— Exponential Integrate-and-Fire

§ 01

Interactive Simulation

State var 1

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State var 2

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Input I

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Firing rate

0.0

Spike count

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Mode

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Time

Playback

Preset

Parameters

Parameter 1—

Parameter 2—

Input I—

T sim (ms)500

Overlays

Grid Nullclines Legend

§ 02

The Idea, Step by Step

▸ From a leaky bucket to a self-launching spike

START — the everyday picture

Picture slowly filling a bathtub whose drain is always open. At first the water just creeps up — that is a neuron's voltage charging toward firing. A plain "leaky integrate-and-fire" (LIF) neuron is exactly this: it fills until it touches a fixed line, then instantly "spikes" and empties. But real brain cells don't snap at a hard line. Once they get close enough, they suddenly take off on their own — like a ball rolling over the crest of a hill and racing down the far side. The exponential integrate-and-fire (EIF) model adds exactly that "rolling-over-the-hill" kick.

BUILD — naming the pieces

The voltage $V$ always leaks back toward a resting value $E_L\approx-65$ mV. Near a special voltage $V_T\approx-50$ mV we add a term that grows like $e^{(V-V_T)/\Delta_T}$. The number $\Delta_T$ (the "slope factor", a few millivolts) sets how sharp the takeoff is. Far below $V_T$ this term is almost zero, so the cell behaves like ordinary LIF. As $V$ creeps up to $V_T$ the exponential reaches 1 and starts to matter; just past it the term doubles every $\Delta_T$ millivolts and the voltage rockets upward into a spike.

DEEPEN — the precise rule and the sliders

The full equation is $C_m\,\dfrac{dV}{dt} = -g_L(V-E_L) + g_L\Delta_T\,e^{(V-V_T)/\Delta_T} + I$. The first term is the leak, the middle term is the spike-generating exponential, and $I$ is the input current. When $V$ shoots past a numerical cutoff $V_{\text{peak}}$, the model records a spike and resets $V$ to $V_r$. The input slider $I$ pushes the whole curve up — more drive means faster, more frequent spikes. Make $\Delta_T$ small and the EIF mimics LIF's hard threshold; make it large and the onset softens into a quadratic (QIF) curve.

CLOSE — try this in the sim above

(1) Raise $I$ from low to high and watch the firing rate climb from silence to steady spiking. (2) Shrink the slope factor toward zero and see the spike onset turn abrupt and LIF-like. (3) Widen it and watch the onset become smooth and gradual. The Phase Plane view makes the exponential "hockey-stick" takeoff near $V_T$ easiest to spot.

§ 03

Equation Derivation

▸ Exponential Integrate-and-Fire Equation

Proposed by Fourcaud-Trocmé et al. (2003), the EIF model adds a single exponential term to the LIF model that captures the sharp Na⁺ channel activation at spike initiation, reproducing the spike initiation zone (SIZ) accurately.

$$\boxed{C_m \frac{dV}{dt} = -g_L(V-E_L) + g_L \Delta_T \exp\!\left(\frac{V-V_T}{\Delta_T}\right) + I}$$ $$\text{if } V \geq V_{\text{peak}}: \quad V \leftarrow V_r, \quad \text{record spike}$$

where Δ_T is the sharpness parameter — controls how sharp the spike initiation is:

$$\Delta_T \to 0: \quad \text{EIF reduces to LIF (hard threshold at } V_T)$$ $$\Delta_T \to \infty: \quad \text{EIF reduces to QIF (Izhikevich-like quadratic)}$$

Exact ISI for constant I:

$$T_{\text{ISI}} = C_m \int_{V_r}^{V_T} \frac{dV}{-g_L(V-E_L)+g_L\Delta_T e^{(V-V_T)/\Delta_T}+I} + t_{\text{ref}}$$

▸ Symbol Table

Symbol	Meaning	Unit	Typical Value
C_m	Membrane capacitance	pF	200 pF
g_L	Leak conductance	nS	10 nS → τ_m = C_m/g_L = 20 ms
E_L	Leak reversal (≈ resting potential)	mV	−65 mV
V_T	Threshold potential (spike initiation zone)	mV	−50 mV
Δ_T	Slope factor — sharpness of Na⁺ activation	mV	2 mV (sharp) to 20 mV (broad)
V_peak	Numerical spike cutoff (reset trigger)	mV	+20 to +40 mV
V_r	Reset voltage after spike	mV	−65 mV

▸ Derivation Steps

STEP 1 — The Spike Initiation Problem with LIF

In the standard LIF model, threshold crossing is instantaneous: the membrane is integrating passively up to V_thresh, then abruptly fires. Real neurons have a gradual spike initiation zone (SIZ) where Na⁺ channels begin to open at ~−55 mV and produce a positive feedback that accelerates dV/dt exponentially before the full spike. The LIF misses this graded initiation — it cannot reproduce the correct phase-plane trajectory near threshold or the empirically observed spike initiation speed.

