Imagine slowly turning up a dimmer knob versus flipping a wall switch. Some neurons behave like the dimmer: as you push more current into them, they start firing very slowly and then speed up smoothly, beginning from essentially zero. Others behave like the switch — below a certain push they stay completely silent, and then, the instant you cross threshold, they jump straight to a brisk firing rate. That one difference, a smooth dial versus a sudden jump, is the whole idea behind Type I and Type II excitability.
What we actually measure is the firing rate $f$ (spikes per second, in Hz) as we change the injected current $I$ — the "push." Plotting $f$ against $I$ gives the f–I curve. For a Type I neuron this curve rises continuously from zero: just past threshold it might fire at 1 Hz, then 5, then 20. Near onset a good description is $f \propto \sqrt{I - I_{c}}$, where $I_c$ is the threshold current. Try a number: if $I_c = 6$ and $I = 6.01$, then $\sqrt{0.01} = 0.1$ — a tiny but genuinely nonzero rate, so arbitrarily slow firing is allowed. A Type II neuron has no such slow regime: the moment $I$ crosses threshold, $f$ jumps to a minimum $f_{\min} > 0$ (say 12 Hz) and never fires more slowly than that.
The deeper reason lives in the bifurcation — how the resting state loses stability as $I$ rises. Type I passes through a SNIC (saddle-node on an invariant circle): the orbit's period stretches toward infinity at onset, so $f = 1/T \to 0$. Type II passes through a Hopf bifurcation: a limit cycle is born already oscillating at frequency $\omega_0/2\pi$, so $f$ starts finite. In the sim above, the Input I slider is exactly this $I$; Parameter 1 and Parameter 2 reshape the model so you can move it between the two regimes; T sim just sets how long you watch.
TRY THIS IN THE SIM ABOVE
① Set Input I just above threshold and watch whether the rate creeps up from near-zero (Type I) or snaps to a finite value (Type II). ② Sweep $I$ slowly across threshold using the Parameter Sweep tab and look for a smooth rise versus a vertical jump in the f–I curve. ③ Open the Comparison tab to see the all-positive (Type I) versus biphasic (Type II) phase-response curves side by side.
§ 03
Equation Derivation
▸ Rinzel-Ermentrout Classification (1989)
The excitability class of a neuron is determined by the bifurcation at spike onset. This classification, by Rinzel & Ermentrout (1989), predicts synchronisation, coding properties, and network behaviour.
$$\text{Type I (SNIC):}\quad f \propto \sqrt{I - I_{\text{SNIC}}}, \quad f(I_{\text{SNIC}}) = 0, \quad f_{\min} = 0 \;\text{Hz}$$
$$\text{Type II (Hopf):}\quad f \text{ jumps to } f_{\min} = \frac{\omega_0}{2\pi} > 0 \;\text{at onset, then increases with I}$$
Type I PRC: all positive (any perturbation advances the next spike). Type II PRC: biphasic (early perturbations delay; late advance). The PRC shape determines synchronisation: all-positive PRC → neurons cannot synchronise in-phase with common noise input; biphasic PRC → neurons synchronise in-phase.
STEP 1 — Why SNIC Gives Type I
At the SNIC bifurcation, the limit cycle period T → ∞ because the trajectory slows near the ghost of the former saddle node. Near onset: T ≈ π/√(I−I_SNIC) → f = 1/T ∝ √(I−I_SNIC). Starting from 0 Hz, f increases continuously. The phase portrait shows a circle (invariant circle) with a slow "parking" region near the SNIC location.
STEP 2 — Why Hopf Gives Type II
At the Hopf bifurcation, a limit cycle appears at frequency ω₀/(2π) > 0. The cycle does not grow from zero — it has a finite minimum amplitude and frequency. Above I_Hopf, f increases but never reaches 0. The minimum frequency is set by Im(eigenvalue)/τ at the Hopf point.
STEP 3 — PRC Measurement
Perturb a periodically firing neuron with a brief current pulse at different phases φ ∈ [0,1] of its cycle. Record the phase advance Δφ of the next spike. Plot Δφ vs φ → PRC. Type I: all Δφ > 0 (every perturbation advances). Type II: Δφ changes sign — perturbations in early phase delay; late phase advance. The PRC shape is directly related to the limit cycle geometry near the relevant bifurcation.
