LIF Network Playground · Wits Computational Neuroscience

From ion channels to networks. Hodgkin–Huxley built a spike out of four coupled nonlinear ODEs for V, m, h, and n. The leaky integrate-and-fire (LIF) neuron below throws away the ion-channel machinery and keeps only what a downstream neuron actually "sees": a leaky membrane that integrates current until it crosses a threshold, then resets. That single simplification is what makes networks of hundreds of neurons simulable in real time.

$$\tau_m \frac{dV}{dt} = -(V - E_L) + R\,I(t)$$

…while $V < V_{th}$. The instant $V \ge V_{th}$: emit a spike, set $V \leftarrow V_{reset}$, and hold $V$ fixed for a refractory period $t_{ref}$.

Single-neuron drive

Input current, $I$ nA

Membrane time constant, $\tau_m$ ms

Threshold, $V_{th}$ mV

Refractory period, $t_{ref}$ ms

Noise intensity, $\sigma$ mV

Recurrent network

Population size, $N$

Connection probability, $p$

Synaptic weight, $w$ mV

80% of neurons are excitatory ($+w$), 20% inhibitory ($-4w$) — the classic 4:1 ratio that balances mean E/I drive (Brunel, 2000). Set $p = 0$ to isolate single neurons; changing $N$ or $p$ regenerates the network.

Run

The scope scrolls at a fixed pace (not wall-clock real time) so spikes stay legible on a projector.

Advanced parameters ($E_L$, $V_{reset}$, $R$, $\Delta t$) — all adjustable here too

Resting potential, $E_L$ mV

Reset potential, $V_{reset}$ mV

Membrane resistance, $R$ MΩ

$\Delta t = 0.1$ ms, fixed (forward Euler / Euler–Maruyama integration). $R$ is an arbitrary MΩ-like unit: $R\,I$ with $I$ in nA gives mV.

A · Single-neuron membrane trace

V(t) V_th & spikes (ticks to +40 mV) V_reset / E_L

A reference neuron driven by $I$ alone — no synaptic input, so it stays regular even while the network (panel C) synchronises.

B · F–I curve

$$ r_\infty = \dfrac{1000}{\,t_{ref} + \tau_m \ln\!\left(\dfrac{E_L + R\,I - V_{reset}}{E_L + R\,I - V_{th}}\right)}\ \text{Hz}, \qquad E_L + R\,I > V_{th} $$

analytic simulated (batch re-run on parameter change) — dots should sit right on the line when σ = 0, confirming the formula; they scatter around it as noise increases.

C · Population raster & rate (recurrent network)

Synapses are delta-current with no transmission delay — a spike hits its targets in the same step — so recurrent excitation can synchronise the population more sharply than a biological network, where conduction and synaptic delays smear the timing.

D · Challenges

Tune the sim to tick these off automatically.

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Pause & Think

What does the LIF discard compared to Hodgkin–Huxley? When would that loss matter?