## Abstract

The property of a neuron to phase-lock to an oscillatory stimulus before adapting its spike rate to the stimulus frequency plays an important role for the auditory system. We investigate under which conditions neurons exhibit this phase locking below rate threshold. To this end, we simulate neurons employing the widely used leaky integrate-and-fire (LIF) model. Tuning parameters, we can arrange either an irregular spontaneous or a tonic spiking mode. When the neuron is stimulated in both modes, a significant rise of vector strength prior to a noticeable change of the spike rate can be observed. Combining analytic reasoning with numerical simulations, we trace this observation back to a modulation of interspike intervals, which itself requires spikes to be only loosely coupled. We test the limits of this conception by simulating an LIF model with threshold fatigue, which generates pronounced anticorrelations between subsequent interspike intervals. In addition we evaluate the LIF response for harmonic stimuli of various frequencies and discuss the extension to more complex stimuli. It seems that phase locking below rate threshold occurs generically for all zero mean stimuli. Finally, we discuss our findings in the context of stimulus detection.

## 1. Introduction

Phase locking—the firing of spikes at a preferred phase of an oscillatory stimulus—is a universal mechanism in sensory neurons and can be observed for nearly all sensory modalities. Early reports exist for cat retinal ganglion cells (Ogawa, Bishop, & Levick, 1966), single auditory nerve fibers of the squirrel monkey (Rose, Brugge, Anderson, & Hind, 1967), and mechanoreceptive afferents of the monkey hand (Talbot, Darian-Smith, Kornhuber, & Mountcastle, 1968). In the olfactory system, a phase locking of mitral cells to intrinsic subthreshold oscillations is discussed as an essential mechanism for robust odor representation (Desmaisons, Vincent, & Lledo, 1999; Wilson & Mainen, 2006). Finally, different types of noisy phase-locked regimes were observed in the electroreceptor of the paddlefish stimulated by oscillating electric dipole fields (Neiman et al., 1999).

Phase locking and synchronization are fundamental concepts playing an important role
in neuronal information processing systems. Although both effects are related (phase
locking is sometimes termed *synchronization in the statistical
sense*), it is important to clearly distinguish between them: whereas
phase locking means that in a drive response relation, the response occurs at a
preferential phase of the driving signal, synchronization (Pikovsky, Rosenblum,
& Kurths, 2001) is characterized by the
forced matching of spike rate and stimulus frequency or the boundedness of a
suitably defined phase difference. Eliciting a spike at each peak of a sinusoidal
stimulus describes a situation where phase locking and synchronization co-occur.
Spikes riding on the signal peaks but skipping cycles preserve phase locking but
introduce phase slips, thus corrupting synchronization. On the other hand,
randomizing the position of each spike in its respective periodic interval would
erase phase locking but preserve synchronization.

In case of a stimulus with vanishing strength, neither phase locking nor synchronization will be observed. Gradually increasing the stimulus strength often leads to the observation that significant phase locking sets in before a noticeable change of the spontaneous spike rate, that is, before reaching rate threshold. This phase locking below rate threshold is particularly prominent in the auditory nerve (Johnson, 1980; Köppl, 1997). It was also found in a conceptual model aiming to explain the exquisite sensitivity of the auditory system to acoustic stimuli that carry less energy than the background noise (Camalet, Duke, Jülicher, & Prost, 2000, Fig. 6). This raises the question of what properties of a neuron are necessary or sufficient to effect phase locking below rate threshold. Surprisingly, we did not find an answer to this very basic question in the existing literature. Here we try to close the gap through a modeling approach and dwell on the mechanism underlying this effect.

## 2. Methods

We combine two methods to elucidate the conditions under which phase locking below rate threshold occurs: (1) we derive a mathematical relation between the spike rate and the vector strength, the latter being a common measure for phase locking; and (2), we perform diverse numerical simulations of the driven leaky-integrate-and-fire neuron model.

