V. K. Jirsa and J. A. S. Kelso- Spatiotemporal pattern formation in neural systems with heterogeneous connection topologies

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Spatiotemporal pattern formation in neural systems with heterogeneous connection topologies

V. K. Jirsa* and J. A. S. KelsoCenter for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, Florida 33431

Received 25 June 2000

Biological systems like the human cortex show homogeneous connectivity, with additional strongly hetero-

geneous projections from one area to another. Here we report how such a dynamic system performs a macro-

scopically coherent pattern formation. The connection topology is used systematically as a control parameter toguide the neural system through a series of phase transitions. We discuss the example of a two-point connec-

tion, and its destabilization mechanism.

PACS numbers: 87.18.Sn, 47.54.r, 82.40.Ck, 84.35.i

A sheet composed of neurons or neural ensembles pro-vides a medium for spatiotemporal pattern formation of neu-ral activity. In contrast to typical pattern formation in physi-cal or chemical systems 1,2, a neural system has a spatiallyvariant connection topology in which a cortical area is notonly connected to its nearest neighbors, but also has projec-tions to distant areas. By these means the nervous system

accomplishes a directed transfer of activity within a continu-ous sheet in which it would spread out uniformly otherwise.Such projections may not only serve to organize local dy-namics within cortical areas such as synchronization of localrhythms, but also contribute to the macroscopic organizationof neural activity or global dynamics. Neurobiological theo-ries of memory are severely weakened by their inability tospecify how changes in synaptic strength, the presumedagency in encoding, alter complex network operations, thepresumed substrates of memory expression 3. We addressthis problem for the dynamics on the network level such thatany local dynamics is represented by a scalar activity result-ing in a neural ensemble theory 5 8. Local changes in

synaptic weights, however, will alter the connectivity of theneural system, and by these means its global network dy-namics. Here we wish to study the properties and mecha-nisms involved in the spatiotemporal neural pattern forma-tion when the networks connection topology is varied. Asan example we discuss a two-point connection embedded ina spatially continuous, homogeneously connected medium,and explain its underlying destabilization mechanism.

Following Jirsa and Haken 6, we define the spatiotem-poral dynamics of a scalar neural field (x ,t) with space xR

n and time tR as a nonlinear retarded integral equa-tion of the form

x ,tAdX fx,XSX,TIX,T , 1

where f(x ,X) describes a general connectivity function, andS a nonlinear function of at a space point X and a timepoint TtxX/v, delayed by the propagation time overthe distance xX . A denotes the surface area of the me-dium, and v the constant signal velocity. I(x, t) is the inputto the field (x ,t). Variations of this type of integral equa-

tion were widely used in theoretical neuroscience 4 8 todescribe the dynamics of neural activity. However, in allthese cases, as well as generally in physical and chemicalsystems, translational variance f(x ,X)fh( xX) was em-ployed. Here f(x,X) will be typically decomposed into ho-mogeneous contributions fh( xX) and heterogeneous pro-

jections fi(x ,X). We decompose the field (x, t) into spatial

modes gn(x) and complex time dependent amplitudes n( t)such as (x ,t)n

g n(x)n(t). The choice of the spa-

tial basis functions will depend on the surface A and itsboundary conditions, but also on practical considerationsabout the type of connectivity f(x ,X) and inputs I to thesystem. We choose a polynomial representation of the non-linear function S in Eq. 1 such as S(X,T)a 0a1(X,T)a 2

2(X,T)m0

a mm(X,T), where

amR, and I(X,T) is not considered no restriction of gen-erality. By projection of Eq. 1 onto a spatial basis function

gq(x), and restricting its dimension N to be finite, we obtaina set of N coupled integral equations

tv

t

d0 t1 t

2 t

v

t

d0 tLt tNt t , 2

where vector notation has been used: (t)q(t)

T, 0(t)q(t) T, 1(t)

qn (t) , . . . . Formally Lt and Nt represent the lin-

ear and nonlinear temporal evolution operators, and q(t

) are tensor matrices composed of the spatial modesgq(x) and the connectivity function f(x ,X). A linear stabil-ity analysis of Eq. 2 yields the eigenvalue problem

detetL teI0, 3

where I is the identity matrix. The entire complexity of theconnection topology is contained in the tensor matrix 1(t

), with L tv t

d1(t), and the spatial basis

functions. The elements of the tensor matrix 1( t) areqn ( t)a 1(

), where*Email address: [email protected]

