a system from a one mode of behavior to another. Such waves have been studied in many areas of nonlinear physics, biology, chemistry and applied engineering [Scott, 1998; Murray, 1989; Kuramoto, 1989; Brindley et al., 1991]. In particular, they play a significant role in behavior and functions of neurons and neuron assemblies in neurophysiology. Traveling waves of electrical activity have been observed in many cortical areas of the brain. A numbers of experimental studies have reported of waves in somatosensory cortex of rats [Wu et al., 1999], spatio-temporal patterns in turtle olfactory bulbs [Lam et al., 2000], stimuli induced waves and spiral-like patterns in visual cortex [Prechtl et al., 1997], slow waves of neural activity in rat neocortex [Peinado et al., 2000], etc. The cortical waves represent traveling interfaces separating regions of low and high activity and play an important role in the processing of sensory stimuli in the brain. Then, such waves could be important for understanding some pathological forms of behavior (spreading depression, epileptic seizures, migraines, etc.). Another point of interest is the transmission of excitation at the single cell level and between cells. Take, for instance, classical example of propagating action potentials and wave trains along the cell axons [Scott,

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1998], various wave reentries in coupled nerve fibers and in dendritic trees [Brindley et al., 1991], propagation in dendritic spines [Coombes, 2001] leading to the interfaces between different activity levels and so on. In modeling, propagating interfaces typically appear in systems with bistable properties. The bistability means the existence of two attractors corresponding to two distinct levels of activity. Furthermore, spatio-temporal dynamics of the network is defined by inter-neuron connectivity and by geometrical architecture of the system. The models involve various modifications of Wilson-Cowan equations, amplitude-phase models, reaction-diffusion systems [Bressloff, 2001; Bressloff et al.,1999; Borisyuk et al., 1999; Hoppenstadt & Mittelmann, 1997; Osan & Ermentrout, 2002; Osan & Ermentrout, 2001; Golomb & Ermentrout, 1999; Nekorkin et al., 1997; Nekorkin et al., 1998; Keener, 1987; Keener, 2000; Erneux et al., 1993; Zinner, 1991; Anderson & Sleeman, 1995; MacKay & Sepulchre, 1995]. One of the simplest example is a bistable reaction-diffusion lattice taking into account local nearestneighbor connections. It has been found that such system has traveling wave solutions in the form of propagating fronts [Keener, 1987; Keener, 2000; Erneux et al., 1993; Zinner, 1991; Anderson & Sleeman, 1995]. Recent studies have shown that such solutions exist in systems with much more complex types of connectivity and inhomogeneous architecture. When the complexity of local neuron dynamics is taken into account the networks are described by Hodgkin-Huxley-like equations. Such units, being stimulated above the certain threshold produce excitation pulses (spikes or action potential) or sequences of pulses. When 3

inter-connected with local or non-local synaptic coupling the neural networks can reproduce a variety of space-time structures including moving interfaces, striped patterns, spiral-like patterns, etc. observed in experiments [Siegel & Read, 2001; Kudela et al., 1999; Golomb & Amitai, 1997; Ermentrout et al., 1997]. In this paper we investigate propagating interfaces in a multi-layer neural network. The multi-layer architecture of the model is organized with two 1D lattices (layers) of locally coupled bistable units. Such a layer is the simplest structure modelling an interface propagation. We assume that the inter-layer interaction occurs only in a single spatial site (pin contact) (Fig. 1). We focus on how such localized inhomogeneity may affect the interface propagation. At variance with homogenizing models we take into account the “discrete” or localized character of inter-neuron connections and consider slow and small-scale processes (of “cell size” order). As noted in [Nekorkin et al., 2001] homogeneous inter-layer connections may substantially change the characteristics of the front. In particular, for suitable conditions the front can be accelerated, stopped or reversed. In this paper we show how the multi-layer architecture with localized inter-layer connection provides interface delay and propagation failure, and creates lurching-like waves of spreading activity. We shall consider two different types of bistability. (i) The first type involves a stable fixed point and a stable limit cycle in the FitzHugh-Nagumo-like system modeling the evolution of membrane potential. Then, depending on initial perturbation the unit can be either at rest or oscillates generating a periodic sequence of spikes (action potentials). (ii) The second type is provided by two stable 4

fixed points corresponding to low and high levels of neural activity. Together with analytical results and computer simulations we carry out the experimental study of two coupled 1D electrical lattices with the pin contact. The paper is organized as follows. In Sect. II we describe the two-layer network of FitzHugh-Nagumo-like and show how “excitatory” and “inhibitory” interfaces propagate in the single layer and with the pin contact. We discuss the conditions of the problem to be reduced to a simplified (averaged) description. Then, in Sect. III we study propagating interface separating levels of activity in the reduced model defined by a discrete Nagumo equation. Here we give a theoretical explanation of interface behavior at the pin contact. In Sect. IV we construct a two-layer electrical lattice with localized interaction and describe the interface propagation in the experiments.

