Bistability Analysis of Excitatory-Inhibitory Neural Networks in Limited-Sustained-Activity Regime

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1 Commun. Theor. Phys. 56 (2011) Vol. 56, No. 6, December 15, 2011 Bistability Analysis of Excitatory-Inhibitory Neural Networks in Limited-Sustained-Activity Regime NI Yun ( ), WU Liang ( ), WU Dan ( ), and ZHU Shi-Qun (ý ) Basic Course Department, Suzhou Polytechnical Institute of Agriculture, Suzhou , China School of Physical Science and Technology, Soochow University, Suzhou , China (Received June 1, 2011; revised manuscript received September 19, 2011) Abstract Bistable behavior of neuronal complex networks is investigated in the limited-sustained-activity regime when the network is composed of excitatory and inhibitory neurons. The standard stability analysis is performed on the two metastable states separately. Both theoretical analysis and numerical simulations show consistently that the difference between time scales of excitatory and inhibitory populations can influence the dynamical behaviors of the neuronal networks dramatically, leading to the transition from bistable behaviors with memory effects to the collapse of bistable behaviors. These results may suggest one possible neuronal information processing by only tuning time scales. PACS numbers: Sn, Hc, Xt Key words: neural networks, bistable 1 Introduction Mechanisms and functions of neural networks of the brains are extensively discussed in current investigations of biologists, mathematicians, and physicists. [1 2] Neurons forming complex networks with synaptic connections are thought to provide the physiological basis for information processing and mental representations. The underlying network structures support the dynamic emergence of coherent physiological activities and all aspects of nervous-system functions. [3 6] The presence of the functional patterns of neural behaviors requires the systems to operate in a so-called limited sustained activity (LSA) regime. Neither quickly dying out nor global activations throughout the entire network takes place. [7 11] Some pathological excitation such as Parkinson s disease are found to be related to neuronal system behaviors intermittently outside this regime. [12] Recently, the LSA behaviors have been studied [8,13] based on interactions and incorporations among inhibitory and excitatory neurons. A large number of neurons generate spikes and bursts of sequential spikes of the same pace. [14 15] Such intriguing activities, often known as synchronization phenomenon, emerge either spontaneously or due to external stimulus. [16 19] Synchronization is of great importance for biological information processing. [20 23] This is partly because the efficient synaptic connections among pulsecoupled neurons are often based on the synchronization of their firings. Moreover, synchronization among neurons is typically a multi-time-scale phenomenon. [24 27] Both fast and slow neural synchronous behaviors have been observed in real circumstances. However, the problem how the interactions across multi-time-scales influence collective neural behaviors in the regime of the LSA remains poorly understood. In this paper, the influence of neural interactions of the different time scales on the overall collective behaviors is investigated when it is operating in the LSA regime. In Sec. 2, an ensemble of stochastic cellular automata model is presented when the effects of neural interactions [13] rather than that of individual neurons are concerned. In Sec. 3, the stabilities of each one of coexisting metastable state are analytically studied. In Sec. 4, numerical simulations are performed. The analytical results are supported by the numerical calculations. The main conclusions are summarized in Sec Model The neural complex network model is composed of N neurons, g i N and g e N are the numbers of excitatory and inhibitory neurons respectively, with g i + g e = 1. It is feasible that g i N or g e N is integer especially when the size of the simulated neuronal network is large. If either g i N or g e N is not an integer, the method to deal such situation still needs to be developed. Excitatory and inhibitory synaptic signals are transmitted from each of active excitatory and inhibitory neurons, respectively, to its neighboring (targeting) neurons. Supported by the National Natural Science Foundation of China under Grant Nos and and by the Natural Science Foundation of Higher Education Institutions of Jiangsu Province under Grant No. 11KJB Corresponding author, liangwu@suda.edu.cn szhu@suda.edu.cn c 2011 Chinese Physical Society and IOP Publishing Ltd

