Device-level analog / mixed-signal computation modelled on biological neural circuits. Canonical substrate elements: leaky-integrate-and-fire (LIF) neurons — Lapicque 1907, Stein 1965 stochastic generalisation; spike-timing-dependent…
neuromorphic-computing
LIF neuron: τ_m·dV/dt = -(V-V_rest) + R·I(t); fire when V≥V_th, reset to V_r
Lapicque 1907 leaky-integrate-and-fire neuron. Membrane potential evolves as a linear first-order ODE τ_m·dV/dt = -(V − V_rest) + R·I(t),…
STDP: Δw = A₊ exp(-Δt/τ₊) for Δt>0 (LTP), -A₋ exp(Δt/τ₋) for Δt<0 (LTD)
Bi-Poo 1998 spike-timing-dependent plasticity: synaptic weight w between a pre-synaptic neuron and a post-synaptic neuron undergoes…
Memristor: V = M(q)·i; HP 2008 model dx/dt = μ_V·R_on·i/D², 0≤x≤1
Chua 1971 theoretical prediction: a fourth passive two-terminal circuit element M(q) — the memristor — links charge q and magnetic flux φ…
LIF firing rate: r(I) = 1/(τ_ref + τ_m·ln((R·I)/(R·I − (V_th-V_rest)))) for suprathreshold
Closed-form firing rate of an LIF neuron under constant suprathreshold input current I > I_th = (V_th − V_rest)/R. Integrate τ_m·dV/dt =…
STDP integral: ∫₀^∞ A₊ exp(-Δt/τ₊) dΔt = A₊·τ₊; ∫_{-∞}^0 A₋ exp(Δt/τ₋) dΔt = A₋·τ₋
Total (integrated) STDP learning contribution across all pre-post timing offsets: for a flat Poisson-paired firing statistic, total LTP per…
HP memristor I-V loop: V(t) = (R_on x + R_off(1-x))·i(t), pinched at origin
HP memristor I-V characteristic: V(t) = M(x(t))·i(t) with memristance M(x) = R_on·x + R_off·(1−x) interpolating linearly between R_on (at…
Hopfield storage capacity via Hoeffding concentration on pattern overlaps
Hopfield associative-memory capacity via Hoeffding concentration. Setup: an N-neuron Hopfield network storing m random binary patterns…
Echo state property: spectral-radius rho(W) < 1 via Perron-Frobenius on |W|
Echo state property via Perron-Frobenius. Setup: a reservoir-computing network x(t+1) = tanh(W*x(t) + W_in*u(t)) needs the echo state…
Oja rule as PCA power iteration: w(t+1) = w(t) + eta*(y*x - y^2*w); w_inf = first eigenvector
Oja's rule as power iteration on the input covariance. Setup: a single linear neuron y = w*x receiving inputs x with zero mean and…
Hopfield pattern-overlap concentration: P(|m-E[m]|>=1/2) <= 2*exp(-N/2)
Hopfield Hoeffding bound at epsilon = 1/2. From framework F7, the general Hoeffding bound P(|m - E[m]| >= epsilon) <=…
ESN spectral-radius verification: W = [[0, 1/2], [1/2, 0]] has rho = 1/2 < 1 (ESP holds)
Concrete ESN spectral-radius verification via 2x2 example. From framework F8 (Perron-Frobenius), consider the canonical 2x2 symmetric…
Oja convergence: C = diag(2, 1) -> w_inf = (+-1, 0); power-iteration fixed-point
Oja's rule convergence on a 2-dimensional diagonal covariance. From framework F9 (power iteration), take the input covariance C = diag(2,…
Memristor (Chua 1971)
L Chua 1971 4th fundamental circuit-element; HP 2008 TiO2 memristor demonstration; basis of crossbar-arrays + analog-AI accelerators.
SNN (Maass 1997)
W Maass 1997 spiking-neural-networks 3rd generation; modern Loihi-Intel 2018 + IBM-TrueNorth + SpiNNaker-Manchester chips.
PCM (Ovshinsky 1968)
S Ovshinsky 1968 PCM; modern Intel 3D-XPoint 2015; multi-bit + analog modes for in-memory-compute neural-net applications.
Crossbar VMM
Crossbar-array implements vector-matrix-multiply via Ohm's law + Kirchhoff's law; energy-efficient analog AI; modern Mythic + IBM ResRAM…
Loihi (Intel 2018)
Davies 2018 Loihi 130k spiking-neurons + 130M synapses; on-chip learning; modern Loihi-2 2021 + Hala Point 1B-neuron 2024.
DVS (Lichtsteiner 2008)
Lichtsteiner-Posch-Delbruck 2008 dynamic-vision-sensor; us-resolution + low-power; modern Prophesee + Sony IMX sensors.
Memristor (Chua 1971)
L Chua 1971 + Strukov-HP 2008 first physical-memristor; modern modern foundational + 2024 phase-change PCM + RRAM analog-AI accelerator.
STDP (Bi-Poo 1998)
G-Q Bi-M-M Poo 1998 STDP-experimental; modern modern foundational + 2024 surrogate-gradient SNN training Neftci-Mostafa.
Hodgkin-Huxley (1952)
A Hodgkin-A Huxley 1952 (Nobel 1963) action-potential ODE; modern modern foundational text + Izhikevich + LIF + 2024 SNN deep-SNN.
TrueNorth (Merolla 2014)
P Merolla-Modha-IBM 2014 TrueNorth 1M neuron / 256M synapse; modern modern foundational text + Loihi 2017 + 2024 SpiNNaker2 + Akida.
Mead (1989)
C Mead 1989 'Analog VLSI'; modern modern foundational text + neuromorphic engineering + 2024 mixed-signal subthreshold-CMOS…
LSM (Maass 2002)
W Maass 2002 + Jaeger 2001 echo-state liquid-state-machine reservoir; modern modern foundational + 2024 photonic-reservoir + RC analog.