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Firing rate-dependent phase responses of Purkinje cells support transient oscillations

  1. Yunliang Zang
  2. Sungho Hong
  3. Erik De Schutter  Is a corresponding author
  1. Computational Neuroscience Unit, Okinawa Institute of Science and Technology Graduate University, Japan
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Cite this article as: eLife 2020;9:e60692 doi: 10.7554/eLife.60692

Abstract

Both spike rate and timing can transmit information in the brain. Phase response curves (PRCs) quantify how a neuron transforms input to output by spike timing. PRCs exhibit strong firing-rate adaptation, but its mechanism and relevance for network output are poorly understood. Using our Purkinje cell (PC) model, we demonstrate that the rate adaptation is caused by rate-dependent subthreshold membrane potentials efficiently regulating the activation of Na+ channels. Then, we use a realistic PC network model to examine how rate-dependent responses synchronize spikes in the scenario of reciprocal inhibition-caused high-frequency oscillations. The changes in PRC cause oscillations and spike correlations only at high firing rates. The causal role of the PRC is confirmed using a simpler coupled oscillator network model. This mechanism enables transient oscillations between fast-spiking neurons that thereby form PC assemblies. Our work demonstrates that rate adaptation of PRCs can spatio-temporally organize the PC input to cerebellar nuclei.

Introduction

The propensity of neurons to fire synchronously depends on the interaction between cellular and network properties (Ermentrout et al., 2001). The contribution of cellular properties can be measured with a phase response curve (PRC). The PRC quantifies how a weak stimulus exerted at different phases during the interspike interval (ISI) can shift subsequent spike timing in repetitively firing neurons (Ermentrout et al., 2001; Ermentrout et al., 2012; Gutkin et al., 2005) and thereby predicts how well-timed synaptic input can modify spike timing. Consequently, the PRC determines the potential of network synchronization (Ermentrout et al., 2001; Ermentrout et al., 2008; Gutkin et al., 2005; Smeal et al., 2010). However, the PRC is not static and shows significant adaptation to firing rates. In cerebellar Purkinje cells (PCs), their phase responses to weak stimuli at low firing rates are small and surprisingly flat. With increased rates, responses in later phases become phase-dependent, with earlier onset-phases and gradually increasing peak amplitudes. This PRC property has never been theoretically replicated or explained (Couto et al., 2015; Phoka et al., 2010), nor has its effect on synchronizing spike outputs been explored.

On the circuit level, high-frequency oscillations caused by reciprocal inhibition have been observed in many regions of the brain, including the cortex, cerebellum and hippocampus (Bartos et al., 2002; Buzsáki and Draguhn, 2004; Cheron et al., 2004; de Solages et al., 2008). The functional importance of oscillations in information transmission is largely determined by their spatio-temporal scale, which for hard-wired inhibitory connections, is generally assumed to be driven by external input. It is interesting to explore whether firing rate-dependent PRCs can contribute to dynamic control of the spatial range of oscillations based on firing rate changes, because this would have significant downstream effects (Person and Raman, 2012).

To examine the mechanism of rate-dependent PRCs, we use our physiologically detailed PC model (Zang et al., 2018) and a simple pyramidal neuron model to explore the rate adaptation of PRCs. By analyzing simulation data and in vitro experimental data (Rancz and Häusser, 2010), we show that rate-dependent subthreshold membrane potentials can modulate the activation of Na+ channels to shape neuronal PRC profiles. We also build a PC network model connected by inhibitory axon collaterals to simulate high-frequency oscillations (de Solages et al., 2008; Witter et al., 2016). Rate adaptation of PRCs increases the power of oscillations at higher firing rates, firing irregularity and network connectivity also regulate the oscillation level. The causal role of the PRC is confirmed using a simpler coupled oscillator network model. The combination of these factors enables PC spikes uncorrelated at low basal rates to become transiently correlated in transient assemblies of PCs at high firing rates.

Results

PRC exhibits rate adaptation in PCs

PRCs were obtained by repeatedly exerting a weak stimulus at different phases of the ISI. The resulting change in ISI relative to original ISI corresponds to the PRC value at that phase (Figure 1A). All previous abstract and detailed PC models failed to replicate the experimentally observed rate adaptation of PRCs (Akemann and Knöpfel, 2006; Couto et al., 2015; De Schutter and Bower, 1994; Khaliq et al., 2003; Phoka et al., 2010). Our recent PC model was well constrained against a wide range of experimental data (Zang et al., 2018). Here, we explored whether this model can capture the rate adaptation of PRCs under similar conditions. When the PC model fires at 12 Hz, responses (phase advances) to weak stimuli are small and nearly flat for the whole ISI (Figure 1B,D). Only at a very narrow late phase do the responses become phase-dependent and slightly increased. With increased rates, the responses remain small and flat during early phases. However, later phase-dependent peaks gradually become larger (Figure 1C), with onset shifted to earlier phases (Figure 1D). It should be noted that the increased late-peak amplitude may be affected by how the PRC is computed (Equation 1): it is normalized by the ISI, causing the peak amplitude to increase for higher firing rates (smaller ISIs).

Figure 1 with 2 supplements see all
PRC exhibits strong rate adaptation in PC model.

(A) Schematic representation of the definition and computation of PRCs. The current pulse has a duration of 0.5 ms and an amplitude of 50 pA. Different spike rates were achieved by somatic current injection (Couto et al., 2015; Phoka et al., 2010). (B) The rate adaptation of the flat part and the phase-dependent PRC peak. (C) PRC peak amplitudes at different firing rates fitted by the Boltzmann function. (D) Duration of the flat phase at different firing rates.

In agreement with experiments under the same stimulus conditions (Phoka et al., 2010), the peak of PRCs finally became saturated at ~0.06 at high rates. The relationship between normalized PRC peaks and rates can be fitted by the Boltzman function and matches experimental data (Figure 1C, fitted with 1/ (1 + e−(rate−a)/b), a = 49.1, b = 26.4 in the model versus a = 44.1 and b = 20.5 in experiments Couto et al., 2015). PRCs in our model show similar rate adaption with inhibitory stimuli (phase delay, Figure 1—figure supplement 1A). This form of rate adaptive PRCs requires the presence of a dendrite in the PC model (Figure 1—figure supplement 2), but the dendrite can be passive (Figure 1—figure supplement 1B). We also tested the effect of increasing stimulus amplitude on PRC adaptation. Increasing stimulus amplitude consistently shifts onset-phases of phase-dependent peaks to the left and increases their amplitudes (Figure 1—figure supplement 1C).

To unveil the biophysical principles governing rate adaptive PRC profiles, we need to answer two questions: why are responses nearly flat in early phases and why do responses become phase-dependent during later phases?

The biophysical mechanism of rate adaptation of PRCs in PCs

We examined how spike properties vary with firing rates and find that the facilitation of Na+ currents relative to K+ currents, due to elevated subthreshold membrane potentials at high rates, underlies the rate adaptation of PRCs. After each spike, there is a pronounced after-hyperpolarization (AHP) caused by the large conductance Ca2+-activated K+ current, and subsequently the membrane potential gradually depolarizes due to intrinsic Na+ currents and dendritic axial current (Zang et al., 2018). As confirmed by re-analyzing in vitro somatic membrane potential recordings (shared by Ede Rancz and Michael Häusser Rancz and Häusser, 2010), subthreshold membrane potential levels are significantly elevated at high firing rates, but spike thresholds rise only slightly with rates (Figure 2A). This means that the ISI phase where Na+ activation threshold (~ −55 mV for 0.5% activation in PCs Khaliq et al., 2003; Zang et al., 2018) is crossed shifts to earlier phases with increasing rates. Consequently, larger phase ranges of membrane potentials are above the threshold at high rates (Figure 2B).

Figure 2 with 1 supplement see all
Modulated subthreshold membrane potentials account for the rate adaptation of PRCs.

(A and B) Experimental and simulated voltage trajectories in PCs at different rates. All voltage trajectories are shown from trough to peak within normalized ISIs. The model used (Zang et al., 2018) was not fitted to this specific experimental data. Spike thresholds at different rates are labeled in plots. The Na+ activation threshold is defined as −55 mV (stippled line). Right plots show phase dependence of Na+-activation threshold on firing rates. (C) Stimulus-triggered variations of inward ionic currents (solid) and outward ionic currents (dashed) at different phases and rates. Ionic currents are shifted to 0 (grey line) at the onset of stimulus to compare their relative changes. At phase = 0.2, the outward current is still decreasing due to the inactivation of the large conductance Ca2+-activated K+ current at 162 Hz. (D) Larger slopes of the Na+ activation curve at high membrane potentials.

