Nature | Vol 585 | 24 September 2020 | 521of these events to specific temperature responses driven by the Mott
transition (Fig. 2i–k), which provides an illustration of how the physics
of our nanoscale element produced a neuromorphic action potential.
Any new electronic component faces stiff competition from
entrenched transistor circuits for chip-level integration. Here we dem-
onstrate that simple networks of our third-order elements can perform
nonmonotonic operations and transistorless all-analogue computa-
tions. First, we constructed a simple network of three third-order ele-
ments (Fig. 3a) and included a tunable coupling resistor. By holding the
input voltages at one of two levels (to produce spiking or damped spik-
ing), and by carefully tuning the coupling resistor, spiking (or damped
spiking) at both the inputs may propagate such that, the output element
received a high (or low) signal, producing damped spiking (or spiking).
Spiking in only one of the two inputs may propagate to the output as a
high or a low signal depending on the coupling resistor. Thus, by tuning
the coupling resistor (Fig. 3b, c), we obtained behaviours corresponding
to NAND and NOR operations. Boolean operations using simple net-
works of neuromorphic elements have been modelled before and are
not intended as hardware primitives to replace transistor-based digital
logic, which is unlikely to be surpassed in efficiency at large scales^28.
However, the nonmonotonic operations based on the bifurcations of
the spiking are being studied as a basis for transistorless neuromorphic
primitives such as cellular automata^28.
To experimentally demonstrate analogue computing, we con-
structed an integrated array of 24 nanocircuit elements coupled by
an impedance matrix defined by a programmable crossbar array of
non-volatile memristive switches exhibiting pseudo-memcapacitance
(Fig. 4c). In the pseudo-memcapacitors, constructed using a material
stack consisting of two back-to-back metal–insulator–metal structures,
the resistance switching is accompanied by changes in capacitance
(Fig. 4d), and thus the passive crossbar array is programmed with a
problem (represented by a weight matrix), just as arrays of non-volatile
resistive memories are programmed. The oscillators are powered by a
single bias voltage, and their phases are monitored for convergence.
The solution of the problem represented by the connection matrix is
encoded in the phase of the oscillations^29. This system has approxi-
mate similarities to thalamo-cortical computations in the brain, which
occur in networks of oscillating neurons connected either via tunable
synapses or a hub/thalamus that processes and routes neural signals
(Fig. 4a)^30. This leads to synchronization within the dynamics of the
neural oscillations (for example, phase alignment), resulting in spati-
otemporal classification, for instance, natural language and face recog-
nition (Fig. 4b)^31 –^33. Although the idea of using phase synchronization
of oscillators to identify data, and to a limited extent to perform opti-
mization, has been previously demonstrated^33 –^35 , the combination of
third-order elements with pseudo-memcapacitors used here enables a
highly efficient and compact hardware implementation. Most digital or
analogue–digital hybrid approaches to neural networks require clock-
ing of the circuit, explicit feedback in the case of recurrent networks
(that usually involves digital–analogue conversions, amplification, and
so on), and often in the case of oscillator-based computing, construc-
tion of bulky transistor-based oscillators and a connection matrix, all
of which impede scalability^34 –^37. Many of these limitations continue to
exist despite recent efforts on two-terminal-memory-based Boltzmann
machines and Hopfield networks^38 ,^39. The neuromorphic-element and
pseudo-memcapacitor network is a transistorless all-analogue repro-
grammable system with all-to-all connectivity that does not require
clocking or explicit feedback, and is simpler to construct relative to
simulating these functions via more elaborate digital circuits.
To illustrate the system’s operation, we programmed toy instances of
the viral quasispecies reconstruction problem (generically formulated
as a maximum-cut graph problem), which seeks to identify genetic
diversities of intra-host viral populations (Fig. 4g)^40 , and is important
in ensuring the effectiveness of medications especially against viral
species exhibiting diversity^41. The problem is solved by repeatedly
partitioning a graph (network of genomic reads and conflicts) into two
sets of vertices (reads) by maximizing the edges (conflicts) between
the sets (known as a maximum graph cut), so that the eventual solution
consists of sets (inferred mutations) of reads with minimized intra-set
conflicts^42. The problem is nondeterministic polynomial-time (NP)-hard
and is not efficiently addressed by traditional von Neumann computers,
necessitating novel software algorithms^40 ,^42. Generic formulations of
graph-partitioning problems are being used for benchmarking per-
formance of optical, quantum and electronic NP-hard optimization
solvers, enabling future performance comparisons^43. In addition to the
illustrative example chosen here, any problem that can be represented
by an Ising formulation can be programmed on this hardware.
The experimental solution to a subset of the programmed problem
displays phase synchronization consistent with the optimal solution
to division of the corresponding graph (Fig. 4h–j). In a similar fashion
to solving optimization problems with recurrent neural networks,
this approach is prone to converge to incorrect solutions, which may
be only locally optimal (local minimum of a non-convex function).
Transient chaotic dynamics—one of the properties of neuromorphic
elements—can excite the system out of local minima and enable con-
vergence to a global minimum (optimal solution)^3. A single third-order
element exhibits a range of behaviours—including periodic oscillations
and chaotic dynamics—and so biasing the neuromorphic elements in a
region that exhibits chaos (vext = 1.7 V) leads to clearly better statistics in
minimizing the errors in solutions, relative to an operational region that
produces non-chaotic periodic oscillations (vext = 1.3 V) (Fig. 4k, l). This
also enables modulation of the input bias in order to tune the degree
of chaos while approaching convergence, similar to computational
simulated annealing. We further provide two benchmark metrics: time020406080 02040600.00.71.40.00.71.40.00.71.4N 1N 2Noutvin1vin2iin1iin2ioutRoutab iout = iin1 NAND iin2
00110010011011101011000111000c1iout = iin1 NOR iin2iin1(mA)iin2(mA)iout(mA)t (μs) t (μs)Fig. 3 | Experimental demonstration of universal Boolean logic via
nonmonotonic spiking behaviour. a, Schematic of the network of three
neuromorphic elements (N 1 , N 2 , Nout), with each constructed using a third-order
element, an external capacitor and a resistor. b, Temporal dynamics of the
three currents marked in a for Rout = 427 Ω, showing the NAND operation.
c, Temporal dynamics of the three currents marked in a for Rout = 389 Ω,
showing the NOR operation.