520 | Nature | Vol 585 | 24 September 2020
Article
no external series resistor is used); I 1 and I 2 are constants defining
the bounds of the hysteresis; and sgn is the sign function. Additional
details are provided in Supplementary Information section 7, Sup-
plementary Figs. 20–25.
This is a third-order model that contains three state variables, each
of which has a distinct dynamical equation: T, vm and Rmet. Note that
each order of complexity is a physical process with its associated state
variable; for example, the Mott transition dynamics is represented by
Rmet, where equation ( 5 ) is written in integral form instead of differential
form. Although the underlying physics and the electrical behaviour can
be represented in more accurate and elaborate forms, we chose a simple
representation to provide an intuitive illustration that a third-order
system can enable neuromorphic dynamics. Numerical simulations
of this model were in agreement with experimental data (Fig. 2e–g),
thus demonstrating how a Mott insulator, which is not a typical elec-
tronic element, can be the basis for a nonlinear response that produces
dynamical behaviour in a circuit.
An action potential in a biological neuron consists of three events:
up-shoot, lowering and under-shoot in potential, relative to the resting
state of the neuron, driven by ionic transport (Fig. 2h, i)^14. The dynamics
of the current flowing into the neuromorphic element contains similar
shapes within each spike (Fig. 2j). Using our model, we correlate each020406080 100 1200.00.51.01.5im(mA)1.401.451.50vext(V)i0.0 0.2 0.40.00.30.60.9im(mA)Vm (V)0.0 0.30 30.00.51.0im(mA)0 30 3vext = 1.8 V0.3 0.6 0.75 1.000.50 0.75vext = 1.95 V vext = 2.05 V vext = 2.1 Vt (μs)abcdg0.0 0.3 0.00.30120.00.0 0.2 0.4 0.0 0.3 0.25 0.500.30.60.9i (mA)m012efvext = 1.7 V vext = 1.9 V vext = 2.1 V vext = 2.2 VExperimental dataSimulation of modelt (μs)Analogy to a biological neuron–2–100.00.1 0.25001,0001,500Dendrites
Cell
bodyNucleusAxonAxon
terminals
and synapsesPhysico-chemical
stimuliElectrical impulse
(action potential)tPotential0 V12
3
41Inter-cell
transmissionCell
wall
Inside cellOutside cellK+Na+≈ Inactivity ≈ UndercoolingNa+K+Na+K+
≈ Cooling ≈ Heating and Mott transitionhjk123 41Chaos BeatsAction
potentialBeyond-threshold
activityActionPeriodic
oscillationsim(mA)0.00.51.00 3t (μs)
Beyond-threshold
activitypotentialVm (V)im(mA)i (mA)m0 3 0 30 3 0 31 Resting state 2 Depolarization4 Refractory period 3 Re- and hyper-polarizationt (μs)- im
(mA)T (K)Periodic
oscillationsFig. 2 | Experimental measurements and modelling of action potentials.
a, Quasistatic current–voltage behaviour of the circuit element (data repeated
from Fig. 1d). The dashed load-lines correspond to the colour-coded voltage
biases shown in b. The overlaid ellipses indicate biasing regions that exhibit a
range of qualitatively different oscillatory behaviours in the third-order
element. b, Measured temporal dynamics of the element’s current (im) at
different applied external voltages, as labelled. c, Data from b magnified in
time. d, The temporal dynamics of the element’s current at different applied
external voltages, measured in a network of the element with an external
capacitor and resistor. e–g, Simulation results corresponding to a–c, obtained
from the third-order compact model. h, Illustration of a biological neuron,
labelling its parts (left) and their functions (right). Four dynamical events
during the electrical impulse (1–4) are marked. i, Illustration of the biological
origins of the four events 1–4, along with equivalent physical processes in the
third-order element (indicated by ‘≈’). j, Modelled current through the element
(the polarity is reversed because the current in the biological neuron is ionic,
not electronic). k, As in j, corresponding temperature dynamics. The
equivalents of the four events 1–4 are marked as time windows.