Leaky Integrate-and-Fire

Hodgkin-Huxley is detailed and beautiful — four coupled ODEs that capture the full action-potential cycle from molecular gating up. It's also expensive. Each neuron costs four state variables, an integration step of $\sim 10^{-2}\mathrm{ms}$, and a stiff integrator. Simulating a network of ten thousand HH neurons for a single second of biological time is a serious computational task. For most network-scale questions, you don't need that much detail. You need a model that fires when driven, has a refractory period, and integrates over time.

Leaky integrate-and-fire is the smallest model that does those things. One state variable, one ODE, one algebraic spike rule. This post is short by comparison to HH — the interesting parts are (a) understanding why the radical simplification still captures what network-level modeling actually needs, and (b) deriving the firing-rate-vs-current curve in closed form, which HH doesn't admit.

The model

Treat the neuron as a leaky RC circuit. The membrane is a capacitor in parallel with a leak conductance. The leak pulls $V$ back toward the resting potential. Applied current pushes $V$ away from rest. The whole subthreshold equation:

${\tau }_m=R{C}_m$ is the membrane time constant (about 20 ms in cortex). $R$ is the input resistance (around $10\mathrm{M\Omega }$). $V_{\mathrm{rest}}$ is the resting potential (about -70 mV). In steady state under constant current, the membrane settles to:

That's the entire subthreshold story. The membrane is a low-pass filter on the applied current with time constant ${\tau }_m$. So far it's just an RC circuit. The neuron-specific machinery is the spike.

The spike: threshold and reset

LIF makes no attempt to model the action potential. Instead, it imposes a rule: when $V$ crosses a threshold $V_{\mathrm{thresh}}$, declare a spike, reset $V$ to a reset value $V_{\mathrm{reset}}$, and (in most variants) hold $V$ at $V_{\mathrm{reset}}$ for a refractory period ${\tau }_{\mathrm{ref}}$ before resuming integration.

The action potential itself, the millisecond-scale up-and-down of HH's voltage trace, is replaced by a delta function — declare the spike, reset, move on. The biology this represents is "the sodium-channel cascade fires, the spike propagates down the axon, the membrane resets." The full waveform collapses into a single event time.

This is non-physical, on several counts. Real neurons have spike waveforms with finite width. The threshold isn't a sharp value but an emergent property of sodium current's voltage-dependent nonlinearity. The reset to a single fixed voltage hides what's actually a fast repolarization driven by potassium current overshooting slightly. None of these matter for most network-scale questions, because at the network level you mostly care about when each neuron fires, not what each spike's waveform looks like. The synaptic input that the next neuron sees is just a function of "did neuron $i$ spike at time $t_k$" — and LIF gets the spike times almost right.

Common parameter choices for a cortical pyramidal neuron: $V_{\mathrm{thresh}}=-55\mathrm{mV}$, $V_{\mathrm{reset}}=-75\mathrm{mV}$, ${\tau }_{\mathrm{ref}}=2\mathrm{ms}$. Numbers vary by neuron type; the structure doesn't.

Try it

The simulator below integrates the LIF subthreshold equation and applies the threshold-and-reset rule. The top plot is $V(t)$ with the threshold and reset levels marked as dashed guide lines. The bottom plot is a spike raster — one tick per spike. Drag the slider for $I_{\mathrm{app}}$ or click through the presets.

Things to look for. In "Subthreshold", $V$ charges exponentially toward $V_{\infty }=-60\mathrm{mV}$ and parks there — never crosses threshold, no spikes. In "Periodic spiking", you can see the charging trajectory between spikes: the same exponential curve every time, because the dynamics are time-homogeneous once you condition on starting from $V_{\mathrm{reset}}$. In "Refractory limit", the firing rate is hard-capped at $1/{\tau }_{\mathrm{ref}}\approx 500\mathrm{Hz}$ no matter how hard you push $I_{\mathrm{app}}$.

