NMT Single-Neuron Compatibility

Abstract

Task 1 of the Neuromorphic Toolbox proposal (P23) asks us to reproduce a single sodium/potassium spiking neuron’s response to a current step in SNNLab, then compare its voltage and spike times with Toolbox reference traces.

Results

Single-neuron current response

The current step produces 4 spikes. The sodium gate responds faster than the potassium gate; the delayed rise in potassium conductance accompanies repolarisation and the voltage undershoot after each spike (Figure 1).

Methods

1. Implement the proposal’s equations

The proposal defines a leaky membrane with sodium and potassium currents. For each channel $x \in \{\mathrm{Na},\mathrm{K}\}$, the gate voltage $z_x$ follows membrane voltage $V$ with time constant $\tau_x$:

Here $t$ is time in ms, $V$ and $z_x$ are in mV, $\tau_x$ is in ms, $g_x$ is the channel conductance, $\bar g_x$ is its maximum conductance, $\gamma_x$ is the sigmoid slope in mV$^{-1}$, and $b_x$ is its midpoint in mV. The sigmoid $\sigma$ maps the dimensionless argument $u$ to a value between zero and one; $z_x$ itself is a voltage, not a probability.

The membrane equation is

Here $C$ is membrane capacitance, $I_{\mathrm{ext}}$ is injected current, $g_L$ is leak conductance, $V_L$ is the leak reversal potential, and $E_{\mathrm{Na}}$ and $E_{\mathrm{K}}$ are the sodium and potassium reversal potentials. We embed the proposal’s numerical values in nF, µS and nA, which are consistent with mV and ms; the proposal does not explicitly label the capacitance, conductance or current units. Saved values appear in the appendix.

Forward Euler must advance the gates before the membrane. At timestep index $n$, with step length $\Delta t$, we compute

Superscripts denote sampled timesteps; $I_{\mathrm{ext}}^n$ applies over $[n\Delta t,(n+1)\Delta t)$. The gate and conductance updates produce panels B and C, and the membrane update produces panel A. There is no voltage reset.

2. Express the dynamics as SNNLab extensions

The implementation below is loaded directly from this experiment’s custom/dynamics.py. SNNLab supplies the previous tensor state, timestep, configuration and incoming currents through NeuronContext. The callback returns the new state and binary upward-crossing events. The initializer sets voltage and both gates to −70 mV; the synapse passes current unchanged.

import torch


def conductances(voltage_na, voltage_k, cfg):
    return (
        cfg["g_na_us"]
        * torch.sigmoid(cfg["gamma_na_per_mv"] * (voltage_na - cfg["bias_na_mv"])),
        cfg["g_k_us"]
        * torch.sigmoid(cfg["gamma_k_per_mv"] * (voltage_k - cfg["bias_k_mv"])),
    )


def initial_state(ctx):
    voltage = torch.full(
        ctx.shape,
        ctx.config["initial_voltage_mv"],
        dtype=ctx.dtype,
        device=ctx.device,
    )
    g_na, g_k = conductances(voltage, voltage, ctx.config)
    return dict(
        voltage=voltage,
        z_na=voltage.clone(),
        z_k=voltage.clone(),
        g_na=g_na,
        g_k=g_k,
    )


def step(ctx):
    cfg, state, dt = ctx.config, ctx.state, ctx.dt_ms
    voltage = state["voltage"]
    z_na = state["z_na"] + dt / cfg["tau_na_ms"] * (voltage - state["z_na"])
    z_k = state["z_k"] + dt / cfg["tau_k_ms"] * (voltage - state["z_k"])
    g_na, g_k = conductances(z_na, z_k, cfg)
    ionic = (
        cfg["leak_us"] * (voltage - cfg["leak_reversal_mv"])
        + g_na * (voltage - cfg["e_na_mv"])
        + g_k * (voltage - cfg["e_k_mv"])
    )
    current = ctx.excitatory - ctx.inhibitory
    updated = voltage + dt * (current - ionic) / cfg["capacitance_nf"]
    threshold = cfg["spike_threshold_mv"]
    spikes = ((voltage < threshold) & (updated >= threshold)).to(voltage.dtype)
    return {
        **state,
        "voltage": updated,
        "z_na": z_na,
        "z_k": z_k,
        "g_na": g_na,
        "g_k": g_k,
    }, spikes


def inject_current(ctx):
    """Pass the projected signed current unchanged, without decay or clipping."""
    return {"value": ctx.drive.to(ctx.state["value"].dtype)}

The recipe registers these importable callbacks once per process, then uses CustomNeuron and CustomSynapse in the authored network. This registration excerpt exposes the gate voltages and conductances as named state ports:

from snnlab import extensions, lang
from experiments.exp146.custom import dynamics

extensions.register_neuron(
    "pinglab.exp146.toolbox_ei/v1",
    dynamics.step,
    initialize=dynamics.initial_state,
    input_unit="nA",
    state_units={"z_na": "mV", "z_k": "mV", "g_na": "uS", "g_k": "uS"},
    capture_safe=True,
)
extensions.register_synapse(
    "pinglab.exp146.injected_current/v1",
    dynamics.inject_current,
    output_unit="nA",
    capture_safe=True,
)

The network contains one custom neuron and a continuous input connected to its excitatory port through the current extension at unit weight, with direct initialization scaling. The recipe passes the saved neuron settings as the custom neuron’s configuration. Native graph outputs retain voltage, gates, conductances, injected current and crossing events.

3. Run the current-step protocol and check the traces

Voltage and both gates start at −70 mV. We execute 50,000 steps of $\Delta t=0.001$ ms, for a total of 50 ms, with float64 arithmetic on CPU. Injected current is zero before 10 ms and 0.8 nA from 10 ms to the end (panel D). An upward crossing satisfies $V^n<-20$ mV and $V^{n+1}\geq-20$ mV; its recorded time is $(n+1)\Delta t$, without interpolation.

An independent scalar Euler implementation checks the saved trajectory.

The maximum voltage difference was 4.55e-13 mV, with identical spike counts and a maximum crossing-time difference of 0 ms.

The proposed acceptance thresholds are a maximum voltage error below 0.1 mV, crossing-time error below 0.05 ms and identical spike counts. Passing the independent numerical comparison does not establish agreement with the Toolbox: its reference traces have not been supplied.