Atomic switch network devices were characterized using a range of potentiostatic inputs, including constant and ramped DC as well as sinusoidal AC signals. These complex atomic switch networks are shown to exhibit various nonlinear behaviors, depending on the magnitude and timing of both present and prior input signals. Behaviors include both weak (continuous I–V loop hysteresis) and strong (discrete threshold switching) memristance as well as nonlinear frequency response (higher harmonic generation) and persistent fluctuations in conductivity under constant bias (recurrent connectivity); results which were found to agree with a recent theoretical study of current flow in memristor networks [22]. Operation of the device using pulsed voltage stimulation produced network-specific emergent behaviors, as spatially localized conductive channels akin to feedforward subnetworks were formed within the embedding recurrent network. While there are significant differences between these atomic switch networks and biological neural networks (NNs), we demonstrate the physical implementation of high-level NN features in an inorganic structure, including bottom-up self-assembly that is reminiscent of neuronal growth in the brain [23], nonlinear input-dependent conductance response which strongly resembles the function of biological synapses [11], [12], and emergent properties considered fundamental to brain function - recurrent dynamics which gives rise to large persistent, correlated network responses and the activation of feedforward subnetworks [8], [9], [24]–[27].

A great deal of effort has been put towards building biologically inspired computational hardware [35] – [42] , though matching the complexity of the brain in a usable electronic device presents an exceedingly difficult engineering challenge. Fabrication requirements force design concessions, such as approximating the complex, recurrent connectivity between neurons by a simpler network geometry. The amenability of crossbar structures to conventional fabrication techniques has led to their use in neuromorphic hardware, with pre- and post-synaptic CMOS neurons connected by memristive elements at the crosspoints [43] . This is an ideal hardware implementation of a 3-layer neural network model [44] , where input and output neurons are connected by a synaptic “hidden layer” of variable strength, and is also a promising platform for building dense, fast solid-state memory devices [45] . However, the structural simplicity of the crossbar architecture is both a strength, enabling independent control of each synaptic element, and a weakness, since the well-defined grid lacks complex structures with the recurrent connections believed to be essential to brain function [6] , [25] . While it is possible to program these features into a software model implemented on neuromorphic hardware, the physical existence of these complex structures may be essential to successfully generate the requisite spatiotemporal interactions between multiple signals simultaneously traveling through the network [11] , [46] .

These mechanisms collectively produce the memristive switching and synaptic memory functions exhibited by a single atomic switch. Specifically, ‘weak’ memristance resulting from redistribution of Ag + dopant cations across the insulator leads to ‘strong’ memristance characterized by abrupt switching through metallic filaments formed once the Ag + cations reach the cathode and are reduced to metallic silver [13] . TEM studies have shown that the metallic silver filaments formed during switching are surrounded by a sheath of β-Ag 2 S, a conductive phase of silver sulfide normally unstable below 170°C [28] , possessing a body-centered cubic structure with sulfide anions forming channels in which silver cations are delocalized, highly mobile and dynamically correlated [32] , [33] . This non-equilibrium phase transition is attributed to a relaxation of strain induced by lattice mismatch between Ag 0 and α-Ag 2 S, the electrically insulating room temperature phase [34] . In the absence of applied bias, thermodynamic pressures return the Ag 2 S to its room-T α-phase, which drives the dissolution of the Ag o filaments and turns the atomic switch OFF at a rate dependent on the history of applied bias, producing the observed memory effects.

Previous reports on the synapse-like properties of single atomic switches have demonstrated features similar to short-term plasticity and long-term potentiation, where applied bias voltage produced a junction conductance dependent on the history of stimulation (pulse frequency, length) [14] . Individual atomic switches exhibit time-dependent nonlinear conductance due to several related mechanisms: (1) bias induced Ag + migration, (2) electrochemical redox reactions involving Ag + /Ag 0 to produce metallic filaments, and (3) an associated non-equilibrium α/β-Ag 2 S phase transition [28] , which all compete with thermodynamically driven stochastic renormalization to the equilibrium OFF state. Though atomic switches can be configured to operate in an essentially nonvolatile manner similar to memristors—two-terminal circuit elements whose resistance depends on the history of charge passed through them [29] —their volatility indicates that they are more properly classified as “memristive systems” [30] , [31] .