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A

dx gqxfx ,xv tg nxv t . 4

In a spatially invariant medium, the connectivity and thecontribution of the spatial modes factorize, resulting in a

separation of spatial and temporal constraints. However, thisis not given when projecting pathways fi(x,X) are intro-duced into the neural sheet. Then truly spatiotemporal con-straints for the linear stability exist such that Cqn (x0)

Adx gq(x)g n(xx0)fi(x ,xx0) represents a measurewith x 0v(t) for the similarity between the spatialmodes and the heterogeneous pathways. The necessary con-dition for destabilization of a pattern in Eq. 3 is Cqn (x0)0, such that only spatial modes may be destabilized thatare sufficiently similar to the connectivity structure. The suf-ficient condition for destabilization is Re()0 in Eq. 3.

The most elementary heterogeneous connection is a fiberconnecting area x1 to area x2, because it is not only the

simplest heterogeneity, but also the heterogenous connectiv-ity matrix fi(x ,xx0) may be reconstructed by the sum ofsuch two-point connections between areas x i and xj such asfi(x ,xx0) i,jfi j(xx i)(xxjx 0), with the synap-

tic weight fi j as a parameter. Typically such projections arebilateral 9, but not necessarily symmetric: fi j fji . Thenthe similarity measure Cqn (x0) can be written as

Cqn x 0i,j

fi jgqx ignx ix0x ixjx 0 . 5Realistically, the neocortex consists of many interareal two-point connections, leading to hierarchical signal processingshemes in a continuous neural sheet, of which the bestknown are the visual areas 9. The brain provides a range ofvarying connectivity structures allowing flexibility for thesame functions, generally assumed to be represented by net-work operations. Such variablility is found in the reorgani-zation of neural function after brain injuries, which reflectsmajor changes in connectivity 10. However, the brain alsoprovides mechanisms to alter these connectivities, such aslong term potentiation 3, which vary the location of theterminals of pathways. An extraordinary, though pathologi-cal, example of stable and coherent global pattern formationin the brain network is found in epilepsies, which are often

treated by changing its connectivity, i.e., by cutting the link-ing pathways of the hemispheres, the corpus callossum. Inthe following we wish to demonstrate how changes of thelocations of a single pathway systematically control the pat-tern formation of the entire network. We embed a heteroge-neous two-point connection between locations x 1 and x 2 in aone-dimensional homogeneous medium. Its connectivityfunction f(x ,X) shall be given by f(x,X)f( xX)f12(x,X)f21(x,X), where f12 is the link from x 2 to x1,and f21 the link in the opposite direction, as illustrated in Fig.1. The distance between the inhomogeneous contributions ofconnectivity is dxX , which serves as our control pa-rameter.

The dynamics of our example is given by Eq. 1, inwhich (x, t)AdX(X, t) is subtracted in the argumentof the sigmoid S to reduce spatially uniform saturation ef-fects 6,11. Periodic boundaries are imposed. The connec-

FIG. 1. The homogeneous connection topology is illustrated

within a one-dimensional continuous medium whose activity is de-

scribed by (x ,t). A projection from x1 to x2 introduces a hetero-

geneity into the connectivity.

FIG. 2. In the top row, the

similarity measure Cmm (x0) is

plotted for the spatial patterns m

1 and 2, in dependence on x0and d. The plus signs show the

maxima of similarity, and identify

the regions in which a destabiliza-

tion of an activity pattern may oc-

cur. In the bottom row, the eigen-

value of a pattern, as a solution of

Eq. 6, is obtained graphically by

the minima in the contour plots

. As the synaptic strength

f12f21 is increased, the activity

pattern is destabilized when its

real part of the eigenvalue

crosses the dotted zero line.

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tivities are specified as f( xX)(2)1 expxX/,f12(x ,X)1/2f12(xx1)(Xx2) and f21(x ,X)

1/2f21(xx2)(Xx1), where , f12 , and f21 are con-

stant parameters. We choose the spatial basis system to be

spanned by the trigonometric functions sin nkx and cos mkx,

with n,m Z and k2/L , where L is the length of the

one-dimensional closed loop. x10 is not varied, thereby

resulting in a pinning of the spatial modes around x0. The

nonlinear function S in Eq. 1 is assumed to be sigmoidal

6. We study the stability of the origin 00, and expand

S around its deflection point, Snn4/33n 3 . We

consider a spatial basis function g m(x)cos mkx to second

order in m, and truncate the expansion of the sigmoid after

the third order to study small amplitude dynamics. The simi-

larity measure Cmm (x 0) is determined after Eq. 5, and plot-

ted in dependence on x0 and d for the first two spatial pat-

terns m1 and 2 in the top row of Fig. 2. The plus signs

show the maxima of the spatiotemporal similarity, and iden-tify the regions in which a destabilization of a pattern may

