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Analysis of Chaotic Resonance in Izhikevich Neuron Model.

Nobukawa S, Nishimura H, Yamanishi T, Liu JQ - PLoS ONE (2015)

Bottom Line: We found the existence of two distinctive states, a chaotic state involving primarily turbulent movement and an intermittent chaotic state.Through computer simulations, we confirmed that both chaotic states in CR can sensitively respond to weak signals.Moreover, we found that the intermittent chaotic state exhibited a prompter response than the chaotic state with primarily turbulent movement.

View Article: PubMed Central - PubMed

Affiliation: Department of Management Information Science, Fukui University of Technology, Fukui, Japan.

ABSTRACT
In stochastic resonance (SR), the presence of noise helps a nonlinear system amplify a weak (sub-threshold) signal. Chaotic resonance (CR) is a phenomenon similar to SR but without stochastic noise, which has been observed in neural systems. However, no study to date has investigated and compared the characteristics and performance of the signal responses of a spiking neural system in some chaotic states in CR. In this paper, we focus on the Izhikevich neuron model, which can reproduce major spike patterns that have been experimentally observed. We examine and classify the chaotic characteristics of this model by using Lyapunov exponents with a saltation matrix and Poincaré section methods in order to address the measurement challenge posed by the state-dependent jump in the resetting process. We found the existence of two distinctive states, a chaotic state involving primarily turbulent movement and an intermittent chaotic state. In order to assess the signal responses of CR in these classified states, we introduced an extended Izhikevich neuron model by considering weak periodic signals, and defined the cycle histogram of neuron spikes as well as the corresponding mutual correlation and information. Through computer simulations, we confirmed that both chaotic states in CR can sensitively respond to weak signals. Moreover, we found that the intermittent chaotic state exhibited a prompter response than the chaotic state with primarily turbulent movement.

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Related in: MedlinePlus

Dependence of bifurcation on parameter d.(a) Bifurcation diagram of ui. (b) Lyapunov exponents λj (j = 1, 2). (c) Coefficient of variation for inter-spike interval CV (a = 0.2, b = 2, c = −56, I = −99).
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pone.0138919.g004: Dependence of bifurcation on parameter d.(a) Bifurcation diagram of ui. (b) Lyapunov exponents λj (j = 1, 2). (c) Coefficient of variation for inter-spike interval CV (a = 0.2, b = 2, c = −56, I = −99).

Mentions: By fixing the value of I at −99, we investigated the bifurcation of this system by replacing I with d by using a bifurcation diagram consisting of (u1, u2, ⋯ , uN). Fig 4(a), (b), and (c) represent the bifurcation diagrams of ui, λj, and CV, respectively, as functions of d. For d ≲ −11.9, the chaotic trajectory was distributed in the range −103 ≲ ui ≲ −80, and the system exhibited a chaotic state (λ1 > 0) and irregular spiking activity (CV > 0), excluding periodic windows (λ1 = 0), given periodic-1 (CV = 0) and multiple periodic (CV > 0) states. As the value of d increased, those of λ1 and CV decreased. They converged at 0 for d ≳ −11.9, i.e., the system assumed periodic states and exhibited periodic spiking. In order to conduct bifurcation analysis in the system with a state-dependent jump, we used the evaluation method intended to assess the stability of a fixed point u0 = ψl(u0) (l = 1, 2, ⋯) on a Poincaré section. In the literature [29], this stability has been evaluated byμ=∂ϕl∂u0=(01)(00-v˙/u˙1)Φ(tl,t0)(01).(10)In the above, u0 = (v0, u0) indicates the initial value of orbit u = (v, u) at t = t0. ∣μ < 1∣, μ = −1, and μ = 1 represent the stable condition, period doubling bifurcation, and tangent bifurcation, respectively. In Fig 4, the tangent bifurcation at l = 2 arises at d ≈ −11.9. Through this tangent bifurcation, the system transitions into chaos at d ≲ −11.9, as shown in Fig 4(a) and (b).


