Commun. Theor. Phys. (Beijing, China) 43 (2005) pp. 466 470 c International Academic Publishers Vol. 43, No. 3, March 15, 2005 Self-organized Criticality and Synchronization in a Pulse-coupled Integrate-and-Fire Neuron Model Based on Small World Networks LIN Min and CHEN Tian-Lun Department of Physics, Nankai University, Tianjin 300071, China (Received July 6, 2004) Abstract A lattice model for a set of pulse-coupled integrate-and-fire neurons with small world structure is introduced. We find that our model displays the power-law behavior accompanied with the large-scale synchronized activities among the units. And the different connectivity topologies lead to different behaviors in models of integrate-and-fire neurons. PACS numbers: 05.65.+b, 87.10.+e Key words: self-organized criticality, synchronization, small world networks 1 Introduction A few years ago, Bak et al. introduced the concept of self-organized criticality (SOC). [1] It is shown that many large dynamical systems tend to self-organize into a statistically stationary state without intrinsic spatial and temporal scales. This critical state is characterized by a power-law distribution of avalanche sizes, which is regarded as fingerprint for SOC. The brain, which possesses about 10 10 10 12 neurons, is one of the most complex system. Now evidence for some aspects of scale invariance has been found in the central nervous system. [2] In fact, the strong analogies between the dynamics of the SOC model for earthquakes and that of neurobiology has been realized by Hopfield. [3] Some scientists stated that the brain might be operating at, or near, a critical state. [4] It makes us be interested to investigate the mechanism of SOC process in the brain. Besides SOC, another form of collective organized behavior is known to occur in large assemblies of elements with pulse interaction, that is, synchronization. Largescale synchronized patterns of activity in the frequency range of 30 70 Hz have been found in the olfactory system, the visual cortex, and other brain areas. [5,6] Synchronization of neurons is believed to represent the binding of object features, a problem of outstanding significance for information processing in the brain. [7] Since both SOC and synchronization are characterized by the large scale spatiotemporal correlation, we argue in the following that there is a close relationship between SOC and synchronization. Recently, Watts and Strogatz [8] have studied the small world networks. Small world stands for a network whose connectivity topology is placed somewhere between a regular and completely random connectivity. It is well known that network connectivity in the cortex and other brain regions is mainly local, with relatively sparse long-distance projections. From a neuron-biological viewpoint, unlike fully connected artificial neural networks, plausible associative memories must have sparse connectivity, reflecting the situation in the cortex and hippocampus. [9] Indeed, it has been shown that nervous system of C. Elegans shows small world properties. [8] Small world networks of coupled phase oscillators are optimal for producing synchronization. The results may be relevant to the observed synchronization of widely separated neurons in the cat visual cortex. [5] Watts et al. suppose that the brain has a small world architecture. [8] In this paper, we develop a lattice model with small world structure to investigate self-organized criticality in the activity of neural populations. The collective behavior of integrate-and-fire neural networks with different topologies has been intensively investigated under various conditions. [10 12] We study avalanches of activity of integrate-and-fire neurons in small world networks. In our model, it exhibits a power-law behavior over a wide range of the parameters, and the SOC behavior is accompanied by the large-scale synchronized pattern of the activity of the units. And different connectivity topologies play a role in dynamics of networks of integrate-and-fire neurons. 2 The Model We generate the networks following the procedure described in Refs. [13] and [14], which we summarized here: (i) Start with a two-dimensional regular square lattice with L L sites. All bonds present between the nearest neighbor sites. (ii) Then we choose randomly two sites of the lattice and place a bond between them. Self-connections and duplicate links are excluded. And then one of the smaller The project supported by National Natural Science Foundation of China under Grant No. 90203008 and Doctoral Foundation of the Ministry of Education of China E-mail: linminmin@eyou.com
No. 3 Self-organized Criticality and Synchronization in a Pulse-coupled Integrate-and-Fire Neuron Model 467 bonds going to a neighboring site of one of the end points of one long bond is removed. (iii) Repeat step (ii) until the number of bonds rewired is the fraction φ of all bonds of the original lattice, i.e. 2φL(L 1). Here a square lattice represents a sheet of cells occurring in the cortex. Each node represents a neuron, and a connection between two nodes represents a synapse. The situation φ = 0 corresponds to the simple regular lattice and large φ corresponds to the random graph. According to the neuron-dynamical picture of the brain, the dynamics of neurons and synapses can be described as follows. When the membrane potential of a neuron exceeds the threshold, the neuron sends out signals with the form of action potential and then returns to the rest state (the neuron fires). The signal is transferred by the synapses to the other neurons, which has an excitatory or inhibitory influence on the membrane potential of the receiving cells according to whether the synapses are excitatory or inhibitory, respectively. The resulting potential, if it also exceeds the threshold, leads to next