1. Introduction

Synchronized neural activity has an important role in neural information coding and brain information processing¹. The correlated firing among neurons has been observed in visual information processing in the brain^2,3. The authors have studied the chaotic synchronization of the neural spike response and its application to the segmentation of input images⁴; feature linking of the visual image of moving objects⁵; and the decomposition of superimposed chaotic spike sequences⁶. In order to represent multiple information by chaotic synchronized cell assemblies, confirming the coexistence of synchronized assemblies is necessary.

In this study, we constructed the network of the bifurcating neuron⁷ with complete bidirectional coupling and designed a coupling model to achieve selective chaotic synchronization based on the phase response. In Sections 2 and 3, we explain the bifurcating neuron network and coupling model. In Section 4, we describe the numerical simulation that is performed to examine the selective synchronization for the phase shift of the background oscillation. We summarize the results in Section 5.

2. Network model of the bifurcating neuron

The bifurcating neuron was introduced by Lee and Farhat⁷. In our previous research⁶, we defined the bifurcating neuron as a form of the spike response model⁸ to extend the coupling term based on the phase response. In the following sections, we explain the extended bifurcating neuron model and its network. Let the network of bifurcating neurons consist of N neurons, where n⁽ⁱ⁾ is the i-th neuron. The dynamics of the i-th bifurcating neuron is defined as follows:

where u⁽ⁱ⁾, u_rest, η⁽ⁱ⁾, ξ+(i) , ξ−(i) , and ν⁽ⁱ⁾ denote the internal potential of n⁽ⁱ⁾, resting potential, kernel function of internal dynamics, positive coupling term, negative coupling term, and external noise, respectively. The coupling terms are described in the next section.

The kernel function η⁽ⁱ⁾ is defined as

where α is the time constant of the kernel η⁽ⁱ⁾, and tlast(i) is the last firing time of n⁽ⁱ⁾. The internal potential u⁽ⁱ⁾ is linearly increasing by the kernel η⁽ⁱ⁾. The neuron n⁽ⁱ⁾ is firing when u⁽ⁱ⁾ exceeds the threshold, θ, and then u⁽ⁱ⁾ is reset to the initial potential given by the background oscillation, η₀(t, ϕ). The constants, A_η, ω, and ϕ⁽ⁱ⁾ are the amplitude, frequency, and phase shift value of background oscillation, respectively.

The dynamics of a single neuron without the coupling term is the same as with the original bifurcating neuron. The single neuron without coupling exhibits various chaotic dynamics in the inter-spike interval, and the phase of the next firing time is determined by the chaotic one dimensional map of the phase of the last firing time^6,7. The shape of this chaotic map is determined by the phase shift value, ϕ⁽ⁱ⁾. Examples of this chaotic map are shown in Fig. 1. Furthermore, the bifurcation diagram and Lyapunov exponents of this map are shown in Fig. 2.

Fig. 1.

Examples of the chaotic one dimensional map of the phase of a single bifurcating neuron with the phase shift (a) ϕ = π/2 and (b) ϕ = 3π/2.

Fig. 2.

The bifurcation diagram and Lyapunov exponents of a single bifurcating neuron without coupling.

3. Coupling model with phase response

Let the set S(k)={t0(k),t1(k),t2(k),⋯} be a set of firing time of the neuron, n^(k), where tj(k) is the firing time of j-th spike of n^(k). The set firing time of the input spikes to n⁽ⁱ⁾ is denoted as Γ⁽ⁱ⁾. For bidirectional complete coupling, Γ⁽ⁱ⁾ is defined as

The coupling term ξ+(i) and ξ−(i) in Eq. (1) are the summation of the positive and negative phase responses ε+(i) and ε−(i) , respectively, such that The simple phase response is a constant value, such that where β₊ and β_– are non-negative coupling constants. The next firing time is hastened by the positive response and delayed by the negative response. We described coupling with these simple responses as constant positive and negative coupling, respectively.

We designed the phase response to strengthen the chaotic synchronization between coupled neurons with the same phase shift value⁶. The positive phase response based on the predicted next firing time is defined as

where t^next(i) is the predicted next firing time of n⁽ⁱ⁾, such We describe this coupling as adaptive positive coupling. The negative phase response based on the last firing time is defined as We describe this coupling as adaptive negative coupling.

If time, s, is within the range, Δ_ε, from the predicted next firing time t^next(i) , then the phase response is positive, so that the next firing time is hastened. On the other hand, if time, s, is within the range, Δ_ε, from the last firing time, tlast(i) , then the phase response is negative to delay the next firing time. By using the phase responses in Eq. (7) and (9), the decomposition of the superimposed chaotic spike sequences with the different phase shift values is possible⁶.

4. Numerical experiments

5. Conclusion

We have constructed a bifurcating neuron network, and assessed the formation of chaotically synchronized cell assemblies for several types of coupling model. As a result, the chaotically synchronized cell assemblies were observed for adaptive positive coupling of the phase response, where neurons that have the same phase shift value were selectively synchronized. This result indicates the possibility of binding information based on the phase shift value of the background oscillation. Our future work will seek to analyze the effect of adaptive coupling using a different chaotic neuron model, which will further elucidate such information coding.

Acknowledgements

This work was partially supported by MEXT KAKENHI Grant Number 15H05878 and 16K00409.