Analysis of Chaotic Resonance in Izhikevich Neuron Model.

Nobukawa S, Nishimura H, Yamanishi T, Liu JQ - PLoS ONE (2015)

Dependence of bifurcation on parameter d.(a) Bifurcation diagram of ui. (b) Lyapunov exponents λj (j = 1, 2). (c) Coefficient of variation for inter-spike interval CV (a = 0.2, b = 2, c = −56, I = −99).
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Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC4589341&req=5

pone.0138919.g004: Dependence of bifurcation on parameter d.(a) Bifurcation diagram of ui. (b) Lyapunov exponents λj (j = 1, 2). (c) Coefficient of variation for inter-spike interval CV (a = 0.2, b = 2, c = −56, I = −99).
Mentions: By fixing the value of I at −99, we investigated the bifurcation of this system by replacing I with d by using a bifurcation diagram consisting of (u1, u2, ⋯ , uN). Fig 4(a), (b), and (c) represent the bifurcation diagrams of ui, λj, and CV, respectively, as functions of d. For d ≲ −11.9, the chaotic trajectory was distributed in the range −103 ≲ ui ≲ −80, and the system exhibited a chaotic state (λ1 > 0) and irregular spiking activity (CV > 0), excluding periodic windows (λ1 = 0), given periodic-1 (CV = 0) and multiple periodic (CV > 0) states. As the value of d increased, those of λ1 and CV decreased. They converged at 0 for d ≳ −11.9, i.e., the system assumed periodic states and exhibited periodic spiking. In order to conduct bifurcation analysis in the system with a state-dependent jump, we used the evaluation method intended to assess the stability of a fixed point u0 = ψl(u0) (l = 1, 2, ⋯) on a Poincaré section. In the literature [29], this stability has been evaluated byμ=∂ϕl∂u0=(01)(00-v˙/u˙1)Φ(tl,t0)(01).(10)In the above, u0 = (v0, u0) indicates the initial value of orbit u = (v, u) at t = t0. ∣μ < 1∣, μ = −1, and μ = 1 represent the stable condition, period doubling bifurcation, and tangent bifurcation, respectively. In Fig 4, the tangent bifurcation at l = 2 arises at d ≈ −11.9. Through this tangent bifurcation, the system transitions into chaos at d ≲ −11.9, as shown in Fig 4(a) and (b).

Bottom Line: We found the existence of two distinctive states, a chaotic state involving primarily turbulent movement and an intermittent chaotic state.Through computer simulations, we confirmed that both chaotic states in CR can sensitively respond to weak signals.Moreover, we found that the intermittent chaotic state exhibited a prompter response than the chaotic state with primarily turbulent movement.

View Article: PubMed Central - PubMed

Affiliation: Department of Management Information Science, Fukui University of Technology, Fukui, Japan.

ABSTRACT
In stochastic resonance (SR), the presence of noise helps a nonlinear system amplify a weak (sub-threshold) signal. Chaotic resonance (CR) is a phenomenon similar to SR but without stochastic noise, which has been observed in neural systems. However, no study to date has investigated and compared the characteristics and performance of the signal responses of a spiking neural system in some chaotic states in CR. In this paper, we focus on the Izhikevich neuron model, which can reproduce major spike patterns that have been experimentally observed. We examine and classify the chaotic characteristics of this model by using Lyapunov exponents with a saltation matrix and Poincaré section methods in order to address the measurement challenge posed by the state-dependent jump in the resetting process. We found the existence of two distinctive states, a chaotic state involving primarily turbulent movement and an intermittent chaotic state. In order to assess the signal responses of CR in these classified states, we introduced an extended Izhikevich neuron model by considering weak periodic signals, and defined the cycle histogram of neuron spikes as well as the corresponding mutual correlation and information. Through computer simulations, we confirmed that both chaotic states in CR can sensitively respond to weak signals. Moreover, we found that the intermittent chaotic state exhibited a prompter response than the chaotic state with primarily turbulent movement.

No MeSH data available.


Related in: MedlinePlus