It has been shown in previous sections that PVP-Ag nanowire networks evolve towards a definitive state of conductance after becoming fully activated, either by means of repetitive I–V cycles or applying super-threshold voltage over longer times. In the activated network, one or many pathways transport electrical currents. The modelling results (Fig. 2a,d,e) suggest that the pathway that opens first is the shortest-distance topological path between the electrodes. However, the model assumes a homogeneous initial distribution of junction resistance, which may not be the case in the real system, which has a larger interelectrode distance, as well as non-ideal contact geometry between nanowires and between nanowires and electrodes. Additionally, the self-assembled network may exhibit larger homogeneity variations that are not accounted for in the model. This implies that filament formation at the level of individual junctions might not always lead to an efficient activation sequence. For example if one junction in a less optimized pathway change its resistance state first, it may promote the activation of junctions in its immediate vicinity, thus giving rise to a winner-takes-all phenomenon, as recently observed by Manning and colleagues41 in a smaller PVP-Ag network. In any case, once a pathway is active, the network will keep further opening parallel pathways in order to optimize the conductance, or else, minimize power consumption, in order to comply with the laws of conservation of electrical current in a resistor network

Previous studies36 have shown the stability of active networks to last extremely long times as long as the energy pumped from a voltage or current source is high enough. Thus, once in the ohmic regime, a network can remain in this state unless, by means of a higher voltage or current injection, other physical phenomena such as joule heating or electro-migration breaks the individual junctions, thereby resetting them36,52. In our study, we have inspected the dynamics of the active network not only near the threshold voltage, but also in the sub-threshold regime. In this regime, the stability of the networks is still very high, but as the current through individual junctions decreases, the balance between junction formation by means of current injection and junction dissolution triggered by stochastic fluctuations is reduced. Decreasing the voltage in a linear sweep while acquiring time series facilitates the inspection of the dynamical events, signaling network instability by single junction breakdowns. This is assessed by analysis of the PSD power-law exponent (β) distributions in Figs. 5d,e. The β distribution in Fig. 5e shows that most of the magnitudes are centered around 1, with a small tail of values rounding up to a magnitude of 2. Typically, a 1/f power-law spectrum is indicative of a system in which scale-free dynamics is at play. To better understand how the network changes its dynamical behavior, we dissected the downward voltage ramp of the I–V cycle of Fig. 5c and examined the conductance time series at a voltage in which this dynamical balance was evident. Figure 6a shows that the network maintains a very stable (small fluctuations) conductance near 2.6 × 10−5 S up to approximately 15 s, after which it suddenly becomes unstable, exhibiting sharp and irregular conductance drops interspersed with short plateaus of unequal duration and magnitude, as well as large spikes. This unstable regime last for 3 s, after which the network automatically recovers the previous stable state, only this time with a slightly higher conductance. The level of fluctuations in this new stable regime appears to be slightly lower. Figure 6a also shows β values calculated at every one second interval in the conductance time series. This analysis reveals that when the network is in a dynamical regime that is stable against fluctuations, β values are closer to 1, whereas they approach 2 when the network succumbs to instability. Figure 6b shows PSDs calculated for a stable and unstable region in the conductance time series in Fig. 6a, indicated by the black triangle and pink diamond, respectively. The PSDs clearly distinguish the different dynamical regimes: the PSD is steeper in the unstable regime due to the stronger low-frequency fluctuations. Upon further inspection of Fig. 6a, with reference to the full distribution of sliced time series in Fig. 5c, it is evident that the onset of the unstable regime is not triggered by the negative voltage sweep, nor it is related to changes in the network appearing immediately after one voltage step is performed in the downward ramp, but instead appears randomly and unpredictably, and with an irregular duration. The unstable regime ends when the network recovers a stable conductance state, as signaled by reduced noise fluctuations. Therefore, the network not only adapt its conductive pathways to the nanowire junction topology whenever energy is supplied either from current or voltage sources, but can also adapt dynamically to random changes in the network, reconfiguring the underlying connectivity to find a new optimal pathway.

Figure 6 Reconfiguration dynamics in PVP-Ag nanowire networks. (a) Conductance time series selected at the 0.31 V step during the downward ramp of the IV cycle, with an overplot of the slope (β) of the power-law fit to the PSD computed for 1 s intervals (blue diamonds). The different spectral distributions of power in the self-organized dynamical regime (indicated by a black triangle) with the unstable regime (indicated by a pink diamond) are compared in (b). In each case, a linear fit is overplotted on the PSD. Full size image