2 2.1

Propagating interfaces in the oscillatory network Model

We consider a two-layer neural network with localized inter-layer interaction (pin contact) in the following form  (1) (1) (1) (1) (1) (1) (2) (1)  u˙ j = f (uj ) − vj + d(uj−1 − 2uj + uj+1 ) + hδjk (uk − uk ),     (1) (1) (1)   v˙ = ǫ(g(uj ) − vj − I),    j (2)

(2)

(2)

(2)

(2)

u˙ j = f (uj ) − vj + d(uj−1 − 2uj      (2) (2) (2)   v˙ = ǫ(g(uj ) − vj − I),    j

(2)

(1)

(2)

+ uj+1 ) + hδjk (uk − uk ),

(1)

j = 1, 2, . . . , N.

The superscripts

(1)

and

(2)

correspond to the variables of the two layers. The uj -variable

describes the evolution of the membrane potential of a neuron (or a colony of neurons) located at the jst spatial site, vj describes the dynamics of outward ionic currents (the 5

recovery variable) [Scott, 1998]. The function f has a cubic shape, f (u) = u − u3 /3, the function g is taken piece-wise linear with g(u) = αu, if u < 0, and g(u) = βu, if u ≥ 0. The parameters α and β control the dynamics of the recovery variable. The parameter ǫ defines the time scale of excitation pulses and parameter I is a constant current stimulus. The coefficient d accounts for the strength of the intra-layer coupling. We assume that the units are coupled electrically, hence the couplings are expressed by difference terms in Eqs. (1) . δjk is the Croneker symbol, hence the inter-layer connection occurs in a given spatial site, k, 1 < k < N , with the strength accounted by the parameter h. We impose Neumann boundary conditions for both layers, u0 = u1 , uN +1 = uN .

2.2

“Limit cycle-fixed point” bistability

Let us first consider the dynamics of the single neuron. Setting d = 0, h = 0 in Eqs. (1)) we come to the following two-dimensional system (

u˙ = f (u) − v,

v˙ = ǫ(g(u) − v − I).

(2)

At variance with classical FitzHugh-Nagumo model for neuron excitability the model (2) takes into account nonlinear properties of the recovery variable, v, depending on membrane potential, u [Kazantsev, 2001]. With such modification the unit yields a rich variety of different dynamic behaviors. For our purpose in this paper we are interested in the following regimes illustrated in Fig. 2. (A) Figure 2 (a) shows the excitable dynamics. The unit has three fixed points O1 , O2 and O3 . The point O1 is stable node or focus and corresponds to the 6

neuron rest state, the point O2 is saddle with the incoming separatrix defining the excitation threshold and O3 is unstable node or focus. Then, if a perturbation of the rest state, O1 , is sufficiently large, i.e. lies below the separatrix, the system responds with an excitation pulse. Otherwise the perturbation decays to the stable rest point O1 . (B) By decreasing the parameter ǫ one can obtain a bistable behavior shown in Fig. 2 (b). The stable limit cycle appears in the result of the separatrix loop bifurcation and the system displays limit cycle - fixed point bistability. Being perturbed above the threshold the unit becomes oscillating and exhibits a periodic sequence of spikes (action potentials). If it is at the limit cycle, then a suitable inhibitory perturbation may bring the unit back to the neighborhood of the rest state. Such an oscillatory bistability is preserved with further changing of ǫ when the unstable limit cycle appears from the smaller separatrix loop bifurcation (Fig. 2 (c)). At variance with Fig. 2 (b), the excitation threshold is now defined by this limit cycle as it separates the basins of the two attractors. (C) Finally, the unstable limit cycle disappears (Andronov-Hopf bifurcation) and the unit becomes oscillating (Fig. 2 (d)).