During early phases of all firing rates, membrane potentials are distant from the Na+ activation threshold of the Na+ channels (Figure 2A,B). The depolarizations to weak stimuli fail to activate sufficient transient and resurgent Na+ channels to speed up voltage trajectories (Figure 2C). Consequently, phase advances in early phases are small and flat. At later phases, membrane potentials gradually approach and surpass the Na+ activation threshold. Stimulus-evoked depolarizations activate more Na+ channels to speed up trajectories in return. Therefore, phase advances become large and phase- (actually voltage-) dependent. Because high-rate-corresponding elevated membrane potentials have larger slopes at the foot of the Na+ activation curve, the same ∆V activates more Na+ channels and, in addition to the normalization, contributes to larger PRC peaks at high rates (Figure 2C,D). Under all conditions (except phase = 0.2, 162 Hz), stimulus-evoked depolarizations also increase outward currents, but this increase is smaller than that of inward currents (mainly Na+) due to the high activation threshold of K+ currents (mainly Kv3) in PCs (Martina et al., 2003; Zang et al., 2018). As the stimulus becomes stronger, it triggers larger depolarizations and the required pre-stimulus membrane potential (phase) to reach Na+ activation threshold is lowered. Thus, increasing the stimulus amplitude not only increases PRC peaks, but also shifts the onset-phases of phase-dependent responses to the left (Figure 1—figure supplement 1C). In the absence of a dendrite (Figure 1—figure supplement 2), the larger amplitude spike is followed by a stronger afterhyperpolarization (Zang et al., 2018) that deactivates K+ currents allowing for an earlier depolarization in the ISI, resulting in a completely different PRC.

We further confirmed that the critical role of subthreshold membrane potentials in shaping PRC profiles is not specific to the PC by manipulating PRCs in a modified Traub model (Ermentrout et al., 2001; Figure 2—figure supplement 1 and accompanying text).

Rate-dependent high-frequency oscillations

The potential effect of firing rate-caused variations of cellular response properties on population synchrony has been largely ignored in previous studies (Bartos et al., 2002; Brunel and Hakim, 1999; de Solages et al., 2008; Heck et al., 2007; Shin and De Schutter, 2006). Here, we examine whether spike rate correlates with synchrony in the presence of high-frequency oscillations that have been observed in the adult cerebellar cortex (Cheron et al., 2004; de Solages et al., 2008). Although axon collateral contacts between PCs were originally described to exist only in juvenile mice (Watt et al., 2009), recent work demonstrated their existence also in adult mice (Witter et al., 2016). We built a biophysically realistic network model composed of 100 PCs with passive dendrites distributed on the parasagittal plane (Witter et al., 2016). Each PC connects to the somas of its five nearest neighboring PCs through inhibitory axon collaterals on each side based on experimental data (Bishop and O'Donoghue, 1986; de Solages et al., 2008; Watt et al., 2009; Witter et al., 2016). Rates of each PC are independently driven by parallel fiber synapses, stellate cell synapses, and basket cell synapses (Figure 3A). More details are in Materials and methods.

High-frequency oscillations show adaptation to cellular firing rates.

(A) Schematic representation of the network configuration. (B) Example of sampled PC voltage trajectories in the network. (C) Example of population rates in the network (time bin 1 ms). (D) The power spectrum of population rates of the network at different cellular rates and firing irregularity (CV of ISIs). (E) Averaged normalized CCGs at different cellular rates.

When the average cellular rate is 116 Hz, PCs in the network tend to fire within interspaced clusters with time intervals of ~6 ms (Figure 3B). However, individual PCs do not fire within every cluster. Therefore, spikes in the network show intermittent pairwise synchrony on the population level rather than spike-to-spike synchrony (Figure 3B). Each peak in Figure 3C corresponds to a ‘cluster’. Based on the power spectrum, the network oscillates at a frequency of ~175 Hz (inverse of the cluster interval,~6 ms), which is independent of cellular firing rates (116 Hz in red and 70 Hz in blue, Figure 3D), because oscillation frequency is mainly determined by synaptic properties (Brunel and Hakim, 1999; Brunel and Wang, 2003; de Solages et al., 2008; Maex and De Schutter, 2003). When cellular firing rates increase from 70 Hz to 116 Hz, the power of high-frequency oscillations significantly increases and the peak becomes sharper. When individual PCs fire at low rates (10 Hz), the network fails to generate high-frequency oscillations and each PC fires independently, as evidenced by the flat power spectrum (Figure 3D). High-frequency oscillations and their firing rate-dependent changes are also reflected in the average normalized cross-correlograms (CCGs) between PC pairs (Figure 3E). When PCs fire at 70 Hz and 116 Hz, in addition to positive central peaks, two significant side peaks can be observed in the CCGs, suggesting correlated spikes with 0 ms-time lag and ~6 ms-time lag. Amplitudes of the peaks increase with cellular firing rates and disappear when they are low (10 Hz).

In Figure 3, the variation of cellular rates was driven by synaptic input to demonstrate the rate adaptation of high-frequency oscillations. However, it is difficult to differentiate the relative contribution of PRC shapes and firing irregularity (measured by the CV of ISIs) since they covary with firing rates (Figure 3D). Therefore, cellular rates were systematically varied by dynamic current injections, which were approximated by the Ornstein–Uhlenbeck (OU) process (Destexhe et al., 2001). This simulation protocol also causes the formation of high-frequency oscillations (Figure 4—figure supplement 1). When PCs fire with low to moderate CV of ISIs, they show loose spike-to-spike synchrony at high rates, and the power peak increases with cellular firing rates. High-frequency oscillations were never observed for low cellular firing rates (Figure 4A, Figure 4—figure supplement 1). With high CV of ISIs, spikes are jittered and the loose spike synchrony is disrupted (Figure 4B). Oscillation changes due to firing properties are also reflected in average normalized CCGs. Both central and side peaks increase with the cellular firing rate and decrease with the spiking irregularity. Our results show that small spiking irregularity supports high-frequency oscillations.

Figure 4 with 2 supplements see all
Effect of cell and network properties on high-frequency oscillations.

(A) Low cellular firing rates decorrelate the network output in the forms of reduced peaks of power spectrums (left) and CCGs (right). CV ISI is ~0.45. Synaptic conductance is 1 nS and radius is 5. (B) Irregular spiking (high CV of ISIs) also decorrelates network. The cellular firing rate is ~141 Hz. Same layout and network properties as in A. (C) Small conductance (cond) of inhibitory synapses decorrelates network output. Same layout and network properties as in A with cellular firing rate ~151 Hz and CV ISI ~ 0.45. (D) Short connection radius also decorrelates network output. Same layout and cellular firing properties as C.

At the circuit level, the strength of inhibitory synapses and connection radius are difficult to determine accurately, but their values are critical for the function of axon collaterals. Within the ranges of experimentally reported synaptic conductance and connection radius (de Solages et al., 2008; Orduz and Llano, 2007; Watt et al., 2009; Witter et al., 2016), the network generates robust high-frequency oscillations (Figure 4C,D). In addition, we find that increasing the conductance of inhibitory synapses or their connection radius increases the power of high-frequency oscillations and make the power spectrum sharper. The increased oscillation power due to connectivity properties is also captured by the larger peaks in CCGs.

Together, our simulation data suggest that the correlation between PC spikes is strong under conditions of low to moderate spiking irregularity, high cellular firing rate, high synaptic conductance, and large connection radius.

High-frequency oscillations are caused by rate-dependent PRCs

Because both oscillation power and PRC are firing rate dependent, a causal relationship is possible. This is supported by the effect of PRC size on oscillations: decreasing its size leads to weaker oscillations and can even cause weaker oscillations at higher spike rates (Figure 4—figure supplement 2). However, it is impossible to manipulate PRC shapes in the complex PC model without greatly affecting other cell and network properties. Therefore, we investigated the effect of rate-dependent PRC shapes in a network of simple coupled oscillators (Kuramoto, 1984; Smeal et al., 2010), where the firing rate specific PRC was used as the coupling term Z(θ) (see Materials and methods). In such a coupled oscillator network, the oscillation power shows a firing rate dependence similar to that of the complex PC network (Figure 5A,B). This finding demonstrates that the firing rate adaptation of the PRC is sufficient to cause firing-rate-dependent oscillations.

Firing-rate adaptation of high-frequency oscillations is caused by the PRC.

(A) Dependence of peak power of high-frequency oscillations in the complex PC network of Figure 3 (cyan) and in the coupled oscillator network (red) on cellular firing rate. (B) The power spectrum of the coupled oscillator network depends on the cellular firing-rate-specific PRC used as coupling term. Inset: firing-rate-dependent coupling factors Z(θ) used. (C) Same as B but with the peak amplitude of Z(θ) set to that of the peak of 30 Hz firing rate. (D) Same as C for the peak of 70 Hz firing rate.

Next, we investigated the specific contribution of the flat part that dominates the PRC at low firing rates versus the late peak with increasing amplitude that appears at higher firing rates. We checked which of these PRC components is responsible for the effect on oscillations by fixing the amplitude of the peak to the value for a specific firing rate. Networks simulated with fixed PRC peak amplitudes show power spectra (Figure 5C,D) that are very similar to that obtained with the actual PRC (Figure 5B). An exception is when peak amplitude is very small (for firing frequencies of less than 30 Hz, not shown). The only significant difference between Figure 5C and D is the peak oscillation frequency, which increases with the firing rate for which the amplitude was taken.

