Neuromorphic computing: Worked examples — Hardware deep dive

This deep dive drills the Neuromorphic computing maths until no input surprises you. We take the two engines from the parent — the Leaky Integrate-and-Fire (LIF) neuron and Spike-Timing-Dependent Plasticity (STDP) — and hit every case class they can produce.

Before we start, one reminder of the core tools we lean on (all derived in the parent note):

Here every symbol means: $R$ = leak resistance (ohms), $C$ = membrane capacitance (farads), $I$ = input current (amps), $V$ = membrane voltage, $V_{t h}$ = firing threshold, $τ_{m} = R C$ = how fast the membrane charges/forgets, $t_{p r e} / t_{p os t}$ = the times two connected neurons spiked, $A_{\pm}$ = maximum learning strength, $τ_{\pm}$ = how quickly timing stops mattering. If any of these feels shaky, re-read the parent's RC derivation first.

The scenario matrix

Every problem this topic throws at you lands in exactly one of these cells. The examples below are labelled with the cell they cover.

#	Case class	What makes it special	Example
C1	Above threshold $R I > V_{t h}$	Neuron fires; $T$ finite	Ex 1
C2	Below threshold $R I < V_{t h}$	Log argument negative → never fires	Ex 2
C3	Exactly at threshold $R I = V_{t h}$	Degenerate: reaches $V_{t h}$ only at $t = \infty$	Ex 3
C4	Limiting: huge input $R I ≫ V_{t h}$	$T \to 0$ as $1/ I$ ; rate grows without bound (only refractory period caps it)	Ex 4
C5	STDP potentiation $Δ t > 0$	Pre before post → strengthen	Ex 5
C6	STDP depression $Δ t < 0$	Pre after post → weaken	Ex 5
C7	STDP degenerate $Δ t = 0$	Simultaneous spikes → boundary	Ex 6
C8	Real-world word problem	Choose $I$ to hit a target rate	Ex 7
C9	Exam twist	Threshold current + rate just above it	Ex 8
C10	Sanity / energy	Why silence = low power (sparsity)	Ex 9

We use one fixed "textbook neuron" wherever possible so numbers stay comparable: $R = 10 M Ω = 1 0^{7} Ω, C = 1 nF = 1 0^{- 9} F, τ_{m} = R C = 10 ms, V_{t h} = 15 mV .$

Look at the figure: the three coloured curves are the same neuron fed three different currents. The orange curve (strong input) crosses the dashed threshold line; the teal one (medium) just reaches it; the plum one (weak) flattens below the line forever. Those three shapes ARE cells C1, C3, C2. Every example is a point on one of these curves.

C1 — Above threshold: it fires

Worked example Example 1 — Time-to-spike and rate (

$R I > V_{t h}$ ) Textbook neuron, constant input $I = 2 nA$ . Find whether it fires, the time to first spike, and the firing rate.

Forecast: guess now — does it fire? Above or below ~10 ms to spike?

Steady-state voltage $R I$ . Why this step? If the ceiling $R I$ is below $V_{t h}$ the neuron can never reach threshold — always check first. $R I = (1 0^{7}) (2 \times 1 0^{- 9}) = 20 \times 1 0^{- 3} V = 20 mV .$ Since $20 > 15$ , it fires. ✅
Time constant. Why? $τ_{m}$ sets how fast the charge-up happens. $τ_{m} = R C = 10 ms$ .
Time to first spike. Why? Plug into the boxed $T$ formula. $T = 10 ms \cdot ln \frac{20}{20 - 15} = 10 ms \cdot ln 4 \approx 13.86 ms .$
Firing rate. Why? Rate is one spike per $T$ (ignoring refractory time). $f = 1/ T \approx 72.1 Hz$ .

Verify: Plug $T$ back: $V (13.86 ms) = 20 (1 - e^{- 13.86/10}) = 20 (1 - 0.25) = 15 mV = V_{t h}$ . ✔ Units: $Ω \cdot A = V$ ✔, $Ω \cdot F = s$ ✔.