step firing, and so on giving an avalanche. We add a kind of integrate-and-fire mechanism into our model and it can be described in the following details. For any neuron sited at position i in the lattice, we give it a dynamical variable V i, which represents the membrane potential of the i-th neuron. V i = 0 and V i > 0 represent the neuron in a rest state and depolarized state, respectively. Here we do not consider the situation of V i < 0, which represents the neuron in the hyperpolarized state. When a neuron s dynamical variable V i exceeds a threshold V th = 1, the neuron i is unstable and it will fire and return to a rest state (V i returns to zero). Each of the nearest neighbors will receive a pulse (action potential) and its membrane potential V j will be changed. Without loss of generality, we assume that the change of V j is proportional to V i. We also consider the slow relaxation of the non-firing neurons to the rest state. Then we get the redistribution of the membrane potentials after the firing of neuron i as V j av j + b/q i V i, V i 0, (1) where a is a constant smaller than 1 denoting the remains of V i due to its slow relaxation after the firing. The term b/q i V i represents the action potential between firing neuron and its neighbors, where b represents the pulse intensity, and q i is the number of neighbors of neuron i. Now we present the algorithm for simulating the above dynamical process of the model in detail. Here we use the open boundary condition: (i) Initialize the membrane potential of each neuron below V th. (ii) Find out the maximal value of all V i, V max, and add V th V max to all neurons. (iii) If there exists any unstable neuron, V i V th, then redistribute the membrane potential V i on the i-th neuron to its nearest neighbors according to Eq. (1). (iv) Repeat step (iii) until all the neurons of the lattice are stable. Define this process as one avalanche, and define the avalanche size as the number of neurons fired once during the process. (v) Apply step (ii) again and another new avalanche begins. Our driving rule is the continuous driving (global perturbation ) rule. It is similar with the global perturbation in the OFC model. We think the continuous driving may be understood as the system is receiving a slow continuous signal from the external or other parts of the brain. So we use the global driving rule. 3 Simulation Results 3.1 Power-Law Behavior and Influence of Density Parameter φ First, we let the size of the lattice be 35 35, where a = 0.98, b = 1 are fixed. In this model three different connectivity patterns have been tested: regular, random, and small world. Here our aim is to investigate avalanche dynamical behaviors for different values of density parameter φ. Fig. 1 The probability of the avalanche size P (S) as a function of S for system size L = 35, a = 0.98, b = 1 with different φ. To prove the SOC of our system, we measure the probability distribution of the size of avalanches. As shown in Fig. 1, the distributions of the avalanche sizes have power-law behaviors, P (S) S τ. For different φ, the avalanches sizes obey different distributions. The data that we present in Fig. 2(a) imply that there is a dependence of the exponents on the density parameter φ
468 LIN Min and CHEN Tian-Lun Vol. 43 of the model. At the same time, we investigate the relation between the mean avalanche size S and parameter φ. In Fig. 2(b), the mean avalanche size S deceases as φ increases. This phenomenon can be explained that the difference of the probability of the large size avalanches (S 300) occurring is not distinct for different φ as shown in Fig. 1. But on the whole, for a certain big avalanche size (S 50), the probability distribution of avalanche size decreases as φ increases. we show the samples of the temporal fluctuations in the average membrane potential per lattice site for φ = 0, 0.01 and 1. Three cases correspond to the three different topological configurations: regular, small world, and random. From Fig. 3, we can see that both the regular and the small world topologies display oscillatory activity, but in the regular network they appear much later. The amplitude in the random network is smaller than in the regular, small world cases. Fig. 2 (a) The power-law exponent τ for the distribution of avalanche size as a function of φ for L = 35, a = 0.98, b = 1. (b) The avalanche average size S as a function of φ. (c) The exponent α as a function of φ. Fig. 4 The power spectrum of the average membrane potential in the model for L = 35, a = 0.98, b = 1 when (a) φ = 0, (b) φ = 0.01, and (c) φ = 1. At the same time, we present the power spectrum S(f) of the signals in the model with φ = 0, 0.01 and φ = 1 in Fig. 4. They display 1/f power law behaviors over a wide range of time scales, S(f) 1/f α. This phenomenon resembles with the wide range of time scales that have observed in the brain, e.g., in EEG brain wave recordings of collective neural activity. [6] The exponent α depends on φ as shown in Fig. 1(c). With the increment of φ, the value of α increases. 3.2 Influence of the Parameter a Fig. 3 The average membrane potential in a network of 35 35, when a = 0.98, b = 1. (a) Regular network (φ = 0). (b) Small world network (φ = 0.01). (c) Random network (φ = 1). To investigate the temporal signature of our model, we focus on the temporal sequence of avalanches. We begin by discussing the fluctuation in the average membrane potential per lattice site. We compute the temporal behavior of V (t) = (1/L 2 ) L 2 i=1 V i(t). It is obtained by averaging over all the nodes of the system after an avalanche, where the time is defined as the number of avalanches. In Fig. 3 Fig. 5 The probability of the avalanche size P (S) as a function of size S for L = 35, b = 1, φ = 0.01 with different a = 0.98, 0.92, 0.85, and 0.80.