In conclusion, the ratio of flat part width to peak width of the firing rate dependent PRC causes the rate dependence of high-frequency oscillations. At low firing rates the dominant flat part suppresses the coupling between oscillators. At high firing rates the coupling increases during the late peak and synchronizes the oscillators, but the strength of oscillation does not depend on peak amplitude in this network.

Transient correlations form cell assemblies

Correlation of spiking has often been proposed as a mechanism to form transient cell assemblies (Abeles, 1982; Hebb, 1949; Singer, 1993). This assumes that oscillations can appear and fade rapidly and that they can appear in networks with heterogeneous firing rates. We have previously simulated networks with a range of homogeneous stable cellular rates. Here, we first test whether rate-dependent synchrony still holds when population rates change dynamically. Population rates of the network approximate the half-positive cycle of a 1 Hz sine wave (peak ~140 Hz) with the duration of each trial being 1 s (Figure 6A). We compute shuffle-corrected, normalized joint peristimulus time histograms (JPSTHs) to reflect the dynamic synchrony (Aertsen et al., 1989; Figure 6—figure supplement 1A). The main and the third diagonals of the JPSTH matrix, corresponding to 0 ms-time lag correlation and 6 ms-time lag correlation respectively, are plotted to show the dynamic synchrony at transiently increased rates (bin size is 2 ms, Figure 6B). At low basal rates, there are no correlations between spikes. Both correlations start to increase ~250 ms after the onset of simulations when the cellular firing rate increases. Closely following rate changes, they decrease again when the cellular rates drop. It demonstrates that axon collateral-caused spike correlations can be achieved transiently to transmit a correlation code conjunctive with temporal cellular firing rate increases.

Figure 6 with 2 supplements see all
Correlations can be transient and robust to heterogeneous spike rates.

(A) Population spike rates of PCs. (B) The 0 ms- and 6 ms-time lag correlations increase with population rates. (C–E) The rate-dependent correlation is robust to heterogeneous cellular rate changes. From (C) to (E), the number of decreased rate cells increases from 10 to 30. (F) Correlations between decreased-rate neurons in the network (n = 30). (G) Correlations between increased-rate neurons and decreased-rate neurons (n = 30 for each group).

Although it remains unclear whether the population of PCs converging onto a same cerebellar nuclei (CN) neuron are homogeneous or heterogeneous (Uusisaari and De Schutter, 2011), simultaneous bidirectional PC rate changes have been observed during cerebellum-related behaviors (Chen et al., 2016; Herzfeld et al., 2015). It is very likely that neighboring PCs show heterogeneous spike rate changes (Hong et al., 2016), which can reduce spike correlations (Markowitz et al., 2008). Therefore, we distributed 10–30 extra cells with decreasing spike rates (Figure 6—figure supplement 1B) in the network to test the effect of heterogeneous neighboring rate changes on transient correlations. They were randomly scattered among the cells with increasing rates. Spike correlations still become larger for the subgroup of PCs showing increased cellular rates, despite a slight decrease of the correlation amplitude when more cells decrease their spike rates (Figure 6C–E). Moreover, the spiking in PCs with decreased firing rates is not correlated (Figure 6F), nor is it correlated with oscillating increased-rate PCs (Figure 6G), making the assembly formation specific to fast spiking PCs. Similar results were obtained for a faster change of population rates (2.5 Hz sine wave, Figure 6—figure supplement 2). The results suggest that a population of PCs with increased spike rates can form a correlated assembly that will strongly affect downstream neurons even when it is surrounded by non-correlated neighboring PCs with decreased spike rates.

Discussion

In this work, we reproduced the firing rate-dependent PRC of PCs and dissected the underlying biophysical mechanisms. Next, we explored the role of these PRCs in synchronizing spikes in cerebellar PCs and how they can support the formation of transient assemblies.

Biophysical mechanisms underlying rate-dependent PRCs

The profiles of neuronal PRCs are regulated by ionic currents (Ermentrout et al., 2012) and they show rate adaptation (Couto et al., 2015; Ermentrout et al., 2001; Gutkin et al., 2005; Phoka et al., 2010; Tsubo et al., 2007). Cerebellar PCs exhibit a transition from small, phase-independent responses to large, phase-dependent type-I responses with increasing rates (Couto et al., 2015; Phoka et al., 2010), but the mechanism was unknown (Akemann and Knöpfel, 2006; Couto et al., 2015; De Schutter and Bower, 1994; Khaliq et al., 2003; Phoka et al., 2010). This work reproduces and explains the experimentally observed rate adaptation of PRCs. Note that the slight increase of PRCs in the very narrow late phase in our model (low rate, Figure 1B) may be annihilated by noise in spontaneously firing neurons (Couto et al., 2015; Phoka et al., 2010).

Compared with previous work emphasizing the slow deactivation of K+ currents in cortical neurons (Ermentrout et al., 2001; Gutkin et al., 2005), here we demonstrate the role of rate-dependent subthreshold membrane potentials and their corresponding activation of Na+channels. In both pyramidal neurons and PCs, spike rates cause significant variation of the subthreshold membrane potential during the ISI (Rancz and Häusser, 2010; Tsubo et al., 2007). In response to a stimulus, both Na+ and K+ currents are activated. In PCs, the main K+ current is high-threshold activated (Martina et al., 2003; Zang et al., 2018); therefore, depolarization-facilitated Na+ currents dominate, causing larger normalized PRCs at high rates (Figure 2). This facilitation may be further boosted in PCs by enhanced excitability, such as SK2 down-regulation reducing the AHP and elevating subthreshold membrane potentials (Grasselli et al., 2020; Ohtsuki and Hansel, 2018). We did not explore possible PRC differences between zebrin-positive and zebrin-negative PCs due to a lack of data (Zhou et al., 2014). Previous PC models (Akemann and Knöpfel, 2006; Couto et al., 2015; De Schutter and Bower, 1994; Khaliq et al., 2003; Phoka et al., 2010) included low-threshold-activated K+ currents, which counteract facilitated Na+ currents. In the original Traub model, slow deactivation of K+ currents and consequent hyperpolarization synergistically reduce the normalized PRC peaks at high rates (Ermentrout et al., 2001; Gutkin et al., 2005). By minimally modifying the Traub model, elevated subthreshold membrane potentials generate larger normalized PRC peaks at high rates (Figure 2—figure supplement 1).

The evidence supporting rate-dependent correlations

Rate-dependent synchrony in the cerebellum has been demonstrated for Golgi cells (van Welie et al., 2016) but not, as yet, for PCs. However, careful analysis of previous experimental data in the cerebellum provides some evidence to support our findings. In the work of de Solages et al., 2008, units with lower average rates (<10 Hz) did not exhibit significant correlations between neighboring PCs, for unknown reasons. This can be explained by the small flat PRCs at low rates. Under extreme conditions, when the PRC is constantly 0 (equivalent to disconnection), no correlations can be achieved (Figures 36). Additionally, the experimental oscillation power increased by the application of WIN 55,212–2, which was intended to suppress background excitatory and inhibitory synapses (de Solages et al., 2008). The increased power could be due to more regular spiking after inhibiting the activity of background synapses (Figure 4B). However, it could also be caused by increased spike rates (Figure 4A), because this agent also blocks P/Q type Ca2+ channels and consequently P/Q type Ca2+-activated K+ currents, to increase spike rates (Fisyunov et al., 2006). Similarly, enhanced oscillations have also been observed in calcium-binding protein gene KO mice, which have significantly higher simple spike rates (Cheron et al., 2004). A more systematic experimental study of the firing rate dependent appearance of loose simple spike synchrony among PCs and its relation to behavior would be required to confirm these predictions.

The rate-dependent correlations observed in this study are different from those reported previously by de la Rocha et al., 2007. In that study, common input-mediated correlation increased rapidly with increasing rate at low firing rates in pyramidal cells (their Figure 1e) and in integrate-and-fire models (their Figure 2c), while the PRC mediated correlations in our study of inhibitory coupling only appear at much higher firing rates (Figure 5A). Moreover, the findings of de la Rocha et al., 2007 are not general, they only apply for neurons with integrator firing properties (Hong et al., 2012).