### 2.1. Relation Between Spike Rate and Vector Strength.

We consider a harmonic stimulus *s*(*t*) = *a* cos(ϕ_{0} + Ω*t*)
with a linearly increasing phase ϕ(*t*) =
ϕ_{0} + Ω*t*, frequency Ω and
strength *a*. The restriction to zero mean harmonic stimuli is
motivated by the filter bank property of the cochlea (Glasberg & Moore, 1990; Hohmann, 2002) and the fact that phase locking below rate
threshold is most prominent in the auditory system. The case of more complex
stimuli is addressed in section 3.2.

*t*. Differences between these spike times constitute the sequence of interspike intervals (ISIs) τ

_{i}_{i}=

*t*−

_{i}*t*

_{i−1}. Moreover, we collect the wrapped stimulus phases at the spike times ϕ

_{i}= ϕ(

*t*)(mod 2π). Binning these spike phases, we construct the so-called period histogram and, after normalization and division by the bin widths, estimate a related probability density

_{i}*p*(ϕ). This phase distribution is a central quantity for a detection of phase locking seen in a concentration at a certain value. The degree of phase locking can be quantified by the so-called vector strength (or synchronization index) (Goldberg & Brown, 1969): This quantity varies between zero for the equidistribution (if

*p*(ϕ) = (2π)

^{−1}), and one for a delta distribution (if

*p*(ϕ) = δ(ϕ − ϕ

_{0})). Using a Fourier decomposition of the 2π-periodic function

*p*(ϕ), with Fourier coefficients, we see immediately that the vector strength is nothing but the modulus of the first Fourier coefficient:

*v*= |

*c*

_{1}| = |

*c*

_{−1}|. Notice that

*c*

_{0}= 1 due to normalization of

*p*(ϕ) and

*c*

_{−k}=

*c**

_{k}since

*p*(ϕ) is real. Moreover, we note that via the phase distribution

*p*(ϕ) =

*p*(ϕ;

*a*, Ω), all Fourier coefficients are functions of the stimulus strength and frequency (i.e.,

*c*=

_{k}*c*(

_{k}*a*, Ω)) and, in particular,

*v*=

*v*(

*a*, Ω).

*r*of the driven neuron. Here, the term

*spike rate*relates to spike counts, not to an ensemble-based instantaneous firing rate (or

*peristimulus time histogram*PSTH).

^{1}Its operative definition requires counting the number

*N*of spikes a neuron fires within the time window

*T*and then computing the ratio Taking the inverse of this relation and noticing that the time window

*T*can be chosen to encompass (

*N*− 1) consecutive ISIs τ

_{1}, τ

_{2}, …, τ

_{N−1}, we find the relation Here we have assumed that downstream neurons compute the inverse spike rate by measuring the time passing between an initial and an

*N*th spike. In case the downstream neuron estimates by counting the number of spikes observed in a preselected fixed time window

*T*, equation 2.5, would require a correction factor where 〈τ〉 denotes the average ISI.

*N*→ ∞ the expression in equation 2.5 can be interpreted as a mean ISI 〈τ〉 where the long-time statistics is described by a probability density

*p*(τ;

*a*, Ω) that covaries with the stimulus parameters

*a*and Ω. Hence, we find In principle we should also account for the initial phase ϕ

_{0}of the stimulus and perform an additional average. However, in the neurons we have in mind, the memory for this initial phase decays and is lost beyond a transient regime (ergodicity assumption).

*a*and Ω), we can rewrite equation 2.13 as For vanishing stimulus strength (

*a*= 0), the phase distribution

*p*(ϕ) is flat, corresponding to

*c*

_{k⩾1}= 0, and ϑ

_{0}= 〈τ〉. Increasing the stimulus strength causes a structuring of the phase distribution that is reflected in the Fourier coefficients

*c*

_{k>0}gaining increasing weight. The spectrum of Fourier coefficients exhibits a decaying profile: the larger

*k*, the smaller the modulus of

*c*. The background of this observation is the reciprocal relation between the effective widths in original ϕ-space and its related Fourier space, well known from quantum mechanical wave packets and the uncertainty relation. Notice that the singular zero-width delta distribution

