Spontaneous and evoked activity of synaptically coupled neuronal fields with axonal propagation...

Spontaneous and evoked activity of

synaptically coupled neuronal fields with

axonal propagation delay for

gamma-distributed connectivity kernels

Axel Hutt∗

Weierstrass Institute for Applied Analysis and Stochastics,

Mohrenstr.39, 10117 Berlin, Germany

E-mail: hutt@wias-berlin.de

Fatihcan M. Atay

Max-Planck-Institute for Mathematics in the Sciences,

Inselstr. 22, 04103 Leipzig, GermanyE-mail: atay@member.ams.org

Abstract

This work studies dynamical properties of spatially extended neu-

ronal ensembles. We first derive an evolution equation from tem-

poral properties and statistical distributions of synapses and somata.

The obtained integro-differential equation considers both synaptic and

axonal propagation delay, while spatial synaptic connectivities ex-

hibit gamma-distributed distributions. This familiy of connectivity

kernels also covers the cases of divergent, finite, and negligible self-

connections. The work derives conditions for both stationary and

nonstationary instabilities for gamma-distributed kernels. It turns out

that the stability conditions can be formulated in terms of the mean

∗Supported by the DFG research center ”Mathematics for key technologies” (FZT86)in Berlin, Germany

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spatial interaction ranges and the mean spatial interaction times. In

addition, a numerical study examines the evoked spatiotemporal re-

sponse activity caused by short local stimuli and reveals maximum

response activity after the mean interaction time at a distance from

stimulus offset location equal to the mean interaction range. These

findings propose new insights to neuronal mechanisms of experimen-

tally observed evoked brain activity.

1 Introduction

Understanding the basic mechanisms of neuronal activity is supposed to yieldinsights to major brain functions such as perception, memory processes ormotor coordination. The presented work attacks this task by studying spa-tiotemporal neuronal dynamics caused by fluctuations and environmentalexternal stimuli.

Fluctuations in space and time are always present due to the large numberof interconnected neurons and are supposed to be responsible for large-scalecoherent phenomena near unstable neural activity states. We mention thephenomena of hallucinations, which frequently result from specific circum-stances such as fatigue or sleep deprivation [1] and which, in some cases,exhbits a shift of the neural state to an instability by increased neuronal ex-citation [2, 3]. For instance, Ermentrout and Cowan [4] introduced a meso-scopical neuronal field theory and explained visual hallucination patterns byloss of stability at bifurcation points.

In contrast, external stimuli may represent sensoric input as auditoryspeech or visual perceptions. For instance, during cognitive experimentsencephalographic measurements reveal coherent evoked brain activity andindicate synchronous neuronal activity on a mesoscopic spatial level [5, 6,7, 8, 9]. In this context, Freeman [10] has shown in an early work thatencephalographic activity relates to mesoscopic dendritic currents. Dipoland current source density models support these findings [11]. Hence, ourwork aims to study neuronal mechanisms on a spatially mesoscopic level.

Many works studying mesoscopic neuronal activity [12, 13, 14, 15, 16, 17,18, 19, 20] treat synaptically coupled neuronal ensembles. Our work followsthe basic field approach of Jirsa and Haken [5], who combined the ensemblemodels of Wilson and Cowan [21] and Nunez [22]. This model considers asingle type of neurons, which are interconnected by axons terminating at

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either excitatory or inhibitory synapses. Though the intrinsic delay due toaxonal propagation had been introduced, it does not affect temporal and spa-tial dynamics. A recent work treating intracortical activity [23, 24] extendsthe model by introducing synaptic response delay and thus adds a furthertime scale. It turns out that the relation between synaptic and propagationdelay affects the stability of the system. In addition to the temporal scales,the synaptic connectivity kernels define the spatial scales of the neuronalfield. In most works, these connectivity kernels exhibit their maximum atzero distances, i.e. strong self-connectivity. Since effects of reduced self-connectivity has not been studied yet in a general form, we shall discuss thefamily of gamma-distributed connectivity kernels which may exhibit infinite,finite, and negligible probability densities of self-connections for diverse pa-rameters. These cases shall be studied in the context of spontaneous andexternally evoked activity.

The paper is organized as follows. The subsequent section presents thederivation of the field equation. In Section 3, stability conditions with respectto spontaneous fluctuations are derived analytically, followed by a numericalstudy of evoked responses caused by external stimuli. Section 4 summarizesthe results and closes the work.

2 Model derivation

The present work treats a three-section model of synaptically coupled neu-ronal ensembles. Here, one section represents the ensemble set of synapses,which convert the incoming presynaptic activity to postsynaptic potentials(PSP). The adjacent model section contains the ensemble set of trigger zonesat somata converting PSPs to axonal pulse activity, while the final one rep-resents the set of axonal fibres linking trigger zones to distant synapses. Inthe following, these model sections are discussed in some detail.

2.1 Model sections

Chemical synapses convert incoming action potentials to postsynaptic cur-rents by emission of neurotransmitters [25]. Most excitatory synapses (e)emit neurotransmitters called glutamate, which enhance the activity of thepostsynaptic cells, while the neurotransmitter γ-aminobutyric (GABA) emit-ted by inhibitory synapses (i) diminishes the postsynaptic cell activity. In

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addition, synapses may bind to dendrites or the soma of the postsynapticneuron. Hence, the efficacy of synapses is very diverse. While excitatorysynapses are often located at dendrites and act faster than inhibitory ones,the latter are often located closer to the soma and thus have larger effects tothe neuron.