Down-stream effects of PC assemblies

PCs inhibit their target neurons in the CN, which in turn form the only cerebellar output. It is difficult to finely regulate CN firing rates with inhibition only, because it operates over the narrow voltage range between resting potentials and GABAA reversal potentials. Two solutions for this problem have been proposed. The first is that synchronized pauses of PC firing will release CN neurons from inhibition, leading to rebound firing (De Schutter and Steuber, 2009; Lee et al., 2015). There is strong evidence that this mechanism works in controlling the onset of movement in the conditioned eyeblink reflex (Heiney et al., 2014) and in saccade initiation (Hong et al., 2016). The other solution provides a more continuous rate modulated CN output by time-locking of CN spikes to PC input. Several experimental studies have demonstrated that partial synchronization of afferent PC spiking can time-lock the spikes of CN neurons to their input (Gauck and Jaeger, 2000; Person and Raman, 2012). The ability to rapidly increase the correlation level within a subgroup of PCs with increased firing rates (Figure 6) is therefore predicted to have a strong effect on CN spiking. Moreover, this does not require strong synchronization. Similar results were observed when jitter higher than the few ms predicted by our network model (Figure 3B) was applied to the synchronous PC input (Gauck and Jaeger, 2000). Previous evidence has demonstrated neocortical oscillations can entrain cerebellar oscillations (Ros et al., 2009). Though high-frequency oscillations (Figure 3) don’t rely on common input, they can still be regulated by cortical inputs and drive neurons in the thalamocortical circuit (Timofeev and Steriade, 1997) and cerebral cortex (Popa et al., 2013).

Advantages of transient PC assemblies

The actual convergence and divergence of PC axons onto CN neurons remains a controversial topic in the literature. There are roughly ten times more PCs than CN neurons and PC axons branch extensively leading to computed convergence values ranging from 20 to over 800 (Uusisaari and De Schutter, 2011), although many authors have recently converged on the compromise of ~50 (Person and Raman, 2012). If CN neurons just average the activity of all afferent PCs, much of the potential information generated by the large neural expansion in cerebellar cortex would be lost. Our PC network with parameters that fall within physiological ranges can rapidly generate and disrupt oscillations based on the cellular firing rates (Figures 3 and 6), with no need of increasing afferent input correlation. Note that rate-related synchrony can also be achieved via common synaptic inputs (Heck et al., 2007), gap junctions (Middleton et al., 2008), and ephaptic coupling (Han et al., 2018), when connections are weak. This means that transiently correlated PC assemblies can form and disappear quickly. Such assemblies, even if consisting of only a few PCs (Person and Raman, 2012), can finely control spiking in CNs. Because the assemblies can consist of variable subsets of afferent PCs to a CN neuron, this greatly expands the information processing capacity of the cerebellum.

Conclusion

We have shown that firing-rate dependent PRCs can cause firing-rate dependent oscillations at the network level. Such a mechanism supports the rapid formation of transient neural assemblies in cerebellar cortex.

Materials and methods

The detailed PC model and the interconnected network model were implemented in NEURON 7.5 (Carnevale and Hines, 2006). The Traub model was implemented in MATLAB.

PRC computations

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Our recently developed compartment-based PC model was used (Zang et al., 2018). To compute the PRCs in Figure 1, brief current pulses with a duration of 0.5 ms and an amplitude of 50 pA were administered at the soma at different phases of interspike intervals. The resulting perturbed periods were then used to calculate phase advances by Ermentrout et al., 2001:

(1) PRC=(ISIISIperturb)/ISI

This is the same equation used in experimental studies (Couto et al., 2015; Phoka et al., 2010), to facilitate comparison. Different cellular rates were achieved by somatic holding currents (Couto et al., 2015; Phoka et al., 2010). To compute PRCs in response to negative stimuli, the amplitudes of the pulses were changed to -50 pA. To compute PRCs of our PC model with passive dendrites, only H current and leak current were distributed on the dendrites with the same parameters as in the active model (Zang et al., 2018). The Traub model (Traub et al., 1999) was implemented according to the work of Ermentrout et al., 2001; Gutkin et al., 2005. In the modified version of this model, the conductance of the kdr current was reduced from 80 to 40. Activation and deactivation rates of this current were shifted to the right by 30 mV, αn(v)=0.032*(v+22)/(1-exp(-(v+22)/5)); βn(v)=0.5*exp(-(v+27)/40); the conductance of AHP current was increased from 0 to 0.1.

Network simulations

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We implemented our recurrent inhibitory PC layer network using the Watts-Strogatz model (Watts and Strogatz, 1998) to avoid boundary effects. To reduce simulation time, we used the PC model with passive dendrites, which exhibits similar rate-dependent PRCs to the PC model with active dendrites (Figure 1—figure supplement 1B). In the baseline version of the network, 100 PCs were distributed on the parasagittal plane (Witter et al., 2016), corresponding to 2 mm of folium with a distance of 20 μm between neighboring PC soma centers. 100 PCs are within the estimated range of PCs converging to a same cerebellar nuclei neuron (Person and Raman, 2012). Each PC was connected to its nearest 2*radius neighboring PC somas and connections had 0 rewiring probability. The PCs were interconnected, according to anatomical data showing collaterals present toward both the apex and the base of the lobule with only slight directional biases (Witter et al., 2016). The baseline value of radius was 5 within the range of experimental estimates (Bishop and O'Donoghue, 1986; de Solages et al., 2008; Watt et al., 2009; Witter et al., 2016). The inhibitory postsynaptic current (IPSC) was implemented using the NEURON built-in point process, Exp2Syn. G = weight * (exp(-t/τ2) - exp(-t/τ1)), with τ1 = 0.5 ms (rise time) and τ2 = 3 ms (decay time). The reversal potential of the IPSC was set at −85 mV (Watt et al., 2009). The conductance was 1 nS (de Solages et al., 2008; Orduz and Llano, 2007; Witter et al., 2016). The delay between onset of an IPSC and its presynaptic spike timing was 1.5 ms (de Solages et al., 2008; Orduz and Llano, 2007; Witter et al., 2016). To test the effect of rate-dependent PRCs on high-frequency oscillations, we varied the cellular rates in two paradigms. In the first paradigm (Figure 3), each PC is contacted by 4000 excitatory parallel fiber synapses (PF, on spiny dendrites), 18 inhibitory stellate cells (STs, on spiny dendrites) and four inhibitory basket cells (BSs, on the soma). Activation of excitatory and inhibitory synapses in each PC was approximated as an independent Poisson process with different rates. We simulated five conditions: PC rate = 10 Hz when PF rate = 0.27 Hz, ST rate = 14.4 Hz, BS rate = 14.4 Hz; PC rate = 47 Hz when PF rate = 1.62 Hz, ST rate = 28.8 Hz, BS rate = 28.8 Hz (used in Figure 5); PC rate = 70 Hz when PF rate = 2.16 Hz, ST rate = 28.8 Hz, BS rate = 28.8 Hz; PC rate = 93 Hz when PF rate = 2.7 Hz, ST rate = 28.8 Hz, BS rate = 28.8 Hz (used in Figure 5); PC rate = 116 Hz when PF rate = 3.24 Hz, ST rate = 28.8 Hz, BS rate = 28.8 Hz.

To more systematically explore different factors regulating network outputs, we used a second paradigm (Figure 4, Figure 4—figure supplements 1 and 2). Cellular rates of each PC were manipulated by injecting stochastic currents on the soma. The stochastic current was approximated by the commonly used Ornstein-Uhlenbeck random process (Destexhe et al., 2001), τdIdt=-I+στηi(t). σ represents the amplitude of the fluctuation; ηi represents uncorrelated white noise with unit variance; τ=5 ms. In this paradigm, we systematically varied the rates and firing irregularities of PCs (CV of ISIs) to explore their importance for network output. Due to the intrinsic relationship between CV of ISIs and firing rates, a larger σ is required for higher firing rates to get the same CV of ISI. Phase response is a result of input current and response gain of the cell. We reduce the phase response by halving the input current (synaptic conductance) to achieve a smaller response at high firing rates (Figure 4—figure supplements 2). The conductance of inhibitory synapses was tested with the values of 0.75, 1.0, 1.25 and 1.5 nS in Figure 4C. We also explored the effect of connection radius with the values of 3, 5, 7 and10 in Figure 4D.

To test a spatio-temporally increased correlation, we randomly distributed extra 10–30 PCs with decreased cellular rates into the original network (Figure 6, Figure 6—figure supplement 2), including 100 increased-rate cells. Similar with Figure 4, each PC receives dynamic current injections approximated by an Ornstein-Uhlenbeck random process. Their mean population firing rates are shown in Figure 6—figure supplement 1B.

Coupled oscillator model

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The model comprises 100 neurons that are randomly connected to each other with connection probability of p=0.75 (Figure 5). The ‘subthreshold dynamics’ of individual neurons is given by the phase equation

dθdt=1T+Z(θ)(snet(t)+sind(t)),

where θ is a phase variable ranging from 0 to 1. T is an intrinsic period of the oscillation. Z(θ) is a PRC. sind and snet are the individual and network input, respectively. At θ=1, the model cell “spiked.” Then, θ was reset to θ-1 and the spike was added to the spike train variable (see below).

Z(θ) is given by

Z(θ)={Acif 0θ1δ,A(c+Bsin(π(θ+δ1)/δ))if 1δ<θ1.