_{k}*p*(ϕ) = δ(ϕ − ϕ

_{0}) corresponds to the infinite-width spectrum

*c*= 1 for all

_{k}*k*. In line with this observation, a weak stimulus (

*ws*) approximation is expressed as (

*c*

_{k⩾2}≈ 0) and means that the initial deviation of

*p*(ϕ) from the flat distribution is nothing but a harmonic modulation with the period 2π, that is, and with the peak located at the phase γ

_{1}. Because of the nonnegativity of the phase distribution, formula 2.17 must necessarily be restricted to values

*v*⩽ 0.5. Moreover, for weak stimuli the relation between spike rate

*r*and vector strength

*v*, equation 2.16, can be approximated as Now we can see how an increase in vector strength, say up to a value of 0.5, can occur without being accompanied by a change in spike rate. This phase locking below rate threshold stems from the fact that ϑ

_{0}remains effectively unchanged and |ϑ

_{1}| ≪ ϑ

_{0}. Equation 2.16 shows that extending the second property to |ϑ

_{k}| ≪ ϑ

_{0}for all

*k*⩾ 1 even yields a sufficient criterion for phase locking below rate threshold without restriction to weak stimuli.

### 2.2. Noisy Leaky-Integrate-and-Fire Neuron.

One of the simplest and most widely used models for neuronal spike generation is the leaky-integrate-and-fire (LIF) model (Stein, 1965). While the integrate-and-fire model is usually ascribed to Lapicque and his often cited 1907 paper (Lapicque, 1907; Brunel & van Rossum, 2007b), it has recently been pointed out (Brunel & van Rossum, 2007a) that it was in fact introduced by theorists much later.

*I*into a resistive leak current and a capacitive current , where

_{s}*u*(

*t*),

*R*, and

*C*are membrane voltage, resistor, and capacitance, respectively. Rearranging the equation

*I*=

_{s}*I*+

_{R}*I*, we can write with the membrane time constant τ

_{C}_{m}=

*RC*. One component of the synaptic current results from a bombardment of the LIF neuron with spikes arriving from a multitude of presynaptic neurons. Under certain assumptions (Lansky & Sato, 1999) its effect can be modeled by a mean current

*I*

_{0}plus gaussian white noise ζ(

*t*). Differing from this frequently considered case, gaussian red (or correlated) noise accounts for synchronous presynaptic firing; its effects on the ISI statistics and serial correlations of a LIF model were recently investigated by Schwalger and Schimansky-Geier (2008). As a second component of the synaptic current, we add a harmonic stimulus with strength , frequency Ω, and initial phase ϕ

_{0}= π (cf. the ergodicity assumption above). Insertion in equation 2.19yields Next we add a spike triggering voltage threshold

*u*

_{θ}and a reset voltage

*u*. We complete our LIF model neuron by endowing it with an absolute refractory time τ

_{r}_{a}= 0.5 during which integration ceases. Choosing

*u*as the voltage reference and going to a rescaled membrane voltage

_{r}*x*(

*t*) = [

*u*(

*t*) −

*u*]/

_{r}*u*

_{θ}, we finally arrive at Here the first term describes the deterministic relaxation: in the absence of second (σ = 0) and third (

*a*= 0) terms, the voltage variable

*x*(

*t*) relaxes to the value

*x*

_{∞}within a time of the order of a few τ

_{m}. The second term in equation 2.21 reflects rescaled voltage fluctuations in the form of standard zero mean additive gaussian white noise, of intensity σ

^{2}. The first two terms in equation 2.21 make up what is called an Ornstein-Uhlenbeck process (Gardiner, 2004). The third term in equation 2.21 expresses the rescaled harmonic stimulus.

By choice of *x*_{∞}, we can arrange two different operational modes of our model
neuron. Placing the resting voltage *x*_{∞} slightly below the firing threshold *x*_{θ} (case A) leaves a gap *x*_{θ} − *x*_{∞} that has to be traversed with the aid of random
fluctuations. This situation corresponds to noise-mediated spontaneous spiking
activity. On the other hand, fixing *x*_{∞} above *x*_{θ} (case B) leads to tonic spiking, with the width of the ISI
distribution being controlled by the noise intensity σ.