2.3

“Limit cycle - fixed point” interface in the single layer

Due to the bistability the single layer of the network (1) may display propagating interfaces. The interface represents a kink-like traveling wave sequentially “switching” the units from one stable state (fixed point or limit cycle) to another. To observe such interfaces we initially set a part of the units to their stable fixed points while the other part is oscillating near the

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limit cycle. Then, switching on the inter-unit coupling, 0 < d ≪ 1, we find that depending on the control parameter ǫ the interface propagates in the layer setting either the oscillatory state or the stable rest state (Fig. 3 (a), (b)). The level of grey color in the space-time diagrams corresponds to the values of the variable uj (t) with the white color referring to the oscillation peaks and the black one to the rest state. In the first case the interface plays the role of an excitatory stimulus that perturbs initially unexcited units above the excitation threshold (“excitatory” interface). In the result each unit displays a sequence of pulses (spikes or action potentials) (Fig. 3 (a)). In the opposite case the interface inhibits the units and their spiking activity becomes damped (Fig. 3 (b)) (“inhibitory” interface). Figure 4 shows the dependence of the interface velocity on control parameter ǫ. Together with the possibility of propagation in both directions (kink and anti-kink wave) there is a propagation failure in a certain range of values of ǫ. The parameter ǫ controls the ratio of attraction basins of the fixed point and the limit cycle, hence defines the direction and the velocity of interface propagation. Note, that since the intra-layer coupling is taken sufficiently small, then the interface is moving with rather low velocity (relative to the characteristic time scale of local oscillations). In the following sections we shall show that main characteristics of the interfaces can be approximately described by an averaged model with simpler bistability.

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2.4

Propagating interfaces in the two-layer network

Let the “excitatory” interface be propagating in the first layer while the second layer evolves near its rest state. When the front comes to the pin contact site there are three possible behaviors: • For a weak inter-layer coupling, h < h∗1 , the front overcomes the pin contact site while the second layer stays slightly perturbed but unexcited. • For increasing h∗1 < h < h∗2 and sufficiently low unit excitation threshold, the pin contact yields an excitatory input to the second layer (Fig. 5 (a)). Then, the second layer displays lurching-like wave composed of two interfaces propagating in opposite directions. In the results all units of the two layers become oscillating. • For larger values of h, h > h∗2 , there is a front propagation failure when the activity is inhibited by the pin contact. In the result the interface is locked in the first layer while the second layer stays unexcited (Fig. 5 (b)). Note, that different dynamics of the pin contact essentially depends on the ratio of attraction basins of the rest point and the limit cycle. The critical values h∗1 and h∗2 defining the three different outcomes of the system will be explored in Sect. III in the frame of the averaged model.

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2.5

Average description

To explain the effects observed we reduce the model to an averaged simplified description. As we have supposed the inter-neuron coupling to be sufficiently small, d ≪ 1, then the characteristic time scale of the interface propagation, Tf ∼ 1/d, is much larger than the local oscillation time scale given by the limit cycle period T . In other words, at the time scale Tf , the propagation is conditioned by an “integral” influence of local oscillations. 2.5.1

Averaging

Let us characterize local dynamics by average oscillation “intensity”. We introduce a positively definite quantity 1 Z t+T |u(t) − u∗ |dt u¯(t) = T t

(3)

with u∗ being the coordinate of the unit rest point O1 (u∗ , v ∗ ). To define the dynamics of the variable u¯(t) one should introduce an average recovery variable v¯(t) and obtain averaged equations from the two dimensional system (2). In the bistable oscillator mode (Fig. 2) depending on initial conditions (excluding the points corresponding to the threshold set) with t → ∞ all trajectories approach either the rest state or the stable limit cycle. Therefore, with t → ∞ the variable u¯ has two locally stable fixed points, corresponding to the stable limit cycle, u¯ = uc , and to the stable rest point, u¯ = 0. Then, at time scales Tf ≫ T the dynamics of the FitzHugh-Nagumo unit is defined by the low and the high levels of activity. Note, that for similar bistable oscillatory lattices composed of Van der Pol oscillators in a hard

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excitation mode [Nekorkin et al., 1997; Nekorkin et al., 1998] the averaging procedure (3) can be carried out analytically. In the result one can obtain the amplitude-phase equations. At variance, the FitzHugh-Nagumo like bistable units (2) display much more complex essentially non-isochronous behavior (Fig. 2 (b), (c)) with relaxation properties. 2.5.2