Here, c represents the flat part of the Purkinje cell PRC and the other term represents a 'bump' around θ = 1. We found that the bump width is ~3 ms in time regardless of the firing rate, and set δ = 3 ms/T. We also used c = 0.08 and A = 12.

In the case when the model PRC scales as the PC PRC (Figure 5B), B = famp(1/T) where famp(r) is a normalized PRC amplitude given a baseline firing rate r in Figure 1C. In Figure 4C and D with no amplitude scaling of the PRC, B = famp(30 Hz) and B = famp(70 Hz) are used, regardless of T, respectively.

sind(t) is given by the Ornstein-Uhlenbeck (OU) process, tsind=-sind/τ+σ0ζ, where ζ is a Wiener process based on the standard normal distribution. We used τ = 3 ms and σ0 = 0.2.

snet(t) is given by

dsnetdt=snetτsyn+i(t),i(t)=gjoj(td),

where j represents other neurons connected to each cell, and oj(t) is a spike train of the cell j. d = 1.5 ms is a synaptic delay. g = -20 is a connection weight, and τsyn= 3 ms is a decay time for the synaptic current.

We used the forward Euler method with a time step of 0.025 ms to integrate the subthreshold equation, while we also confirmed that our results did not change if we use 0.0125 ms. The OU processes were integrated with the same time step and backward Euler method.

Data analysis

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The power spectrum of the spike trains of the network was estimated by Welch’s method, which calculates the average of the spectra of windowed segments (window size 128 points). In each trial under each specific stimulus condition, the length of the signal was 2 s, with a time resolution of 1 ms. The final result was the average of 14 trials.

To compute the CCGs under each specific stimulus condition, we first computed pairwise correlations between the spike trains of two neurons and then corrected them by shift predictors, which removed the ‘chance correlations’ due to rate changes. Then correlations were divided by the triangular function Θτ=T-|τ| and λiλj. T was the duration of each trial and τ was the time lag. Θτ corrects for the degree of overlap between two spike trains for each time lag τ. λi was the mean firing rate of neuron i (Kohn and Smith, 2005). Finally, the CCGs between all pairs in the network were averaged to reflect the population level spike correlations. Thus, similar with previous work (Heck et al., 2007), the computed CCGs reflect the ‘excess’ correlation caused by axon collaterals in our work.

To measure the dynamic correlation over the time course of the stimulus, we computed JPSTHs (Aertsen et al., 1989). We first picked two neurons from our network and aligned their spike-count PSTHs to stimulation onset with 2 ms time bins in each trial (larger time bins annihilated the positive peaks due to the significant negative correlations in paired spikes, see CCGs in Figures 3 and 4). We constructed the JPSTH matrix by taking each stimulus trial segment and plotting the spike counts of one cell on the horizontal and one on the vertical. If there is a spike from neuron i at time x, and a spike from neuron j at time y, one count will be added to the matrix index (x,y). By repeating this process for different trials, we got a raw matrix for a cell pair i and j. Then by the shift-predictor (repeated previous steps with shuffled stimulation trials), we removed correlations due to co-stimulation caused firing rate changes. Next step, we normalized the JPSTH by dividing with the product of standard deviations of the PSTHs of each neuron. To measure the correlation of the assembly, we averaged JPSTH between all non-repeated cell pairs in the defined ‘assembly’ of our network (Oemisch et al., 2015). The corrected matrix values become correlation coefficients, with values between −1 and +1. The main diagonal of the JPSTH matrix provides a measure of time-varying 0 ms time lag correlations and the third main diagonal (2 ms time bin) provides a measure of 6 ms time lag correlations. Due to the small-time bin we used, we simulated 1992 trials (for Figure 6B–E) to compute JPSTH between PC pairs and smoothed the JPSTHs for visualization purpose. Due to the small number of decreased-rate neurons in the network, we simulated 26112 trials to compute Figure 6F,G (30 decreased-rate neurons). When decreased-rate neuron numbers are 10 and 20 (Figure 6C,D), we did not compute their correlations due to the computational challenge. For Figure 6G, we randomly picked 30 from 100 increased-rate neurons to make pairs with 30 decreased-rate neurons.

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Decision letter

  1. Ronald L Calabrese
    Senior and Reviewing Editor; Emory University, United States
  2. Bard Ermentrout
    Reviewer; University of Pittsburgh, United States

In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.

Acceptance summary:

This modeling study addresses an important problem in cellular and network dynamics. First, it provides a biophysical mechanism underlying the changes in the shape of the phase response curve (PRC) of Purkinje cells observed when they change their (simple spike) firing rate. Second, it demonstrates that this mechanism depends on the subthreshold voltage trajectory between spikes, which in turn is governed by the intrinsic biophysics of the neurons. Finally, the authors show that firing rate-dependent changes in the PRC of individual neurons can drive rate-dependent changes in network synchrony. This new mechanism is likely to regulate synchrony of high-frequency oscillations not just in the cerebellum but also in many other circuits in the brain.

The study is carefully designed and performed, and the manuscript is well written. Overall it makes an important contribution to how specific features of single-cell biophysics can help to determine network-level dynamics.

Decision letter after peer review:

[Editors’ note: the authors resubmitted a revised version of the paper for consideration. What follows is the authors’ response to the first round of review.]

Thank you for submitting your work entitled "Firing rate-dependent phase responses of purkinje cells support transient oscillations" for consideration by eLife. Your article has been reviewed by a Senior Editor, a Reviewing Editor, and three reviewers. The following individuals involved in review of your submission have agreed to reveal their identity: Bard Ermentrout (Reviewer #1).

Our decision has been reached after consultation between the reviewers. Based on these discussions and the individual reviews below, we regret to inform you that your work will not be considered further for publication in eLife.

There was a strong agreement on the importance of PRCs for understanding neuronal activity and for the ambitions of the work presented. Several substantive concerns about the execution of the work arose, and there was a concern that the relationship of the PCR to network function was not established limiting the impact of the findings. At the end of this decision letter you will see the de-identified consultation among reviewers for this manuscript. This consultation in conjunction with the written reviews could serve as the basis for a complete revision of the current manuscript that could serve as a new submission to eLife.

Reviewer #1:

I was a bit surprised at this result until I realized what the authors had done with respect to the computation of the PRC. In the infinitesimal limit,the PRC is the normalized nullspace of the adjoint linear operator. The usual normalization is that the inner product is 1. This is how the theory that the authors allude to is done. Here the authors use a different normalization where they divide the adjoint by the period. This produces a different scaling and of course will make the PRC that occurs near the bifurcation (where the period is large) to appear much smaller in amplitude than it would with the standard scaling. If you use the standard scaling on the Traub model (which is readily available) and use the authors modifications, you find that with the standard normalization, higher frequencies have smaller PRCs which is according to the theory. If you divide by the period, then you get the authors results at least for currents that are not too big. For frequencies of about 110 Hz, even with the scaling, I found the PRC to be smaller than the lower frequencies.

Turning to their PC model, it is clear from Figure 1C, that the amplitudes with the usual normalization will follow the theory. That is, higher-frequencies have smaller amplitudes. (Specifically, if you divide the amplitudes shown by their respective frequencies, the graph will be decreasing.)

The main motive for this study is the PRC measurements from Couto et al. which was published in PLoS comp bio where the same result was found experimentally. To me the amplitude is not so much the point of interest here but rather the extreme change in the shape of the PRC. The Traub model does not show the shape dependence in fact it is the opposite with little effects until the late phases for low frequencies, an effect that was explained in the paper by Pascal et al. as coming from the adaptation.

The main result of interest is that at higher rates, oscillations synchronize/correlate better than an low rates So, there is some practical effect of this.

The authors set out to explain several effects. One of them is the largely unresponsive phase after the spike. As noted above, this effect was already explained in the old work of Pascal et al.

In Figure 3D, while the amplitude goes up in this model, the phase dependence is very flat only in the low rate versions. Thus, I think that the authors explanations are not really general. They need to make these explanations in a more rigorous mathematical manner.

The latter part of the paper is the most interesting where they demonstrate rate dependence in the correlation of the networks. It is a shame that they do not directly relate this to the PRCs. It would have been nice to perform some sort of analysis of a network of neurons that viewed as phase oscillators and connected via their phase response curves. I should point out that when coupling, whether or not locking occurs depends on whether or not you rescale the heterogeneties or not. I illustrate this in the following example. Let I be the baseline current to a bunch of neural oscillatora and let I_j be the heterogeneous current (assumed small) Let T be the period and suppose simple pulse coupling such that is neuron j fires, then a delta function current J_{ij} is added to the postsynaptic current. Let Delta be the conventional PRC (normalized to be msec/mv). Then the Kuramoto/Ermentrout/Kopell/ etc reduction yields for the phase:

dtheta_i/dt = 1 + Delta (theta_i)[ I_i/C + sum J_{ij}/C delta(t-t_j)] mod T

with C the membrane capacitance. Theta is a time-like variable at this point. To make it a phase-like variable (as the authors do), then let theta = T phi. Then one obtains

dphi_i/dt = omega + (1/T) Delta(T phi_i) [ I_i/C + sum J_{ij}/C delta(t-t_j)] mod 1

Conditions for synchrony etc are the same as the first model if, say you reduce this to a phase-difference model by replacing the delta(t) with delta(phi).