Similar to the electroforming step usually required to activate single atomic switches and memristors [52] , the observed transition from weak to strong memristive behavior is assigned to two related mechanisms. In poorly conducting regions comprised mainly of Ag 2 S, anodic silver dissolves into and travels across the electrically insulating sulfide as Ag + , decreasing resistance and producing a weakly memristive effect. In regions of higher Ag + dopant concentrations, mobile cations reach the cathode and are reduced to Ag 0 , creating metallic filaments across the insulator that cause an abrupt change to an ON state with a sharp increase in conductance at V t associated with the electrochemical process of filament formation. At the network level, the bias-induced creation of additional memristive junctions and filament formation across existing ones combine to produce the theoretically predicted transition of network I–V behavior to a strongly memristive phase (schematically illustrated in Figure 2b–d ) as the proportion of switching elements in the network exceeds the percolation threshold (50%) [22] . Having undergone this transition, the continuously swept network operates as a hard switching memristor shown in Figure 2a (inset). All further data presented was acquired from devices following activation.

(a) Representative example of initial bias sweeps (0–5 V sweep at 1 V/s) applied to a pristine device which steadily activate higher percentages of atomic switches, resulting in increased current. After 11 sweeps, the device resistance decreases from ∼10 MΩ to ∼500 Ω. Subsequent ±1.5 V bipolar sweeps result in repeatable pinched hysteresis behavior (inset: R OFF = 25 kΩ, R ON = 800 Ω), and bistable switching. (b–d) Schematic representation of the mechanism producing the I–V characteristics shown in (a). The network initially consists of weakly memristive junctions and ohmic contacts (b). Continued application of unipolar bias voltage (c) drives the dissolution of silver into silver sulfide, increasing the number of memristive elements, while cation migration across extant memristive junctions leads to filament formation and the onset of hard switching behavior. (d) After the proportion of strong memristors exceeds the percolation threshold (ρ>0.5), the network functions reliably in the hard switching regime.

Theoretical analysis of current flow in memristor networks during bias voltage sweeps indicated the possibility of a phase transition in device behavior from ‘weak’ to ‘strong’ memristive regimes [22] . Initial voltage sweeps of these network devices ( Figure 2a and S1 ) typically demonstrated smooth, pinched hysteresis loops characteristic of weakly memristive systems followed by an abrupt, nearly discontinuous jump to a distinct, high conductance ON state occurs at an activation bias voltage (V a ). This behavior represents activation of the network and is shown as an illustrative example of a network device undergoing a behavioral phase transition similar to the bias-driven forming step required to activate single resistive switches. Following network activation, devices subjected to repeated bias sweeping generally exhibit robust, strong memristive behavior, typified by hard switching (inset). Robust switching over 10,000 cycles was demonstrated at an operational threshold voltage (V t ) of reduced magnitude ( Figure S2 ) as compared to the formation bias voltage, a general phenomenon in resistive switches [52] . While the specific magnitude of V a and V t differ significantly between devices due to inherent variability in the solution-phase methods used to fabricate them, the qualitative transition from weak to strong memristive behavior was observed regularly, consistent with theoretical predictions [22] .

Based on the view that recurrent connectivity is essential to brain-like function, we have built, characterized and operated devices using massively interconnected (10 9 junctions/cm 2 according to analysis of SEM images), silver nanowire networks functionalized with interfacial Ag|Ag 2 S|Ag atomic switches. These nanowire networks were prepared through self-assembly without pre-patterning of the network topology using the electroless deposition of Ag from Cu inside the SU-8 reaction well of an I/O device platform [18] , [47] . Specifically, spontaneous oxidization of metallic copper through reaction with dilute aqueous solutions of AgNO 3 produces a metallic silver structures with variable morphologies depending on the concentration of Ag + and distribution of Cu [48] – [50] . Dendritic silver nanowires with minimum feature sizes <100 nm seen in Figure 1b were produced by using lithographically patterned Cu posts shown in the inset of Figure 1a . Control over the size and distribution of Cu seeds increased device yield by ensuring the formation of conductive pathways between the Pt device I/O electrodes as seen in Figure 1b . Ag|Ag 2 S|Ag interfaces were formed spontaneously within the network during gas phase sulfurization [51] . Following optimization of fabrication protocols, a total of 96 networks were used for the device characterization described below.

Network-specific properties

While weak and strong memristive behavior can be exhibited by single resistive switches, the most interesting features of this complex atomic switch device are its network-specific properties. In order to confirm that the entire network was involved in processing the input signals, devices were imaged using an IR camera with 20 mK sensitivity to track Joule heating from current flow during slow bias sweeps. The IR images revealed power dissipation occurring across the network, indicating that the phase change in network I–V behavior was not attributable to the formation of a single maximum conductivity pathway of switches arranged in series between the active electrodes [18]. The distribution of activity indicates that the observed I–V characteristics are due to the sum of parallel current flow, meaning that network structure and connectivity are actively influencing device function.