*t*), which is relaxing with time constant τ

_{θ}toward

*x*

_{θ}(case C): As before, the dynamics of

*x*(

*t*) is governed by equation 2.21. The very moment that

*x*(

*t*) hits θ(

*t*) from below,

*x*(

*t*) is reset to

*x*whereas θ(

_{r}*t*) is set to θ

_{0}+

*W*[θ(

*t*), α]. While in Chacron et al. (2003),

*W*was required to be a monotonically increasing function of both arguments obeying

*W*[θ(

*t*), 0] = 0, here we will consider only the special case

*W*[θ(

*t*), α = 1] = θ(

*t*) and θ

_{0}= 1. This means that each time a spike is fired, the threshold is incremented by unity. We note that the standard LIF with constant threshold

*x*

_{θ}= 1 corresponds to θ

_{o}=

*x*

_{θ}and α = 0.

## 3. Results

### 3.1. Numerical Simulation Results of the Driven LIF Model Neuron.

In the following we present simulation results of the driven noisy LIF for the
stimulus frequency Ω = 2π and amplitudes ranging from *a* = 0.001, …, 10 on a decadal logarithmic
scale. The other parameters for the cases *A*[*B*]{*C*} are as given in
the caption of Figure 1. The mean spike
rate *r*_{0} in the absence of the stimulus is roughly 0.12 for all cases;
hence, the mean ISI is roughly 8.3 times larger than the period of the stimulus *T _{s}* = 2π/Ω = 1. This means that the transition
from the undriven neuronal dynamics via phase locking to the (1:1)
synchronization (one spike per stimulus period) regime is accompanied by an
uprise of the spike rate—to synchronize, the neuron has to speed up. The
stimulus-induced increase of vector strength

*v*and spike rate

*r*, computed for all three cases from numerical simulations of 20,000 spikes per amplitude, are displayed in the three panels of Figure 2, respectively.

*r*

_{0}≈ 0.12). Quantifying the rate threshold as and the degree of phase locking at rate threshold by

*v*(

*a*), we find for case A,

_{t}*a*≈ 0.1,

_{t}*v*(

*a*) ≈ 0.4; for case B,

_{t}*a*≈ 0.25,

_{t}*v*(

*a*) ≈ 0.85; and for case C:

_{t}*a*≈ 2,

_{t}*v*(

*a*) ≈ 0.81. In the viewgraphs of Figure 2, these items are indicated by the black dashed line.

_{t}To gain further insight into the phenomenon and the structure of the curves, we also show plots of the ISI and phase histograms in Figure 3. Notice that the gray lines trace the mean ISI (or inverse spike rate) and the vector strength, respectively.

### 3.2. More Complex Stimuli.

Studying phase locking to more complex stimuli first raises the question of how
to generalize the phase concept for nonharmonic or even nonperiodic stimuli. As
discussed in the context of phase synchronization (Freund, Schimansky-Geier,
& Hänggi, 2003), several phase
definitions (natural, linear interpolating, Hilbert phase) may qualify. After an
appropriate phase definition ϕ(*t*) has been chosen, a
general stimulus may be written as *s*(*t*)
= 〈*s*〉 + *a*(*t*)cos ϕ(*t*). The
harmonic case considered above corresponds to 〈*s*〉
= 0, *a*(*t*) = *a*,
ϕ(*t*) = Ω*t*.

The case demonstrates two things: first, that the appropriate phase,
here ϕ(*t*) = 2Ω*t*, can
deviate from the naive expression Ω*t* suggested by the
definition and, second, that nonlinearities may generate a nonvanishing
mean, here 〈*s*〉 = *a*/2,
which covaries with the amplitude of the harmonic oscillatory component. A
nonvanishing mean breaks the balance between depolarizing and hyperpolarizing
stimulus segments; if
∂|〈*s*〉|/∂*a*>0,
the stronger the stimulus. This balance was underlying the pure shifting of
spikes observed for the unbiased harmonic stimuli considered above. From this
reasoning, it is straightforward to anticipate that the harmonic but biased
stimulus changes the spike rate before noticeable phase locking sets
in. Through numerical simulations shown in the top panel of Figure 4 we found that for this stimulus, phase
locking below rate threshold cannot be observed.