My point is, that the amplitude dependence on frequency is partially a consequence of how you normalize the PRC. If you look at Tsubo Figure 4, the amplitudes are all the same in their normalized version. The theory predicted by the theta model is that the period-normalized PRCs should have constant amplitudes.

I think the results are interesting, but I also think that they need to do some more sophisticated math analysis and also relate their PRCs more directly to the synachronization/correlation effects. This could be done with a weak coupling analysis after convolving their PRCs with the synapses to obtain the interaction function. The most important facet of PRCs is not som much their amplitude (which can be compensated for by changing the coupling), but their shape. I'd like more theoretical understanding of the shape dependence.

Reviewer #2:

This manuscript explores how average firing rate in Purkinje cells affects their PRC and high-frequency oscillations. Overall, there are issues with presentation, especially for clarity and the significance of the findings.

Essential revisions:

1) The manuscript, as presented, is disjointed. The first half proposes a mechanistic factor that contributes to rate modulation in Purkinje cell PRCs, and has a clear conclusion (although not many other mechanisms are entertained). The second half describes how the average firing rate increases the power of high frequency oscillations in the cerebellum. But there is not a strong link between these halves. In particular, it is not clear if it is the same mechanism underlying both types of rate adaptation.

2) It is not clear why the results of the study are significant, especially considering the relatively broad readership of eLife. The authors did not link rate adaptation to Purkinje cell or cerebellar function, and the claim that their results hold "for any type of neuron" (subsection “General Effect of Subthreshold Membrane Potentials on Shaping PRCs”) doesn't quite work because only two particular models of two types of neuron were studied.

3) In the second half of the manuscript, the relationship between the form of loose synchrony studied here, and correlations is unclear. This is important because it is well established that correlations increase with firing rate (de la Rocha, 2007, not cited in the manuscript). Does this explain what the authors are observing? (Particularly in the last figure).

4) Figure legends are too brief and do not include the relevant information to understand the figure.

5) The writing, especially of the Introduction and Discussion section, are overly specialized. I do not think that a reader unfamiliar with the PRC would find this manuscript accessible.

Reviewer #3:

This modeling study addresses an important problem in cellular and network dynamics. First, it provides a biophysical mechanism underlying the changes in the shape of the phase response curve (PRC) of Purkinje cells observed when they change their (simple spike) firing rate. Second, it demonstrates that this mechanism depends on the subthreshold voltage trajectory between spikes, which in turn is governed by the intrinsic biophysics of the neurons. Finally, the authors show that firing rate-dependent changes in the PRC of individual neurons can drive rate-dependent changes in network synchrony. This new mechanism is likely to regulate synchrony of high-frequency oscillations not just in the cerebellum but also in many other circuits in the brain.

The study is carefully designed and performed, and the manuscript is well written. Overall, it makes an important contribution to how specific features of single-cell biophysics can help to determine network-level dynamics.

Essential revisions:

1) The authors mention briefly in subsection “PRC Exhibits Rate Adaptation in PCs” that "Rate-adaptive PRCs require the presence of a dendrite in the PC model (not shown), but the dendrite can be passive (Figure 1—figure supplement 1B)", without going into further detail. In their explanation of the biophysical mechanism of rate adaptation of PRCs in PCs that follows (in subsection “PRC Exhibits Rate Adaptation in PCs”), dendrites are not mentioned, however, as if the dendrite was not relevant to the mechanism at all. Did the authors follow this somewhat contradictory strategy because they consider the role of the dendrite too simple or too complicated to explain? Could the main consequence of the presence of an active or passive dendrite be its (capacitive and Ohmic) load, leading to a depolarizing shift of the somatic voltage threshold of spikes (Bekkers and Hausser, 2007; Zang et al., 2018)? With the consequence that, in a well-tuned, physiologically detailed PC model like that of the authors, removal of the dendrite would lead to a shift in the spike threshold in the hyperpolarized direction, artificially interfering with the mechanism illustrated by the authors in Figure 2A and B?

If the importance of the dendrite is in fact due to an effect of this (or a similar) kind, then not only would the reader benefit from a brief explanation, but the authors could also make experimentally testable predictions of what happens to the PRC at different firing rates when the (passive or active) dendrite is pinched, i.e. isolating the some from the dendrite.

2) More generally, can the authors propose other experimentally testable predictions resulting from their biophysical mechanism of rate-dependent PRCs? This would help strengthen the study.

3) I am somewhat baffled by the words "just the passive depolarization" in the explanation (subsection “The Biophysical Mechanism of Rate Adaptation of PRCs in PCs”) that "During early phases of all rates, membrane potentials are distant from the Na+ activation threshold (Figure 2A,B). The depolarizations to weak stimuli fail to activate sufficient Na+ channels to speed up voltage trajectories, and phase advances are caused by just the passive depolarizations (Figure 2C). Consequently, phase advances in early phases are small and flat (or phase independent)." In the bottom (12 Hz) PRC in Figure 1B, there is an (admittedly broad) local maximum near phase 0.2. If these phase advances due to stimuli in an early phase of the PRC were indeed caused just by the resulting passive depolarizations, then the amplitude of these depolarizations should decay with the passive membrane time constant, leading to smaller PRC amplitudes at early phases (such as 0.2) than later phases (such as 0.8). The 12 and 27 Hz PRCs in Figure 1B show the opposite effect, suggesting that the membrane potential of a passive soma is not the only relevant state variable governing the approximately flat part of the PRC. Which other state variables (e.g. dendritic membrane potential, calcium concentration, activation or inactivation state of ion channels) could explain the shape of the 'foot' of the PRC at low rates?

[Editors’ note: further revisions were suggested prior to acceptance, as described below.]

Thank you for submitting your article "Firing Rate-dependent Phase Responses of Purkinje Cells Support Transient Oscillations" for consideration by eLife. Your article has been reviewed by 3 peer reviewers, and the evaluation has been overseen by Ronald Calabrese as the Senior Editor, a Reviewing Editor, and three reviewers. The following individuals involved in review of your submission have agreed to reveal their identity: Bard Ermentrout (Reviewer #2).

The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.

We would like to draw your attention to changes in our revision policy that we have made in response to COVID-19 (https://elifesciences.org/articles/57162). Specifically, we are asking editors to accept without delay manuscripts, like yours, that they judge can stand as eLife papers without additional data, even if they feel that they would make the manuscript stronger. Thus the revisions requested below only address clarity and presentation.

Summary:

This modeling study addresses an important problem in cellular and network dynamics. First, it provides a biophysical mechanism underlying the changes in the shape of the phase response curve (PRC) of Purkinje cells observed when they change their (simple spike) firing rate. Second, it demonstrates that this mechanism depends on the subthreshold voltage trajectory between spikes, which in turn is governed by the intrinsic biophysics of the neurons. Finally, the authors show that firing rate-dependent changes in the PRC of individual neurons can drive rate-dependent changes in network synchrony. This new mechanism is likely to regulate synchrony of high-frequency oscillations not just in the cerebellum but also in many other circuits in the brain.

Essential revisions:

Reviewer #3:

The reviews by the three other referees have already appropriately summarized the findings and commented on the modeling aspects of the study. For this reason, I would like to restrict myself to a brief discussion of cell physiological aspects of the work. Overall, the study is well done, and I believe that this work will be important to the field of cerebellar physiology, with further reaching implications in the neurosciences regarding the impact of neuronal oscillations.

1) It should be stated in the results section whether the modeling focuses on Purkinje cells in adult animals, or during development. This is crucial information, keeping in mind that the nature of Purkinje cell – Purkinje cell interactions changes during development (see Watt et al., 2009; cited).

2) Results section: the cell's responsiveness and spike output (in response to synaptic drive) appear to change with the state of the AHP, not only the amplitude of synaptic input (Ohtsuki et al., 2018). Does the model predict how the oscillatory phase affects synaptically driven spike firing?

3) Are resurgent Na conductances (Raman and Bean, 1997) critical for the occurrence of these oscillations or specific parameters?

4) Does the model account for differences in spike firing frequencies in zebrin-positive and zebrin-negative cerebellar modules (Zhou et al., 2014)? This is suggested by the findings of Schonewille and others (Grasselli et al., 2020) that in SK channel knockout mice these firing properties are differentially affected, which highlights the role of the AHP in sub-and suprathreshold modulation.

5) Are oscillations in the cerebellum coherent with high-frequency oscillations in other brain areas? This is suggested by the observation that input from the cerebellar nuclei regulates gamma frequency oscillations in thalamocortical networks (Timofeev and Steriade, 1997).