Poincare map analysis

To define how the units evolve between the two levels of activity we shall use the approach based on construction of Poincare section. Let us introduce the section half-line S : {v = v ∗ , u > u∗ } (Fig. 2). Assuming that stable rest point O1 has a focus type we obtain that at the section S the flow given by Eqs. (2) defines a point map, Π : S → S, for all points excluding the one, S0 , corresponding to the intersection of the incoming saddle separatrix with the half-line S. This point never returns to the cross-section (Fig. 2 (b)). For the bistability with the unstable limit cycle (Fig. 2 (c)) we also must exclude all the pre-images of the point S0 , Sn = (Π−1 )n S0 , n = 1, 2, . . . , ∞ . Then, the Poincare map Π can be written as w(n + 1) = w(n) + P (w(n)),

(4)

n = 1, 2, . . . , ∞, with w(n) accounting for (u−u∗ )-coordinates of the points at the Poincare section. The map is invertable as it corresponds to the flow of two-dimensional dynamical system and defined in the interval [0, ∞) excluding the point S0 or the countable set of points Sn . The shape

11

of the curve P (w) calculated numerically for the bistable modes of the FitzHugh-Nagumo unit is shown in Fig. 6. The map has a cubic-like shape with two stable fixed points. When the separatrix defines the unit excitation threshold (Fig. 2 (b)) the attraction basins are separated by the point S0 corresponding to a discontinuity of the map at the point S0 . For the unstable limit cycle defining the excitation threshold (Fig. 2 (c)) the map has an unstable fixed point separating the attraction basins and the discontinuity at the point S0 . The trajectories of the map represent sequences of points monotonically approaching to the fixed points. Note, that at the Poincare section we obtain an accurate description of local oscillations at different time scales. The shape of the curve P (w) shows that the map trajectories very rapidly (one iteration of the map) jumps to the neighborhoods of the two stable fixed points. Then, at the large time scales the dynamics of the map is defined by the shape of the curve P (w) near the fixed points. Let us approximate the piece-wise continuous function P (w) with a continuous function P¯ (w) providing qualitatively the same behavior (Fig. 6) and defined for all points [0, ∞). Then, we introduce a continuous time τ and describe the difference term in (4) by the time derivatives

w(n + 1) − w(n) ∼ τ0

dw , dτ

where τ0 ≪ 1 is some parameter that describes the characteristic time scales of the approximated continuous system. Then, the local dynamics of the FitzHugh-Nagumo unit can be

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qualitatively described by the following equation dw = Φ(w), dτ

(5)

with Φ(w) = P¯ (w)/τ0 . Accordingly, the equation (5) has two stable fixed points corresponding to the rest u¯ = 0 and the excited u¯ = uc states of the unit. Note, that the small values of τ0 yields the fast evolution of the variable w. Then, we note that the interaction between units in the network (1) is defined by the difference term ∆uj = (uj+1 (t) − uj (t)). For d ≪ 1 we assume that • the dynamics of the variable uj (t) is approximately defined by a single unit system (zero approximation order), • at the time scale of propagating interface uj (t) can be expressed with its average “intensity”, u¯(t), • the “average activity” evolves very fast (5) approaching to one of the two distinct levels, then for the average variable u¯ the coupling term at the interface has three extreme values ∆¯ uj = (uc , 0, −uc ) and rapidly “jumps” between them, hence can be expressed as the variable difference ∆¯ uj = (¯ uj+1 (t) − u¯j (t)). Therefore, the whole neural network (1) can be approximately modeled by a discrete Nagumo equation for the average neural activity with the bistable units (5) and the twolayer architecture as considered in the following sections. 13

3

Propagating interfaces in two-layer Nagumo network

The dynamics of a two-layer network of bistable units with a pin contact is given by the following equations   w˙ (1) = Φ(w (1) ) + D(w (1) − 2w (1) + w (1) ) + Hδjk (w (2) − w (1) ), j j j−1 j j+1 k k

 w˙ (2) = Φ(w (2) ) + D(w (2) − 2w (2) + w (2) ) + Hδ (w (1) − w (2) ), jk j j j−1 j j+1 k k

