Temporal coding is found virtually everywhere in the brain, underlying numerous sensory, cognitive, and motor functions of the nervous system (Paton and Buonomano, 2018). Well-timed neuronal spiking is essential for vision (Rucci et al., 2018), audition (Grothe et al., 2010; Heil and Peterson, 2015; Yin et al., 2019), balance (Cullen, 2012), olfaction (Gire et al., 2013), gustation (Hallock and Di Lorenzo, 2006), touch (Saal et al., 2016), electrosensation (Carr and Friedman, 1999; Baker et al., 2013), as well as spinal pattern generation (Catela et al., 2015), motor control (Lehman and Bartussek, 2017; Sober et al., 2018), cerebellar motor learning (De Zeeuw et al., 2011), and memory formation and consolidation (Dragoi, 2020). Loss of temporal precision in neural activity is related to cognitive and behavioral deficits (Balci et al., 2009). When action potentials of a neuron occur preferentially at a certain phase of a periodic stimulus, such a neuronal spiking pattern is called “phase-locked” (see Carr and Friedman, 1999; Ashida et al., 2010, for typical examples). In the auditory system, phase-locking to tonal stimuli plays a fundamental role, for example, in detecting the location of a sound source (Grothe et al., 2010; Ashida, 2015; Yin et al., 2019).

Phase-locking of a spike train (Figure 1A) can be captured in a phase histogram (Figure 1C) that visualizes the spiking rate at each phase of the periodic reference signal (see Figure 1B for the schematic construction of a phase histogram). In order to quantify the degree of phase-locking, a metric called the “vector strength” (VS: Goldberg and Brown, 1969; Fisher, 1993; van Hemmen, 2013) has been widely used (for examples, see Joris et al., 2004; Ashida et al., 2010; Heil and Peterson, 2015, 2017). Here, each spike is converted into a vector on a unit circle with a corresponding phase; all these phase vectors are summed up to a mean spike vector, whose length is the value of VS. Mathematically, VS (sometimes called “synchronization index” or “synchronization coefficient”) is equal to the absolute value of the Fourier component of the spike train at the reference signal frequency (f) normalized by the total number of spikes (Johnson, 1974; Ashida et al., 2010). By definition (detailed in Materials and Methods), the value of VS is between zero and one: from no to perfect periodicity at f. Certain types of neurons in the cochlear nucleus, for instance, exhibit enhanced phase-locking compared to their peripheral auditory nerve inputs, which is manifested in an increase of VS (Joris et al., 1994; Köppl, 1997; Wei et al., 2017).

In addition to VS, another metric called the “correlation index” (CI: Joris et al., 2006) was more recently introduced to quantify neuronal synchrony. The value of CI is defined as the height of the central peak of a normalized shuffled autocorrelogram (SAC; Figure 1E), whose construction is depicted in Figure 1D (a more detailed description is given in Materials and Methods). If a set of spike trains has no temporally reproducible spiking pattern across trials, its CI value equals one. A large CI value indicates highly reproducible spiking responses over repeated trials (Joris et al., 2006). In contrast to VS, which is narrowband (i.e., only the spectral component at frequency f is considered), CI is generally broadband, as it is a collective measure of multiple frequency components (Parida et al., 2021). SACs and CIs can be applied to both periodic and aperiodic signals and were used for characterizing the temporal coding properties of auditory nerve (AN) fibers (Dreyer and Delgutte, 2006; Heinz and Swaminathan, 2009; Huet et al., 2018; Heeringa et al., 2019), various neurons in the cochlear nuclei recorded in vivo (Gai and Carney, 2008; Steinberg and Peña, 2011; Recio-Spinoso, 2012; Keine et al., 2016) and in vitro (Street and Manis, 2007; Kreeger et al., 2012), and neurons in the auditory midbrain (Shackleton et al., 2009; Zheng and Escabí, 2013).

While both VS and CI represent the degree of synchronized neuronal spiking activity, it is not known how these two measures are related to each other. Previous analyses of physiological data suggested that they should covary according to some monotonic, non-linear relationship (Dreyer and Delgutte, 2006; Joris et al., 2006). Parida and colleagues mathematically related peristimulus time histograms (PSTHs) to VS and CI under very general conditions (Parida et al., 2021). In the present study, we derive a mathematically rigorous relationship between VS and CI under certain assumptions for phase-locking. Our theoretical results will be validated with spike train data that were either recorded from different types of auditory neurons or simulated with available auditory neuron models. These results provide a foundation for comparing past and future experimental data quantified with these measures of phase-locking.