Few nanowire network models have addressed stochasticity in the sub-threshold regime21. Our modelling of the activation and deactivation cycles (cf. Figs 2 and 3) provides new insights into the process leading to these changes in network. When the applied voltage is much lower than the threshold voltage, the transport hierarchy is revealed through the different irregular decay curves (Fig. 3b), attributed to stochastic junction breakdown. In reducing the voltage from threshold to sub-threshold level in small steps and long waiting times, we were able to track the transition from the stable state down to the regime in which random fluctuations have a larger influence on the dynamics. Whenever β is close to 1, individual junctions within clusters of high network connectivity are continuously switching off and on, with three different dynamical process (potentiation, inhibition and random breakdown) interacting and evolving in a self-organized way such that the connectivity backbone of the network is not compromised. Therefore, a junction that breaks in a cluster of high connectivity is propagated as a small change in conductance, and, depending on the local topology of these clusters, induces the breakdown of neighboring junctions, as the current in them is instantaneously reduced. But at the same time, when these secondary junctions are broken, the backbone of the network needs to carry more current to maintain conductance, so if a region is locally connectivity-depleted, another region will have to sustain a larger current, and thus, will potentiate the switching-on of junctions over that area. However, the more junctions are switched on, the higher the probability of breaking junctions in these newly connected areas. In this way, self-organized feedback produces the scale-free power-law distribution, as fast individual junction switching leads to changes in conductance for clusters of junctions over a slower time scale, and changes in cluster conductance produce changes in network pathway connectivity on an even slower time scale. This hierarchy is responsible for the characteristic 1/f spectrum in the PSD. From this interpretation, we can infer that the steepening of the power-law spectrum associated with abrupt changes in conductance (cf. Fig. 6a) occurs when a critical junction (one bridging areas of larger connectivity) breaks down, and splits a network pathway, reducing the total network conductance (by as much as half in the ideal case of two equivalent parallel pathways). Then, self-organized switching is disrupted by larger junction connectivity-depletion, and the connectivity of the network is spontaneously remapped until the broken pathway, or another parallel pathway, is switched back on. This remapping (or reconfiguring) process produces more low-frequency power in the PSD, as clusters of junctions or secondary pathways collectively redistribute connections, which occurs on longer time scales (lower frequencies) than the higher frequency switching fluctuations of individual junctions. When the network transitions back on to a highly conductive state, the conductance is slightly different, implying this complex dynamical process produces long-lasting changes in the backbone connectivity of the network. We can draw a parallel with long-term retention memory in the brain, which requires not only synaptic potentiation in neural connectivity but also persistent structural reconfigurations in connectivity, referred to as plasticity53.

Further insight into the relation between dynamics and connectivity can be gleaned by encompassing the neuromorphic nanowire network within a broader class of systems, optimal transport networks. In general, these are networks that can be represented by a weighted graph (directed or undirected), with edges representing the conductance (or resistance) between the different parts of the network, and whose solution can be obtained by solving a generalized version of Kirchhoff’s equations, that is, flow and mass conservation. Examples of such systems vary from artificial systems, such as power or water supply networks, to natural systems, e.g. vascular networks in plants or animals54. The optimality condition, though not always solvable, is computed by considering that transport is optimized whenever the connectivity map (weight between edges) minimizes power consumption. For example, for a fixed resistor network, current is hierarchically and homogeneously distributed in proportion to the equivalent resistance of the different parallel plus series pathways. In contrast, in vascular systems, which adapt flow to pressure differences54, dissipation is minimized by concentrating flow on the edges with larger conductance, thus transport is regulated by the formation of tree-like structures55,56. Neuromorphic networks clearly show this adaptive behavior, with the system evolving to create few, large connectivity pathways rather than many different pathways, as our activation curve and other recent research shows41. However, once the network is connected, parallel pathways, most likely in the form of closed loops or circuits that assist or reinforce the main pathway also appear. This further reduces global power dissipation. Interestingly, research on optimal transport structures suggests that the combination of a fixed unique backbone for transport with different recursively nested loops or cycles is the preferred connectivity map for adaptive networks, and appears once variations in the transport dynamics are introduced in the form of defects or fluctuating currents in the junctions, sources or sinks55,56,57,58.

Finally, it is worth considering whether the network dynamics observed in our neuromorphic nanowire system are characteristic of biological neural systems. In particular, we always observe a continuously fluctuating state, as if the system needs to be constantly oscillating, searching the phase space for a solution that guarantees long-term stability. It has been suggested that this edge-of-chaos state could be not only an optimal state of the brain, but a necessary condition for consciousness to exist43,59. However, although neural dynamics research has tended to focus on wave-like collective oscillations (e.g. in EEG signal acquisitions), other time series analyses continue to be developed50,60,61. Power-law PSD spectra, such as that observed in our nanowire networks, are indeed also observed in neural dynamics (e.g. neuronal avalanche size distribution), although whether these spectra can be interpreted as evidence for the brain operating at or near criticality remains a contentious issue62. Changes in power-law slopes, as observed in our nanowire network system, have also been observed in brain measurement data, where a subject performed simple tasks60. In our neuromorphic nanowire network, large changes in the power-law dynamics arise as random fluctuations break critical junctions. Introducing a control mechanism to disrupt selected regions of the network could force this remapping mechanism, that ultimately will allow us to harness the natural nonlinear feedback dynamics, thus navigating the large solution space of the network to solving different optimization or recognition tasks.