^{*}

The authors have declared that no competing interests exist.

Conceived and designed the experiments: JC SM. Performed the experiments: JC JAM. Analyzed the data: JC JAM SM. Wrote the paper: JC SM.

Nanotechnology produces basic structures that show a significant variability in their individual physical properties. This experimental fact may constitute a serious limitation for most applications requiring nominally identical building blocks. On the other hand, biological diversity is found in most natural systems. We show that reliable information processing can be achieved with heterogeneous groups of non-identical nanostructures by using some conceptual schemes characteristic of biological networks (diversity, frequency-based signal processing, rate and rank order coding, and synchronization). To this end, we simulate the integrated response of an ensemble of single-electron transistors (SET) whose individual threshold potentials show a high variability. A particular experimental realization of a SET is a metal nanoparticle-based transistor that mimics biological spiking synapses and can be modeled as an integrate-and-fire oscillator. The different shape and size distributions of nanoparticles inherent to the nanoscale fabrication procedures result in a significant variability in the threshold potentials of the SET. The statistical distributions of the nanoparticle physical parameters are characterized by experimental average and distribution width values. We consider simple but general information processing schemes to draw conclusions that should be of relevance for other threshold-based nanostructures. Monte Carlo simulations show that ensembles of non-identical SET may show some advantages over ensembles of identical nanostructures concerning the processing of weak signals. The results obtained are also relevant for understanding the role of diversity in biophysical networks.

The route to functional nanoelectronics is mined with weak signals, thermal noise and significant diversity between nominally identical units. This hardware variability may be undesirable for most applications. In particular, the threshold voltage mismatching of individual electronic transistors constitutes a serious problem in voltage-driven applications

Frequency-based signal processing is characteristic of the neural populations in the brain and concerns the transduction of external information into patterns of neural activity, discernible rhythms, and synchronization processes

We explore here different information processing schemes with heterogeneous groups of non-identical nanostructures that make use of the above biological concepts (diversity, rate and rank order coding, synchronization). To this end, we simulate the integrated response of ensembles of single-electron transistors (SET) whose threshold potentials suffer from a high individual variability

This work builds and extends upon recent studies by us on signal processing schemes using heterogeneous

In all cases, we use a mixed continuum-Monte Carlo approach where the nanostructure ensembles are not modeled as replicates of the same unit with constant physical properties but as statistical distributions of physical parameters characterized by average and width distribution values. A significant novelty with respect to previous work

Models describing how neural spike trains convey sensory information can be relatively complex

In particular, parallel arrays of resistance-single electron transistors (R-SET) can be used for frequency-based image processing _{i}_{i}_{i}_{225} (diameter 1.8 nm) is 0.6 aF approximately _{40} (diameter 1.2 nm) show a capacitance as low as 0.35 aF. The current–voltage curves of a single ligand-stabilized NP obtained by scanning tunneling spectroscopy can be described by SET equivalent circuits that give tunneling resistances and effective capacitances in the range 100 MΩ – 10 GΩ and 0.1 – 10 aF, respectively

The basic information processing unit is an ensemble of _{i}

All simulations are conducted at non-zero temperature (

The building block _{i}_{i}_{i}

At low temperatures, the Coulomb blockade gives the relaxation oscillation of _{i}

and depends on the time _{i}C_{i}_{i}_{i,}_{th} = _{i}_{i}_{,th}, the NP potential tends asymptotically to

Tunneling can be described as a stochastic process of rate _{i}_{h}

To simulate the charging–tunneling process, a mixed continuum–Monte Carlo approach is used _{i}_{i}_{,th} and the charging process is resumed. The tunneling event is considered to be instantaneous with respect to the average charging time. The periodic sequence of charging-tunneling events produces the electrical potential spikes of

The simulations take into account the experimental fact that nanostructures show statistical distributions of physical parameters characterized by their experimental average and width distribution values. Our objective is to obtain the integrated response of an ensemble of R-SET showing a significant heterogeneity in the individual physical properties. The nanostructure variability is incorporated by considering random distributions of relative width

The frequency of tunneling events (the spikes of _{black} + _{in}_{white} (_{in} is the grey level of the pixel (_{in} = 0 for black and _{in} = 1 for white), _{black} is the potential for a black pixel, and _{black} + _{white} is the potential for a white pixel. This input potential is then applied to the array of parallel R-SET of _{i}_{,th}. The frequency of the spikes generated by this particular R-SET is proportional to the input potential

Two methods have been considered to process the grey level image differing in the processing times allowed. The first method is reminiscent of the _{grey} times on average for the maximum input potential (_{black} + _{white}), where _{grey} is the number of grey levels used to reproduce the image.