*a*and ±

*a*. The strong anharmonicity of in the time domain corresponds to a significant weight placed on the higher harmonics in the spectral domain. Defining the instantaneous phase by linear interpolation between the regular maxima amounts to ϕ(

*t*) = 2π

*t*. Simulation results with and , shown in the middle and bottom panels of Figure 4, respectively reveal that in spite of their pronounced asymmetry, these zero mean stimuli nonetheless induce phase locking below rate threshold.

### 3.3. Evaluation of the Relation Between Spike Rate and Vector Strength.

*c*of the phase distribution 2.3 and of ϑ

_{k}_{k}, the Fourier coefficients of the conditional mean ISI, which occur in the expansion, and are defined in equation 2.14. While the ϑ

_{k}are related to

*p*(τ ∣ ϕ;

*a*, Ω) via equation 2.15, the Fourier coefficients

*c*can be connected to the same conditional ISI probability density via Fourier coefficients of higher conditional moments ϑ

_{k}^{n}

_{k}. This connection is not explicated here for the reason that a closed analytic expression for the basic ingredient

*p*(τ ∣ ϕ;

*a*, Ω) is, to the best of our knowledge, not available for the driven noisy LIF. Results obtained by Burkitt and Clark (2001) and Kempter, Gerstner, van Hemmen, and Wagner (1998) in closely related investigations approach desired expressions, but finally these also rely on numerical evaluation. A detailed elaboration may be feasible for the perfect integrator with random threshold (Gabbiani & Koch, 1996); this will be left to future research.

*p*(τ ∣ ϕ;

*a*, Ω) through numerical simulations, the results of which are displayed in Figure 5. A pronounced structure in the (τ, ϕ)-plane (with period

*T*along the τ-axis) emerges for intermediate stimulus strength, that is, at rate threshold (middle panel). This structure is reflected in the Fourier coefficients

_{s}*c*and, in particular, in a significant vector strength

_{k}*v*= |

*c*

_{1}|. To elaborate the connection, one has to express the stationary phase distribution

*p*(ϕ;

*a*, Ω) as the eigenfunction to the eigenvalue 1 of the transition kernel (Plesser & Geisel, 1999):

The coefficients ϑ_{k} are also illustrated in Figure 5.
The black and gray lines show projections of the conditional mean ISI
〈τ ∣ ϕ; *a*, Ω〉 for the
driven and undriven LIF model, respectively. For *a* = 0
(bottom panel), both coincide, and, as expected, no dependence on ϕ is
visible. The position of the gray line is thus given by
ϑ^{−1}_{0}(*a* = 0, Ω). As *a* approaches the rate threshold, the black line deviates from the gray line within
the bound formulated by equation 3.1. From the middle panel of Figure 5, we see that the black line still exhibits no dependence on
ϕ. This is the graphical equivalent of the fact that |ϑ_{k⩾1}| ≪ ϑ_{0} and that
the black line is solely determined by ϑ^{−1}_{0}(*a*, Ω). This separation of scales remains
valid for intermediate values of stimulus strength where ϑ_{0} starts to decline; even beyond rate threshold, we find that up to a good
approximation, *r*(*a*, Ω) ≈
ϑ^{−1}_{0}(*a*, Ω). Only for very strong stimuli (top
panel of Figure 5) do we see a pronounced
modulation of the black line corresponding to a nonnegligible Fourier
coefficient ϑ_{1}(*a*, Ω). However, this is
far beyond rate threshold in the regime of frequency and phase synchronization
and not plain phase locking.

## 4. Discussion

### 4.1. Phase Locking and ISI Correlations.

The essence of phase locking below rate threshold can be traced back to the fact
that while the probability density *p*(τ ∣
ϕ; *a*Ω) is a highly structured function of
ϕ the average 〈τ ∣ ϕ; *a*,
Ω〉 (see equation 2.15) is nearly independent from ϕ. This lack of sensitivity
also explains why a substitution of the structured *p*(ϕ; *a*, Ω) in equation 2.9 by the equidistribution
[2π]^{−1} does not make a big difference, thus
justifying the approximation *r*(*a*, Ω)
≈ ϑ^{−1}_{0}(*a*, Ω) valid for weak to intermediate
stimuli (to see this connection, specify equation 2.14 for *k* = 0).