No. 3 Self-organized Criticality and Synchronization in a Pulse-coupled Integrate-and-Fire Neuron Model 469 We vary a and set b = 1, φ = 0.01 fixed, as shown in Fig. 5. When a = 0.80, the probability decays exponentially with the size of the avalanches, which means there are only localized behaviors. As a is increased, the transition from localized to SOC behavior occurs. The critical exponents τ for a = 0.98, 0.92, and 0.85 are obtained as 1.22, 1.52, and 2.06, respectively. With the decrease of a, the value of the exponent τ increases. 3.3 Influence of the Pulse Discharging Intensity b In the avalanche process, the parameter b of Eq. (1) is very important. We vary b and setting a = 0.98, φ = 0.01 fixed, as shown in Fig. 6. We find that the power-law behavior gradually degenerate with the decrease of b. When b = 0.18, the probability decays exponentially with the size of the avalanches. So it is a localized behavior. As b is increased, the transition from localized to SOC behavior occurs and the distribution satisfies the power-law P (S) S τ. In Fig. 6, the critical exponents τ are 1.22, 1.55, and 1.97 for b = 1, 0.90, and 0.70, respectively. We also find that the exponent τ increases as b decreases. is, synchronization occurs even in frozen disorder cases. It is very clear that the peak in the SOC state is much higher than that in the state with only localized behavior, which indicates that in the SOC state there are many more units at the same active level after an avalanche. We call the activities of neurons synchronized if their difference in membrane potential are smaller than 0.02. [15] In Fig. 7, there are about 56% units in the SOC state whose activities are synchronized. From this point of view, the SOC process has been accompanied with the large-scale synchronization among the units. Thus our system finds a compromise between synchronization and SOC. This close relationship between SOC and synchrony has also been found in the other systems. [11,16] Fig. 7 The distribution membrane potential after an avalanche for L = 35, φ = 0.01 is shown, where the system is separately in SOC state (a = 0.98, b = 1) and the state having only localized behavior (a = 0.80, b = 1). Fig. 6 The probability of the avalanche size P (S) as a function of size S for L = 35, a = 0.98, φ = 0.01 with different b = 1, 0.90, 0.70, and 0.18. 3.4 Synchronization To observe synchronous activity, we calculate the distribution of the membrane potential after an avalanche for L = 35, φ = 0.01, when the system is separately in SOC state (a = 0.98, b = 1) and the state having only localized behaviors (a = 0.80, b = 1). In Fig. 7, we can see these distributions both concentrate around a peak. It is similar with the result in Ref. [15]. In Ref. [15], the network topology is regular, but our network topology is small world. The result agrees with that in Ref. [11], that 4 Conclusion In this paper, we provide a two-dimensional lattice system with small world structure to investigate scaleinvariance behavior in the activity of neural populations. The model consists of a set of pulse-coupled integrate-andfire neurons. We find a power-law distribution behavior of avalanche sizes in our model and the SOC process is associated with the large-scale synchronization occurring among the elements. More importantly, we find there are different avalanche dynamical behaviors for different topology of the network. This work is just a preliminary study. It should be noted that our model is only a very simple simulation of brain and many details of neurobiology are ignored. For this reason, many other important questions concerned with the model will be studied in future works.
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