2 1156 Communications in Theoretical Physics Vol. 56 Rules: The state of each neuron j at time t is determined by the following rules. An inactive neuron is activated spontaneously with a probability f. Such spontaneous activations may arise from intrinsic stochastic features of neurons and further enhanced by noisy environment in virtue of stochastic-resonance mechanism. [28 32] In addition, the couplings among neurons is in the form of the integrated input V j where V j = N i=1 a jia i J i represents the net excitation received by a neuron over a short period of integration time. The element of the directed matrix {a ij } is 1 when there is a synaptic connect from node j to i, and 0 otherwise. The parameter A i represents the active state of node i, A i = 1 and 0 for an active and inactive neuron respectively. The quantity J i represents the neuron type, J i = 1 and 1 for an excitatory and inhibitory neuron respectively. An inactive neuron can be activated with a probability µ by an over-threshold integrated input (postsynaptic potential) V j Ω. An active neuron can be deactivated and stop firing with a probability µ if its postsynaptic potential V j becomes lower than the threshold Ω. The schematic representation of the neural network model is shown in Fig. 1. Fig. 1 (Color online) Schematic representation of a part of a neural network. Active excitatory, inactive excitatory, active inhibitory and inactive inhibitory neurons are colored in bright red, pink, black and gray respectively. Neurons are coupled with each other through the integrated inputs. An excitatory neuron i is inactive at time t. The neuron has nine neighboring neurons. The integrated input of neuron i is V i = 9 j=1 AjJj = ( 1) ( 1) = 4. This is a discrete-time model. The probability φ i (t) that neuron i is active at time t is φ i (t + 1) = A i (t) + [f + µh(v i Ω)δ Ai(t),0] µh(ω V i )δ Ai(t),1, (1a) where H is the unit heaviside step function, δ is the Kronecker delta, A i (t) is a Bernoulli random variable defined in the state space {0, 1} as follows, φ i (t), A i (t) = 1, P(A i (t)) = 1 φ i (t), A i (t) = 0, (1b) 0, A i (t) {0, 1}, where P is the Bernoulli distribution. The collective behaviors of the networked neuronal systems are studied on a statistical ensemble of classical equilibrium directed network, with a given number of neurons N in each ensemble member and a given probability that two neurons have a directed synaptic connect. If c is the mean in-degree, i.e., the number of in-links of a neuron on average, and P n (c) is the in-degree distribution describing the probability that a randomly chosen neuron has n in-links, P n (c) follows the poisson distribution P n (c) = c n e c /n! in the thermodynamic limit N. Therefore, P k (g e ρ e (t)c) P l (g i ρ i (t)c) is able to express the probability that a randomly chosen neuron has k excitatory inputs and l inhibitory inputs at time t. The quantities ρ e (t) and ρ i (t) are the probabilities that a randomly chosen neuron is active at time t. The parameter Ψ ρe(t),ρ i(t) is used to denote the probability that a neuron has an over-threshold postsynaptic potential at time t. The summation over all possible combinations of k and l satisfies k l Ω with Ψ(ρ e (t), ρ i (t)) = k Ω k Ω l=0 =e geρe(t)c k Ω P k (g e ρ e (t)c) P l (g i ρ i (t)c) (g e ρ e (t)c) k k!(k Ω)! Γ(k Ω+1, g i ρ i (t)c), (2) where Γ(k, x) is the upper incomplete gamma function. The rate equation describing the time evolution of the probability ρ e,i (t) that a randomly chosen neuron is active at time t can be written as ρ e (t) = f e (1 ρ e (t))+µ e (1 ρ e (t))ψ(ρ e (t), ρ i (t)) µ e ρ e (t)[1 Ψ(ρ e (t), ρ i (t))], (3) ρ i (t) = f i (1 ρ i (t)) + µ i (1 ρ i (t))ψ(ρ e (t), ρ i (t)) µ i ρ i (t)[1 Ψ(ρ e (t), ρ i (t))]. (4) The first term on the right hand side of Eqs. (3) and (4) represents the spontaneous activation. The second term represents the activation due to an over-threshold postsynaptic potential. The third term represents the inactivation caused by an under-threshold postsynaptic potential at time t. The probability ρ e (t) = 0 denotes the absence of active excitatory neuron while ρ e (t) = 1 represents that all the excitatory neurons are active. 3 Theoretical Analysis The neural network system can be decomposed into two mutually coupled subsystems of excitatory and inhibitory neurons according to Eqs. (3) and (4). They are coupled with others through Ψ(ρ e (t), ρ i (t)) of Eq. (2).