C2 — Below threshold: it never fires

C3 — Exactly at threshold: the degenerate boundary

Worked example Example 3 — Right on the edge (

$R I = V_{t h}$ ) Choose $I$ so the ceiling equals the threshold exactly.

Forecast: if $R I$ lands exactly on $V_{t h}$ , when (if ever) does it spike?

Solve for the critical current. Why? We want $R I = V_{t h} = 15 mV$ . $I_{cr i t} = \frac{V _{t h}}{R} = \frac{15 \times 1 0 ^{- 3}}{1 0 ^{7}} = 1.5 \times 1 0^{- 9} A = 1.5 nA .$
Plug into $T$ . Why? Check the limiting behaviour. $T = τ_{m} ln \frac{15}{15 - 15} = τ_{m} ln (\frac{15}{0})$ . The argument $\to + \infty$ , so $ln \to + \infty$ , so $T \to \infty$ .
Interpret. Why? $V (t) = 15 (1 - e^{- t / τ_{m}})$ approaches $15 mV$ but reaches it only in the infinite-time limit. So the neuron effectively never fires in finite time — the boundary belongs to the "silent" side.

Verify: At $t = 100 ms = 10 τ_{m}$ : $V = 15 (1 - e^{- 10}) = 15 (1 - 0.0000454) = 14.99932 mV$ — still under $15$ . Confirms $T = \infty$ . ✔ Any current above $1.5 nA$ fires; at or below, it does not.

C4 — Limiting case: huge input

Worked example Example 4 — Blast it with current (

$R I ≫ V_{t h}$ ) Same neuron, $I = 100 nA$ .

Forecast: does $T$ keep shrinking without limit as $I$ grows, or hit a floor?

Ceiling. Why? $R I = (1 0^{7}) (100 \times 1 0^{- 9}) = 1 V = 1000 mV$ . Massively above $15 mV$ .
Time to spike. Why? Plug in. $T = 10 ms \cdot ln \frac{1000}{1000 - 15} = 10 ms \cdot ln (1.01523) \approx 10 ms \times 0.015113 \approx 0.1511 ms .$
Limiting behaviour. Why? For $R I ≫ V_{t h}$ , $ln \frac{R I}{R I - V _{t h}} \approx \frac{V _{t h}}{R I}$ (small-argument log). So $T \approx τ_{m} \frac{V _{t h}}{R I} = \frac{C V _{t h}}{I}$ . It keeps shrinking as $1/ I$ , so the firing rate $f = 1/ T \approx I / (C V_{t h})$ grows without bound as $I \to \infty$ — there is no leak-imposed ceiling. Only a real neuron's refractory period (a minimum dead time after each spike) would cap the rate. $T \approx \frac{C V _{t h}}{I} = \frac{( 1 0 ^{- 9} ) ( 15 \times 1 0 ^{- 3} )}{100 \times 1 0 ^{- 9}} = 1.5 \times 1 0^{- 4} s = 0.15 ms .$

Verify: Exact $0.1511 ms$ vs approximation $0.15 ms$ — match within ~1%, and the approximation is the "capacitor charges linearly at first" limit ( $V \approx I t / C$ ). Double $I$ and $T$ roughly halves — pure $1/ I$ scaling, no floor. ✔

C5 & C6 — STDP: strengthen vs weaken

Worked example Example 5 — Both STDP branches at once

Learning params $A_{+} = 0.1$ , $A_{-} = 0.12$ , $τ_{+} = τ_{-} = 20 ms$ . (a) Pre spike at $5 ms$ , post at $8 ms$ . (b) Pre at $8 ms$ , post at $5 ms$ .

Forecast: which case strengthens the synapse, and will their magnitudes be equal?

Case (a) sign. Why? The sign of $Δ t = t_{p os t} - t_{p r e}$ picks the branch. $Δ t = 8 - 5 = + 3 ms > 0$ → potentiation (pre helped cause post).
Case (a) magnitude. Why? Use the top branch. $Δ w = 0.1 e^{- 3/20} = 0.1 \times 0.86071 = + 0.0861$ .
Case (b) sign. Why? $Δ t = 5 - 8 = - 3 ms < 0$ → depression (pre came too late).
Case (b) magnitude. Why? Bottom branch: $Δ w = - A_{-} e^{+ Δ t / τ_{-}} = - 0.12 e^{(- 3) /20} = - 0.12 \times 0.86071 = - 0.1033$ .

