Figure 1

Analog electric circuit diagram of a simple excitable threshold unit (McCulloch-Pitts neuron [9]). The unit consists of external signal inputs including sensory signals and signals from the other modules or rings, external noise input, a summation of these components, a successive comparator, and an output port. The negative clipper is inserted between the comparator and output port in order to insure that an output with asymmetric voltage is obtained. $R_1$ and $R_2$ are 10 and $100\mathrm{k}\Omega$, respectively. We used quad operational amplifiers, TL084 (Texas Instruments, Inc.). The positive and negative electric power source used for the amplifiers was $\pm 12\mathrm{V}$.Reuse & Permissions

Figure 2

(a) An excitable unit is composed of the following four parts: signal input ports (an external sensory input, the signals from the other units, and external noise), summation of these input signals, a successive comparator, and an output port. The dimensionless output amplitude of the comparator is set to unity. (b) A summing parallel network module (number of units $N$) of the excitable units of (a), with time delay. This is a version of the so-called Collins-type excitable network [20] with time delay. The input signal is fed into the input port of each unit. Signal averaging is performed prior to the delay. (c) A ring circuit consists of $M$ directionally coupled modules (b). The value $ϵ$ denotes the coupling constant between rings, which scales the output voltage of the rings. (d) Variation of transients of the output signal of the ring circuit of $M=4$ as a function of the number of parallels in a module $N$. The other parameters used for the simulation were as follows and summarized in Table in the Appendix. $ϵV_m$ ($V_m$ is the output amplitude): 0.16, threshold amplitude of comparators $\left(V_m^{\mathrm{Th}}\right)$: 0.10, noise amplitude $\left(D\right)$: 0.10, signal delay at each excitable unit: 16, input signal function: $\cos (2\pi t∕16)$, input pulse width $\left(t_{\mathrm{pw}}\right)$: 128, and time step for all simulations: 1. Simulated $(ϵ,D)$ phase diagrams for relaxation rate (e) and signal-to-noise ratio (SNR) (f) for the ring with $(M,N)=(4,100)$ are shown. $V_m^{\mathrm{Th}}$ was 0.20. “NF” at the scale bar of (e) stands for “nonfiring.” Note that the scale bar of (f) is nonlinear.Reuse & Permissions

Figure 3

(Color online) (a) A circuit diagram of the attractor switching device. A detailed electric circuit design is presented in Sec. 2A. (b) The experimental result of the device with $(M,N)=(4,1)$ rings. The firing rates of the fourth module in both rings was monitored as functions of the input signal and noise amplitudes. The parameters for the experiment were as follows. $ϵV_m^{\mathrm{A}}$: $0.16\mathrm{V}$, $ϵV_m^{\mathrm{B}}$: $0.25\mathrm{V}$, $V_{\mathrm{A}}^{\mathrm{Th}}$: $0.03\mathrm{V}$, and $V_{\mathrm{B}}^{\mathrm{Th}}$: $0.06\mathrm{V}$. $D$ was varied from $1.6\text{to}2.8\mathrm{V}$ with a step of $0.2\mathrm{V}$. See Sec. 2C for the details of the experiments.Reuse & Permissions

Figure 4

(Color online) (a) The simulated mean firing rate for the devices with $(M,N)=(4,1)$. The parameters for the simulation were as follows. $V_{\mathrm{A}}^{\mathrm{Th}}$: 0.05, $V_{\mathrm{B}}^{\mathrm{Th}}$: 0.20, $ϵV_m^{\mathrm{A}}$: 0.30, and $ϵV_m^{\mathrm{B}}$: 0.30. $D$ was set to 0.05, 0.10, 0.11, 0.12, and 0.13. The other parameters were set to values as indicated in Fig. 2 and also in Table in the Appendix. The inhibitory connection was implemented by varying the threshold amplitude $V_{\mathrm{A}}^{\mathrm{Th}}=0.05$ at the disinhibition state and 2.00 at the inhibition state. (b) The response of the device was explored as a function of $N$ while $D$ was fixed at 0.11. (c) The frequency response of the device with $N=100$ and $D=0.11$ is shown. The moderate sweep frequency was necessary for hysteresis behaviors.Reuse & Permissions

Figure 5

(Color online) A circuit diagram of the attractor switching device. The output signals of all the modules in ring B were fed into the signal input ports of all the modules of ring A.Reuse & Permissions

Figure 6

(Color online) The simulated mean firing rate for the canonical model of class 1 neurons as a function of noise amplitude $\left(D\right)$ (a), the number of $N$ (b), and input frequency $\left(\omega \right)$ (c). $(M,N)=(4,4)$, $r_A=-0.008$, and $r_B=-0.020$. The excitatory coupling constants (${ϵ}_{\mathrm{ex}}^A$ and ${ϵ}_{\mathrm{ex}}^B$) were 0.5 and the inhibitory connection $\left({ϵ}_{\mathrm{inh}}\right)$ from ring B to ring A was 3.0. For (a) and (b), the input sinusoidal signal is $0.019\times \sin \left(\omega t\right)$, where $\omega ∕2\pi =1.0\times {10}^{-3}$. For (b) and (c), $D=1.0\times {10}^{-3}$. For (a) and (c), $N=50$ The network is composed of slowly connected class 1 neurons with the time constant ${\kappa }_A={\kappa }_B=1.5$. Details of the simulation are given in Sec. 2E and Table in the Appendix.Reuse & Permissions