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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 in case of regular spiking (RS).(a) Time evolution of v(t). (b) Typical trajectory, including state-dependent jump, in the (v, u) phase plane (a = 0.02, b = 0.2, c = −65, d = 8, I = 10 [3]).
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pone.0138919.g001: System behavior in case of regular spiking (RS).(a) Time evolution of v(t). (b) Typical trajectory, including state-dependent jump, in the (v, u) phase plane (a = 0.02, b = 0.2, c = −65, d = 8, I = 10 [3]).

Mentions: The Izhikevich neuron model [3, 4] is a two-dimensional (2D) system of ordinary differential equations of the formv′=0.04v2+5v+140-u+I,(1)u′=a(bv-u).(2)In the above equation, v and u represent the membrane potential of a neuron and the membrane recovery variable, respectively. v and time t are measured in [mV] and [ms], respectively. When the membrane potential v > 30 [mV], the model fires; v is set to c, and u is set to u + d, which is called the resetting process. I is the direct current (DC) input. The parameters a and b describe the time scale and the sensitivity of u, respectively. Spiking behavior, such as regular spiking, intrinsically bursting, and fast spiking can be reproduced using this model. As an example of regular spiking (a = 0.02, b = 0.2, c = −65, d = 8, I = 10) [3], Fig 1 (a) and (b) show the time evolution of v(t) and the system trajectory in a phase plane (v, u), respectively. Due to the resetting process, when v(t) exceeds 30 [mV], the system state (v, u) (discontinuously) jumps to point ((v, u) ≈ (−65, 0.3)), as shown in Fig 1 (b). In our simulation, we numerically analyzed this model through non-linear differential/algebraic equation solvers by using the backward differentiation formula method [28] to achieve sufficient numerical precision in order to evaluate chaotic spiking activity. This method is more precise than Euler’s method, which was adopted to reproduce only periodic spiking in [3].


Analysis of Chaotic Resonance in Izhikevich Neuron Model.

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

System behavior in case of regular spiking (RS).(a) Time evolution of v(t). (b) Typical trajectory, including state-dependent jump, in the (v, u) phase plane (a = 0.02, b = 0.2, c = −65, d = 8, I = 10 [3]).
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4589341&req=5

pone.0138919.g001: System behavior in case of regular spiking (RS).(a) Time evolution of v(t). (b) Typical trajectory, including state-dependent jump, in the (v, u) phase plane (a = 0.02, b = 0.2, c = −65, d = 8, I = 10 [3]).
Mentions: The Izhikevich neuron model [3, 4] is a two-dimensional (2D) system of ordinary differential equations of the formv′=0.04v2+5v+140-u+I,(1)u′=a(bv-u).(2)In the above equation, v and u represent the membrane potential of a neuron and the membrane recovery variable, respectively. v and time t are measured in [mV] and [ms], respectively. When the membrane potential v > 30 [mV], the model fires; v is set to c, and u is set to u + d, which is called the resetting process. I is the direct current (DC) input. The parameters a and b describe the time scale and the sensitivity of u, respectively. Spiking behavior, such as regular spiking, intrinsically bursting, and fast spiking can be reproduced using this model. As an example of regular spiking (a = 0.02, b = 0.2, c = −65, d = 8, I = 10) [3], Fig 1 (a) and (b) show the time evolution of v(t) and the system trajectory in a phase plane (v, u), respectively. Due to the resetting process, when v(t) exceeds 30 [mV], the system state (v, u) (discontinuously) jumps to point ((v, u) ≈ (−65, 0.3)), as shown in Fig 1 (b). In our simulation, we numerically analyzed this model through non-linear differential/algebraic equation solvers by using the backward differentiation formula method [28] to achieve sufficient numerical precision in order to evaluate chaotic spiking activity. This method is more precise than Euler’s method, which was adopted to reproduce only periodic spiking in [3].

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.