NMT Half-Centre Oscillator

Abstract

Task 2 of the Neuromorphic Toolbox proposal (P23) asks us to couple two sodium/potassium neurons through reciprocal graded inhibition, apply an asymmetric current onset and compare their voltages and spike times with Toolbox reference traces.

Results

Response to asymmetric current onset

Neuron 1 produces 20 spikes and neuron 2 produces 20 spikes during the 200 ms recording. After equal drive begins at 10 ms, their recorded spike events alternate. Figure 1 shows the voltages, reciprocal inhibition and joint voltage trajectory.

Methods

1. Couple the proposal’s ionic neurons with graded inhibition

Each cell uses the Task 1 ionic model. For neuron index $i\in\{1,2\}$ and channel $x\in\{\mathrm{Na},\mathrm{K}\}$,

Here $t$ is time in ms, $V_i$ is membrane voltage in mV, $z_{x,i}$ is filtered voltage in mV, $\tau_x$ is the channel time constant in ms, $g_{x,i}$ and $\bar g_x$ are channel and maximum conductance in µS, $\gamma_x$ is the sigmoid slope in mV$^{-1}$, and $b_x$ is its midpoint in mV. The argument $u$ is dimensionless.

For the graded synapse from neuron $j$ to neuron $i$, with $j\ne i$,

Here $z_{j\to i}$ is the synaptic gate voltage in mV, $\tau_s$ its time constant in ms, $g_{j\to i}$ and $\bar g_s$ the synaptic and maximum conductance in µS, $\gamma_s$ the slope in mV$^{-1}$ and $b_s$ the midpoint in mV. The membrane equation gains the inhibitory conductance term:

Here $C$ is capacitance in nF, $I_i$ injected current in nA, $g_L$ leak conductance in µS, and $V_L$, $E_{\mathrm{Na}}$, $E_{\mathrm{K}}$ and $E_s$ are leak, sodium, potassium and inhibitory reversal potentials in mV. These units preserve the proposal’s numerical equations; its C/g/I units require confirmation against the Toolbox.

At every step of length $\Delta t=0.001$ ms, all ionic and synaptic gates advance by forward Euler using the preceding membrane voltages. New gate values determine conductances; both membranes then advance using those new conductances and the old voltages. The cells have no voltage reset.

2. Represent both synapses as native graph connections

SNNLab 0.24 lets a custom synapse declare input_unit="mV" and output_unit="uS". Each projection reads pre.voltage at the beginning of the timestep and delivers the newly updated conductance to post.inhibitory. Dimensionless projection weights are one; the physical maximum conductance belongs inside the synapse, after the nonlinear gate. Zero added delay avoids shifting the proposal’s coupling by an extra timestep.

The following callback implementation is loaded directly from this experiment:

import torch


def neuron_initial_state(ctx):
    voltage = torch.full(
        ctx.shape, ctx.config["initial_voltage_mv"], device=ctx.device, dtype=ctx.dtype
    )
    return dict(
        voltage=voltage,
        z_na=voltage.clone(),
        z_k=voltage.clone(),
        current=torch.zeros_like(voltage),
        step_index=torch.zeros(ctx.shape, device=ctx.device, dtype=torch.int64),
    )


def neuron_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 = cfg["g_na_us"] * torch.sigmoid(
        cfg["gamma_na_per_mv"] * (z_na - cfg["bias_na_mv"])
    )
    g_k = cfg["g_k_us"] * torch.sigmoid(
        cfg["gamma_k_per_mv"] * (z_k - cfg["bias_k_mv"])
    )
    # Use integer intervals so current changes exactly at 5 and 10 ms.
    index = state["step_index"]
    current = torch.where(
        (index >= cfg["kick_start_step"]) & (index < cfg["kick_stop_step"]),
        torch.full_like(voltage, cfg["kick_na"]),
        torch.where(
            index >= cfg["drive_start_step"],
            torch.full_like(voltage, cfg["drive_na"]),
            torch.zeros_like(voltage),
        ),
    )
    outward = (
        cfg["leak_us"] * (voltage - cfg["leak_reversal_mv"])
        + g_na * (voltage - cfg["e_na_mv"])
        + g_k * (voltage - cfg["e_k_mv"])
        + ctx.inhibitory * (voltage - cfg["synapse_reversal_mv"])
    )
    updated = voltage + dt / cfg["capacitance_nf"] * (current - outward)
    spikes = (
        (voltage < cfg["spike_threshold_mv"]) & (updated >= cfg["spike_threshold_mv"])
    ).to(voltage.dtype)
    return {
        **state,
        "voltage": updated,
        "z_na": z_na,
        "z_k": z_k,
        "current": current,
        "step_index": index + 1,
    }, spikes


def synapse_initial_state(ctx):
    gate = torch.full(
        ctx.shape, ctx.config["initial_gate_mv"], device=ctx.device, dtype=ctx.dtype
    )
    value = ctx.config["g_max_us"] * torch.sigmoid(
        ctx.config["gamma_per_mv"] * (gate - ctx.config["bias_mv"])
    )
    return dict(gate=gate, value=value)


def synapse_step(ctx):
    gate = ctx.state["gate"] + ctx.dt_ms / ctx.config["tau_ms"] * (
        ctx.drive - ctx.state["gate"]
    )
    value = ctx.config["g_max_us"] * torch.sigmoid(
        ctx.config["gamma_per_mv"] * (gate - ctx.config["bias_mv"])
    )
    return dict(gate=gate, value=value)

Each neuron declares conductance input ports. Because native custom neuron ports share one input unit, its prescribed injected current is calculated inside the neuron callback from saved schedule parameters and an integer step counter. The counter is retained in native continuation state. This implements the fixed offline protocol; it does not provide an interactive current input.

3. Run one HCO and validate its saved trajectory

Both membrane voltages and every gate start at −70 mV. Neuron 1 receives 1.0 nA for $5\leq t<10$ ms, then 0.8 nA; neuron 2 receives 0.8 nA for $10\leq t<200$ ms. All other input intervals have zero current. One HCO runs for 200 ms in float64 on CPU; no batch benchmark is performed. Spikes are sampled upward −20 mV crossings, timed at the end of the crossing timestep without interpolation. Strict alternation after 10 ms means every consecutive event belongs to the other cell, with no simultaneous events.

Against independent forward-Euler equations, maximum voltage differences are 1.73e-10 mV for n1 and 1.20e-10 mV for n2. Both cells have matching reference spike counts.

The proposal’s thresholds are maximum voltage difference below 0.1 mV, spike-time difference below 0.05 ms and identical counts for each neuron. Toolbox reference traces have not been supplied, so this comparison establishes agreement with the written equations only.