Simulation of Cortical Functional Area FormationBased on Synaptic Plasticity

Li Zhou, Liu, Di,

Department of Systems Science, Beijing Normal University, Beijing,100875, ChinaCenter for Complexity Research, Beijing Normal University, Beijing, 100875, China

* Email:Wangdh@bnu.edu.cn

Abstract- In this paper, we provide a model to simulatethe cortical functional area formation process based on Spike-timing Dependent synaptic Plasticity (STDP). We put neuronson 2D spherical lattices. The membrane voltage of neuronfollows FitzHugh-Nagumo (FHN) model. The strength of synapsechanges according to the activity of pre and post synapticneurons. Some certain neurons are stimulated by external signals,and others are stimulated by inner noise signal. There aredense and uniform connections between neurons at first, however,spatial pattern of connectivity and cluster of connection, whichindicate functional area, emerge at last.

I. INTRODUCTIONAnatomy and experiments told us that there are many func-

tional areas in the cortex of brain, such as motor area, visualarea, auditory area and so on [I]-[3]. These functional areasare important material structures to help brain accomplish itsfunction. For example, if one's visual cortex was damaged forsome cause, he can see nothings even that the light reflectedby things entered his eyes, transformed into spike train, andthe signals were transferred into cortex. It is obvious thatfunctional area is not only determined by gene but also the fol-lowing development process. It is affirmative that experienceis an important factor for the formation of functional area. Ac-tually, even that cortex could develop into many different areaswithout experience, those areas are only morphological areasand have not any function [4]. The story of Wolf Child Kamalahad illustrated this point vividly. So, the formation of corticalfunctional areas depends on one's experience, furthermore,the functional areas are continuously modified by experience[5]-[7]. Up to now, these understandings are acquired fromexperiments and some casual cases. We had not completelyunderstand why and how these functional areas emerged incortex. Fortunately, computational neuroscience and complexnetwork research shed some lights on this problem. Fromthe point of view of complex network research [8]-[10],the cortex could be looked as a complex network embeddedin 2D spherical surface, neuron as vertex, synapse as edge,synaptic strength as weight. Neuron's dynamics, which followsFitzHugh-Nagumo(FHN) model or Hodgkin-Huxley equation,are coupled through the synapse connection [II]. Therefore,there is a coupled dynamic system on the complex network.The dynamic system interacts with the complex network.Dynamic system is constrained by the structure of the network.At the same time, the structure of complex network willchange along with the evolution of dynamic system, especially,

the synaptic strength evolves according to the behavior ofpre and post synaptic neurons. As a result of interactionbetween dynamic system and network structure, it will emergesome certain spatial patterns of synaptic connections on themacroscopic level. The structure with these spatial patterns canaccomplish some function, so we can say that the functionalarea emerges from a morphological dense connectivity cortex.

In the following section, we will propose our model toobserve how the complex network interacts with dynamicsystem on itself. Computer simulation and results are presentedin section III. Finally, we give a conclusion and discussion ofour work.

II. THE MODELA. Initial Connection of Complex Network

In the brain of infant prior to any experience, there are alarge of redundant connections, and these redundant connec-tions will be shaped in its later life experience. So, we startfrom a dense connection complex network which mimics theinitial dense anatomic cortex without experience.The complex network is embedded in 2D lattice (x, y) on

spherical surface. If two vertices are nearest neighbors, thereare two edges connected from each one to the other one. Theprobability of vertices (x1, yi) connected to its non-nearestneighbor other vertex (x2, Y2) is the function of distance:

P(x1,y1)-*(X2,y2) (lxi-x2 1 ± lY2Dy (1)

where p is the parameter describing the denseness of con-nection. a is the parameter indicating the length of axon ofneuron. If a is smaller, the axon is longer. Now, we hope thatspatial pattern emerges from this dense connected complexnetwork.

B. Dynamics of VertexFor computational reasons, we take the FHN model for each

vertex [15]:

E'Vi = Ii,ion (t) + Ii,syn (t) + Ii,ext (t)i = vi -wi-b

Ii,ion = vi(vi -a)(1- vi) -w

(2)

(3)

(4)

where vi is a across membrane voltage-like fast variable,e << Iwi is a slow recovery variable. I is the ionic current

0-7803-9422-4/05/$20.00 (©2005 IEEE1921

through membrane and Ii,ex,t is the external current stimulus.Ii,syn is the synaptical current and equals to the sum of currentfrom excitatory or inhibitory pre-synaptic neurons:

Ii,syn(t) = (gij(t)( -v- i(t)) + 9ijj(t)(V-i(t))) (5)jis

where V(V) is the excitatory(inhibitory) synaptical reversalpotential and gi,j(t)(b,jj(t)) is the excitatory(or inhibitory)synaptical conductance from neuron j to i. If the synapse ofneuron j to i is excitatory, gi,j (t) = 0, or vice versa.

C. Synaptical PlasticityPlasticity is an important feature of synapse. Synaptical

conductance will increase or decrease according to the activityof pre- and post-synaptical neurons. In this paper, we adoptthe spike-timing dependent synaptic plasticity mechanism.

For excitatory synapse, synaptical strength will increaseif presynaptical neuron spikes before postsynaptical neuron,namely, gi,j -+ gi,j + Ag(6t), where L\g(6t) is related withtime difference between pre and postsynaptical spike 6t =tpost - tpre (in this paper, neuron i spikes when vi > 0.7). Itis modified by the following equation [12]:

Ale- t > Oi\g(6t) = O

St*A2e r2

St = OSt < 0

(6)

where the parameters A1, A2 determine the maximum amountof synaptic modification, and rT, T2 are determined by thetemporal window of the spike intervals. It has been shownexperimentally that in most situations, A1 > A2,T1 < T2As for inhibitory synapse, the experiments had not shown us

a coincident result [13]. So, we assume that inhibitory synapticcoupling strengths vary at constant if pre and postsynapticalneuron spike during the time window, namely, ij -9 i,j +Ag, and Ag is a constant.