(6)

j = 1, 2, . . . , N,

where the variables w(1) , w(2) describes an average neural activity normalized at 1, the dot denotes the time derivatives with respect to τ , the parameters D and H are the effective coupling coefficients, D ∼ d, H ∼ h. They should take sufficiently small values for the slowly propagating interfaces. For simplicity, we approximate the function Φ(w) by the cubic polynomial Φ(w) = w(1 − w)(w − a), 0 < a < 1. The parameter a controls the ratio between the attraction basins of the rest and the excited states. We have also re-normalized the average activity at the excited state with uc = 1. At variance with Eqs. (1) the system (6) has no more the recovery terms, what makes its analysis much simpler. The propagating interface in the single layer of the system (6) represents a wave front solution (kink) traveling with a constant velocity [Keener, 1987; Keener, 2000; Erneux et al., 1993; Zinner, 1991]. Let the kink be propagating in the first layer while the units of the (2)

second layer are near the rest state, wj ≈ 0. Then, at the pin contact we obtain essentially similar effects to those found in the original system (1) including (i) propagation failure when the interface is locked at the contact (Fig. 7 (b)), (ii) excitation of the second layer spreading 14

from the place of contact (Fig. 7 (a)) and (iii) the front propagation for sufficiently small inter-layer coupling. The black and white color correspond to the rest and the excited state, respectively.

3.1

Propagation failure

To explain the effect of propagation failure let us consider the dynamics of the unit of the first layer at pin contact site k. It is given by the following equation taken from the system (6) (1)

(1)

(1)

(1)

(1)

(2)

(1)

w˙ k = Φ(wk ) + D(wk−1 − 2wk + wk+1 ) + H(wk − wk ).

(7)

Since d ≪ 1 the neighboring units evolve in the small neighborhood of the rest and the excited states. We can approximately set (1)

(1)

(2)

wk+1 ≈ 0, wk−1 ≈ 1, wk ≈ 0. (1)

Then, the dynamics, w = wk , of the unit k is defined by the first order equation dw = Ψ(w), dτ

(8)

Ψ(w) = Ψ1 (w) = Φ(w) + D(1 − 2w) − Hw,

(9)

with the effective nonlinearity,

and initial condition: w(t = 0) = 0. Two possible evolutions of the initial condition depending on the coefficient H are illustrated in Fig. 8. For sufficiently small values of H the equation (8) has one stable fixed point corresponding to the excited level. Then, the unit 15

evolves to the excited state and the front overcomes the pin contact site. With increasing H two additional fixed points (stable and unstable) appear and the trajectory is attracted by the stable one near the rest state. The unit stays unexcited and the interface is locked at the pin contact site (Fig. 7 (b)). The boundary, Hsup , of the propagation failure region is given by the equality Hsup : Ψ1 (wmin ) = 0,

(10)

with wmin =

1+a−

√

1 − a + a2 − 6D − 3H 3

The curve Hsup is plotted in Fig. 9 by solid line. Corresponding dashed curve obtained in numerical simulations of Eqs. (6) shows a good estimation by the reduced model (8) for small values of H. The non-vanishing difference between the curves with H → 0 is explained by the fact that D is fixed with a non-zero value. It should be taken sufficiently large to sustain the propagation in the single layer while the model was obtained in zero approximation order.

3.2

Excitatory action of the pin contact

The conditions of excitation of the second layer can be estimated in the similar way. For the second layer at the place of contact we have (2)

(2)

(2)

(2)

(2)

(1)

(2)

w˙ k = Φ(wk ) + D(wk−1 − 2wk + wk+1 ) + H(wk − wk ).

16

(11)

Assuming that in the first layer the interface has arrived to the pin contact site in zero approximation order on D and H we set (1)

(2)

(2)

wk ≈ 1, wk+1 ≈ 0, wk−1 ≈ 0. (2)

Then, the dynamics of the unit, w = wk , is given by (8) with the effective nonlinearity

Ψ(w) = Ψ2 (w) = Φ(w) − 2Dw + H(1 − w),

(12)

and the initial condition w(t = 0) = 0. Then, the boundary Hinf of the front excitation in the second layer corresponds to the equality

Hinf : Ψ2 (wmin ) = 0.

(13)

For H < Hinf the equation (8) has three fixed points and the unit stays unexcited (Fig. 8). For H > Hinf the two fixed points disappear and the unit goes to the excited state. Therefore, two wave fronts spreading from the place of contact appear in the second layer. The curve Hinf and its corresponding curve obtained numerically are shown in Fig. 9. Thus, the curves Hsup and Hinf bound the three regions corresponding to different behaviors of the interface at the pin contact. Note, the curves has an intersection point. At this point the units of the two layers at pin contact site have equivalent effective potentials ( Ψ(w)dw) relative to the “excitatory” interface. Then, the interface behavior near this R

point become very sensitive to possible fluctuations. Note, that the existence of the two critical values of inter-layer coupling has been confirmed in numerical simulations of the original 17

FitzHugh-Nagumo system (1). Thus, the reduced model provides a complete qualitative view on the behavior of interfaces at the pin contact.

4

Fronts in electronic experiment

In this section, thanks to an equivalent nonlinear electrical lattice [Nekorkin et al., 2001; Morfu et al., 2002; Binczak et al., 1998], we show experimentally the different behaviors obtained theoretically and numerically in the previous sections. We consider the two-layer nonlinear electrical network (Fig. 10) with the inter-layer interaction localized at the cell number k. In each line the cells are coupled by intraline linear resistor R, while the linear resistor Rh coupling the kst cells of each line provides the interline interaction localized at the spatial site k. Each cell consists of a linear capacitor in parallel with a nonlinear resistor whose current voltage characteristic obeys to the following cubic law:

IN L = V (V − m1 )(V − m2 )/(R0 m1 m2 ),

(14)

where m1 and m2 represent the roots of the cubic characteristic and R0 a weighting resistor. The Kirchhoff laws applied to the lattice give straightforwardly the set of differential equations  dUj Uj Uj 1 Uj 1      dτ ′ = RC (Uj+1 + Uj−1 − 2Uj ) − R C (1 − m )(1 − m ) − R C δjk (Uk − Vk ), 0 1 2 h  dVj 1 Vj Vj Vj 1    = (Vj+1 + Vj−1 − 2Vj ) − (1 − )(1 − )− δjk (Vk − Uk ),  ′

dτ

RC

R0 C

m1

m2

(15)

Rh C

where τ ′ corresponds to the time, j = 1, 2, . . . , 22, and k = 11 is the spatial site of the pin (1)

contact. Setting wj

(2)

= Uj /m2 , wj

= Vj /m2 , D = R0 a/R, H = R0 a/Rh , a = m1 /m2 , 18

τ ′ = R0 Caτ , the system (15) appears as an analog simulation of the dimensionless Nagumo equation (6). The voltage of each cell is collected every 64 µs as an analog luminance signal, where black corresponds to the state close to V = 0 (unexcited state), while white corresponds to the state close to V = m2 (excited state). A parallel to serial converter allows to visualize on a monitor the spatiotemporal evolution of each electrical line in grey scale, with the time growing from top to bottom and the cell number from left to right. Furthermore, the threshold of each nonlinear resistor is adjusted to a = 0.29 ± 0.01, tuning m1 = 0.46 V and m2 = 1.59 V . Since the value of the weighting resistor R0 and the intralayer coupling resistor R are R0 = 3.1kΩ and R = 15kΩ, the intra-layer coupling is set to D = 0.060 ± 0.004. A kink is initiated in the first layer, from a step-like initial condition fixing the two first cells in excited state, while other cells of the first line and all cells of the second one are initially in unexcited state. After maintaining such initial conditions the system evolution is free. Let us investigate the behavior of the system versus the control parameter H (namely the inter-layer resistor Rh ). • For sufficiently small values of H, H ≤ Hinf = 0.052, (Fig. 11) with H11 = 0.045 ± 0.003), a dark grey vertical band showing that V11 < m1 in the line 2. Nevertheless, the interface spreading in the first line overcomes the pin contact, its instant velocity slightly decreasing.

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• Increasing the parameter H in the interval ]Hinf = 0.052, Hsup = 0.082], (namely decreasing Rh ), induces the rise at the pin contact m = 11 of a kink and an antikink spreading in the second line with opposite velocities, while the kink overcomes the place of contact in the first line with only small troubles (Fig. 12 with H = 0.060 ± 0.004). • When H exceeds the critical value Hsup = 0.082 (Fig. 13), the interface is locked at the pin contact, the second line remaining quasi-unchanged from its initial rest condition. Thus, the two-layer electrical lattice with the pin contact displays the three types of behavior observed in the original bistable oscillator network (1) and studied in the reduced model (6). To confirm the theoretical explanation of the effects given in Sect. III we use potentiometer to find experimentally the values of Rh , that define the two critical values Hinf and Hsup for which the system behavior changes versus the nonlinearity threshold a. These experimental results (crosses) are compared in Fig. 9 to the numerical and theoretical results (dotted and solid lines respectively) with a qualitative good agreement, the main discrepancies being directly imputable to the component uncertainties and the current voltage characteristic that does not match exactly a cubic law. Note that the experimental data obtained for H = 0, that is the cross at the right bottom of Fig. 9, represents the experimental critical value (a = 0.35, D∗ = 0.06) under which standard propagation failure occurs in the single Nagumo chain [Comte et al., 2001].

