Analyzing of dynamic characteristics for discrete S-PCNN Rencan Nie1, Shaowen Yao2, Dongming Zhou1and Haiying Deng1,a

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1 Analyzing of dynaic characteristics for discrete S-PCNN Rencan Nie, Shaowen Yao, Donging Zhouand Haiying Deng,a School of Inforation Science, Yunnan University, Kuning 659, China; Graduate School, Yunnan University, Kuning 659, China a d_haiying@6.co Keywords: Pulse Coupled Neutral Network; dynaic characteristic; capture period Abstract: Pulse Coupled Neutral Network (PCNN) is a new artificial neural network with biological echanis, and has been widely applied in any areas such as iage processing and so on, but its dynaic characteristics have not been analyzed effectively. In this paper, the dynaic characteristics about pulse period, refractory period and capture period in a siplified PCNN are discussed by atheatical analysis. Siulation results verify the correctness of the analysis conclusions. Introduction In 99, Eckhorn, et al [] studied the phenoenon that the visual cortex neurons in cats brain can burst synchronous pulses, and brought forward a linking odel for this phenoenon, and then Johnson, et al [] presented Pulse Coupled Neutral Network (PCNN) odel by siplifying Eckhorn odel. PCNN has been widely applied in the areas such as iage segentation [3], iage fusion [4], obect detection [5], optiization calculation [6] and so on. In a PCNN, the neurons exchange essages by pulse-streas. he independent neuron fires periodical pulses, but when receiving coupled pulses fro its neighbor neurons, the neuron will fire a pulse in advance. his is to say the neuron will be captured by neighbor neurons. So the neurons with siilar external stiulus will burst synchronous pulses through the utual capture effect. he effectiveness for PCNN in practical applications is closely related to odel paraeters, but the dynaic characteristics for PCNN have not been studied effectively. Experiental ethod is the ost coon way to set the odel paraeters. In literature [7], the pulse period was analyzed in continuous PCNN, not any literatures about capture characteristics by atheatical analysis can be searched. In this paper, the atheatical analysis about the dynaic characteristics in ters of pulse period, refractory period and capture period for a siplified discrete PCNN (S-PCNN) odel is discussed, and the siulation results verify the correctness of the analysis conclusions. Siplified-PCNN Model Due to the high degree of nonlinear and coplicated interaction of PCNN, it is very difficult for a qualitative analysis. Currently a popular siplified PCNN odel (S-PCNN) is described with the following equations ()-(5): F( n) S () ( n) V WkYk( n ) () k U ( n) F( n) + β ( n) (3) α ( ) ( )e θ n θ n + Y( n ) V (4), U ( n) > θ ( n) Y ( n) (5), otherwise he S-PCNN consists of three parts: the receptive field, the odulation field and the pulse generator. In the receptive field, the neuron receives input pulses fro neighbor neurons, the pulses 76

2 are transitted through the feedback input channel F and link input channel. W is the linking weight atrix of synapses between neurons, S is the external stiulation signal, V is agnitude paraeter of the channel. In the odulation field, U ( n ) is the internal activity by odulation between channel and F channel, β is the link constant. In the pulse generator, when U ( n) is greater than the dynaic threshold θ ( n), the neuron will fire a pulse, and then the threshold will increase to a high level, and then attenuates exponentially also, α and paraeter and agnitude paraeter respectively. Analysis of dynaic characteristics for S-PCNN V are its attenuation Pulse period: suppose the neuron fires a pulse at n tie, and again fires a pulse at n tie without coupled pulses inputs, or again fires a pulse at n tie with coupled pulses inputs, then the interval( n, n] is called non-excitation pulse period, and( n, n] is called excitation pulse period, obviously, n n. Refractory period and capture period: within the interval( n, n ], if the neuron receives the ' ' ' external pulses inputs at n n, n, and then its internal activity U will increase, then the neuron ' ' will fire a pulse in advance, the interval n, n can be called capture period. Due to the threshold of the neuron attenuates exponentially, ' ' ' n > n, n n. he interval n+, n is called refractory period, it eans that the neuron won t fire even if it receives the external pulses at this interval.. Pulse period or pulse phase Suppose at n n, neuron fire a pulse at the first tie. henyn ( ), and the neuron satisfies ) > θ( n) through the iteration, that isun ( ) θ( n). And suppose at n n, the neuron fires a pulse for the second tie. Because of ) θ( n), the following equations can be derived: α ( n n) α ( θ ( ) + ) (6) S n V e e ( ) ( ) n n α n n α α ( n n) α + + θ( n ) θ( n ) e V e ( θ( n ) V e ) e n α θ ( n ) + V e n + ln α ) (7) (8) he neuron fires a pulse for the th ties at n, and n ( is the instantaneous period: α θ ( n ) ln + V e n n + α α θ ( n ) + V e n ( n n ln α (9) () When the neuron receives coupled pulses, θ( n ) U( n ) F( n ) S, and ) Fn ( ) S, and then the non-excitation pulse period can be derived: 77