Besides modification by the activity of pre and postsynapti-cal neurons, the synaptical strength will evolve along time asfollowing:

gsyngi,j = -9i,j (7)and

T8syngi,j = -9i,j (8)Above two equations about synaptical strength mean that

the synapse will degenerate without using. For simulation, wetake the parameters as the following values as in [15]: p =1, a = 1, A1 = 0.01,A2 = 0.006,i-1 = 1.0, r2 = 2.0, a =0.5, b = 0.12, = 0.05, V = 0.7, V 0.0, Tsyn = ;y= 0.2,and A\g = 0.01.

III. SIMULATION AND RESULTSWe start from a dense connected N xN lattices on spherical

surface. Because of connection rule, the degree of initialnetwork obeys normal like distribution(Fig. 1), and the weightof each edge is 0.1. During simulation, we stimulate someneurons by external dc-current, Iext = 0.4, which is supra-threshold stimulus for spontaneous generation of spike. These

stimuli mimic the signals from sense organ. Other neurons arestimulated by inner noise, which obey the uniform distributionin (-0.9, 0.9), at each simulation step. After thousands periodsof relaxation by STDP, some population of synaptical strengthare decreased near to zero and, therefore, the synapse becomeinvalid and be cut off from the network, while the other popu-lation of synaptical strength maintain some certain value. As aresult, the morphological structure is changed by the dynamicsystem and some functional structure emerges from originalmorphological connection. In Fig.1, initial 7 x 7 network aredensely connected(Fig.1-a), whereas, redundant connectionshave been cut off after periods of simulation(Fig.1-b). Afterredundant edge being cut off, the degree distribution of thenetwork changes, which is shown in Fig.2.

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Fig. 1. Intuition illustration of cortical functional area formation. Circlesdenote neuron, green lines denote- synapse. We split the spherical surface intoflat plan in pink rectangle with periodical boundary. Those circles out of thepink rectangle is mirror image in the rectangle. Four corner points of thepink rectangle are connected to the same point in spherical surface. a, initialdense connection. b, the redundant connections have been cut off by dynamicsystem and there is a cluster of connections.

For quantitative illustration the spatial pattern of connection,

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3e- rB * Finalstateo InitMal state

0027- 0

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0 moo

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0- > °° n"'CiS,0--

0 10 20 30 40 00 00 70 80Degree

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0.Fig. 2. Change of the degree distribution. Because of cutting off theredundant edges, the degree distribution shifts from the right to the left. Initialdistribution is symmetry about mean value, whereas, the final distribution isobviously skewed to the left.

O 100 200 31M0 400 501

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Fig. 4. Coherent oscillation of clustering neuron. a. The average v of 49neurons in the cluster. There are 28 neuron staying at steady state( but not atrest state), other neurons are oscillating. In this figure, We plot the average

vi(t) of these 49 neurons,V(t) - 49 V which is defined like[14] had done. b, this figure is the power distribution of V(t). It show thatV(t) is combination of several dominant oscillation and many other 1 noise.

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Fig. 3. Neuron strength spatial distribution. There are 2 x 2 neurons inthe spherical surface. Nine neurons at the center of graph are stimulated byexternal dc-current Ie_t = 0.4, and others are stimulated by inner noise.Along with those redundant edges being cut off by STDP, spatial patternof neuron strength emerges from random initial state. a. Although eachsynapse has equal strength, the neuron strength is random because of randomconnection. b, The final neuron strength distribution with spatial pattern after20 thousands periods simulation. The clustering connection emerge near tothe stimulated neurons, which is consistent with the facts that was observedin anatomy

we investigate the synaptical strength, which is defined by thesum of all its synaptical strength. This definition is similarto vertical weight by the sum of connected edge weight incomplex network. Neuron strength, as a simple variable, couldreveal the importance of neuron in the signal transferring andinformation process at some certain extent, therefore, it isrelated with the function of brain and we could investigatefunctional area by exploring neuron strength spatial distribu-tion. In Fig.3, the initial and final spatial distribution of neuronstrength are shown as contour graph. The figures demonstratethat clustering connection and spatial pattern emerge frominitial random dense connection. In our opinion, it is functionalarea emergence from a morphological cortex. We also findthat the cluster is not far away from the neuron stimulated byexternal signal.We also investigate that the collective behavior of the final

big cluster in Fig.3. We select one neuron at the center ofcluster shaped by STDP and stimulate it by external dc-currentI, = 0.4 during time 50 - 250 mesec. The initial value v ofall neurons is set at 0, and w is a random number between 0and 1. We find that the cluster could reach coherent oscillation,which is an important feature that brain exhibits during itprocess information and accomplish other tasks [14].

IV. DIsCusSION AND CONCLUSION

In this work, we present a model to simulate the functionalarea emergence from the morphological dense connectivitycortex. It is STDP mechanism that cuts off the synapse faraway from the neuron which receive external stimulus. Along

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with synapse being cut off, the cluster near to stimulatedneuron emerges. This process mimics the evolution of sensoryfunctional area. we believe that this path could be appliedto motor area. In this paper, only one area's neurons arestimulated by external signals, and one cluster of connectionforms. If there are two area's neuron being stimulated at thesame time, two clusters may be formed, so much as the thirdcluster emerges. We believe that those higher functional areas,which process language, emotion, and cognition, form by thesame way.

ACKNOWLEDGMENTThis work is supported by National Natural Science Foun-

dation of China Grant 70471080, 60374010.

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