Enhanced processing in arrays of optimally tuned nonlinear biomimetic sensors: A coupling-mediated Ringelmann effect and its dynamical mitigation

Inspired by recent results on self-tunability in the outer hair cells of the mammalian cochlea, we describe an array of magnetic sensors where each individual sensor can self-tune to an optimal operating regime. The self-tuning gives the array its “biomimetic” features. We show that the overall performance of the array can, as expected, be improved by increasing the number of sensors but, however, coupling between sensors reduces the overall performance even though the individual sensors in the system could see an improvement. We quantify the similarity of this phenomenon to the Ringelmann effect that was formulated 103 years ago to account for productivity losses in human and animal groups. We propose a global feedback scheme that can be used to greatly mitigate the performance degradation that would, normally, stem from the Ringelmann effect.

Figure 1

(a) Circuit of Takeuchi and Harada. (b) The complete measuring system.

Figure 2

Transfer function of the model for a set of parameters: $q=0.25$, $q=0.5$, $q=1.0$, and $q=2.0$.

Figure 3

Bifurcation diagram of the Takeuchi and Harada sensor (shown schematically). The parameter $V$ represents the amplitude of the voltage oscillations in the $L_0{C}_0$ resonance circuit. The parameter $q$ depends on $R_1$. $q=0$ corresponds to the Hopf bifurcation.

Figure 4

Possible setup of the adaptive system.

Figure 5

Input-output characteristics of the adaptive system. The periodic input signal is $x=acos\mathrm{\Omega }t$. $A$ is the amplitude of the main harmonic in the output. It is found as $A=\sqrt{A_1^2+B_1^2}$, where $A_1=\frac{1}{\pi /\mathrm{\Omega }}{\int }_0^{2\pi /\mathrm{\Omega }}zcos\left(\mathrm{\Omega }t\right)dt$ and $B_1=\frac{1}{\pi /\mathrm{\Omega }}{\int }_0^{2\pi /\mathrm{\Omega }}zsin\left(\mathrm{\Omega }t\right)dt$ via simulations of Eqs. (2), (3), (6), and (7). Parameters: $\mathrm{\Omega }=2\pi \times 0.01$, $\delta =0.01$, and $T=\tau =10$. The theoretical solution was obtained with Eq. (8).

Figure 6

Organization of the 2D sensory arrays: (a) a single sensor, $N=1$; (b) $N=4$ and (c) $N=9$. The dots denote locations of the sensory units. The parameter $L$ is the intersensor interval.

Figure 7

Ringelmann effect in an array of sensors. The signal-to-noise ratio ${\mathrm{\Gamma }}_N$ as a function of the number $N$ of sensors in the array. The array is organized into the square lattices (see Fig. 6) with the intersensor intervals $L$. The target field is the weak periodic signal $x=asin\left(\mathrm{\Omega }t\right)$, where $a=0.001$ and $\mathrm{\Omega }=2\pi \times 0.01$. The noises ${\xi }_i\left(t\right)$ are independent OU stochastic processes. The theoretical dependence is shown with the dashed line. It was found with Eq. (23). It is easy to see that the obtained results are always below the theoretical capacity. This is a sign of the Ringelmann effect in our coupled array. The inset shows a clear increase in the summed SNR response (for fixed $N=36$) as the sensor separation in the array increases, corresponding to a lower coupling strength.

Figure 8

Ringelmann effect in an array of sensors. Same parameters as Fig. 7 but with $a=0.01$.

Figure 9

Synergetic effect of the coupling on the gain in the array. Here $A$ is the output amplitude of the array. The periodic input signal is $x=acos\mathrm{\Omega }t$. Parameters: $\mathrm{\Omega }=2\pi \times 0.01$, $a=0.01$, $\delta =0.01$, $T=\tau =10$, and $L=0.18$. The expected theoretical dependence was obtained with Eq. (8) and assumption that the amplitude is $A=N\times A_0$, where $A_0$ is the amplitude of the unit (of the array) performing alone.

Figure 10

There are the two major causes of productivity losses in groups working on additive tasks. The portion between the dashed line and the dotted line represents motivation loss (social loafing), and the portion between the dotted line and the solid line represents coordination loss. “Pseudogroups” are defined as groups of individuals who were actually working alone, but thought they were working as part of a group. The design of the figure was adopted from Ref. [16].

Figure 11

(a) Circuit of the sensory unit. Since the parameter $q$ should depend on $R_1$, in the figure we show that the variable resistor $R_1$ is a target for the feedback of the adaptation (self-tuning mechanism). (b) The complete system with four units. The blocks marked with $\sum$ are summators with different weights as required by Eq. (30).

Figure 12

Mitigating the Ringelmann effect in an array of sensors. Same parameters as Fig. 8 with separation $L=0.18$.

Figure 13

(a) Outer hair cell (OHC) interacting with the basilar membrane–tectorial membrane system and with the central nervous system (CNS) [35]. The OHC can schematically be split into the sensor and motor components. The dotted green loop shows that the sensory part of the OHC is adaptive subsystem [36]. (b) The coupling in the auditory system of mammals [37]. The afferents are shown with dash-dotted red arrows targeted to the CNS, and the efferents are shown with dashed blue arrows. Mechanical coupling between the OHCs, and inner hair cells (IHCs) and the basilar membrane–tectorial membrane system are shown with solid black arrows.