Furthermore, there is evidence that the dendritic morphology, such ascompactness of arborization and branching patterns, affects the stability ofconnected neurons [14, 26], e.g. due to propagation delays along spatiallyextended dendrites [27, 28]. In a simplified model, dendrites are electricconductors which exhibit passive spread of current through the dendritictissue. According to this approach, Freeman [29] was one of the first toshow that the response of chemical synapses to incoming action potentialsa(t) is approximately equivalent to the convolution of a(t) with an impulseresponse function he,i(t). The presented approach accounts for this findingand neglects shunting effects.

In experimental practice single cell activation is measured frequently asthe number of action potentials exceeding a certain threshold potential duringthe time interval ∆t . 2ms and a sampling procedure at rates νs ≈ 10 −100kHz. Hence it is reasonable to assume action potentials at discrete timest+k/νs, k = 1, . . . , [νs∆t] and the mean pulse rate at time t gives the numberof action potentials in the interval between t and t+∆t. Here, [..] denotes thenearest integer function. In addition, our model introduces spatial patchesΓ(x) at location x each containing N synapses, that is, the activity discussedis coarse grained in space. Consequently, postsynaptic potentials averagedover time and space obey

V e,i(x, t) =1

∆t

[νs∆t]∑

k=1

1

N

N∑

l=1

g(l)e,i

∫ t+k/νs

−∞

dτhe,i(t − τ )a(xl, τ )

≈∫ t

−∞

dτ he,i(t− τ )1

∆t

[νs∆t]∑

k=1

1

N

N∑

l=1

g(l)e,ia(xl, τ + k/νs)

= ge,i

∫ t

−∞

dτ he,i(t− τ ) Pe,i(x, τ ) , (1)

with xl ∈ Γ(x), g(l)e,i denotes the efficacy of excitatory and inhibitory synapses

at location xl and Pe,i(x, t) represents the presynaptic pulse rate coarsegrained in time and space. Equation (1) assumes presynaptic pulse activ-ity which is fast compared to the slow synaptic response on a time scale of

4

5−10ms. We remark that the replacement of sequences of actual spike trainsa(t) is only justified if quantities relevant for the network dynamics are insen-sitive to trial-to-trial fluctuations, i.e. time coding of spikes is not relevant.This is given in case of uncorrelated single action potentials, which is assumedhere. The introduced spatial patches Γ(x) represent neuronal assemblies orneuronal pools [13, 21, 22, 30] which build mesoscopic functional entities inthe brain and have been found experimentally both in cortical [31, 32] andsubcortical [33] areas.

Essentially, we assume variations of synaptic properties in the consideredneuronal population [34]. Thus PSPs V e,i(t) at single neurons become ran-dom variables with the corresponding probability distributions pe

S(V e − V e)and pi

S(V i− V i). Since excitatory and inhibitory PSPs sum up at the triggerzone of each neuron [35, 36], the probability density function of the effectivemembrane potential V = V e − V i is

pS(V − V ) =1

2π

∫

dzφeS(z)φi

S(−z)e−izV , (2)

where V = V e − V i and φeS , φi

S are the characteristic functions of the corre-sponding probability density functions of pe

S , piS.

We now aim to discuss the synaptic properties in more detail. Accordingto [10], the impulse response function from (1) reads

h(t) =α1α2

α1 − α2

(e−α1t − e−α2t), α1, α2 > 0 .

In case of α1 = α2 = α, the function is the well-known alpha function h(t) =α2t exp(−αt). For maximum impulse response after a delay of ∼ 5ms [37],it is α = 200Hz. Further, h(t) represents a Green function for the temporaloperator

L = (∂

∂t+ α1)(

∂

∂t+ α2) (3)

with Lh(t) = α1α2δ(t) and δ(t) being the Dirac δ function.The latter synaptic and dendritic features define the first model section.

In the present concept of neuronal ensembles, it converts incoming presynap-tic pulse activity to effective somatic membrane potentials.

Now, the adjacent model section focuses on the conversion of membranepotentials to pulse activity. A single neuron generates an action potential

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a(t) at time t, i.e. it fires if the membrane potential V (t) at the trigger zoneexceeds a certain threshold Vth. Thus, the probability of a single neuron tofire is Θ(V (t) − Vth), where Θ(z) denotes the Heaviside function. For anensemble of neurons at spatial location x, there is a distribution of firingthresholds D(Vth − Vth, t) and thus the expected number of firing neurons is

N(t) =

∫ ∞

−∞

dV pS(V − V (t))

∫ ∞

−∞

dVthΘ(V − Vth)D(Vth − Vth, t)

=

∫ ∞

−∞

dw

∫ w+V (t)−Vth

−∞

du pS(w)D(u, t).

where Vth denotes the mean firing threshold and pS is taken from Eq. (2).However, single neurons show refractory time periods and adaption [13, 35],which change the threshold distribution in time. Hence, the time-averagedpulse activity at location x is given by

N(x, t) =

∫ t+∆t

t

dτ

∫ ∞

−∞

dw pS(w)

∫ w+V (x,τ )−Vth

−∞

du D(u, τ )

≈∫ ∞

−∞

dw pS(w)

∫ w+V (x,t)−Vth

−∞

du

∫ t+∆t

t

dτD(u, τ )

=

∫ ∞

−∞

dw

∫ w+V (x,t)−Vth

−∞

du pS(w)D(u) (4)

for the time interval ∆t sufficiently small and the time-averaged distributionof firing thresholds D(u). Here, N(x, t) represents the average number offiring neurons in a time interval between t and t+∆t in a neuronal ensembleat location x. Since N depends on the membrane potential V , Eq. (4) givesthe general definition of the so-called transfer function.

