Study on Simulator for Biological Neural NetworksHoujin Chen Baozong YuanSchool of Electrical InformationNorthern Jiaotong UniversityBeijing 100044, P. R. CHINA

E-mail: hjchen@public.bta.net.cn

Abstract-Neurons are connected with synapses to formneural networks. Neural networks can process informationand generate a specific pattern of electrical activity. It isdifficult to understand the neurons and neural networksthrough limited experimental data. Therefore, informaticsand biology were integrated, and experimental data andmathematical models were combined. One simulator forbiological neural networks was studied and realized to verifythe known and explore the unknown so as to disclose neuralbiophysical and biochemical structure. It is very critical tobiology. In this paper, novel system architecture was designed,efficient data representation was provided, and accuratecomputational method was described. The simulator wasrealized in platform-independent JAVA language and hasbeen widely applied in the world.Key words: Biological Neural network; Computer

simulation; Data representation; Computational method.

I. INTRODUCTION

Neurons are specific cells. Neural networks composedof them through the synaptic connectivity can processinformation and generate a specific pattern of electricalactivity. In biology it is very significant to analyze andcognize its intrinsic biophysical and biochemical structure,the process of information generation and transmission,and its exterior response properties. The specific role thatany one process plays in the overall behavior of thenetwork can be difficult to access due to such factors asinteracting nonlinear feedback loops and inaccessibility ofthe process for experimental manipulation. One way tosolve the problem is to incorporate informatics into biology,and to establish mathematical modeling according toknown data and simulate the biophysical and biochemicalproperties. In this paper one simulator for biological neuralnetworks was studied and realized. It enables biologists toexplore processes that underlie neural functions. And it isnecessary tool not only for theoreticians but also forexperimentalists who wish to explore the properties ofcomplicated neural systems.

Baxter, D. A. Byrne, J. H.Medical School

University of Texas at HoustonTexas 77225, U.S.A.

E-mail: Douglas.Baxterguth.tmc.edu

11. LAYER-BASED SYSTEMARCHITECTURE

As the complexity of individual neural networks, thissimulator should be flexible and easy to add or delete somemodule. In order to examine and control the individual ormultiple variables and parameters in neural network, thesimulator is designed to be open, controllable andmeasurable. To achieve those properties, layer-basedsystem architecture (LBSA) shown in Figure 1 is designedand individual layer is independent.

Fig.1. Layer-based System Architecture

III. ELECTRICAL CIRCUIT EQUIVALENT

Though the functional organization of the nervoussystem differs profoundly between some animals andhumans, the organization of individual neuron differs less.The main parts of one neuron are the dendrites forreceiving electrical signals from other neurons; the soma orcell body for summing electrical potentials from manydendrites and also containing the nucleus; and the axon forconducting electrical signals and transmitting them to other

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cells.The significant variable for information transmission in

a neuron is the electrical potential across the membrane ofthe cell's axon. It is determined by the intracellular andextracellular concentrations of three single-element ions:potassium (K+), sodium (Na+), and chloride (Cl-), alongwith some compound ions. The ionic currents aregenerated due to that potential. Neural network iscomposed of neurons connected with synapses. There aretwo kinds of synapses, electrical synapses and chemicalsynapses.

The electrical circuit equivalent for a neuron ismodeled according to its biophysical & biochemicalstructure and its external electrical properties. Thiselectrical circuit primarily describes the electricalproperties ofmembrane potential for a neuron. The neuronhas a membrane potential V and a membrane capacitanceCm. Each ionic current is determined by the membranepotential and its corresponding ionic conductance Gio. AsGi.0 is dependent to membrane potential, so it isvoltage-dependent and with nonlinear property. Anelectrical circuit equivalent for a neuron is illustrated inFigure 2.

Fig. 2. Electrical Circuit Equivalent of a Neuron

Where Ei., is the equilibrium potential; ECS is the synapticreversal potential; Vr-Vn represent the membrane potentialsof the coupled cells; I,i is the external current injection.