As recent theoretical models predict passive generation of second harmonics in both singular memristors and in random networks, the distribution of switch function throughout the network was examined through analysis of the device's frequency response [22], [53]. Simulation of current flow in memristor networks indicate that 2nd harmonic generation will occur under an applied sinusoidal voltage in networks whose percentage of hard switching junctions exceeds the percolation threshold [22]. Further, the relative magnitude of higher harmonics is predicted to increase with the relative number of hard switching junctions. Following activation, device response to a 10 Hz sinusoidal voltage signal varying in strength from 250 mV to 4 V shows a large increase in higher frequency components after functionalization (Figure 3b). The proportion of higher harmonics generated increases with signal amplitude (Figure 3c), with the largest increase occurring between 250 and 500 mV. A larger degree of higher harmonic generation is consistent with an increased number of memristive junctions operating in the hard switching regime above V t (∼0.5 V). Both the distributed power dissipation [18] and harmonic generation are characteristic of activity distributed throughout the network.

PPT PowerPoint slide

PowerPoint slide PNG larger image

larger image TIFF original image Download: Figure 3. Frequency Response – distributed conductance. (a) Amplitude spectrum from a Fourier transform of a control device's response to a 2 V, 10 Hz sinusoidal input signal compared to (b) that of a functionalized device which shows enhanced overtones of the input signal with respect to (a). (c) Plot of 2nd and 3rd harmonic generation in current response as a function of bias voltage in both functional (black) and control (gray) networks. Harmonic magnitudes are represented as percentage of the fundamental for a 10 Hz sinusoidal input signal. https://doi.org/10.1371/journal.pone.0042772.g003

Having characterized atomic switch operation in an interconnected complex network, we examined the device for emergent behaviors specific to its brain-like recurrent structure. Structurally, the atomic switch network is recurrent in the sense that there exist pathways such that electrical signals produced at one junction may lead to (delayed) feedback at the same junction. Here we present experimental evidence of spatially distributed and correlated network dynamics, which are attributed to such recurrent connectivity. These recurrent dynamics are presented as an emergent property of the atomic switch network.

Applying a constant 1 V DC bias (Figure 4a) produced persistent, bidirectional fluctuations—both increases and decreases—in network conductivity of large magnitudes (∼20–150%) over a range of time scales (seconds-hours). In the absence of recurrent structures within the network, the filamentary mechanism of an atomic switch implies that conductivity would increase monotonically under constant DC bias. The applied voltage leads to the thickening of filaments until the potential drop across the junctions is insufficient to reduce more silver cations [13]. However, large bidirectional fluctuations (ΔI greater than 100% on the scale of hours) in the current response persisted for several days under constant applied voltage, demonstrating that the complex network connectivity inherently resists localized positive feedback that would lead to the serial formation of a single, dominant high conductivity pathway between electrodes. Rather, recurrent loops in the network create complex couplings between switches, resulting in network dynamics that do not converge to a steady state even under constant bias. A single switch turning ON does not simply lead to an increased potential drop across the next junction in a serial chain, but redistributes voltage across many recurrent connections that can ultimately produce a net decrease in network conductivity. This behavior represents a network-scale analog of defect-defect interactions that have been observed to produce current fluctuations in metal nanobridges [54]. The nanoscale switch filaments couple these interactions with electrochemical redox processes, leading to significant changes in the conductivity state of the entire network.

PPT PowerPoint slide

PowerPoint slide PNG larger image

larger image TIFF original image Download: Figure 4. DC Response – recurrent dynamics. (a) Time traces of current response to 1 V DC bias show large current increases and decreases at all time scales around a mean of 5.81 µA (172 kΩ); shorter time traces (ii–iii) are subsets of (i). Representative device parameters: R OFF >10 MΩ, R ON <20 kΩ, V T = 3 V during activation (b) Fourier transforms of DC bias response for Ag control (grey) and functionalized Ag-Ag 2 S (black) networks. The power spectrum of the functionalized network displays 1/fβ power law scaling (β = 1.34). https://doi.org/10.1371/journal.pone.0042772.g004

