A neuron model is a set of equations describing how a neuron integrates its inputs and decides when to fire. The choice of model is a trade-off along a single axis: at one end sit richly biophysical models that reproduce the fine detail of a real cell but are expensive to simulate, and at the other sit stripped-down abstractions that capture only "integrate inputs, cross a threshold, emit a spike, reset" but run fast enough for networks of millions of neurons. The models below span that spectrum, and which one is appropriate depends entirely on whether the goal is biological fidelity or large-scale computation.
Leaky Integrate-and-Fire
The leaky integrate-and-fire (LIF) neuron is the workhorse of large-scale spiking network theory. It treats the membrane as a leaky capacitor: input current is integrated onto the membrane while the voltage continuously leaks back toward its resting value. The subthreshold dynamics are a single linear differential equation,
where τm = RC is the membrane time constant, typically around 10 ms. Crucially, the model does not generate the spike itself: when V reaches a fixed threshold ϑ, a spike is emitted by fiat, the voltage is reset to a value Vr below threshold, and (optionally after a refractory pause) integration resumes. This makes LIF extremely cheap, on the order of five arithmetic operations per millisecond of simulation (Izhikevich, 2004). The price of that thrift is dynamical poverty: with a single state variable it produces only regular tonic firing and cannot reproduce bursting, rebound spikes, subthreshold oscillations or an adaptive threshold. Dropping the leak term entirely gives the simpler integrate-and-fire neuron, a perfect integrator whose lineage traces back to Lapicque's 1907 model of nerve excitation (Brunel & van Rossum, 2007).
Hodgkin–Huxley
At the opposite extreme is the Hodgkin–Huxley model, the foundation of computational neuroscience. Derived from voltage-clamp experiments on the squid giant axon, it is a conductance-based model in which the membrane current is the sum of a capacitive term and voltage- and time-dependent sodium, potassium and leak currents (Hodgkin & Huxley, 1952):
The gating variables m, h and n (sodium activation, sodium inactivation and potassium activation) each follow their own first-order kinetics with empirically fitted, voltage-dependent rates. Because its parameters are physically real conductances and reversal potentials, the model actually generates the action potential and reproduces essentially every single-neuron phenomenon; the work earned Hodgkin and Huxley a Nobel Prize. The cost is severe: Izhikevich estimates around 1,200 operations per millisecond, over two hundred times more than LIF, so only tens of such neurons can be simulated in real time.
Izhikevich Model
The Izhikevich model was designed to resolve exactly this tension: to keep the qualitative richness of Hodgkin–Huxley dynamics at nearly integrate-and-fire cost. Using bifurcation theory, it reduces the dynamics to two variables: a quadratic equation for the membrane potential v coupled to a linear recovery variable u (Izhikevich, 2003).
u′ = a(bv − u)
if v ≥ 30 mV, then v ← c, u ← u + d
Just four parameters (a, b, c and d) tune the recovery dynamics and the after-spike reset, and by choosing them the same equations reproduce a remarkable catalogue of cortical firing patterns: regular spiking, intrinsic bursting, chattering, fast spiking, low-threshold spiking and more. At roughly thirteen operations per millisecond it is cheap enough for large-scale cortical simulation, which is why Izhikevich recommends it as the default choice when both efficiency and dynamical realism matter.
Adaptive Exponential Integrate-and-Fire
The adaptive exponential integrate-and-fire (AdEx) model adds two biologically important ingredients to LIF (Brette & Gerstner, 2005). An exponential term captures the sharp upswing of spike initiation and gives the neuron a soft, dynamic threshold, while a coupled adaptation current w produces spike-frequency adaptation and a variety of firing patterns.
τw dw/dt = a(V − Vrest) − w, and at each spike w → w + b
Here ΔT sets the sharpness of spike initiation and ϑrh the rheobase threshold, while the pair a and b govern subthreshold and spike-triggered adaptation respectively. Despite being only a two-variable system, AdEx predicts the spike times of detailed conductance-based models remarkably well (around 96% of spikes to within a couple of milliseconds) with parameters that can be fitted from simple current-clamp recordings, making it a favourite for realistic yet tractable network models.
Spike Response Model
The Spike Response Model (SRM) generalises LIF in a different direction. Instead of a differential equation, it writes the membrane potential directly as a sum of response kernels, integrating the effect of past spikes and input current (Gerstner et al., 2014):
The kernel η describes the stereotyped spike and after-hyperpolarisation added following each of the neuron's own output spikes, and κ filters input current into a voltage deflection; a spike is fired when V reaches a possibly time-dependent threshold. LIF is recovered when both kernels are exponential, but the kernel formulation additionally accommodates adaptation and a dynamic threshold without extra state variables. Because it maps naturally onto generalised linear models and its kernels can be fitted directly to recorded data, the SRM is a powerful tool for statistical prediction of spike times.
Other Models
Several further models occupy the middle ground. The quadratic integrate-and-fire neuron, equivalent to the theta neuron of Ermentrout and Kopell under a change of variables, is the canonical model for Class 1 excitability (any Type I neuron near its firing onset reduces to it) and captures arbitrarily low firing rates and spike latencies at only about seven operations per millisecond. Reduced conductance-based models such as FitzHugh–Nagumo and Hindmarsh–Rose sit higher up the cost scale (roughly 70 and 120 operations per millisecond) and are valued for the geometric insight they give into excitability, resonance and bursting.
Plausibility versus Cost
The models form a clear frontier between biological realism and computational cost. Izhikevich (2004) tabulated this trade-off explicitly, measuring each model's neuro-computational repertoire against its cost per millisecond of simulation.
| Model | Operations / 1 ms | Biophysical detail |
|---|---|---|
| Leaky integrate-and-fire | ~5 | None; spike is imposed |
| Quadratic IF / theta | ~7 | Canonical Type I excitability |
| Izhikevich (2003) | ~13 | Qualitative, Hodgkin–Huxley-like |
| FitzHugh–Nagumo | ~72 | Reduced biophysical |
| Hindmarsh–Rose | ~120 | Reduced, bursting |
| Hodgkin–Huxley | ~1,200 | Full biophysical |
The moral of the table is that plain LIF is the cheapest but, in Izhikevich's words, "one of the worst models to use" if realism matters, while Hodgkin–Huxley is the most faithful but hundreds of times more costly. The Izhikevich and AdEx models sit near the efficient frontier, offering a Hodgkin–Huxley-like range of behaviours at close to integrate-and-fire cost, which is why, for the spiking networks benchmarked on this site, the integrate-and-fire family and its adaptive cousins remain the pragmatic default.