The structure in *p*(τ ∣ ϕ; *a*Ω) and the independence of 〈τ
∣ ϕ; *a*, Ω〉 from ϕ are caused
by a precise balance of spikes advanced or delayed by the stimulus to match a
favorable stimulus phase. Phase locking below rate threshold thus means
eliciting the same average number of spikes that would occur in the absence of
the stimulus and establishing this balance by shifting (i.e., advancing or
delaying) these spikes. In synchronization research, this mechanism is also
known as *modulation* and contrasted with *synchronization* (Pikovsky et al., 2001). In particular, this means that subsequent
spikes can be shifted independently, which raises the question of what kind of
correlations prevent phase locking below rate threshold.

To see how interspike correlations can affect the phase-locking properties of a neuron, we imagine all spikes to be stiffly coupled, like the teeth of a comb, with a strong tendency to maintain fixed interspike intervals. Shifting the first spike to match a preferred phase generically would lead to a mismatch of some of the following spikes. Phase locking—the excess match of spikes with a preferred stimulus phase—could then be arranged only by excess insertion (or excess deletion) of spikes, that is, a change of the spike rate. Notice that pure spike shifting to the preferred phase (i.e., phase locking below rate threshold) can always be viewed as a balanced deletion (at the mismatch phase) and insertion (at the preferred phase).

Correlations between subsequent spikes are usually discussed in the context of
so-called serial ISI correlations that have been observed in spike trains of
sensory neurons (e.g., Bahar et al., 2001). A prominent anticorrelation between subsequent ISIs—a
short ISI is followed by a long one and vice versa—was reported for the *P*-type electroreceptor of the weakly electric fish (Ratnam
& Nelson, 2000; Chacron, Longtin,
St.-Hilaire, & Maler, 2000). Suitable
models reproducing these effects are the LIF with threshold fatigue (Chacron et
al., 2003) or a perfect integrator with a
noisy threshold and a reset by a constant decrement (Chacron, Lindner, &
Longtin, 2004; Lindner, Chacron, &
Longtin, 2005). Simulation results of a
LIF with threshold fatigue are depicted in the bottom panel of Figure 2. The serial correlation coefficient
ρ_{1} ≈ −0.5 in the inset clearly expresses a
pronounced anticorrelation. Surprisingly, this strong coupling between
subsequent ISIs does not prevent phase locking below rate threshold but, on the
contrary, impedes an increase of the spike rate.

### 4.2. Dependence on Stimulus Frequency.

Phase locking and frequency synchronization in our periodically driven model
neuron are based on an interplay of diverse timescales: the intrinsic absolute
refractory time τ_{a}; the intrinsic relaxation time τ_{m}, which together and in conjunction with the noise intensity
σ^{2} and the gap *x*_{θ} − *x*_{∞} control the average ISI; in the LIF with threshold fatigue
the intrinsic relaxation time τ_{s} of the threshold; and, finally, the external stimulus period *T*_{Ω} = 2π/Ω. Keeping all intrinsic
timescales fixed, we varied the stimulus frequency Ω and computed the
rate threshold *a _{t}* (see equation 3.1) and
the vector strength at rate threshold

*v*(

*a*). Simulation results for both cases A and B are depicted in Figure 6. To interpret the spectral profiles, we note the following facts: there is a general trend of a rising rate threshold

_{t}*a*with increasing stimulus frequency Ω, meaning that fast-oscillating stimuli do not change spike counts so easily. While the tonic spiking mode (case B) displays resonant structures—elevated rate thresholds at ratios Ω:2π

_{t}*r*

_{0}= (1:1), (1:2), (2:1)—the noise-activated spiking mode (case A) exhibits barely any structure (besides the aforementioned mentioned trend). The vector strength at rate threshold is, on the one hand, correlated with the rate threshold curve: the higher

*a*, the higher

_{t}*v*(

*a*). On the other hand, for strong stimuli, there is a tendency of declining vector strength with increasing stimulus frequency. The reason for this decline is found in the noise-induced temporal jitter Δ

_{t}*t*of spikes, which translates into a dispersion of related phases via Δϕ = ΩΔ

*t*. Following Hill, Stange, and Mo (1989), one can derive a parabola

*v*(Ω) = 1 − (ΩΔ

*t*)

^{2}/2 that approximately describes this high-frequency decline of the vector strength (up to ). For the low-frequency branch, there is a reduction of the vector strength due to peak splitting of the phase histogram (Köppl, 1997) caused by more than just one spike being fired at the slowly changing preferred stimulus phase.