3 No. 6 Communications in Theoretical Physics 1157 Figure 2 shows the contour plot of Ψ on the parameter plane (g i ρ i,g e ρ e ) for different values of the mean indegree c. For Fig. 2(a) a relatively large value of c is chosen. When there is no active inhibitory neurons, namely g i ρ i = 0 in the x-axis, Ψ is as small as less than 10 3 for g e ρ e < T 1 = While, each neuron surely has an over-threshold postsynaptic potential, Ψ = 1, for g e ρ e > T 2 = It is also seen that both the thresholds, T 1 and T 2, increase monotonically with g i ρ i. Fig. 2 (Color online) The contour plot of Ψ on the parameter plane (g iρ i, g eρ e) for different values of the mean indegree c with Ω = 20. (a) c = 100, (b) c = 40. A typical LSA often occurs in the region between the two thick countour lines. This region raises as c is decreased. In Fig. 2(b) the case of sparse connectivity is considered, the mean in-degree is c = 30. It is found that the contours are raised as a whole compared with Fig. 2(a). The line of T 2 even disappears, indicating that it becomes impossible for neurons to surely have an over-threshold input. The parameter region on the upper right side of the red dashed line is biophysically meaningless because an inequality g i ρ i + g e ρ e 1 has to be satisfied. The behavior of two subsystems is similar because they follow the rate equations of the same form, as seen in Eqs. (3) and (4). Hence, by using ν a = f a + µ a and F a = f a /(f a + µ a ) for a = e, i, Eqs. (3) and (4) can be rewritten in a simpler form. 1 ρ a = F a ρ a + (1 F a )Ψ(ρ e, ρ i ), (5) ν a where the last term is the coupling between the two subsystems. For stationary solutions of Eq. (5), one sets ρ = 0, F e = F i = F. In the simple case, we have ρ e = ρ i = ρ. Then the following self-consistent equation is derived. [33] ρ = F + (1 F)Ψ(ρ e, ρ i ), (6) Figure 3(a) plots the left and right hand side (LHS and RHS) of Eq. (6) for an intermediate value of activation parameter F. The networked neural system has two attractive fixed points A and B. Their coexistence indicates that the behavior of the neuron network described by Eq. (6) is similar to a bistable system. [34] It has two distinct behaviors, one is a local oscillation around one of the two fixed points and the other is a jumping behavior between the two fixed points. [31] Figure 3(b) shows the dependence of the locations of the fixed points on the parameter F. Three different regions are clearly seen. The spontaneous activation characterized by F is so weak in region I (F < F1) that neurons rarely have an over-threshold postsynaptic potential. As a result the upper fixed point B is absent in this region. A non-zero ρ of point A mainly due to the spontaneous activation. In contrast, the lower fixed point A disappears in region III for F > F2. The activation in this case is mainly contributed to by over-threshold inputs. The system operating in region II is typically bistable. The stability analysis is performed to the system operating in this region by introducing δρ to the two fixed points respectively, δρ ρ. Therefore two coupled equations with respect to ρ e and ρ i are derived by the linearization of Eq. (6). dδρ a (t) ν a dt = δρ a (t) + D ae δρ e (t) + D ai δρ i (t), (7) where D ab (1 F a )( Ψ a (ρ e, ρ i )/ ρ b ) for a, b = e, i. A solution in the form of δρ a (t) = σ a e γt is substituted into Eq. (7) where σ a is a coefficient and γ is a complex exponent. Solutions exist only if the determinant of Eq. (7) equals zero. Hence we get the expression for γ, γ = ν e {1 D ee +α(1 D ii )±[(1 D ee α(1 D ii )) 2 +4αD ei D ie ] 1/2 }/2, where α = ν i /ν e. As ν e and ν i in Eq. (6) determine the time scale of evolution of activation strengths of excitatory and inhibitory populations respectively, their ratio α represents the difference of the two time scales.

4 1158 Communications in Theoretical Physics Vol. 56 Fig. 3 (Color online) (a) Left and right side of Eq. (6), A and B are two fixed points corresponding to the stationary solutions. F = (b) The dependence of the two fixed points on the activation parameter F. Bistable behaviors are only presented in the region II. Real and imaginary parts of the characteristic exponent γ calculated for point A in (c) and for point B in (d). Im( γ A) and Im( γ B) become non-zero for α < α A1 = 0.85 and α < α B2 = 0.89, respectively. When α < α B1 = 0.48 the bistable behavior collapses due to the point B in the region of Re( γ B) < 0. Blue dotted line shows Re( γ B) Re( γ A). In Fig. 3(c) both the real and imaginary parts of γ = (2/ν e )γ are plotted for the lower fixed point A when an intermediate value of F is chosen. It is seen that Re( γ A ) is always greater than zero no matter what the parameter α is, indicating point A is always stable. In addition, oscillatory behavior characterized by a non-zero value of Im( γ A ) emerges for α < α A1. In contrast, as shown in Fig. 3(d) with respect to the fixed point B, Re( γ B ) is less than 0 when α < α B1. This implies that the state becomes unstable and can not stay around the upper point B when a small value of ratio α is chosen. The physical meanings of the different analytic results between A and B is explained in the following section. 4 Numerical Simulations Numerical simulations according to the rules of the states of each neuron described in Sec. 2 are performed on neuronal networks of finite sizes, N = Figures 4(a) 4(e) show time traces of ρ e,i for different values of α. For comparison between theoretical analysis and numerical calculations, the value of the parameter F is chosen to be the same as in Figs. 3(c) and 3(d). The two dotted lines in each panel mark the two points A and B. All Neurons are set to be inactive at t = 0, and all are reset to be active at t = 3000 (νe 1 ). In Figs. 4(a) and 4(b), it is seen that the state of the system always stays at either one of the two fixed points depending on the initial state. This is a typical memory effect, exhibiting no jumping events. In addition, fluctuations behavior around point B after t = 3000 is more strong in Fig. 4(b) than in Fig. 4(a). In Fig. 4(a), Im( γ B ) is zero (see in Fig. 3(c)), therefore the fluctuations behavior characterized by a non-zero Im( γ) is absent. The observed tiny fluctuations behavior arises only from the finite size effects. In contrast, the oscillatory behavior characterized by a non-zero Im( γ B ) (see in Fig. 3(d)) rules over the fluctuations behavior in Fig. 4(b). When α is reduced further to α = 0.7 as shown in Fig. 4(c), the system can stay at either of the two points. This is because both the two metastable states are still stable in the sense that the real parts of γ with respect to both points are positive. However, the fluctuations behavior characterized by a non-zero Im( γ B ) may induce the system to jump out of the vicinity of B occasionally. When α is 0.5, Re( γ B ) is very close to 0 as shown in Fig. 3(d). So system rarely stays around point B in Fig. 4(d) for the weak stability of B. Nevertheless, at about t = 1500 the traces of ρ e,i jump up suddenly from A to B and even stay around B for a short period of time (about 500 (νe 1 )). This sudden jump event occurs due to the combination of two effects. On one hand, Re( γ B ) is still positive though it is close to zero. On the other hand, the random stochastic disturbances of relatively high frequency can occasionally push the traces out of the vicinity of A.

