Reliable Logic Circuit Elements that Exploit Nonlinearity in the Presence of a Noise Floor

The response of a noisy nonlinear system to deterministic input signals can be enhanced by cooperative phenomena. We show that when one presents two square waves as input to a two-state system, the response of the system can produce a logical output ($\mathrm{NOR}/\mathrm{OR}$) with a probability controlled by the noise intensity. As one increases the noise (for fixed threshold or nonlinearity), the probability of the output reflecting a $\mathrm{NOR}/\mathrm{OR}$ operation increases to unity and then decreases. Changing the nonlinearity (or the thresholds) of the system changes the output into another logic operation ($\mathrm{NAND}/\mathrm{AND}$) whose probability displays analogous behavior. The interplay of nonlinearity and noise can yield logic behavior, and the emergent outcome of such systems is a logic gate. This “logical stochastic resonance” is demonstrated via an experimental realization of a two-state system with two (adjustable) thresholds.

Figure 1

For the system (1): (left) the piecewise linear function $g(x)$ and (right) the effective potential it gives rise to, with thresholds (a) $x_l^{*}=-1$, $x_u^{*}=1$ (solid line) and (b) $x_l^{*}=-1.5$, $x_u^{*}=0.75$ (dashed line). Equal upper and lower thresholds (i.e., $x_l^{*}=x_u^{*}$) give rise to symmetric wells. $I=0$, $\alpha =1.8$, $\beta =3.0$.

Figure 2

Response of the system (1), with $\alpha =1.8$, $\beta =3.0$, and thresholds set to $(x_u^{*},x_l^{*})=(1.3,-0.5)$. Each panel has 2 curves. Solid curves represent the solution $x(t)$ from numerical simulations for additive noise intensities $D$ equal to (top to bottom) 0.01, 0.5, and 1. The input $I$ is the sum of randomly switched square pulse trains (see text), and is the same (dashed line) in each panel. For an optimal noise intensity (center panel) a reliable $\mathrm{OR}/\mathrm{NOR}$ gate is obtained.

Figure 3

(left) Probability of obtaining the NAND logic operation (see text), $P(\mathrm{NAND})$, with $(x_l^{*},x_u^{*})=(-1.3,0.5)$. A similar result is obtained (right) for $P(\mathrm{NOR})$ taking $(x_l^{*},x_u^{*})=(-0.5,1.3)$.

Figure 4

Density map of $P(\mathrm{\text{logic}})$ (see text) for the (left) NOR and (right) NAND logic operation, for fixed noise ($D=0.7$), in the space of upper threshold $x_u^{*}$ ($x$ axis) and lower threshold $x_l^{*}$ ($y$ axis). The light portion of this plot indicates the nonlinearity [determined by $x_u^{*}$ and $x_l^{*}$ in (1) for which a logic output (NOR (left) and NAND (right)] is obtained reliably.

Figure 5

$P(\mathrm{NOR})$ (left) and $P(\mathrm{NAND})$ (right) as functions of the noise intensity $D$ ($x$ axis) and asymmetrizing dc input $C$ ($y$ axis). Thresholds are fixed: $(x_l^{*},x_u^{*})=(-0.5,1.3)$. Reliable NAND performance accrues for $C\sim -1$; for reliable NOR performance, $C\sim 0$. Optimal performance in both cases is obtained for $D$ values lying within the optimal window (Fig. 3). Note that, for $D$ outside the optimal window, one can still, for any $D$, optimize $\mathrm{NAND}/\mathrm{NOR}$ performance by adjusting $C$ as well as the thresholds ($x_l^{*}$, $x_u^{*}$) (this amounts to an adjustment of the nonlinearity for a given system noise floor).