occur. Following the eigenvalue problem in Eq. 3, we ob-

tain a transcendental equation for the linear stability of the

mth spatial mode,

imm220 imm0

2m2k2v2 ,

6

1d1e(imm)d/v imm000,

where d12/L(f

12f

21)cos mkx

1cos mkx

2. The real part

m of its eigenvalue and its frequency can be determinedgraphically from Eq. 6 as functions of the control param-eter d, and are shown in the bottom row of Fig. 2 for increas-ing synaptic strength f12f21 .

The destabilization of a particular state m depends on d1and d. For d1 ,d0, a purely homogeneous system is ob-tained in which the origin is the only stable state for suffi-ciently small . The introduction of a heterogeneous projec-tion fi j(x,X) may cause a desynchronization of theconnected areas, and thus a destabilization of the respectivespatial mode, resulting in a phase transition. As seen in Fig.2 bottom row, the complex conjugate pair of eigenvaluescross the dotted zero line, and their real parts become posi-

tive, for increasing synaptic strength f12f21 , causing aHopf bifurcation.

To investigate this dynamics numerically, we prepare thesystem in a well-defined state, and change the connectiontopology by decreasing the control parameter dx 1x2 .

FIG. 3. The transition behavior of the system in Eq. 1 is plotted in a space-time diagram horizontal time, vertical space. The originally

stable spatiotemporal pattern is destabilized by moving the location of the heterogeneous projection from x2B to x2C. The units used

are the number of time steps and the arbitrary amplitude, respectively.

FIG. 4. The dominant spatial

patterns for varying d are dis-

played in a spatiotemporal bifur-

cation diagram. The same amount

of change in d, i.e., in the connec-

tivity, can have entirely different

effects depending on where thesechanges occur. For instance, an

increase of d from 15 to 20 units

does not show any qualitative

changes in the dynamics, but an

increase of d from 22 to 27 units

causes a phase transition. Directly

above, the amplitude full circles

and temporal frequency donuts

of the dominating pattern are plot-

ted over the distance d. In the top

right corner, the power spectra of

the dominating patterns are plot-

ted for all values of d.

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Initially the system is prepared in a coherent oscillatory sta-tionary state. Then the connectivity is changed by movingthe terminal of the inhomogeneity at x2B to a new locationx 2C, whereas the other terminal stays constant at x1A .The change in the connection topology destabilizes the initialstationary dynamics, and the system undergoes a transition toa new stationary state. Without the inhomogeneous connec-tions the zero-activity state is stationary and stable. The pa-

rameters used for the integration of Eq. 1 are v3, , a2.5, 0.7, 120.1, and f12f211. Thetime step is dt0.02, the space is divided into 100 units seeFig. 3.

We investigate the dynamics in more detail by varying thecontrol parameter d systematically from 0 to its maximalvalue d. Due to the periodic boundary conditions, avalue d/2 reduces the relative distance between the inho-mogeneities. The inhomogeneity at x 1 is kept constant, andx 2x 1d is varied. For each distance d the system dynam-ics becomes stationary, and then the spatial pattern at maxi-mum amplitude is extracted and plotted over d in a bifurca-

tion plot. These patterns are normalized with respect to the

largest amplitude. Under variation ofd the system undergoesa series of spatiotemporal bifurcations. Note that these pat-terns are not harmonic due to the inhomogeneous connectiontopology. Shifts in the frequency spectra plotted on the rightfor all distances d) are observed, even though the spatialpattern remains qualitatively the same. Hysteresis effects arefound, resulting in an asymmetry of the bifurcation path be-

low and above d /2. Above the bifurcation plot the cor-responding amplitude full circles of the location x 1 and itstemporal frequency are plotted donuts over d see Fig. 4.

We described macroscopic coherent pattern formation ina spatially continuous neural system system with a heteroge-neous connection topology. Local changes of connectingpathways may guide the neural system through a series ofglobal spatiotemporal bifurcations.

This research was supported by NIMH and The HumanFrontiers Science Project. V.K.J. wishes to thank HermannHaken for interesting and helpful discussions.

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V. K. Jirsa and J. A. S. Kelso- Spatiotemporal pattern formation in neural systems with heterogeneous connection topologies