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Building phase synchronization equivalence between coupled bursting neurons and phase oscillators To cite this article before publication: Fabiano Ferrari et al 2018 J. Phys. Commun. in press https://doi.org/10.1088/2399-6528/aaa853
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AUTHOR SUBMITTED MANUSCRIPT - JPCO-100523.R1
Fabiano A. S. Ferrari
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Building phase synchronization equivalence between coupled bursting neurons and phase oscillators Universidade Federal dos Vales do Jequitinhonha e Mucuri, Instituto de Engenharia, Ciˆencia e Tecnologia, Jana´ uba/MG, 39440-000, BR E-mail:
[email protected]
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Ricardo L. Viana
Universidade Federal do Paran´ a, Departamento de F´ısica, Curitiba/PR, 81531-980, BR November 2017
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Abstract. Bursting neurons are characterized by an alternation of active and inactive state, during the active state they exhibit train of spikes and quiescent behavior in the inactive state. When bursting neurons are coupled they can synchronize depending on the coupling strength. If the coupling strength is increased above a critical value, a transition from an asynchronous state to a partial synchronized state can be observed. This transition has the same type observed in coupled Kuramoto oscillators. When the natural frequency distribution of the oscillators is properly defined, the bursting neurons and the Kuramoto oscillators describe the same phase transition for synchronization.
1. Introduction
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Bursting activity is observed in many neuronal firing patterns, it consist of an alternation between active and inactive state. During the active state bursting neurons show train of spikes and in the inactive state they are silent [1, 2]. When bursting neurons are coupled, depending on topology and coupling strength, they can exhibit phase synchronization. Phase synchronization is observed at experiments in vitro and numerical simulations [3, 4, 5]. To explain the emergence of phase synchronization a model of coupled phase oscillators was proposed by Kuramoto [6]. In the Kuramoto model, phase linear oscillators are coupled through a nonlinear term. To determine phase synchronization is defined an order parameter, called Kuramoto Order Parameter. When the Kuramoto Order Parameter is 1 means all oscillators are synchronized in phase and when the oscillators are not synchronized to each other the order parameter is 0. For this model, depending on frequency distribution, the critical value where the transition from
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asynchronous state to partial synchronized state happens can be determined analytically [7]. The observed phase synchronization profile for coupled bursting neurons are similar to those observed in Kuramoto oscillators. Our analysis show that if the frequency distribution of Kuramoto oscillators is properly defined, then both models can describe the same phase synchronization transition [8]. In this paper, we describe in details the method to generate such equivalence. 2. Rulkov Model
One simple model to describe neuronal bursting is the Rulkov model. It is a two dimensional map described by [9]: xn+1 = α/(1 + x2n ) + yn ,
(1)
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yn+1 = yn − η(xn − σ),
(2)
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where η = 0.001 and σ = −1 are fixed parameters and α is the bifurcation parameter [9]. Different values of α result in different dynamical behaviors. One example of the bursting oscillations produced by this model is shown in the Figure 1. For the Rulkov model a burst starts when the y-variable reaches its maximum value. Considering the maximum value of the y-variable we can define a phase ϕ, such that [10] (n − nk ) , (3) ϕn = 2πk + 2π (nk+1 − nk ) where nk is the initial time of the k-th burst, nk = y(k-th maximum point). The correspondence between this phase transformation and the burst oscillations is shown in the Figure 1 (red line). 2
2
1
xn
1
0
0
-1
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cos( θ )
-1
-2
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n
-2 90000
Figure 1. Example of bursting oscillations and its associated phase produced by the Rulkov model. Black color represents the time series of the x variable (left side axis in the graph) and red color the cosine of the associated phase (right side axis in the graph). This time series was produced after 88000 transient iterations, here α = 4.2.
The average bursting frequency Ω can be defined as ϕn ′ − ϕ0 Ω= , n′
(4)
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where n′ is the time series length, the equation (4) is a good approximation for regular bursting. The bursting frequency Ω increases linearly with parameter α, when it is varied from 4.1 to 4.3, as shown in Figure 2. 0.035
0.03
Ω 0.025
0.015 4.1
4.15
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0.02
4.2
α
4.25
4.3
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Figure 2. Average bursting frequency Ω as function of parameter α. The average was calculated over 100 distinct initial conditions and 10 000 iterations were considered (after 80 000 transient iterations).
3. Generalized Kuramoto Model
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The original version of the Kuramoto model describes globally coupled phase oscillators [6] N dθi ε X = ωi + sin(θj − θi ), dt N j=1
(5)
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where i = 1, ..., N , and N is the network size. The parameter ωi is the natural frequency and ε is the coupling strength. A more powerful version of the Kuramoto model is called: Generalized Kuramoto model [11]. This version allow us to study phase synchronization of different network topologies N X dθi = ωi + ε Aij sin(θj − θi ), dt j=1
(6)
where Aij is the connection matrix. When the natural frequencies ωi obey a symmetric and unimodal frequency distribution g(ω) is possible to determine the critical value εC for the onset of phase synchronization [12]
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εc =
2 hki , πg(0) hk 2 i
(7)
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where hki and hk 2 i are the mean degree and square mean degree of the network. For finite network sizes fluctuations in the Kuramoto Order Parameter can be observed. For large networks sizes, N → ∞, the onset of phase synchronization is the moment in which the Kuramoto Order Parameter starts to increase from zero towards one. 4. Networks
To verify if the phase synchronization in the Kuramoto and Rulkov model are really equivalent we analyzed three different types of network: Erd¨os-R´enyi, Small-World and Scale-Free. 4.1. Erd¨os-R´enyi Network
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This type of network consists of a random network. It is generated from N initially uncoupled nodes. The uncoupled nodes are connected by NK random links, the links are randomly according to an uniform probability p [13]. This probability must to satisfy the relation NK p= . (8) N (N − 1) In the example of Figure 3 (a) is used a Erd¨os-R´enyi network with Nk = 5000 links. 4.2. Small-World Network
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Small-world networks are a type of networks where every node has few connections but the distance (in the number of sites) between any two sites in the network is small [14]. There are several methods to construct a small-world network, here we will consider the Newman-Watts procedure [15]. In this procedure, we start with a regular network of size N , every site is connected with K first neighbors. According with a p uniform probability distribution pN K new connections are added in the network. In the example of Figure 3 (b) is used a small-world network with K = 20 and p = 0.1. 4.3. Scale-Free Network
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To build this type of network we start with an initial Erd¨os-R´enyi network of size N0 , then we insert a new site performing two connections, one random connection and another with the most connected site in the initial network. This procedure is repeated until the network reaches the desirable size N [16]. In the example of Figure 3 (c) is used a scale-free network with N0 = 23 and N0k = 23.
