Adaptive Exponential I-F (AdEx)

Clap as fast as you can. The first few claps are quick and sharp, but your hands tire and you naturally slow down. A neuron does almost the same thing. Give it a strong, steady push and it fires a burst of fast electrical "spikes" — then it gets "tired" and the spikes spread further apart. That slowing-down is called adaptation, and AdEx is a small set of rules that captures exactly this rhythm.

Two numbers describe the cell at any instant. The first is the membrane voltage $V$ — how charged-up the neuron is. The second is an adaptation current $w$ — the "tiredness" that builds with every spike. A steady input current $I$ charges the cell. When $V$ climbs past a threshold the neuron spikes; then $V$ snaps back to a reset value and $w$ jumps up by a fixed amount $b$. Each jump in $w$ subtracts from the next push, so the cell takes a little longer to reach threshold again. For instance, if every spike adds about $b=60$ pA of "brake" while the input is only $200$ pA, after a few spikes the brake eats most of the drive and the firing settles into a slower, steadier beat.

AdEx writes this as the two coupled equations in Section 3. The voltage equation carries an exponential term $g_L\Delta_T e^{(V-V_T)/\Delta_T}$ that switches on sharply near the threshold $V_T$ — it manufactures the fast upstroke of a real spike, with $\Delta_T$ setting how sharp it is. The second equation, $\tau_w\,dw/dt = a(V-E_L)-w$, lets $w$ slowly track voltage through the subthreshold coupling $a$, while the reset rule $w\leftarrow w+b$ adds the per-spike kick. Just two knobs, $a$ and $b$, plus the timescale $\tau_w$, reproduce regular-spiking, bursting, fast-spiking and rebound neurons. On the sliders here, Input I sets the drive, Parameter 1 and Parameter 2 stand for the coupling $a$ and the spike increment $b$, and T sim sets how long you watch.

Push Input I up and watch more spikes appear — notice how they crowd together at the start and spread out later. Raise Parameter 2 (the spike increment $b$) and the firing should slow more steeply: strong adaptation. Drop it back toward zero and the spacing stays even, like a fast-spiking cell that never tires. Lengthen T sim to give the slow adaptation time to settle into its final rhythm.

a (nS): subthreshold adaptation coupling. a>0: resonance (w tracks V). a<0: Class 3. b (pA): spike-triggered increment → adaptation. Large b: strong adaptation (RS). b<0: facilitatory bursting (TC). τ_w: adaptation timescale. Standard HBP model; parameters directly fit from current-clamp recordings.

Brette & Gerstner (2005): RS (a=-0.5, b=7pA, τ_w=100ms), IB (a=2, b=0.02nA), FS (a=2, b=0, τ_w=10ms), LTS (a=2, b=0), TC (a=0, b=-0.06nA, rebound). AdEx reproduces all Izhikevich patterns with biophysically interpretable parameters vs Izhikevich's phenomenological a,b,c,d.

Multi-objective fitting: minimise MSE in spike threshold, AHP, ISI adaptation, f-I curve. Protocol: frozen noise current injection (Gaussian, σ=200–500 pA), record V(t), optimise 9 parameters. AdEx achieves R²≈0.7–0.9 for cortical neurons. Used in NEST simulator (standard Human Brain Project neuron).