Verify: Same $∣Δ t ∣ = 3 ms$ , same exponential factor $0.86071$ , but magnitudes differ ( $0.0861$ vs $0.1033$ ) because $A_{-} > A_{+}$ . The STDP curve is intentionally asymmetric — depression often stronger to keep weights from exploding. Units of $Δ w$ : dimensionless weight change. ✔

The figure plots $Δ w$ against $Δ t$ . Right of zero (orange) = strengthen, decaying as timing loosens; left of zero (teal) = weaken. The dot at $Δ t = + 3$ is Ex 5a; the dot at $Δ t = - 3$ is Ex 5b — note the left dip is deeper because $A_{-} > A_{+}$ .

C7 — STDP degenerate: simultaneous spikes

Worked example Example 6 — Exactly together (

$Δ t = 0$ ) Pre and post both fire at $t = 7 ms$ .

Forecast: strengthen, weaken, or nothing?

Compute $Δ t$ . Why? $Δ t = 7 - 7 = 0$ — the boundary between the two branches.
Approach from each side. Why? The rule is defined for $Δ t > 0$ and $Δ t < 0$ but not at $0$ , so check the limits. As $Δ t \to 0^{+}$ : $Δ w \to A_{+} e^{0} = + 0.1$ . As $Δ t \to 0^{-}$ : $Δ w \to - A_{-} e^{0} = - 0.12$ .
Interpret. Why? The two limits disagree ( $+ 0.1$ vs $- 0.12$ ) — a jump discontinuity at $Δ t = 0$ . Simultaneity is causally ambiguous ("who caused whom?"), so real chips just set $Δ w = 0$ or use whichever spike is logged first. There is no single "correct" value; it is a modelling convention.

Verify: Limits differ → discontinuity confirmed. This is the one cell where the formula genuinely has no defined output, matching the C3 lesson that boundaries need explicit handling. ✔

C8 — Real-world word problem

Worked example Example 7 — Design a 100 Hz neuron

A neuromorphic vision chip needs a pixel-neuron to fire at exactly $f = 100 Hz$ under a steady light current. Using the textbook neuron ( $τ_{m} = 10 ms$ , $R = 10 M Ω$ , $V_{t h} = 15 mV$ ), what constant input current $I$ is required?

Forecast: more or less than the $2 nA$ that gave $72 Hz$ in Ex 1?

Target period. Why? Rate $\to$ period: $T = 1/ f = 1/100 = 10 ms$ .
Invert the $T$ formula for $I$ . Why? We know $T$ , want $I$ . Start from $T = τ_{m} ln \frac{R I}{R I - V _{t h}}$ : $e^{T / τ_{m}} = \frac{R I}{R I - V _{t h}} \Rightarrow R I - V_{t h} = R I e^{- T / τ_{m}} \Rightarrow R I (1 - e^{- T / τ_{m}}) = V_{t h} .$ $I = \frac{V _{t h}}{R ( 1 - e ^{- T / τ_{m}} )}$
Plug numbers. Why? $T / τ_{m} = 10/10 = 1$ , so $e^{- 1} = 0.36788$ , and $1 - e^{- 1} = 0.63212$ . $I = \frac{15 \times 1 0 ^{- 3}}{1 0 ^{7} \times 0.63212} = \frac{15 \times 1 0 ^{- 3}}{6.3212 \times 1 0 ^{6}} = 2.373 \times 1 0^{- 9} A \approx 2.37 nA .$

Verify: Feed $2.373 nA$ back: $R I = 23.73 mV$ ; $T = 10 ln \frac{23.73}{23.73 - 15} = 10 ln (2.7181) = 10 \times 1.0000 = 10.0 ms$ → $100 Hz$ . ✔ And $2.37 > 2 nA$ , matching the forecast (faster firing needs more current). ✔

C9 — Exam twist: threshold current and the soft turn-on

Worked example Example 8 — Smallest spiking current, and rate just above it

An exam asks: "For the textbook neuron, what is the smallest current $I$ that produces any spiking, and what firing rate does a current $10%$ above that threshold current give?"

