522 | Nature | Vol 585 | 24 September 2020
Article
to solution (or convergence) and the accuracy (percentage of attempts
converging on the optimal solution), as a function of the problem size,
N (Fig. 4m). A comprehensive performance comparison to competing
digital and hybrid technologies crucially awaits widespread reporting
of generic benchmark metrics such as the energy required to reach a
solution or the number of solutions obtained for a joule of energy.
Additional information is provided in Supplementary Information
section 9, Supplementary Figs. 26–31.
In conclusion, we have demonstrated that it is possible to incorporate
the Mott transition in NbO 2 as an additional dynamical process to con-
struct an isolated nanoscale electronic circuit element with third-order
complexity. The element can be designed to produce optimal inter-
actions among its constituent electrical and thermal components,
such that it produces neuromorphic action-potential behaviours when
powered by a constant voltage source. We demonstrate two nonmono-
tonic and complete logic operations using one simple network of our
elements, and we further demonstrate a transistorless all-analogue
network of neuromorphic elements to solve computationally hard
problems that have far-reaching applications in alleviating the von
Neumann bottleneck of present digital computers. This result enables
extremely compact and highly functional neuromorphic computing
primitives.Online content
Any methods, additional references, Nature Research reporting sum-
maries, source data, extended data, supplementary information,
acknowledgements, peer review information; details of author con-
tributions and competing interests; and statements of data and code
availability are available at https://doi.org/10.1038/s41586-020-2735-5.- Mead, C. Neuromorphic electronic systems. Proc. IEEE 78 , 1629–1636 (1990).
- Mainzer, K. & Chua, L. Local Activity Principle (Imperial College Press, 2013).
- Kumar, S., Strachan, J. P. & Williams, R. S. Chaotic dynamics in nanoscale NbO 2 Mott
memristors for analogue computing. Nature 548 , 318–321 (2017). - Chua, L. Memristor, Hodgkin–Huxley, and edge of chaos. Nanotechnology 24 , 383001
(2013).
0.0 0.1 0.20.3050Vo(mV)t (μs)01020Frequency (a.u.)Err (%)0.1 (^21020)
1
10
tsol
(μ
s)
N
0.01
0.1
1
ps
(^0) –2 02
30
60
Cc
(pF)
Vc (V)
l
0
Cc (pF)
vext = 1.3 V
0 60
Gc (mS)
1
vext = 1.7 V
Err= 0
i j
h
m
e f
A A′
N = 8
cd
vext
Oscillators
Connection matrix
≈
A
0
1 0
1 1 0
0 1 1 0
0 1 1 1 0
t
a
Spatio-temporal synchronization
k
g
Viral genome Reads C-graphMutations
Con ict
Neuron groups (cortical columns)
Hub (thalamus)-mediated neural
connections
Thalamo-
cortical computing
Computing with neur
omorphic elements
b
Potential
Fig. 4 | Experimental demonstration of neuromorphic analogue
computing. a, Illustration of simplified thalamo-cortical neural topology in a
brain. b, Illustration of the phase synchronization of neuron oscillations
resulting from the thalamo-cortical network. c, Schematic illustration of the
experimental system with the neuromorphic oscillators and the connection
matrix formed by a crossbar array of pseudo-memcapacitors. d, Quasistatic
capacitance as a function of the applied voltage of a prototypical
pseudo-memcapacitor. e, f, Conductance, Gc (e) and capacitance, Cc (f) maps of
a crossbar array of linear size N = 24. g, Depiction of the viral quasispecies
reconstruction problem, where reads of a viral genome are represented by a
conf lict graph (C graph), the graph partitions of which produce the inferred
mutations. h, Oscillations from five neuromorphic oscillators connected to the
part of the crossbar array represented by the dashed squares in e and f. Two
phase groups (A and A′) are identified. Colours of the data correspond to the
node colours of the graph in i. i, The problem graph corresponding to the
matrix within the dashed squares in e and f. The minimum conf lict partitioning
solution—also represented by the phase group A in h—is marked with a pink
dashed ellipse. j, The adjacency matrix representing the graph in i.
k, Illustration of minimization of errors (Err, percentage error from an optimal
solution). The blue curve with no perturbations converges to a local minimum,
whereas the orange curve with perturbations escapes all local minima to
converge to the global minimum (Err = 0). l, Error distribution histograms of
two sets of solutions obtained by applying two different vext to the oscillators.
m, Time to solution (tsol, time for phase differences to settle) and probability of
solution ps (fraction of convergences to the globally optimal solution), as a
function of N. The shaded regions are the ranges over 10 to 60 experiments at
different N. a.u., arbitrary units.