Additionally, the change of pulse activity subject to the membrane po-tential plays an important role concerning the temporal stability of neuronalensembles [22, 38] and reads

∂N

∂V=

∫ ∞

−∞

dw pS(w)D(w + V − Vth). (5)

Equations (4) and (5) are general expressions for the well-known sigmoidalconversion and the corresponding conditional pulse probability density or

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nonlinear gain [38], respectively, which are subjected to synaptic and so-matic statistical properties.To be more specific, in case of normal-distributed synaptic probability dis-tributions pe,i

S , the effective membrane potentials obey a normal distributionpS ∼ N (0, σ2

S). Additionally, for Gaussian-distributed firing thresholds

D(u) =Pmax√2πσT

e−u2/2σ2

T ,

the transfer function and nonlinear gain read

N (x, t) =Pmax

2(1 + erf(

V (x, t)− Vth√2σ

)) (6)

∂N(x, t)

∂V=

Pmax√2πσ

e−(V (x,t)−Vth)2/2σ2

,

respectively, where σ2 = σ2S + σ2

T , erf(x) represents the Gaussian error func-tion [39] and Pmax denotes the maximum firing rate. Equation (6) showsaccordance to previous results [40].

The present work approximates the conversion of effective postsynapticpotentials V to mean pulse rates N from Eq. (6) by the logistic function [29,38, 41, 42, 43]

N(x, t) = PmaxS(V (x, t)) =Pmax

1 + e−c(V (x,t)−Vr)(7)

with parameters c, Vr defined in a later section. Equation (4) gives the majorrelation of this second model section.

To close the conversion chain of state variables, the final model sectioncontains axonal fibres linking trigger zones and dendritic structures of ter-minal neurons. In mathematical terms, pulse activity generated at spatiallocation y propagates along axons by speed ve,i and sums up at terminal ex-citatory and inhibitory synapses at location x according to the correspondingaxonal connectivity distributions fe(x, y) and fi(x, y), respectively. More pre-cisely, these distributions are the probability density of axonal connectionsbetween two spatial locations x and y. Thereby, the nonlocal interactionsyield temporal propagation delays in case of finite axonal propagation speeds.In combination with Eq. (1), this approach is similar to the voltage-basedmodel of Hopfield [44]. Hence, the presynaptic pulse activity reads

Pe,i(x, t) =

∫

V

dx′fe,i(x, x′)N(x′, t− |x − x′|ve,i

) + µe,iI(x, t) (8)

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with coupling factors µe,i. The additional pulse activity I(x, t) introduces anexternal input, e.g. from other cortical regions or from the midbrain [22].

In most previous works [4, 23, 45], neuronal fields exhibit axonal connec-tions which are maximal at zero distance and monotonically decreasing for in-creasing distance. Then, the combination of excitatory and inhibitory axonalnetworks may yield four different spatial interactions, namely pure excitation,pure inhibition, local excitation-lateral inhibition and local inhibition-lateralexcitation. In contrast, the current work picks up an interestimg result ofNunez [22], who estimated the distribution of axonal cortico-cortical fiberlengths in humans, i.e. the spatial range of excitatory connections, based ondistributions in mice. He found that excitatory connections in humans mayobey a gamma distribution with maximum at some centimeters. A similarproblem has been addressed by Rinzel et al. [46], who found new propagationpatterns in inhibitory networks with vanishing self-connections. Since thereis also strong anatomical evidence for self-connections in inhibitory networksin cat visual cortex [47], we set the corresponding axonal distribution to adecreasing exponential.

2.2 The field equation

Now, all model sections have been defined and we combine the major results(1), (3), (7) and (8) to

LV (x, t) =

∫

V

aefe(x, y)S(V (x′, t− |x − y|ve

))

−aifi(x, y)S(V (y, t − |x − y|vi

)) dy + µI(x, t)

with ae,i = ge,iPmax, µ = µe − µi and

fe(x, y) =1

2rpeΓ(p)

|x − y|p−1e−|x−y|/re , fi(x, y) =1

2rie−|x−y|/ri (9)

where p ∈ <, p > 0 is a parameter of the gamma distribution, Γ(p) de-notes the gamma function and re, ri are the spatial interaction range ofexcitatory and inhibitory connectivity kernels, respectively. After scalingt → t

√α1α2, x → x/re, ve,i → ve,i/(re

√α1α2) the final field equation reads

LV (x, t) =

(

∂2

∂t2+ γ

∂

∂t+ 1

)

V (x, t)

8

=

∫ ∞

−∞

aeKe(x − y)S(V (y, t − |x − y|ve

))

−aiKi(x − y)S(V (y, t − |x − y|vi

))dy + µI(x, t) (10)

with

Ke(x) =1

2Γ(p)|x|p−1e−|x| , Ki(x) =

1

2re−r|x| , (11)

γ = (α + 1α) ≥ 2 and r = re/ri. Equation (10) describes the dynamics of

a coarse grained field of neuronal ensembles coupled by nonlinear, nonlocalinteractions. It is damped by synaptic delay dynamics and subjected todelayed spatial interaction and external input. The mean spatial interactionranges of excitatory and inhibitory connections, respectively, are given by

ξe =

∫ ∞

−∞

dxKe(x)|x| = p, ξi =

∫ ∞

−∞

dxKi(x)|x| = 1/r

using Eq. (11). In Figure 1, both kernels are plotted for various parametersp, r and we observe singular local excitations for p < 1 (Fig. 1, left panel). Ata first glance, this singularity of the probability density Ke may appear un-physical. However, this effect occurs even in much more simple processes andwe mention the standard Brownian motion exhibiting a singular probabilitydensity of sojourn times [48, 49].