### 4.3. Phase Locking as a Detection Mechanism.

*a*and the related vector strength

_{t}*v*(

*a*) are clearly defined quantities, the question of what to call phase locked and what not is not so easy to answer. A value of, say, 0.4 is not specific for a narrow single-peaked phase distribution and ambiguously may result from a variety of differently shaped distributions. Nevertheless, this value still might be sufficient for a task like sound localization. All that matters is whether the computed empirical vector strength, is safely above values that can be expected from the statistics in the absence of a stimulus (null hypothesis). Under the assumption of

_{t}*N*independent and identically equidistributed phases ϕ

_{1}, …, ϕ

_{N}(i.e., for undriven neurons modeled by a renewal process), this statistics is described by the Rayleigh distribution . The probability of observing a value of is thus given by exp(−

*N*0.16) (Naundorff & Freund, 2002), which crosses the 5% (1%) significance level after

*N*= 20(30) spikes, respectively. Of course, still larger values, say, , allow rejection of the null hypothesis of stimulus absence after 5(7) spikes, respectively.

^{2}

## 5. Summary and Outlook

The starting point of our investigation was the question of what specific ingredients
of a neuron (model) are necessary to observe phase locking to an oscillatory
stimulus before significant changes of the spike rate. Through numerical
simulations, we found that even in a very basic neuron model, the noisy LIF, phase
locking below rate threshold actually does occur without the need to tune
parameters. Moreover, we found that it is not tied to a certain spiking mode but
happens generically for both noise-induced spiking and tonic spiking. From analytic
considerations, we were able to trace phase locking below rate threshold back to the
neuron's inherent freedom to shift subsequent spikes independently. This insight led
us to conjecture that pronounced interspike correlations might be a mechanism to
preclude phase locking below rate threshold. Surprisingly, we found that significant
anticorrelations between subsequent interspike intervals (ρ_{1} ≈ −0.5), as they occur in the LIF with threshold fatigue, are not
enough to prevent phase locking but rather stiffen the spontaneous spiking rate.
Also, a variation of the stimulus frequency never led to a drastic collapse of the
vector strength at rate threshold, as one might expect for driving the tonic spiking
neuron at half the spontaneous rate. Paradoxically it seems that specific
ingredients of a neuron model are required not to guarantee but, on the contrary, to
prevent phase locking to zero mean stimuli below rate threshold. Finally, we
discussed the ambiguous relation between phase distributions and moderate values of
the vector strength that still might qualify for a detection scheme.

In order to fully exploit the analytic relation between vector strength and spike
rate explicated in this letter, it seems desirable to find a neuron model that
allows deriving a closed-form expression for the core piece *p*(τ|ϕ; *a*, Ω). A good
candidate seems to be the perfect integrator with stochastic threshold and reset;
however, detailed elaboration is left to future research. Eventually further studies
are needed to construct a reasonably realistic neuron model where phase locking to
zero mean stimuli occurs only in combination with a significant variation of the
spike rate.

## Acknowledgments

J.A.F. acknowledges support by the DAAD (ARC D/06/12701). N.G.S. acknowledges support by grant EP/DO51894. We enjoyed fruitful discussions with S. Beulig, M. Buschermöhle, U. Feudel, R. P. Morse, and L. Schimansky-Geier.

## Note

^{1}

An account of the differences is given in Rieke, Warland, de Ruyter van Steveninck, and Bialek (1997).

^{2}

We note that the Rayleigh distribution results from the central limit theorem;
hence, for too few spikes, the statement has to be taken *cum grano
salis.*

## References

*P*-type electroreceptors

*Tyto alba*