EIF voltage traces and a recurrent network raster

The exponential integrate-and-fire (EIF) neuron is the natural next step up from LIF: same one-variable spirit, but with an explicit exponential term that models the upswing of a spike rather than a hard threshold-crossing rule.

The subthreshold dynamics are

with a peak-and-reset rule

Compared to LIF, the threshold $V_T$ is no longer a discontinuity: it is the soft inflection point at which the exponential term takes over and the membrane potential blows up on its own. The slope factor $\Delta_T$ controls how sharply this transition happens — small $\Delta_T$ approximates LIF, larger $\Delta_T$ smooths the spike initiation.

The code integrates with forward Euler at $\Delta t = 0.1\,\text{ms}$,

using $\tau_m = 10\,\text{ms}$, $V_\text{rest} = -65\,\text{mV}$, $V_\text{reset} = -70\,\text{mV}$, $V_T = -50\,\text{mV}$, $\Delta_T = 2\,\text{mV}$, $V_\text{peak} = 0\,\text{mV}$, $R_m = 10\,\text{M}\Omega$, and tonic input $I = 2.5\,\text{nA}$.

With the same tonic input as nb000, the EIF neuron fires at a different rate than the LIF neuron — the exponential term accelerates spike initiation, but the upswing itself takes time, so the period between spikes shifts.

The network is structurally identical to nb000: 200 all-excitatory neurons, sparse $p_\text{conn} = 0.1$ random connectivity, weights $w = 0.1$, exponentially-decaying synaptic input with $\tau_\text{syn} = 5\,\text{ms}$, noisy per-neuron bias $I_i^\text{bias} \sim \mathcal{N}(2.2, 0.4^2)\,\text{nA}$, and a $2\,\text{ms}$ refractory period after each spike. The only difference is that every neuron now obeys the EIF dynamics above.

Same plumbing as nb000 — neuron eif and neuron enet produce the figures, the runner copies them and writes numbers.json.

EIF run parameters

EIF network run parameters