In most works treating spatial structures in neuronal fields, excitatoryand inhibitory connectivity kernels are of the same functional type such asexponentials or Gaussians. In these models, the excitation and inhibitioncomes in by a different sign of the kernel functions and by different spatialscales, say re, ri. In consequence, the spatial interaction ranges can be scaledto ξe = 1 and ξi 6= 1. That is, the single parameter ξi reflects the relationof the excitatory and the inhibitory spatial scale and defines the spatial in-teraction. In contrast, Eq. (9) reveals an additional parameter p yieldingtwo variables ξe, ξi which define the spatial interaction ranges. The addi-tional parameter proposes novel dynamic properties which are extracted inthe subsequent sections.

3 Stability analysis

This section aims to study the stability of a stationary state V0 which isconstant in space. In case of constant external input µI(x, t) = I0, Eq. (10)

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becomes the implicit equation

V0 = (ae − ai)S(V0) + I0. (12)

and Fig. 2 illustrates its different solutions for ae > ai.For deviations u(x, t) = V (x, t)− V0 = u0e

λt+ikx, Eq. (10) reads

L(λ) = s

∫ ∞

−∞

dz(

aeKe(z)e−λ

ve|z| − aiKi(z)e

− λ

vi|z|

)

eikz (13)

with the nonlinear gain s = δS/δV at V = V0.When s = 0, one has L(λ) = λ2 + γλ + 1 = 0, so that Rλ < 0 and theperturbations u are damped out. It follows that V0 is asymptotically stablefor all small s, since the values (λ, k) satisfying the dispersion relation (13)depend continuously on s. However, increasing s further may result in a lossof stability; in this critical transition one has Rλ = 0. Thus setting λ = iωfor some ω ∈ R in (13), we get

1 − ω2 + iωγ = s

∫ ∞

−∞

dz(

aeKe(z)e−iω|z|/ve − aiKi(z)e−iω|z|/vi

)

eikz. (14)

Comparing the magnitudes of both sides,

√

(1 − ω2)2 + γ2ω2 ≤ s

∫ ∞

−∞

dz (ae|Ke(z)|+ ai|Ki(z)|) . (15)

By simple calculus and the fact that γ ≥ 2,

(1 − ω2)2 + γ2ω2 ≥ 1 for all ω ∈ R. (16)

Also, by (11),∫ ∞

−∞

dz|Ke,i(z)| = 1. (17)

Using (16) and (17) in (15), we obtain the necessary condition for loss ofstability

1 ≤ s(ae + ai). (18)

In order to investigate the dynamics of the nonlinear equation (10), it isuseful to classify the different ways in which V0 may lose its stability subjectto the parameter s in case of spontaneous fluctuations.

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3.1 Spontaneous stationary instability

For stationary bifurcations, ω = 0 and the threshold for s becomes

sc =1

aeKe(kc) − aiKi(kc)=

1

K(kc),

where Ke and Ki are the characteristic functions of Ke and Ki, respectively,and

K(k) =ae√

1 + k2p cos(p arctan(k)) − air

2 1

r2 + k2.

In case of a constant bifurcation, the stationary state loses stability due tospontaneous fluctuations for

s >1

K(0)=

1

ae − ai.

Figure 3 shows the corresponding bifurcation diagram for various parametersae, ai.

However, increasing s from zero, a non-constant bifurcation may emergefor (ae +ai) > K(kc) > (ae−ai) with kc > 0 and ae > ai. The correspondingsufficient condition reads for gamma-distributed kernels

∂2K(k)

∂k2|k=0 > 0 ⇒ ξ2

i >ae

2aiξe(ξe + 1) (19)

with

∂2K(k)

∂k2= − aep(p + 1)

√1 + k2

p+2 cos((p + 2) arctan(k)) + 2air2 r2 − 3k2

(r2 + k2)3,

which ensures a local maximum of K(k) at finite kc > 0, cf. Fig. 4, 5and 6. The corresponding bifurcation has been found first by Turing innon-equilibrium activator-inhibitor systems [50, 51].

For ξe < 1, Turing patterns are present for ξi > ξe (Fig. 5), whilefor ξe ≥ (ae/ai)/(2 − ae/ai), ae ≤ ai they occur for ξi ≤ ξe as well. Thislatter case reflects a larger mean excitation range than mean inhibition rangeand has not been found in previous works. Figure 6 shows the effectivekernel aeKe(x)−aeKi(x) and corresponding Fourier transform K(k) for both

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ξe = 1 and ξe = 2.0. In contrast to kernels with ξe = 1, which exhibitlocal excitation-lateral inhibition interaction for ξi > 1, Turing instabilitiesmay also occur for ξe = 2 although the kernel elicits local inhibition-lateralexcitation interaction.Eventually recalling the findings of Nunez [22], experiments indicate ξe =3 and ae > ai for excitatory cortico-cortical connections, i.e. long-rangeinteraction with r � 1 and thus ξi � 1. According to Eq. (19) and Fig. 4,Turing patterns may not occur for these parameters and also have not beenfound in experiments.

3.2 Spontaneous non-stationary instability

This type of bifurcation is characterized by a solution pair (λ, k) of (13) withλ = iω 6= 0. We shall show that such bifurcations are possible only if thetransmission speeds ve,i are not too large, and obtain an estimate to quantifythis statement.

Considering the imaginary part of (14),

ωγ = −s∫ ∞

−∞dz (aeKe(z) sin(ω|z|/ve) − aiKi(z) sin(ω|z|/vi)) cos(kz)

+s∫ ∞

−∞dz (aeKe(z) cos(ω|z|/ve) − aiKi(z) cos(ω|z|/vi)) sin(kz).

Note that the integrand in the first integral is an even function of z while thatin the second integral is an odd function. Thus the second integral vanishes,and we have

ωγ = −2s

∫ ∞

0

dz (aeKe(z) sin(ωz/ve) − aiKi(z) sin(ωz/vi)) cos(kz)

which yields

|ω|γ ≤ 2s

∫ ∞

0

dz (ae|Ke(z) sin(ωz/ve)| + ai|Ki(z) sin(ωz/vi)|) .

Using the fact that | sinx| ≤ |x| for all x, we obtain

|ω|γ ≤ 2s

∫ ∞

0

dz (ae|Ke(z)ωz/ve| + ai|Ki(z)ωz/vi|)

and since ω 6= 0 at a non-stationary bifurcation,

γ ≤ s

(

ae

ve

∫ ∞

0

dz 2Ke(z)z +ai

vi

∫ ∞

0

dz 2Ki(z)z

)

.

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Note that the integrals are the definitions of the mean spatial interactionranges ξe and ξi for excitatory and inhibitory connections, respectively. Hencewe define

τe =1

ve

∫ ∞

−∞

dz Ke(z)|z| = ξe/ve, τi =1

vi

∫ ∞

−∞

dz Ki(z)|z| = ξi/vi (20)

as the mean delay times respectively for the excitatory and inhibitory infor-mation transmission in the field. In this way, we can express a necessarycondition for non-stationary bifurcations to occur, namely

s ≥ sc =γ

aeτe + aiτi

. (21)

Since all quantities are positive, it is clear that at least one of the velocitiesve or vi must be finite for the occurence of non-stationary bifurcations. Forthe particular case when the distributions are given by (11), we have

γ

s≤ aep

ve

+ai/r

vi

by using the mean values of the gamma and exponential probability den-sity distributions. This relation elicits a decreased threshold sc for increasedvalues of p and 1/r, i.e. the larger p and 1/r the easier nonstationary bifur-cations may occur.

With Eq. (21) and the previous condition 1/(ae + ai) < s < 1/(ae −ai) for nonconstant bifurcations , the parameter regime for nonstationarybifurcation is given by

γ

τeae

ai

+ τi< ais <

1ae

ai

− 1,

1ae

ai

+ 1< ais

As can be shown by simple calculus, there is a threshold

ae/ai =γ + τi

γ − τe, τe < γ

beyond which no nonstationary bifurcations occur, while τe > γ allows non-stationary bifurcations for all ae/ai ≥ 1. Figure 7 illustrates these findings.

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3.3 Evoked stimulus response

Finally, we study Eq. (10) numerically and aim to extract its spatiotemporalproperties for different excitatory spatial ranges ξe. In experimental practice,neural tissue is stimulated by local external input during a finite time interval.We choose the external stimulus to be

I(x, t) = I0 +

{

Ilocal : x0 ≤ x ≤ x0 + ∆x , 0 ≤ t ≤ ∆T0 : otherwise

For the numerical investigations parameters are chosen to be physiologicallyreasonable as c = 1.8, Vr = 3.0 [52] and α1 = α2 = 200Hz [37, 43], i.e. γ = 2.From [22], we obtain spatial ranges re = 20mm, ri ≈ 1mm, i.e. ξi = 0.05,and ξe = 3. The spatial field exhibits 400 discrete elements each of lengthdx = 1mm, while the time period is discretized by 100 time steps each ofduration dt = 0.4ms. In addition, the propagation speed along excitatoryaxonal connections is ve = 8m/s, while the delay corresponding to short-range inhibitory connections is neglected.

The subsequent temporal integration procedure applies an Euler algo-rithm while the spatial integration algorithm obeys

∫ L

0

f(z)dz ≈N

∑

i=1

1

2(f(zi) + f(zi+1)) dx (22)

for an integrand f , here with N = 400 and L = 400mm. Furthermore,periodic boundary conditions are set, yielding

∫ ∞

−∞

K(|x − y|)f(y)dy ≈∫ L

0

K(L/2 − |L/2 − |x− y||)f(y)dy.

Initial values V 0(x, t) = V0 for −L/ve ≤ t ≤ 0 guarantee negligible initialtransients. For synaptic parameters ae = 25, ai = 5 and constant externalstimulus I0 = 0.1, it is V0 = 0.23 and the condition (18) for asymptoticstability holds. The additional local stimulus of strength Ilocal = 5, width∆x = 10mm and duration ∆T = 8ms evokes spatiotemporal activity, whichis shown as space-time plots in Fig. 8 for various excitatory spatial rangesξe = p. During stimulation no activity spreads into the field, while after stim-ulus offset activation travels from the stimulus center. Graphical evaluationsreveal that the activation propagates with the axonal speed ve and exhibitsits maximum at a distance ξe · re from stimulus offset location, i.e. at the

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mean excitatory interaction range. Hence, the expected time for maximumstimulus response is the mean interaction time τe from Eq. (20). These find-ings coincide for all parameters ξe = p in Fig. 8. Though further analyticalexaminations are necessary to validate these first numerical findings, theywould exceed the major aim of the paper and shall be discussed in futurework.