In this section, we summarize our theoretical formulations and describe the methods for validating our mathematical results. A full derivation of the equations is provided in Supplementary Materials.

The main goal of the present study is to establish a clear relationship between VS and CI. After a brief mathematical description of their relationship, we test the validity of our theoretical formulation with published auditory neuron models and physiologically recorded in vivo data. Finally, we investigate potential causes for deviations between theoretical and empirical results.

In the era of large-scale recording and simulation, the ability to compare data has become a fundamental issue in various scientific fields, including biology (Cao et al., 2005; Vogt, 2009; Mallott et al., 2019) and neuroscience (Mulugeta et al., 2018; Bzdok et al., 2019). A number of practical suggestions have been made and implemented toward effective integration of multiple datasets (e.g., Poldrack and Gorgolewski, 2014; Zehl et al., 2016; Tratwal et al., 2020). These ongoing efforts, however, do not necessarily guarantee that neurophysiological data that were published decades ago can be compared with more recent data quantified with different methods. As reviewed in the Introduction, VS has served as a practical measure of phase-locking, since it was introduced more than half a century ago (Goldberg and Brown, 1969). More recently, the use of CI has been increasingly common, since SACs and CIs can be defined for a wider variety of stimuli than VS, including non-periodic acoustic stimulation (Joris et al., 2006). To facilitate comparisons between phase-locked spiking data quantified with VS and those with CI, the present study established a mathematical relationship between these measures (Figure 2F). Our mathematical formulation assumed a von Mises distribution characterized with the concentration parameter κ that systematically affects both VS and CI (Figures 2D,E).

Our theoretical predictions generally agreed with measured (Figures 3C,D) or simulated (Figures 3A,B) spiking data of several different auditory neurons, even though spike timings in real neurons do not necessarily follow the von Mises distribution (Peterson and Heil, 2019). Real neurons and modeled AN fibers, for example, have refractory periods, while von Mises distributed spike trains do not. Recordings of auditory nerve fibers showed that refractoriness can affect VS especially at low frequencies (below 800 Hz; Avissar et al., 2013; Peterson and Heil, 2019). Furthermore, spiking timing of auditory nerve fibers and cochlear nucleus neurons are generally more regular than pure Poissonian spike trains (Rothman et al., 1993; Heil and Peterson, 2017). The AN and GBC models we used (Bruce et al., 2018; Ashida et al., 2019) include refractory periods and show more regular spiking than the Poisson process. Nevertheless, our simulation results with these models showed a good agreement with the curve for the von Mises distribution (Figure 3B), suggesting that the violations of these mathematical assumptions may only have limited effects on the theoretical VS-CI relationship. More systematic analyses on the effects of refractoriness and regularity on SAC would be a topic of future studies.

Deviations of data points from the theoretical curve may originate from a skewed (Figures 4A2,B2) or bimodal (Figures 4C2,D2) phase distribution (Peterson and Heil, 2019). In auditory nerve recordings in vivo, such bimodal distributions were observed, for example, with low-frequency, high-intensity acoustic stimulation (Johnson, 1980; Wei et al., 2017) which induces considerable harmonic distortions in the peripheral response (Peterson and Heil, 2019, also see references therein for more examples). In addition to these relatively apparent cases, the value of CI can be higher than what is expected from VS, even if the phase distribution resembles a von Mises distribution. In our example (Figure 4E2), spikes across trials were more likely to occur in the same or neighboring stimulus cycles than naively expected from the period histogram that only retains the phase information of spikes in each period (Figures 5B,C). These results indicate that the applicability of our theoretical results is limited to unimodal (von-Mises-like) phase distributions without additional trial-to-trial coincidences due to variations of the stimulus parameters. A more appropriate fitting of skewed or bimodal phase distribution would require modifications of the fitting function by, e.g., summing multiple unimodal distributions or introducing additional shape parameters to create asymmetry (Gatto and Jammalamadaka, 2007; Umbach and Jammalamadaka, 2009; Abe and Pewsey, 2011; Kim and SenGupta, 2013). In the cat auditory nerve, for example, additional one or two distortion components would be necessary to account for the observed spiking patterns (Peterson and Heil, 2019).