STEP 2 — Exponential Term as Na⁺ Activation Approximation

Near threshold, the Na⁺ channel activation function m∞(V) is approximately exponential: m∞(V) ≈ exp((V−V_T)/Δ_T). The inward Na⁺ current ≈ g_Na m∞³(V)(V−E_Na) ≈ g_Na exp(3(V−V_T)/Δ_T)(V−E_Na). In the EIF, this is further simplified to g_L Δ_T exp((V−V_T)/Δ_T) — the coefficient g_L Δ_T is chosen to match the threshold dynamics of HH. The parameter Δ_T = V_T/k where k is the steepness of the Na⁺ activation curve, directly measurable from voltage-clamp data.

STEP 3 — Connection to AdEx Model

The Adaptive Exponential I&F (AdEx) model adds a slow adaptation current w to EIF: C_m dV/dt = −g_L(V−E_L) + g_L Δ_T exp((V−V_T)/Δ_T) + I − w; τ_w dw/dt = a(V−E_L) − w. After each spike: w → w + b. This adds spike-frequency adaptation (b) and subthreshold resonance (a) to the EIF. The AdEx is the current gold standard for spiking neuron models that balance biophysical accuracy with analytical tractability — it is used in NEURON as the standard single-compartment model.

STEP 4 — Numerical Integration

The EIF ODE has a singularity at V_T (where the exponential grows rapidly). Use forward Euler with dt = 0.025 ms near threshold, or exponential Euler: for the linear part (without the exp term), use the exact exponential update; add the exponential term as a perturbation. The singularity is controlled by setting V_peak: once V exceeds V_peak, reset to V_r. V_peak = 0 or +20 mV produces realistic-looking spike shapes when plotted.

▸ Worked Example — Threshold Sharpness Effect

Use consistent units throughout: $C_m=200$ pF, $g_L=10$ nS (so $\tau_m=C_m/g_L=20$ ms), $E_L=-65$ mV, $V_T=-50$ mV, $\Delta_T=2$ mV, and a constant drive $I=0.5$ nA $=500$ pA. Note that nS×mV = pA and pA/pF = mV/ms. Look at the rate $dV/dt$ at $V=-52$ mV (2 mV below threshold):

$$\text{LIF: } \frac{dV}{dt} = \frac{-g_L(V-E_L) + I}{C_m} = \frac{-10(13) + 500}{200} = \frac{-130+500}{200} = +1.85 \;\text{mV/ms}$$ $$\text{EIF extra term: } \frac{g_L\Delta_T\,e^{(V-V_T)/\Delta_T}}{C_m} = \frac{10\times 2 \times e^{(-52+50)/2}}{200} = \frac{20\,e^{-1}}{200} \approx +0.037\;\text{mV/ms}$$

So 2 mV below threshold the exponential adds almost nothing. Right at $V=V_T=-50$ mV it is $20\,e^{0}/200=+0.1$ mV/ms, and it then explodes: at $V=-46$ mV it is already $20\,e^{2}/200\approx0.74$ mV/ms, and by $-40$ mV about $20\,e^{5}/200\approx15$ mV/ms. That runaway acceleration above $V_T$ is the smooth, self-reinforcing spike onset that LIF's hard threshold completely misses.