▸ Worked Example — Identifying Excitability Type from f–I Data
Given f–I data: I=5→0 Hz, I=6→0 Hz, I=7→12 Hz, I=8→35 Hz, I=10→60 Hz. Does f start from 0 or jump?
$$\text{At I=6: f=0. At I=7: f=12 Hz. Jump from 0 to 12 Hz → \textbf{Type II (Hopf)}}$$
$$\text{Verify: fit } f = \sqrt{I-I_{\text{SNIC}}} \;\text{near onset. At I=7: }\sqrt{7-6}=1\neq 12. \;\text{Fit fails.}$$
$$\text{Fit Hopf: f_onset ≈ 12 Hz, I_Hopf ≈ 6.5, then f increases linearly above onset → Type II confirmed}$$
▸ References
[RE89]
Rinzel & Ermentrout — Analysis of neural excitability, in Methods in Neuronal Modeling, MIT Press, 1989
[IZH07]
Izhikevich — Dynamical Systems in Neuroscience, MIT Press, 2007. Ch. 6–7. Free: dynamicalsystems.org
🔬 SimulationWhat do the f–I Curve and PRC tabs show?▼
The f–I Curve tab shows firing rate vs injected current for both Type I (continuous, starting from 0) and Type II (discontinuous jump). Drag the model selector to switch between SNIC (Type I) and Hopf (Type II) and compare the curves directly. The minimum firing frequency for Type II is shown as a dashed horizontal line. The PRC tab shows the phase response curve Z(φ): for Type I (all-positive, camel-hump shape), perturbations only advance the spike. For Type II (biphasic, S-shaped), early perturbations delay and late ones advance.
Key takeaway: Type I f–I: starts from 0, wide dynamic range. Type II: jumps to f_min, narrow range. PRC shape directly distinguishes them: all-positive = Type I; biphasic = Type II.
🧠 ConceptualWhy is Type II better for coincidence detection and Type I better for rate coding?▼
Type I neurons can fire at any rate from 0 to f_max — the firing rate precisely encodes input strength over a wide dynamic range. They are ideal rate encoders. Type II neurons jump to a minimum frequency at threshold — below a critical input, they are silent; above, they fire at f_min or higher. They respond in an all-or-nothing way to input strength near threshold. This makes them sensitive to synchronous inputs: if two Type II neurons receive the same timing of inputs, they tend to synchronise their firing (biphasic PRC enables in-phase synchrony). Type I neurons resist synchronisation because their all-positive PRC means any phase perturbation just shifts the spike earlier.
Key takeaway: Type I = frequency encoder (all-positive PRC, no preferred phase). Type II = coincidence detector (biphasic PRC, prefers in-phase synchrony). Network function depends critically on which type dominates.
🌍 AppliedHow does excitability type affect auditory processing and speech perception?▼
Auditory nerve fibres and bushy cells in the cochlear nucleus are predominantly Type II (Hopf). This is functionally important: sound localisation requires microsecond-precision synchrony between left and right auditory nerve inputs to the medial superior olive (MSO). Type II neurons with biphasic PRCs can phase-lock to sound waveforms and synchronise their firing with submillisecond precision across the population — Type I neurons cannot achieve this. The minimum firing frequency of Type II also provides a 'squelch' effect: background noise below threshold produces no response, improving signal detection in noisy environments. Cochlear nucleus octopus cells are extreme Type II — they fire a single precise spike at stimulus onset with sub-millisecond jitter.
Key takeaway: Auditory neurons are predominantly Type II because submillisecond precision synchrony is required for sound localisation and speech processing. Type I neurons are too imprecise for microsecond-level timing.
💡 Non-ObviousCan a single neuron switch between Type I and Type II excitability?▼
Yes — neuromodulators can switch neurons between excitability types by shifting the operating point near a codimension-2 bifurcation. Dopamine in striatal neurons shifts them from Type I (D2 receptor activation hyperpolarises, shifting to SNIC geometry) to Type II (D1 activation depolarises, shifting toward Hopf). Acetylcholine in cortical pyramidal neurons closes K⁺ channels (muscarinic), depolarising V_rest, which can shift a Type I pyramidal cell toward Type II behaviour. This modulation changes the neuron's role in the network: the same cell can switch from a rate encoder (Type I) to a coincidence detector (Type II) depending on the neuromodulatory context. This dynamic reconfiguration is a key feature of cortical processing during attention, learning, and different cognitive states.
Key takeaway: Neuromodulators can switch neurons between Type I and II by moving the operating point near a codimension-2 bifurcation. The same neuron acts as a rate encoder or coincidence detector depending on modulatory state.