Finally, we examine the activity response evoked by a different stimulus

I(x, t) = I0 +

{

5.0 · cos(k0x) : 0 ≤ t ≤ ∆T0 : otherwise

for the same parameters as above. Figure 9 shows results for (a) k0 =2π/20mm−1 and (b) k0 = 2π/200mm−1. We observe diverse periodic spa-tiotemporal patterns, which reveal the spatial periodicity of the stimulus andthe mean excitatory interaction time.

4 Conclusion

The present work introduces a three-section model for non-local interact-ing neuronal ensembles. It considers synaptic and axonal propagation de-lay effects besides general gamma-distributed excitatory and inhibitory ax-onal connectivities. In a derivation step, a general somatic transfer functionis derived from statistical synaptic and somatic properties. The obtainedfield equation is studied with respect to its stability towards spontaneousfluctuations and external stimuli. Conditions for Turing and nonstationaryinstabilities exhibit only a few parameters. That is, mean excitatory andinhibitory spatial interaction ranges define the sufficient parameter regimeof spontaneous Turing patterns, while mean excitatory and inhibitory in-teraction times determine the necessary parameter regime of spontaneousnonstationary phenomena. Interestingly, these parameters also appear toplay an important role in the evoked response activity caused by local exter-nal stimuli. In the numerical study, evoked activation propagates at axonalpropagation speed and its maximum occurs at a distance from stimulus on-set equal to the mean spatial interaction range after the corresponding meaninteraction time. This rather intuitive finding gives new insights to stimulusresponses in nonlocally interacting fields and, especially, to information pro-cessing in local brain areas during cognitive processing of external stimuli.

15

Future work will discuss evoked responses analytically subjected to diversestationary and nonstationary spatiotemporal stimuli.

Acknowledgements

The authors thank T.Wennekers for valuable hints and for reading the manuscript.

16

References

[1] J. R. Brasic, Hallucinations, Percep. Motor Skills 86 (1998) 851

[2] S. H. Isaacson, J. Carr, A. J. Rowan, Cibroflocacin-induced complex par-tial status epilepticus manifesting as an acute confusional state, Neurol.43 (1993) 1619

[3] P. Tass, Oscillatory cortical activity during visual hallucinations,J. Biol. Phys. 23 (1997) 21

[4] G. B. Ermentrout, J. D. Cowan, A mathematical theory of visual hal-lucination patterns, Biol. Cybern. 34 (1979) 137

[5] V. K. Jirsa, H. Haken, Field theory of electromagnetic brain activity,Phys. Rev. Lett. 7 (5) (1996) 960

[6] C. Uhl (Ed.), Analysis of Neurophysiological Brain Functioning,Springer-Verlag, Berlin, 1999.

[7] A. Hutt, H. Riedel, Analysis and modeling of quasi-stationary multivari-ate time series and their application to middle latency auditory evokedpotentials, Physica D 177 (2003) 203

[8] H. Haken, Principles of Brain Functioning, Springer, Berlin, 1996.

[9] W. J. Freeman, Neurodynamics: An Exploration in Mesoscopic BrainDynamics (Perspectives in Neural Computing), Springer-Verlag, Berlin,2000.

[10] W. J. Freeman, A model for mutual excitation in a neuron populationin olfactory bulb, IEEE Trans. Biomed. Engin. 21 (1974) 350

[11] J. C. Mosher, P. S. Lewis, R. M. Leahy, Multiple dipol modeling andlocalization from spatio-temporal meg-data, IEEE Trans. Biomed. Eng.39 (6) (1992) 541

[12] P. A. Robinson, P. N. Loxley, S. C. O‘Connor, C. J. Rennie, Modalanalysis of corticothalamic dynamics, electroencephalographic spectraand evoked potentials, Phys. Rev. E 63 (2001) 041909.

17

[13] W. Gerstner, Time structure of the activity in neural network models,Phys. Rev. E 51 (1) (1995) 738

[14] P. Bressloff, S. Coombes, Physics of the extended neuron,Int. J. Mod. Phys. B 11 (20) (1997) 2343

[15] B. Ermentrout, Neural networks as spatio-temporal pattern-forming sys-tems, Rep. Prog. Phys. 61 (1998) 353

[16] W. M. Kistler, R. Seitz, J. L. van Hemmen, Modeling collective excita-tions in cortical tissue, Physica D 114 (1998) 273

[17] R. Ben-Yishai, R. L. Bar-Or, H. Sompolinsky, Theory of orientationtuning in visual cortex, Proc. Natl. Acad. Sci. 92 (1995) 3844

[18] T. Wennekers, Dynamic approximation of spatio-temporal receptivefields in nonlinear neural field models, Neural Comp. 14 (2002) 1801

[19] K. J. Friston, Transients, metastability and neuronal dynamics, Neu-roImage 5 (1997) 164

[20] J. Eggert, J. L. van Hemmen, Unifying framework for neuronal dynam-ics, Phys. Rev. E 61 (2) (2000) 1855

[21] H. R. Wilson, J. D. Cowan, Excitatory and inhibitory interactions inlocalized populations of model neurons, Biophys. J. 12 (1972) 1

[22] P. L. Nunez, Neocortical dynamics and human EEG rhythms, OxfordUniversity Press, New York - Oxford, 1995.

[23] A. Hutt, M. Bestehorn, T. Wennekers, Pattern formation in intracorticalneuronal fields, Network: Comput. Neural Syst. 14 (2003) 351

[24] F. M. Atay, A. Hutt, Stability and bifurcations in neural fields with ax-onal delay and general connectivity, Preprint 857, Weierstrass-InstituteBerlin, Germany (2003).

[25] J. C. Eccles, M. Ito, J. Szentagothai, The Cerebellum as a NeuronalMachine, Springer-Verlag, New York, 1967.

