LapicqueNeuron¶

Module: sc_neurocore.neurons.models.lapicque

Primary source: Lapicque (1907); English translation DOI 10.1007/s00422-007-0189-6

Interpretive companion: Brunel and van Rossum (2007), DOI 10.1007/s00422-007-0190-0

Identity boundary¶

Lapicque's paper treats nerve excitation as the first attainment of a polarization threshold in a leaky-capacitor circuit. It does not define an automatic post-event reset or a repetitive spike generator. SC-NeuroCore therefore exposes two deliberately separate profiles:

Python

from sc_neurocore.neurons.models import LapicqueNeuron, SCLapicqueLIFNeuron

source = LapicqueNeuron.lapicque_1907()  # counted source identity
compat = SCLapicqueLIFNeuron()           # count-neutral SC hard-reset LIF
legacy = LapicqueNeuron()                # preserved alias of the SC profile

The zero-argument legacy constructor remains compatible with existing network, training, and user code. In the compiled NetworkRunner, the exact canonical name LapicqueNeuron selects the source profile; Lapicque, SCLapicqueLIF, and SCLapicqueLIFNeuron select the retained SC profile. Python Population("LapicqueNeuron", ...) follows the same canonical source route. Existing calls that pass SC-only parameters such as tau, resistance, v_rest, or v_reset remain on the compatibility profile.

Lapicque 1907 source profile¶

With source voltage $V$, series resistance $R$, polarization resistance $\rho$, capacitance $K$, and polarization $v$, the maintained source equation is

$$K\frac{dv}{dt}=\frac{V-v}{R}-\frac{v}{\rho}.$$

For a constant pulse over one timestep,

$$ v_{n+1}=v_\infty+(v_n-v_\infty)e^{-\Delta t/\beta},\qquad v_\infty=\frac{V\rho}{R+\rho},\qquad \beta=\frac{KR\rho}{R+\rho}. $$

The first candidate with $v_{n+1}\geq v_\mathrm{threshold}$ emits one event and latches excited=True. Polarization continues evolving; it is not reset. Calling reset() explicitly re-arms a new experiment.

Lapicque's strength-duration relation follows directly:

$$ V(t)=\frac{\alpha}{1-e^{-t/\beta}},\qquad \alpha=v_\mathrm{threshold}\frac{R+\rho}{\rho}. $$

The maintained defaults $K=1.1$, $R=10$, $\rho=1$, $\Delta t=0.01$ ms, and $v_\mathrm{threshold}=1$ give $\beta=1$ ms and $\alpha=11$. They are a normalized reproducibility point, not claimed experimental constants from the paper. Input to step() is source voltage for this profile.

Preserved SC profile¶

SCLapicqueLIFNeuron retains the historical exact-flow hard-reset recurrence

$$ \tau\frac{dv}{dt}=-(v-v_\mathrm{rest})+RI, $$

with constant-current exact flow and reset $v\to v_\mathrm{reset}$ at the threshold. This profile supports repetitive events, but that reset convention is not attributed to the complete 1907 experiment. See SC exact-flow hard-reset LIF.

Execution contract¶

simulate_complete(n_steps, drive, backend=...) returns aligned post-step float64 polarization/voltage and uint8 event arrays. Python, Rust/PyO3, Julia, Go, and Mojo accept the complete profile and parameter packet. Every batch validates fully before caller-visible state commits; Go and Mojo also validate before writing either C-ABI output buffer.

The independent source receipt uses $V=22$ for 2,000 steps. It records one event at zero-based index 69, preserves the complete polarization and event digests, and separately re-derives five strength-duration points. The 100,000-step controlled benchmark reports the same complete event vector in all five runtimes; maximum measured state difference is 1.222e-15.

Hardware boundary¶

The source-specialized sc_lapicque_1907 core implements the normalized exact flow in Q32.32. Co-simulation preserves the complete event vector at source voltages 5.5, 11, 12, and 22; the two suprathreshold cases emit at indices 248 and 69, and maximum state error stays below 7e-8. Yosys coarse synthesis reports 11,511 cells. A depth-20 SymbiYosys/Z3 job proves reset hygiene, permanent excitation latch, and absence of repeated events after latching.

The old sc_lapicque Q16.16 path remains the separate SC compatibility core. Timing, PPA, target-device, board, physical-silicon, and universal fixed-point equivalence evidence remain open; the source profile therefore stays at the honest H2 boundary.

Evidence¶

Surface	Durable evidence
Primary-source identity and schemas	`tests/test_model_lapicque_source_contract.py`
Independent oracle and receipt	`tests/test_reference_lapicque_source_receipt.py`; `src/sc_neurocore/neurons/reference_receipts/lapicque_1907.json`
Five-runtime complete parity	`tests/test_lapicque_backend_parity.py`; `tests/test_lapicque_engine_binding.py`
C-ABI failure atomicity	`tests/test_lapicque_backend_c_abi.py`
Source and SC co-simulation	`tests/test_cosim_lapicque.py`
Synthesis and formal	`hdl/reports/yosys_lapicque_1907_q3232_2026-08-30.json`; `hdl/formal/catalogue/sc_lapicque_1907.sby`
Controlled measurement	`benchmarks/results/bench_lapicque.json`; `tests/test_bench_lapicque.py`