In addition to the violation of theoretical assumptions, multiple sources of errors can affect the calculation of CI. While the length D of each trial would not affect the value of CI (Figure 6), the size of the SAC time bin ω can constitute a major error source (Figure 7). If ω is too large, the resulting CI value becomes smaller than the theoretical estimation, because the SAC peak is reduced by averaging within the time bin. A similar effect was shown for the calculation of VS (Ashida and Carr, 2010). For small ω, the calculation of CI can be error-prone, especially when the number of data points is not uniform across SAC bins (Figures 7D,F). In other words, the size of SAC bins should be selected in a way that the number of time steps in each bin becomes constant (Figures 7B,C). Considering these factors, we support the selection of ω = 50 μs (or some similar value), which was used by the original study of Joris et al. (2006), provided that the recording time step is sufficiently small (e.g., 10 μs or less). In our in vivo recording data, however, the spike times were usually sampled at a lower rate (48 kHz, corresponding to a 20.8 μs time step) due to technical restrictions. Hence, the discrepancy between the theoretical prediction and empirical data (Figure 3C) could be (at least partly) due to the effect of the SAC bin size. Furthermore, the number of repeated trials needs to be large enough to have sufficiently many spike coincidences in each SAC time bin. Otherwise, the empirical CI value may differ from the theoretical prediction, which is exemplified by larger error bars for small ω (Figure 7B). Moreover, for an estimation of CI from VS, even a small error in VS may result in a large error of CI, especially for VS>0.95. This is because the value of VS is bounded by one, while the value of CI can be infinitely large (Figure 2F).

Comparisons of multiple measures may reveal hidden structures of spike trains. As demonstrated in Figures 5B,C, a comparison of the VS-CI relationship with the theoretical prediction may help us discover additional spiking patterns that may not be found with a period histogram or SAC alone. While CI contains broadband spectral information, VS is a measure at a specific frequency (Parida et al., 2021). Expanding the concept of VS is possible by introducing multiple reference frequencies (van Hemmen and Vollmayr, 2013). For spectrally rich acoustic stimuli, such as natural sounds, locking to more than one frequencies may have to be considered (Joris et al., 2004). Even though the use of natural stimuli is increasing common in neuroscience, response patterns to simple stimuli still provide fundamental information about the functionality of the sensory system (e.g., Rust and Movshon, 2005). It is not always clear, however, how neuronal responses to simple stimuli can be related with those to complex stimuli. Future investigation on the relationship between CI and frequency-dependent VS under physiologically reasonable mathematical conditions might be useful to bridge between fundamental spiking patterns of auditory neurons and responses to broadband acoustic stimulation.

A number of measures have been developed to quantify the degree of correlation between spike trains, including SACs and its variations (reviewed in Cutts and Eglen, 2014). Nevertheless, quantitative relationships between these measures are not always clear. Gai and Carney (2008) compared CI, the Victor-Purpura spike distance metric, and mutual information to quantify time-locked spiking patterns recorded in the cochlear nucleus. Our theoretical method used in this study may also be applied to temporal measures other than VS and CI. Such an analysis would not only reveal the mathematical relations behind these measures but also provide practical information on which measure to use for quantifying temporal information processing in the nervous system.

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author. Matlab program code for some of the simulations and analyses in this study is available online at https://github.com/pinkbox-models/CI-VS-2021.

Ethical review and approval was not required for the animal study because only existing data collected in previously published studies were re-analyzed.

DK and GA: conceptualization, formal analysis, numerical simulation, data analysis, interpretation, visualization, and drafting manuscript. CC and GA: physiological data collection. CC and JK: resources. CC, JK, and GA: funding acquisition. JK and GA: project administration and supervision. DK, CC, JK, and GA: reviewing and editing manuscript and approving submission. All authors contributed to the article and approved the submitted version.

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC2177/1 (Project ID 390895286), Associate Junior Fellowship from the Hanse-Wissenschaftskolleg (GA), and NIH/NIDCD Grant R01DC019341 (CC).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.