[26] P. S. Katz, W. N. Frost, Intrinsic neuromodulation: altering neuronalcircuits from within, Trends Neurosci. 19 (1996) 54

18

[27] W. Rall, Theory of physiological properties of dendrites,Ann. N.Y. Acad. Sci. 96 (1962) 1071

[28] W. Rall, Cable properties of dendrites and effects of synaptic location,in: P. Andersen, J. K. S. Jansen (Eds.), Excitatory Synaptic Mecha-nisms, Universitetsforlaget, Oslo, Norway, 1970, pp. 175

[29] W. J. Freeman, Mass Action in the Nervous System, Academic Press,New York, 1975.

[30] J. Eggert, J. L. van Hemmen, Modeling neuronal assemblies: Theoryand implementation, Neural Comput. 13 (9) (2001) 1923

[31] V. B. Mountcastle, Modality and topographic properties of single neu-rons of cat’s somatic sensory cortex., Neurophysiol. 20 (1957) 408

[32] D. H. Hubel, T. N. Wiesel, Receptive fields of cells in striate cortex ofvery young, visually unexperienced kittens, J. Physiol 26 (1963) 994

[33] K. J. Sanderson, The projection of the visual field to the lateral genic-ulate and medial interlaminar nuclei in the cat, J. Comp. Neurol. 143(1971) 101

[34] B. Katz (Ed.), Nerve, Muscle and Synapse, McGraw-Hill, New York,1966.

[35] W. J. Freeman, Tutorial on neurobiology: from single neurons to brainchaos, Int. J. Bif. Chaos 2 (3) (1992) 451

[36] W. J. Freeman, Characteristics of the synchronization of brain activ-ity imposed by finite conduction velocities of axons, Int. J. Bif. Chaos10 (10) (2000) 2307

[37] F. H. L. da Silva, A. Hoeks, A. Smits, L. H. Zetterberg, Model of brainrhythmic activity, Kybernetik 15 (1974) 24

[38] W. J. Freeman, Nonlinear gain mediating cortical stimulus-response re-lations, Biol. Cybern. 33 (1979) 237

[39] E. Zeidler (Ed.), Teubner-Taschenbuch der Mathematik, Teubner-Verlag, Leipzig, 1996.

19

[40] D. J. Amit, Modeling brain function: The world of attactor neural net-works, Cambridge University Press, Cambridge, 1989.

[41] W. A. Little, The existence of persistent states in the brain, Math. Biosc.19 (1974) 101

[42] J. J. Wright, D. T. J. Liley, Simulation of electrocortical waves, Biol. Cy-bern. 72 (1995) 347

[43] P. A. Robinson, C. J. Rennie, J. J. Wright, Propagation and stability ofwaves of electrical activity in the cerebral cortex, Phys. Rev. E 56 (1)(1997) 826

[44] J. Hopfield, Neurons with graded response have collective computationalproperties like those of two-state neurons, Proc. Nat. Acad. Sci. 81(1984) 3088

[45] H. R. Wilson, J. D. Cowan, A mathematical theory of the functionaldynamics of cortical and thalamic nervous tissue, Kybernetik 13 (1973)55

[46] J. Rinzel, D. Terman, X. J. Wang, B. Ermentrout, Propagating activitypatterns in large-scale inhibitory neuronal networks, Science 279 (1998)1351

[47] G. Tamas, E. H. Buhl, P. Somogyi, Massive autaptic self-innervation ofgabaergic neurons in cat visual cortex, J. Neurosci. 17 (16) (1997) 6352

[48] P. Levy, Sur certains processus stochastique homogenes, Comp. Math.7 (1939) 283

[49] W. Feller, An introduction to probability theory and its applications,Wiley, New York, 1966.

[50] A. Turing, The chemical basis of morphogenesis, Phi-los. Trans. R. Soc. London 327B (1952) 37

[51] V. Castets, E. Dulos, J. Boissonade, P. D. Kepper, Experimental-evidence of a sustained standing turing-type non-equilibrium chemical-pattern, Phys. Rev. Lett. 64 (1990) 2953

20

[52] J. J. Wright, D. T. J. Liley, A millimetric-scale simulation of electrocor-tical wave dynamics based on anatomical estimates of cortical synapticdensity, Biosyst. 63 (2001) 15

21

Figure 1: Excitatory and inhibitory kernels for various parameters. All ker-nels are finite except for Ke(x) for p = 0.5. In case of p > 1, kernel Ke(x)exhibits a maximum far from the origin.

Figure 2: Illustration for the detection of constant stationary solutions ofEq. (12). For I0 < 1.32, there are three solutions, while for I0 > 1.32 thereis only a single one at large values of V0. Simple iteration studies near thesolutions reveal their stability: filled circles represent stable fixed points,while empty circles illustrate unstable fixed points. At the critical valueI = 1.32, there is a saddle node solution (hatched circle) synchronous to astable fixed point at large values of V0, cf. Fig. 3. Here, ae = 10, ai = 5 andc = 1.8, Vr = 3.0 [52].

Figure 3: Bifurcation diagram and nonlinear gain for constant bifurcationsfor ae > ai. Left panel: stability of the stationary state V0 with respect toexternal input I0. For ae − ai > 4/c = 2.22, both stable (solid line) andunstable branches (dashed line) exist, while for ae − ai ≤ 2.22 there is onlya single stable solution. Right panel: the nonlinear gain s with respect tothe constant state V0. The horizontal line s = 1/(ae − ai) separates stablefrom unstable states and determines the critical values of V0. In both panels,c = 1.8, Vr = 3.0.

Figure 4: Parameter regimes of Turing patterns for spatial interaction rangesξi and ξe and various values of ae/ai > 1. The thin solid line denotes ξi = ξe,while filled and empty squares, triangles and diamonds mark different casesdiscussed in Fig. 5 and Fig. 6.

Figure 5: Function K(k) for ξe < ξi. Parameter values are chosen accordingto Fig. 4. Both values of ξe allow a local maximum of K(k) for k > 0 andthus facilitates Turing patterns. Here, ae = 10, ai = 5.

22

Figure 6: The kernel function aeKe(x) − aiKi(x) and the function K(k) forξe ≥ ξi. Parameter values are chosen according to Fig. 4. In case of ξe = 1.0,local excitation-lateral inhibition (ξi = 1.10) yields a local maximum of K,i.e. Turing patterns, while local inhibition-lateral excitation (ξi = 0.90)prohibits Turing patterns. In constrast, ξe = 2.0 exhibits local inhibition-lateral excitation for both values of ξi, while K shows a local maximum forξi = 1.96. Here, ae = 10, ai = 5.

Figure 7: Necessary parameter regime for nonstationary instabilities for di-verse mean excitatory interaction time τe and synaptic delay constant γ.Valid parameters (hatched area) are constrained by the threshold of Eq. (21)plotted as solid line, the threshold of constant bifurcation (dotted line) andthe threshold of asymptotic stability (dashed line).

Figure 8: Spatiotemporal stimulus response to a local stimulation for variousparameters ξe = p. Left column: Space-time plots of field activity, while thegreyscale encodes positive (white) and negative (black) deviations from thestationary state (grey). The activity V has been clipped for |V − V0| > 0.05due to illustrative reasons, i.e. equal-greyscaled areas occur due to clipping.Activity propagation speeds are computed for all p along estimated lineswhich are similar to the dashed line plotted for p = 3. Right column: severaltime samples of spatial activity for the corresponding values of p.

Figure 9: Spatiotemporal stimulus response to a spatially periodic stimulus.The greyscale encoding is the same as in Fig. 8.

23

0 1 2 3 4 5location x

0

0.2

0.4

0.6

0.8

p=0.5p=1.0p=1.5p=2.0

excitatory kernel Ke(x)

0 1 2 3 4 5location x

0

0.2

0.4

0.6

0.8

r=0.7r=1.0r=2.0

inhibitory kernel Ki(x)

Figure 1

24

0 1 2 3 4 5 6 7stationary field V0

0

2

4

I0=0.15I0=1.00

I0=1.32I0=1.80

Figure 2

25

0 0.5 1 1.5 2 2.5input I0

1

2

3

4

5

V0

bifurcation diagram for ae>ai

0 2 4 6 8 10V0

0

0.2

0.4

c/4

nonl

inea

r ga

in s

nonlinear gain ae-ai=5.0 ae-ai=3.0

ae-ai=2.22

ae-ai=1.0

1/(ae-ai)

s=1/(ae-ai)

s=1/(ae-ai)

Figure 3

26

0 0.5 1 1.5 2 2.5mean excitatory spatial range ξe

0

1

2m

ean

inhi

bito

ry s

patia

l ran

ge ξ

i

Turing pattern

ae/ai=5.0

ae/ai=1.0

ae/ai=2.0

Figure 4

27

0 2 4 6wavenumber k

1

2

3

4

5

6K

(k)

ξe=0.5<

ξe=0.7

ξi=0.95

ξi=0.83

ξi=1.66

ξi=1.02

Figure 5

28

1 2 3

-1

0

kern

el a

eKe(x

)-a iK

i(x)

ξi=1.96

ξi=1.66

ξi=1.10

ξi=0.90

1 2 3

-3

-2

-1

0

K(k

)

ξe=2.0

<

ξe=1.0

ξe=1.0

ξe=2.0

x

k

Figure 6

29

1 2 3 4 5 6ae/ai

0

1

2

a is

1 2 3 4 5 6ae/ai

0

1

2

a is

1 2 3 4 5 6ae/ai

0

1

2

a is

τe< γ τe> γ

nonstationary instability possible

nonstationary instability possible

Figure 7

30

0.2

0.3

0.4

0.2

0.3

0.4

field

resp

onse

0.2

0.3

0.4

0 100 200 300 400space [mm]

0.2

0.3

0.4

t=9.2ms

t=17.2ms

t=30.0ms

t=6.8ms40

200 400

20

time

[ms]

space [mm]

0.3

0.4

0.5

0.3

0.4

0.5

field

resp

onse

0.3

0.4

0.5

0 100 200 300 400space [mm]

0.3

0.4

0.5

t=11.2ms

t=12.8ms

t=14.4ms

t=16.0ms

40

200 400

20

time

[ms]

space [mm]

0.2

0.3

0.4

0.2

0.3

0.4

field

resp

onse

0.2

0.3

0.4

0 100 200 300 400space [mm]

0.2

0.3

0.4

t=6.4ms

t=17.2ms

t=26.4ms

t=36.4ms

40

200 400

20

time

[ms]

space [mm]

0.3

0.4

0.3

0.4

field

resp

onse

0.3

0.4

0 100 200 300 400space [mm]

0.3

0.4

t=11.6ms

t=15.6ms

t=17.2ms

t=19.6ms

40

200 400

20

time

[ms]

space [mm]

p=1

p=3

p=6

p=9

prop

agat

ion

dire

ctio

n

Figure 8

31

400

40

20

40

20

time

[ms] 0

0

space [mm]200

(b)

(a)

Figure 9

32