We consider heterogeneous and homogeneous ensembles of R-SET processing the sinusoidal input signal_{0} is the minimum applied potential and _{1}/2 and

The ensemble response to

Neuronal activity in the brain involves the synchronization of neuronal ensembles as a response to a common input signal. We study here the synchronization of ensembles with identical and non-identical R-SET coupled by a common resistance _{c}. When a coupling resistance _{c} is incorporated between the electrode at potential _{ij}

The complex order parameter

_{i}C_{i}

Processing of a grey image by homogeneous (

The results obtained with the two coding schemes are shown in _{white} = 100 mV and _{black} = 60 mV, except for case of the rate coding by identical R-SET (_{black} to 75 mV in order to avoid the potential region where the ensemble shows no response (between 50 mV and 65 mV in the inset of _{grey} = 16 grey levels in the processed image out of the 256 levels in the original image. The processing time allowed in the rank order coding (only 30 ns) is shorter than in the rate order coding because a

The oscillating phenomena observed in R-SET ensembles are reminiscent of those found in models of integrate-and-fire neurons

To better understand the effects of nanostructure variability on signal processing, _{0} = 60 mV. We consider two different periods (_{1} = 100 mV and _{1} = 18 mV) in Eq. (4) to allow for slow and fast variations in the

The responses of ensembles with _{1} = 100 mV) that is higher than the average threshold potential _{th}≈80 mV for most of the time (_{min}(_{max}(

The same problem shown in Fig. 3 but now for _{1} = 18 mV, so that the ensemble operate most of the time in the

The effect of threshold distributions in the optimization of information transmission with ensembles of noisy elements through suprathreshold stochastic resonance has been studied previously by McDonnell, Stocks, Pearce, and Abbott.

To illustrate the different individual responses of the R-SET in the ensembles, _{min}) and the maximum (_{max}) number of spikes obtained during the processing time (see the two central rows). In the case of non-identical nanostructures, these curves correspond to the highest and the lowest threshold potentials in the ensemble, respectively. Finally,

The shaded regions in _{max} (R-SET with the minimum threshold potential) spikes more often than the potential _{min} (R-SET with the maximum threshold potential). Because identical nanostructures have the same threshold potential, the differences between _{max}(_{min}(

If we compare the time dependence of the total number of spikes (

When the frequency of the signal is increased (

_{1} = 18 mV and the number of R-SET in the ensemble has been increased to 64 to improve the resolution (the rest of parameters have the same values as in _{min}(_{max}(

The situation changes dramatically when we consider the heterogeneous ensemble. Although the individual responses of R-SET are still unreliable (

The dynamic formation of ensemble synchronized states can also be used as a temporal coding mechanism because firing synchrony is a collective characteristic which is robust to individual failures. Synchronization could then be used to detect particular events and process temporal data series. Therefore, it is useful to compare the synchronization and desynchronization processes in two ensembles of _{c} = 0 to a maximum value _{c} = 20 (_{c} = 0. The synchronization and desynchronization processes of two ensembles of 20 identical (

Synchronization of R-SET ensembles for an applied potential _{c} (and then the coupling strength _{c}) changes with time following a triangular signal (B). The modulus of the order parameter _{c} from 20 (Fig. 5C and E) to 40 (Fig. 5D and F). We assume that the ensembles are initially (

The degree of synchronization is evaluated from the phases of the individual oscillators that can be calculated from the time between two consecutive maxima (_{i}_{c} increases from zero, _{c} tends to zero again (_{c} = 0 (

Synchronization occurs over a central time window which is more extended for the identical (_{c} = 20 (_{c} = 10 (data not shown), full synchronization is not achieved.

Full synchronization (

It is tempting now to relate some of the above results with those found in biophysical networks. Synchronization of neural networks is usually analyzed using integrate-and-fire models that incorporate coupling schemes more complex than that considered here

However, synchronized states may constitute stable attractors difficult to reset _{c} increases but they are also difficult to desynchronize again when _{c} decreases. If the

Nanotechnology is bound to produce ensembles of structures that show a significant variability in their individual physical properties. This fact constitutes a serious problem because nominally identical units are required in many practical applications. In particular, this is the case of information processing. However, the results obtained in

A possible experimental realization of a SET is the nanoparticle-based transistor that mimics some of the spiking characteristics of neurons. We have simulated the integrated response of ensembles of SET that suffer from a high individual variability. The ensembles are not modeled as replicates of the same nanostructure with constant physical properties but as statistical distributions of physical parameters characterized by some average and width distribution values. We consider simple but general information processing schemes to draw conclusions that should be of relevance for other threshold-based nanostructures. Monte Carlo simulations suggest that ensembles of heterogeneous nanostructures whose physical parameters follow statistical distributions approximately controlled could be more efficient than those with identical units because of the extended dynamic response. The integrated ensemble response is based on collective phenomena robust to individual failures and variability.

The three case studies considered here permit to understand the effects of nanostructure variability in a broad range of information processing schemes that could also be relevant for biophysical systems. In particular, the neural variability and its functional significance, together with the role of stochastic resonance and noise in the nervous system, have been discussed recently

Suggestions from Dr. Vladimir Garcia-Morales, Technical University of Munich, are acknowledged.