IV. MATHEMATICAL MODELS

From the electrical circuit equivalent of a neuron, itsmathematical model is established according to 'circuittheory. -The activity of a neuron is described by itsmembrane potential. It is defined by the followingdifferential equation (3-1).

dV1dt

m n n pIstim-IIiony - Eiesik ICEISikl

j=l k=l k=l 1=1

CMi(3-1)

where i, j, k, I are the indices for the neuron, the ioniccurrents, the electrical synaptic currents, and the chemicalsynaptic currents, respectively; Cm is the membranecapacitance; n is the total number of neurons in theneural network.

Currents Ii. ',ICS are determined by the mathematicalmodels of their conductance Gion, Ges, GCS. One of theversatile features of this simulator is to allow the user tochoose more than one way to model a given property of anetwork, so that it reflects the differences of neurons. Onlypart of mathematical models are presented in this paper.

A. Ionic Current Iion

Based on circuit equivalent, the calculation for all ioniccurrents is expressed as equation 3-2.

iion,j = Gioni (Vi -Eon# ) (3-2)

For the calculation of voltage-dependent ionicconductance Gi., five mathematical models are made inthe following equations to account for the ionicconductance specialty of individual neurons.

Gion y =ion* A* Bi f [REGq]q-1

s~ ~~Gio.i 4= in J7*J* i If[RREG q]q=lGiny=gion y. M*hi n f[REG]Gioni =goy mi fE q]

q=lGiony= g-ion y Al f [REG]

= q=l q]q=1

(3-3)

(34)

(3-5)

(3-6)

(3-7)

Where is the maximal conductance, A and B are thedimensionless voltage-dependent activation andinactivation terms using a time constant method. m and hare the equivalent symbols for A and B when using a rateconstant method; f [REG] represents the conductancemodulation by a regulator.

B. Electrical Synaptic Current Ies

The currents due to electrical synapses are calculated

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iex

by the equation 3-8

esik = eesik .(Vi -Vk), (3-8)

where ke is the maximal conductance, Vi and Vk are

the membrane potentials of presynaptic neuron andpostsynaptic neuron, respectively.

C. Chemical Synaptic Current Ic,The currents due to chemical synapses are described

by one of the following equations:

(3-9)Cs= cs * (A t ).(V - Ecs )I< = kcs *f(Av,t)-(V-Ecs).

saves a amount of memory space and facilitates thesimulation, because file shared results in its results shared.For example, when calculating ionic currents,voltage-dependent inactivation term B is needed. Here B isdesigned as a modular file. If there are 20 neurons in anetwork, and there are 5 ionic currents for each neuron,then total number of ionic currents is 100. If 40 ioniccurrents share one B file, another 60 ionic currents shareanother B file, then only two B files need generating andcalculating, it is unnecessary to generate and calculate 100B files, which greatly save system resources and improvecomputation efficiency. In the simulator, eighteen modularfiles are defined and their graphical interfaces are designedfor file edit and management.

(3-10)

Where is the maximal conductance, f(At) andf(Avt) are functions of time-dependent activation andvoltage- and time-dependent activation, respectively.

V. OBJECT-ORIENTED DATA REPRESENTATION

In one network with n neurons, there are m ioniccurrents, n electrical synaptic currents and nxp chemicalsynaptic currents. Its total number of conductances isnx(m+n+nxp). As each conductance is related to onemathematical model, and each model includes manyparameters and initial conditions, so each simulation willdeals with a great amount of data. Without efficient datarepresentation, it is difficult for simulator to control andrun the simulation. One scheme is to put all data into asingle file. It has three shortcomings: (1) it is difficult foroperators to generate all data required. As the type, thenumber and the order of parameters are different for aspecific simulation, it is impossible to design one graphicalinterface to edit those parameters; (2) it is hard foroperators to modify data in a great amount of parameters.(3) data cannot be shared in a simulation and eachsimulation needs to renew its data.

In order to overcome those shortcomings,object-oriented data representation (OODR) is provided. Inthe data representation, all parameters are decomposed of aseries of modular files according to their mathematicalmodels. Each file corresponds to a specific equation ormodel. They are organized in hierarchical way according totheir dependency. As each modular file is with fixed format,it is easy to generate, debug, and edit in graphicalinterfaces. Moreover, it can be shared and reusable, which

