ExpIFNeuron

ExpIFNeuron is the deterministic exponential integrate-and-fire model from Fourcaud-Trocmé, Hansel, van Vreeswijk and Brunel (2003). The maintained model has one voltage state plus an optional post-spike refractory remainder.

Primary source: N. Fourcaud-Trocmé et al., Journal of Neuroscience 23(37), 11628–11640, 2003, doi:10.1523/JNEUROSCI.23-37-11628.2003.

Equation and units

The source current balance is

[ C\frac{dV}{dt} = -g_L(V-V_L)

• g_L\Delta_T\exp!\left(\frac{V-V_T}{\Delta_T}\right) + I. ]

After division by (g_L), the implementation evaluates

[ \tau\frac{dV}{dt} = -(V-V_{rest})

• \Delta_T\exp!\left(\frac{V-V_{rh}}{\Delta_T}\right)
• I_{norm}, \qquad \tau=C/g_L. ]

The public current argument is therefore (I_{norm}=I/g_L), expressed in the same millivolt-equivalent scale as the other terms. It is not an unscaled current in picoamperes.

v_rh is the soft threshold of the exponential current. v_threshold is a separate finite numerical cutoff used to register a spike. Confusing the two changes the response towards a leaky integrate-and-fire model; the source specifically discusses this cutoff dependence.

Maintained defaults

Field	Default	Unit	Meaning
v	−65.0	mV	initial membrane voltage
v_rest	−65.0	mV	leak reversal voltage
v_reset	−68.0	mV	post-spike reset voltage
v_threshold	+30.0	mV	finite numerical spike cutoff
v_rh	−59.9	mV	soft exponential threshold
delta_t	3.48	mV	exponential slope factor
tau	10.0	ms	membrane time constant
dt	0.02	ms	RK4 macro-step
refractory_period	0.0	ms	reset hold loaded after a spike
refractory_remaining	0.0	ms	runtime reset-hold remainder

The voltage, reset, soft threshold, slope factor and membrane time constant are the source's fitted EIF values. The paper used a +30 mV finite cutoff for this fit and a 1.7 ms refractory interval in its Wang–Buzsáki comparison protocol. SC-NeuroCore keeps refractory_period=0.0 as the deterministic schema-to-RTL contract; set it to 1.7 to reproduce that particular protocol facet.

Numerical contract

Each macro-step is candidate-first classical RK4:

k1 = f(v)
k2 = f(v + dt*k1/2)
k3 = f(v + dt*k2/2)
k4 = f(v + dt*k3)
candidate = v + (dt/6)*(k1 + 2*k2 + 2*k3 + k4)

RK4 stage voltages are bounded at v_threshold before evaluating the exponential. This is an event-surface bound: it leaves every pre-cutoff stage unchanged and avoids evaluating an irrelevant divergent post-event voltage. It is not the former arbitrary exponential-argument clip.

If candidate >= v_threshold, the step emits 1, stores v_reset, and loads refractory_period. While refractory_remaining > 0, voltage stays at v_reset, the remainder decreases by dt, and the step emits 0. Otherwise the step stores the candidate and emits 0.

Construction and runtime checks fail closed for non-finite values, non-positive scales, invalid threshold relationships, voltage at or above the cutoff between steps, or refractory state outside its declared interval. Native C-ABI kernels validate the complete run before writing their output buffers.

Python API

from sc_neurocore.neurons.models.expif import ExpIFNeuron

neuron = ExpIFNeuron()
trace, spikes = neuron.simulate(1_000, current=20.0, backend="auto")
assert spikes == 2
assert trace.shape == (1_000,)

trace[t] is the post-step voltage, including reset and refractory-held samples. A successful run commits its final voltage and refractory remainder to the instance. A rejected compiled run leaves both fields unchanged.

Accepted backend selectors are auto, python, rust, julia, go, and mojo.

Backend	Public contract
Python	complete state and parameter contract
Rust engine	factory-default contract; explicit rejection otherwise
Julia	complete state and parameter contract
Go C ABI	complete state and parameter contract
Mojo C ABI	complete state and parameter contract

The Rust safety kernel is additionally maintained and tested as a fail-closed native surface. auto follows the measured Julia → Go → Mojo → compatible Rust → Python order recorded by benchmarks/bench_model_expif.py. The Rust lane is considered only when the instance matches its factory contract.

Reproducibility and parity

The descriptor's reference configuration is 1,000 default-parameter steps at current=20.0. It emits two spikes, ends at v=-59.00168087076545, and hashes the float64 voltage trace to cd3fc0cd092a9924e7477426c3d8622510ea01c37ed51b83188b38f7e026a1c6.

The enrolled 1,000-step event goldens are:

Normalised current	0	5	10	20	50	100
Spike count	0	0	1	2	5	9

The hand recurrence, TOML schema and JSON schema have exact event parity and a float64 state envelope of at most 2e-10. The generated RTL is enrolled at Q32.32 with the same event counts over this operating set. Q16.16 is not claimed: the steep source exponential crosses its active fixed-point range and does not preserve these event counts.

Evidence anchors:

• tests/test_expif_backends.py — executable Python/Rust/Julia/Go/Mojo parity, complete ABI state, dispatch and rejection paths;
• tests/test_cosim_exp_if.py — independent source recurrence, paired schemas and Q32.32 RTL event parity;
• tests/test_reference_exp_if.py — DOI-bound driven reference trace;
• src/sc_neurocore/neurons/model_descriptors/ExpIFNeuron.toml — reproducible digest and dual-axis readiness facets;
• hdl/formal/catalogue/sc_exp_if.sby — bounded formal safety job.

Benchmark evidence

benchmarks/bench_model_expif.py measures the public simulate() path for all five backends under one workload. It records source hashes, event and trace parity, CPU affinity, host load, governor state and runtime versions. The committed result is benchmarks/results/local_python_2026-06-16_expif_rk4.json; its timings are a local regression record, not a general throughput or hardware-isolation claim.

Scope limits

• The model represents exponential spike initiation in a point-neuron voltage equation; it does not reproduce ionic-channel waveforms.
• The zero-refractory default is chosen for deterministic co-simulation and is not the source paper's entire fitted comparison protocol.
• Fixed-point validation currently proves Q32.32 event parity only at the declared operating points and horizon.
• Higher silicon readiness requires separate synthesis, timing and hardware evidence.