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The membrane equation

The leaky integrate-and-fire (LIF) neuron is the workhorse of computational neuroscience and the primary building block of THRINDEX models. Its dynamics are described by a first-order linear differential equation: τmemdVdt=−V(t)+I(t)\tau_{\text{mem}} \frac{dV}{dt} = -V(t) + I(t) Where:
  • V(t) is the membrane potential at time t
  • τ_mem (tau_mem) is the membrane time constant — how quickly the potential decays without input
  • I(t) is the input current (the weighted sum of incoming spikes)
When V(t) reaches the threshold θ, the neuron emits a spike and resets.

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Discrete-time form

In simulation and on digital hardware, time is discretized into steps of size Δt (dt). The continuous equation becomes a recurrence: V[t+1]=α⋅V[t]+(1−α)⋅I[t]V[t+1] = \alpha \cdot V[t] + (1 - \alpha) \cdot I[t] α=e−Δt/τmem\alpha = e^{-\Delta t / \tau_{\text{mem}}} The factor α (alpha) is the leak factor: how much of the current potential survives to the next timestep. For τ_mem = 20ms and dt = 1ms, α ≈ 0.951. For smaller τ_mem, the neuron forgets faster.
THRINDEX computes α exactly as exp(-dt / tau_mem) at compile time and stores the resolved constant in the .thx artifact. The Python model stores tau_mem and dt; the artifact stores alpha. You never set alpha directly.

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Parameters

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Reset modes

After a spike, the membrane potential is reset. THRINDEX supports two modes: Subtract (reset="subtract", default): the threshold is subtracted from the current potential. V←V−θV \leftarrow V - \theta If the potential exceeded the threshold by 0.3, the post-spike potential is 0.3, not zero. This preserves information about the excess charge. Zero (reset="zero"): the potential is set to zero unconditionally after a spike. V←0V \leftarrow 0 Simpler, but discards the overshoot. Use subtract mode unless you have a specific reason to zero the potential.

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Synaptic dynamics

A single snn.LIF layer in THRINDEX couples a dense layer (the synapse) with the LIF integrator. The input current is the output of the preceding linear transformation:
If tau_syn is specified, THRINDEX adds a first-order synaptic filter before the membrane integrator — the input current itself decays with time constant tau_syn: I[t+1]=β⋅I[t]+W⋅s[t]I[t+1] = \beta \cdot I[t] + W \cdot s[t] β=e−Δt/τsyn\beta = e^{-\Delta t / \tau_{\text{syn}}} Most practical models use the default (tau_syn not set), which means input is injected directly into the membrane. Add tau_syn when the application requires temporal smoothing of the input current.

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What a LIF layer computes, step by step

For each timestep t = 0, 1, ..., T-1:
  1. Compute input current: I[t] = W · x[t] + b (the preceding Dense layer)
  2. Update membrane: V[t] = α · V[t-1] + I[t]
  3. Check threshold: spike[t] = 1 if V[t] ≥ threshold else 0
  4. Apply reset: V[t] = V[t] - threshold · spike[t] (subtract mode)
The output is the spike raster spike[0..T-1] — a binary matrix of shape [T, n_neurons].

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In code

The membrane state is maintained across the T dimension automatically. It resets to zero at the start of each new sample in the batch.