Spike train

A spike train records when a neuron fires. It keeps the timing of events while setting aside the detailed voltage waveform of each action potential.

Intuition: timing and counting answer different questions

In a raster, each short vertical mark is one event. The illustrative figure contains four spikes in a 100 ms interval. A count divided by interval duration gives 40 spikes per second. That rate would be identical if all four events were evenly spaced, yet the temporal pattern would differ.

Event notation

A list of event times $\{t_1,t_2,\ldots\}$ can be written as

$$s(t)=\sum_k\delta(t-t_k).$$

The Dirac delta is a distribution with unit integral, not a finite-height voltage pulse. Integrating $s$ over a time interval counts the events in it. A windowed count rate is

$$\hat r=\frac{N_{[a,b)}}{b-a}.$$

The half-open interval convention counts an event at $a$ and excludes one at $b$, avoiding double-counting across adjacent bins. If time is measured in seconds, the resulting rate is in spikes/s.

Information lost by averaging

For our illustrative times 10, 20, 65 and 90 ms, the successive interspike intervals are 10, 45 and 25 ms. Their average is not the same calculation as the count rate over the full observation window: the observation also includes time before the first and after the last event.

A shorter counting window gives finer temporal resolution but more variable counts. Smoothing with a kernel has a related tradeoff. Trial-averaged rate, time-averaged rate and the rate of a population are distinct summaries; say which you mean.

Limits of the abstraction

Recording a neuron's events requires a detection or sorting procedure in real data. A simulated threshold event already has an exact model time. Do not silently equate a model raster with experimentally isolated single-neuron activity.

Where you will see this

The lower panel of the LIF explorable shows events generated by a threshold model. The upper voltage curve and the raster describe related but different quantities.

References

• Neuronal Dynamics: spike-train variability.
• Neuronal Dynamics: mean firing rate.