These fluctuations are of a magnitude significantly greater than what can be considered noise. An internal control experiment compared Fourier transformed current responses (Figure 4b) of the devices to constant voltage before and after functionalization. The formation of atomic switch junctions expands the degree of correlation in current fluctuations, producing 1/f-like behavior across the entire sampled range, far exceeding that of control devices (unsulfurized silver network, grey line in Figure 4b), which flattens to white noise and some high energy, high frequency fluctuations attributed to arcing between neighboring wires. Functionalization with atomic switch elements increases the influence of past events on the present state of the network, in accordance with their memristive characteristics [14], [15], [55]. This results in an expanded degree of correlation in the measured frequency response. Similar 1/f spectra have been observed along with current fluctuations in other resistive switching systems, exhibiting relative resistance changes ΔR/R ranging from <0.002 for metallic filaments to an experimental and theoretical maximum of 0.5 in the semiconducting high resistance OFF state [56]. The network device of Figure 4 is operating in an intermediate state with an average resistance of 172 kΩ (compared to R OFF >10 MΩ) and fluctuations of ΔR/R∼1. In order to produce relative resistance changes of such high magnitude, switching events within the network must be correlated. While stochastic processes may be involved in the correlation of these fluctuations [55], [57], their magnitude and persistence is an emergent feature of recurrent connectivity in the device architecture that has not been observed in simpler atomic switch geometries.

Inside the generally recurrent structure of the brain's neural network, there is evidence for the existence of feedforward subnetworks utilized for the fast propagation of certain signals [24]. In this device, persistent fluctuations in current under constant DC bias are produced by the recurrent network architecture, creating operational dynamics that resist the feedfoward activation of serial chains of switches. However, by altering the form of the input signal, we were able to independently operate conductance channels between different pairs of electrodes within the same device. The application of a single, large voltage pulse (±3 V, 1 s) selectively switched connections between electrode pairs ON and OFF (Figure 5a) with a R ON /R OFF ratio greater than 30. In the example shown, the conductive paths between the two channels overlap spatially, yet are switched independently, indicating that local sub-regions of the network can transition to distinct operational modes despite being embedded within a highly interconnected, largely metallic structure. This is analogous to the presence of feedforward subnetworks within the recurrent architecture of the cortex. Single pulses of sufficient magnitude overwhelm the recurrent dynamics and induce feedforward activation of local sub-regions along a path connecting the involved pair of electrodes without significantly altering the conductivity of other spatially intertwined channels within the same nanowire network.

PPT PowerPoint slide

PowerPoint slide PNG larger image

larger image TIFF original image Download: Figure 5. Distributed Memory Storage from Network-scale Switching. (a) The device operates as a 2-bit non-volatile memory device. The resistance states across two channels (i–iii and ii–iv) are monitored. ON/OFF switching of each channel is induced using super-threshold pulses (3 V, 1 s in duration); the threshold voltages for each channel are ∼1.5 V. The resistances are measured every 5 s with a sub-threshold 200 mV, 100 ms pulses. (b) Although the device operates with a four state output (both channels ON, 1 ON/1 OFF, etc), the network's internal configurations show diverse correlated patterns, from no correlation (blue) to total correlation (yellow). The figure shows correlation coefficients of channel resistances for all 6 pairwise electrode combinations. The correlation coefficients are calculated during each of the 4 network switching configurations; the black and red bars (insets) show the channels that are ON in the switching state. https://doi.org/10.1371/journal.pone.0042772.g005

The degree to which pulse-mode channel creation influences overall network connectivity can be visualized in electrode resistance cross-correlation matrices (Figure 5b). In this case, net electrode resistance is calculated from the pair-wise resistances to be a representative measure of the overall connectivity of a given electrode to the network. The correlation strength (denoted by color) represents the degree to which a pair of net electrode resistances fluctuate in unison, interpreted as a measure of the number of shared network sub-regions connected to both electrodes (Supporting Information). Correlation strength increases strongly between electrodes connected by an ON channel, and decreases again when the channel is switched OFF, with a varying degree of influence on electrodes not directly involved in the switching. This implies that spatially central regions of the network can be selectively associated with particular pairs of electrodes without globally increasing the network connectivity. However, when conductive channels exist between all four electrodes, the overall magnitude of correlation in the network is correspondingly large, as fluctuations are spread evenly throughout the increasingly metallic network. This simple example of the interaction between local and global operational characteristics is a promising indicator of the possibility for the creation of a brain-like hierarchy of distinct functional regions within a single network where the functional connectivity of the network itself is both dynamic and self-organized [58].