▸ References

[FT03]	Fourcaud-Trocmé et al. — How spike generation mechanisms determine the neuronal response to fluctuating inputs, J. Neurosci. 23:11628, 2003
[BG05]	Brette & Gerstner — Adaptive exponential integrate-and-fire model, J. Neurophysiol. 94:3637–3642, 2005
[Ger14]	Gerstner et al. — Neuronal Dynamics, Cambridge, 2014. Ch. 5–6. neuronaldynamics.epfl.ch

§ 04

Frequently Asked Questions

🔬 SimulationWhat does the Δ_T slider control, and what happens as Δ_T → 0?▼

Δ_T (delta-T, the slope factor) controls the sharpness of spike initiation. Large Δ_T (10–20 mV): the exponential term activates gradually, producing a soft-threshold that resembles QIF/Izhikevich — the neuron charges up smoothly and fires when the exponential overcomes the leak. Small Δ_T (1–3 mV): the exponential term is negligible below V_T but grows explosively at V_T — the model approaches the LIF hard-threshold. Drag Δ_T from 0.5 to 10 mV and observe: the spike onset changes from abrupt (LIF-like) to smooth (QIF-like). Real cortical neurons have Δ_T ≈ 2–5 mV, measurable from the phase portrait (dV/dt vs V) — the exponential acceleration is visible as a hockey-stick shape at spike onset.

Key takeaway: Δ_T measures how sharply Na⁺ channels activate at threshold. Δ_T → 0 = LIF (hard threshold), Δ_T → ∞ = QIF (quadratic). Real neurons: Δ_T ≈ 2–5 mV.

🧠 ConceptualWhat is the spike initiation zone (SIZ) and why does it matter?▼

The SIZ is the region of the axon initial segment (AIS) where the high density of Na⁺ channels creates the lowest threshold for spike initiation — typically 20–50 μm from the soma. In the EIF model, V_T represents the voltage at which Na⁺ channel activation in the AIS becomes regenerative. The Δ_T parameter reflects the sharpness of this activation. The SIZ matters for neural coding because: (1) it determines the effective threshold for sensory detection; (2) the sharpness of spike initiation determines how precisely a neuron can detect the timing of synchronous inputs; (3) the SIZ distance from the soma affects how back-propagating APs interact with dendritic synaptic inputs for STDP.

Key takeaway: The SIZ is the axon initial segment where Na⁺ channel density is highest. V_T and Δ_T in the EIF model characterise SIZ properties — directly measurable from phase-plane analysis of recorded APs.

🌍 AppliedHow is the AdEx (EIF + adaptation) used in large-scale brain models?▼

The AdEx (Brette & Gerstner, 2005) is the standard model in the Human Brain Project's Brain Simulation Platform and in many large-scale cortical circuit models. It can reproduce all major cortical firing patterns (RS, IB, FS, LTS, TC — the same as Izhikevich but with biophysically interpretable parameters). The parameters have direct experimental correlates: Δ_T measured from phase-plot of recorded spikes; V_T from extracellular single-unit recordings; τ_w from adaptation measurements; b from AHP analysis. The AdEx is programmed in the NEST simulator (standard for Human Brain Project models) as its default neuron type.

Key takeaway: AdEx = EIF + slow adaptation current w. It is the standard model in Human Brain Project simulations and NEST simulator. Parameters are directly measurable from electrophysiology.

💡 Non-ObviousWhy is the EIF's f–I curve fundamentally different from LIF near threshold?▼

LIF has a rheobase I_rh = (V_T−E_L)/R_m — below this, the neuron never fires (V∞ < V_T). Above it, the f–I curve is smooth and analytic. The EIF has no hard rheobase — because the exponential term can always provide some additional depolarisation, the effective threshold is not a fixed voltage but a current-dependent voltage. For I below I_rh_eff, the neuron is in the 'excitable' state (stable fixed point, no firing). But noise can trigger spikes because the exponential provides positive feedback near V_T. The EIF f–I curve is therefore more sensitive to input fluctuations near threshold — explaining why cortical neurons are much more sensitive to input fluctuations than the LIF model predicts. This was Fourcaud-Trocmé et al.'s key result: spike initiation sharpness (1/Δ_T) determines the neuron's sensitivity to input fluctuations.

Key takeaway: EIF has no hard rheobase — the exponential term makes threshold sensitivity, making the neuron more responsive to fluctuating inputs than LIF predicts. Δ_T sets this fluctuation sensitivity.