VI. HYBRID NUMERICAL COMPUTATION METHOD

From neuron equivalent circuit, the mathematicalmodels for the neural network are nonlinear and timevariant. They are described by a set of differentialequations due to the feedbacks and thresholds during thetime process. The best way to simulate the neural networkis to compute it by numerical integrations. In order to gethigh calculation accuracy, the step size of numericalintegrations must be small enough. As the neural networkis large and complicated, so its computation complexity ishuge. Therefore, the step size is critical to accuracy andefficiency. There are several numerical integrationschemes.One numerical integration is with fixed step size. It is

required that integration step size is kept fixed during thecomputation process. The commonly used is Euler method.Euler method is the simplest and the fastest for numericalintegration. However, Euler method is not recommendedfor practical use among them, because it is inaccuracy andonly with first order accuracy. Another numericalintegration is with adaptive step size. It requires the stepsize is frequently modified. Theoretically it can achievesome predetermined accuracy. However, it is not practicalfor neural networks with a great number of variablesdependent on one another. For example, if there are twovariables vl(t) and v2(t), variables v1(t) is dependent on v2(t).When the adaptive step size was respectively applied tovariables v1(t) and v2(t), it is not possible to keep themcomputed in the identical time sequences. However, if vl(t)at time ti needs to be calculated, v2(t) at the same time tshould be calculated in advance due to the dependency.Therefore, when there exists the dependency among

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variables, adaptive step size is not practical.To improve the computation efficiency, hybrid

numerical computation method (HNCM) was presented tomeet the special requirements of neural networks. Itadopted different integration methods for differentvariables, but with the same step size for all variables. Asthe data representation is object-oriented, so thecomputation is also performed object by object and eachobject took corresponding integration method according tothe characteristics of its mathematical model. In HNCM,for the stable and slow objects, Euler integration methodwas applied; for spike objects, two-order or four-orderRunge-Kutta (R-K) method was applied. As differentobjects were calculated with different integration methodsaccording to their model characteristics, so computationaccuracy and efficiency were achieved for the simulator.

VII. RESULTS AND APPLICATIONS

One network with three neurons was simulated (Fig. 3)and comparison of computation accuracy and efficiencywas illustrated in table 1.

80.000 V&. Nsw)

40000300.000

.0.000:-20im-43000:-60M)0

0.6 03*0 O6bo 2.400 3.4bo0 40tiraelivrl

80.000 VIC .J-(svr)60.000

40.00040.000-601)0

o.0O bo 4)0o t 2400 3.b0 40t{mw3v}

efficiency (about 60% time complexity). Obviously, thebigger the neural network, the greater the computationcomplexity for HNCM.

Table I

COMPUTATION ACCURACYAND EFFICIENCY

Computation Method Euler R-K Hybrid

Accuracy (step size) 50 (ns) 125 (ns) 125 (ns)

Efficiency (time) 5.685(s) 5.268(s) i 3.436(s)

The simulator for neural networks was implemented inplatform independent language JAVA. It is widespread useand is benefiting a large segment of scientific community.The simulator is freely available from web site(http://snnap.uth.tmc.edu). Between March, 2003 and May,2005, over 8000 copies of the software have beendownloaded. In addition, search of databases and of theinternet have identified at least 47 scientific publicationsand 17 college courses that either describe or use this BNNsimulator. Thus, this simulation system has been applied inmany countries for biomedical research and education.

REFERENCES

[1] Robert E. Keen, James D. Spain, Computer Simulation in Biology,Wiley-Liss, Inc. 1992.

[2] Christof Koch, Biophysics ofComputation-Information Processing inSingle Neurons, Oxford University Press, New York, 1999.

[3] Christof Koch, Idan Segev, The role ofsingle neurons in informationprocessing, Nature neuroscience supplement, Vol.3, November 2000.

[4] Daniel S. Levine, Introduction to Neural and Cognitive Modeling,New Jersey, 2000.

[5] Houjin Chen, et al. Parallel computation in computer simulation forneural networks. Proceedings ofIEEE TENCON, 2002.

Fig. 3. Simulation Result With Three Neurons

Although Euler method is simple and fast, its step sizemust be smaller to guarantee its stabilization and accuracyfor all objects and its computation complexity is thehighest. TIhie step size for four order R-K method can belarger, but all the objects take the same method regardlessof smooth objects, thus its computation complexity is closeto Euler's. HNCM takes the same step size as four orderR-K's. However, it applies corresponding method toindividual objects, and gains the greatest accuracy and

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