The f-I curve, in closed form

One of LIF's main advantages over HH: the firing rate as a function of applied current has a closed-form expression. Integrating the subthreshold equation from $V_{\mathrm{reset}}$ to $V_{\mathrm{thresh}}$ under constant $I_{\mathrm{app}}$ gives the charging time:

That's the time the membrane spends climbing exponentially from reset back up to threshold. Add the refractory hold and the interspike interval is:

And the firing rate is just the inverse:

This is the LIF f-I curve. It's zero for $V_{\infty }\le V_{\mathrm{thresh}}$ — the membrane never reaches threshold. Right at threshold the logarithm diverges, so $f\to 0$ from above, giving the characteristic "soft" onset that real cortical neurons show. At very large $I_{\mathrm{app}}$, $V_{\infty }$ is much larger than $V_{\mathrm{thresh}}$, the logarithm collapses, and the firing rate saturates at $1/{\tau }_{\mathrm{ref}}$. Real neurons have softer saturation because the sodium-channel inactivation kinetics that LIF threw away actually impose a smoother high-current ceiling. LIF's hard cap is one of the model's clean failure modes.

What LIF captures and what it doesn't

LIF captures: firing-rate dependence on current, refractory limit, integration of subthreshold inputs as an exponential low-pass filter with time constant ${\tau }_m$, threshold behavior. Those are the pieces you need to model networks where the relevant computation is "neuron $i$ fires when its accumulated synaptic input crosses a threshold."

LIF misses: the action potential waveform itself (delta function vs real shape), spike-frequency adaptation (real neurons slow down during sustained drive, LIF doesn't), bursting (LIF can only do regular firing, not doublets or bursts), subthreshold resonance and oscillations (real neurons have voltage-dependent currents that LIF lacks), and any kind of inactivation beyond the refractory clamp.

The cheap fixes

Three standard extensions, each adding minimal machinery for a major increase in expressiveness.

Adaptive LIF. Add one slow adaptation variable $w$ that jumps up at each spike and decays exponentially between spikes. The variable acts as a negative current on $V$, slowing the next spike. One extra ODE and one extra parameter gets you spike-frequency adaptation — the slow decay of firing rate during sustained input that's visible in almost every real cortical neuron.

Exponential LIF. Replace the linear leak with a term that pulls $V$ upward as it approaches threshold. The hard threshold becomes a soft one, and the spike onset becomes more realistic. Used in the AdEx (adaptive exponential) variant — adaptation plus exponential threshold — which is the workhorse for "detailed but tractable" network simulations in current research.

Izhikevich neuron. A 2D quadratic system with a piecewise reset that produces regular spiking, bursting, chattering, fast spiking, low-threshold spiking, and a half-dozen other firing patterns by tuning four parameters. Designed specifically to be computationally cheap — no transcendentals in the dynamics — while reproducing the firing-pattern diversity of cortical neuron types. The dominant choice for large-scale simulations that want a richer neuron repertoire than plain LIF.

Why LIF is the network workhorse

For network-scale modeling — tens of thousands to millions of neurons — LIF and its variants are the de facto standard. Computational efficiency is the obvious reason: a million-neuron LIF network simulates in real time on a single GPU; a million-neuron HH network is a supercomputer job. Every major simulator (Brian, NEST, GeNN, Auryn) takes LIF as the default neuron type.

The less-obvious reason: at the network level, the questions you typically want to answer — does this circuit synchronize, does it produce gamma oscillations, what's the firing-rate distribution, how does it respond to noisy input — depend on spike timing and rate, not on action-potential shape. LIF gets the spike timing approximately right and the rate-vs-input curve closed-form. That's almost always enough for network behavior. The biophysics LIF throws away comes back in through synaptic models, which are bolted on separately.

Future posts in this section go both directions from here. Down toward biophysics: compartmental models that resolve the spatial structure of the dendrite and axon, where HH-like channel dynamics matter at every compartment. Up toward networks: AdEx and Izhikevich for richer single-neuron behavior, then population-level descriptions like Wilson-Cowan and mean-field reductions where you don't model individual neurons at all. LIF sits at the middle of both directions and is the right starting point for either.