Addressing these points will shed more light on the consequences of these oscillations for cerebellar output functions and will thus hopefully further enhance the impact of this interesting paper.

https://doi.org/10.7554/eLife.60692.sa1

Author response

[Editors’ note: the authors resubmitted a revised version of the paper for consideration. What follows is the authors’ response to the first round of review.]

Reviewer #1:

I was a bit surprised at this result until I realized what the authors had done with respect to the computation of the PRC. In the infinitesimal limit,the PRC is the normalized nullspace of the adjoint linear operator. The usual normalization is that the inner product is 1. This is how the theory that the authors allude to is done. Here the authors use a different normalization where they divide the adjoint by the period. This produces a different scaling and of course will make the PRC that occurs near the bifurcation (where the period is large) to appear much smaller in amplitude than it would with the standard scaling. If you use the standard scaling on the Traub model (which is readily available) and use the authors modifications, you find that with the standard normalization, higher frequencies have smaller PRCs which is according to the theory. If you divide by the period, then you get the authors results at least for currents that are not too big. For frequencies of about 110 Hz, even with the scaling, I found the PRC to be smaller than the lower frequencies.

We use a standard normalization for PRCs computed for experimental data or, in our case, complex model data where it is close to impossible to compute the adjoint (for more than 1000 coupled ODEs). Equation 1 we used to compute the PRC is identical to the equation in Ermentrout, Pascal and Gutkin (2001). As far as we know, the usual normalization the reviewer alludes to can only be applied to very simple models. We now mention explicitly that the use of Equation 1 facilitates comparison with experimental data (subsection “PRC Computations”).

We do understand the broader argument that this standard normalization of the PRC, dividing the difference in spike timing by the period, promotes a firing rate-dependence of the peak amplitude of the PRC. This has never been explicitly acknowledged in previous experimental literature that described this phenomenon, namely Phoka et al., 2010 and the confirmation by Couto et al., 2015. Therefore, we mention this issue now explicitly in subsection “PRC Exhibits Rate Adaptation in PCs” of the new manuscript.

It should be noted that, by itself, this normalization does not always increase the peak amplitude of the PRC as demonstrated in Figures 1—figure supplement 2 and Figure 1—figure supplement 1A where, using Equation 1, peak amplitude decreases with firing rate.

Turning to their PC model, it is clear from Figure 1C, that the amplitudes with the usual normalization will follow the theory. That is, higher-frequencies have smaller amplitudes. (Specifically, if you divide the amplitudes shown by their respective frequencies, the graph will be decreasing.)

The main motive for this study is the PRC measurements from Couto et al. which was published in PLoS comp bio where the same result was found experimentally. To me the amplitude is not so much the point of interest here but rather the extreme change in the shape of the PRC.

This is a very astute remark. We show indeed later in the new manuscript that the important PRC property is largely the phase width of the flat part relative to the phase width of the peak (new Figure 5 and subsection “High-frequency Oscillations are Caused by the Rate-dependent PRC”).

The Traub model does not show the shape dependence in fact it is the opposite with little effects until the late phases for low frequencies, an effect that was explained in the paper by Pascal et al. as coming from the adaptation.

The main result of interest is that at higher rates, oscillations synchronize/correlate better than an low rates So, there is some practical effect of this.

The authors set out to explain several effects. One of them is the largely unresponsive phase after the spike. As noted above, this effect was already explained in the old work of Pascal et al.

In Figure 3D, while the amplitude goes up in this model, the phase dependence is very flat only in the low rate versions. Thus, I think that the authors explanations are not really general. They need to make these explanations in a more rigorous mathematical manner.

We thank the reviewer for agreeing with the importance of firing rate-dependent PRCs on the better oscillations at higher rates by ‘there is some practical effect of this’. Traub model is less emphasized in the new manuscript and has completely been moved to supplementary material. We corrected statements about the model according to these comments.

The latter part of the paper is the most interesting where they demonstrate rate dependence in the correlation of the networks. It is ashame that they do not directly relate this to the PRCs. It would have been nice to perform some sort of analysis of a network of neurons that viewed as phase oscillators and connected via their phase response curves. I should point out that when coupling, whether or not locking occurs depends on whether or not you rescale the heterogeneties or not. I illustrate this in the following example. Let I be the baseline current to a bunch of neural oscillatora and let I_j be the heterogeneous current (assumed small) Let T be the period and suppose simple pulse coupling such that is neuron j fires, then a δ function current J_{ij} is added to the postsynaptic current. Let Δ be the conventional PRC (normalized to be msec/mv). Then the Kuramoto/Ermentrout/Kopell/ etc reduction yields for the phase:

dtheta_i/dt = 1 + Δ(theta_i)[ I_i/C + sum J_{ij}/C δ(t-t_j)] mod T

with C the membrane capacitance. Theta is a time-like variable at this point. To make it a phase-like variable (as the authors do), then let theta = T phi. Then one obtains

dphi_i/dt = omega + (1/T) Δ(T phi_i) [ I_i/C + sum J_{ij}/C δ(t-t_j)] mod 1

Conditions for synchrony etc are the same as the first model if, say you reduce this to a phase-difference model by replacing the δ(t) with δ(phi).

We thank the reviewer for agreeing this part ‘is the most interesting’ and for this excellent suggestion. We implement a coupled oscillator model and show the results in new Figure 5 and accompanying new subsection “High-frequency Oscillations are Caused by the Rate-dependent PRC”. Thanks to the coupled oscillator model, we can now formally show that the rate-dependent PRC directly causes the rate-dependent oscillations and that the flat part of the PRC plays an important role in this property.

My point is, that the amplitude dependence on frequency is partially a consequence of how you normalize the PRC. If you look at Tsubo Figure 4, the amplitudes are all the same in their normalized version. The theory predicted by the theta model is that the period-normalized PRCs should have constant amplitudes.

Explicitly mentioned in the new manuscript in subsection “PRC Exhibits Rate Adaptation in PCs”.

I think the results are interesting, but I also think that they need to do some more sophisticated math analysis and also relate their PRCs more directly to the synachronization/correlation effects. This could be done with a weak coupling analysis after convolving their PRCs with the synapses to obtain the interaction function. The most important facet of PRCs is not som much their amplitude (which can be compensated for by changing the coupling), but their shape. I'd like more theoretical understanding of the shape dependence.

See comments above about new Figure 5 and the importance of PRC shape.

Reviewer #2:

This manuscript explores how average firing rate in Purkinje cells affects their PRC and high-frequency oscillations. Overall, there are issues with presentation, especially for clarity and the significance of the findings.

Essential revisions:

1) The manuscript, as presented, is disjointed. The first half proposes a mechanistic factor that contributes to rate modulation in Purkinje cell PRCs, and has a clear conclusion (although not many other mechanisms are entertained). The second half describes how the average firing rate increases the power of high frequency oscillations in the cerebellum. But there is not a strong link between these halves. In particular, it is not clear if it is the same mechanism underlying both types of rate adaptation.

We agree with this criticism, as argued in subsection “High-frequency Oscillations are Caused by Rate-dependent PRCs” this was the best we could do with the complex PC network model we used. Based on a suggestion of reviewer #1 we have now also implemented a coupled oscillator model that allows us to demonstrate a causal link between the rate-dependent PRC and the rate-dependent oscillations. This is described in a new section “High-frequency Oscillations are Caused by the Rate-dependent PRC” (subsection “High-frequency Oscillations are Caused by Rate-dependent PRCs” and new Figure 5). The manuscript has also been reorganized to focus mostly on the cerebellum and to emphasize this causal link (we moved the work on the Traub model to supplementary material).

2) It is not clear why the results of the study are significant, especially considering the relatively broad readership of eLife. The authors did not link rate adaptation to Purkinje cell or cerebellar function, and the claim that their results hold "for any type of neuron" (subsection “General Effect of Subthreshold Membrane Potentials on Shaping PRCs”oesn't quite work because only two particular models of two types of neuron were studied.

Reviewer #3 does believe the paper ‘addresses an important problem in cellular and network dynamics’. But we agree that the original manuscript did a rather poor job of explaining, based on the work of Person and Raman, (2012), why transient synchrony can be important in the cerebellum. We now explicitly refer to the formation of transient assemblies, discuss how such correlated assemblies can change the firing of downstream neurons and why their transient nature expands the information processing capacity of the cerebellum (Discussion section). We also expanded the analysis of the 1 Hz wave data of Figure 6 and generated new data (2.5 Hz wave, Figure 2—figure supplement 2) to support the hypothesis of transient assemblies.

3). In the second half of the manuscript, the relationship between the form of loose synchrony studied here, and correlations is unclear. This is important because it is well established that correlations increase with firing rate (de la Rocha, 2007, not cited in the manuscript). Does this explain what the authors are observing? (Particularly in the last figure).