📐 ComputationalWhat numerical method should be used for EIF, and how to handle the exponential singularity?▼

The exponential term g_L Δ_T exp((V−V_T)/Δ_T) grows rapidly near V_T. Above V_T, it grows without bound — the numerical solution diverges in finite time. The fix: set a numerical spike cutoff V_peak (e.g., +20 mV). When V exceeds V_peak, record a spike and reset. Use forward Euler or exponential Euler with dt = 0.025 ms — smaller than LIF requires because of the exponential acceleration. Do NOT use RK4 near threshold — the exponential blowup between timesteps makes RK4 worse than Euler for this singularity. The correct approach: explicit Euler + event detection. For the AdEx version, use implicit methods (Crank-Nicolson) because the adaptation variable w makes the system mildly stiff.

Key takeaway: Use forward Euler at dt ≤ 0.025 ms with V_peak cutoff. Do not use RK4 near the exponential singularity — explicit Euler + event detection is the correct and more stable choice for EIF.

🎓 DeepHow does the slope factor Δ_T connect to Na⁺ channel biophysics?▼

Δ_T is directly related to the slope of the Na⁺ channel activation curve. From HH theory, m∞(V) ≈ 0.5(1 + tanh((V−V_half)/k)). Near threshold, tanh ≈ exp, so m∞ ≈ exp((V−V_T)/Δ_T) where Δ_T ≈ 2k (k = activation slope factor). For cortical pyramidal neurons, k ≈ 5–8 mV from voltage-clamp data, giving Δ_T ≈ 1.5–3 mV. Fourcaud-Trocmé et al. (2003) showed that the precision of spike timing (jitter in response to a frozen noise input) is inversely proportional to Δ_T: sharp threshold (small Δ_T) → precise timing; broad threshold → imprecise. This predicts that neurons with sharper Na⁺ activation (e.g., axon initial segment channels, which have Δ_T ≈ 2 mV) encode timing more precisely than somatic channels (Δ_T ≈ 5 mV).

Key takeaway: Δ_T ≈ 2k where k is the Na⁺ channel activation slope factor from voltage-clamp data. Smaller Δ_T = sharper Na⁺ activation = more precise spike timing. Directly measurable from phase plots of recorded spikes.

§ 04 Best Resources

Izhikevich — Dynamical Systems in Neuroscience, MIT Press, 2007. Free: dynamicalsystems.org
Gerstner et al. — Neuronal Dynamics, Cambridge, 2014. neuronaldynamics.epfl.ch
Strogatz — Nonlinear Dynamics and Chaos, Westview, 2015
Ermentrout & Terman — Mathematical Foundations of Neuroscience, Springer, 2010

§ 05

Misconceptions & Common Errors

Sub-block A — Conceptual Misconceptions

❌"EIF is just LIF with a different threshold — V_T plays the same role as V_thresh in LIF."

✅V_T in EIF is the spike initiation threshold where the exponential term equals the leak, but it is NOT a hard firing threshold. The neuron does not fire when V reaches V_T — it fires when the exponential term grows large enough to drive V exponentially upward. The actual firing is determined by V_peak (the numerical cutoff), which can be set anywhere above V_T. The real threshold (where dV/dt becomes self-aining) is slightly above V_T. In LIF, V_thresh is both the detection threshold and the firing point — they coincide. In EIF, the threshold is a smooth unstable fixed point, not a discontinuity.

📖 Fourcaud-Trocmé et al. (2003) J. Neurosci. 23:11628; Gerstner et al. — Neuronal Dynamics, Ch. 5

Sub-block B — Numerical Errors

❌Setting V_peak = V_T in EIF — resetting the voltage as soon as the exponential starts, before the spike completes.

✅V_peak must be set significantly above V_T (typically +20 to +40 mV). V_T is where the exponential term begins to contribute — not where the spike peak occurs. If V_peak = V_T, the model resets immediately at threshold without a spike and behaves exactly like LIF (no benefit from the exponential term). Set V_peak = 0 to +20 mV for physiologically realistic spike shapes, or V_peak = +40 mV if you want the full AP shape approximation.

🔍 Why: V_peak is confused with V_T — forgetting that the exponential drives V well above V_T before the biological spike peaks.

§ 05 References

Izhikevich — Dynamical Systems in Neuroscience, MIT Press, 2007
Gerstner et al. — Neuronal Dynamics, Cambridge, 2014. neuronaldynamics.epfl.ch
Ermentrout & Terman — Mathematical Foundations of Neuroscience, Springer, 2010
Strogatz — Nonlinear Dynamics and Chaos, Westview, 2015