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5

Conclusion

We have investigated the dynamics of slowly propagating interfaces in two-layer neural networks with localized inter-layer connections (pin contacts). We have found that such a local inhomogenity may dramatically affect the activity of the whole network. There is an effect of interface locking at the pin contact site leading to the propagation failure. Then, the localized inter-layer connection may provide local excitatory input to the rest layer leading to the appearance of lurching-like wave of spreading activity. We have used the FitzHughNagumo-like oscillatory units for the description of local neuron (or population of neurons) activity. Then the activity is characterized by periodic sequence of spikes (action potentials). We have chosen the simple (local) intra-layer connectivity. Then the single layer is a simple tool to sustain the interface propagation. Assuming that the local behavior is much faster then the characteristic time scale of the interface, we have shown that all the effects appearing then at the pin contact can have a good qualitative explanation with simpler discrete Nagumo lattice. Indeed, the waves of neural activity found in experiments on cortical slices are very slow. They have about 10 times longer time scales compared with propagating action potentials (0.06 ms−1 against 0.5 ms−1 for an axon) [Bressloff, 2001]. Then, at the time scale of interface propagation, they can be characterized by average levels of local activity. We have constructed a two-layer electrical network with localized connections. The experiments have shown a good qualitative and quantitative agreement with theoretical and

21

numerical results. Finally, the effects of interface dynamics at the pin contact have been found (theoretically and experimentally) for two originally different models in quite wide range of parameters. They are indeed robust and appear in systems with multi-layer architecture that have (i) a local bistability and (ii) a weak intra-layer connectivity.

Acknoledgments This research was supported in part by Russian Foundation for Basic Research (grants 03-0217135), by grant of President of Russian Federation (MK 4586.2004.2). V.B.K. acknowledges Russian Science Support Foundation for financial support.

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neural network” Physica D 155, 83–100. Bressloff, P.C. & Coombes, S. [1999] “Travelling waves in chains of pulse-coupled integrateand-fire oscillators with distributed delays” Physica D 130, 232–254. Brindley, J., Holden, A.V. & Palmer, A. [1991] Nonlinear Wave Processes in Excitable Media, (Ed. By A.V. Holden et al.), (Plenum Press, N. Y.). Comte, J.C., Morfu, S. & Marqui´e, P. [2001] “Propagation failure in discrete bistable reaction-diffusion systems: theory and experiments” Phys. Rev. E 64 027102 (2001) Coombes, S. “From periodic traveling waves to traveling fronts in spike-diffuse-spike model of dendritic waves” [2001] Mathematical Biosciences 170, 155–172. Ermentrout, B., Chen, X. & Chen, Z. [1997] “Transition fronts and localized structures in bistable reaction-diffusion equations” Physica D 108, 147–167. Erneux, T. & Nicolis, G. [1993] “Propagating waves in discrete bistable reaction-diffusion systems” Physica D 67 237–244. Golomb, D. & Amitai, Y. [1997] “Propagating Neuronal Discharges in Neocortical Slices: Computational and Experimental Study” J. Neurophysiol. 78, 1199–1211. Golomb, D. & Ermentrout, G.B. [1999] “Continuous and lurching traveling pulses in neuronal networks with delay and spatially decaying connectivity” Proc. Natl. Acad. Sci USA 96 (23), 13480–13485. 23