📐 ComputationalHow do you measure the PRC experimentally or in simulation?▼
Experimental protocol: inject just enough current to drive regular firing (I just above threshold). At phase φ of the firing cycle (φ = 0 at spike, φ = 1 just before next spike), inject a brief (1–2 ms) current pulse of amplitude δI. Measure the time shift Δt of the next spike. Normalise: Z(φ) = Δt/T₀ where T₀ is the unperturbed period. Repeat for many phases to trace out the PRC. In simulation: exactly the same protocol, with sub-ms precision. The PRC is equivalent to the iPRC (infinitesimal phase response curve) for small perturbations, derivable analytically via adjoint method: L*Z = 0 where L* is the adjoint of the linearised flow operator along the limit cycle.
Key takeaway: PRC measurement: inject brief pulses at different phases, record spike-time shift. Z(φ) = Δt/T₀. Analytically: solve the adjoint equation L*Z = 0 along the limit cycle. All-positive PRC = Type I; biphasic = Type II.
🎓 DeepWhat is the connection between PRC shape and network synchrony — Kuramoto model?▼
The Kuramoto model of weakly coupled oscillators shows that synchrony is determined by the odd part of the PRC convolution. Two neurons with PRCs Z₁, Z₂ synchronise in-phase if H(0) − H(π) > 0, where H(φ) = ∫Z(φ+θ)Z(θ)dθ (interaction function). For all-positive PRC (Type I): H is nearly constant → in-phase and anti-phase are both stable → no preferred phase → poor synchrony. For biphasic PRC (Type II): H(0) > H(π) → in-phase stable, anti-phase unstable → robust synchrony. This is why Type II neurons (FS interneurons with biphasic PRCs) generate coherent gamma oscillations, while Type I pyramidal cells do not self-synchronise without the Type II interneuron drive.
Key takeaway: PRC determines synchrony via the interaction function H(φ). Type II biphasic PRC → H(0) > H(π) → in-phase synchrony stable. This is why FS interneurons (Type II) generate gamma and pyramidal cells (Type I) don't self-synchronise.
§ 04 Best Resources
Gerstner et al. — Neuronal Dynamics, Cambridge, 2014. neuronaldynamics.epfl.ch
Dayan & Abbott — Theoretical Neuroscience, MIT Press, 2001
Izhikevich — Dynamical Systems in Neuroscience, MIT Press, 2007. dynamicalsystems.org
Hille — Ion Channels of Excitable Membranes (3rd ed.), Sinauer, 2001
§ 05
Misconceptions & Common Errors
Sub-block A — Conceptual Misconceptions
❌"Type I neurons always fire at lower rates than Type II — Type II is 'faster.'"
✅Excitability type determines the onset firing rate (0 Hz vs f_min), not the maximum rate. A Type I neuron can fire at 200+ Hz with sufficient current; a Type II neuron may fire at rates below its minimum only briefly before adapting. Fast-spiking (FS) interneurons are Type II with f_min ≈ 50 Hz but can fire at 200 Hz — not because of their Type II classification but because of their fast membrane time constant (τ_m ≈ 5 ms). Regular spiking (RS) pyramidal neurons are Type I but typically fire at 5–40 Hz in vivo. Maximum firing rate is set by refractory period; Type I/II classification is about onset frequency only.
❌Fitting a √(I−I_0) curve to any f–I data and calling it 'Type I' — without checking whether f actually starts from 0.
✅√(I−I_0) fitting is valid for Type I only when f(I_0) = 0 (neuron is silent just below I_0). If the f–I data shows a discontinuous jump (f=0 for I<I_0, then f>0 for I≥I_0+ε), this is Type II — do not fit a square-root curve. Additionally, near a Type I SNIC, the f–I curve is approximately √(I−I_SNIC) only very close to I_SNIC; further above, it deviates from square-root. The definitive test: is there a current I* where arbitrarily low firing rates are achievable? Yes → Type I. No → Type II.
🔍 Why: Applying Type I curve-fitting to Type II data because both look like 'increasing' functions near threshold — the key diagnostic is whether f can be made arbitrarily small.
§ 05 References
Gerstner et al. — Neuronal Dynamics, Cambridge, 2014. neuronaldynamics.epfl.ch
Dayan & Abbott — Theoretical Neuroscience, MIT Press, 2001
Hille — Ion Channels of Excitable Membranes (3rd ed.), Sinauer, 2001
Izhikevich — Dynamical Systems in Neuroscience, MIT Press, 2007