From neurons to brain waves

One population, equal contributions

Every neuron contributes equally to the population signal. When their rhythms align, their contributions reinforce and produce a larger wave. When phases are scattered, positive and negative deviations partly cancel, producing a smaller wave. The current of each neuron oscillates about its mean; every source is added with the same positive weight.

Each dot is a simplified phase oscillator representing a neuron’s rhythmic current source. The hand and phase wheel show its position within a cycle. A raster mark records an upward crossing of a full cycle. The raster is an illustrative timing display; the model does not compute an action-potential waveform.

• Independent neurons: start with scattered phases and zero coupling. Increase coupling while keeping the natural rhythm fixed, and compare the phase wheel, raster, signal amplitude, and spectrum.
• Lock together: coupling attracts phases toward a common phase. Use Scatter phases and watch the population align again. Lower coupling or increase frequency spread and noise to weaken coordination.
• Common input: with independent neurons, deliver a brief idealized phase reset. Watch the initially aligned rhythms drift apart as their different natural frequencies accumulate different phases.

Adding neuronal currents into an EEG-like signal

Each source contributes the smooth waveform pᵢ = cos θᵢ. The population signal is y = (1/N) Σ pᵢ: add every current in the same way, then divide by the number of neurons. This normalization keeps the comparison focused on timing as the neuron count changes. The dark line is the population average; six faint blue traces show representative individual currents at their full per-neuron amplitude.

Real scalp EEG reflects neuronal transmembrane currents, their spatial arrangement, and electrical conduction through the head. Here, source geometry is held fixed and every neuron has the same positive measurement weight. The app illustrates how synchronized current fluctuations generate a population rhythm; its arbitrary units are a teaching approximation to EEG, rather than calibrated scalp voltages. Frequency is adjustable in hertz independently of phase alignment.

Phase dynamics and measurements

The phase equation is dθᵢ = [2πfᵢ + K R sin(ψ − θᵢ)]dt + √(2D)dWᵢ, where R exp(iψ) = (1/N) Σ exp(iθⱼ). Natural frequencies span the selected percentage above and below the mean. Phase-noise increments are independent across neurons. Phase alignment is 100R percent; read it together with the wheel. The model uses ordinary phase-attracting coupling.

Common input is an idealized instantaneous phase reset. Scatter phases randomizes timing while preserving the natural frequencies. Both actions retain time and prior observations. Frequency, coupling, spread, and noise affect subsequent dynamics; a displayed time window or spectrum can include observations from before and after an adjustment. Presets, neuron-count changes, and Restart begin again with six seconds of history.

Integration uses a fixed 0.002 s step, independent of playback speed, with signal samples every 0.004 s. Up to 12 s of history are retained; time plots display the last 2 or 6 s. The Hann-windowed, mean-subtracted spectrum uses up to 1024 samples (4.096 s; 0.244 Hz bin spacing) and a fixed power scale. It shows one-sided power spectral density in a.u.²/Hz. Field RMS and mean phase-crossing rate use the last 2 s. The rate can vary during transients even when the natural-frequency mean is unchanged.

Background: Hansel, Mato & Meunier (1993), Clustering and slow switching in globally coupled phase oscillators; Hagen et al. (2018), Multimodal modeling of neural network activity with LFPy 2.0. Compare individual-neuron dynamics in the FitzHugh–Nagumo network simulator.