5 No. 6 Communications in Theoretical Physics 1159 This panel is different significantly from the above four panels. The metastable state point B losses stability because Re( γ B ) < 0. Meanwhile, the point A is still stable. Therefore, the bistable behavior of the neuron system collapses. The intrinsic stochastic behavior is able to almost regularly push the traces out of the attractive field of A. Moreover, a phase-shift between the populations of excitatory and inhibitory neurons is clearly seen. More excitatory neurons are activated firstly, followed by inhibitory neurons being activated. More active inhibitory neurons leads to a sudden deactivation of excitatory neurons while the number of active inhibitory neurons is reduced relatively slowly. Such deactivation process last till to a new ground of bursting behavior being triggered again. The five panels are different obviously though all the parameters are the same except for α. This suggests that the manipulation of neural system behavior for information transmission processing is possible by appropriate real-time tuning of the ratio of the time scales between the two different populations of neurons. The emergence of phase shift is a natural indicator of the collapse of neural system bistable behavior. [35 37] Fig. 4 Simulation results on a classical random network composed of N = 10 4 neurons for different values of the time-scale ratio α. The state of neuron at time t, active or inactive, is determined by the rules (see the second paragraph of Sec. 2). Two horizontal lines in each panel mark the stationary solutions, points A and B respectively. Numerical results in this figure show a good agreement with analytical results in Fig. 3. Parameters are c = 140, Ω = 20, ν e = 0.1, F = 0.233, g i = The most important changes of the system dynamics emerge when α is reduced further. In Fig. 4(e), α = Conclusions In this paper, the stability of the two metastable states is analyzed in the thermodynamical limit. It is shown that the system activation behavior change dramatically with the difference of time scales of the two neuron populations. Numerical simulations on neuronal network models of finite size shows a good agreement with the theoretical results in the thermodynamical limit. These results may suggest one of simple mechanisms for the manipulation of the information processing in neural systems by tuning only time scales appropriately. Acknowledgments We thank Alexander V. Goltsev for his many useful conversations. References [1] G. Deco, V.K. Jirsa, and A.R. McIntosh, Nature Reviews Neuroscience 12 (2010) 43. [2] D.D. Bock, W.C.A. Lee, A.M. Kerlin, et al., Nature (London) 471 (2011) 177. [3] A.L. Barabasi and Z.N. Oltvai, Nature Reviews Genetics 5 (2004) 101. [4] D.J. Watts and S.H. Strogatz, Nature (London) 393 (1998) 440. [5] E. Bullmore and O. Sporns, Nature Reviews Neuroscience 10 (2009) 186. [6] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.U. Hwang, Physics Reports 424 (2006) 175. [7] M. Bazhenov, I. Timofeev, M. Steriade, and T.J. Sejnowski, Nature Neuroscience (1999). [8] J.M. Beggs and D. Plenz, The Journal of Neuroscience 23 (2003) [9] R.C. Muresan and C. Savin, Journal of Neurophysiology 97 (2007) [10] M. Kaiser and C.C. Hilgetag, Frontiers in Neuroinformatics 4 (2010) 8. [11] D. Guo and C. Li, Neural Networks, IEEE Transactions on 21 (2010) 895. [12] P.A. Tass, Biological Cybernetics 87 (2002) 102. [13] A.V. Goltsev, F.V. de Abreu, S.N. Dorogovtsev, and J.F.F. Mendes, Phys. Rev. E 81 (2010) [14] I. Belykh, E. de Lange, and M. Hasler, Phys. Rev. Lett. 94 (2005)

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