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Murray, J.D. [1988] Mathematical Biology (Springer-Verlag, Berlin). Nekorkin, V.I., Kazantsev, V.B., Rabinovich, M.I. & Velarde, M.G. [1998] “Controlled disordered patterns and information transfer between coupled neural lattices with oscillatory states” Phys. Rev. E 57, 3344–3351. Nekorkin, V.I., Kazantsev, V.B., Morfu, S., Bilbault, J.M. & Marqui´e, P. [2001] “Theoretical and experimental study of two discrete coupled Nagumo chains” Phys. Rev. E 64, 036602-1–036602-13. Nekorkin, V.I., Makarov, V.A., Kazantsev, V.B. & Velarde, M.G. [1997] “Spatial disorder and pattern formation in lattices of coupled bistable elements” Physica D 100, 330– 342. Osan, R. & Ermentrout, B. [2001] “Two dimensional synaptically generated traveling waves in a theta-neuron neural network” Neurocomputing 38 795–795. Osan, R. & Ermentrout, B. [2002] “The evolution of synaptically generated waves in oneand two-dimensional domains” Physica D 163, 217–235 (2002); Peinado, A. “Traveling Slow Waves of Neural Activity: A Novel Form of Network Activity in Developing Neocortex” [2000] J. Neurosci 20, RC54,1-6. Prechtl, J.C., Cohen, L.B., Pesaran, B., Mitra, P.P. & Kleinfield, D. “Visual stimuli induce waves of electrical activity in turtle cortex” [1997] Proc. Natl. Acad. Sci. USA 94, 25

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Figure captions Fig. 1 Two-layer architecture of with a pin contact. Fig. 2 Qualitative view of the phase plane for different dynamic behaviors of the FitzHughNagumo unit (2). (a) The excitable mode. (b) Limit cycle - fixed point bistability with separatrix threshold. (c) The bistability with unstable limit cycle. (d) Oscillatory behavior. Fig. 3 Propagating interfaces in the single layer network defined by Eqs. (1). Parameter values: α = 0.5, β = 2, I = 0.225, d = 0.01, h = 0. (a) “Excitatory” interface, ǫ = 0.36. (b) “Inhibitory” interface, ǫ = 0.53. Units are arbitrary. Fig. 4 The dependence of the interface velocity, c, on control parameter ǫ in the single layer network (1). Parameter values: α = 0.5, β = 2, I = 0.225, d = 0.01, h = 0. Units are arbitrary. Fig. 5 The behavior of propagating interface of neural activity at the pin contact. Parameter values: α = 0.5, β = 2, I = 0.225, ǫ = 0.36, d = 0.01, N = 100. (a) Excitatory input to the second layer. Creation of lurching-like wave spreading from the pin contact site, h50 = 0.05. (b) Propagation failure at the pin contact, h50 = 0.1. Units are arbitrary. Fig. 6 The shape of the function P (w) calculated from Eqs. (2) corresponding to the bistable oscillator mode of the FitzHugh-Nagumo unit (Fig. 2 (b)). Black and white 27

circles correspond to stable and unstable fixed points of the map (4), respectively. Parameter values: α = 0.5, β = 2, I = 0.225, ǫ = 0.36. Units are arbitrary. Fig. 7 Propagating interface at the pin contact in the two-layer Nagumo network. Parameter values: a = 0.3, D = 0.06, N = 100. (a) Excitatory input to the second layer. Kink and anti-kink waves are excited. h50 = 0.09. (b) Propagation failure at the pin contact. h50 = 0.12. Units are arbitrary. Fig. 8 Qualitative picture of two different behaviors defined by Eq. (8). Fig. 9 Parameter regions corresponding to three different behaviors of the interface at the pin contact in two-layer Nagumo lattice. Parameter values: D = 0.06, N = 22, k = 11. Solid curves: boundaries Hsup and Hinf calculated from (10) and (13). (− • −) signs: Hinf and Hsup obtained by numerical simulations of Eqs. (6). Experimental results obtained with a two-layer electrical lattice are shown by crosses with their uncertainties. The vertical solid line for α = 0.4 limits the parameter region (in hatched line) where classical propagation failure takes place in a single Nagumo lattice (see [Comte et al., 2001] for an analytical expression). Fig. 10 Sketch of the two-layer nonlinear electrical lattice. Fig. 11 The kink in the first line overcomes the pin contact for H < Hinf . Parameters: a = 0.29 ± 0.01, C = 22nF , D = 0.060 ± 0.004, H = 0.045 ± 0.003, k = 11.

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Fig. 12 Kink and antikink rising in the second line at the pin contact for H ∈]Hinf , Hsup ], when the kink spreading in the first line overcomes the pin contact. Parameters: a = 0.29 ± 0.01, C = 22nF , D = 0.060 ± 0.004, H = 0.060 ± 0.004, k = 11. Fig. 13 Pinning of the kink at the pin contact in the first line for H > Hsup . Parameters: α = 0.29 ± 0.01, C = 22nF , D = 0.060 ± 0.004, H = 0.100 ± 0.006, k = 11.

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