5.3 Phase locking

For sinusoidal stimuli below about 5 kHz, an auditory-nerve fiber does not spike at random within each cycle of the stimulus. It tends to spike near the same phase of the cycle, every cycle, for as long as the cycle lasts. This is phase locking. It is, perhaps, the most remarkable single fact about auditory-nerve coding.

A fiber phase-locked to a 500 Hz tone will not necessarily spike on every cycle of the tone (its overall rate is limited; it cannot spike 500 times per second sustained), but the spikes it does emit will cluster at the same phase. The histogram of spike phases, computed over many cycles, looks like a narrow distribution at one preferred phase, much narrower than would be expected by chance.

The standard measure of phase locking is vector strength (Goldberg & Brown, 1969):

\text{VS} = \left| \frac{1}{N} \sum_{i=1}^{N} e^{i \phi_i} \right|.

▶ Derivation: vector strength as the magnitude of the mean unit vector

Each spike fires at some phase $\phi_i$ of the stimulus cycle (a number in $[0, 2\pi)$ ). Represent each spike as a unit vector pointing in the direction $\phi_i$ in the complex plane: $e^{i\phi_i}$ .

If all spikes are at the same phase $\phi^*$ , all unit vectors point in the same direction. Their sum is $N\, e^{i\phi^*}$ , and the magnitude of the average is $|N e^{i\phi^*}/N| = 1$ .

If spikes are uniformly distributed over phases, the unit vectors point in all directions and cancel. As $N$ grows, the magnitude of their average approaches zero (by the law of large numbers applied to circular variables).

So define vector strength as the magnitude of the mean unit vector:

\text{VS} = \left| \frac{1}{N} \sum_i e^{i\phi_i} \right| \in [0, 1].

VS = 1 means perfect phase locking; VS = 0 means none. Real auditory-nerve fibers at low frequencies achieve VS ≈ 0.9. ∎

If all spikes occur at the same phase, $\text{VS} = 1$ . If they are uniformly distributed across phases, $\text{VS} = 0$ . Real auditory-nerve fibers can achieve $\text{VS}$ values of 0.9 at low frequencies — extraordinarily sharp phase locking.

Above about 1.5 kHz, $\text{VS}$ begins to fall. By 5 kHz, $\text{VS} \approx 0.05$ — barely phase-locked. By 6 kHz, almost no phase locking is detectable. The cellular reason is that the IHC’s membrane capacitance starts to low-pass-filter the receptor potential, and the synapse cannot resolve cycle-to-cycle phase past a certain rate.

The interactive below shows three auditory-nerve fibers — one each at CF = 500 Hz, 1.5 kHz, and 4.5 kHz — responding to a single tone.

stimulus frequency

894 Hz

stimulus level

50 dB SPL

fiber type: