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Conceived and designed the experiments: BTN YY. Wrote the paper: BTN IS YY.

The authors have declared that no competing interests exist.

It is commonly accepted that the Inferior Olive (IO) provides a timing signal to the cerebellum. Stable subthreshold oscillations in the IO can facilitate accurate timing by phase-locking spikes to the peaks of the oscillation. Several theoretical models accounting for the synchronized subthreshold oscillations have been proposed, however, two experimental observations remain an enigma. The first is the observation of frequent alterations in the frequency of the oscillations. The second is the observation of constant phase differences between simultaneously recorded neurons. In order to account for these two observations we constructed a canonical network model based on anatomical and physiological data from the IO. The constructed network is characterized by clustering of neurons with similar conductance densities, and by electrical coupling between neurons. Neurons inside a cluster are densely connected with weak strengths, while neurons belonging to different clusters are sparsely connected with stronger connections. We found that this type of network can robustly display stable subthreshold oscillations. The overall frequency of the network changes with the strength of the inter-cluster connections, and phase differences occur between neurons of different clusters. Moreover, the phase differences provide a mechanistic explanation for the experimentally observed propagating waves of activity in the IO. We conclude that the architecture of the network of electrically coupled neurons in combination with modulation of the inter-cluster coupling strengths can account for the experimentally observed frequency changes and the phase differences.

There is a profound interest in the dynamics of neuronal networks and the simulation of network models is a prevalent approach to study these dynamics. Generally, network models contain neurons that are connected mostly through chemical synapses to form either a completely regular topology (such as nearest neighbor connections), a completely random topology, small-world networks or scale-free networks. We investigate the dynamics of an atypical network, inspired by the Inferior Olive (IO) network, a brain structure located at the end of the brainstem that is responsible for timely execution of motor commands. This network is atypical in the sense that it has neurons in a clustered topology, which are connected solely by electrical synapses. The dynamics in the IO are enigmatic as the membrane voltage of some neurons can oscillate at the same frequency while maintaining phase difference with other neurons. It has also been demonstrated that propagating waves of activity occur spontaneously in this network. Using computer simulations we unraveled the mechanism underlying these previously enigmatic experimental observations. In so doing, we stress the importance of investigating more realistic network topologies to explore complex brain dynamics.

There is a profound interest in the dynamics of neuronal networks and the simulation of network models is a prevalent approach to study these dynamics. One aspect of network dynamics is the generation of oscillatory activity. It has been hypothesized that oscillations subserve brain-wide communications. For instance, “binding” to connect distinct sensory streams in the brain

The Inferior Olive (IO) nucleus is the exclusive provider of cerebellar climbing fibers. Neurons in the IO form a network solely through electrical connections (gap junctions) between them. This electrically coupled network of neurons generates subthreshold voltage oscillations, which were observed both

There are two observations in relation to the function of the IO as a timekeeper. The first observation is that the frequency of the subthreshold oscillation shifts from time to time

In this work we address both frequency modulation and the generation of phase differences in the IO network. To this end we built a network model of the IO consisting of basic conductance-based model neurons _{l}) and low-threshold Ca^{2+}-conductances (g_{Ca}, see _{1} = V_{2}/V_{1}, CC_{2} = V_{2}/V_{1}, and see

These data constrain the model's architecture to a topology in which similar neurons (in terms of conductance densities) are clustered together and are densely connected via gap junctions. The anatomical clustering of dendrites leads to sparse connectivity between a given cluster and all other clusters, i.e., neurons from one cluster are connected to neurons in one or a few other clusters but not necessarily to all other clusters. Thus, major constraints on the network architecture are imposed by the connectivity scheme, the limited number of connections per neuron, and the weak coupling coefficient between cell pairs.

We demonstrate that network models which obey these experimental constraints, and in which electrical-coupling strength is subject to modulation, are sufficient to account for frequency changes and for the generation of phase differences across frequencies. The robustness of the results is discussed and the key mechanisms that support the observed network dynamics are highlighted. We also discuss a prediction based on our theoretical study.

The aforementioned constraints still leave several free parameters. The exact number of neurons in a cluster is bounded by biological data (8 to 12 neurons per cluster _{l}-g_{Ca} space is shown in

A: Model neurons only contain leak and Ca^{2+} currents and spontaneously oscillate at frequencies determined by the exact density of the associated conductances. Colors of the g_{l}-g_{Ca} plane indicate the frequency at which a model with the corresponding density of conductances oscillates; in the white region model neurons do not oscillate spontaneously. The network itself consists of individual neurons (red squares) grouped in clusters (colored ellipses; color not related to the frequency). Neurons inside the cluster are connected to 4 neighbors. When two clusters are connected (black arrows) each neuron from one cluster is connected to a random neuron in the other cluster. All connections are gap-junctions. B: Resulting coupling coefficients of all connections in the network. This specific network is used throughout the manuscript for demonstration purposes.

In our reference network, the conductance densities of twenty-six out of forty-eight model neurons are such that they oscillate spontaneously (

A: Raster plot containing all neurons in the network; peaks of the oscillation are denoted by a dot. Without connections only 26 out of 48 neurons oscillate (left panel). When the intra-cluster connections are added, 3 out of 4 clusters show synchronized oscillations within the clusters (center panel). After adding the inter-cluster connections as well, the whole network reaches a synchronized oscillation of 9.2 Hz. B: Detail of the membrane potential of one neuron from each cluster indicating that the network can sustain stable subthreshold oscillations. Colors of the membrane trace and the ellipses in panel A are matching. C: Detail of the membrane potential of all neurons in one cluster (C0). D&E: Stable oscillations in the proposed network architecture are robust to changes in the number of clusters and the number of cluster per neuron. In D, networks with a varying number of clusters but a fixed cluster size (10 neurons) and a randomized connectivity scheme were tested. In E, networks with 4 clusters and a varying cluster size were tested (while the connectivity scheme was fixed as in the reference network. Therefore, the “4 clusters×10 neurons” from panel D and E are not the same). Boxplots indicate the median and the boxes extend from the lower to the upper quartile. It follows that robust synchronized oscillations can be generated by a variety of networks and that each network can achieve a range of frequencies.

It is important to stress that the network dynamics are robust with respect to the free network parameters (i.e., the exact number of clusters and the cluster size), as long as the resulting connectivity pattern meets the anatomical and physiological constraints outlined before. Namely, we can obtain different networks composed of various numbers of clusters and cluster sizes that exhibit synchronized oscillations. To support this claim, we simulated two sets of pseudo-random network. In the first set, we simulated networks consisting of 10 neurons per cluster and varied the number of clusters from 4 to 8. The inter-cluster connectivity scheme was also sampled randomly, with each cluster connecting to 1–3 other clusters. In the second set of simulations, we varied the number of neurons inside each cluster between 8 and 16, while keeping the number of clusters constant, and using a fixed inter-cluster connectivity scheme as in the reference “4 clusters×12 neurons” network. The resulting frequencies at which these networks exhibited spontaneous oscillations are shown in

We also want to stress that roughly 50% of neurons in our “4 cluster×12 neurons” reference network oscillate spontaneously. Evidently, the mechanism we presented for generating synchronized oscillations also holds in networks with a higher proportion of spontaneously oscillating neurons (e.g., 85%, as in

Two model IO neurons are known to be able to oscillate synchronously when they are connected with a suitable coupling strength _{l}-g_{Ca} plane (

A: Membrane potential of one neuron per cluster just before and after manually changing the inter-cluster connection strength in the reference network. The change in inter-cluster strength caused a shift in the synchronized oscillation frequency from 6.3 Hz to 10.9 Hz. B: Short-term Fourier transformation of the membrane potential of one neuron in the network indicates the shift in frequency. C: Fourier transformation of the membrane potential of one neuron of each cluster. All clusters oscillate at the same frequency and are subject to the same shift. D: Histogram of frequencies at which the same network with pseudo-random inter-cluster connections strengths can oscillate in synchrony. Only changing the inter-cluster coupling strength (within realistic ranges, i.e., CC<20%) can be sufficient to bring the network to a state of synchronized oscillations with frequencies between 6 and 11 Hz.

Thus, we identified a robust mechanism to change the frequency of the synchronized oscillations by means of (small) changes of the inter-cluster strengths that in turn change the weighted-average neuron that dictates the frequency of the synchronized oscillation.

An emergent feature of the proposed clustered network architecture is that such networks display a phase difference between neurons (^{2+}-conductance is faster. As a result, these high Ca^{2+}-conductance neurons oscillate at a higher frequency when uncoupled. In the coupled case, the faster voltage build-up leads to their advance in phase over neurons with less Ca^{2+}-conductance. During the period directly after the peak, the current flowing between both neurons reverses and causes both neurons to remain in pace with each other. When the coupling strength is sufficient, it is this mechanism that binds the two connected neurons to the same frequency. The same principle holds for networks with clusters of similar neurons: the cluster with highest concentration of Ca^{2+}-conductance is advanced in phase over clusters with less Ca^{2+}-conductance. ^{2+}-density (the colors of the traces match the colors of the clusters in

A: Focus on the normalized membrane potential of one neuron per cluster reveals that clusters with higher Ca^{2+}-conductance are advanced in phase with respect to other clusters (traces have colors matching with ^{2+}-density indicates a higher resting membrane potential that causes the neuron to lead in the phase. B: Phase-map color coding the phase-difference between all neurons in the network. Phase differences are given in degrees relative to the inter-peak-interval; the phase of the bottom left neuron is taken as reference (0°). Neurons within the same cluster have similar phases due to similar resting potentials, while larger phase-differences arise between clusters that are farther apart in terms of their conductances. The maximum phase-difference between two neurons was 72° in the demonstration network. C: Cross-correlation of the peak times between (one neuron from the) four clusters computed for 5 s traces confirms that the phase-differences are stable over time. D: The amplitude and phase difference is proportional to the amount of g_{Ca}-conductance a neuron contains. The y-axis denotes the peak voltage and the x-axis indicates the conductance density. The color-coding is the same as in A while the size represents the phase-difference (as measured between the neuron at the bottom left and any other neuron).

The implication that neurons advanced in their phase also have higher voltage amplitude (because of the larger g_{Ca}) can be verified using _{Ca}-density. The size of the data points indicates the phase difference relative to the reference (0° phase difference). Hence, larger data-points in

The observed phase difference also provides an explanation for the “propagating waves of activity” found experimentally _{Ca}. This sequential activation can be observed as a propagating wave (see Supporting

Thus, our model also successfully reproduces the experimental observation of phase differences, and provides a mechanistic explanation for this phenomenon.

In this work we proposed a plausible model of the IO network that provides an explanation for timing and timekeeping within the IO. The activity in the IO is crucial for the proper function of the olivo-cerebellar circuit, and as such it is at the focus of many studies. Different models of IO neurons have been proposed to explain single-cell subthreshold oscillations

We purposely used minimalistic model neurons, as the focus of this work was the dynamics of the subthreshold oscillations in the IO network. The model neuron contains only a leak and a Ca^{2+}-current because these currents are most prominent in the subthreshold voltage oscillation regime ([−65 mV,−50 mV]) ^{2+} spikes). These currents could be added in the future in large-scale models of the olivo-cerebellar circuit. Despite its limitations, our model is elegant in its minimalistic, yet biologically rooted approach.

In this work we re-evaluate a finding from an earlier work in which it was shown that two IO model neurons that are not necessarily oscillatory in isolation can be connected in such a way that they oscillate synchronously _{l}-g_{Ca} plane where a single neuron would oscillate spontaneously

Having shown that the inter-cluster coupling strength determines the frequency of oscillation, it is straightforward to see that changes in the inter-cluster coupling strength change the oscillatory frequency in the network. We note that the intra-cluster coupling strength does not contribute to the network frequency because inside a cluster all neurons are electrically similar and hence the average neuron that represents a single cluster is very stable; only the inter-cluster connections can change the frequency. We also note that in the clustered network as we propose it, the synchronized oscillations can cease in two ways. First, the virtual, weighted average (neuron) can be moved to a region in the g_{l}-g_{Ca} space were no oscillations occur (i.e., the white space in

Changes in the functional coupling strength can be induced by the GABAergic inputs coming from the deep cerebellar nuclei (DCN). DCN inputs to the IO are co-located at the sites of the gap junction

Blocking of GABAergic inputs has been reported to have the effect of increasing the size of the group of synchronously oscillating neurons

We found that basic neuron models including one active component (Ca^{2+} T-type current) in combination with a clustered network with differential inter-cluster electrical connections can account for synchronized network oscillations, the modulation of the frequency and the emergence of phase differences, which in turn lead to propagating waves of activity. There is a great deal of theoretical literature related to synchrony in neural network ^{2+} current. However, it remains unclear what the minimal conditions are for realistic, synchronized subthreshold oscillations in our network. The minimal conditions depend on what is functionally relevant for the network. For instance, shifts between 1 and 4 Hz have been observed experimentally

Many network models are devised to address a particular question dealing with a part of the natural, experimentally observed dynamics. To model different dynamics in the same system, a new model is constructed in the present study that can accommodate diverse sets of dynamics. We have shown that our network model, which successfully reproduces subthreshold oscillations, also accounts for the experimentally observed frequency changes and phase differences. Moreover, based on current data from the DCN

The results presented in this study also give rise to a testable prediction about the IO. Our prediction addresses the possibility of modulating IO oscillation frequencies by changing the inter-cluster coupling strength. This prediction could be tested in an

In conclusion, we present the first anatomically and physiologically plausible (albeit reduced) network model of the IO that provides a biophysical explanation for previously unexplained experimental observations. As such, we believe that our model is suitable to test future hypotheses about the origin of the subthreshold oscillations and their role in timing.

We use conductance-based model neurons based on the model presented in ^{2+} current. Formally, the dynamics of the model neurons are described by:_{l} and E_{Ca} are the reversal potentials for the leak and low-threshold Ca^{2+} current, respectively. g_{l} and g_{Ca} are the maximum conductances of these currents. m and h are the gating variables for the time and voltage dependent T-type current and follow

In all presented simulations, E_{L} = −63 mV while g_{l} and g_{Ca} vary between [0.15,0.4] mS/cm^{2} [0.2,1.4] mS/cm^{2} _{l} and g_{Ca} can exhibit spontaneous oscillations over a range of frequencies as illustrated in Figure S2 in _{l}) and calcium (g_{Ca}) conductance. Depending on the exact density of g_{l} and g_{Ca} the neuron can be i) a spontaneous oscillator and oscillate at different frequencies (

We create the network model by connecting selected neurons through electrical coupling (gap-junctions). The effect of a gap-junction on a single neuron can be represented by an additional current that mimics the current flowing between two connected cells proportionally to the difference in membrane potential in both cells: _{c1} and R_{c2} are of little importance as they depend on the actual input resistance of a neuron. A more useful measurement of coupling through gap-junction is the coupling coefficient: CC_{1} = V_{2}/V_{1} = R_{2}/(R_{2}+R_{c1}) and CC_{1} = V_{2}/V_{1} = R_{1}/(R_{1}+R_{c2}) as it directly assesses the electrical impact of one neuron on the other. Note that the voltages V_{1} and V_{2} are not the same in the calculation of CC_{1} and CC_{2} because they are measured from two separated experiments; one in which the current is injected in the first neurons and another experiment in which the current is injected in the second neuron. Due to the dependence on the input resistances, CC_{1} and CC_{2} also do not need to be the same.

Based on anatomical and physiological data the network architecture has to satisfy three interconnected constraints. First, neurons similar in terms of their conductances densities are clustered together and connected more densely to neurons inside the same cluster than to neurons belonging to different clusters. Second, the number of connections per neurons is between 1 and 38

We generated pseudo-random networks in which we manually set the meta-parameters of the network, namely the number of neurons per cluster (12), the number of clusters (4), the number of connected neighbors inside a cluster (4), the overall connectivity scheme between clusters (_{l} and g_{Ca}, respectively) to get set the actual values for the conductances of the model neurons inside that cluster. The networks in

We implemented all simulations in PyNEURON

Additional information about the robustness of the model, modulation of network frequencies and the range of frequencies at which (a pair of) IO model neurons can oscillate.

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Video illustrating propagating waves of activity in a “20 clusters×20 neurons” network.

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We thank Albert Gidon and Avi Libster for useful discussions.