Forecast: at the threshold current itself, is the rate zero or infinite-period?

Threshold current. Why? Spiking requires $R I > V_{t h}$ , so the infimum is $R I = V_{t h}$ : $I_{0} = V_{t h} / R = 1.5 nA$ (this is the C3 boundary — at exactly $I_{0}$ , $T = \infty$ , rate $0$ ).
Ten percent above. Why? $I = 1.1 I_{0} = 1.65 nA$ , so $R I = 1.1 \times 15 = 16.5 mV$ .
Time to spike. Why? Plug in. $T = 10 ms \cdot ln \frac{16.5}{16.5 - 15} = 10 ln (11) = 10 \times 2.3979 = 23.98 ms$ .
Rate. $f = 1/ T = 1/0.02398 \approx 41.7 Hz$ .

Verify: Just above threshold the rate is small but positive ( $41.7 Hz$ ), then rises steeply as $I$ grows — the classic "soft turn-on" of an LIF f–I curve. Check: $V (23.98 ms) = 16.5 (1 - e^{- 2.398}) = 16.5 (1 - 0.0909) = 16.5 \times 0.9091 = 15.0 mV = V_{t h}$ . ✔

C10 — Sanity / energy check

Worked example Example 9 — Why silence saves power

Suppose a neuromorphic layer of $1000$ neurons is fed a sparse input where only $30$ neurons cross threshold; the other $970$ sit at $R I < V_{t h}$ (Ex 2 situation). If each spike costs $E_{s p ik e} = 10 pJ$ and idle neurons cost effectively $0$ , what is the energy per timestep, and how does that compare to a dense chip that "computes" all $1000$ ?

Forecast: roughly what fraction of a dense chip's work is done?

Active count. Why? Only firing neurons draw dynamic energy — this is the event-driven principle from the parent's in-memory contrast. Active $= 30$ .
Neuromorphic energy. Why? Sum only active spikes. $E = 30 \times 10 pJ = 300 pJ$ .
Dense-equivalent energy. Why? A dense accelerator pays for all $1000$ units each step: $E_{d e n se} = 1000 \times 10 pJ = 10, 000 pJ$ .
Ratio. Why? $300/10000 = 0.03 = 3%$ .

Verify: The chip does $3%$ of the energy for the same layer because $97%$ of neurons were sub-threshold (C2) — the C2 "never fires" case is not a bug, it is the entire energy story. Units: $pJ$ throughout, dimensionless ratio. ✔

Recall Self-test — one per matrix cell

Does a neuron with $R I = V_{t h}$ ever fire in finite time? ::: No — $T \to \infty$ ; the boundary is on the silent side (see C3). What single quantity do you always compute first in an LIF problem, and why? ::: $R I$ vs $V_{t h}$ — if $R I \leq V_{t h}$ the neuron never fires and the $T$ formula's log goes negative/undefined (C2). For $R I ≫ V_{t h}$ , how does time-to-spike scale with $I$ ? ::: $T \approx C V_{t h} / I$ , i.e. $\propto 1/ I$ ; the rate grows without bound, only a refractory period caps it (C4). Pre at 5 ms, post at 8 ms: strengthen or weaken? ::: $Δ t = + 3 > 0$ → potentiation, strengthen (C5). What happens in STDP when $Δ t = 0$ ? ::: Jump discontinuity between $+ A_{+}$ and $- A_{-}$ ; convention-defined, usually $0$ (C7). To double the firing rate you must (increase/decrease) $I$ ? ::: Increase — higher $R I$ shortens $T$ (C8/C9).

See also: Neuromorphic computing (parent), Spiking Neural Networks (SNN), Hebbian learning, Memristors and ReRAM, RC circuits, Von Neumann architecture.