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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.

No MeSH data available.


System behavior at the Poincaré section. Time series of ui (left). Return map of (ui, ui + 2) (right).The solid line represents the orbit of ui, the dotted line shows the solution to ui + 2 = ψ2(ui), and the dashed line depicts ui + 2 = ui. (a) d = −11, (b) d = −12, (c) d = −13, (d) d = −16 (a = 0.2, b = 2, c = −56, I = −99).
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pone.0138919.g006: System behavior at the Poincaré section. Time series of ui (left). Return map of (ui, ui + 2) (right).The solid line represents the orbit of ui, the dotted line shows the solution to ui + 2 = ψ2(ui), and the dashed line depicts ui + 2 = ui. (a) d = −11, (b) d = −12, (c) d = −13, (d) d = −16 (a = 0.2, b = 2, c = −56, I = −99).

Mentions: Following this, we evaluated in greater detail the behavior of the system shown in Fig 5 by using the Poincaré section method. Fig 6 (left) shows the time series of ui as system behavior on the Poincaré section Ψ. As shown in Fig 6(a) (left), the value of ui remained fixed (ui ≈ −98.6). At d = −12 (Fig 6(b) (left)), the value of ui began to oscillate with a focus on ui ≈ −98.6. This oscillation expanded to −102 ≲ ui ≲ −90, following which the value of ui reverted to approximately −98.6. The periodic oscillation disappeared gradually as the value of d decreased, as shown in Fig 6(b) (d = −12), (c) (d = −13), and (d) (d = −16). In order to focus on the oscillatory behavior of ui, we used the return map (ui, ui + 2). Fig 6 (right) shows the orbit of ui (solid line), and the solutions to ui + 2 = ψ2(ui) (dotted line) and ui + 2 = ui (dashed line). When d = −11 ((a)), the orbit of ui remained at the intersection (≈ (−98.5,−98.5)) of ui + 2 = ψ2(ui) and ui + 2 = ui, and there were two unstable fixed points on both sides of this stable fixed point at ui ≈ −101.5 and −91.5. As noted above, the tangent bifurcation arose at d ≈ −11.9, i.e., a pair consisting of an unstable fixed point and a stable one destroyed each other. Through this tangent bifurcation, at d = −12 ((b)), the orbit of ui exhibited sluggish movement (called laminar state) in the region where the slope of ψ2 was approximately 1.0 (−102 ≲ ui ≲ −94), and irregularly active movement (called turbulent or burst state) in regions with larger slopes (≫ 1). Such chaotic, dynamic alternation between laminar and turbulent states is called intermittency chaos [34, 35]. Note that the term of burst is not used in this paper to avoid confusion between the chaotic movement and the neural spike patterns in neurodynamics such as intrinsically bursting and chattering bursting. As the value of d decreased, the area of the region producing the laminar state, where the slope of ψ2 was approximately 1.0, shrunk as well. The turbulent state was subsequently dominant in the dynamics, i.e., the system transitioned from intermittency chaos to chaos involving primarily turbulent movement, as shown in Fig 6 (b)–(d).


Analysis of Chaotic Resonance in Izhikevich Neuron Model.

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

System behavior at the Poincaré section. Time series of ui (left). Return map of (ui, ui + 2) (right).The solid line represents the orbit of ui, the dotted line shows the solution to ui + 2 = ψ2(ui), and the dashed line depicts ui + 2 = ui. (a) d = −11, (b) d = −12, (c) d = −13, (d) d = −16 (a = 0.2, b = 2, c = −56, I = −99).
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Related In: Results  -  Collection

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pone.0138919.g006: System behavior at the Poincaré section. Time series of ui (left). Return map of (ui, ui + 2) (right).The solid line represents the orbit of ui, the dotted line shows the solution to ui + 2 = ψ2(ui), and the dashed line depicts ui + 2 = ui. (a) d = −11, (b) d = −12, (c) d = −13, (d) d = −16 (a = 0.2, b = 2, c = −56, I = −99).
Mentions: Following this, we evaluated in greater detail the behavior of the system shown in Fig 5 by using the Poincaré section method. Fig 6 (left) shows the time series of ui as system behavior on the Poincaré section Ψ. As shown in Fig 6(a) (left), the value of ui remained fixed (ui ≈ −98.6). At d = −12 (Fig 6(b) (left)), the value of ui began to oscillate with a focus on ui ≈ −98.6. This oscillation expanded to −102 ≲ ui ≲ −90, following which the value of ui reverted to approximately −98.6. The periodic oscillation disappeared gradually as the value of d decreased, as shown in Fig 6(b) (d = −12), (c) (d = −13), and (d) (d = −16). In order to focus on the oscillatory behavior of ui, we used the return map (ui, ui + 2). Fig 6 (right) shows the orbit of ui (solid line), and the solutions to ui + 2 = ψ2(ui) (dotted line) and ui + 2 = ui (dashed line). When d = −11 ((a)), the orbit of ui remained at the intersection (≈ (−98.5,−98.5)) of ui + 2 = ψ2(ui) and ui + 2 = ui, and there were two unstable fixed points on both sides of this stable fixed point at ui ≈ −101.5 and −91.5. As noted above, the tangent bifurcation arose at d ≈ −11.9, i.e., a pair consisting of an unstable fixed point and a stable one destroyed each other. Through this tangent bifurcation, at d = −12 ((b)), the orbit of ui exhibited sluggish movement (called laminar state) in the region where the slope of ψ2 was approximately 1.0 (−102 ≲ ui ≲ −94), and irregularly active movement (called turbulent or burst state) in regions with larger slopes (≫ 1). Such chaotic, dynamic alternation between laminar and turbulent states is called intermittency chaos [34, 35]. Note that the term of burst is not used in this paper to avoid confusion between the chaotic movement and the neural spike patterns in neurodynamics such as intrinsically bursting and chattering bursting. As the value of d decreased, the area of the region producing the laminar state, where the slope of ψ2 was approximately 1.0, shrunk as well. The turbulent state was subsequently dominant in the dynamics, i.e., the system transitioned from intermittency chaos to chaos involving primarily turbulent movement, as shown in Fig 6 (b)–(d).

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.