Based both on mechanism (common excitatory input for de la Rocha model, inhibitory coupling reinforced by firing-rate dependent PRC in our model) and actual shape of the firing rate dependence (‘square root’ like in de la Rocha model, ‘exponential’ like for our model, now shown in new Figure 5A), we believe that our work describes a fundamentally different phenomenon. This is now discussed in subsection “The Evidence Supporting Rate-dependent Correlations”.

4) Figure legends are too brief and do not include the relevant information to understand the figure.

We improved several legends.

5) The writing, especially of the Introduction and Discussion section, are overly specialized. I do not think that a reader unfamiliar with the PRC would find this manuscript accessible.

Introduction and Discussion section were largely rewritten.

Reviewer #3:

This modeling study addresses an important problem in cellular and network dynamics. First, it provides a biophysical mechanism underlying the changes in the shape of the phase response curve (PRC) of Purkinje cells observed when they change their (simple spike) firing rate. Second, it demonstrates that this mechanism depends on the subthreshold voltage trajectory between spikes, which in turn is governed by the intrinsic biophysics of the neurons. Finally, the authors show that firing rate-dependent changes in the PRC of individual neurons can drive rate-dependent changes in network synchrony. This new mechanism is likely to regulate synchrony of high-frequency oscillations not just in the cerebellum but also in many other circuits in the brain.

The study is carefully designed and performed, and the manuscript is well written. Overall, it makes an important contribution to how specific features of single-cell biophysics can help to determine network-level dynamics.

We thank the reviewers for acknowledging the importance of our work.

Essential revisions:

1). The authors mention briefly in subsection “PRC Exhibits Rate Adaptation in PCs” that "Rate-adaptive PRCs require the presence of a dendrite in the PC model (not shown), but the dendrite can be passive (Figure 1—figure supplement 1B)", without going into further detail. In their explanation of the biophysical mechanism of rate adaptation of PRCs in PCs that follows (in subsection “PRC Exhibits Rate Adaptation in PCs”), dendrites are not mentioned, however, as if the dendrite was not relevant to the mechanism at all. Did the authors follow this somewhat contradictory strategy because they consider the role of the dendrite too simple or too complicated to explain? Could the main consequence of the presence of an active or passive dendrite be its (capacitive and Ohmic) load, leading to a depolarizing shift of the somatic voltage threshold of spikes (Bekkers and Hausser, 2007; Zang et al., 2018)? With the consequence that, in a well-tuned, physiologically detailed PC model like that of the authors, removal of the dendrite would lead to a shift in the spike threshold in the hyperpolarized direction, artificially interfering with the mechanism illustrated by the authors in Figure 2A and B?

If the importance of the dendrite is in fact due to an effect of this (or a similar) kind, then not only would the reader benefit from a brief explanation, but the authors could also make experimentally testable predictions of what happens to the PRC at different firing rates when the (passive or active) dendrite is pinched, i.e. isolating the some from the dendrite.

As already shown in Zang et al., 2018 the dendrite extensively changes the spike shape, without dendrite there is a stronger afterhyperpolarization that significantly changes the subthreshold trajectory. As a consequence, the PRC without dendrite looks completely different. See new Figure 1—figure supplement 2 and subsection “Biophysical Mechanisms Underlying Rate-dependent PRCs”.

2) More generally, can the authors propose other experimentally testable predictions resulting from their biophysical mechanism of rate-dependent PRCs? This would help strengthen the study.

We now added suggestions to the Discussion section.

3) I am somewhat baffled by the words "just the passive depolarization" in the explanation (subsection “The Biophysical Mechanism of Rate Adaptation of PRCs in PCs”) that "During early phases of all rates, membrane potentials are distant from the Na+ activation threshold (Figure 2A,B). The depolarizations to weak stimuli fail to activate sufficient Na+ channels to speed up voltage trajectories, and phase advances are caused by just the passive depolarizations (Figure 2C). Consequently, phase advances in early phases are small and flat (or phase independent)." In the bottom (12 Hz) PRC in Figure 1B, there is an (admittedly broad) local maximum near phase 0.2. If these phase advances due to stimuli in an early phase of the PRC were indeed caused just by the resulting passive depolarizations, then the amplitude of these depolarizations should decay with the passive membrane time constant, leading to smaller PRC amplitudes at early phases (such as 0.2) than later phases (such as 0.8). The 12 and 27 Hz PRCs in Figure 1B show the opposite effect, suggesting that the membrane potential of a passive soma is not the only relevant state variable governing the approximately flat part of the PRC. Which other state variables (e.g. dendritic membrane potential, calcium concentration, activation or inactivation state of ion channels) could explain the shape of the 'foot' of the PRC at low rates?

We no longer use “passive depolarization” as it may be a confusing terminology. We emphasize the importance of activation of transient versus persistent sodium channels (subsection “Subsection “The Biophysical Mechanism of Rate Adaptation of PRCs in PCs”).

[Editors’ note: what follows is the authors’ response to the second round of review.]

Essential revisions:

Reviewer #3:

The reviews by the three other referees have already appropriately summarized the findings and commented on the modeling aspects of the study. For this reason, I would like to restrict myself to a brief discussion of cell physiological aspects of the work. Overall, the study is well done, and I believe that this work will be important to the field of cerebellar physiology, with further reaching implications in the neurosciences regarding the impact of neuronal oscillations.

We thank the reviewer for agreeing with the importance of this work.

1) It should be stated in the Results section whether the modeling focuses on Purkinje cells in adult animals, or during development. This is crucial information, keeping in mind that the nature of Purkinje cell – Purkinje cell interactions changes during development (see Watt et al., 2009; cited).

We added text to the Results section to clarify that we simulated adult cerebellar cortex and the evidence to support the presence of collateral connections in adults.

2) Results section: the cell's responsiveness and spike output (in response to synaptic drive) appear to change with the state of the AHP, not only the amplitude of synaptic input (Ohtsuki et al., 2018). Does the model predict how the oscillatory phase affects synaptically driven spike firing?

Previous theoretical work cited in the Introduction established that the PRC predicts how the oscillatory phase affects spike firing caused by synaptic input. As the model reproduces the experimental PRC, it should also make this prediction. The paper has been cited and discussed in Subsection “Biophysical Mechanisms Underlying Rate-dependent PRCs”.

3) Are resurgent Na conductances (Raman and Bean, 1997) critical for the occurrence of these oscillations or specific parameters?

Yes, resurgent Na conductance is critical for PRCs and then for oscillations. We used a state model of sodium channel gating from Raman and Bean, 2001, which incorporates transient, resurgent and persistent current in one model. We now mentioned resurgent component in subsection “The Biophysical Mechanism of Rate Adaptation of PRCs in PCs”.

4) Does the model account for differences in spike firing frequencies in zebrin-positive and zebrin-negative cerebellar modules (Zhou et al., 2014)? This is suggested by the findings of Schonewille and others (Grasselli et al., 2020) that in SK channel knockout mice these firing properties are differentially affected, which highlights the role of the AHP in sub-and suprathreshold modulation.

Due to lack of data, we can’t account for differences in zebrin-positive and zebrin-negative cerebellar modules. We have clearly stated this as a limitation. These papers have been cited and discussed. Subsection “Biophysical Mechanisms Underlying Rate-dependent PRCs”.

5) Are oscillations in the cerebellum coherent with high-frequency oscillations in other brain areas? This is suggested by the observation that input from the cerebellar nuclei regulates γ frequency oscillations in thalamocortical networks (Timofeev and Steriade, 1997).

We discuss coherence of oscillations in subsection “Down-stream effects of PC assemblies”.

https://doi.org/10.7554/eLife.60692.sa2

Article and author information

Author details

  1. Yunliang Zang

    Computational Neuroscience Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa, Japan
    Contribution
    Conceptualization, Data curation, Software, Formal analysis, Validation, Investigation, Visualization, Methodology, Writing - original draft, Writing - review and editing
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0001-8999-1936
  2. Sungho Hong

    Computational Neuroscience Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa, Japan
    Contribution
    Software, Investigation, Visualization, Methodology, Writing - review and editing
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-6905-7932
  3. Erik De Schutter

    Computational Neuroscience Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa, Japan
    Contribution
    Supervision, Writing - original draft, Project administration, Writing - review and editing
    For correspondence
    erik@oist.jp
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0001-8618-5138

Funding

Okinawa Institute of Science and Technology Graduate University

  • Erik De Schutter

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

Acknowledgements

The authors thank for helpful suggestions from Drs. Eve Marder, Tomoki Fukai and Sergio Verduzco to improve the manuscript and for the language editing by Steven Douglas Aird.

Senior and Reviewing Editor

  1. Ronald L Calabrese, Emory University, United States

Reviewer

  1. Bard Ermentrout, University of Pittsburgh, United States

Publication history

  1. Received: July 3, 2020
  2. Accepted: August 20, 2020
  3. Version of Record published: September 8, 2020 (version 1)

Copyright

© 2020, Zang et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.

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