3 θ () ln, α S n ( α V e ln +, α S () When not receiving coupled pulses, the neuron will start to fire pulses with a stable period fro the second tie. he n at which the neuron fires a pulse for the th ties can be derived: n θ () ln α S, α θ () V e ln +(-) ln, + α S α S (). Refractory period and capture period of neuron Suppose the neuron fires a pulse at n, if there are no external pulses inputs, the neuron will fire a pulse again at n +. Suppose coupled pulses are input to the neuron, it will fire a pulse in advance whent ( n, n + ). θ( n + ) θ( n + ) t n + + ln n + + ln α Ft ()( + βt ()) α S( + βt ()) (3) he tie span of refractory period D and the tie span of capture period C are: θ ( n + ) D ( t ) n ln ( (4) α S + βv Yi ( t ) Wi ) θ( n + ) θ( n + ) C ( n+ n D + ln ln ( α S (5) α S + βv Yi ( t ) Wi ) he top integral function is generally ignored in equation (5): C + ln βv Yi( t) W + i α (6) Siulation results and analysis 3. Validation test of pulse period analysis Fig. teporal phases of neuron firing pulses Fig. the changes of neuron pulse period 78

4 Setα., V, I., θ ().4, teporal phases are shown in Fig.. here are two kinds of pulse teporal phase. One is calculated by iterations, the other is obtained by the equation (). In order to distinguish between the two pulses, the agnitudes are set to and.5. It can be observed the calculation results are consistent with the actual results. After firing for the second tie, the neuron fires pulse with a stable period. he pulse period, 8, 4 can be obtained. 3. Validation test of capture characteristics analysis Set W [ ; ; ], α., V 3, V, I.3, θ ().4 β.3. Suppose that if the neuron receives external pulses as inputs, then it can receives pulses fro its 8 neighbor neurons at the sae tie (i.e. Y () t W 6). As shown in Fig.3, after the neuron firing a pulse, the i i threshold will increase rapidly then attenuate exponentially. If the neuron doesn t receive external pulses, its internal activity U will be a constant. Otherwise, it will increase to a high level. An attenuation interval of threshold θ is consisted of capture and refractory period. In the capture period, threshold θ has decayed below U. And if the neuron receives pulses as inputs, it will fire a pulse in advance. In the refractory period, threshold θ is still above U, so it won t fire any pulse in advance even if it receives pulses as inputs. he tie span of capture period is unrelated to the external stiulation signal. Fig.3 the changes of firing tie in the cases of non-excitation and U or θ Stiulation Pulse Non-stiulation Pulse Refractory Period Capture Period θ non-situlation θ stiulation U stiulation U non-situlation Conclusions his paper analyzed the dynaic characteristics for S-PCNN. he equations of the pulse period and pulse phase were deduced. It s showed that neuron will start to fire pulses with a stable period fro the second tie. he equations of the tie span of refractory period and capture period were deduced. When the accuulation value of coupled pulses is fixed, the tie span of capture period is also fixed. he experiental results showed that the calculation results by the equations are consistent with the iterations. Acknowledgents Our work is supported by the National Natural Science Foundation of China (6658), Natural Science Foundation of Yunnan Province (No.FD3) and Yunnan Province Education Departent Natural Science Fund of China (No.Y47). References [] R.Eckhorn, H.J. Reitboeck, M.Arndt, P.Dicke. Feature linking via synchronization aong distributed asseblies: Siulation of result fro cat visual cortex [J]. Neutral Coput, 99, (3):

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