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5. Phase synchronization The transition from asynchronous to a partial synchronized state can be analyzed through the Kuramoto Order Parameter. The temporal Kuramoto Order Parameter
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1 X iϕn e |, rn = | N i=1
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(9)
and to analyze phase synchronization we use the average order parameter ′
n 1 X R= ′ rn , n n=1
(10)
where n′ is a sufficient large number of iterations.
6. Comparison Between Kuramoto and Rulkov models Considering coupled Rulkov neurons, their equations are (i)
= α /(1 +
2 (x(i) n ) )
+
N X
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(i) xn+1
yn(i)
+ε
Aij x(j) n ,
(11)
j=1
(i)
yn+1 = yn(i) − η(x(i) n − σ),
(12)
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in which Aij is the connection matrix, ε is the coupling strength, η and σ are the same of the uncoupled case. If α ∈ [4.1 : 4.3], each α will have a distinct bursting frequency Ω. For this reason, if the α(i) parameters are chosen from a given probability density function, then the associated distribution of bursting frequencies Ω will satisfy the same probability density function. When the natural frequencies ω of Kuramoto oscillators and the bursting oscillation frequency Ω of Rulkov neurons, obey the same probability distribution function then we can build a network of Kuramoto oscillators with the same phase synchronization transition than the network of Rulkov neurons. To show this property we will consider that the frequencies obey an uniform distribution, in this case ( 1 for −a ≤ ω ≤ a 2a , (13) g(ω) = 0 otherwise
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where a is the range, ω ∈ [−a : a]. For this g(ω) the critical value of the Kuramoto network will be [12] εc =
4a hki , π hk 2 i
(14)
this equation can be rewriten as
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a=
πεc hk 2 i . 4 hki
(15)
In this case, the bursting frequencies Ω of the Rulkov neurons will uniformly vary from 0.0175 to 0.03, as shown in Figure 2. The equation (15) allow us to build a network of Kuramoto oscillators with the same onset for phase synchronization than a specific network of Rulkov neurons with a known εc . To generate this equivalent network we just need to determine the a value.
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(a)
ε
1 0,8 0,6 R 0,4 0,2 0,002 0,004 00
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1 0,8 0,6 R 0,4 0,2 0,005 0,01 00
ε
(c)
0,02
ε
0,04
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1 0,8 0,6 R 0,4 0,2 0 0
(b)
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Consequently, they will have the same εc and the same phase synchronization transition. Examples using the three mentioned networks are shown in the Figure 3. For this example, the Kuramoto oscillators were integrated with classical Runge-Kutta method with step h = 0.01, the average order parameter was calculated over 104 iterations per site after 16 × 104 transient iterations and 20 distinct random initial conditions were used. For the Rulkov neurons, the order parameter is an average of 104 iterations per site after 8 × 104 transient iterations and 50 distinct random initial conditions.
Figure 3. Comparison in the order parameter for the three different network types: (a) Erd¨os-R´enyi, (b) Small-World and (c) Scale-free. Black color indicates the result for coupled Rulkov neurons and blue color for Kuramoto oscillators. Dashed lines indicate the critical coupling strength εc . The network size is N = 1000.
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As networks of Figure 3 are finite size networks, the critical value εc , that will generate the equivalence between the models, is not the coupling value where the Kuramoto Order Parameter starts to increase above the fluctuation but an intermediate value, as indicated by dashed lines in Figure 3. The more abrupt is the phase synchronization transition or the larger is the network size the better is the equivalence between both models.
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7. Final Considerations
When the natural frequencies of Kuramoto oscillators and the bursting frequencies of Rulkov neurons obey the same probability distribution function, it is possible to adjust the range of frequencies in the Kuramoto network such that both models can have the same phase synchronization transition. We have shown only the results for uniform distribution, but any other unimodal, symmetric and truncated distribution can provide similar results (truncated Cauchy distribution, for example) [8]. Equivalences for other types of distribution still need to be study. The presented method works well for neurons with regular bursting. Although we have not compared with experimental results, we expect this method can be used to describe phase synchronization for in vitro neuronal networks, with bursting patterns
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similar to those described by the Rukov model. The equivalence between Rulkov neurons and Kuramoto oscillators remains the same for different types of networks. Unfortunately, cases where bursting frequency is not regular the method fails. We expect in future works to develop a more general method to include irregular bursting neurons. Acknowledgments
This work was made possible through financial support from the Brazilian Research Agencies, CAPES and CNPq. References
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